Gerade components

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... 

Unless (v iG)spin = ( i E)spin. then the spin component is zero and the transition is spin-forbidden. Nevertheless, spin-forbidden transitions are observed as weak features (as in Fig. 2.18) typically with 10 -10 the intensity of fully allowed transitions. This is because of the interaction between the electron spin magnetic moment and the magnetic moment due to the orbital motion of the electron (spin-orbit coupling). The La-porte selection rule, furthermore, states that only transitions between wave functions with one having gerade and the other ungerade character are allowed (hence all d-d transitions are Laporte forbidden). This arises since the spatial component can be further broken down ... 

The first selection rule concerns the multiplicity of states Since the components of p are ungerade or odd, the integrand becomes gerade or even only provided the product of the two spin functions is ungerade, i.e., if their multiplicity or the total spin quantum number S does not change, or in other words if AS=0. Thus singlet-triplet transitions are normally forbidden. 

Both singlet and triplet CT states are split, and this splitting can be determined experimentally by, e.g., optical absorption spectroscopy if both transitions from the respective GS spin component to each of the split components of the CT state are electric-dipole (ED) allowed. This is the case if the dimer does not have a center of symmetry. In that case the +j— would correspond to g/u (gerade/ungerade) combinations and only the g<- u transition would have ED intensity. 

Problem 6.2 Show that in a group containing the inversion centre the dipole moment components must have ungerade representations. Hence, demonstrate that Moi must be zero for any gerade representation. 

The upper two wavefunctions belong to gerade crystal symmetries a.g and bg and are thus forbidden transitions, whereas the lower two states are of mgerade symmetry, optically allowed. However, the au component will be visible with a polarization parallel to the b-crystal axis whereas the b component will be polarized within the ac-plane of the crystal. The corresponding energies can be then written as following ... 

Furthermore, the two components of G produced by the symmetry factoring of G will be called the A component (symmetric, gerade, f = 0) and the B component (antisymmetric, ungerade, f = 1). Specialization of the Davidson rules outlined in the previous section leads to the following rules for determining the two components of G ... 

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... 

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