Molecular Electronic Wavefunctions

Still another notation designates /3-spin by a bar over the wavefunction, a functions being unbarred for example, 7i+ = 7T+/3, or —1 = 7t /3 (Field et al., 1972). 

The electronic wavefunction, constructed from the j i(i = a,b.n) spin-orbitals, is written in the form of a determinant (or a sum of determinants), 

This determinants form expresses the antisymmetry of the wavefunction with respect to interchange of two identical particles. The determinants wavefunction will always be specified by an abbreviated notation that lists only the diagonal of the determinant, 

When a configuration consists exclusively of closed shells, values of A = 0 and 5 = 0 are the only ones possible, giving rise to a single 1E+ electronic state. For example, 

Consider first the trivial case of the 7r2 configuration. The following table gives all possible distributions of these two electrons among the two subshells 7r+ and tt (the right superscript specifies whether A 0 or A 0 the spin projection is not yet specified). 

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Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

If I write possible atomic orbitals for hydrogen atom A as Xa possible atomic orbitals for hydrogen atom B as the molecular electronic wavefunction will be... 

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. 

I am going to leave you to prove for yourself that the wavefunction corresponding to this infinite-distance H2 problem is a product of two hydrogen atom wavefunc-tions. Physically, you might have expected this the two atoms are independent and so the electronic wavefunctions multiply to give the molecular electronic wavefunction. 

K. Ruedenberg, M. W. Schmidt, M. M. Gilbert, S. T. Elbert, Chem. Phys. 71, 41 (1982). Are Atoms Intrinsic to Molecular Electronic Wavefunctions I. The FORS Model. 

Before leaving the discussion of this area, let us consider a specific chemical example. The water molecule has C2V symmetry, hence its normal vibrational modes have A, Ai, B, or B2 symmetry. The three normal modes of H2O are pictorially depicted in Fig. 6.3.1. From these illustrations, it can be readily seen that the atomic motions of the symmetric stretching mode, iq, are symmetric with respect to C2, bending mode, i>2, also has A symmetry. Finally, the atomic motions of the asymmetric stretching mode, V3, is antisymmetric with respect to C2 and This example demonstrates all vibrational modes of a molecule must have the symmetry of one of the irreducible representations of the point group to which this molecule belongs. As will be shown later, molecular electronic wavefunctions may be also classified in this manner. 

Approximate Natural Orbitals and Natural Expansion Coefficients of Atomic and Molecular Electronic Wavefunctions. II. Decoupling of the Pair Equations and Calculation of the Pair Correlation Energies for the Be and LiH Ground States. R. Ahlrichs and W. Kutzelnigg, Theor. Chim. Acta, 10, 377 (1968). Ah initio Calculations on Small Hydrides Including Electron Correlation. I. The BeHz Molecule in Its Ground State. 

Ruedenberg, K., Schmidt, M.W., Gilbert, M.M. and Elbert, S.T. (1982) Are atoms intrinsic to molecular electronic wavefunctions. 1. The fors model. Chem Phys, 71, 41-49. 

Basic studies on plasmons and their related materials will influence wider research areas in fundamental and applied fields. Among them, applications of plasmonic optical fields to photochemical reactions have a large impact in photo-and material-sciences. For instance, the interaction between localized optical (or plasmon) fields with molecular electronic wavefunctions may enhance photochemical reaction rates, which is sometimes forbidden under the far-field irradiation of light. It has a potential to open up new chemical reaction routes beyond the dipolar approximation. Such novel photochemical reactions shed new light on photo- and material-sciences. 

The Schrodinger equation has not been solved exactly for electrons in molecules larger than the H2 ion the interactions of multiple electrons become too complex to handle. However, the eigenfunctions of the Hamiltonian operator provide a complete set of functions, and as mentioned in Sect. 2.2.1, a linear combination of such functions can be used to construct any well-behaved function of the same coordinates. This suggests the possibility of representing a molecular electronic wavefunction by a linear combination of hydrogen atomic orbitals centered at the nuclear positions. In principle, we should include the entire set of atomic orbitals... 

Nuclear magnetic moments are small enough to have an almost ignorable effect on atomic and molecular electronic wavefunctions. On the other hand, the electronic structure has a measurable influence on the energies of the nuclear spin states. In this situation, it is an extremely good approximation to separate nuclear spin from the rest of a molecular wavefunction. The electronic, rotational, and vibrational wavefunctions of a molecule can be accurately determined while ignoring nuclear spin, as has been done so far. Then, the effect of the electrons and the effect of vibration and rotation can be incorporated as an external influence on the nuclear spin states. 

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