16.1 Equations-of-Motion (EOM) Method for N ± 1 Electrons
245
as the scalar product of the respective eigenvector and a vector of of transition
moments for the basis operators, 0 |{ ˆ
O
†
J , c p }| 0 . Here anticommutators can be
used as a result of the KC relation (16.24).
The explicit forms (16.21), (16.22) of the final states make apparent that the EOM
qualifies as a particular ISR approach, in which the treatment of the (N −1)- and
(N +1)-electron systems is combined. In fact, the EOM secular equations can be
obtained explicitly as a Fock-space state representation of a generalized hamiltonian, as will be demonstrated in the ensuing Sect. 16.2.
Orthonormalization and Approximation Schemes
We note that the EOM secular matrix M is hermitian, and the likewise hermitian overlap matrix S consists of two positive definite sub-blocks. Accordingly, the eigenvalue
problem can be transformed into a regular hermitian one. This of course touches upon
the issue of the appropriate orthonormalization. As we have seen in Sect. 12.1, the
excitation class Gram-Schmidt orthonormalization of the CE states is crucial with
regard to the canonical order relations and the separability of the secular matrices. This finding applies also to the EOM schemes. In the given form, based on
non-orthonormal CE states, the EOM secular matrices are neither “canonical” nor
separable. This can be seen by inspecting, for example, the matrix elements of the
1h/3h-2 p coupling blocks of S and M, displaying non-vanishing first-order contributions (Exercise 16.1),
M
(1)
i,abjkl = i
−δ i j
V kl[ab]
a + b − k − l
+ δ ik
V jl[ab]
a + b − j − l
− δ il
V jk[ab]
a + b − j − k
(16.26)
Similar first-order contributions arise in the overlap matrix element, S
(1)
i,abjkl , which
indicates that the problem can be traced to the fact that the EOM basis states are not
orthonormal. Again, symmetric orthonormalization based on S
−1/2 is not expedient,
but the ECO-Gram-Schmidt procedure described in Chap. 11 can be extended to the
EOM case [8, 10], which procures the desired properties.
In its general form, the EOM method is an exact approach to (N ±1)-electron
states and energies, though obviously not a practicable one. To devise viable approximation schemes, one has to truncate the operator expansion manifold in appropriate
ways and replace the exact ground state with consistent finite PT expansions in
the expressions (16.13), (16.14) for the secular matrix elements. An example is the
consistent third-order EOM approximation scheme discussed in Ref. [6]. Here the
explicit operator manifold comprises the 1h, 1p, 2h-1 p, and 2 p-1h excitation classes;
the PT expansions of the secular matrix elements in the respective sub-blocks are
such that the energies of the 1h and 1 p main states are treated consistently through
third order.
Like the Dyson equation discussed in Chap. 8, the EOM method entangles the
treatment of N −1 electrons with that of N +1 electrons. Obviously, this procedure
is to be seen as a mathematical device rather than being physically motivated. The
advantages and disadvantages of the (N ±1)-electron coupling with regard to comparable separated procedures have been addressed in Chap. 10. The coupled schemes
245
as the scalar product of the respective eigenvector and a vector of of transition
moments for the basis operators, 0 |{ ˆ
O
†
J , c p }| 0 . Here anticommutators can be
used as a result of the KC relation (16.24).
The explicit forms (16.21), (16.22) of the final states make apparent that the EOM
qualifies as a particular ISR approach, in which the treatment of the (N −1)- and
(N +1)-electron systems is combined. In fact, the EOM secular equations can be
obtained explicitly as a Fock-space state representation of a generalized hamiltonian, as will be demonstrated in the ensuing Sect. 16.2.
Orthonormalization and Approximation Schemes
We note that the EOM secular matrix M is hermitian, and the likewise hermitian overlap matrix S consists of two positive definite sub-blocks. Accordingly, the eigenvalue
problem can be transformed into a regular hermitian one. This of course touches upon
the issue of the appropriate orthonormalization. As we have seen in Sect. 12.1, the
excitation class Gram-Schmidt orthonormalization of the CE states is crucial with
regard to the canonical order relations and the separability of the secular matrices. This finding applies also to the EOM schemes. In the given form, based on
non-orthonormal CE states, the EOM secular matrices are neither “canonical” nor
separable. This can be seen by inspecting, for example, the matrix elements of the
1h/3h-2 p coupling blocks of S and M, displaying non-vanishing first-order contributions (Exercise 16.1),
M
(1)
i,abjkl = i
−δ i j
V kl[ab]
a + b − k − l
+ δ ik
V jl[ab]
a + b − j − l
− δ il
V jk[ab]
a + b − j − k
(16.26)
Similar first-order contributions arise in the overlap matrix element, S
(1)
i,abjkl , which
indicates that the problem can be traced to the fact that the EOM basis states are not
orthonormal. Again, symmetric orthonormalization based on S
−1/2 is not expedient,
but the ECO-Gram-Schmidt procedure described in Chap. 11 can be extended to the
EOM case [8, 10], which procures the desired properties.
In its general form, the EOM method is an exact approach to (N ±1)-electron
states and energies, though obviously not a practicable one. To devise viable approximation schemes, one has to truncate the operator expansion manifold in appropriate
ways and replace the exact ground state with consistent finite PT expansions in
the expressions (16.13), (16.14) for the secular matrix elements. An example is the
consistent third-order EOM approximation scheme discussed in Ref. [6]. Here the
explicit operator manifold comprises the 1h, 1p, 2h-1 p, and 2 p-1h excitation classes;
the PT expansions of the secular matrix elements in the respective sub-blocks are
such that the energies of the 1h and 1 p main states are treated consistently through
third order.
Like the Dyson equation discussed in Chap. 8, the EOM method entangles the
treatment of N −1 electrons with that of N +1 electrons. Obviously, this procedure
is to be seen as a mathematical device rather than being physically motivated. The
advantages and disadvantages of the (N ±1)-electron coupling with regard to comparable separated procedures have been addressed in Chap. 10. The coupled schemes