Explicitly correlated wave function

A method that avoids making the HF mistakes in the first place is called quantum Monte Carlo (QMC). There are several types of QMC variational, dilfusion, and Greens function Monte Carlo calculations. These methods work with an explicitly correlated wave function. This is a wave function that has a function of the electron-electron distance (a generalization of the original work by Hylleraas). 

Explicitly correlated wave functions have been shown to give very accurate results. Unfortunately, these calculations are only tractable for very small molecules. 

With an appropriate /(r12) function, e.g., in the original linear form f(r-[2) — C12, the operator product r firu) is no longer singular. Such cancellation is not possible with Slater determinants alone and this is what allows explicitly correlated wave functions to achieve accurate correlation energies with relatively small basis sets. With the single explicitly correlated term, therefore, we effectively include a linear combination of an infinite set of Slater determinants, but without the need to solve an infinite set of equations to determine the corresponding amplitudes. The R12 method constructs wave functions that are more compact and computationally tractable than naive Slater-determinant-based counterparts. 

Rychlewski, J. (ed.), Explicitly Correlated Wave Functions in Chemistry and Physics, Kluwer Academic Publishers, Dordrecht, 2003. 

This paper presents a brief review of the use of explicitly correlated wave functions in molecular quantum chemistry computations. This review is restricted mainly to the direct variational approaches. Special attention will be given to two-electron molecular systems. The possible direction of extending the use of the correlated wave function for three- and four-electron molecular systems as well as the accuracy of the results will be discussed. 

The simplest molecular system exhibiting effects of electron correlation is the hydrogen molecule. For this molecule the explicitly correlated wave function has been applied in the early days of quantum mechanics (James and Coolidge, 1933), It was later generalised by Kolos and Wolniewicz (Kcrfos and Wolniewicz, 1965) and successfully used to solve variety of problems in the ground and excited states of the hydrogen molecule. This wave function, called there the Kolos-Wolniewicz function (Kolos and Wolniewicz, 1965) is assumed in the form of an expansion ... 

Explicitly correlated wave functions described above have been specifically designed for two-electron molecular systems. As it was demonstrated in the previous section these functions give the energies which appear to be superior to the variational energies reported. Therefore several attempts have been made to extend this approach to many-electron molecules. The James-Coolidge (JC) type of function has been extended to three- and four-electron diatomic molecules by Clary and... 

Table 5 gives the errors for a DFT method, four hybrid DFT methods, and AMI. Although hybrid DFT is very affordable, it lacks the accuracy of multicoefficient semi-empirical methods based on explicitly correlated wave functions. Nevertheless the mPWlPW91/MG3S and AMI methods have performance/cost characteristics that put them near the envelope of best performance in Figure 1. AMI is valuable for larger systems where the other methods in the figure are not affordable. 

DPT has been applied to two kinds of explicitly correlated wave functions for the He-isoelectronic series [19, 10], the H2 molecule [10], and the H3-ion [96]. These calculations give some information on the coupling between relativistic and correlation effects. Only the leading relativistic corrections of 0(c ) were considered. For higher orders in 0(c ) problems arise if one wants to account for the correct behavior near ri2 = 0 (see section... 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

VARIATIONAL METHODS USING EXPLICITLY CORRELATED WAVE FUNCTION ( ) p. 584... 

The explicitly correlated wave function fwhich we wiU explain in a moment j has the inter-electronic distance built in its mathematical form. We may compare this to making the electrons wear spectacles. Now they avoid each other. One of my students said that it would be the best if the eleetrons moved apart to infinity. Well, they cannot. They are attracted by the nucleus... 

VARIATIONAL METHODS USING EXPLICITLY CORRELATED WAVE FUNCTION... 

Variational Methods Using Explicitly Correlated Wave Function... 

Key words Helium atom - Electron correlation -Explicitly correlated wave functions - Hylleraas expansion... 

Already in 1929 Hylleraas found that the orbital expansion of the wave function of helium converges extremely slowly. The problem could be overcome by including terms in the wave function that depend explicitly on interelectronic coordinates [6, 7]. The proposed explicitly correlated wave function was of the form... 

Rychlewski J (2003) Explicitly correlated wave functions in chemistry and physics — theory and applications. Kluwer Academic, Dordrecht... 

Marchette O, Werner H-J (2008) Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions. Phys Chem Chem Phys 10 3400-3409... 

---