Molecular symmetry wavefunctions

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

The projection-operator technique will be employed in several examples presented in the following chapter and Chapter 12. For. the quantitative interpretation of molecular spectra both electronic and vibrational, molecular symmetry plays an all-important role. The correct linear combinations of electronic wavefunctions, as well as vibrational coordinates, are formed with the aid of the projection-operator method. 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

Extraction of information from p may not be as elegant as from P. For example, the Woodward-Hoffmann rules follow fairly transparently from the symmetries of molecular orbitals (wavefunctions), but deriving them from p requires using a dual descriptor function [1]. 

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. 

With the above symmetry considerations one can explain the unexpected trend in the tetrasubstituted TEE series by the molecular symmetry elements without knowing the wavefunctions themselves. It was assumed that the dipolar term dominates and that the negative and two-photon term are of secondary importance. As all the nonlinearities of the tetrasubstituted TEEs are positive the negative term can assumed to be small. 

The PI group operations are defined by their effect on the space-fixed coordinates of the atomic nuclei and electrons. Since our molecular wavefunctions are written in terms of the vibrational coordinates, the Euler angles and the angle p, we must first determine the effect of the PI group operations on these variables. In the case of inversion this can lead to certain problems both in the understanding of the concepts of molecular symmetry and in the proper use of group theoretical methods in the classification of the states of ammonia. 

In calculations based on the MO-LCAO technique [32-34], the one-electron Kohn-Sham equations Eq. (11) are solved by expanding the molecular orbital wavefunctions V i(r) in a set of symmetry adapted functions Xj(r), which are expanded as a linear combination of atomic orbitals i.e. 

This four-CSF wavefunction contains the full molecular symmetry even... 

This increase of CSF expansion length upon transformation to symmetry-adapted orbitals potentially affects any of the expansion forms that attempt to describe electron correlation in terms of localized orbitals and that are not invariant to transformations that mix the different localized orbitals. All of the product and direct product expansion forms (including the RCI, PPMC, PPGVB and SOGVB expansions) are potentially of this type. It often happens that these wavefunctions do have the full molecular symmetry even though they are described in terms of localized orbitals and not symmetry-adapted orbitals. The localized orbital description that results from these wavefunction optimizations is therefore both an asset and a liability it aids the chemical interpretability and results in more compact CSF expansions but the computations must be performed in an orbital basis that does not possess the full molecular symmetry. This is computationally important since many steps of the MCSCF wavefunction optimization can exploit such orbital symmetry when it is present. 

Why does symmetry matter Why must there be a conservation of orbital symmetry Recall from Chapter 14 that molecular orbitals are just convenient creations of an approximation to the Schrodinger equation. Yet, orbital wavefunctions do have inherent symmetries. Wavefunctions of different symmetry cannot mix, or stated more accurately, their mixture does not give any net interaction. As we will explain further below, it is fruitless to mix them. To see this better, we now turn to an analysis of the mixture of total wavefunctions, where symmetry again is of paramount importance. Here, the theoretical basis for why symmetry matters is more clear. 

Molecular symmetry provides tools that make it easier for us to describe the electronic and nuclear wavefunctions of the molecule. Now we return to the molecular Hamiltonian, and for this chapter we just look at the contributions to the electronic energy term which determines the effective potential energy for our nuclei in Eq. 5.10 ... 

By keeping symmetry in mind, it is possible to construct appropriate combinations of atomic orbital wavefunctions to approximate molecular orbital wavefunctions that cover, or span, the entire molecule. The use of symmetry is the first real restriction we have placed on linear combinations, but it makes sense. After all, it serves... 

The lowest molecular singlet transition for 6T is from a state of Ag symmetry to a Bu-state. Due to the Cah symmetry of the crystal point group the synunetry representations are the same as in the molecular symmetry frame and for clarification we will use lower case letters with respect to the crystal framework. Since the site occupied by a 6T molecule has no special symmetry operations, the site synunetry is Ci. Within the unit cell (space group P2i/ ) there are 4 molecular sites which are related to each other by different symmetry operations 1 identity, 2 inversion, 3 glide plane, 4 two-fold axis. These symmetry operations correspond to the factor group whose irreducible representations are displayed in Table 2 [110]. The molecular wavefunctions have thus to be written as follows, to require the synunetry properties of the crystal (see characters in Table 2) [110]. 

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