Bom-Oppenheimer Hamiltonians

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

The Bom-Oppenheimer Hamiltonian is not dilatation analytic if only the electronic co-ordinates are scaled since relS — R vanishes for a continuous range of 0 3 r = R and r.R = cos(0) /80/. The conceptual and numerical difficulties associated with complex scaling of both the nuclear and electronic coordinates and solution of the combined problem with a plethora of rotational, vibrational and electronic thresholds have prompted the formulation of (see ref. 22) two different schemes by McCurdy and Rescigno /81 / and Moiseyev and Corcoran... 

Application of the Hartree-Fock Method. - Since numerical Hartree-Fock programs dealing with complex numbers are available in many research groups, it seemed natural to apply this scheme also to the scaled Bom-Oppenheimer Hamiltonian (4.15). As a consequence, some numerical results were obtained before the theory was developed, and - as we have emphasized in the Introduction - some features seemed rather astonishing. 

A symmetry operator leaves the potential energy unchanged if it moves each electron so that after the motion it is the same distance from each nucleus or the same distance from another nucleus of the same charge as it was prior to the motion. If this is the case, the symmetry operator commutes with the Bom-Oppenheimer Hamiltonian. There is another way to see if the symmetry operator will commute with the electronic Hamiltonian. Apply it to the nuclei and not to the electrons. If the symmetry operator either leaves a nucleus in the same position or places it in the original position of a nucleus of the same type, it belongs to the nuclear framework. The symmetry operators that belong to the nuclear framework will commute with the electronic Hamiltonian when applied to the electrons. 

The nonrelativistic, spin-free, n-electron Bom-Oppenheimer Hamiltonian of Eq. 12 is self-adjoint, bounded from below, and below some energy has a discrete energy spectrum it belongs to an elliptic differential equation with coefficients analytic almost everywhere within its domain G, 3 C (that is, analytic everywhere in G except a set of measure zero) that leaves the rest Go of the domain G-connected. 

Abstract Some previous results of the present author are combined in order to develop a Hermitian version of the Chemical Hamiltonian Approach. In this framework the second quantized Bom-Oppenheimer Hamiltonian is decomposed into one- and two-center components, if some finite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones, which become exact only in the limit of complete basis sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and diatomic energy terms which were introduced in our previous chemical energy component analysis scheme. The corresponding modified Hartree-Fock-Roothaan equations are also derived they do not contain any three- and four-center integrals, while the non-empirical character of the theory is conserved. This scheme may be useful also as a layer in approaches like ONIOM. 

The LCAO version of the Bom-Oppenheimer Hamiltonian has a standard form in terms of the creation and annihilation operators referring to the Lowdin-orthogonalized basis [16,17] ... 

Figure 20.10 depicts the hydrogen molecule, consisting of two hydrogen nuclei at locations A and B and two electrons at locations 1 and 2. The distances between the particles are labeled in the figure. The Bom-Oppenheimer Hamiltonian is... 

Due to the Bom-Oppenheimer approximation, the inter-nuclear distances (the Rab terms) are constant for a particular nuclear configuration. Consequently it is convenient to express the Bom-Oppenheimer Hamiltonian as only the operator parts in terms of the electrons,, and add the... 

In the general case of an electronic Hamiltonian for atoms or molecules under the Bom-Oppenheimer approximation,... 

Perturbation terms in the Hamiltonian operator up to still lead to the uncoupling of the nuclear and electronic motions, but change the form of the electronic potential energy funetion in the equation for the nuclear motion. Rather than present the details of the Bom-Oppenheimer perturbation expansion, we follow instead the equivalent, but more elegant procedure of M. Bom and K. Huang (1954). 

The nonrelativistic electronic Hamiltonian for N electrons in the field of M point charges (nuclei) under the Bom-Oppenheimer approximation is given by... 

To understand the main idea behind DFT, consider the following. In the absence of magnetic fields, the many-electron Hamiltonian does not act on the electronic spin coordinates, and the antisymmetry and spin restrictions are directly imposed on the wave function (r j, v j,..., rvyv). Within the Bom-Oppenheimer approximation,... 

The time-independent nonrelativistic electronic Hamiltonian, under the Bom-Oppenheimer approximation can be written as [ 1 ]... 

Here G(vj, v2, v3) is the level energy in wave number units (as far as possible we follow the notation of Herzberg, 1950) and the constants in Equation (0.1) are given in Table 0.1. As usual the vs are the vibrational quantum numbers of S02 and rather high (above 10) values can be reached using the SEP technique. Equation (0.1) provides a fit to the observed levels to within an error below 10 cm 1, which is almost the experimental accuracy. We need, however, to be able to relate the parameters in this expansion directly to a Hamiltonian. The familiar way of doing this proceeds in two steps. First, the electronic problem is solved in the Bom-Oppenheimer approximation, leading to the potential for the... 

Treating the electronic Schrodinger equation in the usual clamped nuclei (Bom-Oppenheimer) approximation, [20] we have (in atomic units) the Hamiltonian, H, and the spectrum of eigenvalues and eigenvectors, and fife,... 

Later we will discuss conventional Born-Oppenheimer calculations used in conjunction with the current work. Thus, for completeness, we will cover here the integrals needed for these calculations. These integrals are quite similar to the ones used in the non-Bom-Oppenheimer calculaions, as will be shown below. We show first the integrals over the Born-Oppenheimer Hamiltonian ... 

We show here how we may take the information obtained above and use it as a starting point for non-Bom-Oppenheimer calculations. The five-particle systems of non-Bom-Oppenheimer and its isotopomers were transformed via separation of the center-of-mass Hamiltonian to four-pseudoparticle systems as described above. The resulting total position vector is... 

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