Jastrow factor

Of course the cusp can be represented by including the interparticle distances in a trial wave functions, most simply by means of Jastrow factors, exponentials of the inter-particle distances. But the problems of integral evaluation with such fac-... 

The function T is not provided by the QMC method. It need not be highly accurate in all regions in space, but it must have accurate nodal surfaces in 3n-dimensional space, and it must be accurate near singularities of V. In practice, simple SCF wave functions provide sufficiently accurate nodal surfaces, and symmetrical Jastrow factors [ll] that serve to satisfy... 

Now let us look at the paper. Eqn. (1) gives the form of the transcorrelated wave function C0, where C = li >jfiri,rj) is a Jastrow factor, and is a determinant. This compact wave function includes the effects of electron correlation through the introduction of r in C. The form C0 is taken as the trial wave function in quantum Monte Carlo (QMC) molecular computations today. Indeed the explicit form for/(r r,j is most often used by the QMC community. The transcorrelated wave function was obtained by solving (C //C - W) = 0, which Boys called the transcorrelation wave equation. Because C //C is a non-Hermitian operator, it was important to devise independent assessments of the accuracy of the wavefunction C0. 

For metallic hydrogen we have described a parameter-free trial function which does not need optimization. However, if we use the pair proton action both for molecular or LDA orbitals, we are left with free parameters in the Jastrow factor and with the width of the gaussians for molecular orbitals. Optimization of the parameters in a trial function is crucial for the success of VMC. Bad upper bounds do not give much physical information. Good trial functions will be needed in the Projector Monte Carlo method. First, we must decide on what to optimize and then how to perform the optimization. There are several possibilities for the quantity to optimize and depending on the physical system, one or other of the criteria may be best. 

A suitable trial wave function can be constructed from a Hartree-Fock wave function multiplied with a suitable correlation function, often taken as a Jastrow factor 7(r). 

Although the product form of a SJ wave function has typically been used with a single global Jastrow factor, recently [111] the benefit of separate Jastrow terms for each molecular orbital of the antisymmetric function was demonstrated. The latter approach improves treatment of local electron correlation by facilitating adjustment to the local molecular environment. Also, the nodal structure of the trial wave function can better reflect the parameters of multiple Jastrow functions than the single global Slater-Jastrow wave function. This modification notably improves the nodal structure of trial wave functions and FN-DMC energies. 

Wave function Single or Multi-determinant Jastrow-Slater wave function and their linear combinations are available. Jastrow factors of Schmidt-Moskowitz type are available. 

He atoms within a van der Waals cluster must be treated as Fermions, while He atoms in a cluster are Bosons. Rather than the Hartree ansatz of a product of single-mode wavefunctions (as in Eq. (2)), one should employ a determinantal Hartree-Fock ansatz in the case of He atoms. The Hartree-Fock equations for the single-particle orbtials include exchange terms, as in the case of elecfrons in atoms. Other atoms in the cluster should be treated on a Hartree footing. For He clusters, a symmetrical ansatz for the wavefunction should be used, as required for Boson calculations. All these considerations do not affect the Jastrow factor. A formulation of the HF-SCF and Jastrow/HF-SCF for clusters such as Xe( He) Xe( He) will be given in Ref. 18. Calculations we carried out for Xe( He)2, Xe(" He)2 will be briefly discussed in the next section. 

The Slater determinants are often taken from HF, DFT or MC-SCF/ CASSCF calculations. The Jastrow correlation function is often parametrized in the Schmidt-Moskowitz form where U is expanded in powers of the scaled distances r, and fy,f = r/ + ar). This expansion goes back to Boys and Handy. Other expansions such as Pade-type or expansions in terms of unsealed distances with cutoff parameters have been used successfully. The Jastrow factor does not have to be rotationally invariant. Riley and Anderson demonstrated that a directional Jastrow term can improve the VMC energy of LiH considerably. The parameter vector p of U x p) can be determined efficiently by minimizing the variance of the local energy in the form... 

After optimization of the Jastrow correlation factor the orbitals which have been taken from HF or KS-DFT calculations may no longer be optimal. Forty years ago. Boys and Handy developed the transcorrelated method, a SCF method that determines variationally optimal orbitals like HF-SCF, but with a simple correlation factor present. Umezawa and Tsuneyuki have adapted the transcorrelated method to VMC to obtain variationally optimal orbitals in the presence of a Jastrow factor." This method was applied to first row atoms in a series of papers. VMC energies were greatly improved when using the transcorrelated orbitals instead of HF or B3LYP orbitals, but, unfortunately, the DMC energies could not be improved." ... 

Similarly, Riley and Anderson optimized Cl coefficients directly in the presence of a preoptimized Jastrow factor by solving the generalized eigenvalue problem based on matrix elements of the H and S matrices that were calculated by Monte Carlo integration. The method was applied to the Be atom and required a considerably larger sample size than a variance minimization method. 

Sorella and coworkers devised a similar VMC energy optimization method that is based on their stochastic reconfiguration approach originally developed for lattice systems. This method has been extended successfully to atoms and molecules by Casula and Sorella. They optimized the parameters of AGP wave functions with Jastrow factors for atoms up to phosphorus and molecules such as Li2 and benzene with resonating valence bond wave functions. ... 

A second orbital-dependent expression, originally introduced for use with the Hartree-Fock scheme, is the Colle-Salvetti (CS) correlation functional [23]. The starting point for the derivation of the CS functional is an approximation for the correlated wavefunction F(ricri,... rjvajv). The ansatz for. .. r aA consists of a product of the HF Slater determinant and Jastrow factors,... 

The Cu (001) surface is exposed. This truncation of the bulk lattice, as well as adsorption, leads to drastic changes in electronic correlation. They are not adequately taken into account by density-functional theory (DFT). A method is required that gives almost all the electronic correlation. The ideal choice is the quantum Monte Carlo (QMC) approach. In variational quantum Monte Carlo (VMC) correlation is taken into account by using a trial many-electron wave function that is an explicit function of inter-particle distances. Free parameters in the trial wave function are optimised by minimising the energy expectation value in accordanee with the variational principle. The trial wave functions that used in this work are of Slater-Jastrow form, consisting of Slater determinants of orbitals taken from Hartree-Fock or DFT codes, multiplied by a so-called Jastrow factor that includes electron pair and three-body (two-electron and nucleus) terms. 

A second, more accurate step, takes the optimised Jastrow factor as data and carries out a diffusion quantum Monte Carlo (DMC) calculation, based on transforming the time-dependent Schrodinger equation to a diffusion problem in imaginary time. An ensemble in configuration space is propagated to obtain a highly accurate ground state. 

Electron correlation is introduced via a Jastrow factor which can be optimised by Variational Monte Carlo methods. 

This optimisation procedure generates a correlation.data file containing the optimised numerical parameters for the electron-electron and electron pair-nuclear contributions to the Jastrow factor. 

In these calculations, our trial wave function will be of Slater-Jastrow form. The Slater determinants will contain orbitals taken from density functional theory (DFT) calculations. The Jastrow factor is an explicit function of electron-electron distance, enabling a highly accurate and compact description of electron correlation. The Jastrow factor consists of polynomial expansions in electron-electron separation, electron-nucleus separation, in which the polynomial expansion coefficients are optimisable parameters [21]. These parameters were determined by minimising the VMC energy. 

RQMC and diffusion Monte Carlo were employed to calculate the potential energy curve of helium dimer from a single-determinantal wave function with a large basis set (19s9p8d/8s7p6d) and a three-body-Jastrow factor [36]. This van der Waals system presents well-known challenges both to experimentalists... 

The simplest QMC calculations of excited states have been performed without reoptimizing the determinantal part of the wave function in the presence of the Jastrow factor. It has recently become possible to optimize in VMC both the Jastrow and determinantal parameters for excited states, either in a state-specific or a state-average approach [6,7,9,10,12,14,15]. Although this leads to very reliable excitation energies, reoptimization of the orbitals in VMC can be too costly for large systems. 

The excitation energies obtained from the CAS(6,5) wave functions depend very little on whether (a) they are calculated in MCSCF, VMC or DMC, (b) the state-average or the state-specific approach is employed, and (c) fhe CSF and orbital coefficients are reoptimized or not in the presence of the Jastrow factor. In contrast, the excitation energies obtained from CAS(2,2) wave functions do depend on all of the above and, in particular the reoptimization of the CSF and orbital coefficients in the presence of the Jastrow factor significantly improves the VMC and DMC excitation energies, to 3.80(2) and 3.83(l)eV, respectively. The importance of reoptimizing in VMC the CAS(2,2) expansions but not the CAS(6,5) expansions suggests that the Jastrow factor includes important correlation effects that are present in CAS(6,5) but not in CAS(2,2). 

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