Orbitales hidrogenoides y molecula de hidrógeno (formato pdf)

> restart:
> with(plots):
Warning, the name changecoords has been redefined
Orbitals hidrogenoids:
Orbital 1s (unitats atòmiques): f_1s( r, θ, φ ) = N e
> f:=N*exp(-r);
>
f := N e
( −r )
( −r )
>
f_int:=Int(r^2*f^2,r=0..infinity)*Int(sin(theta),theta=0..Pi)
*Int(1,phi=0..2*Pi);
∞
2π
⌠ 2 2 ( −r ) 2 ⌠ π
⌠ 1 dφ
f_int := 
r
N
(
e
)
d
r
sin
(
θ
)
d
θ




⌡0
⌡0
⌡0
>
f_int:=int(r^2*f^2,r=0..infinity)*int(sin(theta),theta=0..Pi)
*int(1,phi=0..2*Pi);
f_int := N 2 π
> solutions:=[solve(f_int=1,N)];
1
1 
,−
solutions := 

 π
π 

> f:=subs(N=solutions[1],f);
f :=
> plot(f,r=0..5);
e
( −r )
π
> plot(4*Pi*r^2*(f)^2,r=0..5);
> p1:=int(4*Pi*r^2*f^2,r=0..infinity);
p1 := 1
> p1:=int(4*Pi*r^2*f^2,r=0..1);
p1 := −5 ( e
( -1 )
2
) +1
> evalf(p1);
.3233235835
> solve(diff(4*Pi*r^2*f^2,r)=0);
0, 1
> fsolve(int(4*Pi*r^2*f^2,r=0..a)=0.9,a);
2.661160169
>
plot3d([2.66,theta,phi],phi=0..2*Pi,theta=0..Pi,coords=spheri
cal,scaling=constrained);
Orbital 2s (unitats atòmiques): f_2s( r, θ, φ ) = N ( 2 − r ) e
> f:=N*(2-r)*exp(-r);
>
f := N ( 2 − r ) e
( −r )
( −r )
>
f_int:=Int(r^2*f^2,r=0..infinity)*Int(sin(theta),theta=0..Pi)
*Int(1,phi=0..2*Pi);
∞
π
2π
2
⌠ 2 2
( −r )
2
⌠
⌠

f_int :=  r N ( 2 − r ) ( e ) dr  sin( θ ) dθ  1 dφ

⌡0
⌡0
⌡0
>
f_int:=int(r^2*f^2,r=0..infinity)*int(sin(theta),theta=0..Pi)
*int(1,phi=0..2*Pi);
f_int := N 2 π
> solutions:=[solve(f_int=1,N)];
1
1 
solutions := 
,−

 π
π 

> f:=subs(N=solutions[1],f);
(2 − r) e
f :=
π
> plot(f,r=0..5);
> plot(4*Pi*r^2*(f)^2,r=0..5);
( −r )
> p1:=int(4*Pi*r^2*f^2,r=0..infinity);
p1 := 1
> p1:=int(4*Pi*r^2*f^2,r=0..1);
p1 := −3 ( e
( -1 )
2
) +1
> evalf(p1);
.5939941501
> solve(diff(4*Pi*r^2*f^2,r)=0);
Warning, computation interrupted
> fsolve(int(4*Pi*r^2*f^2,r=0..a)=0.9,a);
>
plot3d([3.99,theta,phi],phi=0..2*Pi,theta=0..Pi,coords=spheri
cal,scaling=constrained);
Orbital 2pz (unitats atòmiques): f_2pz( r, θ, φ ) = N cos( θ ) r e
> f:=N*r*cos(theta)*exp(-r);
>
( −r )
f := N r cos( θ ) e
> fr:=Nr*r*exp(-r);
>
fr := Nr r e
> fang:=Nang*cos(theta);
>
( −r )
fang := Nang cos( θ )
> f_int:=int(r^2*fr^2,r=0..infinity);
3
f_int := Nr 2
4
> solutions:=[solve(f_int=1,Nr)];
2
2
solutions := 
3, −
3 
3
3


( −r )
> fr:=subs(Nr=solutions[1],fr);
2
( −r )
fr :=
3 re
3
> int(r^2*fr^2,r=0..infinity);
1
>
f_int:=int(sin(theta)*fang^2,theta=0..Pi)*int(1,phi=0..2*Pi);
4
f_int := Nang 2 π
3
> solutions:=[solve(f_int=1,Nang)];
1
1 3
,−
solutions := 
2
2 π
3
π




> fang:=subs(Nang=solutions[1],fang);
1 3 cos( θ )
fang :=
2
π
> int(sin(theta)*fang^2,theta=0..Pi)*int(1,phi=0..2*pi);
π
π
> f:=fr*fang;
>
f :=
> plot(fr,r=0..5);
re
( −r )
cos( θ )
π
> plot(r^2*(fr)^2,r=0..5);
>
p1:=int(r^2*fr^2,r=0..infinity)*int(sin(theta)*fang^2,theta=0
..Pi)*int(1,phi=0..2*Pi);
p1 := 1
>
p1:=int(r^2*fr^2,r=0..1)*int(sin(theta)*fang^2,theta=0..Pi)*i
nt(1,phi=0..2*Pi);
p1 := −7 ( e
( -1 )
2
) +1
> evalf(p1);
.0526530169
> solve(diff(r^2*fr^2,r)=0);
2, 0, 0, 0
>
p2:=int(r^2*fr^2,r=0..infinity)*int(sin(theta)*fang^2,theta=0
..Pi/4)*int(1,phi=0..2*Pi);
1
1
p2 := −
2 +
8
2
> p2:=evalf(2*p2);
p2 := .6464466095
> fsolve(int(r^2*fr^2,r=0..a)=0.9,a);
3.996794793
>
plot3d(3.99*cos(theta)^2,phi=0..2*Pi,theta=0..Pi,coords=spher
ical,scaling=constrained);
>
plot3d({3.99*cos(theta)^2,2.66},phi=0..2*Pi,theta=0..Pi,coord
s=spherical,scaling=constrained);
>
Diatomic molecules
Hydrogen molecule
> dist:=2.0;
>
dist := 2.0
> f1:=Pi^(-1/2)*exp(-abs(r-dist/2));
( − r − 1.000000000
e
f1 :=
π
> f2:=Pi^(-1/2)*exp(-abs(r+dist/2));
( − r + 1.000000000
e
f2 :=
π
> plot(0.5*(f1+f2)^2,r=-3..3);
)
)
> plot(0.5*(f1-f2)^2,r=-3..3);
> fcart1:=Pi^(-1/2)*exp(-((x^2+y^2+(z-dist/2)^2)^(1/2)));
fcart1 :=
e
( − x 2 + y 2 + ( z − 1.000000000 ) 2 )
π
> fcart2:=Pi^(-1/2)*exp(-((x^2+y^2+(z+dist/2)^2)^(1/2)));
fcart2 :=
e
( − x 2 + y 2 + ( z + 1.000000000 ) 2 )
π
> plot3d((subs(x=0,fcart1)+subs(x=0,fcart2))^2,y=-2..2,z=3..3);
>
> g:=(2^(-1/2)*(fcart1+fcart2));
( − x 2 + y 2 + ( z + 1.000000000 ) 2 ) 
 ( − x2 + y 2 + ( z − 1.000000000 )2 )
e

1
e

g :=
2 
+


2
π
π


> g2:=(2^(-1/2)*(fcart1-fcart2));
( − x 2 + y 2 + ( z + 1.000000000 ) 2 ) 
 ( − x 2 + y2 + ( z − 1.000000000 )2 )
e

1
e

g2 :=
2 
−


2
π
π


> with(plots):
> implicitplot3d(g^2=0.001,x=-4..4,y=-4..4,z=4..4,scaling=constrained,grid=[20,20,20]);
> implicitplot3d(g2^2=0.001,x=-4..4,y=-4..4,z=4..4,scaling=constrained,grid=[20,20,20]);
>