Download 1.2.1. LCAO

1.2.1. LCAO
Covalent bondings in solids are dominated by nearest neighbor interactions. As an
introduction, consider a diatomic molecule with a single bonding electron ( see
Fig.1.2 ). Assuming the nuclei to be stationary, the total Hamiltonian is
2
Ze2
Z e2
ZZ e2
(1.1)
H 
2 


2m
4 0rA 4 0rB 4 0 R
where all quantities are defined in Fig.1.2. The corresponding Schrodinger equation
is
(1.2)
H  mo  E mo
Eq (1.2) is separable, and hence exactly soluble, in terms of ellipsoidal coordinates.
However, this approach is applicable only to diatomic molecules. More useful for
our purposes is another approximate, but easily generalized, approach based on the
Linear Combinations of Atomic Orbitals (LCAO). For example, the simplest
approximation to the ground state of (1.1) is the ansatz
(1.4)
  cA A  cB B
where  i is the atomic ground state of the bonding electron of atom i. The
expectation value of  with respect to  is
 dr  H 
E 
 dr  
*
*


 H

c A A  cB B H c A A  cB B
c A A  cB B c A A  cB B
c A  A H  A  cB  B H  B  c*AcB  A H  B  cB* c A  B H  A
2

2
c A  cB  c*AcB  A  B  cB* c A  B  A
2
2
cA H AA  cB H BB  2 Re  c*AcB H AB 
2

(1.3)
2
cA  cB  2 Re  c*AcB S AB 
2
2
(1.6)
where
Sij   dr  i*  j   i  j
(1.5a)
H ij   dr  i*H  j   i H  j
(1.5b)
If the atomic orbitals  i are normalized, then Sii  1 . For i  j , we expect
0
Sij  
1
if

R
0
The best approximation of  to the ground state is obtained by minimizing E  by
varying c A and c B so that
E  E 

0
(1.7)
c A cB
However, the mathematics are much simpler if we minimize  H 
subject to the
constraint    constant . With the help of the Lagrange multiplier E  , this is
tantamount to the variational problem
    H  E        i  ci*  H  E       H  E  ci   i
0
i
which requires
ci*  H  E   0
  H  E ci  0
and
for all i
or simply
 H  E 
0
(1.7a)
  H  E  0
(1.7b)
and
Left-multiplying (1.7a) by  i

gives

 i  H  E    c j  j   0

j

or
 H
ij
 E Sij  c j  0
for all i
(1.8)
j
The reader should verify that the same result is obtained using (1.7b).
Equations (1.1) and (1.8) can be easily generalized to the case of an aggregate of an
arbitrary number of atoms each contributing an arbitrary number of bonding electrons.
All we need to do is to treat each H ij in (1.8) as an ni  n j matrix, where nk is the
number of atomic orbitals from atom k.
A necessary condition for eq(1.8) is the secular equation
det H  ES  0
(1.8a)
where H and S are matrices with submatrices H ij and S ij , respectively.