Quantum Monte Carlo method correlation energy

We also note that there are other, more advanced techniques for the treatment of electron correlation such as the r 2 approaches [101], geminal techniques [101], or quantum Monte Carlo methods [102]. Although these approaches are undeniably very powerful treatments for obtaining accurate ground-state energies, they are still awaiting implementations of analytic derivatives and therefore are outside the scope of this chapter. 

The analytical form for the correlation energy of a uniform electron gas, which is purely dynamical correlation, has been derived in the high and low density limits. For intermediate densities, the correlation energy has been determined to a high precision by quantum Monte Carlo methods (Section 4.16). In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula, and such formulas have been constructed by Vosko, Wilk and Nusair (VWN) and by Perdew and Wang (PW), and are considered to be accurate fits. The VWN parameterization is given in eq. (6.36), where a slightly different spin-polarization function has been used. 

Alternative strategies have also been proposed for estimating correlation energies, including quantum Monte Carlo methods (see reference 67 and references therein), MP2 schemes, either canonical " " or based on the Laplace transform algorithm, and the molecular-like incremental method applied by Stoll.However, none of these methods seems to have arrived at a sufficiently advanced stage of development to be of general use to the scientist at the moment. 

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. 

The cornerstone of the field (the "Hartree-Fock" of Density Functional Theory) is the Local Density Approximation (LDA) also called the Local (Spin) Density (LSD) method Here the basic information on electron correlation, how electrons avoid each other, is taken from the uniform density electron gas Essentially exact calculations exist for this system (the Quantum Monte Carlo work of Ceperley and Alder) and this information from the homogeneous model is folded into the inhomogeneous case through the energy integral ... 

Many theoretical determinations have been proposed for small clusters. In Monte Carlo calculations, we showed the crucial role of the three-body interactions to describe clusters with N=2 and 3 (59hj). For ab initio quantum chemistry calculations, it is seen that small basis sets systematically overestimate the clustering energies (9, 52). In this case, the Zero-Point Energy correction is large (9), and seems less important with larger basis sets (63b). The role of the correlation contribution is not clear and seems to depend on the clusters considered (52, 63b, 69). The other determinations, from semi-empirical quantum chemical methods (8, 55) or analytical models (22, 50, 67a) are of variable accuracy. 

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