310
Appendix
can be treated in a similar way, using here Eq. (A.6.31). This establishes the order
relations (11.20) or (14.4) for the transition amplitudes, being
f I,rs ∼ O([I ] − 1)
(A.6.43)
in the N -electron case.
Order Structure of the ISR Eigenvalue Matrix
As an immediate consequence of the order structure (12.1) of the ISR secular matrix
M, the canonical order relations apply to the eigenvector matrix X as well (see
Eq. 12.15),
X J n = = ˜
J | n ∼ O(|[J ] − [n]|)
(A.6.44)
Here, it is assumed that the final state | n has a PT origin in a specific HF configuration, say | I , of class [I ]:
| n ← | I
(A.6.45)
Accordingly, | n is said to belong to the excitation class [n] := [I ].
The emergence of the order relations (A.6.44) can be explained as follows (for a
different, more stringent derivation see Ref. [13]). Let X n be the eigenvector for a
state | n of class [n],
M X n = ω n X n
(A.6.46)
and assume that | n derives from a HF configuration | I , [I ] = [n]. Now we
consider an eigenvector component X J n , where J denotes an excitation from another
excitation class, [J ] = [I ]. Applying a simple matrix perturbation theory (MPT),
where the diagonal matrix elements, M K K , furnish the “zeroth-order” level, and the
non-diagonal elements, M K L , K = L, define the perturbation, one obtains a “firstorder” contribution to X J n of the form
X J n ←
M J I
M J J − M I I
(A.6.47)
This contribution results from the coupling of the configurations | I and | J via
the secular matrix element M J I . The actual PT order of this term is equal to the order
of M J I ,
M J I
M J J − M I I
∼ M J I ∼ O(|[J ] − [I ]|) = O(|[J ] − [n]|)
(A.6.48)
since the denominator is of zeroth order, M J J − M I I ∼ O(0). In fact, there are no
contributions of lower order than that. Consider an “interaction path” J ← K ← I
encountered in “second order” of MPT. The corresponding contribution is given by
Appendix
can be treated in a similar way, using here Eq. (A.6.31). This establishes the order
relations (11.20) or (14.4) for the transition amplitudes, being
f I,rs ∼ O([I ] − 1)
(A.6.43)
in the N -electron case.
Order Structure of the ISR Eigenvalue Matrix
As an immediate consequence of the order structure (12.1) of the ISR secular matrix
M, the canonical order relations apply to the eigenvector matrix X as well (see
Eq. 12.15),
X J n = = ˜
J | n ∼ O(|[J ] − [n]|)
(A.6.44)
Here, it is assumed that the final state | n has a PT origin in a specific HF configuration, say | I , of class [I ]:
| n ← | I
(A.6.45)
Accordingly, | n is said to belong to the excitation class [n] := [I ].
The emergence of the order relations (A.6.44) can be explained as follows (for a
different, more stringent derivation see Ref. [13]). Let X n be the eigenvector for a
state | n of class [n],
M X n = ω n X n
(A.6.46)
and assume that | n derives from a HF configuration | I , [I ] = [n]. Now we
consider an eigenvector component X J n , where J denotes an excitation from another
excitation class, [J ] = [I ]. Applying a simple matrix perturbation theory (MPT),
where the diagonal matrix elements, M K K , furnish the “zeroth-order” level, and the
non-diagonal elements, M K L , K = L, define the perturbation, one obtains a “firstorder” contribution to X J n of the form
X J n ←
M J I
M J J − M I I
(A.6.47)
This contribution results from the coupling of the configurations | I and | J via
the secular matrix element M J I . The actual PT order of this term is equal to the order
of M J I ,
M J I
M J J − M I I
∼ M J I ∼ O(|[J ] − [I ]|) = O(|[J ] − [n]|)
(A.6.48)
since the denominator is of zeroth order, M J J − M I I ∼ O(0). In fact, there are no
contributions of lower order than that. Consider an “interaction path” J ← K ← I
encountered in “second order” of MPT. The corresponding contribution is given by