Ungerade states

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

Selection rules for M so (<—> coupling allowed < > no interaction u signifies ungerade states)... 

To obtain a rough estimate of the excitation transfer cross section, consider first the adiabatic approximation, neglecting the Coriolis coupling (the so-called rotating-atom approximation ). The transfer is then associated with mere interference between gerade and ungerade states. As an example, the 2 time-dependent wave function can be written as... 

Platt s notation for molecular states is N M lt,pl,c lyStateSymmelry, where N labels the first, second, third, etc., states of the same type, the multiplicity is (25 + 1), A and B states are non-degenerate and degenerate, respectively, and g and u identify symmetric (gerade) and antisymmetric (ungerade) states, respectively. 

For example, in a doubly-excited N=3 linear chain with a center of inversion (D,) symmetry), (10) yields zero for ungerade states and y for gerade states. 

The state eneigies refer to the lower one of the tunneling pairs, which is usually the g( except for the 3 state where the ungerade state has a lower energy. All energies in cm... 

The symmetry of the spin-orbit operator was derived in section 10.3. The spatial part of this operator transforms as the vector of rotations R = (Rx, Ry, Rz)- This means that the spin-orbit operator will connect states of different spatial symmetry. In C2v, for example, states of all spatial symmetries are connected by the spin-orbit operator. In D2h, the gerade states are all connected by the spin-orbit operator, and likewise the ungerade states, but there is no connection between gerade and ungerade states because the spin-orbit operator commutes with the inversion operator (it is an even operator). The spin operator transforms as a spherical tensor of rank 1 it is essentially a triplet operator. Therefore, it can connect states whose S and Ms values differ by 0 or 1. 

Figure 8. Davydov-splitting for the lowest singlet state (L-polarized) in 6T according to simple dipole calculations (Table 2). Note the degeneracy of the gerade and ungerade states [110].

Coriolis interaction parameters (cm" ), first between the 4 ungerade states, and second between the 3 gerade states (all values in cm ). 

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