Limit of Separated Atoms

In this section, we examine in some detail the case when the off-diagonal ma.-trix elements of the core operator 0rs are small in comparison with the electron repulsion integrals jrr- This corresponds to the situation of separated atoms, but it should be realized that other matrix elements also depend on the relative positions of the nuclei, both through the core operator and the symmetric orthonormalization procedure of the orbitals used in the expansion Eq. (11.10). In the following, the constant term of H PPP) is omitted, the third term is considered as a perturbation, while the second and fourth parts constitute the unperturbed hamiltonian for this case. 

in the limit of separated atoms, we write the hamiltonian as 

The only nonvanishing anticommutation relation of the basis field operators is 

Within each manifold of states characterized by the eigenvalues of the operators Nr, there will be a unique correspondence between the operators r and ordinary spin operators and they will be interpreted as spin operators for atoms. The different spin components for an atom axe all constants of the motion in the unperturbed case. Spin coupling becomes important when the perturbation terms are introduced. 

Before considering spin coupling details, we examine the Heisenberg equation of motion for the electron annihilation operators in the unperturbed case. Hubbard found that the two components nr-uttru and (1 — rir-v)arv of the 

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As a guide to this calculation, we consider the limit of separated atoms, for which we may write... 

The two types of elementary excitation operators, nr- o i/ and l-rir-u)arw, to describe the limit of separated atoms form the basis also for the discussion of interacting atoms. We introduce the particular linear combinations... 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

The electronic structure of atoms shows a similar stability since the shell structure remains constant over a wide range of experimental environments. However, with molecules this picture must be modified. The electronic structure of a diatomic molecule varies with bond length, the limit being that of a pair of separated atoms. Accordingly, the ranks of the blocks in the fc-matrix description vary with bond length. 

There are three different schemes for building up the electronic states of diatomic molecules (a) from separated atoms, (b) from the united atom, and (c) from the molecular orbitals of the diatomic molecule itself. It is the correlation between the electronic states of the diatomic molecule as built up from the separated atoms and as determined from the molecular orbitals of the diatomic which is most valuable for any general consideration of reactions and excited states. The correlation of molecular states obtained by these two methods is not limited solely to diatomic molecules but also forms a valid approach for polyatomic molecular systems. The correlation of separated atoms with the hypothetical united atom has value for diatomics and has been applied to simple polyatomic molecules, especially those with a heavy atom or two and a number of hydrogen atoms. However, it is conceptually less appealing even for simple polyatomic molecules and completely inapplicable for complex polyatomic molecules. 

An alternative approach to extend the molecular orbital method is offered by the work of Hubbard for the study of narrow energy bands in solids with the aim to study magnetism. The main idea of this work is to analyze the many-electron problem for the case of separated atoms, which means the limit of zero bandwidth. 

SEQUENCE AND STABILITY OF MOLECULAR STATES FROM THE LIMITING CASE OF SEPARATE ATOMS... 

Figure 29. Electron energy spectra from He-He collisions at collision energies of (a) 200 eV and (b) 500 eV. Electrons are due to autoionization of quasimolecular states, correlating to two singly excited He atoms in the separated atom limit. The separated atom states are indicated in part (b).

The fact that the separated-atom and united-atom limits involve several crossings in the OCD can be used to explain barriers in the potential energy curves of such diatomic molecules which occur at short intemuclear distances. It should be noted that the Silicon... 

Here, Wj is the number of neighboring atoms at the distance d,. Because the atoms separated by more than twice the shortest interatomic distance cannot be counted as neighboring, the summation is Hmited only to the interatomic distances less than twice dg. Without this limitation, calculation of the parame-... 

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