Quantum Monte Carlo method precision

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. 

The answer to this question as well as the question of the precise meaning of the term ab initio itself in the context of quantum chemistry seems to differ considerably according to the particular researcher that one might consult.3 Some authors I have questioned claim that the two terms are used interchangeably to mean calculations performed without recourse to any experimental measurement. This would include Hartree-Fock, and many of the DFT functionals, along with quantum Monte Carlo and Cl methods. 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

The fits of Janev et al. [12] stem from a compilation of the results obtained with different theoretical approaches (i) semi-classical close-coupling methods with a development of the wave function on atomic orbitals (Fritsch and Lin [16]), molecular orbitals (Green et al [17]), or both (Kimura and Lin [18], (ii) pure classical model - i.e. the Classical Trajectory Monte Carlo method (Olson and Schultz [19]) - and (iii) perturbative quantum approach (Belkic et al. [20]). In order to get precise fits, theoretical results accuracy was estimated according to many criteria, most important being the domain of validity of each technique. 

In contrast, the first class of applications can require very precise solutions. Increasingly, computers are being used to solve very well defined but difficult mathematical problems. For example, as Dirac [1] observed in 1929, the physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known and it is only necessary to find precise methods for solving the equations for complex systems. In the intervening years fast computers and new computational methods have come into existence. In quantum chemistry, physical properties must be calculated to chemical accuracy (say, 0.001 Rydberg) to be relevant to physical properties. This often requires a relative accuracy of 10s or better. Monte Carlo methods are used to solve the electronic... 

In this short review we have pointed out only very few of the basic issues involving the simulation of chemical systems with Quantum Monte Carlo. What has been achieved in the last few years is remarkable very precise calculations of small molecules, the most accurate calculations of the electron gas, silicon and carbon clusters, solids, and simulations of hydrogen at temperatures when bonds are forming. New methods have been developed as well high-accuracy trial wavefunctions for atoms, molecules, and solids, treatment of atomic cores, and the generalization of path-integral Monte Carlo to treat many-electron systems at positive temperatures. 

There are still few investigations of PAH systems with wave function methods although some of them have been reported in this review. One could then expect that the computing facilities and original implementation of these methods will continue to provide very quantitatively precise data in the next few years. Less standard methods like Quantum Monte Carlo [208] could also be used as high level reference calculations. 

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