Quımica cuantica 1: Orbitales hidrogenoides Prof. Jesus Hernandez Trujillo -Fac. Quımica, UNAM

Orbital 1s

(%i1) psi1s(r,theta,phi):=1/sqrt(%pi)*exp(-r);

(%o1) psi1s (r, θ, φ) :=exp (−r)√

π

(%i2) psi1sxyz(x,y,z):=1/sqrt(%pi)*exp(-sqrt(x^2+y^2+z^2));

(%o2) psi1sxyz (x, y, z) :=exp

(−√z2 + y2 + x2

)√π

Perfil de 1s:

(%i3) wxplot2d(psi1s(r,0,0),[r,0,5])$

(%t3)

1s en el eje x:

(%i4) wxplot2d(psi1sxyz(x,0,0),[x,-5,5]);

1

(%t4)

(%o4)

1s en el plano xy:

(%i5) wxplot3d(psi1sxyz(x,y,0),[x,-5,5],[y,-5,5])$

(%t5)

2

1s en los planos z=0,1 y 2:

(%i6) wxplot3d([psi1sxyz(x,y,0),

1+psi1sxyz(x,y,1),2+psi1sxyz(x,y,2),

[x,-5,5],[y,-5,5]])$

(%t6)

Parte esferica de 1s

(%i7) wxplot3d (psi1s(0.1,theta,phi), [theta, 0, %pi],[phi, 0, 2*%pi],

[transform_xy, spherical_to_xyz], [grid,30,60])$

3

(%t7)

Orbital 2s

(%i8) psi2s(r,theta,phi):=1/sqrt(32*%pi)*(2-r)*exp(-r/2);

(%o8) psi2s (r, θ, φ) :=(2− r) · exp

(−r2

)√

32 · π

Perfil 2s

(%i9) wxplot2d(psi2s(r,0,0),[r,0,15])$

4

(%t9)

2s tiene 1 nodo en r=2:

(%i10) solve(psi2s(r,theta,phi)=0,r);

(%o10) [r = 2]

Grafica de 2s en el plano xy

(%i11) wxplot3d(1/sqrt(32*%pi)*(2-sqrt(x^2+y^2))*exp(-sqrt(x^2+y^2)/2),[x,-9,9],[y,-9,9],

[z,-0.05,0.25]);

5

(%t11)

(%o11)

Orbital 2pz

(%i12) psi2pz(r,theta,phi):=1/sqrt(32*%pi)*r*exp(-r/2)*cos(theta);

(%o12) psi2pz (r, θ, φ) :=r · exp

(−r2

)· cos (θ)

√32 · π

(%i13) wxplot3d(abs(psi2pz(1,theta,phi)),[theta, 0, %pi],[phi, 0, 2*%pi],

[transform_xy, spherical_to_xyz], [grid,30,60])$

6

(%t13)

Orbital 2px

(%i14) psi2px(r,theta,phi):=1/sqrt(32*%pi)*r*exp(-r/2)*sin(theta)*cos(phi);

(%o14) psi2px (r, θ, φ) :=r · exp

(−r2

)· sin (θ) · cos (φ)√

32 · π

(%i15) wxplot3d (sqrt(psi2px(10,theta,phi)^2), [theta, 0, %pi],

[phi, 0, 2*%pi],

[transform_xy, spherical_to_xyz], [grid,30,60])$

7

(%t15)

Orbital 3dz2

(%i16) psi3dz2(r,theta,phi):=r^2*exp(-r/3)*(3*cos(theta)^2-1);

(%o16) psi3dz2 (r, θ, φ) := r2 · exp

(−r3

)·(

3 · cos (θ)2 − 1

)(%i17) wxplot3d (sqrt(psi3dz2(10,theta,phi)^2), [theta, 0, %pi],

[phi, 0, 2*%pi],

[transform_xy, spherical_to_xyz], [grid,30,60])$

8

(%t17)

Funcion de distribucion radial

R10:

(%i18) r1s(r):=(1/sqrt(%pi))*exp(-r);

(%o18) r1s (r) :=exp (−r)√

π

(%i19) wxplot2d(r^2*r1s(r)^2,[r,0,5]);

9

(%t19)

(%o19)

El maximo esta en r=1:

(%i20) solve(diff(r^2*r1s(r)^2,r)=0,r);

(%o20) [r = 0, r = 1]

R21 y R32:

(%i21) r2p(r):=1/(2*sqrt(6))*r*exp(-r/2);

(%o21) r2p (r) :=r · exp

(−r2

)2 ·√

6

(%i22) r3d(r):=4/(81*sqrt(30))*r^2*exp(-r/3);

(%o22) r3d (r) :=4 · r2 · exp

(−r3

)81 ·√

30

(%i23) wxplot2d([r^2*r2p(r)^2,r^2*r3d(r)^2],[r,0,20]);

10

(%t23)

(%o23)

R30:

(%i24) r3s(r):=2/(9*sqrt(3))*(3-2*r+2*r^2/9)*exp(-r/3);

(%o24) r3s (r) :=2 ·(

3− 2 · r + 2·r29

)· exp

(−r3

)9 ·√

3

(%i25) wxplot2d(r^2*r3s(r)^2,[r,0,18]);

11

(%t25)

(%o25)

Las tres juntas:

(%i26) wxplot2d([r^2*r2p(r)^2,r^2*r3s(r)^2,r^2*r3d(r)^2],[r,0,20]);

(%t26)

12

(%o26)

13