Hylleraas methods

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

In this section a practical point of view is emphasized. This means that we disregard methods which, in our opinion, are not perspective for one reason or another. The drawback of the Hylleraas method is its applicability only to the two-electron problem. We also omit here the methods based on the concept of separated electron pairs (geminals)54-57 because these methods are inherently incapable of accounting for the interpair correlation energye We shall also not discuss here the Bethe-Goldstone equations since, from the calculations reported7,58 for BH and H20, it appears that they are computationally not suitable for chemical applications. 

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 first application of the Hylleraas method to the H2 molecule by James and Coolidge appeared in the first volume of J. Chem. Phys. [44]. The accuracy was later pushed further by Kolos and Wolniewicz, with whose names the theory of the H2 molecule is intimately linked [45, 46]. The first triumph of this group was a slight disagreement with the experimental dissociation energy of Herzberg et al. [47], which made Herzberg reconsider his analysis of the spectra, and confirm the theoretical prediction. The most recent calculations are really spectacular [48, 49]. However, also for H2, Rychlewski et al. [36]... 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

Yan and Drake [19] proposed the use of these funetions in combinations yielding eigenfunctions of the total angular momentum of quantum numbers L, M, presenting their analyis in a form appropriate to the calculations by the Hylleraas method. 

For the 5 states of three-body systems, the Hylleraas-method correlation factor rj2 leads to integrals that can be easily evaluated in r,y coordinates without the need to introduce an expansion. A recent paper by Drake et al. [61 ]... 

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. 

Notice that O5 and Og are two-electron functions, which cannot be factorized into one-electron functions. By calculating all matrix elements and solving the 6 x 6 eigenvalue problem, Hylleraas, in 1928, obtained, without comparison, the best description of the helium atom with the energy -2.903329 H, compared to the earlier best value of -2.86 H. With the help of modern computers, it was recently possible to determine the ground state energy with more than accurate 20 decimal places (-2.903724 H) using essentially the Hylleraas method. 

The Hylleraas method gives a very good description of the electronic states, and is the only method that reproduces the cusp for 2 5. The model needs great computer resources and can only be used for small molecules and atoms. Werner Kutzelnigg, the German quantum chemist, has carried out particularly accurate and important work. 

Recently, impressive calculations using Hylleraas wave functions have been done for the H2 molecule by the Hylleraas method [44,63], the Iterative Complement Iteration (ICI) [36], and Explicitly Correlated Gaussian (ECG) [12] methods. Few molecules have yet been calculated using Hylleraas-type wave functions HeH+ and some other species [72] using the Hylleraas method, the helium dimer He2 interaction energy [46] and the ground state of the BH molecule [7], both using the ECG method. 

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. 

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