Orbitales hidrogenoides Prof. Jesús Hernández Trujillo

Quı́mica cuántica 1: Orbitales hidrogenoides Prof. Jesús Hernández Trujillo Fac. Quı́mica, UNAM Orbital 1s
-->
psi1s(r,theta,phi):=1/sqrt(%pi)*exp(-r);
-->
psi1sxyz(x,y,z):=1/sqrt(%pi)*exp(-sqrt(x^2+y^2+z^2));
1s es función propia de H:
-->
-(1/2)*((1/r^2)*diff(r^2*diff(psi1s(r,th,ph),r),r))-psi1s(r,th,ph)/r;
factor(%)/psi1s(r,th,ph);
Perfil de 1s:
-->
wxplot2d(psi1s(r,0,0),[r,0,5])$
1s en el eje x:
-->
wxplot2d(psi1sxyz(x,0,0),[x,-5,5]);
1s en el plano xy:
-->
plot3d(psi1sxyz(x,y,0),[x,-5,5],[y,-5,5])$
1s en los planos z=0,1 y 2:
-->
plot3d([psi1sxyz(x,y,0),
1+psi1sxyz(x,y,1),2+psi1sxyz(x,y,2),
[x,-5,5],[y,-5,5]])$
Parte esférica de 1s
-->
plot3d (psi1s(0.1,theta,phi), [theta, 0, %pi],[phi, 0, 2*%pi],
[transform_xy, spherical_to_xyz], [grid,30,60])$
Orbital 2s
-->
psi2s(r,theta,phi):=1/sqrt(32*%pi)*(2-r)*exp(-r/2);
Perfil 2s
-->
plot2d(psi2s(r,0,0),[r,0,15])$
1
2s tiene 1 nodo en r=2:
-->
solve(psi2s(r,theta,phi)=0,r);
Gráfica de 2s en el plano xy
-->
plot3d(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]);
Orbital 2pz
-->
psi2pz(r,theta,phi):=1/sqrt(32*%pi)*r*exp(-r/2)*cos(theta);
-->
plot3d(abs(psi2pz(1,theta,phi)),[theta, 0, %pi],[phi, 0, 2*%pi],
[transform_xy, spherical_to_xyz], [grid,30,60])$
Orbital 2px
-->
psi2px(r,theta,phi):=1/sqrt(32*%pi)*r*exp(-r/2)*sin(theta)*cos(phi);
-->
plot3d (sqrt(psi2px(10,theta,phi)^2), [theta, 0, %pi],
[phi, 0, 2*%pi],
[transform_xy, spherical_to_xyz], [grid,30,60])$
Orbital 3dz2
-->
psi3dz2(r,theta,phi):=r^2*exp(-r/3)*(3*cos(theta)^2-1);
-->
plot3d (sqrt(psi3dz2(10,theta,phi)^2), [theta, 0, %pi],
[phi, 0, 2*%pi],
[transform_xy, spherical_to_xyz], [grid,30,60])$
Función de distribución radial R10:
-->
r1s(r):=(1/sqrt(%pi))*exp(-r);
-->
plot2d(r^2*r1s(r)^2,[r,0,5]);
El máximo está en r=1:
-->
solve(diff(r^2*r1s(r)^2,r)=0,r);
R21 y R32:
2
-->
r2p(r):=1/(2*sqrt(6))*r*exp(-r/2);
-->
r3d(r):=4/(81*sqrt(30))*r^2*exp(-r/3);
-->
plot2d([r^2*r2p(r)^2,r^2*r3d(r)^2],[r,0,20]);
R30:
-->
r3s(r):=2/(9*sqrt(3))*(3-2*r+2*r^2/9)*exp(-r/3);
-->
plot2d(r^2*r3s(r)^2,[r,0,18]);
Las tres juntas:
-->
plot2d([r^2*r2p(r)^2,r^2*r3s(r)^2,r^2*r3d(r)^2],[r,0,20]);
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