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The Journal of Chemical Physics

Development of a new variational approach for thermal density matrices

Tapta Kanchan Roy, M. Durga Prasad

Published Jun 2, 2011• 16 citations
Publisher

Abstract

A McLachlan-type variational principle is derived for thermal density matrices. In this approach, the trace of the mean square of the differences between the derivatives of the exact and model density matrices is minimized with respect to the parameters in the model Hamiltonian. Applications to model anharmonic systems in the independent particle model show that the method can provide thermodynamic state functions accurately (within 5% of the converged basis set results) and at the same level of accuracy as the results using Feynman-Gibbs-Bogoliubov variational principle at this level of approximation.

DOI
10.1063/1.3592777

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A McLachlan-type variational principle is derived for thermal density matrices. In this approach, the trace of the mean square of the differences between the derivatives of the exact and model density matrices is minimized with respect to the parameters in the model Hamiltonian. Applications to model anharmonic systems in the independent particle model show that the method can provide thermodynamic state functions accurately (within 5% of the converged basis set results) and at the same level of accuracy as the results using Feynman-Gibbs-Bogoliubov variational principle at this level of approximation.
Variational principles for lower bounds to the energy, or free energy for T > 0°, of many-body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy—or free energy—for the model Hamiltonian HM. Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H − HM approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs-Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.
We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.
A general variational approach to the quantum statistics of many-particle systems is developed, based on the Gibbs-Bogoliubov variational principle and the ``thermodynamically equivalent Hamiltonian'' method first applied to the BCS model by Bogoliubov, Zubarev, and Tserkovnikov [Sov. Phys.---Dokl. 2, 535 (1957)] and subsequently generalized by Wentzel [Phys. Rev. 120, 1572 (1960)]. The method retains a judiciously chosen discrete set of momentum transfers in the second-quantized interaction Hamiltonian. It is expected to be applicable to a variety of systems including both crystalline solids and quantum fluids such as liquids and dense plasma phases.
A variational formalism for the determination of the spinless single-particle density matrix $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator X=Y+iZX=Y+iZX=Y+iZ, where $Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})$ is a translation operator, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator WWW employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.
We show the existence of the limiting free-energy density of inhomogeneous ( site-dependent coupling ) mean-field models in the thermodynamic limit, and derive a variational formula for this quantity. The formula requires the minimization of an energy term plus an entropy term as a functional depending on a function with values in the one-particle state space. The minimizing functions describe the pure phases of the system, and all cluster points of the sequence of finite volume equilibrium states have unique integral decomposition into pure phases. Applications are considered; they include the full BCS-model, and random mean-field models
Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.