^{1,a)}, Wolfgang Hackbusch

^{1}and Heinz-Jürgen Flad

^{2}

### Abstract

We have studied an iterative perturbative approach to optimize Jastrow factors in quantum Monte Carlo calculations. For an initial guess of the Jastrow factor we construct a corresponding model Hamiltonian and solve a first-order perturbation equation in order to obtain an improved Jastrow factor. This process is repeated until convergence. Two different types of model Hamiltonians have been studied for both energy and variance minimization. Our approach can be considered as an alternative to Newton’s method. Test calculations revealed the same fast convergence as for Newton’s method sufficiently close to the minimum. However, for a poor initial guess of the Jastrow factor, the perturbative approach is considerably more robust especially for variance minimization. Usually only two iterations are sufficient in order to achieve convergence within the statistical error. This is demonstrated for energy and variance minimization for the first row atoms and some small molecules. Furthermore, our perturbation analysis provides new insight into some recently proposed modifications of Newton’s method for energy minimization. A peculiar feature of the analysis is the continuous use of cumulants which guarantees size-consistency and provides least statistical fluctuations in the Monte Carlo implementation.

The authors gratefully acknowledge Professor A. Savin and Professor J. Toulouse (Paris) for useful discussions.

I. INTRODUCTION

A. Jastrow perturbation theory

B. QMC optimization of Jastrow factors

II. JPT

III. PERTURBATIVE ENERGY AND VARIANCE MINIMIZATION OF JASTROW FACTORS

A. Partitioning of the Hamiltonian

B. JPT equations for energy minimization

C. JPT equations for variance minimization

IV. LEAST STATISTICAL FLUCTUATIONS FOR JPT EQUATIONS VIA CUMULANTS

V. PERFORMANCE OF JPT AND NEWTON’S METHOD IN TEST CALCULATIONS

A. Convergence tests for energy minimization

B. Test calculations for variance minimization

VI. CONCLUSIONS AND OUTLOOK

### Key Topics

- Wave functions
- 49.0
- Perturbation theory
- 22.0
- Correlation functions
- 10.0
- Optimization
- 10.0
- Perturbation methods
- 6.0

## Figures

Effect of cumulant representations on the standard error of matrix elements for energy and variance minimization. Results are shown for the Ne atom with Hartree–Fock reference wave function and a matrix element with respect to the two basis functions Eq. (41) vs the number of Monte Carlo sample points. (a) Upper part: relative standard error of the matrix element Eq. (35) (solid line), Eq. (36) (dotted line) and Eq. (37) (dashed line) for Newton’s method. Lower part: relative standard error of the matrix element Eq. (38) (solid line), Eq. (39) (dotted line) and Eq. (40) (dashed line) of the JPTE-I equation. (b) Relative standard error of the corresponding matrix element for JPTV-I (30) (solid line), JPTV-II (31) (dotted line) and Newton’s method (32) (dashed line).

Effect of cumulant representations on the standard error of matrix elements for energy and variance minimization. Results are shown for the Ne atom with Hartree–Fock reference wave function and a matrix element with respect to the two basis functions Eq. (41) vs the number of Monte Carlo sample points. (a) Upper part: relative standard error of the matrix element Eq. (35) (solid line), Eq. (36) (dotted line) and Eq. (37) (dashed line) for Newton’s method. Lower part: relative standard error of the matrix element Eq. (38) (solid line), Eq. (39) (dotted line) and Eq. (40) (dashed line) of the JPTE-I equation. (b) Relative standard error of the corresponding matrix element for JPTV-I (30) (solid line), JPTV-II (31) (dotted line) and Newton’s method (32) (dashed line).

Ratios of variational correlation energies for Jastrow factors with respect to the exact ground state (upper part) and fixed-node DMC (lower part) energies. Energy optimized correlation functions from the first iteration step of the JPTE-I , JPTE-II and Newton’s method are compared with fully converged FOPIM based on the JPTE-I equation.

Ratios of variational correlation energies for Jastrow factors with respect to the exact ground state (upper part) and fixed-node DMC (lower part) energies. Energy optimized correlation functions from the first iteration step of the JPTE-I , JPTE-II and Newton’s method are compared with fully converged FOPIM based on the JPTE-I equation.

Convergence behavior of JPTE-I (○), JPTE-II (◻), and Newton’s method (◇) for energy optimization with respect to the number of iterations for the (a) Ne atom and (b) .

Convergence behavior of JPTE-I (○), JPTE-II (◻), and Newton’s method (◇) for energy optimization with respect to the number of iterations for the (a) Ne atom and (b) .

Convergence of FOPIM based on the JPTE-I equation, in the case of the Ne atom, for an initial reference wave function with prescribed electron-electron cusp (47). For comparison convergence is shown for a Hartree–Fock wave function as an initial guess.

Convergence of FOPIM based on the JPTE-I equation, in the case of the Ne atom, for an initial reference wave function with prescribed electron-electron cusp (47). For comparison convergence is shown for a Hartree–Fock wave function as an initial guess.

Convergence to the ground state energy of for different bond lengths starting from a RHF reference wave function. Results are shown for FOPIM with JPTE-I, JPTE-II, and Newton’s method. The bond length varies from the equilibrium bond length up to . For each method, results are shown only for those bond lengths at which convergence has been achieved.

Convergence to the ground state energy of for different bond lengths starting from a RHF reference wave function. Results are shown for FOPIM with JPTE-I, JPTE-II, and Newton’s method. The bond length varies from the equilibrium bond length up to . For each method, results are shown only for those bond lengths at which convergence has been achieved.

Removal of polar configurations by the Jastrow factor for at bond length . (a) RHF and (b) Jastrow wave functions are plotted along the bond axis, i.e., . The nuclei are located at .

Removal of polar configurations by the Jastrow factor for at bond length . (a) RHF and (b) Jastrow wave functions are plotted along the bond axis, i.e., . The nuclei are located at .

Convergence of the standard deviation of the local energy for the Ne atom with different basis sets for the correlation function. Results are shown for FOPIM with JPTV-I (○), JPTV-II (◇), and Newton’s method . Three different basis sets have been studied: (a) 7 term basis, (b) 17 term basis, (c) 92 term basis. The 7 and 17 term bases were taken from Ref. 33. A Hartree–Fock reference wave function has been taken as an initial guess, except for Newton’s method where in (b) and (c) the JPTV-I result from the first iteration step was chosen.

Convergence of the standard deviation of the local energy for the Ne atom with different basis sets for the correlation function. Results are shown for FOPIM with JPTV-I (○), JPTV-II (◇), and Newton’s method . Three different basis sets have been studied: (a) 7 term basis, (b) 17 term basis, (c) 92 term basis. The 7 and 17 term bases were taken from Ref. 33. A Hartree–Fock reference wave function has been taken as an initial guess, except for Newton’s method where in (b) and (c) the JPTV-I result from the first iteration step was chosen.

## Tables

Variational energies (Hartree) and standard deviations of the local energy for first row atoms and the molecules and . Jastrow factors have been optimized by FOPIM for energy and variance minimization based on JPTE-I, JPTV-I, and JPTV-II, respectively. Hartree–Fock orbitals have been used for the Slater determinant. For comparison we have also listed fixed-node DMC and “exact” energies.

Variational energies (Hartree) and standard deviations of the local energy for first row atoms and the molecules and . Jastrow factors have been optimized by FOPIM for energy and variance minimization based on JPTE-I, JPTV-I, and JPTV-II, respectively. Hartree–Fock orbitals have been used for the Slater determinant. For comparison we have also listed fixed-node DMC and “exact” energies.

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