The Two-Electron Problem

A characteristic feature of the two-electron problem is that the total wave function may be factorized into a space part and a spin part ... 

Convergence Radii. Applications to Mathieu Functions and to the Two-Electron Problem. 

Application to the two electron problem.545 If we take suitable system of units, the Hamiltonian operator of the two-electron problem is given by... 

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. 

Table XV. Complete Wave Functions for the Two Electron Problem...

Many problems involving the electronic states of organic molecules can be dealt with by considering only the optical electrons, especially if they are located in the highest occupied orbital. Thus, in some cases an adequate description can be obtained by solving a two or, if there is a doubly degenerate orbital involved, a four-electron problem. The latter is just more tedious than the former, so in this section some pertinent aspects of the two electron problem will be discussed. 

This formula looks like trouble. After the ionization, there is only a single electron in the molecule, while here, some electron-electron repulsion (integral J) appears But everything is fine because we still use the two-electron problem as a reference, and e relates to the two-electron problem, in which ei — hn + Jn-Hence,... 

The evaluation of R[C], Eq. (32), requires the same computational work as a single SCF iteration, and the iterative solution of the Cl equations (complete in the given basis) requires the same work as an SCF treatment since no integral transformation is required. The present formulation of the two-electron problem is not only formally simple, but it is also ideally suited for applications. These advantages are basically a consequence of the transformation properties (24)-(26), which result from the special ansatz (19) for the wavefunction. 

Let us now proceed with the two-electron problem and address the next question, which is that, after having determined which symmetry species are present, we should like to know what the corresponding two-electron wavefunctions look like, i.e. we should like to construct the SALCs. This construction does not pose any new problems the projection operators that were introduced in Sect. 4.5 will do the job perfectly well. Some notation is important here. The product function will be written as ... 

In the last part of his classic book. Das also provides some material regarding the quantum electrodynamical treatment of the two-electron problem. 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

As a final remark we may comment on the fact that we need to study the two-electron problem in the attractive external potential of an atomic nucleus, hence, as a bound-state problem. It is immediately seen that this affects, for instance, the expansion in terms of zeroth-order state functions in Eq. (8.22), where bound states of the one-electron problem rather than free-particle states become the basis for the construction of the wave function (and wave-function operators in second quantization). The situation is, however, more delicate than one might think and reference is usually made to the discussion of this issue provided by Furry [216] (Furry picture). Of course, it is of fundamental importance to the QED basis of quantum chemistry. However, as a truly second-quantized QED approach, we abandon it in our semi-classical picture and refer to Schweber for more details [165, p. 566]. Instead, we may adopt from this section only the possibility to include either the Gaunt or the Breit operators in a first-quantized many-particle Hamiltonian. 

Finally we can construct in principle, a set of two-electron continuum states which have an incident boundary condition that two electrons are scattering from the nucleus. We can now make a crucial statement. The exact solution of the correlated two-electron wave function at a given total energy will in general be expressible as some linear combination of the channel states. As the best that we can hope for, when we diagonalize the two-electron problem to obtain the set that these pseudostates approximately represent the projection of the exact correlated wave function at energy %, we must assume that Xm is not pure. Its character is neither that of any single ionization channel, nor a double ionization channel. How then are we to proceed ... 

This perturbation energy is about 14-16% of the total energy and 9-10% of the electronic energy in the neighbourhood of the minimum. This means that our eigenfunction of zeroth order is still a bad approximate solution of the two-electron problem, and that the corresponding value for the energy is not very precise. 

The considerations given above which relate to the two-electron problem can easily be extended to the case of an atom which is completely filled up to a position, for instance in the K-shell that has been emptied by an electronic collision... 

The parameter 5 is the same root of the quadratic equation that determines Sp], its value is obviously smaller - indeed negative -for the divergent solution and positive for the convergent one. The variation of C(p) is assumed here to be slow, as appropriate to a WKB procedure. The treatment summarized here was developed for the two-electron problem but is equally relevant to all the examples described in Sec. 2. 

XIll. Resonance and the Structure op Complex Molecules Spin Theory and Bond Eigenfunctions, 232. Evaluation of the Integrals, 240. The Two-Electron Problem, 244. The Four-Electron Problem, 245. The Concept of Resonance, 248. The Resonance Energy of Benzene, 249. The Resonance Energy of Benzene by the Molecular Orbitals Method, 254. 

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