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 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.
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.
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%.
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.
An appropriate iterative scheme for the minimization of the energy, based on the variational Monte Carlo (VMC) technique, is introduced and compared with existing stochastic schemes. We test the various methods for the one-dimensional Heisenberg ring and the two-dimensional $t\text{\ensuremath{-}}J$ model and show that, with the present scheme, very accurate and efficient calculations are possible, even for several variational parameters. Indeed, by using a very efficient statistical evaluation of the first and the second energy derivatives, it is possible to define a very rapidly converging iterative scheme that, within VMC, is much more convenient than the standard Newton method. It is also shown how to optimize simultaneously both the Jastrow and the determinantal part of the wave function.
Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.
Abstract It is difficult to imagine that density functional theory (DFT) could be useful in improving quantum Monte Carlo methods, but it is easy to imagine that QMC could be useful in improving density functional methods. Since QMC can produce accurate wavefunctions and since wavefunctions lead to all other properties one should be able, in principle, to use QMC to predict the exchangecorrelation energy functional Exc[n(r)], the key (unknown) quantity required in DFT. In practice, that is not so easy, but this paper shows how QMC can take steps in that direction by providing insights gained from comparisons of results from QMC and DFT. The authors considered the case of the strongly inhomogeneous electron gas using variational Q:tvIC to obtain wavefunctions 1/J>. and energies E[7/J>.] for the ground state of the Hamiltonian iI>- = T +AVee+V >. associated with the coupling constant A in the range 0 to 1. Optimization of a Slater-Jastrow many-body wavefunction was made by minimizing a penalty function given by a weighted sum of the variance in local energies and a sum of squares of electron density differences. The variational parameters in 7/J>. and v>. were repeatedly varied to reach convergence followed by additional calculations to obtain the local values of the exchange-correlation energies e c and the exchange-correlation holes n c from which Exe and nxc could be determined.