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Physical Review Letters

Variational Approach to Configuration Interaction

Richard R. Chasman

Published Dec 20, 2005• 5 citations
Publisher

Abstract

A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.

DOI
10.1103/physrevlett.95.262501

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A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.
Pair vibrations are studied for a Hamiltonian with neutron–neutron, proton–proton and neutron–proton pairing. The spectrum is found to be rich in strongly correlated, low-lying excited states. Changing the ratio of diagonal to off-diagonal pairing matrix elements is found to have a large impact on the excited-state spectrum. The variational configuration interaction (VCI) method, used to calculate the excitation spectrum, is found to be in very good agreement with exact solutions for systems with large degeneracies having equal T=0 and T=1 pairing strengths.
A method is proposed for determining the ground state of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k}{\ensuremath{\epsilon}}_{k}{N}_{k}\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{\mathrm{ijkl}}{V}_{\mathrm{ijkl}}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}$when the residual interaction is such that $〈{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}〉={〈{N}_{i}{N}_{j}(1\ensuremath{-}{N}_{k})(1\ensuremath{-}{N}_{l})〉}^{\frac{1}{2}}{〈{N}_{k}{N}_{l}(1\ensuremath{-}{N}_{i})(1\ensuremath{-}{N}_{j})〉}^{\frac{1}{2}}$is a good approximation. The method is applied to pairing interactions involving one type of nucleon as well as to the generalized pairing problem containing neutron-neutron, neutron-proton, and proton-proton interactions.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function Q(N)Q(N)Q(N) depending on the number of particles, NNN. The function Q(N)Q(N)Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.
A method is described to construct the ground state of a general pairing Hamiltonian for a finite nuclear systemas a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of themethod are provided that include comparisonswith exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. The method is seen to work well leading, under some circumstances, to solutions that are basically exact.
We develop a general quantum many-body theory in configuration space by extending the traditional coupled-cluster method to a variational formalism. A set of distribution functions are introduced to evaluate the expectation value of the Hamiltonian. An algebraic technique for calculations of these distribution functions via a set of self-consistent set of equations is given. As a demonstration, we apply this method to a quantum antiferromagnetic spin model. It is shown that one of the lowest-order approximations within this new technique agrees with the traditional coupled-cluster method. Comparison with the method of correlated basis functions is also made.
A method is implemented wherein numerical approximations to the ground and first few excited states of a quantum mechanical N -body 1D harmonic oscillator are found through variational methods, representing the states as a linear combination of normalized pseudo-states which are themselves linear combinations of non-orthogonal Slater determinants. These states are then used as a low energy basis for configuration interaction. An expression is derived for an analytical matrix derivative of the energy functional, in order to improve the speed of the variation. The speed and accuracy using the analytical derivative is compared to that of the numerical derivative, and a number of different gradient descent methods are tried and compared.