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Part 2
Atomic Orbital Based Response Theory
Tr ∆S = Tr ∆exp( iκ SS ) exp( −iSκ ) SS
−1 T −1 T −1
−1
T T −1
= Tr ∆exp( iκ S) exp( −iκ SS )
= Tr ∆S ,
(2.13)
where we have used the relation
−1 −1
B exp( A) B = exp( B AB ) . (2.14)
The same relation may be used to show the idempotency relation
−
( i ) ( i ) ( i ) ( i )
T T T −1 T
( iSκ ) ∆ ( iκ S ) ( iκ S) S ∆ ( iκ S )
T −1 T
( iSκ ) ∆S ∆exp
( iκ S )
T
T
( iSκ ) ∆ ( iκ S ) ∆
−1 T T 1 T T
∆S ∆
= exp − Sκ ∆exp κ S S exp − Sκ ∆exp
κ S
= exp − exp exp − exp
= exp −
= exp − exp = .
(2.15)
We can therefore conclude that ∆ fulfils Eqs. (B-7) and exp( iκ ˆ) 0 is therefore a legitimate
normalized single-determinant wave function. It can be shown that all matrices fulfilling Eqs. (B-7)
can be obtained from an appropriate choice of κ, so the transformation of Eq. (2.8) is a complete
parameterization.
2.2.2 The Linear Response Function
We will now use the parameterization of Eq. (2.8) for an arbitrary single-determinant wave function
to describe a Hartree-Fock wave function in an external, time-dependent field. The parameters in κ
will become time-dependent and we will in the following develop equations for obtaining these
parameters. The time-dependent Hamiltonian can be written as
H = H0 + Vt
, (2.16)
where H 0 is the Hamiltonian for the unperturbed system, and V t is a first-order perturbation. The
perturbation will be turned on adiabatically, and V t can be expressed as
∞
−∞
Vt
= ∫ dωVω
exp( ( − iω + ε ) t)
, (2.17)
where ε is a positive infinitesimal that ensures V t → 0 as t → -∞. The perturbation is required to be
Hermitian, so we have the relation
†
ω
V
= V . (2.18)
−ω
To determine the linear response function, we begin by considering the time dependence of the
expectation value 0
A 0 of a one-electron operator A. We need only expand the wave function
0 of Eq. (2.8) to first order in the external perturbation to obtain the linear response:
(1) (2)
t t
ˆ κ = ˆ κ + ˆ κ +. (2.19)
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