Hylleraas expansion

Unlike the r 2 singularity, the 2s — 2p "degeneracy" is handled well by the configuration interaction method and poorly by the Hylleraas method. Perhaps the two-configuration ls 2s — ls 2p function would be a good starting point for a Hylleraas expansion. 

The Hylleraas expansion is well known to be an efficient tool for obtaining accurate results for two-electron systems. The complex-rotation transformation does not effect the angular properties of this expansion Its capability of representing the angular electron correlation effects to the infinite order is preserved. Computations for helium-like atoms employing the Hylleraas-type expansion have been performed for two decades and have given numerous accurate results (6,98-111). 

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

The Hylleraas expansion for He-like atoms was extended by Kinoshita who introduced negative powers,... 

Pekeris succeeded to employ the Hylleraas expansion on a large scale by using Laguerre-polynomials, and nanohartree accuracy was obtained with a wave function consisting of 1078 terms. It is quite remarkable, however, that this accuracy was obtained also by C. Schwartz from a 189-term Hylleraas expansion of the form... 

Table 2 He Atom 5 Ground State Energy (in h) from the Hylleraas Expansion (equation 12)...

Hylleraas expansion with /, -i- 2m, - - , < N m- Number of terms in the Hylleraas expansion. Optimized exponent. 

Although the Hylleraas expansion seems to yield highly accurate approximate wave functions for He-like atoms, it was found that logarithmic terms are also required. Morgan proved that the expansion by Fock, ... 

Fig. 1 Illustration of the slow convergence of orbital expansions for He in the vicinity of the cusp. The figure shows a normalized value of the wavefunction ri, ri2) for ri = luo- All values were normalized to the value r, 0) = 0.032406 of the Hylleraas expansion, which shows the correct cusp behavior. The plotted wavefunctions are (from top to bottom as seen near the origin) (1) a 56-term Cl expansion (2) a 286-term Cl expansion (3) an 816-term Cl expansion (4) a 1540-term Cl expansion and (5) a 346-term Hylleraas expansion. Reprinted with permission from Ref 10. Copyright 2012 American Chemical Society.

Having examined the behaviour of the exact wave function in the ground-state helium atom, let us now consider the description of this atom by approximate electronic wave functions. We begin by examining the standard expansion of Cl theory and then go on to investigate how this model may be amended to allow for a better description of the short-range electronic interactions. We conclude this section with a discussion of the Hylleraas expansion. 

A great deal of attention has been paid to the question of the necessity of having logarithmic terms in the expansion (Gronwall 1937, Bartlett 1952, 1955, Fock 1954, Hylleraas 1955), but Kino-shita has pointed out that although such terms may be convenient from the. numerical point of view (Hylleraas and Midtdal 1956), they are not necessarily required by the form of the Schrodinger equation itself. 

We note that the power series expansion III. 119is a direct generalization of the Hylleraas form III. 114 to which it should go over in the limiting case Rab = 0. James and Coolidge obtained a value of the electronic energy, —1.17347 at.u., in excellent agreement with the experimental results available, and their work forms even today the best basis for our understanding of the electronic structure of the chemical bond. 

E. A. Hylleraas, Z Phys. 65 (1930), 209 note 6, p. 279. Note that if(2) can alternatively be expressed as an infinite expansion in the unperturbed eigenfunctions but the Hylleraas variation-perturbation expression (1.5d) is generally more useful for practical numerical applications. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

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

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