## Regularity and approximability of electronic wave functions

From 4. Its counterpart is the ionization threshold. Therefore, by the definition 5. At this place, the particular properties of the given potential enter. By Theorem 4. As one knows from , and as we will show in the fourth section, it is closely linked to the spectral properties of the Hamilton operator 4.

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As we will see later, this assumption implies that the minimum energy 5. The condition thus means that the nuclei can bind all electrons, which evidently does not always need to be the case, but of course holds for stable atoms and molecules. The condition 5. It is, however, possible to replace bilinear forms like 4. Eigenvalues and eigenvectors or eigenfunctions in concrete applications are defined in weak sense, in the same way as weak solutions of differential equations.

Definition 5. The spectrum obviously contains the eigenvalues but can be much larger, which is the case with the bilinear forms induced by the Hamilton operators of atoms and molecules. A first lower bound can be given in terms of the constants from 5.

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Theorem 5. It can be shown that A is self-adjoint and that the spectrum of A and of the bilinear form coincide. The operator A and the bilinear form determine each other. In the case in that we are mainly interested, A is the self-adjoint extension of the given Hamilton operator discussed in Sect. We will not utilize these facts here. The resolvent is an open and the spectrum a closed set.

Hence the resolvent is open and the spectrum correspondingly closed. Since G is 5. The isolated eigenvalues of finite multiplicity form the discrete spectrum, the other values in the spectrum the essential spectrum. The discrete spectrum is of special importance in the study of atoms and molecules. As we will see, it fixes the energies of the bound states and with that the frequencies of the light that the atom or molecule emits and absorbs, its spectrum. There remains the subproblem on E0. Since the spectrum is a closed subset of R, this proves also the second proposition.

Hence the fn converge weakly to zero and nothing is left to be done. As every bounded sequence in a Hilbert space contains a weakly convergent subsequence, we can assume that the f n converge weakly in H0 to a limit element f.

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Corollary 5. Choosing the un proportional to G fn with the fn from Theorem 5. In particular the minimum of the Rayleigh quotient is the minimum eigenvalue in finite space dimensions. The goal is to transfer these properties to the infinite dimensional case. The situation is much more subtle there because it is not even a priori clear whether the Rayleigh quotient attains its minimum. The most general result, at the same time demonstrating that the spectrum is never empty, is: Theorem 5. The range of G is a dense subspace of H1. The hydrogen eigenvalues are calculated in Sect.

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Our next theorems aim at such situations. They form the mathematical basis of the Ritz method to compute the eigenvalues corresponding to the ground state and the excited states of atoms and molecules.

By Theorem 5. Let u1 ,. Let E j be the subspace spanned by the vectors u1 ,. It has the advantage of being based on minimal, very general assumptions and produces optimal solutions in terms of the approximation properties of the underlying trial spaces. The theory of the Rayleigh-Ritz method has to a large extent been developed in the context of finite element methods, see [8,9], or .

A recent convergence theory and a survey of the current literature can be found in . We start from the same abstract framework as in the preceding section and from assumptions as in Theorem 5. This already fixes the method, 72 5 Spectrum and Exponential Decay which replicates the weak form of the eigenvalue problem and is completely determined by the choice of the subspace S replacing the original solution space.

### Regularity and Approximability of Electronic Wave Functions

The proof is a simple consequence from the min-max principle. Let Vk be the k-dimensional subspace of H1 spanned by u 1 ,. In finite element methods, Pu is the approximate solution. By the min-max principle from Theorem 5. The eigenvalues are thus much better approximated than is possible for the eigenvectors. For the minimum eigenvalue the estimate 5. The problem here is that in general there is no unique correspondence between the original eigenvectors and their discretized counterparts and that a multiple eigenvalue can split into a cluster of discrete eigenvalues.

The following theorem reflects this: 5. The proposition follows from the orthogonality properties of the different terms. The choice 5. The natural norm associated with the problem is the energy norm induced by the bilinear form. This error norm is considered in the following theorem which applies to eigenvectors for eigenvalues that are well separated from their neighbors: Theorem 5.

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Denoting by u the given projection of the eigenvector u from Theorem 5. Combining the estimate from Theorem 5. The Rayleigh-Ritz method in this respect fully exhibits the approximation properties of the trial spaces, however these are chosen, and is in this sense optimal. The estimate 5. This holds, for example, for certain spectral methods, for wavelets, and in the finite element case, there at least under some restrictions on the underlying grids [14, 17].

It is not astonishing that a similar error estimate holds for the higher eigenvalues, at least for those that are sufficiently well separated from the eigenvalues below them: Theorem 5. Denoting by u the given projection of u from Theorem 5.

It should further be noted that in the finite-element context one gains, depending on the regularity of the problem, up to one order of approximation in the H0 -norm compared to the H1 -norm. The results of the previous two sections transfer to this case if one replaces the given bilinear form by a shifted variant as in 4.

We recall the definition 4. The aim of this section is to translate our basic assumption 5. Since the bilinear form 4. By the corollary from Theorem 5. The subspace spanned by the eigenfunctions for the eigenvalues in the discrete spectrum is then dense in the solution space as can be seen applying Theorem 5. The next lemma shows that the limit 5.