We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.
We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
The Fock space Hamiltonian H has a simpler structure than its projection Hn to n-particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: WD=WD+, c:σD=(ln W)D=0. The transformation of operator products to normal product form is achieved by means of a generalization of Wick’s theorem to spinfree quantum chemistry. The contributions to the various orders of L, W, and σ are illustrated by means of diagrams. Only variant c guarantees a connected-diagram expansion of the Fock space energy operator L. None of the three variants satisfies a generalization of the Wigner (2n+1) rule, but with a slight change in the normalization from b to b′ or c to c′ (that does, in the case of c′, not affect the connectedness of the diagram expansion) L(2n) and L(2n+1) are expressible in terms of the W(k), k⩽n. A generalization of the variation principle for the Fock space energy operator is given and its consequences are discussed. Generalized Fock space Brillouin conditions are derived and a variational scheme for the construction of the energy operator is proposed. We finally compare the present Fock space approach with the traditional degenerate perturbation theory in n-particle Hilbert space. If the diagonal part ΩD of an operator Ω is defined in the spirit of the theory of the universal wave and energy operators, then the two approaches lead to the same results (for the same normalization), but the Fock space approach looks simpler and more transparent.
Effective Hamiltonians and effective operators produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. We obtain degenerate and quasidegenerate perturbative expressions for the particularly useful canonical effective operator ÂC through second order in perturbation theory. The corresponding ÂC diagrammatic expressions are derived for the case where ÂC acts in a complete finite space. Our first order results have been used previously for ab initio computations of dipole and transition dipole moments in diatomic hydrides and for testing the assumptions in semiempirical methods for dipole properties. A brief discussion is also provided on the computational labors required by first and second order ÂC many-body calculations, the derivation of ÂC diagrams when ÂC acts in an incomplete finite space, and on the derivation of diagrammatic rules for ÂC in arbitrary perturbation order.
For a quantum many-body problem, effective Hamiltonians that give exact eigenvalues in reduced model space usually have different expressions, diagrams, and evaluation rules from effective transition operators that give exact transition matrix elements between effective eigenvectors in reduced model space. By modifying these diagrams slightly and considering the linked diagrams for all the terms of the same order, we find that the evaluation rules can be made the same for both effective Hamiltonian and effective transition operator diagrams, and in many cases it is possible to combine many diagrams into one modified diagram. We give the rules to evaluate these modified diagrams and show their validity.
By use of basic results of diagrammatic perturbation theory and combinatorial arguments, a formula is derived for the number of topologically distinct, connected, time ordered Goldstone diagrams contributing to the nth order perturbation energy. A general perturbation consisting of one- and two-particle operators is considered. The result is extended to an effective one-particle operator that implicitly includes all equal-time contractions and to multiple one-particle perturbations that enter only a restricted number of times in any order; the former is important for a Hartree–Fock zero order function and the latter for a system with external or internal field perturbations. It is pointed out that the method for computing the number of disconnected diagrams is quite general and independent of the nature of the perturbation and that the connected topologies which possess unusual symmetry (i.e., those for which all sets of vertex space–time labels are not distinct) are of a very specific and easily identified type. Numerical results for the various cases considered are given through 5th order.
A general coupled-cluster method valid for arbitrary multideterminantal reference states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and applied by Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and collaborators. The connected nature of the cluster operators and the effective interaction is proven in the case when the reference space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete reference spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approximate schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.