Irreducible representations, symmetric states

If the system contains symmetry, there are additional Cl matrix elements which become zero. The symmetry of a determinant is given as the direct product of the symmetries of the MOs. The Hamilton operator always belongs to the totally symmetric representation, thus if two determinants belong to different irreducible representations, the Cl matrix element is zero. This is again fairly obvious if the interest is in a state of a specific symmetry, only those determinants which have the correct symmetry can contribute. 

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. 

To see why this is so, let us attempt to apply the procedure of Section II.B to a bound-state wave function. This is illustrated schematically in Fig. 19. It is clear immediately that we cannot construct an unsymmetric in the double space, because each bound-state eigenfunction must be an irreducible representation of the double-space symmetry group. Thus a bound-state function in the double space is necessarily symmetric or antisymmetric under R2k, and is thus either a Fq or a Fn function. For a Fq function, we have Fn = 0 (since and Fn cannot form a degenerate pair), which implies [from Eq. (6)] that... 

Separability between electronic and nuclear states is fundamental to get a description in terms of a hierarchy of electronic and subsidiary nuclear quantum numbers. Physical quantum states, i.e. wavefiinctions 0(q,Q), are non-separable. On the contrary, there is a special base set of functions Pjt(q,Q) that can be separable in a well defined mode, and used to represent quantum states as linear superpositions over the base of separable molecular states. For the electronic part, the symmetric group offers a way to assign quantum numbers in terms of irreducible representations [17]. Space base functions can hence be either symmetric or anti-symmetric to odd label permutations. The spin part can be treated in a similar fashion [17]. The concept of molecular species can be introduced this is done at a later stage [10]. Molecular states and molecular species are not the same things. The latter belong to classical chemistry, the former are base function in molecular Hilbert space. 

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 die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

However, the problem of variational collapse typically prevents an equivalent SCF description for excited states. That is, any attempt to optimize the occupied MOs with respect to the energy will necessarily return the wave function to that of the ground state. Variational collapse can sometimes be avoided, however, when the nature of the ground and excited states prevents their mixing within the SCF formalism. This simation occurs most commonly in symmetric molecules, where electronic states belonging to different irreducible representations do not mix in the SCF, and also in any situation where the ground and excited stales have different spin. 

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. 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

For a vibrational mode of the molecule to induce coupling between adiabatic electronic states p(r R), the direct product of the irreducible representations of < >.,(r R), p(r R), and the vibrational mode must contain the totally symmetric representation of the molecular point group,... 

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