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Appendix B
The Density Matrix in the Atomic Orbital Basis
In this appendix we will briefly review the density matrix in the atomic orbital basis and derive the
most important relations. For convenience consider a single-determinant wave function with n
molecular orbitals occupied. The expectation value of a one-electron operator may then be written
as a sum over occupied spin-orbitals
0 hˆ
0
n
= ∑ h . (B-1)
i=
1
ii
Explicitly introducing the MO-AO transformation matrix C allow us to write the expectation value
as
0 hˆ
0
=
n
i=
1
ii
N n
⎛
∗
∑ hµν ∑Cµ iCν
i
µν , = 1 i=
1
N
h
⎞
= ⎜ ⎟
⎝ ⎠
=
∑
∑
h
D
µν µν
µν , = 1
,
(B-2)
where N is the number of AO basis functions and we have introduced D as
D
n
µν C ∗
µ iCνi
i=
1
= ∑ . (B-3)
It is of interest to study the relation between D and the expectation values ∆ of Eq. (2.10). To
accomplish this we consider the second quantization expression for 0 h ˆ 0 in the nonorthogonal
atomic orbital basis. According to ref. 46 one obtains
N
0 hˆ
0 =
0 0
µν , = 1
N
µν , = 1
N
h
1 1 †
aµ a
µν ν
= ∆
=
− −
∑ ( S hS )
−1 −1
∑ ( S hS )
∑
µν
−1 −1
( S ∆S )
µν
µν µν
µν , = 1
.
(B-4)
By comparing Eqs. (B-4) and (B-2) we have the identification
−1 −1
D = S ∆S . (B-5)
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