We derive perturbation expansions for the mapping operators (k,l) that transform a full Hilbert space time-independent Hamiltonian H and operator A into, respectively, a finite (multidimensional) space effective Hamiltonian h and effective operator a. The eigenvalues of h are identical to some of those of H, and a produces exact matrix elements of A for the corresponding states. Our derivations are substantially both more general and simpler than most literature ones and yield simple linear recursive expressions for k and l. Both these mapping solutions and new identities involving h, a, k, and l straightforwardly produce new recursive relations for h and the first known recursive a expressions. We apply our results to the Bloch, Kato, and all norm-preserving formalisms, including the canonical one. The new h and a identities are also shown to be suitable for iterative and multireference coupled cluster approaches.
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.
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.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.
This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.