16.3 EOM Treatment of N -Electron Excitations
253
admixture of 3 p-3h configurations in the exact final state. This makes explicit what
was inferred in Sect. 15.1 from the PT analysis of the excitation energy, namely that
the RPA accounts for the effect of the 3 p-3h admixtures.
Furthermore, the first-order state expansion reveals the main shortcoming of the
RPA, that is, the neglect of 2 p-2h admixtures. As a result, the RPA does not account
for the important relaxation and polarization effects (see Sect. 15.1). For a more
appropriate treatment of excitations in the N -electron system, one has to include
the 2 p-2h excitation operators together with their 2h-2 p de-excitation counterparts,
as is the case in the second-order polarization propagator approximation (SOPPA)
scheme [12, 13].
The considerations in Sect. 16.1 concerning the order relations and the separability
of the EOM secular matrices apply to the N -electron case as well. In the form of
Eqs. (16.46), (16.47), M is neither canonical nor separable, and it requires again
an orthonormalization procedure of ECO-Gram-Schmidt type to afford the latter
properties [8].
The N -electron EOM treatment features a non-hermitian secular problem combining physical excited configurations and an equally large manifold of unphysical
de-excitation configurations. As in the (N ±1)-electron case above, one may weigh
advantages and disadvantages of the EOM concept as compared to the plain ISR
schemes of the type presented in Chap. 14. Besides the size of the EOM secular
matrix, being here exactly twice the size of comparable plain ISR schemes, the fact
that the EOM treatment involves a non-hermitian RPA-type secular problem constitutes another disadvantage.
Exercises
16.1 Verify the expression (16.26) for the first-order contribution to the EOM secular
matrix element M i,abjkl ; evaluate the analogous contribution, S
(1)
i,abjkl , for the
EOM overlap matrix.
16.2 Field operators in coordinate space representation:
(a) Revisit the abstract definition (2.3) of the creation operator c
†
q and “translate” it to the coordinate space representation. Let (ξ 1 , . . . , ξ N ) denote an
antisymmetrized and normalized wave function of N electrons (N even). Write
the (N +1)-electron wave functions c
†
q (ξ 1 , . . . , ξ N ) in terms of a properly
antisymmetrized linear combination of products of the type φ q .
(b) Write c q (ξ 1 , . . . , ξ N ) as an expansion of (N −1)-electron basis states n
in analogy to Eq. (2.7).
(c) Let φ q be a complex-valued orbital. The coordinate space representation
according to (a) and (b) allows one to define “complex” field operators, c
∗†
q
and c
∗
q . Verify that c
∗†
q = (c
†
q )
∗ and c
∗
q = (c q )
∗ for a real-valued wave
function .
16.3 EOM state representation in case of complex-valued orbitals:
Suppose a complex one-particle representation and modify the definition
(16.27) of Fock-space states according to | I = ( ˆ
O
∗†
I + ˆ
O I )| 0 , where the
253
admixture of 3 p-3h configurations in the exact final state. This makes explicit what
was inferred in Sect. 15.1 from the PT analysis of the excitation energy, namely that
the RPA accounts for the effect of the 3 p-3h admixtures.
Furthermore, the first-order state expansion reveals the main shortcoming of the
RPA, that is, the neglect of 2 p-2h admixtures. As a result, the RPA does not account
for the important relaxation and polarization effects (see Sect. 15.1). For a more
appropriate treatment of excitations in the N -electron system, one has to include
the 2 p-2h excitation operators together with their 2h-2 p de-excitation counterparts,
as is the case in the second-order polarization propagator approximation (SOPPA)
scheme [12, 13].
The considerations in Sect. 16.1 concerning the order relations and the separability
of the EOM secular matrices apply to the N -electron case as well. In the form of
Eqs. (16.46), (16.47), M is neither canonical nor separable, and it requires again
an orthonormalization procedure of ECO-Gram-Schmidt type to afford the latter
properties [8].
The N -electron EOM treatment features a non-hermitian secular problem combining physical excited configurations and an equally large manifold of unphysical
de-excitation configurations. As in the (N ±1)-electron case above, one may weigh
advantages and disadvantages of the EOM concept as compared to the plain ISR
schemes of the type presented in Chap. 14. Besides the size of the EOM secular
matrix, being here exactly twice the size of comparable plain ISR schemes, the fact
that the EOM treatment involves a non-hermitian RPA-type secular problem constitutes another disadvantage.
Exercises
16.1 Verify the expression (16.26) for the first-order contribution to the EOM secular
matrix element M i,abjkl ; evaluate the analogous contribution, S
(1)
i,abjkl , for the
EOM overlap matrix.
16.2 Field operators in coordinate space representation:
(a) Revisit the abstract definition (2.3) of the creation operator c
†
q and “translate” it to the coordinate space representation. Let (ξ 1 , . . . , ξ N ) denote an
antisymmetrized and normalized wave function of N electrons (N even). Write
the (N +1)-electron wave functions c
†
q (ξ 1 , . . . , ξ N ) in terms of a properly
antisymmetrized linear combination of products of the type φ q .
(b) Write c q (ξ 1 , . . . , ξ N ) as an expansion of (N −1)-electron basis states n
in analogy to Eq. (2.7).
(c) Let φ q be a complex-valued orbital. The coordinate space representation
according to (a) and (b) allows one to define “complex” field operators, c
∗†
q
and c
∗
q . Verify that c
∗†
q = (c
†
q )
∗ and c
∗
q = (c q )
∗ for a real-valued wave
function .
16.3 EOM state representation in case of complex-valued orbitals:
Suppose a complex one-particle representation and modify the definition
(16.27) of Fock-space states according to | I = ( ˆ
O
∗†
I + ˆ
O I )| 0 , where the