Hydrogen Atom
Potential Energy
qe qn
 Ze 2
V (r ) 

4 o r 4 o r
Kinetic Energy
m
Kˆ  Kˆ n  Kˆ e
M  m    m
 Kˆ  
2
2M
 2R  
Kˆ (r )  
2
2m
2
2m
r
C
 r2
+
M
R
 2r
20_01fig_PChem.jpg
Hydrogen Atom
2
2
Ze
Hˆ  Kˆ (r )  Vˆ (r )  
 2r 
2m
4 o r
1 d 2 d
1
d
d
1
d2
  2
r
 2
sin 
 2 2
r dr dr r sin  d
d r sin  d 2
2
r
2


1
d
d
1
d
2
2
ˆ
L  
sin 
 2
2
sin

d

d

sin

d



2
2
ˆ
d
d
L
Ze


2
ˆ
H 
r


2 
2
2mr  dr dr  2mr
4 o r
2
Radial
Angular
Coulombic
20_01fig_PChem.jpg
Hydrogen Atom
 ( r , ,  )
will be an eigenfunction of
Hˆ  (r , ,  )  E  (r , ,  )
 (r , , )  Rn (r )Yl .m ( , )
Hˆ , Lˆ2 & Lˆz
Separable
2

Lˆ2
Ze2 
d 2 d 
r


 Rn (r )Yl .m ( ,  )  En Rn (r )Yl .m ( ,  )

2 
2
4 o r 
 2mr  dr dr  2mr
2
 2 d 2 d 
l (l  1)
Yl .m ( , )
r
R
(
r
)

Y
(

,

)
Rn (r )
l .m
 n
2 
2
2mr  dr dr 
2mr
Ze 2
Yl .m ( ,  )
Rn (r )  En Rn (r )Yl .m ( ,  )
4 o r
2
  2 d 2 d 

l (l  1) Ze 2
r

 En  Rn (r )  0


2 
2
2mr
4 o r
 2mr  dr dr 

20_01fig_PChem.jpg
Hydrogen Atom
 1  d 2 d  l (l  1)
En 2m 
Zme 2
 2
 2  Rn (r )  0

 2  r
2
r
2  o r
 r  dr dr 

Recall
2
 2 d
1d 2 d  1 d
d2
2 d
r

 2
  2  2r  r
2 
2 
r  dr dr  r  dr
dr  r dr dr
Bohr Radius
4 o
ao 
me 2
2
 0.0529 nm
 2 d d 2  l (l  1) 2Z En 2m 
 2 

 2  Rn (r )  0

2
r
ao r
 r dr dr 

20_01fig_PChem.jpg
Hydrogen Atom
Assume
R1 (r )  0 as r  
Let’s try
R1 (r )  e r
 2 d d 2 l (l  1) 2Z E1 2m   r
 2

 2 e  0

2
r
ao r
 r dr dr

 2
l (l  1) 2Z E1 2m   r
2
 

 2 e  0

2
r
ao r
 r

 1  2 E1 2m 
1  2Z
l (l  1) 2  
 2      2   0
r  ao

r 
E1 2m
2Z
2
l (l  1)  0;
 2  0; &   2  0
ao
20_01fig_PChem.jpg
It is a ground
state as it has no
nodes
Hydrogen Atom
l (l  1)  0;
E1 2m
2Z
2
 2  0; &   2  0
ao
Z
 l  0;   ; &
ao
E1  
2
 me2 
Z
Z
 E1   2  


2a0 m
2m  4 o 2 
2
2
2

2
2
2m
2
Z 2 me 4
 4 o  2
2
2
The ground state as it has no nodes n=1, and since l=0
and m = 0, the wavefunction will have no angular
dependence
  (r , , )  R (r )Yl ,m ( ,  )  CR1 (r )  Ce
l
n
0
20_01fig_PChem.jpg

Zr
a0
Z2 2
E1   2
2a0 m
Hydrogen Atom
l
4 Z (n  l  1)!  2 Zr   na0 2l 1  2 Zr 
l
 Rn (r )   4 3
 e Lnl 

3 
n a0 [(n  l )]  na0 
na
 0
3
In general:
n  1
Laguerre

Polynomials
n  2


2 l 1  2 Zr 

Lnl 


n  3
 na0 




Zr
l  0 L11 ( x)  1
2Zr
x
na0
l  0 L12 ( x)  (2!)(2  x)
l 1
1S- 0 nodes
2S- 1 node
L33 ( x)  (3!)
l 0
L13 ( x)  (3!)  3  3 x  0.5 x 2 
l 1
L33 ( x)  (4!)(4  x)
l2
L55 ( x)  (5!)
20_01fig_PChem.jpg
3S-2 nodes
Energies of the Hydrogen Atom
In general:
En  
1
 4 o 

2
1
 4 o 
Z2
 2
2n
me4 Z 2
2 2 n2
e2 Z 2
2a0 n 2
kJ/mol
Hartrees
e2
EH 
 27.2 eV
4 o a0
 627.51 kcal / mol  2625.5 kJ / mol
Wave functions of the Hydrogen Atom
In general:
(r , , )  Rnl (r )Yl ,m ( , )
l
4 Z (n  l  1)!  2 Zr   na0 2l 1  2 Zr 
l
Rn (r )   4 3
 e Lnl 

3 
n a0 [(n  l )]  na0 
na
 0
3
Zr
1
m
Yl ,m ( , ) 
Clm Pl (cos( )) eim
2
 2r 
L11    1
 a0 
Z=1, n = 1, l = 0, and m = 0:
P0 (cos( ))  1
C00 
0
Y0,0 ( , ) 
1
1
2
R10 (r ) 
3
0
(r , , )  R (r )Y0,0 ( , ) 

2
3
0
a
e
r
a0
a
2 
0
1

2
e
r
a0
1
2 


1
a
3
0
e
r
a0
Wave functions of the Hydrogen Atom
Z=1, n = 2, l = 0, and m = 0:
Y0,0 ( , ) 
 r 

r 
L12      2!  2  
a0 
 a0 

1
2 
(r , , )  R20 (r )Y0,0 ( , )

r
2 a0

r 

1

3 
2
a
2 2 a0 
0 
e
R (r ) 
0
2

1
3
0
2a
e
r
2 a0

r 
1 

2
a
0 

Z=1, n = 2, l = 1
m = 0:
1 2 r
 2,1,0 (r , , ) 
e
3
8  a0 a0
Hydrogen Atom

r
2 a0
m = +1/-1:
cos
r
1 1 r  2 a0

 2,1,1 (r , , ) 
e
sin

e
8  a03 a0
r
1
1 1 r  2 a0
 2,1, x (r , , )    2,1,1 (r , ,  )   2,1,1 (r , ,  )  
e
sin  cos 
3
2
8  a0 a0
r
1
1 1 r  2 a0
 2,1, y (r , , )    2,1,1 (r , , )   2,1,1 (r , , )  
e
sin  sin 
3
2i
8  a0 a0
-
+
-
+
_
+
+
+
+
-
+
Radial Distribution Functions
P( R)   *n ,l ,m (r , , ) n,l ,m (r , , )r 2 sin  drd d
  Rnl* (r )Yl*,m ( , ) Rnl (r )Yl ,m ( , )r 2 sin  drd d
  Rnl* (r ) Rnl (r )r 2 dr  Yl*,m ( , )Yl ,m ( , )sin  d d
For radial distribution functions we integrate over all angles only
 2
P(r )  Rnl* (r ) Rnl (r )r 2   Yl*,m ( , )Yl ,m ( , )sin  d d
0 0
 Rnl* (r ) Rnl (r )r 2
Prob. density as a function of r.
20_06fig_PChem.jpg
Radial Distribution Functions
R (r ) 
0
1
R (r ) 
0
2
2
3
0
e
2
r

a0
4r
P1,0,0 (r )  R (r ) R (r )r  3 e
a0
0*
1
a
3
0
2a
2


1
0
1
e
r
2 a0

r 
1 

2
a
0 

r
a0
e
P2,0,0 (r )  R (r ) R (r )r  3
2a0
0*
2
20_09fig_PChem.jpg
0
2
2

2r
a0
 2 r3 r4 
r  

a
4
a
0
0 

Probability Distributions
P(r , , )   *n ,l ,m (r , , )  n ,l ,m ( r , ,  ) r 2 sin 
 Rnl*, (r ) Rnl , (r )r 2Yl *,m ( , )Yl ,m ( ,  )sin 
 Pn,l (r ) Pl ,m ( , )
r

r4
a0
2
P2,1 (r ) P1,0 ( ,  ) 
e
cos
 sin 
5
32 a0
Pn,l (r )
Z
Yl ,m ( , )
Y
X
20_08fig_PChem.jpg
Atomic Units
Set:
e2
 1, me  1, &
1
4 0
Z 2 me 4
Z2
En  
 2
2
2 2
2n
 4 o  2 n
4 o
 ao 
me 2
2
 1 a.u.
Hartrees
2
ˆ2
d
d
L
Ze


2
Hˆ  
r


2 
2
2me r  dr dr  2me r
4 o r
2
2 Z


2 r
1  d 2 d  Lˆ2 Z
 2 r
 2 
2r  dr dr  2r
r
1s 
Z3
 a03

e
Zr
a0

Z3

e  Zr

r
2 a0
a.u.

r
2

r 
Z 3e  r 
 2s 
1

1  
3 
2 2 a0  2a0  2 2  2 
Z3e
Much simpler forms.
20_12fig_PChem.jpg
Atoms
Potential Energy
qe qn
 Ze 2
Z
Ven (ri ) 


4 o ri 4 o ri
ri
1
Vee (r ) 


4 o rij 4 o rij
rij
V  Ven (ri )  Vee (rij )   
i
i
j 1
i
Z
1

ri
ij rij
Kinetic Energy
Kˆ  
2
2M
me
C
 2R   
i
 Kˆ (ri )   
i
e2
qi q j
i
2
2me
2
2me
i2
 
2
i
 r2i
M
=r12
 2ri
me
Helium Atom
2
 Z


1
i
Hˆ  Kˆ  Vˆ   
     
2
i
i 
 ri j i rij 
i2 Z
1
 
 
2 ri i j rij
i
1
ˆ
  Hi  
i
i ,i  j rij
1
ˆ
ˆ
 H1  H 2 
r12
Hydrogen like 1 e’
Hamiltonian
i2 Z
Hi  

2 ri
C
M
=r12
Cannot be separated!!!
i.e. r12 cannot be expressed as a
function of just r1 or just r2
What kind of approximations can be
made?
me
me
Ground State Energy of Helium Atom
Ionization Energy of He
EFree
E2
I1 = 24.587 ev
E1
Eo
E2
E1
Eo
I2 = 54.416 ev
Eo=- 24.587 - 54.416 ev
=- 79.003 ev
=- 2.9033 Hartrees
1
ˆ
ˆ
ˆ
H  H1  H 2 
r12
1
1
Hˆ 0  Hˆ 1  Hˆ 2
Ĥ 
 0 (r1 , r2 )   0 (r1 ) 0 (r2 )
r12
1  r1
Hˆ 110 (r1 )  E10 10 (r1 )
10 (1s ) 
e

Perturbation Theory
 r1 2
Z1
0
H1  

2
r1
2
2
Z
2
E10   1 2  
 2
2
2n1
 2 1
Ground State Energy of Helium Atom
Hˆ 0  Hˆ 1  Hˆ 2
 0 (r1 , r2 )   0 (r1 ) 0 (r2 )
Hˆ 0  0 (r1 , r2 )   Hˆ 1  Hˆ 2   0 (r1 ) 0 (r2 )
 Hˆ 1 0 (r1 ) 0 (r2 )  Hˆ 2  0 (r1 ) 0 (r2 )
  0 (r2 ) Hˆ 1 0 (r1 )   0 (r1 ) Hˆ 2  0 (r1 )
  0 (r2 ) E1 0 (r1 )   0 (r1 ) E2  0 (r1 )
  E01  E02   0 (r2 ) 0 (r1 )  E 0  0 (r1 , r2 )
Z12
E   2  2  E20
2n1
0
1
E  E  E  2  2  4 H
0
0
1
0
2
Not even close.
Off by 1.1 H, or
3000 kJ/mol
Therefore e’-e’ correlation, Vee, is very significant
Ground State Energy of Helium Atom
1
0
ˆ
ˆ
H H 
r12
Hˆ 0  Hˆ 1  Hˆ 2
1
1
Ĥ 
r12
Hˆ (r1 , r2 )  E(r1 , r2 )
 0 (r1 , r2 )   0 (r1 ) 0 (r2 )
Hˆ 110 (r1 )  E110 (r1 )
 (r1 , r2 )   0 (r1 , r2 )  1 (r1 , r2 )
E  E 0  E1  E10  E20   0  r1 , r2  Hˆ 1  0  r1 , r2 
1
 0  r1 , r2  Hˆ 1  0  r1 , r2      0*  r1 , r2   0  r1 , r2  dV1dV2
r12
S1 S2
Ground State Energy of Helium Atom
E    r1 , r2 
1
0
 (1s ) 
0
i
2    2  

   
0 0 0 0 0 0
Z3

2 2

 1s (1)1s (2)
1 0
1
0
0*
ˆ
H   r1 , r2       r1 , r2    r1 , r2  dV1dV2
r12
S1 S2
e
    0*  r1   0*  r2 
 Zri
S1
e  Zr1
2 2

e  Zr2
S2
1 0
  r1   0  r2  dV1dV2
r12
1 2 2  Zr1 2 2  Zr2 2 2
e
e r1 r2 sin 1 sin  2 dr1 dr2 d1d 2 d1d2
r12 

1
5Z 5
1s(1)1s(2) 

r12
8
4
5
E  E  E  4   2.75H
4
0
1
E1 1.25

 31.5%
0
E
4
Closer but still far off!!!
Perturbation is too large for PT
to be accurate, much higher
corrections would be required
Variational Method
The wavefunction can be optimized to the system to
make it more suitable
Consider a trail wavefunction
t
and
 exact
t
 exact
Is the true wavefunction, where:
Hˆ  exact  E0  exact

n ,exact

is a complete set
Then
 t Hˆ  t
t t
 E0
The exact energy
is a lower bound
Assume the trial function can be expressed in terms of
the exact functions
   ci  i
H  i  Ei  i
i
We need to show that
 t Hˆ  t  E0  t  t   t Hˆ  E0  t  0
t
Variational Method
 t Hˆ  E0  t 
c 
i
i
Hˆ  E0
i
Variational Energy
 t ( ) Hˆ  t ( )
 t ( )  t ( )
 Evar ( )
j
j
j
  ci*c j  i Hˆ  E0  j
i
j
  ci*c j   i Hˆ  j  E0  i  j 
i
j
  ci*c j  Ei ij  E0 ij 
Evar()
i
d
Evar ( )  0
d
j
  ci*ci  Ei  E0   0 Since
i
ci*ci  0 &  Ei  E0   0
min
E0
c 
d2
Evar ( )  0
2
d

 t Hˆ  t
t t
 E0
Variational Method For He Atom
1s(i )  1,0,0 (ri ) 
Z3

e  Zri
Let’s optimize the value of Z, since the presence of a second electrons
shields the nucleus, effectively lowering its charge.
 (ri ) 
1s(i )  1,0,0
3
Z eff

Evar 
e
3
 Z eff
ri
(r1 , r2 )  1s(1)1s(2)
 (r1 , r2 ) Hˆ  (r1 , r2 )
 (r1 , r2 )  (r1 , r2 )
 (r1 , r2 ) (r1 , r2 )  1s(1) 1s(1) 1s(2) 1s(2)  1
Evar
1
ˆ
ˆ
 1s(1)1s(2) H1  H 2  1s(1)1s(2)
r12
Variational Method For He Atom
Evar
1
ˆ
ˆ
 1s(1)1s(2) H1  H 2  1s(1)1s(2)
r12
 1s(1)1s(2) Hˆ 1 1s(1)1s(2)  1s(1)1s(2) Hˆ 2 1s(1)1s(2)
 1s(1)1s(2)
1
1s(1)1s(2)
r12
 1s(1) Hˆ 1 1s(1) 1s(2) 1s(2)  1s(2) Hˆ 2 1s(2) 1s(1) 1s(1)
 1s(1)1s(2)
1
1s(1)1s(2)
r12
1
ˆ
ˆ






 1s (1) H1 1s (1)  1s (2) H 2 1s (2)  1s (1)1s (2)
1s(1)1s(2)
r12
Variational Method For He Atom
1
Evar  1s(1) Hˆ 1 1s(1)  1s(2) Hˆ 2 1s(2)  1s(1)1s(2)
1s(1)1s(2)
r12
Z
Z Z eff Z eff
ˆ
ˆ
ˆ





1s (1) H1 1s (1)  1s (1) K1  1s (1)  1s (1) K1  

1s(1)
r1
r1
r1
r1
Z eff
1
ˆ



 1s (1) K1 
1s (1)   Z eff  Z  1s (1) 1s(1)
r1
r1
Z eff
ˆ
ˆ
H1  K1 
r1
1s(1) 
 1s(1) Hˆ 1 1s(1)  
Z eff2
21
Z eff3

e
3
 Z eff
r1
  Z eff
Hˆ 11s(1)  
Z eff2
2
1
 Z  1s(1) 1s(1)
r1
1s(1)
En  
Z eff2
2n 2
Variational Method For He Atom
1s(1) Hˆ 1 1s(1)  
  2


1
  Z eff  Z  1s(1) 1s(1)
21
r1
Z eff3
1

1s (1) 1s(1)    
r1
0 0 0
Z eff3
Z eff2


re
e
3
 Z eff
r1
3
2 Z eff
r1
1
r sin 1dr1d1d1
dr1   sin 1d1d1
3
2 Z eff
r1
1
 2Z 
2


dr1
1
au 
ue
du

(
au

1)
e
0
 a 2

0


0
au


  2Z    1 e 2 Zeff    2Z  0  1 e a 0 
eff
eff


eff
3
 4Z eff

e
3
 Z eff
r1 2
1
0 0
 4 Z eff3  r1e
 4Z eff3
Z eff3
 2
0

1
r1
1
1  Z eff
2  
4Z eff
Variational Method For He Atom
1s(1) Hˆ 1 1s(1)  

Similarly
1
  Z eff  Z  1s(1) 1s(1)
21
r1
Z eff2
  Z eff  Z  Z eff
2
1s(2) Hˆ 2 1s(2)  
Recall from PT
Evar
Z eff2
Z eff2
  Z eff  Z  Z eff
2
1
5
1s(1)2s(2)
1s(1)2 s(2)  Z eff
r12
8
1
ˆ
ˆ






 1s (1) H1 1s (1)  1s (2) H 2 1s (2)  1s (1)1s (2)
1s(1)1s(2)
r12
Z eff2
Z eff2
5

  Z eff  Z  Z eff  
  Z eff  Z  Z eff  Z eff
2
2
8
Variational Method For He Atom
Evar   Z
2
eff
5
2


 2  Z eff  ZZ eff   Z eff
8
5
 Z eff2  2 ZZ eff  Z eff
8
d
5
5
Evar  2Z eff  2Z   0  Z eff  Z 
dZ eff
8
16
 Z eff
5 27
 2 
16 16
2
Evar
 27 
 27  5  27 
    2(2)       2.8479 H
 16 
 16  8  16 
Much closer to -2.9033 H
(D E= 0.055 H =144.4 kJ/mol error)
Variational Method For He Atom
Z eff
27

 1.69
16
1s(i ) 
1

 27 
3
 27   16  ri
 e
 16 
3
27
27
r1  r2
1  27   16
(r1 , r2 )  1s(1)1s(2)    e e 16
  16 
Optimized wavefunction
Variational Method For He Atom
Z eff
27

16
1s(i ) 
1

 27 
3
 27   16  ri
 e
 16 
3
27
27
r1  r2
1  27   16
(r1 , r2 )  1s(1)1s(2)    e e 16
  16 
Optimized wavefunction
Other Trail Functions
Z Z  

 (r , r ) 
1
2

3
2
e  Z r1 e  Z r2  e  Z r1 e  Z r2 
Optimizes both nuclear charges simultaneously
Z   1.19 & Z   2.18
Evar  2.8757 H
(D E= 0.027 H =71.1 kJ/mol error)
Variational Method For He Atom
Other Trail Functions
1  Z ( r1  r2 )
 (r1 , r2 )  e
(1  br12 )
N
Z’, b are optimized. Accounts for dependence on r12.
Z   1.849 & b  0.364
Evar  2.8920 H
(D E= 0.011 H =29.7 kJ/mol error)
In M.O. calculations the wavefunction used are designed to give the most
accurate energies for the least computational effort required.
The more accurate the energy the more parameters that must be optimized
the more demanding the calculation.
Variational Method For He Atom
Experimental -79.003 ev
-2.9003 H
-2.862879 H
-2.862871 H
-2.84885 H
In M.O. calculations the wavefunction used are designed to give the most
accurate energies for the least computational effort required.
The more accurate the energy the more parameters that must be optimized
the more demanding the calculation.
The H2+ Molecule
One electron problem
Two nuclei
Define electron position,
i..e. internal coordinates,
w.r.t. nuclear positions.
 Zi e2 Zi
Vne (ri ) 

4 o ri
ri
Z A Z B e2 Z A Z B
Vnn ( RAB ) 

4 o RAB
RAB
Kˆ  Kˆ nuclear  Kˆ electronic  
2
 2RA 
2
 2RB  
2
 2rA 
2
2M A
2M B
2me
2me
1
1
1 2 1 2
2
2

 RA 
 RB    rA   rB
2M A
2M B
2
2
 2rB
The H2+ Molecule
Hˆ  Kˆ  Vˆ
Vˆ  Vˆnn ( RAB )  Vˆen (rA )  Vˆen (rB )

1
1 1
 
RAB rA rB
Since ZA=1 and ZB=1
Kˆ  Kˆ N ( RA )  Kˆ N ( RB )  Kˆ e (rA )  Kˆ e (rB )
1
1
1 2 1 2
2
2

 RA 
 RB    rA   rB
2M A
2M B
2
2
Hˆ 
1
1
1 2 1 2
1
1 1
2
2
 RA 
 RB    rA   rB 
 
2M A
2M B
2
2
RAB rA rB
The H2+ Molecule
Hˆ 
1
1
1
1 2 1 2
1 1
2
2
 RA 
 RB 
  rA   rB   
2M A
2M B
RAB 2
2
rA rB
Nuclear
Electronic
Hˆ  Hˆ N (R)  Hˆ e (r; R)
The electronic part is determined by the nuclear positions
Hˆ  ( R, r )  W  ( R, r )
W- Total Energy
 ( R, r )   (r; R)  (R)
The nuclear positions determine the electronic wavefunction
Assume electronic motion is much faster than nuclear
motion, implies that the nuclear positions are essentially
static
Separable??
The H2+ Molecule
Hˆ ( R, r )   Hˆ N (R)  Hˆ e (r; R)  (r; R)  (R)  ET (r; R)  ( R)
 Kˆ N  Vˆnn  Kˆ e  Vˆne   (r; R)  (R)  ET  (r; R)  (R)


Nuclear
 Kˆ N  Vˆnn   (R)  EN  (R)


Electronic
 Kˆ e  Vˆne   (r; R)  Ee  (r; R)


 Kˆ N  Vˆnn  Ee   (R)  ET  (R)


Potential energy surface.
Of primary interest
Linear Variational WFctns.
Suppose the trial wavefunction can be expressed in terms of an expansion of
an appropriate set of functions, not necessarily othonormal
 (r )   cii (r )
i
Evar (r ) 
c
2
i
i  j
1
i
 (r ) Hˆ  (r )
 (r )  (r )
c
Hˆ
i i

1 i j
 Sij  
 Sij i  j
c 
j
i
j
c c 
i i
j
i
j
 c c H

 c c S
i
i
i
j
ij
i
j
ij
j
j
j
 c c
i

i
j
j
 c c
i
j
i Hˆ  j
i
j
j
i  j
Linear Variational WFctns.
Evar  ci c j Sij   ci c j H ij
i
j
i
j
Find the optimum coefficients, that minimize Evar.
For each ci


 Ei  ci c j Sij    2ci c j H ij
j
 j

d
d
Ei  ci c j Sij 
dci
dci
j
 2c c H
i
j
d
Evar  0
dci
ij
j
1
1

dEi 
d
d
ci c j Sij  
ci c j H ij
  ci c j Sij   Ei 
dci  j
j dci
j dci

c H
j
j
ij
 c j Sij Ei  0
Linear Variational WFctns.
c
j
 H11  Ei
H  E S
i 21
 21


 H j1  Ei S j1


 H n1  Ei S n1
j
 H ij  Sij Ei   0
 H  EiS   i
H12  Ei S12
H 22  Ei
H1 j  Ei S1 j
H 2 j  Ei S 2 j
H j 2  Ei S j 2
H jj  Ei
H n 2  Ei S n 2
H nj  Ei S nj
ˆ   Hˆ 
H
i
j
S  i  j
0
H1n  Ei S1n   ci1 
H 2 n  Ei S 2 n  ci 2 
 
 
  0
H jn  Ei S jn  cij
 
 
 
H nn  Ei   cin 
1 i j

 Sij i  j
Need to diagonalize matrix, to find eigenvalues and eigen vectors:
Hˆ  i  Ei  i
where  i  ci*1 ci*2
cij*
cin* 
Linear Combination of Atomic Orbitals.
Lets use the 1s Hydrogen like orbitals to be a basis for a trial function and
apply variational theory to find the best approximate wavefunction
 i (r )  ciA1s A (r)  ciB1sB (r)
Where 1s (r ) & 1s (r ) are Hydrogen like wavefunction with n=1,
A
B
l=0, centred in nucleus a and b, resp.
1 2 1 2
1 1
ˆ
H    rA   rB   
2
2
rA rB
 i (r )  ciA1s A (r)  ciB1sB (r)
Hˆ  i  Ei  i
Linear Combination of Atomic Orbitals.
Hˆ ciA 1s A  ciB 1sB   Ei ciA 1s A  ciB 1sB 
 H  EiS   i
0
1 2 1 2
1 1
ˆ
H  1sk H 1sl  1sk   rA   rB    1sl  Hˆ kl
2
2
rA rB
 i  c
*
iA
c 
*
iB
 H AA  Ei
H  E S
i BA
 BA
1
S  1sk 1sl  
 Slk
H AB  Ei S AB   ciA 
0



H BB  Ei  ciB 
lk
lk
Linear Combination of Atomic Orbitals.
1 2 1 2 1 1
ˆ
H kk  1sk 1H 1sk  1sk   rA   rB   1sk
2
2
rA rB
1 1   1 
      1   e 2 R
2 R  R
1
ˆ
H kl  1sk H 1sl   Skl   R  1 e  R  1sl Hˆ 1sk
2
Skl  1sk 1sl  1sl 1sk
2


R
R
 Slk  e   R  1
 3

 H AA  Ei
H  E S
i AB
 AB
H AB  Ei S AB   ciA 
0



H AA  Ei  ciB 
 H AA  H BB
 H AB  H BA
 S AB  S BA
Linear Combination of Atomic Orbitals.
H AA  Ei
H AB  Ei S AB
H AB  Ei S AB
0
H AA  Ei
 H AA  Ei    H AB  Ei S AB   0
2
2
2
2
Ei2 (1  S AB
)  2 Ei ( H AB S AB  H AA )  H AA
 H AB
0
E
2
2
2
2( H AA  H AB S AB )  4( H AA  H AB S AB ) 2  4(1  S AB
)  H AA
 H AB

2
2(1  S AB
)
2
2
2
2( H AA  H AB S AB )  2 H AB
 2 H AB H AA S AB  H AA
S AB

2(1  S AB )(1  S AB )
( H AA  H AB S AB )  ( H AB  H AA S AB )  H AA  H AB 


(1  S AB )(1  S AB )
(1 S AB )
2
Prediction of the Bond
E
H AA  H AB 


(1 S AB )
E g
1 ( R  1)e 2 R  R ( R  1)e  R  1
 
2

R2  R 
R  1  R 
e  1

3 


Bonding and Antibonding Orbitals of H2+
Density Difference Between MO’s and 1s O’s
23_09fig_PChem.jpg
Electron Densities of Sigma and Pi M.O’.s
1
1
 (r )  1s A (r )  1sB (r )
2
2
1
1
 g (r )  1s A (r )  1sB (r)
2
2
*
u
g=gerade
(same)
u=ungerade
(opposite)
Bonding
Antibonding
1
1
 (r )  2 px , A (r )  2 px , B (r )
2
2
*
g
1
1
 g (r )  2 px , A (r )  2 px , B (r )
2
2
Antibonding
23_11fig_PChem.jpg
Bonding
Other Types of M.O.’s
-13.6 e.v.
Electron population on F is larger, ie. bond in polarized to F,
ie. shows the F is more electronegative.
-18.6 e.v.
-19.6 e.v.
  (0.345)1sH  (0.840)2 pzF
MO’s for the Diatomics
23_13fig_PChem.jpg
Energy Level Diagram For the Diatomics
Electron Configuration for H2 &He2
23_02tbl_PChem.jpg
Electron Configuration of N2
23_17fig_PChem.jpg
Electron Configuration of F2
23_16fig_PChem.jpg
Electron Configurations of the Diatomics
23_18fig_PChem.jpg
Bonding in HF
23_20fig_PChem.jpg