Hylleraas variational method

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 ... 

Variational approximations to the second-order energy E2 are obtained using the Hylleraas variational method outlined in Section 1.3 of Chapter 1. 

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] ... 

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. 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

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. 

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. 

Hylleraas variational principle (p. 246) perturbation (p. 240) perturhational method (p. 240) perturbed system (p. 240)... 

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... 

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

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. 

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). 

The third entry refers to the self-consistent field method, developed by Hartree. Even for the best possible choice of one-electron functions V (r), there remains a considerable error. This is due to failure to include the variable rn in the wave-function. The effect is known as electron correlation. The fourth entry, containing a simple correction for correlation, gives a considerable improvement. Hylleraas in 1929 extended this approach with a variational function of the form... 

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