Nuclear displacement operators

In Sections III—VI the nuclear displacement operators H n) are treated as perturbation operators and response function theory is used to determine the implicit geometry dependence of the wave function to each order in the nuclear displacement ... 

Consider now the case where the perturbation A is a specific nuclear displacement, A"i Xk + AX t. The derivatives of the one- and two-electron integrals are of two types, those involving derivatives of the basis functions, and those involving derivatives of the operators. The latter are given as... 

The mass weighted displacements of the nuclei of a molecule from their equilibrium positions q can be used to generate a representation of the point group to which the molecule belongs. If under the symmetry operation R, the mass weighted nuclear displacements... 

The EO effect is a second-order nonlinear optical (NLO) effect. Only non-centrosymmetrical materials exhibit second-order NLO effects. This non-centrosymmetry is a condition, both at the macroscopic level of the bulk arrangement of the material and at the microscopic level of the individual molecule. All electro-optic modulators that are presently used by telecom operators are ferro-electric inorganic crystals. The optical nonlinearity in these materials is to a large fraction caused by the nuclear displacement in the applied electric field, and to a smaller fraction by the movement of the electrons. This limits the bandwidth of the modulator. The nonlinear response of organic materials is purely electronic and, therefore, inherently faster. 

We may see that all A2" characters of D h which are in the E, C3 and CTv headed columns (the corresponding operations constitute the C v group) are equal to -1-1. It means that these symmetry operations are conserved during any nuclear displacement (such as vibrations) described by IR A2" of Dsh group. The subgroup formed in this way is called kernel or kernel subgroup K(G, A) where G is the parent group and A denotes the IR of this displacement [11], In our case this relation is described by the formula... 

In order to elucidate the selection rules for the minimal value / resulting in a nonzero torque, we present the operators Qrm, as earlier, in the form of a sum of the electronic and nuclear contributions, and we assume again that the operator of vibronic interaction is linear in the nuclear displacements. In this approximation, Eq. (228) can be transformed to the following form ... 

The (Tv operator corresponds to a reflection through the molecule-fixed xz plane. If it operates on the spatial and spin coordinates of all electrons, the nuclear displacement vectors, and the rotational wavefunction (expressed in terms of Euler angles, which specify the orientation of the molecule-fixed coordinate system relative to a laboratory-fixed coordinate system), then the eigenvalues of av label the total parity, , of a rotating molecule basis function,... 

To the exact electronic Hamiltonian operator for the system of electrons in the field of fixed nuclei is now added a perturbation, which represents the small variations of total electronic energy with small nuclear displacements. The Hamiltonian may be expanded in powers of normal coordinates, and the term linear in the displacement Q, is taken to be the perturbation 3C ... 

The coupling coefficient on the right-hand side of Eq- (6.57) restricts the symmetry of the nuclear displacements to the direct square of the irrep of the electronic wave-function. This selection rule is made even more stringent by time-reversal symmetry. The Hamiltonian is based on displacement of nuclear charges, and not on momenta, so as an operator it is time-even or real. For spatially-degenerate irreps, which are... 

Third, as a consequence of the foregoing, the use of IDs is suitable in investigating the dependence of the electronic matrix element for radiationless transition on the nuclear coordinates. This problem can be solved, as has been shown in Chapter 5, by considering the matrix element as one that of an operator that depends upon both electronic and nuclear displacements and by introducing a q-centroid approximation for the electronic factor. The latter is obtained as an average with DSWVO factor. The familiar Condon approximation can be so improved as to write the whole matrix element as a product of a vibrational overlap integral and an electronic factor, the latter being evaluated at some (j-centroid for the nuclear positions. 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

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