Symmetric states group theory

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In the TMM problem, many of the CSFs having partially occupied orbitals correspond to an electronic state symmetry other than that of the totally symmetric irreducible representation, and thus make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, the calculation itself will determine the coefficients of non-contributing CSFs to be zero, but no advantage in efficiency will have been gained). Since this application of group theory involves no approximations, it is one of the best ways to speed up a CAS calculation. 

We now turn to electronic selection rules for symmetrical nonlinear molecules. The procedure here is to examine the structure of a molecule to determine what symmetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then permit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Here we will only pick one very simple point group called 2 and look at some simple examples to illustrate the method. 

In addition to labeling states and determining selection rules, group theory appears in chemistry in many forms. Here we briefly touch on some examples, showing how group theory has its uses outside perfectly symmetric molecules, and even outside physical chemistry. 

An appreciation of electronic state symmetry is particularly useful when studying transition metal compounds. The transition metals are defined by a partially filled d subshell in the ground electronic state of the atom. The five d orbitals are degenerate in the spherically symmetric isolated atom, but that degeneracy is broken when we introduce chemical bonds. The distribution of the orbital energies that results can often be interpreted using the tools of group theory and a little chemical common sense. 

With a little group theory, we can determine whether or not the vibration has a dipole derivative. The same symmetry selection rules apply to vibrations as to electronic transitions for a transition to be allowed, the direct product of the representations for the initial and final states must be one of the representations for the transition moment. The transition moments for electric dipole or infrared selection rules correspond to the functions x, y, and z. For Raman transitions, the transition moments correspond to any of the second-order functions of x, y, and z, such as xz or -I- y. The representation of the ground vibrational state is always the totally symmetric representation, so F, F is equal to Fy for fundamental transitions. Therefore, the selection rule for fundamental transitions is F, (x) Fy = F = F. For example, the group theory predicts that for CO2 the transitions V2 = 0 1 and V3 = 0 1 are infrared-allowed, because those vibrational modes have TTu (x,y) and (z) symmetry, respectively. On the other hand, the symmetric stretch transition Vj = 0 1 is forbidden by infrared selection rules but allowed by Raman selection rules, because that vibrational mode has (x + y, z ) symmetry. Here are the relevant rows from the character table in Table 6.4 ... 

Apart from the energies, computing electronic oscillator strengths, f, provides information about the relative intensity of the different transitions, initially for those states allowed in one-photon (optical) spectroscopy. Group theory indicates that one-photon allowed transitions of the molecule are those in which the direct product of the symmetries of the initial state, the corresponding dipole moment component (x, y or z), and the final state belong to the totally symmetric irreducible representation of the point group of the molecule ... 

Symmetry analysis can provide information about properties of the states of systems. Group theory tells us that a property will have a zero value if the operator associated with that property transforms as other than the totally symmetric representation of the group. Consider the dipole moment vector. The form of the operators of the components of the dipole is that of charge times a position coordinate (Equation 10.66). The symmetry of the component operators is deduced by applying symmetry operators to unit vectors in the directions of the position coordinate. In the case of a molecule with an inversion center, such as N2, applying the inversion operator to x yields -x. Thus, the x operator and thereby the operator for the x-component of the dipole moment transform as an ungerade representation. Since the totally symmetric representation is a gerade representation, we... 

The parity or Laporte rule states that for an optical transition to be allowed, the parity between final and initial states needs to change [53]. Utilising group theory arguments, it can be concluded that, for die transition to be allowed, the totally symmetric irreducible representation Ti (Bethe s symbol, equivalent to Mulhken s symbols AO needs to be contained in the direct product shown in Equation 1.54. 

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

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