Hylleraas function methods

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. 

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 same procedure can be applied to find approximations to the second-order energy E2 of Section 4.2 of Chapter 4 in the context of the Hylleraas variational method (Magnasco, 2007, 2009a), as we shall illustrate in the simple case of two functions. We start from a convenient set of basis functions X written as the (1x2) row vector ... 

In the Hylleraas-CI method. the Hylleraas idea has been exploited when designing a method for larger systems. The electronic wave function is proposed as a linear combination of Slater determinants, and in front of each determinant , (1,2,3,..., N), we insert, next to the variational coefficient c,-, correlational factors with some powers (v, m ) of the interelectronic... 

We obtain the same equation, if in the Hylleraas functional eq. (5.28), the variational function x s expanded as a linear combination (5.29), and then vary d,- in a similar way to that of the Ritz variational method described on p. 202. 

The family of variational methods with explicitly correlated functions includes the Hyller-aas method, the Hylleraas Cl method, the James-Coolidge and the Kolos-Wolniewicz approaches, and the method with exponentially correlated Gaussians. The method of explicitly correlated functions is very successful for 2-, 3- and 4-electron systems. For larger systems, due to the excessive number of complicated integrals, variational calculations are not yet feasible. 

A hierarchy of functionals exists from which the Rayleigh-Schrodinger eneigies of even orders and wave functions of general orders may be obtained by a variational procedure. In the Hylleraas variation method, the energy of order 2n is calculated by a minimization of the Hylleraas functional of order 2n [3] ... 

At the minimum of the Hylleraas functional (to be constructed below), the variational parameters determine the wave function to order n. In some situations, the Hylleraas method represents a useful alternative to standard perturbation theory, as we shall see in the present subsection. 

In some cases, it is better to calculate by the Hylleraas variation method than from the standard expression of perturbation theory. Assume that, instead of the exact wave function we have at our disposal only an approximation ... 

The Hylleraas variation method has the advantage that we can calculate the wave-function corrections from variations in a nonlinear set of parameters that define the electronic state. We are thus not restricted to a linear variational space. As a bonus, the error in <2 ) jg quadratic in the error in the nth-order wave function. The lower-order corrections (C with k < n) must, however, be accurately calculated. 

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). 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

The three basic methods introduced by Hylleraas in his work on the He series have in modern terminology obtained the following names (a) Superposition of configurations (b) Correlated wave functions (c) Different orbitals for different spins. The first two approaches are developed almost to the full extent, whereas the last method is at least sketched in the 1929 paper. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

Unfortunately, extending Hylleraas s approach to systems containing three or more electrons leads to very cumbersome mathematics. More practical approaches, known as explicitly correlated methods, are classified into two categories. The first group of approaches uses Boys Gaussian-type geminal (GTG) functions with the explicit dependence on the interelectronic coordinate built into the exponent [95]... 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

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