Jastrow correlation factor

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." ... 

The nodal surface is independent of the Jastrow correlation factor which therefore determines only the efficiency of the calculation, not the accuracy. High accuracy is already obtained for many systems with one Slater determinant, often built from Kohn-Sham orbitals. For many years, multideterminant wave functions have been employed which are necessary for high accuracy in systems with non-dynamical correlation. Recent examples are papers by Caffarel and coworkers where large complete active spaces have been used." " These authors emphasize the importance of systematic cancellation of the node location error which can be achieved with CASSCF-type wave functions. While full CASSCF wave... 

In Eq. (54) there appears a new Hamiltonian containing the additional Hamiltonians HM and H 2 The latter contain two- and three-particle operators, respectively. Although the new Hamiltonian operates to the right over the single Slater determinant we have on the left the presence of the correlation term C2. For a Jastrow-type correlating factor C = n (l + 4>x ) the energy expression gives rise to a cluster expansion... 

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 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... 

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. 

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. 

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. 

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. 

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). 

---