Total molecular eigenfunction

The Born-Oppenheimer approximation may then be thought of as keeping the electronic eigenfunctions independent and not allowing them to mix under the nuclear coordinates. This may be seen by expanding the total molecular wavefunction using the adiabatic eigenfunctions as a basis... 

Inserting this expression into the total molecular Schrodinger equation, and using the fact that the electronic eigenfunctions are... 

The standard theoretical treatment of chemical reaction dynamics is based on the separation of the total molecular motion into fast and slow parts. The fast motion corresponds to the motion of the electrons and the slow motion corresponds to the motion of the nuclei. The theoretical foundation for the separation of the electronic and nuclear motion was first developed by Born and Oppenheimer. In this approach, the total molecular wave function is expanded in terms of a set of electronic eigenfunctions which depend parametrically on the nuclear coordinates. The expansion coefficients are the... 

The total molecular wave function can be expanded in terms of the electronic eigenfunctions... 

I is in general no direct relation between such functions and ionization energies or electron excitation this is because they are not eigenfunctions of a hamiltonian, hence they cannot be associated with an energy. For that reason, we kept the usual designation localized molecular orbitals but with [ the last word in inverted commas orbitals . However, for the interpretation of some other molecular properties, the minimized residual interactions i between quasi-localized molecular orbitals are not very importaint and, so, the direct use of a localized bond description is quite justified. That is the [ Case for properties such as bond energies and electric dipole moments, as well as the features of the total electron density distribution with which those properties are directly associated. 

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

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