Wavefunctions symmetry representations for

Symmetry Representations for Wavefunctions 276 BIOSKETCH James Rondinelli 280... 

The character table of a point group defines the symmetry properties of a (wave) function as either 1 for symmetric or —1 for antisymmetric with respect to each symmetry operation/ The first row lists all the symmetry operations of the point group and the first column lists the Mulliken symbols of all possible irreducible representations, the symmetry transformation properties that are allowed for wavefunctions. As an example, the character table for the D2h point group is given in Table 4.1. The character tables of all relevant point groups are given in many textbooks.134,273-275 The last column shows the transformation properties of the axes x, y and z, which are used to determine electronic dipole and transition moments (Section 4.5). 

Clary, D.C. (1994) Four-atom reaction dynamics,. 7. Phys. Chem. 98, 10678-10688. Pack, R.T. and Parker, G.A. (1987) Qtianttim reactive scattering in three dimensions tising hypersidierical (APH) coordinates. Theory, J. Chem. Phys. 87, 3888-3921. Truhlar, D.G., Mead, C.A. and Brandt, M.A. (1975) Time-Reversal Invariance, Representations for Scattering Wavefunctions, Symmetry of the Scattering Matrix, and Differential Cross-Sections, Adv. Che.m. Phys. 33, 295-344. 

We now know how to assign an MO to a particular symmetry representation, but MOs describe the wavefunctions of only one electron at a time. For example, in the H2 molecule, we can put two electrons in the la-g MO. Does the resulting wavefunction for both electrons still have the symmetry of the cTg representation How about if we put one electron in the la-g and one in the lc7 —what is the representation for the resulting two-electron wave-function then ... 

When a wavefunction is created by multiplying together a bunch of component wavefunctions, it seems sensible that the symmetry of the total wavefunction is determined somehow by the symmetries of its components. This is indeed the case the symmetry of a product wavefunction is given by the direct product of the representations for the wavefunction s components. The direct product is a representation itself, with its characters obtained by multiplying together—one symmetry element at a time—the characters of the original representations. 

We will find an excitation which goes from a totally symmetric representation into a different one as a shortcut for determining the symmetry of each excited state. For benzene s point group, this totally symmetric representation is Ajg. We ll use the wavefunction coefficients section of the excited state output, along with the listing of the molecular orbitals from the population analysis ... 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

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