14.3 Properties of the ISR-ADC Schemes
217
The order structure of ˜
D (in case of a one-particle operator) is as shown in Fig. 12.2
(with the respective N -electron excitation classes replacing the original ones).
Figure 12.4 shows the (non-separable) block structure of ˜
D generated in the partitioning scheme of the separate fragment model. The conclusions with regard to the
size-consistency of excited-state properties (14.48) and transition moments (14.49)
can be inferred from the corresponding discussion in Sect. 12.2.
The Dipole Sum Rule and the Equivalence of Length and Velocity Forms of the
Transition Moments
In the following, we dispense from supposing atomic units and display the equations
in their more familiar explicit form.
The first spectral moment of the dipole operator, more specifically its z-component,
ˆ
Z =
N
i=1
ˆ
z
(i)
=
r,s
z rs c
†
r c s
(14.54)
is given by
S 1 (Z ) =
n
(E n − E 0 )
n| ˆ
Z | 0
2
(14.55)
As is easily seen, S 1 (Z ) can be expressed as the ground-state expectation value
S 1 (Z ) =
1
2
0 |[ ˆ
Z , [ ˆ
H , ˆ
Z ]]| 0 .
(14.56)
of a double commutator, which can directly be evaluated to give a constant,
[ ˆ
Z , [ ˆ
H , ˆ
Z ]] = [ ˆ
Z , −i ˆ
P z ] /m e = N
2
/m e
(14.57)
Here, ˆ
P z is the z-component of the momentum operator, coming into play as the
result of the commutator
[ ˆ
H , ˆ
Z ] = −i ˆ
P z /m e
(14.58)
Accordingly, the final result for S 1 (Z ) reads
S 1 (Z ) =
1
2
N
2
/m e
(14.59)
which is the well-known Thomas–Reiche–Kuhn (TRK) or dipole sum rule.
In the ADC formulation, the moment S 1 (Z ) can be written in the following
compact form:
S 1 (Z ) = F(Z )
† M F(Z )
(14.60)
Here, F(Z ) denotes the vector of effective transition moments for the dipole operator
ˆ
Z ,
217
The order structure of ˜
D (in case of a one-particle operator) is as shown in Fig. 12.2
(with the respective N -electron excitation classes replacing the original ones).
Figure 12.4 shows the (non-separable) block structure of ˜
D generated in the partitioning scheme of the separate fragment model. The conclusions with regard to the
size-consistency of excited-state properties (14.48) and transition moments (14.49)
can be inferred from the corresponding discussion in Sect. 12.2.
The Dipole Sum Rule and the Equivalence of Length and Velocity Forms of the
Transition Moments
In the following, we dispense from supposing atomic units and display the equations
in their more familiar explicit form.
The first spectral moment of the dipole operator, more specifically its z-component,
ˆ
Z =
N
i=1
ˆ
z
(i)
=
r,s
z rs c
†
r c s
(14.54)
is given by
S 1 (Z ) =
n
(E n − E 0 )
n| ˆ
Z | 0
2
(14.55)
As is easily seen, S 1 (Z ) can be expressed as the ground-state expectation value
S 1 (Z ) =
1
2
0 |[ ˆ
Z , [ ˆ
H , ˆ
Z ]]| 0 .
(14.56)
of a double commutator, which can directly be evaluated to give a constant,
[ ˆ
Z , [ ˆ
H , ˆ
Z ]] = [ ˆ
Z , −i ˆ
P z ] /m e = N
2
/m e
(14.57)
Here, ˆ
P z is the z-component of the momentum operator, coming into play as the
result of the commutator
[ ˆ
H , ˆ
Z ] = −i ˆ
P z /m e
(14.58)
Accordingly, the final result for S 1 (Z ) reads
S 1 (Z ) =
1
2
N
2
/m e
(14.59)
which is the well-known Thomas–Reiche–Kuhn (TRK) or dipole sum rule.
In the ADC formulation, the moment S 1 (Z ) can be written in the following
compact form:
S 1 (Z ) = F(Z )
† M F(Z )
(14.60)
Here, F(Z ) denotes the vector of effective transition moments for the dipole operator
ˆ
Z ,