Appendix
273
As these examples show, the general PT expansion (A.1.8) for | 0 establishes
individual PT expansions for the amplitudes (or CI coefficients)
x J = = J | 0
(A.1.23)
The onset of these expansions, that is, the PT order O(x J ) of the first non-vanishing
contribution, depends on the excitation class [J ] of the respective excitation J . The
amplitudes x abkl are of first order (and that would apply to x ak as well were it not
for Brillouin’s theorem). The x-amplitudes for the triple and quadruple excitations
(excitation classes 3 and 4, respectively) begin in second order; in third order, the
next two excitation classes, ν = 5 and 6, come into play, and so forth. The general
order relations read
O(x J ) =
1
2
[J ], [J ] even
1
2
([J ] + 1), [J ] odd, > 1
(A.1.24)
They follow from the structure of the expansion (A.1.8) and the instance that via the
two-body Coulomb operator contained in ˆ
H I there is a (non-vanishing) coupling of
states of excitation class ν to states of the two higher classes, ν + 1 and ν + 2.
A.1.2 Matrix Algebra
There are some basic matrix algebra techniques which are generally useful and apply,
in particular, to some of the topics treated in this book. While these algebra tools
are simple, they cannot necessarily be considered common knowledge so that the
following brief inspection may be helpful.
Inverse of a Partitioned Matrix
Consider a square matrix
M =
A B
C D
(A.1.25)
consisting as indicated of sub-blocks, where A and D are square matrices, and B
and C are rectangular matrices of corresponding size. The inverse of the partitioned
matrix
A B
C D
−1
=
E F
G H
(A.1.26)
273
As these examples show, the general PT expansion (A.1.8) for | 0 establishes
individual PT expansions for the amplitudes (or CI coefficients)
x J = = J | 0
(A.1.23)
The onset of these expansions, that is, the PT order O(x J ) of the first non-vanishing
contribution, depends on the excitation class [J ] of the respective excitation J . The
amplitudes x abkl are of first order (and that would apply to x ak as well were it not
for Brillouin’s theorem). The x-amplitudes for the triple and quadruple excitations
(excitation classes 3 and 4, respectively) begin in second order; in third order, the
next two excitation classes, ν = 5 and 6, come into play, and so forth. The general
order relations read
O(x J ) =
1
2
[J ], [J ] even
1
2
([J ] + 1), [J ] odd, > 1
(A.1.24)
They follow from the structure of the expansion (A.1.8) and the instance that via the
two-body Coulomb operator contained in ˆ
H I there is a (non-vanishing) coupling of
states of excitation class ν to states of the two higher classes, ν + 1 and ν + 2.
A.1.2 Matrix Algebra
There are some basic matrix algebra techniques which are generally useful and apply,
in particular, to some of the topics treated in this book. While these algebra tools
are simple, they cannot necessarily be considered common knowledge so that the
following brief inspection may be helpful.
Inverse of a Partitioned Matrix
Consider a square matrix
M =
A B
C D
(A.1.25)
consisting as indicated of sub-blocks, where A and D are square matrices, and B
and C are rectangular matrices of corresponding size. The inverse of the partitioned
matrix
A B
C D
−1
=
E F
G H
(A.1.26)