Theorems of Molecular Quantum Mechanics

Group theory can be applied to several different areas of molecular quantum mechanics, including the symmetry of electronic and vibrational wave functions and the study of transitions between energy levels. There is also a theorem which says that there is a correspondence between an energy level and some one of the irreducible representations of the symmetry group of the molecule, and that the degeneracy (number of states in the level) is equal to the dimension of that irreducible representation. 

It follows from the first HK theorem that the non-degenerate ground-state is also uniquely determined by its electron density 1P0 = P0[p0]. Thus, p0(r) represents the alternative, exact specification of the molecular quantum-mechanical ground-state. In other words, there is a unique mapping between P,l and p0, P0 <-> po, so that both functions carry exactly all the information about the quantum-mechanical state of the N electron system. 

This chapter discusses theorems that are used in molecular quantum mechanics. Section 14.1 expresses the electron probability density in terms of the wave function. Section 14.2 shows how the dipole moment of a molecule is calculated from the wave function. Section 14.3 gives the procedure for calculating the Hartree-Fock wave function of a molecule. Sections 14.4 to 14.7 discuss the virial theorem and the Hellmann-Feynman theorem, which are helpful in understanding chemical bonding. 

Molecular quantum mechanics was born in the late 1920s, but for many years the use of perturbation theory in the study of molecules was rather limited. Perturbation theory was used to describe the interactions between molecules at long range, but for the description of the structure and properties of isolated molecular systems, methods based on the variation theorem found favour. The reason for this preference is that many of the series encountered in atomic and molecular quantum mechanical problems were found to be poorly convergent or even divergent. Although as early as 1930, Dirac [5] had recognized that... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

In studying molecular orbital theory, it is difficult to avoid the question of how real orbitals are. Are they mere mathematical abstractions The question of reality in quantum mechanics has a long and contentious history that we shall not pretend to settle here but Koopmans s theorem and photoelectron spectra must certainly be taken into account by anyone who does. 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

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