Jastrow correlation factor for atoms, molecules, and solids

Many-electron wave functions may be accurately and compactly approximated by a product of a small number of Slater determinants and a positive Jastrow correlation factor. The Jastrow factor is an explicit function of the electronelectron separations, so that expectation values calculated with a Slater-Jastrow wave function do not separate in the electron coordinates. Nevertheless, the variational and diffusion quantum Monte Carlo (VMC and DMC) methods permit the use of such explicitly correlated wave functions. In VMC, expectation values are calculated using an approximate trial wave function, the integrals being performed by a Monte Carlo method. In DMC (Refs. 1 and 2) the imaginary-time Schrödinger equation is used to evolve an ensemble of electronic configurations towards the ground state. The fermionic symmetry is maintained by the fixednode approximation,3 in which the nodal surface of the wave function is constrained to equal that of a trial wave function. The DMC method gives the energy that would be obtained in a VMC calculation with the same Slater determinants, but using the best possible Jastrow factor. Although the DMC energy is in principle independent of the Jastrow factor, a poor trial wave function increases the statistical error bars and the time-step and population-control biases. When nonlocal pseudopotentials are used within DMC, the locality approximation4,5 leads to additional errors which are second order in the error in the trial wave function.6 The expectation values of operators that do not commute with the Hamiltonian are often evaluated using extrapolated estimation,2 the accuracy of the extrapolation depending on the quality of the trial wave function. In practice the efficiency and accuracy of both VMC and DMC calculations are critically dependent on the availability of highquality Jastrow factors. Our Jastrow factor is designed for use in atoms, molecules, and solids. We have used it in a variety of systems, and here we report results on the He, Ne8+, Ne, and Ni atoms, the NiO and SiH4 molecules, and crystalline Si in the diamond structure. These systems include all-electron and pseudopotential descriptions of atoms, with the total number of electrons varying from 2 to 216. We pay particular attention to the issue of cutting off terms in the Jastrow factor at finite ranges, which is desirable because of the local nature of the inhomogeneous correlations in many systems, as well as for reasons of computational efficiency in large systems. We obtained the values of the free parameters in our Jastrow factors by minimizing the variance of the energy.7,8 All of our QMC calculations were performed using the CASINO package.9 We use Hartree atomic units "= ue u =me=4pe0=1 throughout this article. The rest of this paper is organized as follows. In Sec. II we describe the general form of our Jastrow factor, while in Sec. III we show how the electron-electron and electronnucleus cusp conditions10 apply to this form. The behavior of the local energy at electron-electron and electron-nucleus coalescence points is discussed in Sec. IV. Section V describes the Jastrow factor in detail. In Sec. VI we make further comments on the form of our Jastrow factor, while in Sec. VII we define our notation for specifying the Jastrow factor and give our criterion for judging its quality. In Secs. VIII–XI we report the results of studies of various systems. Finally, we draw our conclusions in Sec. XII.