Hylleraas function

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Equation (9) can be solved variationally by minimizing the Hylleraas functionals [22,23]... 

The unknown amplitudes (f-amplitudes) in the T operator are determined by minimization of the Hylleraas functional [40] ... 

There are many ways to improve this independent-particle model by incorporating electron correlation in the spatial part Hylleraas function [Hyl29] and the method of configuration interaction (Cl) will be used as illustrations. 

The Hylleraas function, with its improved properties as compared to a Hartree-Fock function, is called a correlated wavefunction, because it takes into account the mutual electron-electron interaction much better, and the motion of electrons beyond a mean-field average is termed correlated motion or the effect of electron correlations. (The definition of electron correlation is used here in the strict terminology. The mean-field average of electron-electron interactions is frequently also called electron correlation.) Comparing equ. (1.20) with equ. (1.16b) one has... 

These preliminary, but important clarifications, lead us to a discussion of the correlated motion within the Cl picture for both electrons in the ground state of helium. For this purpose it is illustrative to analyse the three-parameter Hylleraas function, the correlation properties of which have previously been described, in terms of Cl functions. Looking only for the individual components of orbital angular momenta r = i2 which couple to the desired Se state, one gets [GMM53]... 

As implied by the name, a correlated wavefunction takes into account at least some essential parts of the correlated motion between the electrons which results from their mutual Coulomb interaction. As analysed in Section 1.1.2 for the simplest correlated wavefunction, the helium ground-state function, this correlation imposes a certain spatial structure on the correlated function. In the discussion given there, two correlated functions were selected a three-parameter Hylleraas function, and a simple Cl function. In this section, these two functions will be represented in slightly different forms in order to make their similarities and differences more transparent. 

Due to its importance, the Hylleraas wavefunction introduced in equ. (1.20) will also be discussed in the context of Cl. For simplicity, only the three-parameter Hylleraas function is chosen. The essential quantity, responsible for the correlated motion of the electrons, in this function is the distance rl2 between the electrons. Usually, rl2 is expressed as... 

If this expansion is applied to the three-parameter Hylleraas function, one obtains [GMM53]... 

The F rur2) are rather cumbersome, but completely known, functions which depend on r>, r<, and f (compare the related case of the functions f/ru r2) defined in equ. (7.109b)). They are normalized in such a way that the coefficients c( 2 describe the weight given to the individual (f-components. For the three-parameter Hylleraas function, one has [GMM53] ... 

This result states that the angular momenta ( in the expansion of the Hylleraas function impose the following symmetry properties on the individual angular momenta and (2 attached to the two electrons with spatial directions and r2, respectively ... 

In Paper I of this series [5], the extremal pair functions for the systems He2, Ne, F, HF, H20, NH3 and CH4 were analyzed. We now follow a different line of thought that was also opened in Paper I, namely to use extremal pairs for the construction of correlated wave functions. We have already pointed out that there is a special set of extremal pair functions associated with MP2 (Moller-Plesset perturbation theory of second order). In fact we have shown, that there are two choices for which the Hylleraas functional of MP2 decomposes exactly into a sum of pair contributions. One choice is the conventional one of pairs of canonical spin orbitals, the other one the use of first-order pairs with extremal norm... 

Let us insert (47) into the Hylleraas functional (19) (with H<, the two-electron Fock operator, g = H — H0, and 4>(0) the Hartree-Fock wave function)... 

The ground state energy values compare very well with those obtained by Aquino et al. through a 40-term expansion of generalized Hylleraas functions (see Tables 10 and 11). 

Table 2 Correlation energy (CE) estimated as the difference between the wave function expanded with 40 Hylleraas functions (H-WF) and the HF wave function obtained with optimized exponents, for the lowest singlet state of confined helium atom. All quantities are in hartrees...

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