Hylleraas equation

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. 

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. 

But Equation 11.9 also requires finding (1 /w2). The Hylleraas coordinates give a value of (2 o/3). In an earlier work [5], the potential of Equation 11.17 was simply squared leading to an average value of (1/m2) equal to (ag/2). Using this result and using the global value of tj = 2a, we find... 

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

The differential equation for <1> results from a replacement of each coordinate X in the Hamiltonian operator by iS/Sp. This approach is not at all simple because of the Coulomb potentials, which involve r . Nevertheless, Hylleraas did succeed in solving this equation for the hydrogen atom [35]. 

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. 

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. 

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. 

The presence of the l/ri2 Coulomb repulsion term in Eq. (1) makes the Schrodinger equation nonseparable, and so exact analytic solutions cannot be found. Early in the history of quantum mechanics, Hylleraas [23] suggested expanding the wave function in the form (generalized for states of arbitrary angular momentum L)... 

The second method of dealing with the /x equation is used by Wind [119], but was probably first employed by Hylleraas [120]. Returning to equation (6.474) we expand M(/x) in terms of Legendre functions ... 

We add a few remarks on the wave mechanics of many-body problems, Here of course we are concerned with the solution of a wave equation in many-dimensional space thus the calculation of the helium spectrum needs as many as six co-ordinates, and that of the lithium spectrum nine. It is clear that in these cases an exact solution is not to be looked for, so that we must be content with an approximate solution of the problem. The methods of a highly developed perturbation theory enable us to push this approximation as far as we please the labour involved, however, increases without limit with the order of the approximation. The lowest terms of He, Li+ and Li have already been successfully calculated by this method, with results in good agreement with experiment (Hylleraas, 1930). 

If the phase shifts are given as a function of the energy a unique solution can only be obtained if, in addition, the bound state energies and the normalization constants of the bound state wave function are known (Gelfand and Levitan, 1951). The explicit construction of the potential leads to a complicated solution of a Fredholm integral equation. Other procedures (Agranovich and Marchenko, 1963 Hylleraas, 1963) may be more convenient for a practical application (see Benn and Scharf, 1967, O Brien and Bernstein,... 

One direct method involves a differential Schrodinger-like equation, which contains a Hamiltonian operator obtained from its classical counterpart by replacing each Cartesian component of linear momentum p by a multiplicative operator Pq and each Cartesian component of position q by the corresponding operator q = i(d/dpq). This approach is not at all simple because the q operators transform Coulomb potentials, which involve into fearsome operators. Nevertheless, Egil Hylleraas,... 

The work and conclusions of Hylleraas [38] provide evidence of the state of affairs in 1950, as regards both formal theory and understanding of even the simplest of resonance states in atomic physics and of possibilities of solving reliably Schrbdinger s equation as though the problem belongs to those of the discrete spectrum. 

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