Norberto de Kássio Vieira Monteiro

NOVOS PROTOCOLOS TEÓRICOS UTILIZADOS NOS ESTUDOS DAS

LIGAÇÕES HIDROGÊNIO-HIDROGÊNIO, NA ESTABILIDADE DO

TETRAEDRANO, SEUS DERIVADOS E DAS REAÇÕES DE DIELS-ALDER E NA

QUANTIFICAÇÃO DA AFINIDADE DE FÁRMACOS.

Orientador: Caio Lima Firme

NATAL/RN

2015

Divisão de Serviços Técnicos

Catalogação da Publicação na Fonte. UFRN Biblioteca Setorial do Instituto de Química

Monteiro, Norberto de Kássio Vieira. Novos protocolos teóricos utilizados nos estudos das ligações hidrogênio-hidrogênio, na estabilidade do tetraedrano, seus derivados e das reações de diels-alder e na quantificação da afinidade de fármacos / Norberto de Kássio Vieira Monteiro. – Natal, RN, 2015.

167 f. : il.

Orientador: Caio Lima Firme. Tese (Doutorado em Química) - Universidade Federal do Rio Grande do Norte. Centro de Ciências Exatas e da Terra. Programa de Pós-Graduação em Química.

1. Diels-Alder – Tese. 2. Ligação H-H – Tese. 3. Tetraedrano – Tese. 4. CYP17 – Tese I. Firme, Caio Lima. II. Universidade Federal do Rio Grande do Norte. III. Título.

RN/UFRN/BSE- Instituto de Química CDU 544.142.4:615 (043.3)

AGRADECIMENTOS

Às minhas três mães: Marilú, Vitória e Dona Francisca. Sem elas, nem sei o que seria

de mim.

Ao meu amor, Richele, a qual tenho o privilégio de chamar de esposa, pela ajuda e por

me aguentar nos momentos de ansiedade e estresse nos anos que me dediquei ao

doutorado.

Ao meu gordinho! Filhão, você foi a melhor coisa que já aconteceu na minha vida!

Papai te ama!

À minha querida afilhada Shara Maria, a qual considero como uma filha.

Aos meus padrinhos que sempre estiveram comigo durante toda a minha vida.

Aos alunos de iniciação científica: Cidinha, Luís, Franklin, Fabrício, Bruno, Elizeu e

Valéria.

Aos alunos e amigos da pós-graduação: Gutto, Caio Sena, João Batista, Acássio,

Roberta, William, Tamires e Francinaldo.

Aos meus amigos: Elton, Isabelle, Amisson, Clesiane, Aninha e, em especial, Sérgio,

que além de amigo também é meu padrinho de casamento.

Aos meus sogros, seu Laércio e Dona Rita, que me tratam como um filho.

À minha cunhada Raynara por me dar caronas à UFRN quando eu estava sem moto.

Muito especialmente, agradeço ao meu orientador Prof. Caio de Lima Firme, pela

disponibilidade, paciência, dedicação e profissionalismo e, acima de tudo, por ter

acreditado em mim desde o início.

Aos professores Jones de Andrade, Gorette Cavalcante e Luiz Seixas pelas longas e

produtivas conversas.

Aos professores Carlos Alberto Martinez e Sibele Berenice pela competência em dirigir

o Programa de pós-graduação em Química.

Aos professores componentes da banca de qualificação, Dr. Euzébio Guimarães, Dr.

Davi Serradella e Dr. Caio de Lima Firme, pelas correções e aperfeiçoamento desta tese.

Aos professores componentes da banca de defesa de tese, Dr. Euzébio Guimarães, Dr.

Davi Serradella, Dr. Cláudio Costa dos Santos, Dra. Thereza A. Soares e Dr. Caio de

Lima Firme.

“... com o passar dos tempos, você mudou. Deixou de ser você.

Deixou as pessoas apontarem pra sua cara e dizerem que você não

é bom. E quando as coisas ficaram difíceis, você começou a

procurar alguém para culpar... como uma grande sombra. O mundo

não é um mar de rosas. É um lugar ruim e asqueroso... e não

importa o quão durão você é... ele te deixará de joelhos e te

manterá assim se permitir. Nem você, nem eu, nem ninguém baterá

tão forte quanto à vida. Mas isso não se trata de quão forte

você pode bater. Se trata de quão forte pode ser atingido e

continuar seguindo em frente. Quanto você pode apanhar e

continuar seguindo em frente. É assim que a vitória é

conquistada!”

Rocky Balboa

“Empty your mind, be formless, shapeless —

like water. If you put water in a cup, it

becomes the cup; You put water into a bottle

it becomes the bottle; You put it in a teapot

it becomes the teapot. Water can flow or it

can crash. Be water, my friend”.

Lee Jun Fan (Bruce Lee).

RESUMO

Esta tese foi realizada em quatro capítulos, todos a nível teórico, enfocados

principalmente na densidade eletrônica. No primeiro capítulo, nos descrevemos a

aplicação de um minicurso para estudo das reações de Diels-Alder na Universidade

Federal do Rio Grande do Norte. Utilizando ferramentas de química computacional, os

estudantes puderam construir um determinado conhecimento e puderam associar

importantes aspectos da físico-química e da química orgânica. No segundo capítulo,

estudamos um novo tipo de interação química envolvendo átomos de hidrogênio de

carga neutra, denominada de ligação hidrogênio-hidrogênio (H-H). Nesse estudo

realizado com alcanos complexados, fornece novas e importantes informações acerca

das suas estabilidades envolvendo esse tipo de interação. Mostramos que a ligação H-H

desempenha um papel secundário na estabilidade de alcanos ramificados em

comparação aos seus isômeros menos ramificados ou lineares. No terceiro capítulo,

estudamos a estrutura eletrônica e a estabilidade do tetraedrano, de tetraedranos

substituídos e seus parentes de silício e germânio. Avaliamos o efeito do substituinte na

gaiola de carbono dos derivados do tetraedrano e os resultados indicam que fortes

grupos retiradores de elétrons (GRE) causam pequena instabilidade na gaiola carbônica,

em contrapartida, fracos GRE resulta em grande instabilidade. Mostramos que na

aromaticidade-σ, GRE e grupos doadores de elétrons (GDE) resultam em um

decréscimo e em aumento, respectivamente, dos índices de aromaticidade NICS e

D3BIA. Em adição, outro fator pode ser utilizado para explicar a estabilidade do tetra-

tert-butiltetraedrano, assim como a ligação H-H. GVB e ADMP também foram

utilizados para avaliar o efeito de substituintes na gaiola do tetraedrano. No quarto

capítulo, nós realizamos uma investigação teórica do efeito inibitório do fármaco

abiraterona (ABE), utilizado no tratamento do câncer de próstata, frente à enzima

CYP17, comparando as energias de interação e densidade eletrônica desse fármaco com

o substrato natural, a pregnenolona (PREG). Dinâmica molecular e docking foram

utilizados para obter os complexos CYP17-ABE e CYP17-PREG. Á partir da dinâmica

molecular, foi obtido que a ABE possui uma maior tendência de difusão água sítio de

interação da enzima CYP17, quando com a mesma tendência da PREG. Com o método

ONIOM (B3LYP:AMBER), encontramos que a energia de interação da ABE é 21,38

kcal mol-1

mais estável em comparação à PREG. Os resultados obtidos através da

QTAIM indicam que essa estabilidade se dá devido a uma maior densidade eletrônica

das interações entre a ABE e a CYP17.

Palavras-chave: Diels-Alder; Ligação H-H; Tetraedrano; CYP17.

ABSTRACT

This thesis was performed in four chapters, at the theoretical level, focused mainly on

electronic density. In the first chapter, we have applied an undergraduate minicourse of

Diels-Alder reaction in Federal University of Rio Grande do Norte. By using

computational chemistry tools students could build the knowledge by themselves and

they could associate important aspects of physical-chemistry with Organic Chemistry.

In the second chapter, we studied a new type of chemical bond between a pair of

identical or similar hydrogen atoms that are close to electrical neutrality, known as

hydrogen-hydrogen (H-H) bond. In this study performed with complexed alkanes,

provides new and important information about their stability involving this type of

interaction. We show that the H-H bond playing a secondary role in the stability of

branched alkanes in comparison with linear or less branched isomers. In the third

chapter, we study the electronic structure and the stability of tetrahedrane, substituted

tetrahedranes and silicon and germanium parents, it was evaluated the substituent effect

on the carbon cage in the tetrahedrane derivatives and the results indicate that stronger

electron withdrawing groups (EWG) makes the tetrahedrane cage slightly unstable

while slight EWG causes a greater instability in the tetrahedrane cage. We showed that

the sigma aromaticity EWG and electron donating groups (EDG) results in decrease and

increase, respectively, of NICS and D3BIA aromaticity indices. In addition, another

factor can be utilized to explain the stability of tetra-tert-butyltetrahedrane as well as H-

H bond. GVB and ADMP were also used to explain the stability effect of the

substituents bonded to the carbon of the tetrahedrane cage. In the fourth chapter, we

performed a theoretical investigation of the inhibitory effect of the drug abiraterone

(ABE), used in the prostate cancer treatment as CYP17 inhibitor, comparing the

interaction energies and electron density of the ABE with the natural substrate,

pregnenolone (PREG). Molecular dynamics and docking were used to obtain the

CYP1ABE and CYP17-PREG complexes. From molecular dynamics was obtained that

the ABE has higher diffusion trend water CYP17 binding site compared to the

PREG. With the ONIOM (B3LYP:AMBER) method, we find that the interaction

electronic energy of ABE is 21.38 kcal mol-1

more stable than PREG. The results

obtained by QTAIM indicate that such stability is due a higher electronic density of

interactions between ABE and CYP17.

Keywords: Diels-Alder; H-H bond; Tetrahedrane; CYP17.

LISTA DE FIGURAS – REFERÊNCIAL TEÓRICO

Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das

equações Kohn-Sham......................................................................................................30

Figura 2. Gráfico molecular do Ferroceno.....................................................................32

Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente

do campo vetorial ∇ρ.......................................................................................................33

Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35

Å......................................................................................................................................37

Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB

e GVB, comparadas com a energia exata não-relativística.............................................37

Figura 6. Representação do método ONIOM de um sistema proteico dividido em duas

camadas. Na camada baixa (low layer), os átomos são representados por estruturas do

tipo arame (wireframe) e na camada alta (high layer) os átomos são representados por

estruturas do tipo bola e bastão (ball-and-stick)..............................................................38

LISTA DE FIGURAS

Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder

cycloadditions by using computational chemistry – an undergraduate experiment.

Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of

Diels-Alder reactions (1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)-

(7)………………………………………………………………………………………48

Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine

and nitroethene (B); and 2-nitrobuta-1,3-diene and ethenamine (C)………………...…50

Figure 3. Optimized geometries of corresponding reactants and products from Diels-

Alder reactions 4 (A) and 7 (B). Selected bond lengths, in Angstroms, are

depicted………………………………………………………………………….…...…51

LISTA DE FIGURAS

Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane

complexes: a DFT, ab initio, QTAIM and ELF study

Figure 1. Virial graphs of C8H18 isomers. (A) Octane, (B) 3-methylheptane, (C) 3,3

dimethylhexane, (D) 2,2,3-trimethylpentane and (E) 2,2,3,3-tetramethylbutane. The

geometry optimization of the molecules were performed at ωB97xd/6-311++G(d,p)

level of theory…………………………………………………………………………..76

Figure 2. Randomized, non-optimized and optimized geometries of methane-methane,

ethane-ethane, propone-propane and butane-butane complexes depicting their

corresponding selected CH--HC dihedral angles, in degrees. The geometry optimization

of the molecules were performed at G4/CCSD(T)/6-311++G(d,p) level of theory. The

dashed lines represent the selected dihedral angles………………………………….…78

Figure 3. Randomized non-optimized initial geometry and optimized geometry of

methane-methane, ethane-ethane, propone-propane, butane-butane, pentane-pentane and

hexane-hexane complexes depicting their corresponding selected H-H interatomic

distances, in angstroms. The geometry optimization of the molecules were performed at

ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the selected

interatomic distances………………………………………………………………...…80

Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p)

level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane

(Pro-Pro) and butane-butane (But-But) along with the charge density, ρ(r), and

Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au. The others

complexes were not optimized due to lack of computational resources……………….81

Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p)

level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane

(Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane

(Hex-Hex) complexes along with the charge density, ρ(r), and Laplacian of the electron

density, ∇2 ρ(r) at H-H bond critical, in au…………………………………………..…82

Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in

kcal/mol, of selected alkanes at ωB97XD/6-311++G(d,p) level of theory…………….83

Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave

function, versus boiling point, in °C, of selected alkanes…………………………...….84

Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(H-

H)) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of selected

steps in the optimization steps……………………………………………………...…..85

Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to H-H bond

(EH) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of

selected steps in the optimization steps………………………………………………...86

Figure S1. Optimized geometries of alkane molecules. The alkane molecules were

optimized at (a) B3LYP/6-311++G(d,p), (b) M062X/6-311++G(d,p), (c) G4 method, (d)

G4/CCSD/6-311++G(d,p) (T) and (e) wB97XD/6-311++G(d,p). (1) Methane, (2)

ethane, (3) propane, (4) butane, (5) pentane and (6) hexane………………………..….95

Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p)

level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane

(Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane

(Hex-Hex) complexes………………………………………………………………..…96

Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p)

level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane

(Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane

(Hex-Hex) complexes………………………………………………………………..…97

Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methane-

methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane

(But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes……98

Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of

optimization steps………………………………………………………………………99

LISTA DE FIGURAS

Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and

Germanium Parents – a DFT, ADMP, QTAIM and GVB study.

Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and

germanium cages (G-I) with corresponding C1-C4, C1-C5, C1-Si, C1-N, Si-Si, Si-H,

Ge-Ge, Ge-H and Ge-C bond lengths (in Angstroms); values of selected angles (in

degree)………………………………………………………………………………107

Figure 2: Virial graphs of substituted tetrahedranes (A-F), silicon and germanium cages

(G-I) with the electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local

energy density, H(r) and relative kinetic energy density, G(r)/ρ at cage-substituent bond

critical, in au. Values of electronic density, ρ(r), Laplacian of the electron density, ∇2

ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization

index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in

the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b………………………….108

Figure 3: Gibbs free energy difference, in kcal mol-1

, from diazocompound (as a

reference) to cyclobutadiene and tetrahedrane structures, along with nitrogen, with

respect to the syntheses of 3a (A), 3b (B), 3f (C) and 3e (D)…………...……………109

Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of

oxygen of nitro (-NO2) functional group substituent (LP1, LP1’, LP2 and LP2’) (A and

B) and singly occupied GVB orbitals of C-C sigma bond [VB(C-C)1, VB(C-C)2 and

VB(C-C)’] (B)………………………………………………………………………...113

Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS

versus D3BIA for 3a-3f, 4a, 4b and 5a……………………………………………….115

Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond

[VB(C-C)1, VB(C-C)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D)…...117

Figure 7: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,

of tetrahedrane (3a) using ADMP method. Some structures are depicted at the

corresponding selected points of the trajectory…………………………………….…118

Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,

of 3c using ADMP method. The structure at the end of trajectory is also depicted…..119

Figure 9. Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,

of 3e using ADMP method. A couple of structures are depicted at the selected

corresponding points of the trajectory………………………………………………...119

LISTA DE FIGURAS

Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation

of inhibitory effect of Abiraterone on CYP17.

Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT

complex undergoes phosphorylation and is translocated to the nucleus in the form of

dimer. Inside the nucleus, AR-dimer binds to DNA coregulators leading to transcription

activation……………………………………………………………………………...125

Figure 2. Chemical structure of ABE……………………………..………………….126

Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP

(C)……………………………………………………………………………………..128

Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17-

PREG (B) complexes …………………………………………………………………133

Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE

complex of each protonation state of His 373. HISD (A), HISE (B) and HISH

(C)………………………………………………………………………………..……135

Figure 6. Plots of RMSD (A) and χ (B) of the His373 of each protonation state. The

data represent the average of five 10-ns snapshots for each protonation state expressed

as mean standard deviation. It was considered a p value less than 0.05………..…….135

Figure 7. The binding poses of PREG in the active site of the enzyme CYP17….….136

Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG

complex……………………………………………………………………………….136

Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG,

heme group and CYS442 of CYP17-PREG complex (B)…………………………….137

Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of

the CYP17-ABE (A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the

ONIOM (B3LYP:AMBER) level……………………………………………………..139

Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B)

complexes…………………………………………………………………………….139

Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in

the triplet and quintet states …………………………………………………………157

Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes

before (yellow) and after (red) molecular dynamics simulation. The picture was

rendered using VMD……………………………………………………….…………158

Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The

picture was rendered using VMD, “blue” indicates high, “white” indicates intermediate

and “red” low values. High temperature factors imply exibility and low ones

rigidity……………………………………………………………………………...…158

LISTA DE TABELAS – REFERÊNCIAL TEÓRICO

Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos........32

LISTA DE TABELAS

Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder

cycloadditions by using computational chemistry – an undergraduate experiment.

Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1

, and entropy, S, in

cal mol-1

K-1

of all studied molecules…………………………………………………..48

Table 2. Gibbs free energy change, ∆G, Enthalpy change, ∆H, entropy change and

temperature product, in kcal mol-1

K-1, T∆S, entropy change, in kcal mol

-1, ∆S, and

equilibrium constant logarithm, ln K, of Diels-Alder reactions (1)–(7)………………..49

Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap

between HOMO and LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the

Diels-Alder reactions (1) – (7)……………………………………………….…………50

Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in

Angstroms, of the studied Diels-Alder reactions…………………………….…………52

LISTA DE TABELAS

Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane

complexes: a DFT, ab initio, QTAIM and ELF study

Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb)

in branched alkanes…………………………………………………………………….75

Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes

(Gcomplex), two-fold Gibbs energy of the corresponding isolated alkane (2*Gisolated), and

complex stability (G(complex-2*isolated)) of the alkane complexes………………………..82

Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density,

∇2 ρ(r) at bond critical point of a selected H-H bond in alkane complexes…………....87

LISTA DE TABELAS

Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and

Germanium Parents – a DFT, ADMP, QTAIM and GVB study

Table 1. Values of Gibbs free energy difference and enthalpy difference for the

corresponding tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to

cyclobutadiene (and derivatives), ΔG1 and ΔH1, from diazocompound to tetrahedrane

(and derivatives), ΔG2 and ΔH2, and from cyclobutadiene (and derivatives) to

tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol-1, respectively. Values of

delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or

germanium atoms in the tetrahedrane (or parent) cage, bond path angle, in degree, from

bonds in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b……………...…110

Table 2. Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole

moment, M(Ω), atomic volume, V(Ω) and localization index, LI of carbon, silicon and

germanium atoms in tetrahedrane, silicon and germanium derivatives, respectively...111

Table 4. Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone

pairs (LP1, LP2, LP1’ and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’

and VB(C-C)2]………………………………………………………………………...113

Table 3. Average DI, DI, of the carbon or silicon or germanium atoms in the

tetrahedrane (or parent) cage, the corresponding DIUs, the degree of degeneracy (δ)

involving the atoms of the tetrahedrane (or parent) cage, the charge density of the cage

critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and

5b……………………………………………………………………………………...115

Table 5. Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (C-

C)2, of 3a, 3b, 3f and cubane. ……………………………………………………...….117

LISTA DE TABELAS

Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation

of inhibitory effect of Abiraterone on CYP17.

Table 1. Binding energies of CYP17 with PREG…………….…………...……….…135

Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal

mol-1

, between CYP17 (p)/water (w) and ligands ABE and PREG…………………..138

Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and

CYP17-PREG complexes……………………………………………………………..138

Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or

pregnenolone) and CYP17 active site……………………………………………..….139

TABLE S1. Electronic energy, in a.u., bond length Fe-NH2 and bond length Fe-NH3, in

angstroms, of triplet and quintet states of the Fe(NH3)2(NH2)2 with different functionals

using 6-311G(d,p) basis set……………………………………………………..…….157

LISTA DE ABREVIATURAS, ACRÔNIMOS E SIGLAS EMPREGADAS.

ρ Densidade eletrônica

𝛁𝟐𝝆 Laplaciano da densidade eletrônica

ΔG Variação da energia livre de Gibbs

AMBER Assisted Model Building with Energy Refinement (Campo de força ou

pacote de programas de simulação )

BCP Bond Critical Point (Ponto Crítico de Ligação)

DFT Density Functional Theory (Teoria Funcional da Densidade)

GVB Generalized Valence Bond (Ligação de Valência Generalizada)

HF Hartree-Fock

Hb Total Energy Density (Densidade de Energia Total)

QM/MM Mecânica Quântica/Mecânica Molecular

QTAIM Quantum Theory of Atoms in Molecules (Teoria Quântica de

Átomos em Moléculas)

RESP Restrained Electrostatic Potential

ABE Abiraterona

PREG Pregnenolona

SUMÁRIO

1. INTRODUÇÃO GERAL ................................................................................... 24

2. REFERENCIAL TEÓRICO ............................................................................. 25

2.1. TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT)

...........................................................................................................................25

2.2. TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of

Atoms in Molecules - QTAIM) ................................................................................ 30

2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization

Function - ELF)....................................................................................................... 34

2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE

VALÊNCIA GENERALIZADA (Generalized Valence Bond – GVB)..................... 36

2.5. CÁLCULOS DE QM/MM ............................................................................ 38

2.6. SIMULAÇÃO MOLECULAR ..................................................................... 39

2.7. DOCAGEM MOLECULAR ......................................................................... 40

3. CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical

Education.................................................................................................................... 41

4. CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry

A. ..................................................................................................................................70

5. CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry. ...... 100

6. CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The

Journal of Physical Chemistry B. .............................................................................. 124

7. REFERÊNCIAS GERAIS ............................................................................... 159

24

1. INTRODUÇÃO GERAL

A química é a ciência que lida com a construção, transformação e propriedades das moléculas. A

química teórica é uma subárea da química onde os métodos matemáticos são combinados com as leis

fundamentais da física com o objetivo de se estudar processos de relevância química. (Clark, 1985)

A química computacional é uma disciplina que se estende muito além dos limites tradicionais que

separam a química, a física, a bioquímica e a ciência da computação, permitindo o estudo aprofundado de

átomos, moléculas e macromoléculas através da ferramenta computacional. A química computacional pode

atuar como ferramenta de apoio na análise e interpretação de dados experimentais, através de informações

que muitas vezes não são possíveis de serem obtidas diretamente de experimentos (Morgon et al., 1995;

Morgon et al., 1997; Morgon e Riveros, 1998; Morgon et al., 2000), ou na previsão de propriedades

diversas. Além disso, a química computacional pode ser aplicada nas mais diversas áreas da química, assim

como: Físico-química, química orgânica, química inorgânica, química analítica, bioquímica, além de ser

utilizada como ferramenta de ensino.

Na química orgânica, por exemplo, existem muitas explicações consolidadas acerca da estabilidade de

alcanos e sua relação o ponto de fusão e ebulição. A química computacional pode nos fornecer novas e

importantes informações acerca do modo de interação dessas moléculas e relacioná-las com as suas

propriedades macroscópicas.

A química computacional também pode ser aplicada no desenvolvimento de moléculas de alta energia de

interesse militar. Moléculas orgânicas de alta tensão em formato de gaiola podem ser usadas como

potenciais explosivos em resposta a estímulos externos. (Katz e Acton, 1973; Zhou et al., 2004)

O estudo teórico na área de bioquímica possui uma gama de aplicações como a análise conformacional

de grandes sistemas moleculares de importância biológica; o estudo da interação enzima-substrato e

principalmente no planejamento racional de novos fármacos. Ainda existe um novo cenário surgindo com o

aparecimento de uma recente estratégia de cálculos híbridos que envolvem a mecânica quântica e a

mecânica molecular, conhecida como QM/MM (Quantum Mechanics/Molecular Mechanics). Assim, por

exemplo, existe a possibilidade de descrição da quebra e formação de ligações químicas em reações que

envolvam grandes sistemas através da mecânica quântica, onde é sabido que a mecânica molecular é falha

na descrição das propriedades eletrônicas.

Portanto, para essa tese, foram realizados quatro estudos teóricos das áreas descritas acima, utilizando

métodos quânticos, de mecânica molecular e híbridos objetivando um estudo mais aprofundado, assim como

fornecer novas perspectivas em cada tema abordado.

25

2. REFERENCIAL TEÓRICO

2.1. TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT)

A teoria funcional da densidade (DFT) é um dos métodos quanto-mecânicos mais populares e bem-

sucedidos da atualidade. (Capelle, 2006) Essa metodologia é rotineiramente aplicada para cálculos de

energia de ligação de moléculas na química e para cálculos de estrutura de bandas de sólidos em física. Por

outro lado, as primeiras aplicações em campos considerados mais distantes da mecânica quântica, assim

como a biologia (Devipriya e Kumaradhas, 2012; Sicolo et al., 2014; Bigler et al., 2015; Jitonnom et al.,

2015) e a mineralogia (Deng et al.; Liu et al., 2013; Mikhlin et al., 2014; Kahlenberg et al., 2015) estão

começando a aparecer.

Para se ter uma ideia do que é a teoria funcional da densidade, se faz necessário algum conhecimento

prévio da mecânica quântica. Na mecânica quântica, toda e qualquer informação relevante a um sistema está

contida na sua função de onda, Ψ. Os graus de liberdade nucleares aparecem somente na forma de energia

potencial, ν(r), atuando nos elétrons devido à aproximação de Born-Oppenheimer.(Woolley e Sutcliffe,

1977) Devido a isso, a função de onda depende apenas das coordenadas eletrônicas. Essa função de onda

não-relativística é calculada a partir da equação de Schrödinger para um elétron se movendo em um

potencial ν(r):

[−ħ2∇2

2𝑚+ 𝜈(𝒓)]𝛹(𝒓) = 휀𝛹(𝒓) (1)

onde ∇2 é o operador Laplaciano definido como a soma dos operadores diferenciais em coordenadas

cartesianas:

∇2=𝜕2

𝜕𝑥2+𝜕2

𝜕𝑦2+𝜕2

𝜕𝑧2 (2)

para um sistema multieletrônico, a equação de Schrödinger se torna:

[∑(−ħ2∇2

2𝑚+ 𝜈(𝒓))

𝑁

𝑖

+∑𝑈(𝒓𝑖, 𝒓𝑗)

𝑖<𝑗

]𝛹(𝒓1, 𝒓2, … , 𝒓𝑁) = 𝐸𝛹(𝒓1, 𝒓2, … , 𝒓𝑁) (3)

onde N é o número de elétrons e U(ri,rj) é a interação elétron-elétron. Para um sistema Coulômbico, temos:

=∑𝑈(𝒓𝑖 , 𝒓𝑗)

𝑖<𝑗

=∑𝑞2

|𝒓𝑖 − 𝒓𝑗| (4)

𝑖<𝑗

e o operador energia cinética é dado por:

26

= −ħ2

2𝑚∑∇𝑖

2

𝑖

(5)

e o potencial de interação elétron-núcleo é dado por:

=∑𝜈(𝒓𝑖) =

𝑖

∑𝑄𝑞

|𝒓𝑖 − 𝑹|𝑖

(6)

Onde Q é a carga nuclear (Pople, 1999) e R é a posição do núcleo.

A função de onda Ψ em si, não é uma observável, pois não é mensurável. Entretanto, o quadrado do

valor absoluto da função de onda possui uma interpretação física. (Koch e Holthausen, 2001) Essa

interpretação só é possível porque o produto de um número complexo pelo seu complexo conjugado é real e

não negativo, onde:

|𝛹(1,2,… 𝑁,)|2𝑑1𝑑2…𝑑𝑁 (7)

representa a probabilidade que os elétrons 1, 2, ..., N são encontrados simultaneamente em volumes

𝑑1𝑑2…𝑑𝑁. Desde que esses elétrons sejam indistinguíveis, não deve haver troca de coordenadas de

quaisquer dois elétrons (i e j) se os mesmos são comutados, ou seja:

|𝛹(1,2, … , 𝑖, 𝑗, … , 𝑁,)|2= |𝛹(1,2, … , 𝑗, 𝑖, … , 𝑁,)|

2 (8)

Isso garante que as funções de onda possuam sinais contrários resultando em funções antissimétricas

aplicadas à férmions com spins semi-inteiros. Elétrons são férmions com spin = ½ e Ψ deve ser, portanto,

antissimétrica com respeito a permutação das coordenadas espaciais e spin de quaisquer dois elétrons:

𝛹(1,2, … , 𝑖, 𝑗, … , 𝑁,) = −𝛹(1,2, … , 𝑖, 𝑗, … , 𝑁,) (9)

esse princípio da antissimetria, representa uma generalização quanto-mecânica denominada de princípio da

exclusão de Pauli (onde dois elétrons não podem ocupar o mesmo estado). Uma consequência lógica da

interpretação probabilística da função de onda, é que integrando a eq. (7) em todo o intervalo sobre todas as

variáveis, o resultado é igual a um. Em outras palavras, a probabilidade de encontrar os N elétrons em

qualquer lugar do espaço corresponde a uma unidade,

∫…∫|𝛹(1,2,… 𝑁,)|2𝑑1𝑑2…𝑑𝑁 = 1 (10)

a função de onda que satisfaz a eq. (10) é dita ser normalizada.

A interpretação probabilística da eq. (10) da função de onda leva diretamente à densidade eletrônica,

ρ(r), que é definida como a integral múltipla sobre todas as coordenadas spin e sobre todas as variáveis

espaciais de todos os elétrons:

27

𝜌() = 𝑁∫…∫|𝛹(1,2,… 𝑁,)|2𝑑1𝑑2…𝑑𝑁 (11)

𝜌() determina a probabilidade de encontrar qualquer um dos N elétrons dentro de um volume 𝑑(𝒓𝟏). 𝜌()

é a densidade de probabilidade, comumente denominada de densidade eletrônica.

Diferentemente de uma função de onda, a densidade eletrônica é uma observável e pode ser

mensurada experimentalmente, como por exemplo, através da difração de raios X. (Averkiev et al., 2014) A

teoria funcional da densidade é um método baseado na análise do sistema molecular através da densidade

eletrônica. O primeiro trabalho com a densidade eletrônica foi realizado no início do século XX com o

objetivo de explicar as propriedades térmicas e elétricas dos metais. (Chudnovsky, 2007) Mas foi somente

em 1964, através do trabalho de Hohenberg e Kohn (Hohenberg e Kohn, 1964) que mostra que a densidade

eletrônica, de fato, determina o operador Hamiltoniano e, assim, todas as propriedades do sistema. Ainda

assim, o método precisava buscar os funcionais adequados, os chamados funcionais exatos, necessários para

serem incorporados na expressão da energia total eletrônica. Somente com os trabalhos de Kohn-Sham

(Kohn e Sham, 1965) melhorando o tratamento matemático das equações de densidade e do funcional, que a

teoria funcional da densidade avançou como ferramenta de cálculo teórico para estudo das propriedades

moleculares com aplicações computacionais. (Kohn et al., 1996)

O primeiro teorema de Hohenberg-Kohn afirma que a densidade eletrônica determina o potencial

externo [ν(r)] devido aos núcleos, assim como também determina o número total de elétrons, N, via sua

normalização ∫𝜌(𝒓)𝑑𝑟 = 𝑁. N e ν(r) determinam o operador Hamiltoniano molecular (Hop) [vide eq. (3)]

que, por sua vez, determina a energia do sistema através da equação de Schrödinger 𝐻𝑜𝑝𝛹 = 𝐸𝛹. A

densidade eletrônica determina a energia do sistema e todas as outras propriedades eletrônicas do sistema no

estado fundamental. (Geerlings et al., 2003) O esquema 1 mostra a relação existente entre as variáveis

descritas acima.

Esquema 1:

consequentemente, E é um funcional de ρ:

ν

N

Hop E

28

𝐸 = 𝐸[𝜌] (12)

O segundo teorema de HK estabelece que o funcional FHK é universal, ou seja, o mesmo é usado para

todos os problemas de estrutura eletrônica (Burke, 2007), que é dado por:

𝐹𝐻𝐾 = 𝑇[𝜌] + 𝑉𝑒𝑒[𝜌] (13)

onde T[ρ] é o funcional de energia de energia cinética eletrônica e Vee[ρ] é o funcional de interação elétron-

elétron. Esse funcional é um componente da expressão do funcional da energia (Geerlings et al., 2003):

𝐸[𝜌] = ∫𝜌(𝒓)𝜈(𝒓)𝑑𝒓 + 𝐹𝐻𝐾 (14)

dessa forma, o segundo teorema de HK estabelece um princípio variacional onde 𝐸[𝜌] ≥ 𝐸0.

Para contornar o problema dos teoremas de HK, Kohn e Sham propuseram uma abordagem para a

aproximação dos funcionais de energia cinética e potencial elétron-elétron. (Kohn e Sham, 1965) Eles

introduziram um sistema fictício de N elétrons não-interagentes para serem descritos através de um único

determinante em N orbitais ϕi.

Kohn e Sham reescreveram a equação (14) e explicitaram a repulsão eletrônica de Coulomb e

definiram uma nova função universal G[ρ]:

𝐸𝜈[𝜌] = 𝐺[𝜌] +1

2∫∫

𝜌(𝑟1)𝜌(𝑟2)

|𝑟1 − 𝑟2|𝑑𝑟1𝑑𝑟2 +∫𝜌(𝑟)𝜈(𝑟)𝑑𝑟 (15)

em que

𝐺[𝜌] = 𝑇𝑠[𝜌] + 𝐸𝑥𝑐[𝜌] (16)

onde 𝑇𝑠[𝜌] é o funcional energia cinética de um sistema de elétrons que não interagem e 𝐸𝑥𝑐[𝜌] inclui o

termo de interação elétron-elétron não-clássica (troca e correlação) e a parte residual da energia cinética,

𝑇[𝜌] − 𝑇𝑠[𝜌], em que 𝑇[𝜌] é a energia cinética exata para o sistema de elétrons que interagem.

O Hamiltoniano dos elétrons que não interagem com o potencial efetivo, 𝜈𝑒𝑓 , é dado por:

𝐻𝐾𝑆 = −1

2∇2 + 𝜈𝑒𝑓 (17)

A obtenção da função de onda Kohn-Sham, ΨKS

, do estado fundamental do sistema de elétrons que

não interagem, descrito pelo Hamiltoniano da equação (17), é dada pelo produto anti-simetrizado de N

orbitais de um elétron, ϕi(r), através do determinante de Slater (Morgon e Coutinho, 2007):

29

𝛹𝐾𝑆 =1

√𝑁!||

𝜙1𝐾𝑆(𝑟1) 𝜙2

𝐾𝑆(𝑟1) …

𝜙1𝐾𝑆(𝑟2) 𝜙2

𝐾𝑆(𝑟2) …⋮ ⋮ ⋱

𝜙𝑁𝐾𝑆(𝑟1)

𝜙𝑁𝐾𝑆(𝑟2)⋮

𝜙1𝐾𝑆(𝑟𝑁) 𝜙2

𝐾𝑆(𝑟𝑁) … 𝜙𝑁𝐾𝑆(𝑟𝑁)

|| (18)

Por sua vez, os orbitais Kohn-Sham, 𝜙𝑖𝐾𝑆, são obtidos a partir da equação de Schrödinger para um elétron:

(−1

2∇2 + 𝜈𝑒𝑓)𝜙𝑖

𝐾𝑆 = 휀𝑖𝜙𝑖𝐾𝑆 (19)

A relação existente entre o sistema real e o sistema que não interage é dada por:

𝜌𝑠(𝑟) =∑2|𝜙𝑖𝐾𝑆(𝑟)|

2= 𝜌0(𝑟) (20)

𝑁

𝑖

onde ρ0(r) é a densidade eletrônica no estado fundamental. A energia cinética é dada por:

𝑇𝑠[𝜌] = −1

2∑⟨𝜙𝑖

𝐾𝑆|∇2|𝜙𝑖𝐾𝑆⟩ (21)

𝑁

𝑖

e o potencial efetivo é dado por:

𝜈𝑒𝑓 = 𝜈(𝑟) +𝜌(𝑟1)

|𝑟 − 𝑟1|𝑑𝑟1 + 𝜈𝑥𝑐(𝑟) (22)

em que

𝜈𝑥𝑐(𝑟) =𝛿𝐸𝑥𝑐[𝜌]

𝛿𝜌(𝑟) (23)

A equação (23) é o funcional proveniente da energia de correlação e troca [equação (15)] em relação à

densidade.

Das equações acima, percebe-se que o potencial efetivo é dependente da densidade, ρ(r), a qual

depende dos orbitais Kohn-Sham, 𝜙𝑖𝐾𝑆, que deverão ser encontrados. Para isso, utiliza-se um procedimento

autoconsistente que é mostrado na Figura 1.

Primeiramente é construída a densidade inicial, ρ0, para se calcular o potencial efetivo, 𝜈𝑒𝑓 , para se

obter o Hamiltoniano. Assim, resolve-se a equação de Kohn-Sham para se obter um conjunto de

autofunções, 𝜙𝑖𝐾𝑆, que irão compor o determinante de Slater. Em seguida é obtida a função de onda Kohn-

Sham, ΨKS

, do qual a nova densidade eletrônica, ρ1, pode ser calculada. A densidade recém-calculada, ρn é

comparada com a anterior ρn-1. Se ρn ≠ ρn-1 o ciclo é reiniciado até que a densidade não se altere de um ciclo

30

para outro, ou seja, ρn = ρn-1. Quando essa condição for atingida, significa dizer que essa densidade de carga

minimiza a energia e a convergência foi alcançada.

Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das equações Kohn-Sham.

Fonte: Autor, 2015.

2.2.TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of Atoms in

Molecules - QTAIM)

A teoria quântica de átomos em moléculas (QTAIM) é um modelo quântico considerado eficiente no

estudo da ligação química (Cortés-Guzmán e Bader, 2005) e caracterização das interações intra e

intermoleculares. (Grabowski et al., 2004; Oliveira et al., 2009; Monteiro e Firme, 2014).

A teoria quântica de átomos em moléculas (Bader, 1990), desenvolvida pelo Professor Richard F. W.

Bader e seus colaboradores, as propriedades observáveis dos átomos constituintes de um sistema molecular

estão contidas na sua densidade eletrônica, ρ. As trajetórias da densidade eletrônica são obtidas à partir do

vetor gradiente da densidade eletrônica (∇ρ) (Popelier, 2000), que é dada pela primeira derivada da

ρ0

Cálculo do potencial efetivo

Resolução da equação de KS

Determinante de Slater

Obtenção de ΨKS

Cálculo da nova densidade

ρn é a densidade procuradaSimNão

31

densidade eletrônica sobre todas as coordenadas (Equação 24) e o conjunto de trajetória desse gradiente

formam as bacias atômicas (Ω).

∇𝜌 = 𝒊𝑑𝜌

𝑑𝑥+ 𝒋

𝑑𝜌

𝑑𝑦+ 𝒌

𝑑𝜌

𝑑𝑧 (24)

Um ponto crítico (PC) na densidade eletrônica é um ponto no espaço em que cada derivada do ∇ρ é zero

(∇ρ = 0). Para se distinguir um ponto crítico de mínimo local, máximo local ou um ponto crítico de sela, são

consideradas as derivadas segundas da densidade eletrônica. Existem nove derivadas segundas de ρ(r) que

podem ser dispostas em uma matriz Hessiana, mostrada na equação 25 (Bader, 1990):

∇∇𝜌 =

(

𝜕2𝜌

𝜕𝑥2𝜕2𝜌

𝜕𝑥𝜕𝑦

𝜕2𝜌

𝜕𝑥𝜕𝑧

𝜕2𝜌

𝜕𝑦𝜕𝑥

𝜕2𝜌

𝜕𝑦2𝜕2𝜌

𝜕𝑦𝜕𝑧

𝜕2𝜌

𝜕𝑧𝜕𝑥

𝜕2𝜌

𝜕𝑧𝜕𝑦

𝜕2𝜌

𝜕𝑧2 )

(25)

A matriz Hessiana pode ser diagonalizada através da rotação do sistema de coordenadas r (x, y, z) r

(x’, y’, z’). Os novos eixos coordenados são chamados de eixos principais da curvatura porque a magnitude

das três derivadas da segunda de ρ, calculadas com respeito a esses eixos, são maximizados. A rotação do

sistema de coordenadas é acompanhada de uma transformação unitária, r’ = rU, onde U é uma matriz

unitária construída à partir de uma conjunto de três equações de autovalores (𝛁𝛁𝝆)𝒖𝑖 = 𝜆𝑖𝒖𝑖 (com 𝑖 = 1, 2,

3) em que 𝒖𝑖 é o inésimo autovetor em U. A equação matricial U-1

(𝛁𝛁𝝆)U = Λ transforma a Hessiana na

sua forma diagonal, que pode ser escrita como:

𝛬 =

(

𝜕2𝜌

𝜕𝑥′20 0

0𝜕2𝜌

𝜕𝑦′20

0 0𝜕2𝜌

𝜕𝑧′2)

= (𝜆1 0 00 𝜆2 00 0 𝜆3

) (26)

em que λ1, λ2 e λ3 são as curvaturas da densidade em relação aos três eixos principais x’, y’ e z’.

A soma dos autovalores da matriz Hessiana é conhecida como o Laplaciano da densidade eletrônica

(∇2ρ), que é dado pela equação 27:

∇2𝜌 =𝜕2𝜌

𝜕𝑥2+𝜕2𝜌

𝜕𝑦2+𝜕2𝜌

𝜕𝑧2= 𝜆1 + 𝜆2 + 𝜆3 (27)

32

Um ponto crítico pode ser classificado de acordo com o seu ranking, denotado por ω, que é o número

de autovalores não-zero da ρ(r) de um ponto crítico e pela assinatura, denotada por σ, que é a soma

algébrica dos sinais dos autovalores. Assim, um ponto crítico é caracterizado pelo conjunto de valores (ω,σ),

como pode ser exemplificado na Tabela 1. (Firme, 2007) Existem quatro tipo de pontos críticos estáveis,

possuindo três autovalores não-zero. O primeiro deles é o ponto crítico (3,-3), denominado de atrator nuclear

(Nuclear Attractor – NA), que possui todos os autovalores negativos, sendo a sua densidade eletrônica ρ(r)

um máximo local; o segundo ponto crítico é o (3,-1), denominado de ponto crítico de ligação (Bond Critical

Point - BCP), que possui dois autovalores negativos (λ1 e λ2) e um positivo (λ3) com ρ(r) sendo um máximo

no plano definido pelos autovetores correspondentes e um mínimo ao longo dos três eixos que é

perpendicular a esse plano; o terceiro ponto crítico (3,+1) é o ponto crítico do anel (Ring Critical Point -

RCP) que possui dois autovalores positivos (λ2 e λ3) e um negativo (λ1) com ρ(r) sendo um mínimo e por

último o ponto crítico de gaiola (Cage Critical Point – CCP) (3,+3) que possui todos os autovalores

positivos, sendo a ρ(r) um mínimo local. (Matta et al., 2007) Todos os pontos críticos estão representados na

Figura 2.

Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos.

Nome Acrônimo λ1 λ2 λ3 (ω,σ)

Atrator Nuclear NA - - - (3,-3)

Ponto Crítico da Ligação BCP - - + (3,-1)

Ponto Crítico do Anel RCP - + + (3,+1)

Ponto Crítico da Gaiola CCP + + + (3,+3)

Figura 2. Gráfico molecular do Ferroceno. Fonte: (Firme et al., 2010).

(3,+3)

(3,-1)

(3,-3)

(3,+1)

33

Um ponto crítico (3,-3) age como um atrator do campo vetorial do ∇ρ, ou seja, existe uma vizinhança

aberta do atrator que é invariante ao fluxo de ∇ρ tal que qualquer caminho do gradiente originado nessa

vizinhança aberta termina no atrator. Existem também caminhos de gradiente que terminam ou se originam

nos pontos críticos (3,-1). Os caminhos de gradiente que se originam nos pontos críticos (3,-1) definem os

caminhos de ligação (Figura 2). Os dois caminhos de gradiente definem uma linha através da densidade

eletrônica ligando os núcleos vizinhos ao longo do qual ρ(r) é máximo em relação a qualquer linha vizinha.

(Bader et al., 1979). Essa linha é encontrada entre cada par de núcleos cujas bacias atômicas compartilham

uma superfície interatômica comum e é chamada de interação atômica. A existência do ponto crítico (3,-1) e

a sua linha associada de interação atômica indicam que a densidade eletrônica é acumulada entre os núcleos

que estão ligados. Esse acúmulo de carga é a condição necessária quando as forças de Feyanman que atuam

nos núcleos e elétrons estão em equilíbrio.(Bader e Fang, 2005) Sendo assim, a presença da linha de

interação atômica satisfaz as condições necessárias para que os átomos estejam ligados. Essa linha de

interação é denominada de caminho de ligação e o ponto crítico (3,-1) é chamado de ponto crítico de

ligação. Mais a frente voltaremos a falar do caminho de ligação.

A densidade eletrônica máxima nas posições dos núcleos resulta em uma topologia originada através

das trajetórias da densidade eletrônica são obtidas à partir do vetor gradiente ∇ρ. Essa topologia é

particionada do espaço molecular em regiões mononucleares, Ω, denominadas de bacias atômicas. (Matta et

al., 2007) A superfície de ligação entre duas bacias atômicas é denominada de superfície de fluxo zero

(Figura 3), onde o produto escalar do vetor ∇ρ e o vetor normal [n(r)] se anulam (Equação 28).

Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente do campo vetorial ∇ρ.

Fonte: (Bader, 1990).

∇𝜌. 𝑛(𝑟) = 0 (28)

Superfície de fluxo zero

34

Existe um ponto na superfície de fluxo zero onde os vetores gradiente se anulam (∇ρ = 0), pois esses

vetores apontam para os núcleos atômicos e, consequentemente, ocorre formação do ponto crítico de ligação

(3,-1).

A presença de uma superfície interatômica de fluxo zero entre dois átomos ligados é sempre

acompanhada por uma linha de densidade local máxima, denominada de caminho de ligação. O caminho de

ligação é um indicador universal de todos os tipos de ligação química: interações fracas, fortes, camada

fechada e aberta.(Bader, 1998)

O conjunto de caminhos de ligação que conectam os núcleos dos átomos ligados em uma geometria

de equilíbrio, assim como os seus pontos críticos de ligação, é denominado de gráfico molecular. (Wiberg et

al., 1987) O gráfico molecular é o resultado direto das propriedades topológicas principais da distribuição de

cargas de um sistema em que o máximo local, os pontos críticos (3,-3), ocorrem nas posições dos núcleos e

os pontos críticos (3,-1) que conectam certos pares de núcleos em uma molécula (Figura 2). (Bader, 1990)

Por outro lado, existe outro tipo de gráfico, que é considerado um “espelho” do gráfico molecular. Esse tipo

de gráfico é definido por um conjunto de linhas com densidade máxima de energia potencial negativa. Em

outras palavras, existe uma única linha com densidade máxima de energia potencial negativa conectando os

mesmos núcleos de um caminho de ligação. (Keith et al., 1996) Essa linha de “máxima estabilidade” no

espaço real é denominada de “caminho virial”. O conjunto de caminhos viriais associados aos pontos

críticos constituem o gráfico virial.

2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization Function - ELF)

A função de localização eletrônica (Electron Localization Function – ELF) descrita primeiramente por

Becke e Edgecombe (Becke e Edgecombe, 1990) como uma função local que demonstra o quanto o

princípio da exclusão de Pauli é eficiente em um dado ponto do espaço molecular. Ou seja, o ELF fornece

uma descrição quantitativa do princípio da exclusão de Pauli. O ELF é definido como:

𝜂(𝑟) =1

1 + (𝐷 𝐷ℎ⁄ )2 (29),

em que,

𝐷 =1

2∑|∇ѱ𝑖|

2 −1

8

|∇𝜌|2

𝜌 (30)

𝑁

𝑖=1

e

35

𝐷ℎ =3

10(3𝜋2)2/3𝜌5/3 (31), 𝜌 = ∑|ѱ𝑖|

2 (32),

𝑁

𝑖=1

onde as somas são todos os (spin) orbitais ѱi(r) monocupados.(Savin, 2005) O valor do ELF é confinado ao

intervalo [1,0], onde a tendência para 1 significa que os spins paralelos são amplamente improváveis, e onde

há, portanto, uma alta probabilidade de spins opostos. A tendência para zero indica uma alta probabilidade

de regiões com os mesmos pares de spin. A função de localização eletrônica fornece um rigoroso suporte

para a análise da função de onda e da estrutura eletrônica em moléculas e cristais.(Noury et al., 1999) A

topologia da ELF tem sido amplamente empregada no estudo da ligação química.(Noury et al., 1998;

Chevreau et al., 2001; Chesnut, 2002; Chevreau, 2004; Rosdahl et al., 2005) O particionamento topológico

do campo gradiente da ELF (Häussermann et al., 1994; Silvi e Savin, 1994) fornece bacias de atratores

correspondentes às ligação químicas e aos pares de elétrons isolados. Em uma molécula podemos encontrar

dois tipos de bacias. A primeira como sendo as bacias de caroço que estão em torno dos núcleos com

número atômico Z > 2 denominado de C(A), onde A é o símbolo atômico do elemento. O outro tipo de bacia

é denominado de bacia de valência. As bacias de valência são caracterizadas pelo número de camadas de

valência que a bacia possui, em outras palavras, pelo número de bacias de caroço que os elementos

compartilham nas suas fronteiras. Esse número é chamado de ordem sináptica. Assim, existem bacias

monosinápticas, disinápticas, trisinápticas e assim por diante, especificadas como V(A), V(A, X), V(A, X,

Y) respectivamente. Bacias monosinápticas correspondem aos pares de elétrons isolados do modelo de

Lewis. Bacias polisinápticas correspondem aos pares de elétrons compartilhados do modelo de Lewis.

A função local ELF, η(r), fornece a informação química e ∇η(r) fornece dois tipos de pontos no

espaço real: pontos críticos onde (r)=0 e para os demais pontos, (r)≠0. Os pontos críticos são

caracterizados pelos seus índices que são os números de autovalores positivos da matriz Hessiana de η(r). A

topologia da ELF já foi aplicada no estudo da ligação de hidrogênio.(Fuster e Silvi, 2000; Alikhani e Silvi,

2003; Alikhani et al., 2005; Soler et al., 2005) No complexo FH•••N2 o valor da ELF entre as bacias V(F,H)

e V(N) é 0,047. Enquanto para o complexo FH•••NH3 o valor de (r) é 0.226. Esses valores são atribuídos

às ligações fracas e fortes respectivamente. Nesse trabalho, a ELF foi utilizada como um método

complementar para avaliar a ligação hidrogênio-hidrogênio entre os complexos de alcanos estudados.

36

2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE VALÊNCIA

GENERALIZADA (Generalized Valence Bond – GVB)

A descrição da estrutura eletrônica de átomos e moléculas em termos de orbitais eletrônicos fornece os

fundamentos básicos para teorias mais complexas.(Petersson, 1974) A energia obtida à partir de cálculos

baseados em orbitais possuem geralmente um erro de cerca de 0,015 a.u. (10 kcal mol-1

) para cada elétron.

Embora esses erros compreendam uma pequena fração em relação à energia total, eles são bem

significativos para os padrões químicos. Faz-se, portanto, necessário uma descrição mais adequada da

estrutura eletrônica dos orbitais para se obter dados mais confiáveis.

Uma aproximação comum das funções de onda das moléculas é o método Hartree-Fock,(Hartree, 1947)

em que a função de onda é dada como um produto antissimetrizado (determinante de Slater) das funções

espaciais e spin. O produto antissimétrico assegura que o princípio da exclusão de Pauli esteja satisfeito.

Assim, para a molécula de H2, a função de onda Hartree-Fock é dada por:

Â[𝜙(1)𝛼(1)𝜙(2)𝛽(2)] = 𝜙(1)𝜙(2)[𝛼(1)𝛽(2) − 𝛽(1)𝛼(2)] (33)

onde  é o antissimetrizador (operador determinante), 𝜙 é o orbital espacial duplamente ocupado otimizado,

e α e β são as funções spin. Para o H2, o orbital otimizado 𝜙 é gerado(Goddard e Ladner, 1971) como

mostrado na Figura 4A e pode ser expendido como:

𝜙 = 𝜒𝑙 + 𝜒𝑟 (34)

onde 𝜒𝑙 é a função localizada à esquerda do próton e 𝜒𝑟 é a função localizada à direta deste. A expansão da

parte espacial da função de onda total é dada por:

𝜙(1)𝜙(2) = (𝜒𝑟𝜒𝑟 + 𝜒𝑙𝜒𝑙) + (𝜒𝑟𝜒𝑙 + 𝜒𝑙𝜒𝑟) (35)

a equação (35) se aplica a todas as distâncias internucleares R, enquanto que para grandes distâncias, a

função de onda toma a forma:

(𝜒𝑟𝜒𝑙 + 𝜒𝑙𝜒𝑟) (36)

o que significa que existe probabilidade zero de ambos os elétrons estarem simultaneamente próximos no

mesmo núcleo. A função de onda Hartree-Fock se comporta incorretamente para altos valores de R, como

mostrado na Figura 5, que compara o método Hartree-Fock com a energia exata.

37

Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35 Å. Fonte: (Goddard e

Ladner, 1971).

Assim, o método Hartree-Fock fornece uma descrição pobre com relação à quebra e a formação de

uma ligação que é intolerável em estudos de reações químicas, pois esse método força ambos os elétrons a

ocuparem um único orbital, enquanto, para grandes separações, teremos dois orbitais monocupados. Uma

alternativa à esse problema, é o uso da função de onda da ligação de valência (VB) que já foi mostrada na

equação (36), o que leva a uma função de onda apropriada para R = ∞. Entretanto, essa função de onde

funciona incorretamente para pequenos valores de R (Figura 5).

Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB e GVB, comparadas

com a energia exata não-relativística. Fonte: [(Goddard et al., 1973)].

Assim, nenhum dos métodos (Hartree-Fock e VB), leva a uma descrição satisfatória para todos os

valores de R.(Goddard et al., 1973) Com a finalidade de contornar esses problemas, é usado o formalismo da

função de onda VB, de modo que a mesma se torna:

𝛹𝐺𝑉𝐵 = (𝜙𝑙𝜙𝑟 + 𝜙𝑟𝜙𝑙) (37)

A B

38

essa função de onda é apropriada para altos valores de R, quando resolvida para orbitais utilizando o campo

auto-consistente à cada R tal como no método Hartree-Fock. Isso combina as qualidades tanto do método

Hartree-Fock quanto do método VB e leva a uma função de onda que se comporta apropriadamente quando

os átomos estão separados assim como para pequenos valores de R. Essa abordagem é chamada de ligação

de valência generalizada (generalized valence bond – GVB)(Goddard, 1967; Ladner e Goddard, 1969;

Goddard e Ladner, 1971; Hunt et al., 1972) e difere do método Hartree-Fock pois agora existem dois

orbitais, um para cada elétron, ao invés de um par de elétrons por orbital.

2.5. CÁLCULOS DE QM/MM

Métodos híbridos combinam duas técnicas computacionais em um cálculo e permitem o estudo de

grandes sistemas químicos com uma precisão maior que cálculos de mecânica molecular. (Hall et al., 2008;

Senn e Thiel, 2009; Wang et al., 2012) O método QM/MM é uma técnica hibrida que combina um método

quanto-mecânico (QM) com um método de mecânica molecular (MM).(Warshel e Levitt, 1976; Singh e

Kollman, 1986; Field et al., 1990; Maseras e Morokuma, 1995) O ONIOM(Dapprich et al., 1999a) (N-layer

Integrated molecular Orbital molecular Mechanics) é um dos métodos de QM/MM mais empregados e

permite a combinação de diferentes métodos quanto-mecânicos assim como métodos clássicos de mecânica

molecular. No geral, a região do sistema onde o processo químico (interação química ou reação química)

ocorre é tratado com um método de maior acurácia (mecânica quântica ou semi-empírico), e essa região é

denominada de camada alta (high layer). O restante do sistema é tratado com MM (camada baixa – low

layer) (Figura 6). Muitos estudos com o método ONIOM são focados em sistemas biológicos, assim como

em proteínas(Lin e O’malley, 2008; Alzate-Morales et al., 2009; Kitisripanya et al., 2011; Ugarte, 2014)

onde o sítio ativo é tratado na camada alta, usando frequentemente a teoria funcional da densidade (Density

Functional Theory – DFT) e o restante da proteína é tratado como camada baixa, usando frequentemente o

campo de força AMBER.

Figura 6. Representação do método ONIOM de um sistema proteico dividido em duas camadas. Na

camada baixa (low layer), os átomos são representados por estruturas do tipo arame (wireframe) e na

camada alta (high layer) os átomos são representados por estruturas do tipo bola e bastão (ball-and-stick).

39

O ONIOM obtém a energia do sistema simulado através da combinação das energias computadas pelos

diferentes métodos teóricos,(Dapprich et al., 1999b) em que a camada baixa engloba todo o sistema

molecular incluindo a parte pertencente a camada alta. A equação abaixo apresenta, de modo simplificado,

as considerações realizadas em um sistema de dupla camada calculado com o método ONIOM (Equação

38):

𝐸𝑂𝑁𝐼𝑂𝑀 = 𝐸𝑙𝑜𝑤(𝑅) − 𝐸ℎ𝑖𝑔ℎ(𝑆𝑀) − 𝐸𝑙𝑜𝑤(𝑆𝑀) (38)

Onde a energia final do sistema, EONIOM

, corresponde a energia calculada da região com o método de

maior precisão, Ehigh

(SM), e da energia dessa mesma região calculada com um método de menor precisão,

Elow

(SM), subtraídas da energia do restante do sistema calculado com o método de menor precisão, Elow

(R).

O termo “SM” refere-se a expressão small, ou seja, do sistema pequeno tratado com o método QM. O termo

“R” vem da palavra real e refere-se a todo o sistema.

2.6. SIMULAÇÃO MOLECULAR

A simulação por dinâmica molecular (DM) é uma das técnicas mais versáteis no estudo de

macromoléculas biológicas.(Delano, 2002; Alonso et al., 2006) A DM pode ser caracterizada como um

método para gerar as trajetórias de um sistema de N partículas por integração numérica direta das equações

de movimento Newtonianas(Burkert e Allinger, 1982; Höltje e Folkers, 1996), ou seja, os átomos são

tratados como uma coleção de partículas unidas por forças harmônicas ou elásticas. O conjunto completo

dos potenciais de interação entre as partículas é referido como “campo de força”. (Brooks et al., 1990; Van

Gunsteren e Berendsen, 1990) O campo de força empírico é conhecido como uma função energia potencial e

permite que a energia potencial total do sistema, V(r), seja calculada partir da estrutura tridimensional do

sistema. V(r) é descrito como a soma de vários termos de energia, incluindo os termos para os átomos

ligados (comprimentos e ângulos de ligação, ângulos diedros) e não-ligados (interações de van der Waals e

de Coulomb) e assume a forma geral:

𝑉(𝑟) = 𝑉𝑙𝑖𝑔𝑎𝑑𝑜 + 𝑉𝑛ã𝑜−𝑙𝑖𝑔𝑎𝑑𝑜 (39)

𝑉𝑙𝑖𝑔𝑎𝑑𝑜 = ∑ 𝐾𝑟(𝑟 − 𝑟0)2 + ∑ 𝐾𝜃(𝜃 − 𝜃0)

2 +1

2â𝑛𝑔𝑢𝑙𝑜𝑠𝑙𝑖𝑔𝑎çõ𝑒𝑠

∑ 𝑉𝑛[1 − (−1)𝑛cos (𝑛𝜑 + 𝛾𝑛)

𝑑𝑖𝑒𝑑𝑟𝑜

(40)

𝑉𝑛ã𝑜−𝑙𝑖𝑔𝑎𝑑𝑜 =∑4휀𝑖𝑗[(𝜎𝑖𝑗𝑟𝑖𝑗)

12

− (𝜎𝑖𝑗𝑟𝑖𝑗)

6

]

𝑖,𝑗

+𝑞𝑖𝑞𝑗4𝜋휀0𝑟𝑖𝑗

(41)

40

Vários campos de força são comumente usados em simulações de dinâmica molecular de sistemas

biológicos, incluindo o AMBER(Wang et al., 2004), CHARMM(Brooks et al., 1983) e GROMOS.(Christen

et al., 2005) Estes diferem principalmente no modo de parametrização, mas geralmente fornecem resultados

semelhantes.

2.7. DOCAGEM MOLECULAR

É amplamente aceito que a atividade farmacológica é obtida através da interação de uma molécula

(ligante) em um sítio (pocket) de outra molécula (receptor), comumente uma proteína.(Vieth et al., 1998) O

processor computacional capaz de predizer a orientação geométrica de um determinado ligante candidato à

fármaco dentro do sítio ativo de uma enzima é denominado de docking molecular.(Cavasotto e Orry, 2007)

A ideia geral das técnicas de docking é de gerar um leque de conformações do ligante e ordená-las através

de um score com base em suas estabilidades com o receptor.(Alonso et al., 2006)

41

3. CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical Education.

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder

cycloadditions by using computational chemistry – an undergraduate experiment.

Norberto K. V. Monteiro* and Caio L. Firme

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: norbertokv@gmail.com

ABSTRACT

We have applied an undergraduate minicourse of Diels-Alder reaction in Federal University of Rio

Grande do Norte. By using computational chemistry tools (Gaussian, Gaussview and Chemcraft) students

could build the knowledge by themselves and they could associate important aspects of physical-chemistry

with Organic Chemistry. They have performed a very precise G4 method for the quantum calculations in

this 15-hour minicourse. Students were taught the basics of Diels-Alder reaction and the influence of

frontier orbital theory on kinetics. They learned how to use the quantum chemistry tools and after all

calculations they organized and analyzed the results. Afterwards, by means of the PhD student and teacher

guidance, students discussed the results and eventually answered the objective and subjective questionnaires

to evaluate the minicourse and their learning. We realized that students could understand the influence of

substituent effect on the electronic (and kinetic indirectly), geometric and thermodynamic aspects of Diels-

Alder reactions. Their results indicated that Diels-Alder reactions are exergonic and exothermic and that

although there is an important entropic contribution, there is a linear relation between Gibbs free energy and

enthalpy. They learned that electron withdrawing groups in dienophile and electron donating groups in diene

favor these reactions kinetically, and on the other hand, the opposite disfavor these reactions. Considering

thermodynamically controlled Diels-Alder reactions, the EDG decrease the equilibrium constant, in relation

to reference reaction, unlike the EWG. We believe that this minicourse gave an important contribution for

Chemistry education at undergraduate level.

42

GRAPHIC FOR MANUSCRIPT

KEYWORDS: Diels-Alder reaction; Diene; Dienophile; Computational Chemistry; G4; Frontier Orbital

Theory

INTRODUCTION

Pericyclic reactions[1,2] are one of three major types of reactions, together with polar and radical

reactions. Pericyclic reactions can be divided into four classes, cycloadditions, electrocyclic reactions,

sigma-tropic rearrangements and the less common group transfer reactions. All these categories share a

common feature that is a concerted mechanism involving a cyclic transition state with a concerted

movement of electrons. Concerted periclyclic cycloaddition involves the reaction of two unsaturated

molecules with resulting in the formation of a cyclic product with two new σ-bonds.[3] Examples of

cycloaddition reactions include the 1,3-dipolar cycloaddition (also referred as Huisgen cycloaddition),[4]

nitrone-olefin cycloaddition[5] and cycloadditions like the Diels-Alder reaction which is the most frequently

encountered and most studied of all pericyclic reactions (Scheme 1). The number of π-electrons involved in

the components characterizes the cycloaddition reactions, and for the three abovementioned cases this would

be [2+3], [3+2] and [4+2] respectively.

HOMOdiene

LUMOdienophile

≡

43

In 1950 the Nobel Prize of chemistry was granted to Otto Diels and Kurt Alder by describing an

important reaction that involves the formation of C-C bond in a six membered ring with high stereochemical

control, named Diels-Alder reaction.[6] Since 1928, the Diels-Alder reaction has been used in the synthesis

of new organic compounds, providing a reasonable way for forming a 6-membered systems.[7–13]

The simplest Diels-Alder reaction involves the reagents butadiene and ethane (dienophile), but it needed

200°C under high pressure although it has low yield[14] (about 18%) (Scheme 1A). Butadiene reacts more

easily with acrolein[15], (Scheme 1B). A methyl substituent on the diene, on C1 or C2, the reaction rate

increases, e.g. trans-piperylene reacting with acrolein[16,17] at 130°C (Scheme 1C). These last two

reactions achieved a yield Close to 80%.

Scheme 1

The standard state thermodynamic functions for Diels-Alder reactions are represented by enthalpy

change, Gibbs free energy change and entropy. The exothermic nature of these reactions are the result of

converting two weaker π-bonds into two stronger σ-bonds.[18] Because it is an addition reaction it has

negative entropy change but Diels-Alder reactions are exergonic due to the high negative entalphy change.

44

The reactivity of Diels-Alder reactions can be reasoned by molecular orbital (MO) theory[19] in

which the HOMO of diene donates electrons to the LUMO of the dienophile by overlapping these orbitals.

The addition of an EDG to the diene results in a better electron donor (and a better nucleophile) by

increasing its HOMO energy. Likewise, a EWG-substituted dienophile generates a better electron acceptor

(and a better electrophile) by decreasing its LUMO energy.[20] As a consequence, the frontier molecular

orbital theory accounts for the dependence of the Diels-Alder reaction rate on the substituent effect.[21] In

the transition state of the Diels-Alder reactions, the HOMO of diene interacts with LUMO of dienophile.

The Scheme 2A shows the energy diagram of butadiene and ethene. The Scheme 2B shows the energy

diagram of butadiene and a substituted dienophile with an EWG whose energy gap is lower than that in

Scheme 2A. Even lower energy gap is reached when an EDG is attached to the C1 or C2 of diene (Scheme

2C). It is well-known that the lower the energy gap between HOMO and LUMO from diene and dienophile,

respectively, the faster the corresponding reaction rate.

Scheme 2

One great challenge for chemistry education research is to improve students' understanding of

physical chemical properties of chemical reactions. One very important tool for chemistry education practice

is computational chemistry which improves the understanding of microscopic properties and/or provides

macroscopic properties more easily. Several successful chemistry education papers focused on using

computational chemistry as the main tool for the study of organic reactions. Hessley[22] used the molecular

modeling to study reaction mechanisms for the formation of alkyl halides and alcohols in order to prevent

the students from memorizing these reactions. Other works used computational chemistry for obtaining the

Hammett plots;[23] for studying the controversial Wittig reaction mechanism;[24] for analyzing the origin of

solvent effects on the rates of organic reactions;[25] and for investigating the kinetic and thermodynamic

aspects of endo vs exo selectivity in the Diels-Alder reactions.[26] However, except for ref. 26 none of these

studies have associated the aspects of physical-chemistry of organic reactions of these reactions.

HOMO

LUMO

HOMO

LUMO

(A)

HOMO

LUMO

HOMO

LUMO

HOMO

LUMO

HOMO

LUMO

(B) (C)

45

In this work, we used the computational chemistry tool in an undergraduate minicourse in order to

provide chemistry undergraduate students from Federal University of Rio Grande do Norte better

understanding of the influence of substituent groups on the kinetic and thermodynamic properties of Diels-

Alder reactions. All results from this minicourse were obtained by the students. Due to relatively short time

of the minicourse, each student or pair of students calculated just one reaction out of seven selected reactions

and afterwards all results were collected and analyzed as it is presented in this work.

COMPUTATIONAL DETAILS

Seven Diels-Alder reactions were previously selected and the geometries of all studied species were

optimized by using standard techniques.[27] Frequency calculations were performed to analyze vibrational

modes of the optimized geometry in order to determine whether the resulting geometries are true minima or

transition states. The geometry optimization, frequency calculations and the generation of the HOMO and

LUMO molecular orbitals of all studied species were calculated by G4 method[28] from Gaussian 09

package.[29] Every molecule´s initial geometry was done with GaussView 5[30] and final geometry was

taken from Chemcraft.[31] The G4-based computational method has been successfully used in the study of

pericyclic reactions before.[32]

DESCRIPTION OF THE COMPUTATIONAL CHEMISTRY MINICOURSE

We offered a 15-hour minicourse (divided into 3 hours per day) in July, 2015. Ten students participated

in the minicourse and according to pre-minicourse questionnaire (see Supporting Information) all

participants were undergraduate chemistry students in different grades from Institute of Chemistry of

Federal University of Rio Grande do Norte. All students have completed Organic Chemistry 1 which is the

main prerequisite for this minicourse. About 50% of students did not have any experience with

computational chemistry. This minicourse was part of a teaching project in the Institute of Chemistry in

which the first week at the beginning of the every semester is devoted to a series of minicourses where all

chemistry students must attend and can choose the minicourse(s) that is (are) most convenient. This

minicourse is taught once a year at the beginning of the first or second semester. This project can be part of a

regular Organic Chemistry discipline which includes Diels-Alder reaction or it can be similarly given as

minicourse. All data shown in this paper were provided by the students from calculations performed in the

minicourse. Some time-consuming calculations have lasted nearly one day.

In the first class, the students received a pre-minicourse questionnaire with short-answer questions

(Supporting information) in order to obtain student's academic background prior to the minicourse; After

46

pre-minicourse questionnaire, the students were taught the theoretical content about the Diels-Alder

reactions; the thermodynamic and kinetic properties for analyzing the Diels-Alder reactions; and they were

presented Gaussian 09, GaussView 5 and Chemcraft software. In the second class, the students were

grouped (four pairs of students) and each group or individual student performed the optimization and

frequency calculations of the reagents and products of one selected Diels-Alder reaction shown in Scheme 2

(The G4 calculations occurred during the second class and in the interval between the second and third

classes). In the third class the students analyzed all thermodynamic data in order to obtain student-generated

data in Tables 1-2 and the plots in Figures 1 and 3. In the first half of fourth class the students generated the

HOMO and LUMO molecular orbitals in order to obtain similar MO representations as shown in Figure 2

and to get their energy values and corresponding energy gaps as it is shown in Table 3. In the second half of

fourth class the students analyzed the bond lengths of all reagents and products of the Diels-Alder reactions

in order to obtain data in the Table 4 and Figure 4. In the fifth class, all the results were discussed as

presented in sections 5 to 8 and the students answered the post-minicourse questionnaire (objective and

subjective questions) and the evaluation which are shown in Supporting Information. Eventually, a statistics

(in Supporting Information) were generated with the rates for answers given by the students for objective

questions and student’s evaluation. Three filled post-minicourse subjective questionnaire forms are shown in

Supporting Information.

RESULTS

The Scheme 3 shows all studied Diels-Alder reactions where R1 and R2 represent the substituents of the

diene and dienophile respectively and the products cyclohexene derivatives formed which resulted in seven

Diels-Alder reactions. We used two substituents: one electron donating group (EDG) (-NH2) and one

electron withdrawing group (EWG) (-NO2).

47

Scheme 3

In Table 1 it is depicted the absolute values of enthalpy (H), Gibbs free energy (G) and entropy (S) of all

reagents (dienes and dienophiles) and products (cyclohexene and cyclohexene derivatives) and the

corresponding values of enthalpy change (∆H), Gibbs free energy change (∆G), temperature and entropy

change product (T∆S) and logarithm of equilibrium constant (ln K) for Diels-Alder reactions (1)-(7) are

shown in Table 2. In Figure 1A it is shown the linear relation involving ∆H and ∆G for Diels-Alder

reactions (1)-(7) whereas in Figure 1B there is the linear relation of ln K and ∆H for Diels-Alder reactions

(1)–(7).

Figure 2 depicts the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and

LUMO of ethene, nitroethene and etheneamine (Figure 2A, 2B and 2C respectively). The energy values of

HOMO and LUMO of the reagents of the Diels-Alder reactions (1)–(7), along with the corresponding

energy gap values, are shown in Table 3.

1. R1 = R2 = H;

2. R1 = NH2 and R2 = H

3. R1 = H and R2 = NO2

4. R1 = NH2 and R2 = NO2

5. R1 = NO2 and R2 = H

6. R1 = H and R2 = NH2

7. R1 = NO2 and R2 = NH2

R² = 0.9839

-55

-50

-45

-40

-35

-30

35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0

ΔH

(k

cal

mol-1

)

ln K

B

R² = 0.9839

-40

-38

-36

-34

-32

-30

-28

-26

-24

-22

-20

-52.5 -50 -47.5 -45 -42.5 -40 -37.5 -35

ΔG

(kca

l mol -1)

ΔH (kcal mol-1)

A

48

Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of Diels-Alder reactions

(1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)-(7).

Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1

, and entropy, S, in cal mol-1

K-1

of all

studied molecules.

Molecule G/kcal mol-1

H/kcal mol-1

S/cal mol-1

K-1

Ethene -49286.36 -49270.75 52.31

Ethenamine -84010.42 -83991.90 62.02

nitroethene -177588.33 -177567.34 70.25

Buta-1,3-diene -97825.36 -97806.25 63.98

Buta-1,3-dien-1-amine -132549.78 -132527.46 74.63

1-nitrobuta-1,3-diene -226121.56 -226098.50 77.14

Cyclohexene -147141.70 -147119.59 73.92

Cyclohex-2-en-1-amine -181859.99 -181835.93 80.38

Cyclohex-3-en-1-amine -181861.41 -181837.44 80.09

4-nitrocyclohexene -275447.53 -275420.66 89.78

3-nitrocyclohexene -275446.12 -275419.62 88.53

6-nitrocyclohex-2-en-1-amine -310164.56 -310136.10 95.04

2-nitrocyclohex-3-en-1-amine -310164.17 -310135.86 94.52

49

Table 2. Gibbs free energy change, ∆G, Enthalpy change, ∆H, entropy change and temperature product, in kcal mol-1

K-1, T∆S, entropy change, in kcal mol

-1, ∆S,

and equilibrium constant logarithm, ln K, of Diels-Alder reactions (1)–(7).

Reaction ∆G/kcal mol-1

∆H/kcal mol-1

T∆S/kcal mol-1

∆S/kcal mol-1

K-1

ln K

(1) Buta-1,3-diene + Ethene Cyclohexene -29.98 -42.59 -12.63 -0.0424 50.61

(2) Buta-1,3-dien-1-amine + Ethene Cyclohex-2-

en-1-amine -23.85 -37.72 -13.88 -0.0466 40.26

(3) Buta-1,3-diene + Nitroethene 4-

nitrocyclohexene -33.84 -47.07 -13.25 -0.0445 57.12

(4) Buta-1,3-dien-1-amine + Nitroethene 6-

nitrocyclohex-2-en-1-amine -26.45 -41.29 -14.86 -0.0498 44.65

(5) 1-nitrobuta-1,3-diene + Ethene 3-

nitrocyclohexene -38.21 -50.37 -12.20 -0.0409 64.49

(6) Buta-1,3-diene + Ethenamine Cyclohex-3-en-

1-amine -25.63 -39.29 -13.69 -0.0459 43.26

(7) 1-nitrobuta-1,3-diene + Ethenamine 2-

nitrocyclohex-3-en-1-amine -32.19 -45.46 -13.31 -0.0446 54.34

50

Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine and nitroethene (B);

and 2-nitrobuta-1,3-diene and ethenamine (C).

Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap between HOMO and

LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the Diels-Alder reactions (1) – (7).

Figure 3 shows the C-C bond length of the diene and C-C bond length of the dienophile of reactions

(4) (Figure 3A) and (7) (Figure 3B) in which these bond lengths are represented in the Scheme 4 (r1, r2, r3,

r4, r’1, r’2, r’3 and r’4). The values of C-C bond length for Diels-Alder reactions (1)–(7) are listed in Table 4.

A B C

Reaction EHOMO/a.u. ELUMO/a.u. |EHOMO-ELUMO|/a.u.

(1) -0.2293 0.0173 0.2466

(2) -0.1867 0.0173 0.2040

(3) -0.2293 -0.0923 0.1370

(4) -0.1867 -0.0923 0.0944

(5) -0.2522 0.0173 0.2695

(6) -0.2293 0.0509 0.2801

(7) -0.2522 0.0509 0.3031

51

Scheme 4

Figure 3. Optimized geometries of corresponding reactants and products from Diels-Alder reactions 4 (A)

and 7 (B). Selected bond lengths, in Angstroms, are depicted.

+

A

+

B

52

Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in Angstroms, of the

studied Diels-Alder reactions.

THERMODYNAMIC DATA AND EQUILIBRIUM CONSTANT

Enthalpy change values from Table 2 indicate that all reactions are exothermic because two weaker

bonds in reactants are converted to two stronger bonds in the product. Cycloaddition reactions have

negative entropy change owing to the smaller translational and rotational contributions of entropy in

cyclohexene adducts compared to those values from reactants (diene and dienophile). The negative entropy

does not contribute to the spontaneity of Diels-Alder reacdrptions. However, according to Gibbs free energy

change values in Table 2 these reactions are exergonic due to low entropy change values in modulus (see

Table 2) and high enthalpy change values in modulus and one can see that the modulus of TS is nearly

three times smaller than the modulus of H, in average. Experimental data show that Diels-Alder reactions

are exothermic and exergonic, e.g., the reaction of buta-1,3-diene and ethene to form cyclohexene[33] which

has an H and G of -40.0 and -27.0 kcal mol-1

respectively; the reaction of cyclopentadiene with itself to

form dicyclopentadiene (H and G of -18.4 and -8.2 kcal mol-1

respectively);[34] and the reaction of

cyclopentadiene and maleic anhydride (H = -24.8 kcal mol-1

).[35] Although the reactions are not the same

as those students have calculated, their thermodynamic data is within the same range.

The tendency for a reaction to reach equilibrium at the standard conditions is driven by the standard

Gibbs free energy, ΔG°, which is related to the equilibrium constant[36] according to Eq. 1.

∆𝐺° = −𝑅𝑇𝑙𝑛𝐾 (1)

There is no relationship between the kinetics and thermodynamics for most of chemical reactions,

except for some electron transfer and some radical reactions.[37–39] As previously described, an EDG

attached to the diene and a dienophile substituted with a EWG result in an increase in the rate of a Diels-

Reaction r1/Å r2/Å r3/Å r4/Å r’1/Å r’2/Å r’3/Å r’4/Å

(1) 1.336 1.468 1.336 1.327 1.507 1.333 1.507 1.534

(2) 1.345 1.458 1.339 1.327 1.510 1.332 1.506 1.532

(3) 1.336 1.468 1.336 1.323 1.508 1.332 1.506 1.532

(4) 1.345 1.458 1.339 1.323 1.509 1.334 1.504 1.524

(5) 1.330 1.465 1.335 1.327 1.506 1.333 1.504 1.533

(6) 1.336 1.468 1.336 1.335 1.507 1.332 1.507 1.532

(7) 1.330 1.465 1.335 1.335 1.506 1.333 1.504 1.540

53

Alder reaction. However, these effects are not observed for the thermodynamic properties as shown in Table

2. The EDG (amino group) thermodynamically disfavors the Diels-Alder reactions (2), (4) and (6) resulting

in less exothermic and exergonic reactions and hence lower ln K values compared to the reaction (1). On the

other hand, the reactions with EWG (nitro group) substituent [reactions (3), (5) and (7)] have more negative

values of ∆G and ∆H and higher values of ln K which thermodynamically favor these reactions.

The plot from Figure 1A has a very good correlation coefficient indicating that the trend in enthalpy

change for formation of cyclohexene and derivatives is nearly the same as that from corresponding Gibbs

free energy change, which means that entropy change for these cycloaddition reactions does not alter

significantly both trends, though TS values are not negligible (see Table 2).

Since G for all reactions is negative, there is a direct relation between modulus of G and

equilibrium constant, K, according to expressions in Eq. 2.

𝐾 = 𝑒−∆𝐺𝑅𝑇 ∴ ∆𝐺 < 0 ∴ 𝐾 = 𝑒−

(−∆𝐺)𝑅𝑇 ∴ 𝐾 = 𝑒

∆𝐺𝑅𝑇 (2)

Then, for exergonic Diels-Alder reactions, there is a direct relation between K and |ΔG°|, i.e., the

smaller ΔG° (in modulus) the smaller K. Moreover, there is direct relation between K and reaction yield.

The very good correlation coefficient of the Figure 1B indicates that ΔH can be associated with the K

as well. Thus, we can associate K with stability factors instead of exergonicity. Then, there is a direct

relation between modulus of ΔH (in modulus) and K.

The data in Table 2 indicate that reaction (5) has the highest K value and hence the highest negative

values of ΔG and ΔH. On the other hand, the reactions (2) and (6) have the lowest K values and the lowest

negative values of ΔG and ΔH. The direct proportionality between K and |ΔG| and between K and |ΔH| can

be observed in the decreasing order of K, |ΔG| and |ΔH|: (5) > (3) > (7) > (1) > (4) > (6) > (2).

YIELD AND KINETIC PROPERTIES

The efficiency of a chemical reaction is usually related to reaction yield. In a thermodynamically

controlled reaction, the reaction yield depends on the equilibrium constant and not on the reaction rate. In a

kinetically controlled reaction, yield depends on the reaction rate and the reaction time. In Diels-Alder

reactions, according to frontier molecular orbital theory, the electron-rich diene acts as a nucleophile. The

most important orbital in the diene is the Highest Occupied Molecular Orbital (HOMO). For the formation

of chemical bond, this orbital interacts with the Lowest Unoccupied Molecular Orbital (LUMO) of the

dienophile, which acts as an electrophile. As most Diels-Alder reactions are kinetically controlled, the

54

smaller the HOMO-LUMO energy difference, the greater the charge transfer, the faster the reaction and the

higher its yield.2 The energy diagram (Figure 2A) shows the interaction between the HOMO of the diene

and the LUMO of the dienophile. The energies of HOMO and LUMO depend on the nature of the

substituents in the diene and in the dienophile, in which an EWG decreases the LUMO energy of the

dienophile (Figure 2B) bringing it closer to the HOMO energy of the diene. An EDG-substituted diene

(Figure 2C) further decreases the energy gap between HOMO and LUMO. Thus, EWG-substituted

dienophiles and EDG-substituted dienes result in faster reactions and higher yields in comparison with

unsubstituted dienophiles and dienes.

The Table 3 shows the energy values of HOMO (EHOMO) and LUMO (ELUMO) of the reagents of the

Diels-Alder reactions (1)-(7), and the energy gap between HOMO and LUMO (|EHOMO-ELUMO|). The

reference reaction (1) has 0.2466 a.u. energy gap. The reactions with EDG in diene [reaction (2)] or EWG in

dienophile [reaction (3)] have lower energy gaps as expected and additive effect of both EDG in diene and

EWG in dienophile [reaction (4)] imparts further decrease in energy gap. From the analysis of energy gaps

in Table 3 and assuming that frontier molecular orbital theory can indicate relatively faster or slower

reactions, we could predict lower energy barriers for reactions (2), (3) and (4) which would provide higher

yields for these reactions in comparison with reaction (1). On the other hand, reactions (5), (6) and (7) have

the highest |HOMO-LUMO| values (0.2695, 0.2801 and 0.3031 a.u. respectively) due to inverse electron

demand, i.e. EWG-substituted dienes and/or EDG-substituted dienophiles. Since these reactions have the

highest energy gap, they are the slowest reactions. Therefore, the decreasing order of |HOMO-LUMO| is: (7)

> (6) > (5) > (1) > (2) > (3) > (4).

Figure 2 shows the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and

LUMO of ethene, nitroethene and etheneamine (Figures 2A, 2B and 2C respectively). According to Figures

2B (HOMO of buta-1,3-dien-2-amine and LUMO of nitroethene) and 2C (HOMO of 2-nitrobuta-1,3-diene

and LUMO of etheneamine) some changes occur in the size and shape of their corresponding HOMO and

LUMO in comparison with those orbitals of their parent molecules depicted in Figure 2A (HOMO of buta-

1,3-diene and LUMO of ethene). When an EDG (-NH3) is attached to the diene, its HOMO includes the

nitrogen lone pair participation resulting in a better nucleophile while LUMO in nitroethene includes oxygen

lone pair participation and –NO2 vicinal carbon orbital distortion towards nitrogen atom (Figure 2B). On the

other hand, HOMO of 2-nitrobuta-1,3-diene (Figure 2C) has no nitrogen lone pair participation and 2-

nitrobuta-1,3-diene is probably a weaker nucleophile than buta-1,3-dien-2-amine.

GEOMETRIC CHANGES

Scheme 4 represents a general Diels-Alder reaction in which selected bond lengths of reactants and

product are assigned (r1, r2, r3 for dienes, r4 for dienophiles, and r’1, r’2, r’3, r’4 for cyclohexenes). Figure 3

depicts the optimized geometries of reactants and product of reactions (4) (Figure 3A) and (7) (Figure 3B).

55

Surprisingly, data from Table 4 indicate that amino group increases the double bond length of reactants

more than nitro group. Moreover, data from Table 4 also confirm that double bonds r1, r3 and r4 in reactants

become single bonds in products while the single bond r2 becomes double bond in products.

Except for r’4 in reactions (4) and (7), no other selected bond length of all cyclohexene derivatives is

significantly affected by substituent group. Then, the variations of enthalpy change in the reactions (1) to (7)

are probably accounted for by the reactants than the product. Figure 3 also shows that cyclohexane

derivatives lost the carbon chain planarity from their precursors due to the lost of bonds in the product.

CONCLUSIONS

The computational chemistry allowed to teach the thermodynamic, electronic (indirectly kinetic) and

geometric aspects of Diels-Alder reactions. All of these reactions were exergonic and exothermic. Entropy

change has an important contribution on the Gibbs free energy change, however it did not affect the linear

relation between the latter and enthalpy change. Then, the thermodynamics of these reactions can be

reasoned by the influence of electronic effects on the stability of reactants and product. However, from the

analysis of geometrical parameters, stability is more affected by reactants than by products because the bond

lengths of reactants vary more significantly in comparison with reference compound due to the substituent

effect than those in the corresponding product.

Students could also understand that substituent effects on thermodynamics and kinetics (indirectly

from frontier orbital theory) are distinguished. The EDG in diene makes Diels-Alder reaction less favorable

thermodynamically, whereas EWG substituent in diene results in more favorable reaction in

thermodynamics. On the other hand, EDG in diene and EWG in dienophile favor the Diels-Alder reaction

kinetically.

From the statistics of post-minicourse questionnaire and students evaluation (Supporting information)

of these minicourse we realized that students could better understand thermodynamic and kinetic aspects of

a reaction from the use of computational chemistry. This minicourse also contributed for the Organic

Chemistry knowledge by giving students practical and simple tools to develop the knowledge by

themselves, in which these tools can also be employed in the teaching of Diels-Alder reactions in the

classroom. This minicourse also contributed for the association between physical-chemistry knowledge

(quantum chemistry, thermodynamics and kinetics, indirectly) with organic chemistry by means of Diels-

Alder reaction.

ASSOCIATED CONTENT

Supporting Information

56

Pre-minicourse, Post-minicourse questionnaires (objective and subjective questions) and students

evaluation. This material is available via the Internet at http://www.sciepub.com/

ACKNOWLEDGMENTS

Authors thank Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),

Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional

de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

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[14] Joshel LM, Butz LW. The Synthesis of Condensed Ring Compounds. VII. The Successful Use of

Ethylene in the Diels—Alder Reaction1, J Am Chem Soc, 1941, 63, 3350–1.

[15] Olah GA, Molnár Á. Hydrocarbon Chemistry. Wiley, 2003.

[16] Anh NT. Frontier Orbitals: A Practical Manual. 2007.

[17] Fleming I. Molecular Orbitals and Organic Chemical Reactions. Wiley, 2010.

[18] Williamson K, Masters K. Macroscale and microscale organic experiments. Cengage Learning, 2011.

[19] Palmer DRJ. Integration of Computational and Preparative Techniques To Demonstrate Physical

Organic Concepts in Synthetic Organic Chemistry: An Example Using Diels-Alder Reactions, J

Chem Educ, 2004, 81, 1633.

[20] Fleming S a., Hart GR, Savage PB. Molecular Orbital Animations for Organic Chemistry, J Chem

Educ, 2000, 77, 790.

[21] Houk KN. Frontier molecular orbital theory of cycloaddition reactions, Acc Chem Res, 1975, 8, 361–

9.

[22] Hessley RK. Computational Investigations for Undergraduate Organic Chemistry: Modeling

Markovnikov and anti-Markovnikov Reactions for the Formation of Alkyl Halides and Alcohols, J

Chem Educ, 2000, 77, 794.

[23] Ziegler BE. Theoretical Hammett Plot for the Gas-Phase Ionization of Benzoic Acid versus Phenol: A

Computational Chemistry Lab Exercise, J Chem Educ, 2013, 90, 665–8.

[24] Albrecht B. Computational Chemistry in the Undergraduate Laboratory: A Mechanistic Study of the

Wittig Reaction, J Chem Educ, 2014, 91, 2182–5.

[25] Lim D, Jenson C, Repasky MP, Jorgensen WL. Solvent as Catalyst: Computational Studies of

58

Organic Reactions in Solution. Transit. State Model. Catal., vol. 721, American Chemical Society,

1999, p. 6–74.

[26] Rowley CN, Woo TK, Mosey NJ. A Computational Experiment of the Endo versus Exo Preference in

a Diels–Alder Reaction, J Chem Educ, 2009, 86, 199.

[27] Fletcher R. Practical Methods of Optimization. vol. 53. 2000.

[28] Curtiss L a, Redfern PC, Raghavachari K. Gaussian-4 Theory, J Chem Phys, 2007, 126, 84108.

[29] Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, et al. Gaussian 09,

Revision A.02, Gaussian Inc Wallingford CT, 2009, 34, Wallingford CT.

[30] Roy Dennington TK and JM. GaussView, Version 5, Semichem Inc, Shawnee Mission KS, 2009.

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[31] Zhurko GA, Zhurko DA. Chemcraft Program, Academic version 1.5 (2004) 2004.

[32] Karton A, Goerigk L. Accurate reaction barrier heights of pericyclic reactions: Surprisingly large

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185.

59

Supporting Information

for

Teaching aspects of organic chemistry reaction from Diels-Alder cycloadditions by

using computational chemistry – an undergraduation experiment.

Norberto K. V. Monteiro* and Caio L. Firme

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: norbertokv@ufrnet.br

PRE-MINICOURSE QUESTIONNAIRE

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder

cycloadditions by using computational chemistry – an undergraduate experiment.

FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE

CHEMISTRY INSTITUTE

STUDENT´S NAME:_________________________________________________ DATE:____________

1) What course are you majoring in?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

2) What is your grade?

______________________________________________________________________________________

3) Which organic chemistry disciplines have you done before?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

60

4) Do you have any previous experience in computational chemistry?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

POST-MINICOURSE QUESTIONNAIRE – OBJECTIVE QUESTIONS.

1. Did this minicourse help you to understand Diels-Alder reaction? (a) Very much; (b) Considerably;

(c)Moderately; (d) Hardly ever

2. Did this minicourse help you to understand the frontier orbital theory? (a) Very much; (b)

Considerably; (c)Moderately; (d) Hardly ever

3. Did this minicourse help you to understand the relation between frontier orbital theory, kinetic

approach and yield? (a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever

4. Did this minicourse help you to understand the relation between thermodynamic approach and yield?

(a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever

5. How do you evaluate the importance of computational chemistry for understanding chemical

reactions? (a) Very important; (b) Important; (c) Moderately important; (d) Not important

STUDENTS EVALUATION

Grades (1-minimum to 5-maximum) that correspond to students opinion about features of the

Diels-Alder minicourse.

Features Grade (% of students)

1 2 3 4 5

I had the prerequisite knowledge to complete

this minicourse?

Depth of content

Backgrounds need

61

Visually attractive

Multidisciplinary

Easy to be performed

The objectives of this minicourse were clear?

The computational tool really helped in

understanding?

POST-MINICOURSE QUESTIONNAIRE – SUBJECTIVE QUESTIONS.

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder

cycloadditions by using computational chemistry – an undergraduate experiment.

FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE

CHEMISTRY INSTITUTE

STUDENT´S NAME:_________________________________________________ DATE:___________

1) List the strong points of the minicourse. Give reason(s) in few words.

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

2) Do you agree that the computational chemistry contribute for better understanding thermodynamics

and kinetics (by means of Frontier Orbital Theory) of organic chemistry reactions? Why?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

3) What is the relation between HOMO-LUMO energy difference and kinetics in DA reactions? What

is the relation between Gibbs free energy difference and equilibrium constant?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

62

______________________________________________________________________________________

4) How do the EWG and EDG affect thermodynamic and kinetics of the DA reactions?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

5) What are the main changes in bond lengths and planarity of reactants and products in DA reactions?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

STATISTICAL RESULTS OF POST-MINICOURSE QUESTIONNAIRE (OBJECTIVE

QUESTIONS)

The figures below show de plots of students percentage choice versus multiple choice questions

1. Did this minicourse help you to understand Diels-Alder reaction?

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3. Did this minicourse help you to understand the relation between frontier orbital theory, kinetic

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5. How do you evaluate the importance of computational chemistry for understanding chemical

reactions?

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STATISTICAL RESULTS OF STUDENTS EVALUATION

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I had the prerequisite knowledge to complete this minicourse?

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POST-MINICOURSE QUESTIONNAIRE ANSWERS – SUBJECTIVE QUESTIONS.

68

69

70

4. CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry A.

Monteiro, N. K. V.; Firme, C. L. The Journal of Physical Chemistry A 2014, 118, 1730.

Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio,

QTAIM and ELF study

Norberto K. V. Monteiro and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*E-mail: caiofirme@quimica.ufrn.br

ABSTRACT

The hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen bonding is formed by the interaction between a

pair of identical or similar hydrogen atoms that are close to electrical neutrality and it yields a stabilizing

contribution to the overall molecular energy. This work provides new, important information regarding

hydrogen-hydrogen bonds. We report that stability of alkane complexes and boiling point of alkanes are

directly related to H-H bond, which means that intermolecular interactions between alkane chains are

directional H-H bond, not nondirectional induced dipole-induced dipole. Moreover, we show the existence

of intramolecular H-H bonds in highly branched alkanes playing a secondary role on their increased

stabilities in comparison with linear or less branched isomers. These results were accomplished by different

approaches: density functional theory (DFT), ab initio, quantum theory of atoms in molecules (QTAIM) and

electron localization function (ELF).

Keywords: hydrogen-hydrogen bond; H-H bond, hydrogen-hydrogen bonding; alkane; QTAIM; ELF.

INTRODUCTION

Bader defined bond path as an atomic interaction line (AIL) which is mirrored by its corresponding

virial path, where the potential energy density is maximally negative1,2

. Bader3,4

has stated that bond paths

71

are indicative of bonded interactions, which encompass all kinds of chemical interactions. Bond paths or

virial paths can be used to characterize four types of bonded interactions involving hydrogen atom:

dihydrogen bond, hydrogen bond, hydride bond and hydrogen-hydrogen bond.

The dihydrogen bond is a bonded interaction consisting of electrostatic interaction between two

hydrogen atoms of opposite charges and its formation is restricted to systems that possess a hydridic

hydrogen wherein transition metals and boron atom are typical elements that can interact with this hydridic

hydrogen5. On the other hand, the hydrogen bond involves only one hydrogen atom and it can be

represented by X-H•••Y interaction, where X and Y are electronegative atoms; X-H is named as a proton-

donating bond, and Y is an acceptor center.6

An unconventional type of hydrogen bond is named hydride bond, also known as “inverse hydrogen

bond”. In the hydride bond, a negatively charged hydrogen atom will be electron donor for non-hydrogen

acceptors. The hydride bond consists of a hydride placed between two electrophilic centers.7,8

Recently,9 a new kind of interaction involving hydrogen atoms has been discovered by means of the

quantum theory of atoms in molecule (QTAIM): the hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen

bonding, wherein a bond path links a pair of identical or similar hydrogen atoms that are close to electrical

neutrality, opposing the dihydrogen bonding with positively and negatively charged hydrogens.5,9

It is

demonstrated that these interactions are found to link hydrogen atoms bonded to both unsaturated and

saturated carbon atoms in hydrocarbon molecules and contribute to molecular stabilization, where each H-H

bond makes a stabilizing contribution of up to 10 kcal/mol to the energy of the molecule.

9 These interactions

are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the

C1-C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen

atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups.9

Some experimental10–14

and computational15–18

studies exemplify interactions between two

hydrogen atoms, although not differentiating H-H bond from dihydrogen bond. Nonetheless, the interactions

cited in these works occur between similar or identical hydrogen atoms close to electrical neutrality. Some

studies used the term “non-bonding repulsive interaction” to describe the H-H bond.

The H-H bond in planar biphenyl was questioned.19

However. no other studied molecular system

with hydrogen-hydrogen bond was questioned to the date. Bader20

argued that there is no repulsion in

rotamers such as ethane or biphenyl, only decrease of nuclear attraction. In reply21

to Bader´s statements, it

was used an arbitrary set of MO´s (which is obviously not univocal22

). Nonetheless, QTAIM is dependent

upon the electron density which, in turn, arises from a variational calculation and it represents “the

distribution that minimizes the energy for any geometry and thus the presence of bond paths represent

spatial accumulation of density that lower the energy of the system”, as stated by Bader.20

Then, bond path

72

is an universal indicator of bonded interactions.3 Accordingly, QTAIM has become a method that has

already been successfully applied to understand conventional hydrogen bonds,23,24

C-H•••O bonds25

and

dihydrogen bonds.26,27

In this work, we show that there are very good linear relations between H-H bond in alkane

complexes and their stabilities and between H-H bond in alkane complexes and their boiling points. We also

demonstrate that there are H-H bonds in highly branched alkanes and we show that the H-H bonds play a

secondary role in the stability of highly branched alkanes. All results in this work were obtained by means of

different approaches: the topology of the virial field function based on QTAIM; the critical points of

electron localization function (ELF); ab initio and density functional theory.

COMPUTATIONAL DETAILS AND THEORETICAL METHODS

The geometries of the studied species were optimized by using standard techniques.28

Vibrational

modes of the optimized geometry were calculated in order to determine whether the resulting geometries are

true minima or transition states. Calculations were performed at ωB97XD29

, G430

, G4/CCSD(T),31

M062X32

and B3LYP33,34

methods with the 6-311++G(d,p) basis set by using Gaussian 09 package,35

including the

electron density which was further used for QTAIM calculations. The 6-311++G(d,p) basis set was applied

because the inclusion of the diffuse components in the basis set is desired to describe properly the hydrogen-

hydrogen bond36

. Among the density functionals accounting for dispersion, ωB97XD37,38

contains empirical

dispersion terms and also long-range corrections, exhibiting a van der Waals correction. All topological

information were calculated by means of AIM2000 software39

. The algorithm of AIM2000 for searching

critical points is based on Newton-Raphson method which relies heavily on the chosen starting point.40

Every available alternative for increasing calculation accuracy and choosing different starting points was

done. Integrations of the atomic basins were calculated in natural coordinates in default options of

integration. All integrations yielded 10-3

to 10-4

order of magnitude for the Laplacian of the calculated

atomic basin. Atomic energies were calculated from the atomic virial approach. All calculated bond paths

are mirrored by their corresponding virial paths (where maximum negative potential energy exists between

two atoms). According to atomic virial approach, the bond paths mirrored by virial paths are indicative of

bonded interaction. The evaluation of the electron localization function (ELF) values, (r), and topological

information of bond critical point (BCP) have been carried out with the TopMoD program.41,42

For each

alkane complex, the QTAIM coordinates of bond critical point of a selected H-H bond were used to locate

the corresponding ELF BCP. All ELF BCP ´s were confirmed as an index 1 critical point since their

corresponding [r(QTAIM BCP coordinates)] is 0.0000. Fuster and Grabowski24

obtained a 0.9935 coefficient of

determination involving ELF H--- BCP distance and QTAIM H--- BCP distance, which reinforces our

approach for finding ELF bond critical point of H-H bond from QTAIM coordinates.

73 QTAIM AND VAN DER WAALS INTERACTIONS

The QTAIM is an extension of quantum mechanics to an open system which is made up of atomic

basins in a molecule and it can be developed by deriving Heisenberg´s equations of motion from

Schrödinger´s equation wherein any atomic property in a molecule can be calculated.43

The quantum theory of atoms in molecules was developed by Bader to study the electronic

structure and especially the chemical bond by trespassing the limits of orbital theories when approaching to

the physics underlying atoms in a molecule1,2

. This quantum model is considered innovative in the study of

chemical bonding and characterization of intra- or intermolecular interactions. 44,45

According to the QTAIM

quantum-mechanic concepts,46,47

the calculation of gradient of charge density, ∇ρ, of the studied molecular

system is performed through the electron density's Hessian matrix in order to find critical points of the

charge density.48

The critical points of the charge density are obtained where ∇ρ = 0. The bond critical point

is a saddle point on an electron density curvature being a minimum in the direction of the atomic interaction

line and a maximum in the two directions perpendicular to it.49–51

The BCP’s give information about the

character of different chemical bonds or chemical interactions, for example, metal-ligand interactions, 49,52,53

covalent bonds1,2,54

, different kinds of hydrogen bonds,24,55,56

including dihydrogen bonds26,57,58

and H-H

bond.36,59

The Laplacian of the charge density (∇2ρ) is obtained from the sum of the three eigenvalues of the

Hessian matrix (λ1, λ2 and λ3) of the charge density1,2

. The negative (∇2ρ < 0) or positive (∇2

ρ > 0) sign of

the Laplacian represents concentration and depletion of the charge density, respectively60

.

The H-H bonds exhibit characteristics of closed-shell interaction since they have: low value of

charge density (ρb) (< 6 x 10-2

au); the ratio λ1 / λ3 < 1; ∇2ρ>0; and the positive value for the total energy

density Hb that is close to zero. The H-H bond presents BCP topological values similar to those from weak

hydrogen bonds61–63

.

All properties of matter are somewhat related to its corresponding density of electronic charge ρ(r),

where one can associate to its topology delineating atoms and the bonding between them.9 In molecules, the

nuclei of bonded atoms are linked by a line along which the electron density is a maximum with respect to

any neighboring gradient path and it is termed bond path.9 In truth, according to Bader, a bond path has to

be mirrored by its corresponding virial path, otherwise it cannot be regarded as a bond path. In a virial path

the potential energy density is maximally stabilizing.64

When the molecular system is also in a stable

electrostatic equilibrium, it ensures both a necessary and a sufficient condition for bonding.65

Bader also

defined bond path as a measured consequence of the action of the density operator along the trajectories that

originate in bond critical point and end in both interacting nuclei.66

The bond path is based on theorems of

quantum mechanics that govern the interactions between atoms. Only two forces operate in a molecular

system: Feynman and Ehrenfest, which act on the nuclei and electrons, respectively. They both are united in

the virial theorem,67

which relates these forces to the energy of the molecule.68

This is the essential physics

74

of all bonded interactions that when combined with the properties of the bond critical point, provides all the

features of a chemical bond.

Many studies involving critical points and bond paths in van der Waals and weak interactions were

performed.69–71

The van der Waals interactions were identified as directional bonded interactions because of

the existence of bond paths in solid chloride and N4S4 which are important to understand the structure of

these crystals. Hydrogen-hydrogen bond is also a directional interaction, different from induced dipole-

induced dipole interaction which is not directional. Wolstenholme and Cameron72

showed that crystal

structures of tetraphenylphosphonium squarate, bianthrone, and bis (benzophenone) azine contains a variety

of C-Hδ+•••

+δH-C interactions, as well as a variety of C-H•••O and C-H•••Cπ interactions. This study

established similarities and differences between these weak interactions, leading to a better understanding of

H-H bonds. Slater has shown in his paper73

that there are no fundamental differences between van der Waals

and covalent bonding, because they have the same quantum mechanism.74

According to Feynman,75

the

origin of chemical bond is derived from electron density between nuclei, at the same time the requirement

for occurrence of bond path.

ELF

The electron localization function introduced by Becke and Edgecombe76

is measured from the

Fermi hole curvature in terms of the local excess kinetic energy density due to Pauli repulsion. It has been

shown that ELF is confined to the [1,0] interval, where the tendency to 1 means that parallel spins are highly

improbable, otherwise opposite spin pairs are highly probable; while the tendency to zero indicates a high

probability of same spin pairs. This interpretation gives ELF function a deep physical meaning. The electron

localization function provides a rigorous basis for the analysis of the wave function and electronic structure

in molecules and crystals.77

The ELF topology has been widely applied for study of chemical bonding.78–82

The topological partitioning of ELF gradient field83,84

provides two types of basins of attractors which can

be thought as corresponding to atomic cores, bonds and lone pairs: core basins surrounding nuclei with

atomic number Z > 2 and labeled by C(A) where A is the atomic symbol of the element, and valence basins

which are characterized by its synaptic order85

which is the number of the atomic valence shells in which

they participate. Therefore, there are monosynaptyc, disynaptyc, trisynaptyc basins etc, labeled V(A), V(A,

X), V(A, X, Y), etc, respectively. Monosynaptyc basins correspond to the lone pairs of the Lewis model.

Disynaptyc and trisynaptyc basins correspond to two-center and three center bonds respectively.

The ELF local function, (r), provides the chemical information and (r) provides two types of

points in real space: wandering points where (r)≠0 and critical points where (r)=0. The critical points

are characterized by their index which is the number of positive eigenvalues from hessian matrix of (r).

The topology of the ELF has been applied in the study of hydrogen bond.86–89

In FH•••N2 complex, the value

75

of ELF at the index 1 critical point between the V(F,H) and V(N) basins is 0.047, while for FH•••NH3 the

corresponding (r) is 0.226. These values are attributed to weak and strong hydrogen bonds, respectively.

In this work, the ELF was used as a complementary method for evaluating hydrogen-hydrogen bond for

alkane complexes from methane-methane until hexane-hexane.

RESULTS AND DISCUSSION

We analyzed the H-H bonds as intramolecular interactions in some C8H18 isomers and their

contribution to molecular stabilization. Afterwards, we analyzed the H-H bonds as intermolecular

interactions in the case alkane-alkane complexes and its relation with complex stability and boiling point.

Octane isomers: We investigated the possible relation between the amount of H-H bonds and

branched alkane stabilization based on heat of combustion (ΔHcomb) of the corresponding alkanes. The

isomer with the lowest ΔHcomb is the most stable. It is well known that branched alkanes are more stable than

their linear isomers.90

Among some C8H18 isomers (Table 1), the highly branched isomer, 2,2,3,3-

tetramethylbutane, is the most stable and the linear isomer octane is the least stable. The heats of combustion

were calculated at ωB97xd/6-311++G(d,p) level, since this functional has significantly lower absolute errors

when considering the noncovalent attractions and empirical dispersion terms.91

Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb) in branched

alkanes.

Octane isomers* Branching H-H bonds ΔHcomb(kcal/mol) ΔΔHcomb(kcal/mol)

Octane 0 0 -1647.6 2.6

3-methyl-heptane 1 0 -1647.3 2.3

3,3-dimethyl-hexane 2 0 -1646.9 1.9

2,2,3-trimethyl-pentane 3 2 -1645.8 0.8

2,2,3,3-tetramethyl-butane 4 3 -1645.0 0

*Optimized at ωB97XD/6-311++(d,p) level of theory.

Figure 1 shows the virial graphs of some C8H18 isomers (octane, 3-methylheptane, 3-3-

dimethylhexane, 2-2-3-trimethylpentane and 2-2-3-3-tetramethylbutane). The averaged value of charge

density of the bond critical point, ρb, from the H-H bonds, is 0.0098 au. which is typical for weak hydrogen

bonds and higher than that for van der Waals interactions.61

76

Figure 1. Virial graphs of C8H18 isomers. (A) Octane, (B) 3-methylheptane, (C) 3,3 dimethylhexane, (D)

2,2,3-trimethylpentane and (E) 2,2,3,3-tetramethylbutane. The geometry optimization of the molecules were

performed at ωB97xd/6-311++G(d,p) level of theory.

On going from octane, 3-methyl-heptane to 3,3-dimethyl-hexane, there is a unitary increase of

methyl branching accompanying an increasing branching stabilization on C8H18 system (Table 1). However,

no H-H bond was found in this set of alkanes. Then, other factors contribute for 3-methyl-heptane and 3,3-

dimethyl-hexane stabilizations relative to octane, which is not the focus of this work. In literature, one may

find different explanations for the alkane branching stabilization. Gronert’s work92

suggests that germinal

steric interactions in linear alkane are relieved on branching. Opposing Gronert´s hypothesis, Wodrich and

Schleyer93

attributed branching stabilization to 1,3-alkyl-alkyl stabilizing interaction (so called

(A)(B)

(C) (D)

(E)

1

2

77

protobranching). In a more recent work, Kennitz and collaborators94

attribute alkane branching stabilization

to a greater electron delocalization in branched alkanes relative to linear alkanes.

Nonetheless, when comparing 3,3-dimethyl-hexane, 2,2,3-trimethyl-pentane and 2,2,3,3-tetramethyl-

butane, there is an increasing trend of intramolecular H-H bond and branching stabilization (Table 1). It was

already been shown that these interactions contribute to energy stabilization in hydrocarbons.9,36

The

2,2,3,3-tetramethylbutane is the most stable branched C8H18 isomer and it has three H-H bonds, while the

second most stable C8H18 isomer, 2,2,3-trimethylpentane, has two intermolecular H-H bonds. Moreover, in

the series 3-methyl-heptane to 3,3-dimethyl-hexane, the average stabilization, ΔΔHcomb, is 0.35 kcal/mol,

while for 3,3-dimethyl-hexane to 2,2,3,3-etramethyl-butane series, the average stabilization is 0.95 kcal/mol,

which implies the influence of an additional stabilization factor when leaving the first series and going to the

latter.

Additional evidence on stabilization due to H-H bond of highly branched C8H18 isomer was given

by comparing hydrogen atomic energy belonging or not to H-H bond. The atomic energy of hydrogen atoms

belonging to H-H bond (H1 in Figure 1E) are in average 0.80 kcal/mol lower when compared to the

hydrogen atoms not belonging to H-H bond (H2 in Figure 1E). Then, for highly branched alkanes, H-H bond

can be regarded as secondary stabilization factor.

One possible reason for the non-existence of H-H bond in octane, 3-methylheptane and 3,3-

dimethylhexane can be ascribed to the lack of necessary alignment involving C-H---H-C bonds of vicinal or

germinal methyl groups. Further details on this geometrical requirement will be in the following subsection.

Alkane complexes: In this section we evaluated the H-H bonds in linear bimolecular alkane

complexes: methane-methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane-

hexane. These alkane complexes were obtained from two different optimization steps: firstly, the

optimization of each single alkane molecule (the geometry of each single molecule is in Supporting

Information Figure S1); secondly, the optimization of the whole bimolecular complex built from an initial

geometry where each single optimized molecule was arbitrarily positioned. The optimizations were done in

different levels of theory: ωB97XD/6-311++G(d,p), G4/CCSD(T)/6-311++G(d,p), B3LYP/6-311++G(d,p)

(Figure S2, Supporting Information), and M062X/6-311++G(d,p) (Figure S3, Supporting Information).

Only optimized structures of alkane complexes obtained from G4/CCSD(T)/6-311++G(d,p) and ωB97XD/6-

311++G(d,p) are depicted in Figures 2 and 3, respectively. We emphasize that the single molecules in

Figures 2 and 3 comes from optimized geometries, but both Figures contain randomized non-optimized

initial geometry (to the left) and optimized geometry (to the right) for the complex where individual

molecules are optimized as above-mentioned. Both (columns) are shown for comparison reasons which

depict differences of some geometrical parameters after optimization of the whole complex affording better

understanding of H-H bond.

78

In Figure 2, it is depicted the CH---HC dihedral angle for methane-methane, ethane-ethane,

propane-propane, butane-butane from G4/CCSD(T)/6-311++G(d,p) level of theory. In this case, geometry

optimization proceeded through G4 method. The CCSD(T)/6-311++G(d,p) was used for self-consistent field

and density matrix calculations from the G4-optimized geometry (Figure S4, Supporting Information). Due

to infrastructure limitations, it was not possible to calculate pentane-pentane and hexane-hexane complexes

from G4/CCSD(T)/6-311++G(d,p) level of theory. From Figure 2, one can see that all CH---HC dihedral

angle decreases from non-optimized to optimized structures. Except for ethane-ethane complex, this

decrease is reasonably considerable: in methane-methane (from 17.53 to 0.34o), in propane-propane (from

30.51 to 8.02o), and in butane-butane (from 65.18 to 18.56

o), which tends to point both hydrogen atoms in

nearly the same direction. This is a clear indication that hydrogen atoms tend to form an intramolecular

interaction.

Figure 2. Randomized, non-optimized and optimized geometries of methane-methane, ethane-ethane,

propone-propane and butane-butane complexes depicting their corresponding selected CH--HC dihedral

angles, in degrees. The geometry optimization of the molecules were performed at G4/CCSD(T)/6-

311++G(d,p) level of theory. The dashed lines represent the selected dihedral angles.

RANDOMIZED, NON-OPTIMIZED GEOMETRY OPTIMIZED GEOMETRY

79

In Figure 3, it is depicted selected H-H interatomic distances for methane-methane, ethane-ethane,

propane-propane, butane-butane, pentane-pentane and hexane-hexane from ωB97XD/6-311++G(d,p) level

of theory where all molecular pairs are closer after optimization. Except from methane-methane complex,

the optimized structures of the alkane complexes shown in Figure 3 are notoriously different from those in

Figure 2. This is a consequence in the usage of different methods for optimization calculation. The

optimization with ωB97XD functional tends to impose a conformation which increases the number of

intramolecular H-H bonds in the alkane complex. This is confirmed when comparing the virial graphs of G4

and ωB97XD optimized structures in Figures 4 and 5, respectively.

RANDOMIZED, NON-OPTIMIZED GEOMETRY OPTIMIZED GEOMETRY

80

Figure 3. Randomized non-optimized initial geometry and optimized geometry of methane-methane,

ethane-ethane, propone-propane, butane-butane, pentane-pentane and hexane-hexane complexes depicting

their corresponding selected H-H interatomic distances, in angstroms. The geometry optimization of the

molecules were performed at ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the

selected interatomic distances.

Figure 4 shows the virial graphs of the optimized alkane complexes using G4/CCSD(T)/6-

311++G(d,p) level of theory. There are one H-H bond in methane-methane complex and two H-H bonds in

other alkane complexes. The values of charge density of BCP (ρb) and the Laplacian of ρb (∇2𝜌) indicate

that all H-H bond paths from alkane complexes exhibit the characteristics of closed-shell interactions.

81

Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p) level of theory.

Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro) and butane-butane (But-

But) along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical,

in au. The others complexes were not optimized due to lack of computational resources.

Figure 5 shows the virial graphs of the optimized alkane complexes using ωB97XD/6-311++G(d,p)

level of theory. There is an increasing number of H-H bond as the number of carbon atoms increase in the

alkane complex. The number of H-H bond increase by a unit from methane-methane to butane-butane. From

butane-butane to hexane-hexane, the increase is two units of H-H bond. In ethane-ethane complex there are

also two C-H intramolecular bonds whose virial paths are curved, meaning less attractive and stable

interactions, with a low value of ρb (about 8.5 × 10-6

at 6.7 × 10-3

ua range values). Accordingly, Table 2

shows the number of H-H bonds in the alkane complexes calculated at ωB97XD/6-311++G(d,p) level of

theory, along with the experimental values of the boiling points of the corresponding alkanes and the so-

called complex stability, G(complex-2*isolated), derived from Gibbs energy of complex and two-fold Gibbs

energy of the corresponding isolated alkane, whose equation is: G= Gcomplex - 2Gisolated.

82

Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes (Gcomplex), two-fold

Gibbs energy of the corresponding isolated alkane (2*Gisolated), and complex stability (G(complex-2*isolated)) of

the alkane complexes.

Complexes* Number of

H-H bonds

Boiling Point

(°C)

Gcomplex

(kcal/mol)

2*Gisolated

(kcal/mol) G(complex-

2*isolated)

(kcal/mol)

Met-Met 1 -162 -50848.0 -50847.8 -0.2

Eth-Eth 2 -89 -100184.2 -100182.2 -2.0

Pro-Pro 3 -42 -149522.3 -149520.2 -2.1

But-But 4 0 -198860.6 -198858.2 -2.3

Pen-Pen 6 36 -248201.1 -248196.5 -4.6

Hex-Hex 8 69 -297540.0 -297534.5 -5.4

*Optimized at ωB97XD/6-311++(d,p) level of theory.

Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p) level of theory.

Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-

Eth-EthMet-Met

Pro-Pro

But-ButPen-Pen Hex-Hex

Met-Met Eth-Eth Pro-Pro

But-But Pen-Pen Hex-Hex

0.0000085

0.0000060

0.0050

0.0043

0.0051

0.0034

0.0059

0.0045 0.0067

0.0049

0.0057

0.0045

ρ = 0.00322ρ = −0.0023

83

But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes along with the charge density,

ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au.

We can see that as the number of H-H bonds increase from methane-methane to hexane-hexane, the

G(complex-2*isolated) increases as well. This linear relation is plotted in Figure 6, with a reasonably good

coefficient of determination. In another words, the hydrogen-hydrogen bonds influence on the stability of

alkane complexes.

Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in kcal/mol, of selected

alkanes at ωB97XD/6-311++G(d,p) level of theory.

From Table 2 there is another linear relation: between alkanes’ boiling point and number of H-H

bonds, which is plotted in Figure 7, whose coefficient of determination is 0.9052. The boiling point is

directly dependent upon the magnitude of the intermolecular interactions which, in turn, is directly related to

the complex stability. Each H-H bond in these complexes contribute to decrease on complex electronic

energy (Ecomplex) compared to the sum of electronic energy of the two corresponding isolated alkanes

(Eisolated), leading to an increased complex stability and boiling point. As a consequence, as the number of H-

R² = 0.9428

0.00

0.80

1.60

2.40

3.20

4.00

4.80

5.60

6.40

0 2 4 6 8 10

G

(co

mp

lex

-2*

iso

late

d)(k

cal/

mo

l)

H-H bond paths

84

H bonds increase, the complex stability increases, which leads to an expected increase in the alkane boiling

point. These trends are confirmed with the reasonably good coefficient of determination in Figure 7.

Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave function, versus boiling

point, in °C, of selected alkanes.

It is well known the correlation between boiling/melting point in linear alkanes with increasing

number of carbon atoms.90,95

The intermolecular attractive forces acting in neutral species are known as van

der Waals forces based on non-directional induced dipole-induced dipole interactions. However, the results

from this work strongly indicate that the intramolecular forces in alkane complexes are consequence of

directional attractive H-H bonds.

In order to demonstrate the stabilizing effect of H-H bond, the atomic energies of a hydrogen atoms

belonging and not to H-H bond were obtained from atomic virial theorem of QTAIM. Figures 8 and 9 show

the energy of a hydrogen atom belonging (EH(H-H)) and not belonging (EH) to H-H bond, respectively, versus

energy of the ethane-ethane complex in some selected optimization steps, which includes the first and the

last steps of optimization plus five other points (steps 1, 22, 27, 33, 43, 49 and 52). Figure S5, in

Supporting Information, shows the plot of energy, E, in hartrees, versus optimization steps of ethane-ethane

R² = 0.9052

-200

-150

-100

-50

0

50

100

150

0 2 4 6 8 10

Bo

ilin

g p

oin

t (

C)

H-H bond paths

R² = 0.9052

-200

-150

-100

-50

0

50

100

150

0 1 2 3 4 5 6 7 8

Boil

ing p

oin

t ( C

)

Number of H-H bond paths

Met-met

Eth-Eth

Pro-Pro

• But-But

Pen-Pen

Hex-Hex

85

complex. The decrease of EH(H-H) of hydrogen atom belonging to H-H bond versus E (Figure 8) reveals the

stabilizing effect of this type of interaction in the ethane-ethane complex. On the other hand, there is no

decrease in overall EH of a hydrogen atom not belonging to H-H bond in ethane-ethane complex (Figure 9).

Then, the decreasing trend in alkane complex energy is accompanied with the decreasing trend of hydrogen

atoms from H-H bonds, while no overall change in hydrogen energy not belonging to H-H bond occurs

when alkane complex reaches the minimum in the potential energy surface.

Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(H-H)) in ethane-ethane

complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps.

-0.6225

-0.6220

-0.6215

-0.6210

-0.6205

-0.6200

-159.654 -159.653 -159.652 -159.651 -159.650 -159.649

E (hartrees)

EH

(H-H

)(h

artre

es)

86

Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to H-H bond (EH) in ethane-ethane

complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps.

A complementary ELF analysis of H-H bond in alkane complexes (from methane-methane until

hexane-hexane) was carried out. The coordinates of QTAIM bond critical point from a selected H-H bond in

each alkane complex was used for locating the corresponding ELF BCP. The calculation of gradient of ELF

function at QTAIM BCP coordinates confirmed the ELF BCP for all alkane complexes. These BCP’s are

index 1 critical points, (r), whose values are 0.010, 0.015, 0.021, 0.017, 0.026 and 0.025 for the methane-

methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane-hexane complexes,

respectively (Table 3). These (r) values are nearly half the (r) values for weak hydrogen bond.89

Furthermore, the electron density and Laplacian of electron density values at these points have topological

features of closed-shell interactions. Then, ELF is capable to identify H-H bond as well as QTAIM,

indicating that each H-H bond in alkane complexes is weaker than a weak hydrogen bond.

-0.6224

-0.6222

-0.6220

-0.6218

-0.6216

-0.6214

-0.6212

-0.6210

-0.6208

-0.6206

-0.6204

-159.654 -159.653 -159.652 -159.651 -159.650 -159.649

E (hartrees)

EH

(hartrees)

87

Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at bond

critical point of a selected H-H bond in alkane complexes.

Complex η(r)

ρ(r) 𝛁2

ρ(r)

Methane-methane 0.010 0.0032 0.0093

Ethane-ethane 0.015 0.0053 0.0180

Propane-propane 0.021 0.0051 0.0137

Butane-butane 0.017 0.0050 0.0158

Pentane-pentane 0.026 0.0067 0.0193

Hexane-hexane 0.025 0.0067 0.0197

CONCLUSIONS

This work gives new insights and understanding about intramolecular and intermolecular

interactions in alkanes.

There are H-H bonds in highly branched C8H18 isomers, but not in linear or less branched alkanes.

Then, the hydrogen-hydrogen bond plays a secondary role on the stability of branched alkanes which partly

accounts for the increased stability of highly branched alkanes when compared with linear or less branched

isomers.

We found very good linear relations between H-H bond in complex alkanes and their stabilities and

between H-H bond in alkanes and their boiling points. For all studied molecular systems, the hydrogen-

hydrogen bonds perform a stabilizing interaction. The hydrogen-hydrogen bond plays an important role on

the boiling point of linear alkanes. Intermolecular interactions in alkane complexes are related to H-H bond,

not to non-directional induced dipole-induced dipole.

The electron localization function can also be used for identification of hydrogen-hydrogen bond.

The ELF topological approach enables a clear identification of hydrogen-hydrogen bonds in alkane

complexes whose (r) are smaller than that from a weak hydrogen bond.

ACKNOWLEDGMENTS

Authors thank, Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),

Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional

de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

88

ASSOCIATED CONTENT

Supporting Information

Optimized geometries of alkane molecules; Virial graphs of Alkane complexes optimized at

B3LYP/6-311++G(d,p), M062X/6-311++G(d,p) levels and G4 method; plot of energy optimization of

ethane-ethane complex. This material is available free of charge via the Internet at http://pubs.acs.org.

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94

TABLE OF CONTENTS

R² = 0.9052

-200

-150

-100

-50

0

50

100

150

0 2 4 6 8 10

Bo

ilin

g p

oin

t (

C)

H-H bond paths

R² = 0.9052

-200

-150

-100

-50

0

50

100

150

0 1 2 3 4 5 6 7 8

Bo

ilin

g p

oin

t ( C

)

Number of H-H bond paths

Met-met

Eth-Eth

Pro-Pro

• But-But

Pen-Pen

Hex-Hex

95

Supporting Information

Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio,

QTAIM and ELF study

Norberto K. V. Monteiro1, and Caio L. Firme

1*

Figure S1. Optimized geometries of alkane molecules. The alkane molecules were optimized at (a)

B3LYP/6-311++G(d,p), (b) M062X/6-311++G(d,p), (c) G4 method, (d) G4/CCSD/6-311++G(d,p) (T) and

(e) wB97XD/6-311++G(d,p). (1) Methane, (2) ethane, (3) propane, (4) butane, (5) pentane and (6) hexane.

a)

12

34

5

6

b)

c)

d)

e)

96

Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p) level of theory.

Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-

But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.

Met-Met

But-But

Eth-Eth

Pen-Pen

Pro-Pro

Hex-Hex

Met-Met Eth-Eth Pro-Pro

But-But Pen-Pen Hex-Hex

97

Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p) level of theory.

Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-

But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.

Met-Met

But-But

Eth-Eth

Pen-Pen

Pro-Pro

Hex-Hex

Met-MetEth-Eth Pro-Pro

But-But Pen-Pen Hex-Hex

98

Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methane-methane (Met-Met),

ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen)

and hexane-hexane (Hex-Hex) complexes.

Met-Met

But-But

Eth-Eth

Pen-Pen

Pro-ProMet-Met

Eth-Eth Pro-Pro

But-But Pen-PenHex-Hex

Hex-Hex

99

Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of optimization steps.

-159.6540

-159.6535

-159.6530

-159.6525

-159.6520

-159.6515

-159.6510

-159.6505

-159.6500

0 10 20 30 40 50 60 70

E (

Ha

rtre

e)Step N

100

5. CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry.

Monteiro, N. K. V.; de Oliveira, J. F.; Firme, C. L. New Journal of Chemistry 2014, 38, 5892.

Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and Germanium Parents – a

DFT, ADMP, QTAIM and GVB study

Norberto K. V. Monteiro, José F. de Oliveira and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: caiofirme@quimica.ufrn.br

Abstract

The electronic structure and the stability of tetrahedrane, substituted tetrahedranes and silicon and

germanium parents have been studied at ωB97XD/6-311++G(d,p) level of theory. The quantum theory of

atoms in molecules (QTAIM) was used to evaluate the substituent effect on the carbon cage in the

tetrahedrane derivatives. The results indicate that electron withdrawing groups (EWG) have two different

behaviors, ie., stronger EWG makes the tetrahedrane cage slightly unstable while slight EWG causes a

greater instability in the tetrahedrane cage. On the other hand, the sigma electron donating groups, σ-EDG,

stabilizes the tetrahedrane cage and π-EDG leads to tetrahedrane disruption. NICS and D3BIA indices were

used to evaluate the sigma aromaticity of the studied molecules, where EWGs and EDGs results in decrease

and increase, respectively, of both aromaticity indices, showing that sigma aromaticity plays an important

role in the stability of tetrahedrane derivatives. Moreover, for tetra-tert-butyltetrahedrane there is another

stability factor: hydrogen-hydrogen bonds which imparts a high stabilization in this cage. Generalized

G = -35.9 kcal.mol-1

ΔGf = 20.5 kcal.mol-1

G= -15.5 kcal.mol-1

REF = 0

Graphical Abstract

101

valence bond (GVB) was also used to explain the stability effect of the substituents directly bonded to the

carbon of the tetrahedrane cage. Moreover, the ADMP simulations are in accordance with our

thermodynamic results indicating the unstable and stable cages under dynamic simulation.

Keywords: Tetrahedrane; QTAIM; NICS; D3BIA; GVB; ADMP

1. Introduction

The development of new high energy molecules has aroused great interest of theoreticians and

experimentalists, with applications in the military industry and others high-tech fields.1 Cage strained

organic molecules can be used as potential explosives in response to various external stimuli such heat,

shock or impact.1–4

The highly-strained cage compounds have raised interest due to their high density and

high energy. As a consequence, they can also be used as energetic materials, such as

hexanitrohexaazaisowurtzitane 1,5

and octanitrocubane.6 Similarly, tetrahedrane is also a high-energy

molecule. Despite many attempts to isolate the parent tetrahedron (unsubstituted – C4H4) were not

successful,7 Maier et al

8 in 1978 were able to synthesize the first tetrahedrane derivative, the tetra-tert-

butyltetrahedrane.

Tetrahedrane is one of the most strained organic molecule, with highly symmetrical structure and

unusual bonding being known as a Platonic solid. It has a strain energy of 586 kJ.mol-1

according to Wiberg

et al 9 which makes this compound thermodynamically unstable. As a consequence, their synthesis and

isolation tends to be extremely difficult. Conversely, tetrahedrane has σ-aromaticity10

because it has a large

negative NICS (Nucleus Independent Chemical Shift ) value, which can somehow contribute for decreasing

the high instability of tetrahedrane cage.

Tetrahedrane and its derivatives can be obtained from the corresponding cyclobutadiene or

derivatives. The anti-aromatic and highly-reactive cyclobutadiene has only been isolated at low temperature

in inert matrices11

and immobilized more recently at room temperature in hemicarcerand.12

On the other

hand, structural parameters of substituted cyclobutadienes have been reported showing pronounced bond

alternations, 13,14

although only slight CC bond length alternations (1.464 and 1.482 Å), under room

temperature, of tetrakis (tert-butyl) cyclobutadiene 15

have been reported. In fact, measurements at -150 °C

showed noticeably larger bond alternation (1.441 and 1.526 Å) for tetrakis (tert-butyl) cyclobutadiene as

well.

Tetra-tert-butyltetrahedrane can be formed from tetra-tert-butylcyclobutadiene by photolysis using n-

octane as solvent while the reverse reaction takes place thermally with an activation barrier of 26 kcal mol-1

16. Similarly, an ab initio study of the interconversion of the C4H4 system predicts an activation energy of

about 30 kcal mol-1

.17

Maier and co-workers succeeded in synthesizing tetrakis (tert-butyl)tetrahedrane by

102

photochemical isomerization of the corresponding cyclobutadiene.8 The reason for its stability is attributed

to the voluminous tBu-substituents that avoid the tetrahedrane skeleton from ring-opening due to the van der

Waals strain among them, the so-called “corset effect”. However, this effect is lost if one of the tert-butyl

substituents is replaced with a smaller group. Indeed, phenyl- and methyl-substituted tetrahedrane

derivatives were not detected, even under matrix isolation conditions.18

Recently, it was synthesized the tetrakis (trimethylsilyl) tetrahedrane ((Me3Si)4THD) by

photochemical isomerization of the corresponding cyclobutadiene ((Me3Si)4CBD),19,20

where (Me3Si)4THD

has enhanced thermal stability due the four σ-donating trimethylsilyl groups that electronically stabilize the

highly strained tetrahedrane skeleton.21,22

Indeed, it was also reported the synthesis of

perfluoroaryltetrahedranes with extended σ-π conjugation between the strained tetrahedrane core and the

aromatic ring.23

In addition, the insertion of a heteroatom (N, O, S, Ge, Si or a halogen atom) into the

tetrahedrane core would be interesting because the tetrahedrane σ framework has the potential to interact

with the nonbonding orbitals on the heteroatom, according to molecular orbital theory. However, such

heteroatom-substituted tetrahedrane derivatives have remained elusive because of the synthetic difficulty in

preparing such molecules.24

In this work tetrahedrane and its derivatives where studied from different tools such as theory of

density functional (DFT), quantum theory of atoms in molecules (QTAIM), atom-centered density matrix

propagation (ADMP) and generalized valence bond (GVB) in order to investigate their electronic structures,

thermochemical and dynamical stabilities that eventually lead to the substituents that enhance the stability

of tetrahedrane cage. The results from QTAIM, GVB and the aromaticity indices provided new and

important information about the main structural and electronic effects associated with the tetrahedrane cage

instability and the reasons why some substituents increase its stability. Moreover, this paper also provides

information about the stability of cages containing germanium and silicon parents of tetrahedrane and

tetrakis(trifluoromethyl) tetrahedrane.

2. Computational details

The geometries of the studied species were optimized by using standard techniques.25

Frequency

calculations were performed to analyze vibrational modes of the optimized geometry in order to determine

whether the resulting geometries are true minima or transition states. Calculations were performed at

ωB97XD/6-311++G(d,p)26

level of theory by using Gaussian 09 package,27

including the electronic density

which was further used for QTAIM calculations. ωB97XD functional incorporates an empirical dispersion

term (D) that improves treatment of non-covalent complexes, it uses 100% Hartree Fock (HF) exchange for

long-range interactions and it has an adjustable parameter (X) to include short-range exact exchange.28,29

103

Mohan and co-workers30

have shown that ωB97XD functional exhibited better performance in the

description of bonded and nonbonded interactions in comparison with other DFT methods. All topological

information 31,32

were calculated by means of AIM2000 software.33

The algorithm of AIM2000 for

searching critical points is based on Newton-Raphson method which relies heavily on the chosen starting

point.34

Integrations of the atomic basins were calculated in natural coordinates with default options of

integration.

The valence bond package VB200035

, version 2.5, was used for all generalized valence bond, GVB,

calculations. The GVB singly occupied orbitals were calculated from VB/6-31G level theory and they were

generated by means of Molekel visualization program.36

The package VB2000 generates nonorthogonal

singly occupied orbitals from a general implementation of group function theory (GFT)37

and modern VB

methods based on high efficiency of the algebrant algorithm.35,38

Nucleus Independent Chemical Shift (NICS)39

and Density, Degeneracy, Delocalization-Based Index

Aromaticity (D3BIA)40

were applied to determine the aromaticity in tetrahedrane and its derivatives. D3BIA

is based on density of (homo)aromatic ring, degeneracy and uniformity of the electron density among the

atoms of the aromatic ring. We will use the same D3BIA formula for homoaromatic species41

where the

electron density is associated with the charge density of the ring critical point (for 3c-2e bonding systems,

for example) or of the cage critical point (for tridimensional 4c-2e bonding system). The degeneracy (𝛿) is

associated with the atomic energy of the constituent atoms of the aromatic ring of the molecular system.

Another important electronic feature in D3BIA is the uniformity of the electronic density calculated from

the delocalization index of atoms of the aromatic ring. Then, D3BIA for homoaromatic especies is defined

as:

𝐷3𝐵𝐼𝐴 = 𝜌(3,+3) ∙ 𝐷𝐼𝑈 ∙ 𝛿 (1)

Where ρ is the ring density factor of the cage critical point (3,+3), for cage structures, and 𝛿 is the

degree of degeneracy where its maximum value (𝛿=1) is given when the difference of energy between the

atoms of the cage is less than 0.009 au. The formula of 𝛿 is given in Eq. (2).

𝛿 =𝑛

𝑁 (2)

Where n is the number of atoms whose ∆𝐸(Ω) ≤ 0.009𝑎𝑢. and N is the total number of atoms in the

homoaromatic circuit. The minimum value for n is 1, where there is no atomic pair whose ∆𝐸(Ω) ≤

0.009𝑎𝑢.

The DIU is the delocalization index uniformity among bridged atoms given by Eq. (3):

𝐷𝐼𝑈 = 100 −100𝜎

𝐷𝐼 (3)

104

Where σ is mean deviation and 𝐷𝐼 is the average of delocalization index between C-C, Si-Si or Ge-

Ge bonds of the cage. The DI gives the amount of electron(s) between any atomic pair. 31

The NICS values were calculated with the B3LYP/6-311++G(d,p) through the gauge-independent

atomic orbital (GIAO) method42

implemented in Gaussian 09. The magnetic shielding tensor was calculated

for ghost atoms located at the geometric center of the cage.

The ab initio molecular dynamics simulations involves quantum chemistry calculations aiming to

obtain the potential energy and nuclear forces.43–46

The simulations performed in this work involved the

atom-centered density matrix propagation (ADMP)47–50

in order to study the stability of tetrahedrane and

selected substituted tetrahedranes. The ADMP calculations were performed at B3LYP/6-31G(d) level of

theory in Gaussian 09. A time step of 0.1 fs was used for the ADMP trajectories. The Nosé-Hoover

thermostat51,52

was employed to maintain a constant temperature at 298.15K. The default fictitious electron

mass was 0.1 amu. A maximum number of 50 steps were used in each trajectory.

3. Results and discussion

Scheme 1 shows the reaction route (diazocompounds 1a-1i cyclobutadienes 2a-2i

tetrahedranes 3a-3i) to synthesize tetrahedrane (and derivatives) from its precursors diazocompound and

cyclobutadiene (and derivatives).

Scheme 1

Different substituted tetrahedranes were studied: tricyclo[1.1.0.02,4

]butane, 3a, 1,2,3,4-

tetramethyltricyclo[1.1.0.02,4

]butane, 3b, 1,2,3,4-tetrakis(trifluormethyl)tricyclo[1.1.0.02,4

]butane, 3c,

1,2,3,4-tetrakis(trifluorosilyl)tricyclo[1.1.0.02,4

]butane, 3d, tetra-tert-butyltricyclo[1.1.0.02,4

]butane, 3e, 1-

1 2 3

R1=R2=R3=R4 = -H (2a)

= -CH3 (2b)

= -C(H3C) 3 (2c)

= -CF3 (2d)

= -SiF3 (2e)

R1=R2=R3=H and R4 = -NO2 (2f)

= -NH2 (2g)

= -OH (2h)

= -NO2 (2i)

R1=R2=R3=R4 = -H (1a)

= -CH3 (1b)

= -C(H3C) 3 (1c)

= -CF3 (1d)

= -SiF3 (1e)

R1=R2=R3=H and R4 = -NO2 (1f)

= -NH2 (1g)

= -OH (1h)

= -NO2 (1i)

R1=R2=R3=R4 = -H (3a)

= -CH3 (3b)

= -C(H3C) 3 (3c)

= -CF3 (3d)

= -SiF3 (3e)

R1=R2=R3=H and R4 = -NO2 (3f)

= -NH2 (3g)

= -OH (3h)

= -NO2 (3i)

105

nitro tricyclo[1.1.0.02,4

]butane, 3f, 1,2,3,4-tetraamino tricyclo[1.1.0.02,4

]butane, 3g, 1,2,3,4-tetrahidroxy

tricyclo[1.1.0.02,4

]butane, 3h, and 1,2,3,4-tetranitro tricyclo[1.1.0.02,4

]butane, 3i. In addition, Ge4 and Si4

parents of tetrahedrane, 4a-4b, and tetrakis(trifluoromethyl) tetrahedrane, 5a-5b, were also studied (Scheme

2).

Scheme 2

In scheme 3 shows the structures of tetraamino-substituted tetrahedrane, 3g, tetrahydroxi-substituted

tetrahedrane, 3h, tetranitro-substituted tetrahedrane, 3i, and tetratrifluoromethyl-substituted silicon parent of

tetrahedrane, 4b, that were optimized and yielded the corresponding open structures 3g’, 3h’, 3i’ and 4b’,

which prevented any structural study with these molecules.

Scheme 3

3g 3h 3i

3g’ 3h’ 3i’

4b

4b’

R1=R2=R3=R4 = -H (4a)

= -CF3 (4b)

4 5

R1=R2=R3=R4 = -H (5a)

= -CF3 (5b)

106

The Scheme 3 indicates that 3g, 3h, 3i and 4b do not have stable tetrahedrane cage. However, no

unique pattern can be established to account for this fact. The –NH2 and –OH groups are electron

withdrawing groups (EWG) by inductive effect and electron donating groups (EDG) by resonance effect. On

the other hand, the –NO2 and –CF3 do not have dual behavior and they are EWGs. Moreover, for

tetrahedrane cage four –CF3 do not disrupt the tetrahedrane cage while four –NO2 groups do so. Conversely,

in Si4 tetrahedrane parent, -CF3 disrupts the corresponding cage.

The equilibrium geometries and selected bond lengths, in Angstroms, of tetrahedrane and

derivatives are depicted in Figure 1. The corresponding virial graphs are shown in Figure 2, along with

several topological data of selected bond critical points. Electronic density, ρ(r), Laplacian of the electron

density, ∇2 ρ(r), local energy density, H(r) and relative kinetic energy density, G(r)/ρ values are shown for

critical points cage-substituent and electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local

energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization index, DI, bond order, n, and

ellipticity are shown for critical points of carbon, silicon or germanium atoms in the tetrahedrane (or parent)

cage. From Figure 1, we can see that the alkyl and silyl groups impart the shortening of C-C bonds from the

cage while the –CF3 and –NO2 groups lengthen the corresponding C-C bonds. As a consequence, based only

on geometrical information, we can assume that alkyl and silyl groups behave as EWGs. These EWGs and

EDGs do not disrupt the tetrahedrane cage.

We have analyzed the topology of the electron density of all species using the quantum theory of

atoms in molecules (QTAIM).31,53,54

According to the topological analysis of QTAIM, when ρ of the critical

point is relatively high (×10-1

au.) and ∇2ρ < 0, the chemical interaction is defined as shared shell and it is

applied to covalent bond. However, other parameters are required to describe the nature of the chemical

interaction such as the local energy density, H(r), which is the sum of local kinetic energy density, G(r), and

local potential energy density, V(r), respectively, and the ratio G(r)/ ρ. Cremer and Kraka55

suggested that

H(r) < 0 and G(r)/ρ < 1 are indicative of shared shell (or covalent) interaction. In Figure 2 the values of ρ,

∇2ρ, H(r) and G(r)/ρ of C-C, Si-Si and Ge-Ge bonds of the tetrahedrane cage (or parent) indicate that all C-

C, Si-Si and Ge-Ge bond paths are shared shell interactions for 3a – 3f, 4a, 5a and 5b. The ellipticity (ε)

values in the tetrahedranes indicate no cylindric symmetry on the single C-C bond, while for Si-Si and Ge-

Ge cages the ε is closer to zero indicating symmetry around the distribution of electronic density of these

bonds.

107

Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and germanium cages

(G-I) with corresponding C1-C4, C1-C5, C1-Si, C1-N, Si-Si, Si-H, Ge-Ge, Ge-H and Ge-C bond lengths

(in Angstroms); values of selected angles (in degree).

(3a)

A

1

23

4

5

6

7

8

(3b)

B

(3c)

C

(3d)

D

(3e)

E

(3f)

F

(4a)

G

(5a)

H

(5b)

I

108

Figure 2: Virial graphs of substituted tetrahedranes (A-F), silicon and germanium cages (G-I) with the

electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r) and relative

kinetic energy density, G(r)/ρ at cage-substituent bond critical, in au. Values of electronic density, ρ(r),

Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ,

delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the

tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b.

The virial graph of tetra-tert-butyltetrahedrane, 3c, in Figure 2C indicates a different reason for its

stability rather than the so-called “corset effect” which is a steric repulsion between bulk substituents

preventing the breaking of the cage.8 There are 23 intramolecular hydrogen-hydrogen (H-H) bonds between

tert-butyl groups according to the virial graph of 3c where the average charge density of the corresponding

critical point, ρH-H, is 0.0041 au. The augmented cooperative effect of these small interactions contributes for

ρ = 0.28012ρ = 0.2442

H(r) = -0.280

G(r)/ρ = 0.129

ρ = 0.25392ρ = 0.1515

H(r) = -0.217

G(r)/ρ = 0.256

ρ = 0.25312ρ = 0.1494

H(r) = -0.213

G(r)/ρ = 0.252

ρ = 0.27482ρ = 0.1835

H(r) = -0.244

G(r)/ρ = 0.222

ρ = 0.15672ρ = 0.0748

H(r) = -0.115

G(r)/ρ = 0.259

ρ = 0.29962ρ = 0.1685

H(r) = -0.277

G(r)/ρ = 0.362

ρ = 0.13792ρ = 0.0853

H(r) = -0.110

G(r)/ρ = 0.181

ρ = 0.14692ρ = 0.0683

H(r) = -0.062

G(r)/ρ = 0.595

ρ = 0.13392ρ = 0.0290

H(r) = -0.072

G(r)/ρ = 0.325

(3a)

A(3b)

B (3c)

C

(3d)

D

(3e)

E

(3f)

F

(4a)

G

(5a)

H(5b)

I

47

48

ρ = 0.22302ρ = -0.015

H(r) = -0.174

G(r)/ρ = 0.847

DI = 1.025

n = 1.02

ε = 0.105

ρ = 0.21982ρ = -0.017

H(r) = -0.168

G(r)/ρ = 0.845

DI = 0.980

n = 0.97

ε = 0.089

ρ = 0.21912ρ = -0.017

H(r) = -0.168

G(r)/ρ = 0.842

DI = 0.948

n = 0.94

ε = 0.092

ρ = 0.22412ρ = -0.017

H(r) = -0.175

G(r)/ρ = 0.857

DI = 0.953

n = 0.94

ε = 0.109

ρ = 0.22022ρ = -0.016

H(r) = -0.169

G(r)/ρ = 0.838

DI = 1.016

n = 1.01

ε = 0.213

ρ = 0.23162ρ = 0.0007

H(r) = -0.190

G(r)/ρ = 0.819

DI = 1.012

n = 1.00

ε = 0.099

ρ = 0.07832ρ = 0.0145

H(r) = -0.036

G(r)/ρ = 0.280

DI = 0.918

n = 1.60

ε = 0.039

ρ = 0.06992ρ = -0.0034

H(r) = -0.026

G(r)/ρ = 0.420

DI = 0.951

n = 1.25

ε = 0.063

ρ = 0.06952ρ = -0.0035

H(r) = -0.026

G(r)/ρ = 0.420

DI = 0.912

n = 1.18

ε = 0.043

109

the stabilization of the molecular system.56

An important study based on QTAIM, ELF and ab initio

calculations demonstrated the stability effect of the H-H bonds in alkane complexes and highly branched

alkanes. 57

As to the derivative 3c, according to the values of the hydrogen atomic energy belonging or not

to H-H bond, the hydrogens atoms belonging to H-H bond (e.g., H48 in Figure 2C) are in average 2.65 kcal

mol-1

lower in energy than those not belonging to H-H bond (e.g., H47 in Figure 1C). This result shows the

stability effect of H-H bond in 3c.

Figure 3 shows the Gibbs free energy difference along the coordination reaction from diazo

compound (as a reference) to cyclobutadiene and to tetrahedrane structures (plus nitrogen), as depicted in

Scheme 1.

Figure 3: Gibbs free energy difference, in kcal mol-1

, from diazocompound (as a reference) to

cyclobutadiene and tetrahedrane structures, along with nitrogen, with respect to the syntheses of 3a (A), 3b

(B), 3f (C) and 3e (D).

Data of G1 in Figure 3 indicate that the diazocompounds 1a, 1b, 1f and 1e are higher in Gibbs free

energy than the corresponding cyclobutadiene (or cyclobutadiene derivative), 2a, 2b, 2f and 2e, respectively.

Their corresponding formation enthalpy (H1) follows the same trend (Table 1). In fact, all cyclobutadiene

derivatives (2a – 2f), plus nitrogen molecule, are more stable than the corresponding diazocompound (Table

1). The same trend is shown for predecessors of silicon tetrahedrane parent 4a, but nothing can be said about

ΔG1= -35.9 kcal.mol-1ΔG2= -15.5 kcal.mol-1

ΔG1= -32.3 kcal.mol-1

ΔG2= -9.7 kcal.mol-1

ΔG1= -44.7 kcal/mol

ΔG2= -12.3 kcal.mol-1

ΔG1= -30.6 kcal.mol-1

ΔG2= -33.5 kcal.mol-1

(A) (B)

(D)(C)

ΔGf = 20.5 kcal.mol-1 ΔGf = 32.3 kcal.mol-1

ΔGf = 22.5 kcal.mol-1

ΔGf = -2.9 kcal.mol-1

110

the predecessors of germanium tetraedrane parents 5a and 5b because their corresponding diazocompounds

were not formed. Likewise, the tetrahedrane and all tetrahedrane derivatives, 3a – 3f, are lower in energy

(both enthalpy and Gibbs free energy) than the corresponding diazocompound, as indicated by G2 and H2,

except for H2 related to 3f. The silicon tetrahedrane parent 4a exhibits the same behavior.

Table 1 depicts the values of Gibbs free energy difference and enthalpy difference from

diazocompound to cyclobutadiene (and derivatives), G1 and H1, from diazocompound to tetrahedrane

(and derivatives), G2 and H2, and from cyclobutadiene (and derivatives) to tetrahedrane (and derivatives),

Gf and Hf, in kcal mol-1

, respectively. Table 1 also shows the delocalization index, DI, bond order, n, and

ellipticity, ε, of carbon-carbon, silicon-silicon or germanium-germanium bond in the tetrahedrane (or parent)

cage, besides the bond path angle, in degree, in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b.

The values of n were obtained from the linear relation between formal bond order (n) and delocalization

index,58

which yields very good correlations for carbon-carbon and germanium-germanium bonds, but

presents moderate correlation for Si-Si bond. The ellipticity, ε, of a bond critical point indicate whether the

corresponding bond has elliptical symmetry (when ε=0, for single or triple bonds) or not.31

Table 1: Values of Gibbs free energy difference and enthalpy difference for the corresponding

tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to cyclobutadiene (and derivatives), G1 and H1,

from diazocompound to tetrahedrane (and derivatives), G2 and H2, and from cyclobutadiene (and

derivatives) to tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol-1

, respectively. Values of

delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the

tetrahedrane (or parent) cage, bond path angle, in degree, from bonds in the tetrahedrane (or parent) cage for

3a – 3f, 4a, 5a and 5b.

Molecule ΔG1 /

kcal mol-1

ΔG2 /

kcal mol-1

ΔH1 /

kcal mol-1

ΔH2 /

kcal mol-1

ΔGf /

kcal mol-1

ΔHf /

kcal mol-1

DI / e n Bond path

angle/o

3a -35.92 -15.45 -27.43 -6.24 20.47 21.19 1.025 1.02 0.105 76.9

3b -44.65 -12.30 -33.77 -2.31 32.35 31.46 0.980 0.97 0.089 79.1

3c -12.26 -27.72 -2.90 -16.48 -15.46 -13.58 0.948 0.94 0.092 78.8

3d -37.51 -15.81 -27.04 -5.50 21.70 21.54 0.953 0.94 0.109 78.6

111

(a) The diazocompound was not obtained.

(b) Values from carbon(from cage)-nitrogen(from nitro group).

(c) Average value

Table 2 shows the topological values (AIM atomic charge, virial atomic energy, atomic volume and

localization index) of the carbon, silicon and germanium atomic basins, , of the tetrahedrane (or parent)

cage for 3a – 3f, 4a, 5a and 5b. These values are obtained from the integration of the corresponding atomic

basin.

Table 2: Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole moment, M(Ω), atomic

volume, V(Ω) and localization index, LI of carbon, silicon and germanium atoms in tetrahedrane, silicon and

germanium derivatives, respectively.

Molecule q(Ω) E(Ω) M(Ω) V(Ω)

LI

3a -0.121 -38.0936 0.435 6.09 4.059

3b -0.109 -38.1522 0.292 6.10 3.968

3c -0.089 -38.1732 0.300 6.09 3.934

3d 0.037 -38.0509 0.202 6.10 3.844

3e -0.697 -38.2994 1.48 6.69 4.732

3f 0.235(a)

-37.9671(a)

0.305(a)

5.76(a)

3.691(a)

4a 0.655 -289.3407 0.889 13.30 11.44

5a 0.283 -2074.3391 0.076 31.66 29.74

5b 0.387 -2074.3590 0.097 31.57 29.66

(a) Values of carbon atom attached to the nitro group.

3e -30.59 -33.48 -20.97 -22.71 -2.89 -1.74 1.016 1.01 0.213 72.2

3f -32.28 -9.72 -20.76 3.45 22.55 24.21 1.012(b)

1.00(b)

0.099(b)

80.2(c)

4a -10.27 -8.92

-4.88

-1.11

1.35 3.77 0.918 1.60 0.039 80.6

5a -(a)

-(a)

-(a)

-(a)

7.24 10.60 0.951 1.25 0.063 74.0

5b -(a)

-(a)

-(a)

-(a)

-3.23 11.24 0.912 1.18 0.043 75.3

112

Despite the antiaromatic nature of cyclobutadiene or its derivatives and the σ-aromaticity10

of

tetrahedrane cage, in most cases, tetrahedrane and its derivatives are higher in energy (both enthalpy and

Gibbs free energy) than the corresponding cyclobutadiene as it is indicated by Gƒ and Hƒ. The fact that

the σ-aromaticity10

of tetrahedrane (or derivatives) does not impart smaller energy in comparison to the

corresponding cyclobutadiene (which is antiaromatic) can be attributed to the highly strained cage structure

of tetrahedrane and derivatives, except for 3c and 3e which are more stable than 2c and 2e. As for 3c,

QTAIM analysis indicates that its relative stability is ascribed to a large amount of stabilizing hydrogen-

hydrogen bonds, as aforementioned, while the relative stability of 3e is possibly reasoned by the great

charge density donator effect of the –SiF3 group, according to the more negative values of q(C) and E(C)

plus higher values of V(C) and LI(C) compared to those from 3a. Nonetheless, the relative stability imparted

by hydrogen-hydrogen bond in 3c is nearly 12 kcal mol-1

higher than that bestowed by the donation effect of

–SiF3 group in 3e.

Taking values of any topological property of tetrahedrane 3a as a reference, the DI values of C-C

bonds in the tetrahedrane cage of 3e indicates that trifluorosilyl group partially removes the charge density

of each C-C bond (Table 1 and Figure 2) where ρ = 0.2801 u.a. and 0.2539 u.a. for 3a and 3e respectively.

However, from Table 2, the AIM atomic charge of carbon atom in 3e is 6 times more negatively charged

than that from 3a, which is corroborated by a greater localization index and greater atomic volume of carbon

atom in 3e. In addition, the virial atomic energy of each carbon atom in 3e is 0.2058 au. smaller than that

from 3a. Then, –SiF3 pronouncedly donates charge density to the carbon atoms of the tetrahedrane cage in

3e, although removes partially the charge density in the C-C bonding region. There is also a greater atomic

dipole moment in the carbon atoms of 3e in comparison with those from all other studied tetrahedrane

derivatives or parents, which can be partially related to a larger amount of charge density within carbon

atomic basins in 3e, besides the influence of the electropositive Si(F3) atom bonded to the carbon atom.

Then, –SiF3 in 3e increases the charge density of C atom attached to it but removes the charge density of the

adjacent C-C bond.

Conversely, the methyl and t-butyl groups, in 3b and 3c, respectively, are probably EWGs

according to carbon atomic charge and localization index, where these values are slightly less negative, than

those from 3a. Moreover, methyl and t-butyl groups also partially remove charge density from C-C region

bond of tetrahedrane cage, as indicated by the corresponding charge density of BCP (see Figure 2B and 2C),

the DI, bond order and ellipticity (see Table 1) in comparison with those values from 3a.

The nitro group in 3f has a more pronounced electron withdrawing effect than alkyl groups as can

be easily observed in the positive atomic charge, low LI and higher atomic energy of the carbon atom

attached to the nitro group in 3f with respect to those values for the carbon atoms in 3a (Table 2).

Nonetheless, 3b is more unstable than 3f, but 3f is slightly more unstable than 3a. Then, one nitro group

113

LP1’

LP1

LP2

LP2’

VB(C-C)1

VB(C-C)2

VB(C-C)’

(A) (B)

stabilizes the tetrahedrane cage in comparison with four methyl groups, but four hydrogen substituents (in

3a) still generate a more stable structure than one nitro and three hydrogen substituents (in 3f).

Figure 4 shows the doubly occupied GVB oxygen lone pair of nitro group, LP1 and LP2, and singly

occupied C-C sigma orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3f. Table 4 shows the overlap

integrals between the GVB orbitals of Figure 4. From GVB orbitals of 3f (Figure 4), one of the oxygen lone

pair, named LP2, is directed towards the cage. The average overlap integral between LP2 and C-C bonds in

the cage (LP2-CC1) are three-fold greater than the average overlap integral between LP1 (which is not

directed towards the cage) and C-C bonds in the cage (LP1-CC1) as indicated in Table 4. Then, we can infer

that the influence of both oxygen LP2 lone pairs from four nitro groups may cause a great instability and

may lead to the disruption of tetrahedrane cage in 3i yielding 3i’.

Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of oxygen of nitro (-NO2)

functional group substituent (LP1, LP1’, LP2 and LP2’) (A and B) and singly occupied GVB orbitals of C-C

sigma bond [VB(C-C)1, VB(C-C)2 and VB(C-C)’] (B).

Table 4: Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone pairs (LP1, LP2, LP1’

and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’ and VB(C-C)2].

LP1 LP1’ LP2 LP2’

VB(C-C)1 0.007 0.019 0.054 0.045

VB(C-C)2 0.005 0.005 0.005 0.006

VB(C-C)’ 0.008 0.006 0.034 0.009

114

When comparing topological data between 3d and 3f in Table 2, we observe that the

trifluoromethyl group in 3d has a less electron withdrawing effect than the nitro group in 3f, since carbon

atomic charge is less positive in 3d, carbon atomic energy is more stable in 3d, and LI is greater in 3d. In

addition, the –CF3 group also decreases the charge density in the C-C bonding region as noted from its DI

and bond order (Table 1). In terms of molecular energy, the stability of 3d is quite similar to that from 3a

and it is slightly higher than that from 3f.

To sum up the energetic influence of the studied substituents in tetrahedrane cage we can note that

EWGs such as –CF3 do not affect the relative stability of the tetrahedrane cage, while with a stronger EWG

(e.g., –NO2) the tetrahedrane cage becomes slightly more unstable. On the other hand, slight EWG such as –

CH3 imparts a higher instability in tetrahedrane cage. Conversely, a -EDG such as –SiF3 makes the

tetrahedrane cage more stable, while a -EDG (e.g. –NH2 and –OH) disrupts the tetrahedrane cage. These

trends can be observed in the decreasing order of Gƒ (3c > 3e > 3a > 3d > 3f > 3b) and Hƒ (3c > 3e > 3a

> 3d > 3f > 3b), without the silicon and germanium parents.

The integration of the atomic basins gives the value of the bond path angle of the electron density

function. The bond path angle involving the atoms in the cage of all tetrahedrane derivatives and Si/Ge

parents is higher than the corresponding geometric bond angle (Table 1). The increase of the bond angle in

cyclopropane ring (which is part of the tetrahedrane cage) decreases its ring strain (or cage strain) which is

so-called banana bond. 59

However, there is no linear relation between bond path angle and Gƒ for the

studied series.

The tetrahedrane cage with silicon and germanium atoms, 4a and 5a, were also obtained, as well as

trifluoromethyl tetrasubstituted germanium cage, 5b. However, the trifluoromethyl tetrasubstituted silicon

parent disrupted during optimization procedure (Scheme 3). Formation enthalpy, Hƒ, indicates that they

have intermediate stability between the most stable (3c and 3e) and least stable (3a, 3b, 3d and 3f)

substituted tetrahedranes. Nonetheless, this stability order does not match the aromaticity order, according to

NICS and D3BIA, where the set 4a, 5a and 5b are the least aromatic.

Table 3 shows the topological values (𝐷𝐼 , DIU, δ and ρ(3,+3)) associated with the D3BIA formula

for homoaromatic and cage structures, along with NICS values in the geometrical center of the tetrahedrane

(or parent) cage for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and 5b. The aromaticity of the substituted tetrahedranes,

germanium and silicon parents were investigated with two aromaticity indices: NICS and D3BIA. Hereafter

we mention the increase or decrease of NICS with respect to the aromaticity where aromaticity increases

with higher negative values of NICS and decreases with lower negative values (or higher positive values) of

NICS.

115

Figure 5 shows the linear relations between NICS and D3BIA where Figure 5A lacks only one

molecule, 3f, and Figure 5B has all the studied molecules (3a-3f, 4a, 4b and 5a).

Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS versus D3BIA for 3a-

3f, 4a, 4b and 5a.

Figure 5A shows a very good linear relation between NICS and D3BIA where 3f was excluded

because NICS in 3f is higher (more negative) than it is supposed to be since 3c is 10.81 kcal mol-1

more

stable than 3f but they have the same NICS value. Probably, this is the influence of lone pairs of oxygen

atoms from nitro group that interact with the tetrahedrane cage of 3f (Figure 4 and Table 4) which enhances

its NICS value.

Table 3: Average DI, 𝐷𝐼 , of the carbon or silicon or germanium atoms in the tetrahedrane (or parent) cage,

the corresponding DIUs, the degree of degeneracy (δ) involving the atoms of the tetrahedrane (or parent)

cage, the charge density of the cage critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e,

3f, 4a, 5a and 5b.

Molecule 𝑫𝑰 DIU δ ρ(3,+3)/au D3BIA NICS / ppm

3a 1.025 99.97 1.0 0.182 18.24 -48.53

3b 0.979 99.89 1.0 0.176 17.61 -45.31

3c 0.954 99.99 1.0 0.177 17.68 -44.95

3d 0.961 99.98 1.0 0.182 18.21 -47.45

3e 1.029 99.97 1.0 0.187 18.67 -54.54

3f 0.997 96.77 0.5 0.181 8.77 -45.30

4a 0.933 99.95 1.0 0.046 4.57 -1.03

5a 0.951 99.99 1.0 0.040 4.00 1.85

5b 0.911 99.93 1.0 0.039 3.88 32.33

R² = 0.9142

0

5

10

15

20

-60 -40 -20 0 20 40

D3BIA

NICS / ppm

NICS X D3BIA

R² = 0.7664

0

5

10

15

20

-60 -40 -20 0 20 40

D3BIA

NICS /ppm

NICS X D3BIA

(A)(B)

116

The alkyl groups (in 3b and 3c) decrease both NICS and D3BIA in comparison with 3a. Likewise,

the trifluoromethyl and nitro groups in 3d and 3f, respectively, also decrease both aromaticity indices with

respect to 3a. On the other hand, the trifluorosilyl group increases both NICS and D3BIA in 3e. Then, we

may assume that EWGs decrease the aromaticity of the tetrahedrane cage while the EDG increases its

aromaticity. The decreasing order of formation enthalpy and NICS for 3a-3f is nearly the same (except for

3c where hydrogen-hydrogen bonds play an important role on its stabilization): 3e > 3a > 3d > 3b 3f (for

NICS) and 3e > 3a > 3d > 3f > 3b (for Hƒ). Then, we may assume that the aromaticity (influenced by

EWGs or EDGs) is an important stability factor in the tetrahedrane cage besides the intramolecular

hydrogen-hydrogen bonds in 3c.

As aforementioned, regarding the set 4a, 5a and 5b, there is no relation between their stability and

aromaticity orders. According to NICS, they are not aromatic molecules, but their stability is higher than

that in 3a, 3b, 3d and 3f. Probably, the stabilities of 4a, 5a and 5b are influenced by their predecessors’

stabilities. Regarding 4a, its H1 is the least negative, which means that its corresponding silicon

cyclobutadiene parent is the least stable of all set of substituted cyclobutadienes. Then, the high instability of

predecessor of 4a plays an important role on the large relative stability of 4a.

Figure 6 shows the singly occupied GVB orbitals of sigma C-C bonds, VB(C-C)’, VB (C-C)1 and

VB (C-C)2, of 3a, 3b, 3f and cubane. Table 5 shows the overlap integral between the singly occupied GVB

orbitals represented in Figure 6. Tetrahedrane and cubane are platonic hydrocarbons where only cubane was

synthetically obtained.60

The overlap between vicinal GVB orbitals in cubane, VB(C-C)1 – VB(C-C)2, is

0.224 – nearly four times smaller than the overlap between GVB orbitals of each C-C bond, VB (C-C)1 –

VB(C-C)’. The overlap between vicinal GVB orbitals in tetrahedrane is higher than that from cubane, 0.279,

where we can infer that vicinal C-C bonds tend to increase the electronic repulsion in the tetrahedrane cage,

which could partially explain the instability of tetrahedrane with respect to cubane. The electron

withdrawing effect of alkyl group can also be noticed by the slightly smaller overlap between vicinal GVB

orbitals in 3b (Table 5). Likewise, the higher electron withdrawing effect of the nitro group can also be

noticed from overlap of GVB orbital because this value in 3f is similar to that from cubane. However, as

aforementioned 3f is more unstable than 3a. Then, the electronic repulsion from vicinal C-C bonds within

the cage is not the main instability factor of tetrahedrane cage which means that the main reason for

tetrahedrane instability should be attributed to its angle strain.

117

Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond [VB(C-C)1, VB(C-

C)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D).

Table 5: Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3a, 3b, 3f and

cubane.

VB(C-C)1

3a 3b 3f Cubane

VB(C-C)2 0.279 0.272 0.224 0.224

VB(C-C)’ 0.828 0.826 0.823 0.810

aVB orbitals of 3a bVB orbitals of 3b cVB orbitals of 3f dVB orbitals of cubane

The Figures 7, 8 and 9 shows the plots of total energy (in Hartree) versus the time of trajectory of 5

fs using atom-centered density matrix propagation (ADMP) method for tetrahedrane and substituted

tetrahedranes (3a, 3c and 3e) where the starting structure was previously optimized at ωB97XD/6-

311++G(d,p) level of theory. We used ADMP calculations to test the stability of selected tetrahedrane cages

under dynamic simulation. Some corresponding structures from selected points of each trajectory are also

depicted in Figures 7, 8 and 9. In Figure 7, we can see that no significant changes occur in 3a structure at the

(A) (B) (C)

VB(C-C)1

VB(C-C)2

VB(C-C)’

(D)

118

beginning of the trajectory (0.2 to 0.6 fs). However, at about 1.3 fs the C2-H7 bond is broken. At 2.8 fs the

C3-H5 bond is broken and C4-H8 bond length increases. Moreover, the C-C bond lengths of three

cyclopropane rings of tetrahedrane cage shorten. At about 4.2 fs, the C4-H8 bond is broken and at 5.0 fs the

C3-H5 bond is restored. Then, during the ADMP trajectory, 3a undergoes structural changes in its structure

but it is not restored to its starting optimized geometry at any point. On the other hand, there is no significant

change in the total energy of 3c during the course of its trajectory (Figure 8). Then, 3c has a thermochemical

and dynamical relative stability mainly due to the large amount of stabilizing H-H bonds. Likewise, 3e

presented a similar behavior (Figure 9), but its relative stability is given by –SiF3 group that acts as σ-EDG,

where in 2.5 fs and 4.4 fs occurs a decreasing in two C-Si bond lengths (indicated by red arrows) according

to the nature of donating electrons of trifluorosilyl group.

Figure 7: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of tetrahedrane (3a)

using ADMP method. Some structures are depicted at the corresponding selected points of the trajectory.

119

Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3c using ADMP

method. The structure at the end of trajectory is also depicted.

Figure 9: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3e using ADMP

method. A couple of structures are depicted at the selected corresponding points of the trajectory.

-783.58084

-783.58082

-783.58080

-783.58078

-783.58076

-783.58074

-783.58072

-783.58070

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0T

ota

l E

nerg

y (

Ha

rtr

ee)

Time in Trajectory (femtosec)

-1366.426605

-1366.426600

-1366.426595

-1366.426590

-1366.426585

-1366.426580

-1366.426575

-1366.426570

-1366.426565

-1366.426560

-1366.426555

-1366.426550

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

To

tal

En

erg

y (

Ha

rtr

ee)

Time in Trajectory (femtosec)

120

Conclusions

The influence of substituents on the stability of the tetrahedrane cage reveals that strong electron

withdrawing groups do not affect significantly the relative stability in the tetrahedrane cage, unlike weak

EWG that generate a large instability in tetrahedrane cage. However, substituents that act as sigma electron

donating groups (σ-EDG) stabilize the tetrahedrane cage, although π-EDGs leads to disruption of the

tetrahedrane cage.

According to NICS and D3BIA and their relations with the corresponding formation enthalpy, the

aromaticity is an important factor for the stability of the tetrahedrane cage. The -EDGs increase both

aromaticity and stability of the tetrahedrane cage, whereas the EWGs decrease both aromaticity making and

cage stability, except for 3c in which its stability is due to the intramolecular hydrogen-hydrogen bonds.

Although there is no relation between aromaticity and stability for germanium and silicon tetrahedrane

parents 4a, 5a and 5b their NICS and D3BIA values indicate that these molecules are not aromatic.

According to generalized valence bond method, one of the oxygen lone pair has an electronic

repulsion with C-C bonds in its tetrahedrane cage and probably when four nitro groups are attached to the

tetrahedrane cage the higher electronic repulsion may lead to disruption of corresponding substituted

tetrahedrane. Moreover, the GVB analysis indicate that the instability of the tetrahedrane cage is mainly

imparted by its high strain energy while the C-C electronic repulsion within the cage has a minor influence.

The ADMP analysis indicates that tetrahedrane 3a is dynamically unstable where some C-H bonds

break during the simulation trajectory. Conversely, 3c and 3e remain its structure during the whole

trajectory. Therefore, thermodynamic and dynamic results are in accordance for the relatively stable and

unstable cage structures.

Acknowledgments

Authors thank, Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),

Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional

de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

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124

6. CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The Journal of Physical

Chemistry B.

Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of

Abiraterone on CYP17.

Norberto K. V. Monteiro, Sérgio R. B. Silva and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: caiofirme@quimica.ufrn.br

ABSTRACT

After skin cancer, prostate cancer (PC) is the most common cancer among men in the western world. Once

metastasized, the prostate cancer treatment becomes palliative. The initial treatment is performed with

androgen deprivation therapy (ADT) leading to a decrease in the level of serum testosterone (chemical

castration). However, resistance to castration-therapy can develop, resulting in men with metastatic

castration-resistant prostate cancer (mCRPC). Recently, a novel hormonal agent, Abiraterone acetate

(abiraterone) has been shown to increase the survival rate by 4.6 months when compared to placebo in men

with mCRPC, by inhibiting the key enzyme for the biosynthesis of androgens [17α-hydroxylase-17,20-lyase

(CYP17)]. In this paper, we present a new computational protocol in order to quantify the affinity of

abiraterone (ABE) compared to pregnenolone (PREG). Molecular dynamics and molecular docking were

used to obtain the CYP17-ABE and CYP17-PREG complexes and to obtain the interaction energies of

ligands ABE and PREG. From the ONIOM (B3LYP:AMBER) method, we find that the interaction

electronic energy of ABE is 21.38 kcal mol-1

lowest than PREG. The quantum theory of atoms in molecules

(QTAIM) method was used to quantify the interactions between ABE (or PREG) and CYP17. The results

indicate that ABE has a higher electronic density in their interactions compared to PREG.

Keywords: Prostate cancer; CYP17; Abiraterone; Pregnenolone; Molecular dynamics; ONIOM; QTAIM.

INTRODUCTION

Prostate cancer (PC) is the most common type of cancer in the US and Europe.1 Its development occurs

mainly by the action of androgens testosterone (T) and dihydrotestosterone (DHT), which are synthesized by

125

the testis and adrenal glands through the androgen receptor (AR). The androgen action consists of the

following steps: (a) the entrance of testosterone in the prostatic cells which is converted to

dihydrotestosterone by 5α-reductase; (b) in presence of DHT, AR is released from heat shock protein 90

(HSP90); (c) the AR further undergoes dimerization, phosphorylation and translocation to the nucleus2,3

(Figure 1). The translocated receptor binds to the androgen response elements in the DNA, thereby

activating transcription of AR target genes and thus leading to cell proliferation.4,5

In a reciprocal manner,

after androgen withdrawal, the prostatic cells cease proliferation and undergo apoptosis.6 Then, there is a

dependence of prostatic cells on AR signaling in neoplastic transformation, which is the basis for metastatic

PC therapy.

Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT complex

undergoes phosphorylation and is translocated to the nucleus in the form of dimer. Inside the nucleus, AR-

dimer binds to DNA coregulators leading to transcription activation.

According to the above description, there is a biological close relationship between testosterone and

PC.7 However, epidemiologic studies have shown that the association between testosterone and PC is not

clearly understood.7,8

Solania,9 formulated the time-dependency theory which postulates that the endocrine

biology of prostatic tissue is dependent on the exposure time at a given concentration of sex steroid. This

postulate is a consequence of the results obtained by Solania and collaborators10

where serum testosterone

levels could be stable over time in PC patients.

The most common initial treatment of PC is the androgen deprivation therapy (ADT) resulting in

temporary regression of disease in most patients. This therapy is accomplished by the chemical castration

with luteinizing-hormone releasing-hormone (LHRH) analogs, or surgical castration (orchiectomy).1,11,12

The ADT results in a reduction of about 90% of serum testosterone.13

However, ADT do not affect the

126

production of adrenal and intra-tumoral androgens, which may be relevant in restoring disease14

resulting in

patients with metastatic castration-resistant prostate cancer (mCRPC).15

As aforementioned, androgens T

and DHT play an important role in development of PC and then, potent and specific compounds that inhibit

androgen synthesis may be effective for the treatment of PC.16

A promising target is the 17α-hydroxylase-

17,20-lyase (CYP17) involved in the biosynthesis of T and DHT through of two key reactions, which act

sequentially. First, by converting pregnenolone (PREG) to 17α-hydroxypregnenolone and progesterone to

17α-hydroxyprogesterone; and second by converting 17α-hydroxypregnenolone to dehydroepiandrosterona

and 17α-hydroxyprogesterone to androstenedione.17,18

Furthermore, Auchus19

described an alternative

pathway in the biosynthesis of DHT testosterone-independent wherein CYP17 also acts in this pathway,

emphasizing its importance as a therapeutic target, because its inhibition exerts control over androgen

synthesis. Ketoconazole is known as antifungal which inhibits nonspecifically the CYP17 and reduces

androgen levels, however, its use is associated with several side effects including hepatotoxicity,

gastrointestinal toxicity, adrenal insufficiency and their effects on other cytochrome enzymes.14,20,21

Therefore, is very important to search and development of more potent and specific novel CYP17 inhibitors

as well as the identification of these inhibitors.

Many experimental and computational studies were reported on inhibitory molecules against

CYP17.1,20–24

Recently, several promising agents with different mechanisms of action and therapeutic targets

have demonstrated efficacy and four new drugs (cabazitaxel, sipuleucel-T, denosumab, and abiraterone

acetate) were approved by FDA for the treatment of patients with mCRPC.25–27

Abiraterone (Figure 2) (ABE) is a high selective irreversible CYP17 inhibitor (IC50, 2 to 4 nmol/L),

more potent and selective and less toxic than ketoconazole.12,28,29

Moreover abiraterone reduces serum

testosterone levels below 1.0 ng/dl.30,31

In phase 3 trials conducted in 1195 patients, abiraterone increased

the survival rate to 15.8 months compared with 11.2 months in the control (placebo).32

Figure 2. Chemical structure of ABE

The recognition of ligands by proteins is a key process of many biological functions and systems that

involve the enzymatic catalysis, cell signaling and protein conformational switching.33,34

The binding

affinity is a reflection of the molecular recognition. Prediction of the binding affinity are targets of several

127

computational studies.35–38

Molecular dynamics simulations have been used extensively to predict the

affinity of protein-ligand interactions.39–41

In the present work, molecular dynamics was used to predict the

affinity of the ligands ABE and PREG against CYP17 enzyme.

The main focus of our study is to quantify the affinity of ABE and PRE with the CYP17 enzyme and

investigate the specific interactions protein-ligand. We performed the geometric optimization of the CYP17-

ABE and CYP17-PREG complexes and CYP17 enzyme with ONIOM (B3LYP/6-311G(d,p):AMBER)

method (Figures 10A, 10B and 10C, respectively).

Experimental methods for evaluating the affinity of a specific ligand may require a long working

time and a high monetary cost. The objective of the present work was investigate the inhibitory effect of the

ABE on CYP17. For this study, we compared the stabilities between CYP17-ABE and CYP17-PREG

complexes by using classic molecular dynamics simulations, molecular docking, the ONIOM hybrid method

and, QTAIM method. Our comparative study will provide theoretical information for distinguishing

inhibitors and natural substrates pointing out the direction of predicting new CYP17 inhibitors.

COMPUTATIONAL DETAILS AND THEORETICAL METHODS

PREPARATION OF THE STRUCTURE AND CLASSIC MOLECULAR DYNAMICS SIMULATIONS

The X-ray crystal structure of human CYP17 in complex with abiraterone was obtained from Protein

Data Bank (PDB ID code: 3RUK)42. Protons were added to the protein structure using GROMOS96 43a1

force field43 assuming typical protonation states at pH 7.0. On the other hand, the imidazole side chain of the

histidine residue (His) contains two nitrogen atoms that can be protonated (ND1 and NE2) at pH 7.0, called

HID, HIE and HIP44 (Figure 3). The protonation state of histidine residue close to the heme group (His373),

was systematically varied. The value of the CA-CB-CG-ND1 dihedral angle (χ1) (Figure 3) was also

evaluated because an incorrect protonation of a His residue can result in large fluctuations and rotations of

the His side chain. Different studies about the analysis of the histidine protonation state have been

performed.44–46

In all these studies, the root-mean-square deviation (RMSD) is used to analyze the state of

protonation. To determine the proportion of neutral (HID and HIE) and positively charged (HIP) His on

CYP17 structure, we used the Henderson-Hasselbalch equation.47 Molecular dynamics (MD) simulations

were performed by GROMACS 4.6.5 program48 for each protonation state using GROMOS96 43a1 force

field with SPC water molecules.49,50

CYP17 has 26 aspatic acids, 24 glutamic acids, 34 lysines, 20 arginines

and 1 positively charged histidine (His160), totaling a +5 charge. The histidine close to the heme group

(His373) is hydrogenated in nitrogen NE2 (HIE). Counter ions Na+

and Cl- were added to neutralize charge

128

and maintain the system at a concentration of 150 mM. The system was immersed in a rhombic

dodecahedron box and the minimal distance between the protein and the wall of the cell was 1.2 nm.

Covalent bonds in the protein were constrained using the LINCS algorithm.51

Electrostatic forces were

calculated with the PME52 implementation of the Ewald summation method, and the Lennard–Jones

interactions were calculated within a cut-off radius of 1.2 nm. The system was equilibrated according to

the following protocol: the solvated CYP17-abiraterone complex geometry was minimized twice by the

steepest descent algorithm for 5000 steps with tolerances of 1000 and 500 KJ mol-1

nm-1

followed by two

gradient conjugated algorithm for 50000 steps with tolerances of 250 and 100 KJ mol-1

nm-1

. In order to

adapt the water to the complex due the greatest strain dissipated from the system and couple the system to

the heat bath using NVT ensemble at temperature of 310K, several short MD simulations were performed.

Initially, 50, 100, 150 and 200-ps molecular dynamics were performed where the non-hydrogen atoms of the

complex were kept fixed and the water can move freely. After that, 50, 50 and 37.5-ps MD simulations were

performed where the water and abiraterone move freely, following of 25, 50 and 75-ps MD simulations

where the heme group is released. Then, two unrestrained MD simulations of 50 and 75-ps were performed.

In order to control the pressure coupling of system, 200-ps MD simulation were performed at pressure of 1

bar using isotropic pressure bath with Berendsen pressure coupling algorithm.53 Finally, 50-ns production

MD simulation using NVT ensemble was performed for each protonation state in order to assess the stability

of the system. The RMSD of C-α of His 373 residue and CYP17 structure was measured with respect to the

respective X-ray structure. Snapshots were collected every 10-ns MD simulation and RMSD and dihedral

angle χ1 average values of the His 373 residue of each protonation state were analyzed. All data represent

five snapshots and were expressed as mean ± SD (standard deviation). Differences between groups were

compared by using ANOVA54

and Tukey Test.55

Differences were considered significant when p value was

less than 0.05. Statistical data were analyzed by GraphPad Prism 5.0 software.

Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP (C).

A B C

(ND1)(CG)(CA)(CB)

129 MOLECULAR DOCKING

The automated docking was conducted in order to find the best conformations and energy ranking of

PREG inside the active site of the human CYP17 (previously equilibrated from MD simulation as

aforementioned). The docking simulation was carried out using AutoDock Vina 1.1.2.56 The three-

dimensional structure of PREG was obtained from Protein Data Bank (http://www.rcsb.org). Polar

hydrogens were added and Gasteiger-Marsili partial charges57

were assigned to the PREG and human

CYP17 with the AutoDock Tools (ADT).58

All torsions to PREG were allowed to rotate during docking.

Grid box with size 38 Å was centered on the iron of the heme group. The AutoDock Vina parameter,

Exhaustiveness, that determined how the program searches exhaustively for the lowest energy conformation

was set to 16 (the default value is 8). After docking, resulting conformations of PREG were ranked based on

the interaction energy. The best conformation of PREG in complex with human CYP17 was chosen with the

lowest docked energy and then was submitted to 50-ns MD simulation.

The ABE was also subjected to the same docking methodology as described above for PREG on

CYP17. We found that the conformation of the docked ABE was very similar to that found in the CYP17-

ABE complex in the Protein Data Bank (PDB ID code: 3RUK).

TOPOLOGY

The topology contains all the necessary information to describe a molecule in a molecular

simulation. All this information includes atom types and charges (nonbonded parameters) as well as bonds,

angles and dihedrals (bonded parameters).48

In addition, exclusion and constraints parameters are also

generated by the force field. For this work, the topology was generated for the protein without ligands (ABE

and PREG) through the GROMOS96 43a1 force field. The ligands topology was generated by the

PRODRG59

2.5 automated server.

MOLECULAR DYNAMICS FLEXIBLE FITTING (MDFF) METHOD

In order to analyze conformational changes in the three-dimensional structure of CYP17 complexes,

before and after the molecular dynamics, the molecular dynamics flexible fitting (MDFF)60

method was

applied. The MDFF method consists of a molecular dynamics that includes information from an

experimental or simulated electron microscopy (EM) map for fitting an initial model of biomolecule in the

target density map. The analysis of MDFF were performed with Visual Molecular Dynamics (VMD)61

software.

130

THE TEMPERATURE FACTOR

The temperature factor (or B-factor) describes the displacement of the atomic positions in three-

dimensional protein structures from an average value.62

The B-factor will reflect the flexibility of atoms in

relation to their average positions. In other words, B-factor is an important indicator of the flexibility of the

protein structure. In this work, we used the B-factor to analyze the regions of the CYP17 complex structures

which have undergone flexibility during the course of molecular dynamics.

CALCULATION OF MOLECULAR MECHANICS INTERACTION ENERGIES

The nonbonded interaction energies involves running two MD simulations: one for the ligand in the

protein biding site (𝐸𝑏𝑖𝑛𝑑𝑝

) and the other for ligand in water (𝐸𝑏𝑖𝑛𝑑𝑤 ). The binding energies are composed of

two separate energy terms: The van der Waals (vdW) interaction energy (VLJ) given by the Lennard –Jones

(LJ) potential and the electrostatic interaction energy (VC) given by Coulomb potential.63–66

The computed

interaction energy is the sum of the LJ and Coulomb interaction energies which is given by:

𝐸𝑏𝑖𝑛𝑑𝑝

= (⟨𝑉𝐿𝐽⟩𝑝 + ⟨𝑉𝐶⟩𝑝) (1)

and

𝐸𝑏𝑖𝑛𝑑𝑤 = (⟨𝑉𝐿𝐽⟩𝑤 + ⟨𝑉𝐶⟩𝑤) (2)

where < > denotes MD averages of the nonbonded interactions between ligand and its receptor binding site

or just water. ⟨𝑉𝐿𝐽⟩𝑝 is the LJ term for ligand-protein interaction; ⟨𝑉𝐿𝐽⟩𝑤 is the LJ term for ligand-water

interaction; ⟨𝑉𝐶⟩𝑝 is the electrostatic term for ligand-protein interaction; and ⟨𝑉𝐶⟩𝑤 is the electrostatic term

for ligand-water interaction. For this study, we analyse the nonbonded interaction energies between CYP17

and its ligands ABE and PREG. Two simulations are required to obtain the 𝐸𝑏𝑖𝑛𝑑𝑤 and 𝐸𝑏𝑖𝑛𝑑

𝑝: one with ligand

bound to solvated protein (as described above) and one with the ligand free in solution. The latter was

performed with 1.0-ns using NVT ensemble.

QM/MM CALCULATIONS

131

Hybrid methods combine two computational techniques in one calculation and allow the study of large

chemical systems.67–69

The QM/MM method is a hybrid technique that combine a quantum mechanical

(QM) method with a molecular mechanics (MM) method.70–73

ONIOM74 (N-layer Integrated molecular

Orbital molecular Mechanics) is one of the QM/MM methods most employed and allows the combination of

different quantum mechanical methods as well as classical methods of molecular mechanics. In general, the

region of the system where the chemical process occurs is treated with a more accurate method as well as a

QM theory, and the remainder of the system at low level. Many ONIOM studies are focused on biological

systems, such as proteins,75–78

treating the active site by a high level method, using often density functional

theory (DFT), and the rest of the protein by MM. In a two layer ONIOM calculation, the total energy of the

system is obtained from three independent calculations:

𝐸𝑂𝑁𝐼𝑂𝑀 = 𝐸𝑚𝑜𝑑𝑒𝑙,𝑄𝑀 + 𝐸𝑟𝑒𝑎𝑙,𝑀𝑀 − 𝐸𝑚𝑜𝑑𝑒𝑙,𝑀𝑀 = 𝐸𝑚𝑜𝑑𝑒𝑙,ℎ𝑖𝑔ℎ + 𝐸𝑟𝑒𝑎𝑙,𝑙𝑜𝑤 − 𝐸𝑚𝑜𝑑𝑒𝑙,𝑙𝑜𝑤 (3)

The real system contains all the atoms (including both QM and MM regions) and is calculated only at

the MM level. The model system is the QM region treated at QM level. To obtain the ONIOM energy, the

model system also needs to be treated at MM and be subtracted from real system MM energy.79,80

In this

work, the ONIOM method was applied in geometric optimization of the complexes (ABE or PREG in

complex with human CYP17) from 50-ns production MD simulation. The high layer atoms were treated at

the B3LYP/6-311G(d,p) level81–83

and the low layer was treated by means of the AMBER9484 force field

implemented in the Gaussian 09 package.85

The Restrained electrostatic potential (RESP)86 atomic charges of the abiraterone and pregnenolone

were derived at the HF/6-31G* level by Gaussian03 and AmberTools.87 The heme group, ABE (or

PREG) and Cys442 were included in the high layer region, while the remaining residues were in the low

layer.

The interaction energy of ABE (or PREG) was calculated as below:

𝐸𝐶𝑌𝑃17+𝐴𝐵𝐸𝑐𝑜𝑚𝑝𝑙𝑒𝑥

- 𝐸𝐶𝑌𝑃17 - 𝐸𝐴𝐵𝐸 = 𝐸𝐴𝐵𝐸𝑖𝑛𝑡 (4)

𝐸𝐶𝑌𝑃17+𝑃𝑅𝐸𝐺𝑐𝑜𝑚𝑝𝑙𝑒𝑥

- 𝐸𝐶𝑌𝑃17 - 𝐸𝑃𝑅𝐸𝐺 = 𝐸𝑃𝑅𝐸𝐺𝑖𝑛𝑡 (5)

where 𝐸𝐶𝑌𝑃17+𝐴𝐵𝐸𝑐𝑜𝑚𝑝𝑙𝑒𝑥

and 𝐸𝐶𝑌𝑃17+𝑃𝑅𝐸𝐺𝑐𝑜𝑚𝑝𝑙𝑒𝑥

are the electronic energies of CYP17 in complex with ABE and PREG,

respectively. 𝐸𝐶𝑌𝑃17 is the electronic energy of CYP17; 𝐸𝐴𝐵𝐸 is the electronic energy of ABE; 𝐸𝑃𝑅𝐸𝐺 is the

electronic energy of PREG; 𝐸𝐴𝐵𝐸𝑖𝑛𝑡 and 𝐸𝑃𝑅𝐸𝐺

𝑖𝑛𝑡 are the interaction electronic energies of ABE and PREG,

respectively. The electronic density of the region enclosed by a cut-off radius of 3Å from ABE or PREG

(binding pockets) which was used for QTAIM calculations, was performed at B3LYP/6-31G(d,p) level by

means of Gaussian09 package, where all topological information were calculated using the AIM2000

132

software.88 The binding pockets in the CYP17 active site were generated through VMD software and are

shown in Figure 4.

The electronic energies of CYP17, ABE, PREG, CYP17-ABE, CYP17-PREG complexes, as well as

the electronic interaction energies of ABE and PREG are shown in Table 3. The electronic energy of

complexes were obtained after optimization of its geometries. To obtain the electronic energy of the CYP17,

the ligands (ABE and PREG) were previously removed from complexes and submitted to the single point

calculation at B3LYP/6-311G(d,p) level. Already to obtain ligands electronic energies, these were removed

from their complexes and submitted to the single point calculations with the same level of theory described

earlier.

QUINTET AND TRIPLET STATE OF HEME

The Density Functional Theory (DFT) method was used in this paper because it is already well

employed to porphyrins in general89–93

and B3LYP functional predicts the energy separation of the of FeII-

porphyrin of deoxyheme between quintet and triplet states, with the quintet state 12 kJ mol-1

lowest in

energy94

and this is agreement with experimental data. A study conducted in 1936 by Pauling and Coryell95

measured the magnetic moments of deoxyhemoglobin and oxyhemoglobin. The results showed that the

ferroheme of deoxy state contains four unpaired electrons. In this work, we checked the electronic energy

differences of ferroheme of triplet and quintet states using six functionals (B3LYP,96,97

BP86,98,99

BVWN,100

M062X,101

ωB97XD102

and PBE1PBE103–105

) at 6-311G(d,p) basis set through a simple Fe(NH3)2(NH2)2

complex. We found that the quintet state is about 5.85 kcal mol-1

more stable compared to triplet state using

B3LYP functional (Table S1 and Figure S1, Supporting Information). Therefore, we used the quintet spin

state (S=2) for heme group of CYP17 at the B3LYP/6-311G(d,p) level on the high layer.

QTAIM RATIONALE

The Quantum Theory of Atoms in Molecules (QTAIM) is a quantum model considered efficient in the

study of chemical bond106

and characterization of intra- and intermolecular interactions.107–109

According to the quantum-mechanical concepts of QTAIM, the observable properties of a chemical

system are contained in its electron density, ρ. The trajectories of the electron density are obtained from the

gradient vector of the electron density (∇ρ)110

and the set of these gradient trajectories form the atomic

basins (Ω). The bond critical points [BCP´s, (3,-1)] of the charge density are obtained where ∇ρ = 0111

and

its location is performed through the Laplacian of the charge density (∇2ρ) that is given by the trace of the

Hessian matrix of the electronic density.112

BCPs may give information about the character of different

133

chemical bonds or chemical interaction, for example, metal-ligand interactions,113–115

covalent bonds,112,116

different kinds of H-bonds,117–119

so-called hydrogen-hydrogen bonds108,120–123

as well as other weak

noncovalent interactions.124–126

In addition, several studies show that there is a direct relation between

electronic density of a BCP and the interaction energy between two atomic centers forming the bond, where

there is a linear relation for small ranges of electronic density values.127–130

Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17-PREG (B) complexes.

RESULTS

Molecular dynamics was used in order to achieve the stability of the CYP17-ABE complex and to detect

the protonation state of His 373. The plots of RMSD of the Cα of the CYP17-ABE complex of each

protonation state of His 373 are shown in Figures 5A, 5B and 5C. The RMSD of the Cα atoms of the

A

B

134

complex for each protonation state first increased as far as plateau at 5.0 ns, indicating that equilibrium had

been reached (Figures 5A, 5B and 5C) wherein the superposition of C-α of CYP17 complexes onto the X-

ray coordinates reaching convergence within 50-ns MD simulation. The plateau reached had an average

value of 0.39 nm. There was no difference observed between RMSD´s, since the only difference between

simulations is the protonation state of the His373.

The Figures 6A and 6B shows the plots of RMSD and χ average values, respectively, of His373 for each

protonation state. These average values were obtained every 10-ns snapshots from 50-ns production MD of

CYP17-ABE complex.

In order to obtain the CYP17-PREG interaction, and then the appropriate enzyme-substrate binding

pose, molecular docking using AutoDock Vina was performed. The binding affinity was characterized by

binding energy (ΔG) value (Table 1). The docking poses of PREG within the active site of the CYP17

enzyme are shown in Figures 7A, 7B, 7C and 7D. According to the Autodock Vina binding energies, the

best docking pose (pose 1) has a ΔG value of -12.2 kcal mol-1

(Figure 7A) and then was submitted to 50-ns

MD simulation (Figure 8) where the plateau was reached at 20 ns with superposition of C-α RMSD value

onto the X-ray coordinates of 0.38 nm. The structure obtained after a period of 50 ns simulation was named

CYP17-PREG complex.

0.000

0.100

0.200

0.300

0.400

0.500

0.00 10.00 20.00 30.00 40.00 50.00

nm

Time (ns)

0.000

0.100

0.200

0.300

0.400

0.500

0.00 10.00 20.00 30.00 40.00 50.00

nm

Time (ns)

0.000

0.100

0.200

0.300

0.400

0.500

0.00 10.00 20.00 30.00 40.00 50.00

nm

Time (ns)

(A) (B)

(C)

135

Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE complex of each

protonation state of His 373. HISD (A), HISE (B) and HISH (C).

Figure 6. Plots of RMSD (A) and χ1 (B) of the His373 of each protonation state. The data represent the

average of five 10-ns snapshots from 50-ns production MD of CYP17-ABE complex for each protonation

state expressed as mean standard deviation. It was considered a p value less than 0.05.

Table 1. Binding energies of CYP17 with PREG.

Poses Binding Energy / kcal mol-1

1 -12.2

2 -10.6

3 -10.2

4 -9.5

(A) (B)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

HID HIE HIP

nm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HID HIE HIP

χ(d

egre

e)

136

Figure 7. The binding poses of PREG in the active site of the enzyme CYP17.

Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG complex

The Figure 9A shows the ABE, heme group and residue CYS442 of the CYP17-ABE. Whereas the

Figure 9B shows the PREG, heme group and residue CYP442 of the CYP17-PREG complex. Both

structures were obtained after 50-ns MD simulations. We have found that after 50-ns MD simulations for

CYP17-ABE and CYP17-PREG complexes, there is no formation of a complex hexacoordinate with

octahedral geometry which is imposed by Fe+2

atomic d-orbitals. Conversely, hexacoordinated states are

A B

C D

0.000

0.100

0.200

0.300

0.400

0.500

0.00 10.00 20.00 30.00 40.00 50.00

nm

Time (ns)

137

formed after geometric optimization of complexes by ONIOM calculations (Figures 10A and 10B). This

leads us to infer that classical MD simulations does not adequately describe the d-orbitals geometry. In

addition to the limitations of MD simulations, we can see in Figure 9A and 9B that the porphyrin ring of

heme group is nonplanar where the opposite occurs with the quantum mechanical calculations (Figures 10A

and 10B).

Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG, heme group and

CYS442 of CYP17-PREG complex (B).

The Table 2 shows the values of Lennard-Jones potential for protein-ligand (CYP17-ABE/PREG)

and ligand-water interactions (⟨𝑉𝐿𝐽⟩𝑝 and ⟨𝑉𝐿𝐽⟩𝑤) and Coulomb potential for protein-ligand and ligand-water

interactions (⟨𝑉𝐶⟩𝑝 and ⟨𝑉𝐶⟩𝑤), as well as ⟨𝑉𝐿𝐽⟩𝑝 + ⟨𝑉𝐶⟩𝑝 and ⟨𝑉𝐿𝐽⟩𝑤 + ⟨𝑉𝐶⟩𝑤 are referred as the binding

energies between ligand (ABE or PREG) and CYP17 and between ligand and water respectively.

According the Eq. 1 and Eq. 2 and the data of Table 2, the interaction energy for ABE and the

CYP17 binding site and for ABE and water are -53.00 kcal mol-1

and -35.50 kcal mol-1

respectively.

Likewise, the interaction energies for PREG are -68.90 kcal mol-1

and -60.20 kcal mol-1

(Table 2). The

difference between interaction energies of ligand for enzyme binding site and for water (𝐸𝑏𝑖𝑛𝑑𝑝

− 𝐸𝑏𝑖𝑛𝑑𝑤 )

indicates its diffusion trend of the water enzyme binding site. The more negative energy difference value,

the greater the diffusion trend of the water enzyme binding site.

AB

(N22)

(Fe)

(SG)

(C17)

(SG)

138

Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal mol-1

, between CYP17

(p)/water (w) and ligands ABE and PREG.

Potentials Interaction ABE /

kcal mol-1

Interaction PREG /

kcal mol-1

⟨𝑉𝐿𝐽⟩𝑝 -52,49 -39,55

⟨𝑉𝐶⟩𝑝 -0,51 -29,35

⟨𝑉𝐿𝐽⟩𝑝 + ⟨𝑉𝐶⟩𝑝 -53.00 -68.9

⟨𝑉𝐿𝐽⟩𝑤 -33,53 -23,90

⟨𝑉𝐶⟩𝑤 -1,52 -36,30

⟨𝑉𝐿𝐽⟩𝑤 + ⟨𝑉𝐶⟩𝑤 -35.5 -60.20

The optimized structures of the QM (high layer) and MM (low layer) of the CYP17-ABE and

CYP17-PREG complexes and enzyme CYP17 are shown in Figures 10A, 10B and 10C respectively. The

electronic energy values of CYP17, ABE, PREG, CYP17-ABE and CYP17-PREG complexes are shown in

Table 3. As expected, the interaction energy results of molecular mechanics and QM/MM calculations for

ABE and PREG are not in agreement. This is primarily due to the description of the spin state of the iron

heme from ONIOM calculations, which does not occur in molecular mechanics calculations (more detail see

section discussion).

Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and CYP17-PREG

complexes.

Molecule Ee / kcal mol-1

Interaction energy /

kcal mol-1

ABE -667313.50 -61.75a

PREG -608804.64 -40.37b

CYP17 -2249240.31

CYP17-ABE complex -2916615.57

CYP17-PREG complex -2858085.32

aInteraction energy derived from Eq. (4) aInteraction energy derived from Eq. (5)

139

Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of the CYP17-ABE

(A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the ONIOM (B3LYP:AMBER) level.

The virial graphs of binding pockets of the abiraterone and pregnenolone in the CYP17 active site are

shown in Figures 11A and 11B, respectively. Table 4 shows the charge densities (ρ) of the bond critical

points (BCPs) of the bond paths between abiraterone (or pregnenolone) and CYP17 active site.

Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B) complexes.

Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or pregnenolone) and

CYP17 active site.

Complex BCP ρ (au.) Complex BCP ρ (au.)

CYP17-ABE

26 0.0003

CYP17-PREG

28 0.0018

31 0.0058 31 0.0006

34 0.0011 36 0.0012

A B C

(SG)

(N22)

(C17)

(SG)

(SG)

(Fe)

A B

140

42 0.0027 39 0.0007

47 0.0001 49 0.0017

51 0.0001 51 0.0004

53 0.0021 53 0.0005

60 0.0010 56 0.0008

68 0.0017 58 0.0006

72 0.0026 59 0.0005

75 0.0013 62 0.0002

76 0.0014 65 0.0002

81 0.0012 79 0.0073

82 0.0007 82 0.0026

92 0.0010 86 0.0017

97 0.0033 90 0.0010

110 0.0003 101 0.0005

127 0.0017 107 0.0010

149 0.0009 122 0.0009

150 0.0033 126 0.0012

154 0.0008 127 0.0002

163 0.0012 130 0.0059

168 0.0016 131 0.0023

182 0.0013 142 0.0012

195 0.0003 143 0.0018

196 0.0012 155 0.0021

200 0.0012 155 0.0021

204 0.0015 165 0.0010

206 0.0008 174 0.0004

208 0.0020 185 0.0016

212 0.0003 190 0.0010

219 0.0004 199 0.0014

225 0.0022 202 0.0006

231 0.0018 212 0.0015

233 0.0004 213 0.0004

238 0.0012 217 0.0009

246 0.0012 220 0.0001

258 0.0017 232 0.0010

141

286 0.0022 233 0.0005

300 0.0012 237 0.0010

311 0.0012 253 0.0006

323 0.0015 258 0.0006

458 0.0011 287 0.0007

294 0.0007

300 0.0008

362 0.0005

Total 0.0610 0.0560

DISCUSSION

From Henderson-Hasselbalch equation the proportion found between positively charged and neutral

His was 1:23. Since CYP17 has fifteen His in its structure, only one histidine was protonated on nitrogens

ND1 and NE2, the histidine 160. Molecular dynamics were employed to assess the stability of the CYP-

ABE complex and define the protonation state of His373 situated close to the heme group. According to the

plot shown in Figure 6A, there is no significant difference between RMSD average values of the three

protonation states. The RMSD indicated that there have been no large variations in the positions of C-α of

HID, HIE and HIP protonation states compared to X-ray structure. Likewise, the average values of dihedral

angle showed no significant differences between protonation states (Figure 6B). Therefore, subsequent

theoretical studies involving CYP17 enzyme, its histidine residue close to the heme group (His373), can

assume any protonation state.

There are no significant differences between structures before and after the production MD

simulation of CYP17-ABE and CYP17-PREG complexes as shown in structural fitting (Figures S2A and

S2B, Supporting Information). These data are in accordance with the B-factor62 (that reflects the fluctuation

of an atom about its average position) of complexes (Figures S3A and S3B, Supporting Information).

According to the above results, the differences between the interaction energies of the ABE and PREG

are -17.50 kcal mol-1

and -8.70 kcal mol-1

respectively. Although the PREG has a more negative interaction

energy with the CYP17 binding site (-68.90 kcal mol-1

) compared with ABE (-53.00 kcal mol-1

), PREG also

has a more negative energy interaction energy with water compared to the ABE (-60.20 kcal mol-1

and -35.5

kcal mol-1

respectively). In the other words, ABE has a higher diffusion trend water enzyme binding site

compared to PREG.

According to Eqs. (4) and (5), the electronic interaction energies of ABE and PREG have a value of -

61.75 kcal mol-1

and -40.37 kcal mol-1

respectively (Table 3). In other words, the interaction electronic

142

energy of ABE is 21.38 kcal mol-1

lowest in comparison to the PREG, indicating that the drug for metastatic

prostate cancer has highest affinity. This more negative value of interaction electronic energy of ABE is

reflected in a smaller distance with respect to iron heme (2.70 Å) compared to PREG (5.80 Å) (Figures 10A

and 10B, respectively).

According to the above description, the interaction energy results between ligands (ABE or PREG)

and CYP17 using molecular dynamics and ONIOM are not in agreement. PREG interaction energy by MD

simulations resulted in a more stable interaction with the CYP17 compared to ABE. In contrast, ONIOM

calculations showed that ABE has a higher affinity for the enzyme then PREG. Metalloproteins like

Cytochrome P450 are hard to be characterized computationally due to the presence of the metal cation

which exerts a strong Coulombic forces acting on structure backbone.131,132

Metals with unpaired electrons

of d-shells of atomic orbitals have preferred coordination geometries as well as tetrahedral, octahedral, or

square planar. This preference directly influences the position of the amino acids and therefore in the protein

tertiary structure. In addition, the electronic structure of a metal is closely related to its coordination

geometry. Different electronic configurations may be taken for a metal depending on the nature of its ligand.

Therefore, there is a close relationship between electronic structure of a metal and the tertiary structure of

the protein.132

For that reason, a classical description of a metal through molecular dynamics would be a

failure, because the coordination geometry could undergo changes during the simulation. This is viewed in

the distances between ABE or PREG and iron heme after molecular dynamics (Figures 9A and 9B,

respectively) and after ONIOM calculations (Figures 10A and 10B, respectively) of CYP17-ABE and

CYP17-PREG complexes. After ONIOM calculations, the bond length between aromatic nitrogen of ABE

and iron heme (2.70 Å) and the bond length between gamma-sufur (SG) of CYS442 and iron heme (2.40 Å)

were shortened compared to bond lengths after molecular dynamics (4.72 Å and 4.61 Å, respectively). The

same occurred to the bond lengths between C17 of PREG and iron heme and between SG of CYS442 and

iron heme. Furthermore, classical force fields are parameterized for a metal in specific coordination

geometry and ligated to a specific ligand. The spin state of iron porphyrins is determined by the coordination

geometry and the nature of ligands.133 In heme proteins, iron to be found in different coordination states:

four, five or six-coordinate and exhibits formal charge of +2 or +3,134 where unligated ferroheme can have

three spin states as a singlet, triplet or quintet (S = 0, 1 and 2, respectively). All these aspects about the metal

cations, particularly iron porphyrins, should be approached in terms of electronic structure through the

quantum mechanics calculations.135 The study of heme systems through the quantum-mechanical

computational methods, as well as Hartree-Fock and density functional theory (DFT) contributed

satisfactorily in the interpretation and understanding of the electronic structure of active site of

hemoproteins.136,137

Thereby, hybrid quantum mechanical and molecular mechanical (QM/MM) methods

can be very effective in the study of hemoproteins, as well ONIOM method. The difference in accuracy

143

between classical and hybrid methods was the main reason for the disagreement between interaction energy

results.

The molecular graphs of binding pockets of the ABE and PREG, in Figures 11A and 11B

respectively, shows all the interactions between the ligand (ABE or PREG) and the CYP17 active site. There

are 43 intramolecular interactions between ABE and active site according to the virial graph of ABE binding

pocket where the charge densities (ρ) values of the corresponding critical points are shown in Table 4. On

the other hand, PREG binding pocket has 46 intramolecular interactions (Figure 11B and Table 4). The sum

of all electronic density of interactions of ABE and PREG binding pockets were 0.0610 au. and 0.0560 au.

respectively. This result shows the overall stabilizing effect of the electronic density of interactions of

pockets on the ABE binding pocket.

CONCLUSIONS

In the present study, the molecular dynamics revealed that the histidine residue situated close to

heme group (His373) in the CYP17 enzyme can display any protonation state since it showed no significant

differences both in RMSD average values of Cα as the variation of dihedral angle (χ). With the protonation

state of His373 defined, we have obtained the two protein-ligand complexes used in this study: CYP17-ABE

and CYP17-PREG complexes.

Although the molecular mechanics interaction energy showed that ABE has a higher diffusion trend

(hydrophobicity) water CYP17 binding site compared to the PREG, classical descriptions cannot be

employed for enzymes containing heme group due to the limitations observed in Figures 9A and 9B. The

ONIOM hybrid (B3LYP:AMBER) methods showed that ABE has a higher affinity for the CYP17 because

it has a more negative interaction energy value compared to PREG.

To investigate the intramolecular interactions of ligands ABE and PREG, we used the quantum

theory of atoms and molecules (QTAIM). QTAIM analysis indicates that ABE has a higher electronic

density in their interactions with the active site of the CYP17 enzyme compared to PREG.

By using molecular mechanics (Docking and MD simulations) and quantum mechanics calculations

(ONIOM and QTAIM), we have investigated the inhibitory effect of Abiraterone through the interaction

energies and electronic density. This detailed study proves that the rationality evaluation inhibition

performance may grant valuable information about CYP17-ABE recognition providing new insights to

design the new homologous CYP17 inhibitors.

144

ACKNOWLEDGMENTS

Authors thank, Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),

Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional

de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

ASSOCIATED CONTENT

Supporting Information

Table with the electronic energy, bond length Fe-NH2 and bond length Fe-NH3 of triplet and quintet

states at B3LYP, BP86, BVWN, M062X, ωB97XD and PBE1PBE methods with the 6-311G(d,p) basis set

of the Fe(NH3)2(NH2)2. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet

and quintet states. Fitting structures of CYP17-ABE and CYP17-PREG complexes. B-factors for the

CYP17-ABE and CYP17-PREG complexes. This material is available free of charge via the Internet at

http://pubs.acs.org.

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156

TABLE OF CONTENTS

157

Supporting Information for “Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical

investigation of inhibitory effect of Abiraterone on CYP17”.

TABLE S1. Electronic energy, in a.u., bond length Fe-NH2 and bond length Fe-NH3, in angstroms, of triplet

and quintet states of the Fe(NH3)2(NH2)2 with different functionals using 6-311G(d,p) basis set.

METHOD Multiplicity Electronic Energy

(a.u.)

Bond

length

Fe-NH2

(Å)

Bond

length

Fe-

NH3(Å)

B3LYP Triplet -1488.82340086 1.92212 2.02763

Quintet -1488.83272142 1.88714 2.35259

BP86 Triplet -1489.00912021 1.87322 1.98187

Quintet -1488.99952264 1.87293 2.31036

BVWN Triplet -1492.23892216 1.90792 2.04101

Quintet -1492.23690248 1.89844 2.41968

M062X Triplet Error - -

Quintet Error - -

ωB97XD Triplet -1488.74019588 1.87885 2.01482

Quintet -1488.74695927 1.88793 2.31236

PBE1PBE Triplet -1488.28486454 1.90899 2.00701

Quintet Error - -

Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet and quintet

states.

B3LYP - Triplet B3LYP - Quintet

158

Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes before (yellow) and

after (red) molecular dynamics simulation. The picture was rendered using VMD.1

Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The picture was rendered

using VMD1, “blue” indicates high, “white” indicates intermediate and “red” low values. High temperature

factors imply exibility and low ones rigidity.

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A B

A B

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