Pair product trial function

The unknown functions, Cij r),riij r) in (57) and (58) need to be parameterized in some way. In a first attempt we have chosen gaussians with variance and amplitude as new variational parameters [16]. This form was shown to be suitable for homogeneous electron gas [13]. Approximate analytical forms for ij r) and r]ij r), as well as for the two-body pseudopotential, have been obtained later in the framework of the Bohm-Pines collective coordinates approach [14]. This form is particularly suitable for the CEIMC because there are no parameters to be optimized. This trial function is faster than the pair product trial function with the LDA orbitals, has no problems when protons move around and its nodal structure has the same quality as the corresponding one for the LDA Slater determinant [14]. We have extensively used this form of the trial wave function for CEIMC calculations of metallic atomic hydrogen. 

The pair product trial wave function is the simplest extension of the Slater determinant of single particle orbitals used in mean field treatment of electronic systems (HF or DFT). This is also the ubiquitous form for trial functions in VMC... 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

Over the years there have been important progress in finding trial functions substantially more accurate then the pair product form for homogeneous systems [12,13]. Within the generalized Feynman-Kac formalism, it is possible to systematically improve a given trial function [13,14]. The first corrections to the pair product action with plane wave orbitals are a three-body correlation term which modifies the correlation part of the trial function (Jastrow) and a backfiow transformation which changes the orbitals and therefore the nodal structure (or the phase) of the trial function [14]. The new trial function has the form... 

Currently, the ubiquitous choice for the trial function is of the Slater-Jastrow or pair-product form. It is a linear combination of spin-up and spin-down determinants of one-body orbitals multiplied by a correlation factor represented by an exponential of one-body, two-body, and so on, terms [16, 30] ... 

The typical trial wavefunction for QMC calculations on molecular systems consists of the product of a Slater determinant multiplied by a second function, which accounts to some extent for electron correlation with use of interelectron distances. The trial wavefunctions are most often taken from relatively simple analytic variational calculations, in most cases from calculations at the SCF level. Thus, for the 10-electron system methane," the trial function may be the product of the SCF function, which is a 10 x 10 determinant made up of two 5x5 determinants, and a Jastrow function for each pair of electrons. 

Though not discussed above, in all the studies mentioned the trial wavef unctions included pair correlation functions. J j. as prescribed by Reynolds et al. ( ). Moskowitz et al. (48.49) have shown that the product of a relatively simple multiconfiguration wavefunction with pair correlation functions can provide a rather accurate approximation to the exact wavefunction. In our calculations and in those of Hammond et al. (59) the many-electron local potential, has been obtained by allowing the REP to... 

For certain choices of trial wave functions further simplifications are possible, and this has been exploited in an interesting way in a series of variational calculations on the ground states of He and He, begun by McMillan and continued by the Orsay group. The model is again that of N particles with pair interactions, confined to a box, and with periodic boundary conditions to mimic an infinite system. The wave function has to be able to prevent strong overlaps of the particles, and a popular form for the boson case is therefore simply a product of pair functions. 

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