Hamiltonian eigenfunctions, calculation

In my discussion of pyridine, I took a combination of these determinants that had the correct singlet spin symmetry (that is, the combination that represented a singlet state). I could equally well have concentrated on the triplet states. In modem Cl calculations, we simply use all the raw Slater determinants. Such single determinants by themselves are not necessarily spin eigenfunctions, but provided we include them all we will get correct spin eigenfunctions on diago-nalization of the Hamiltonian matrix. 

The application of the Hamiltonian in Eq. (4) allows the calculation of energies of the eigenfunctions describing the levels arising from the... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

This requires that the eigenfunctions of the Hamiltonian are simultaneously eigenfunctions of both the Hamiltonian and the symmetric group. This may be accomplished by taking the basis functions used in the calculations, which may be called primitive basis functions, and projecting them onto the appropriate irreducible representation of the symmetric group. After this treatment, we may call the basis functions symmetry-projected basis functions. 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

The most efficient method of obtaining approximate eigenfunctions of the electronic Hamiltonian (10) depends on the strength of the interaction between the atoms. This is a qualitative concept but emphasizes the fact that the best method of calculating intermolecular forces, which are generally considered as weak interactions, may be quite different to the best method of calculating valence forces which are strong interactions. 

The power of quantum theory, as expressed in Eq. (4.1), is that if one has a molecular wave function in hand, one can calculate physical observables by application of the appropriate operator in a manner analogous to that shown for the Hamiltonian in Eq. (4.8). Regrettably, none of these equations offers us a prescription for obtaining the orthonormal set of molecular wave functions. Let us assume for the moment, however, that we can pick an arbitrary function, , which is indeed an eigenfunction for the specific case of Eq. (4.2). Since we defined the set of orthonormal wave functions 4, to be complete (and perhaps infinite), the function must be some linear combination of the 4>,, i.e.,... 

It is common for valence-only calculations to use a form of effective hamiltonian which is based on the eigenfunctions for atoms or ions with only one valence electron,83 This is equivalent to choosing a set of core orbitals l which satisfy... 

One of the most valuable features of theoretical methods based on classical VB structures is their ability to calculate the energy of a diabatic state. Contrary to adiabatic states, a diabatic state is not an eigenfunction of the Hamiltonian. Such a state can be a single VB structure, separate VB curves of covalent and... 

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