In the conventional quantum-theoretical treatment of finite many-electron systems, there are two basic tools: firstly, perturbation theory (PT), which, however, as
a computational scheme applies only to the N-electron ground state; and, secondly,
the standard numerical procedure of solving the time-independent Schrödinger
equation, that is, using suitable basis set expansions for the states of interest and
transforming the Schrödinger equation into the secular problem of the corresponding matrix representation of the hamiltonian. The general problem of configuration interaction (CI), as the standard procedure is referred to in quantum
chemistry, is the exponentially increasing dimension of the secular matrix,
d ¼
M
N
, both with the size of the systems, reflected in the number of electrons, N,
and the demand for accuracy, reflected in the number M of one-particle states
underlying the many-electron basis states (CI configurations). This means that a full
CI treatment is not viable except for very small systems and limited one-particle
basis sets, and one has to resort to approximate CI schemes obtained by truncating
the configuration manifold in suitable ways.
Here, however, an unsuspected problem arises which disqualifies the CI as a
means of treating extended electron systems. In the CI secular equations, there is an
interaction (mixing) of configurations that differ exactly by a double excitation,
such as in a singly (S) excited configuration (relative to the reference state) and a
triply (T) excited configuration comprising the former single excitation.
A corresponding S-T secular matrix element is potentially “non-local”; that is, its
magnitude does not decrease or vanish when the involved single and double
excitations can be assigned to distant parts of the system or even to separate
fragments of a composite system. In truncated CI expansions, the presence of these
potentially non-local admixtures causes an uncontrollable error which grows with
the spatial extension of the system and, accordingly, is referred to as
size-consistency error.
The propagator methods, by contrast, do not suffer from this deficiency. As a
common feature, approximation schemes deriving from field-theoretical many-body
theory combine perturbation expansions (of the ground-state type) and eigenvalue
algebra within a generalized secular problem where in particular any potentially
non-local coupling contributions are taken care of in the PT part. As a consequence,
the propagator methods are inherently size-consistent and, moreover, more economical, requiring distinctly smaller explicit configuration manifolds in the secular
problem than in CI expansions of comparable accuracy.
A brief guide to the tour through the five parts of this book may be helpful. The
first two chapters of Part I lay the groundwork for the quantum theory of
many-electron systems, addressing states, operators, the evaluation of matrix elements, and, finally, the use of second quantization. Thereupon, the prototypical
one-particle Green’s function or electron propagator is presented and discussed in
Chap. 3.
vi
Preface
a computational scheme applies only to the N-electron ground state; and, secondly,
the standard numerical procedure of solving the time-independent Schrödinger
equation, that is, using suitable basis set expansions for the states of interest and
transforming the Schrödinger equation into the secular problem of the corresponding matrix representation of the hamiltonian. The general problem of configuration interaction (CI), as the standard procedure is referred to in quantum
chemistry, is the exponentially increasing dimension of the secular matrix,
d ¼
M
N
, both with the size of the systems, reflected in the number of electrons, N,
and the demand for accuracy, reflected in the number M of one-particle states
underlying the many-electron basis states (CI configurations). This means that a full
CI treatment is not viable except for very small systems and limited one-particle
basis sets, and one has to resort to approximate CI schemes obtained by truncating
the configuration manifold in suitable ways.
Here, however, an unsuspected problem arises which disqualifies the CI as a
means of treating extended electron systems. In the CI secular equations, there is an
interaction (mixing) of configurations that differ exactly by a double excitation,
such as in a singly (S) excited configuration (relative to the reference state) and a
triply (T) excited configuration comprising the former single excitation.
A corresponding S-T secular matrix element is potentially “non-local”; that is, its
magnitude does not decrease or vanish when the involved single and double
excitations can be assigned to distant parts of the system or even to separate
fragments of a composite system. In truncated CI expansions, the presence of these
potentially non-local admixtures causes an uncontrollable error which grows with
the spatial extension of the system and, accordingly, is referred to as
size-consistency error.
The propagator methods, by contrast, do not suffer from this deficiency. As a
common feature, approximation schemes deriving from field-theoretical many-body
theory combine perturbation expansions (of the ground-state type) and eigenvalue
algebra within a generalized secular problem where in particular any potentially
non-local coupling contributions are taken care of in the PT part. As a consequence,
the propagator methods are inherently size-consistent and, moreover, more economical, requiring distinctly smaller explicit configuration manifolds in the secular
problem than in CI expansions of comparable accuracy.
A brief guide to the tour through the five parts of this book may be helpful. The
first two chapters of Part I lay the groundwork for the quantum theory of
many-electron systems, addressing states, operators, the evaluation of matrix elements, and, finally, the use of second quantization. Thereupon, the prototypical
one-particle Green’s function or electron propagator is presented and discussed in
Chap. 3.
vi
Preface