Abstract In conventional Hartree-Fock calculations for molecular systems the optimization of a trial wavefunction by energy minimization with respect to the orbitals of a determinantal wavefunction is relatively easy. When Jastrow factors are included with a determinant as for typical QMC trial functions the task becomes more difficult. Optimization by minimizing the variance in local energies has been one alternative. This paper describes another approach which is, in fact, an analog of the Hartree-Fock method and provides an exact solution to the problem of optimizing the orbitals when a Jastrow function is present.
Abstract In conventional Hartree-Fock calculations for molecular systems the optimization of a trial wavefunction by energy minimization with respect to the orbitals of a determinantal wavefunction is relatively easy. When Jastrow factors are included with a determinant as for typical QMC trial functions the task becomes more difficult. Optimization by minimizing the variance in local energies has been one alternative. This paper describes another approach which is, in fact, an analog of the Hartree-Fock method and provides an exact solution to the problem of optimizing the orbitals when a Jastrow function is present.
Abstract The optimization of parameters in trial wavefunctions by minimizing the variance in local energy was used for some of the earliest variationala and diffusionb QMC calculations, and it has been used ever since by most practitioners of QMC. Perhaps the best known examples are given in the paper by Schmidt and Moskowitzc for optimizing their nine-parameter Jastrow functions for first-row atoms. An alternative method of optimization, leading to lower variational energies for these atoms, is reported in this paper.
Abstract The optimization of trial wavefunctions ‘I/Jr for use in QMC calculations can be accomplished with a fixed set of configurations selected with probabilities proportional to ‘1/Jf, often using Metropolis sampling and occasionally using other methods. By varying the coefficients of ‘I/Jr, one can minimize the variance in local energies for the original set of configurations. This is effective except when coefficients affecting the node locations are varied and occasional very high weighting factors ‘I/J}(new)/’1/J}(original) can give large fluctuations in the variance. In this paper, the authors propose the use of probability density functions p to produce nodeless distributions of points replacing the original ‘1/Jf distributions. This has two advantages: it avoids the unbounded weighting problem allowing nodes to be varied, and it facilitates sampling without a Metropolis walk and its associated serial correlation. To be effective, i.e., for high efficiency, p must be carefully selected, it must at least resemble ‘1/Jf, and its distributions must be easily sampled. Tests were performed for the case of CH with optimization of all parameters of a two-determinant-and-Jastrow trial function. These included basis set exponents, MO coefficients, CI coefficient, and SchmidtMoskowitz Jastrow parameters. The resulting variance in local energy was 0.19 h2, versus 0.25 h2 for optimization of the Jastrow coefficients only. More interesting is the improvement in diffusion QMC energy for the complete optimization including node movement. The energy was lowered by 0.004 hartree, or 2.5 kcal/mol, with recovery of correlation energy increased from 92.8 to 94.8%.
A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.
A scheme of optimization of a trial wave function in the variational Monte Carlo calculation is developed. It provides a more precise description of the ground state for strongly correlated electronic systems with highly improved trial wave function. A general type of variational wave function constructed from a linear combination of Slater determinants multiplied by Jastrow correlation factors is proposed. They can explicitly contain low-energy fluctuations and long-range correlations. Using this optimization method, we examine the properties of these wave functions by applying them to a one-dimensional Hubbard model for which the exact solution is available for comparison. It is found that these wave functions prove to be substantial improvements compared to the conventional Jastrow-type wave functions.
We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
Multideterminant wavefunctions, while having a long history in quantum chemistry, are increasingly being used in highly accurate quantum Monte Carlo calculations. Since the accuracy of QMC is ultimately limited by the quality of the trial wavefunction, multi-Slater determinants wavefunctions offer an attractive alternative to Slater-Jastrow and more sophisticated wavefunction ansatz for several reasons. They can be efficiently calculated, straightforwardly optimized, and systematically improved by increasing the number of included determinants. In spite of their potential, however, the convergence properties of multi-Slater determinant wavefunctions with respect to orbital set choice and excited determinant selection are poorly understood, which hinders the application of these wavefunctions to large systems and solids. In this paper, by performing QMC calculations on the equilibrium and stretched carbon dimer, we find that convergence of the recovered correlation energy with respect to number of determinants can depend quite strongly on basis set and determinant selection methods, especially where there is strong correlation. We demonstrate that properly chosen orbital sets and determinant selection techniques from quantum chemistry methods can dramatically reduce the required number of determinants (and thus the computational cost) to reach a given accuracy, which we argue shows clear need for an automatic QMC-only method for selecting determinants and generating optimal orbital sets.