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Part 2
Atomic Orbital Based Response Theory
The parameters of κ may similarly be arranged in a vector
such that
⎛ ⎞ >
() i
κ µν µ ν
⎜ ⎟
() i () i
= ⎜ κµµ
⎟
⎜ () i
κ µ ν ,
µν
∗ ⎟ >
α (2.29)
⎝
⎠
ˆ() i ()
κ = ∑ αm
i Λm
. (2.30)
m
Here the index m on Λ runs over all three classes of operators listed in Eq. (2.28).
The single excitation operators a †
µ aν have by Eq. (2.27)-(2.28) been divided into a set of atomic
orbital excitations, corresponding to µ > ν and a set of atomic orbital deexcitations, corresponding to
µ < ν. As the atomic orbital excitations and deexcitation have the same formal properties, this
division does not have any physical content. However, the division will prove important when the
paired structure of the response equations is investigated in Section 2.2.5. Note that it is not possible
to exclude the number operators a †
µ aµ in the atomic orbital representation, whereas they are
redundant in the standard molecular orbital formulation.
In the presence of the time-dependent perturbation, we introduce the time transformed operator
basis
⎛ Q
⎞
† ⎜ ⎟
Λ
= ⎜ D
⎟ , (2.31)
⎜ † ⎟
⎝Q
⎠
where
and similarly for
†
Q m and D m .
Q = exp( iˆ
κ) Q exp( −iˆ
κ)
(2.32)
m
The time evolution of 0 may now be determined using Ehrenfest’s theorem for the transformed
†
operators of Λ in Eq. (2.31):
d † ∂
0 0 0
† 0 0
†
Λ −
⎛
Λ
⎞
= − ⎡ Λ , 0 + ⎤ 0
dt
2.2.4 The First-order Equation
m
⎜
i H V
∂t
⎟
⎣ t
⎝ ⎠
⎦ . (2.33)
We now expand Eq. (2.33) in orders of the external perturbation, restricting ourselves to terms that
are linear in the amplitudes. Inserting Eq. (2.19) into Eq. (2.33) and collecting the terms linear in the
perturbation, we obtain the first-order time-dependent equation
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