Perturbed moment operators

P-Q-R triplets 225 P-R doublet 225, 249, 250 P-R exchange 135, 256 partial dipole moment operator 231 perturbation theory 5-6, 64-9 accuracy 78-9... 

These treatments of periodic parts of the dipole moment operator are supported by several studies which show that, for large oligomeric chains, the perturbed electronic density exhibits a periodic potential in the middle of the chain whereas the chain end effects are related to the charge transfer through the chain [20-21]. Obviously, approaches based on truncated dipole moment operators still need to demonstrate that the global polarization effects are accounted for. In other words, one has to ensure that the polymeric value corresponds to the asymptotic limit of the oligomeric results obtained with the full operator. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

If one is interested in changes of the solute molecule, or if the structure of the surrounding solvent can be neglected, it may be sufficient to regard the solvent as a homogeneous dielectric medium, as was done in the older continuum theories, and to perform a quantum mechanical calculation on the molecule with a modified Hamiltonian which accounts for the influence of the solvent as has been done by Hylton et al. 18 5>. Similarly Yamabe et al. 186> substituted dipole-moment operators for the solvent in their perturbational treatment of solvent effects on the activation energy in the NH3 + HF reaction. 

The predominant term in the perturbing potential V is of the form er, equal to the electric dipole moment operator. This is the origin of the selection rule that if ( 0, er i) = 0, the perturbed secular equation will not mix the states xpo and t/ i) so that the transition tpo ip i will not occur. 

The term maia a(1) is the first-order correction to the integral of the electric dipole moment operator in the a direction over orbitals a and i. The perturbed integral will depend on the change of the orbitals in the presence of a magnetic field or spin-orbit coupling. 

From Eq. (72) we see that the contribution to the MCD intensity from the perturbation to the transition density can be identified with the MCD due to the mixing of the excited state J with other excited states. The remainder of the MCD intensity from terms and spin-orbit-induced C terms is due to the perturbation of the integrals over the electric dipole moment operator (Eq. 52). The perturbed integrals thus include the contribution to the MCD from the mixing of excited states with the ground state. The perturbed integrals are written in terms of unperturbed orbitals (Eqs. 53 and 54) rather than unperturbed states or transition densities as this form is much easier to compute. With some further effort the contribution to the MCD from the perturbed integrals can also be analyzed in terms of transitions. 

Selection rules also arise on considering the point-group symmetry of tfo(Qeq). In the case of electric dipole radiation the perturbation Y, which describes the interaction with the radiation field, may be expressed in terms of the x, y, z components of the dipole moment operator r. The operators (t) transform as the x, y, or z components of r. 

For a perturbing electric field in the v-direction we have V = W = Dv and W — Y = 0, while for a magnetic field in the v-direction we have for the imaginary magnetic moment operator W = —V = +MV and V + W = 0. A nonzero frequency couples the symmetric and the antisymmetric part of the perturbed density matrix, whereas in the static case the two equations in (16) are not coupled. For comments on the apparent lack of symmetry for the perturbation equations for static electric and magnetic fields see [46]. 

While the chemical interpretation of the e parameters is a matter of real concern to us, there are also several other difficulties which are, however, more apparent than real. Consider the question of the calculation of magnetic properties in transition metal complexes - paramagnetic susceptibilities and e.s.r. g values. In contrast to the study of eigenvalues for optical transition energies, these require descriptions of the wavefunc-tions after the perturbation by the ligand field, interelectron repulsion and spin-orbit coupling effects. In susceptibility calculations it is customary to use Stevens orbital reduction factor k in the magnetic moment operator... 

The first-order correction to the charge density is independent of the choice of origin for the dipole moment operator, as the extra contribution to Do arising from a shift 5R in origin is — e5R<0 /c>, a term which vanishes because of the orthogonality of the zero-order states. The expression for the electric polarizability density (r) using the above expression for the perturbed density is... 

Consider a molecule in its ground state tj/g, an exact eigenstate of the molecular Hamiltonian, subjected to the very short external perturbation M S (Z) (such as caused by a very short radiation pulse, in which case M is proportional to the dipole moment operator). From Eq. (2.74) truncated at the level of first-order perturbation theory... 

To calculate the quantum-mechanical expression, we use perturbation theory. The perturbation operator H corresponding to (13.136) is H = -E /i, where the electric dipole-moment operator p is... 

Is it possible to deteet exeited states by exciting the ground state Well, there is a promising path showing how to do it. From Chapter 2, we know that this requires the time-dependent periodic perturbation —/t S exp ( i(ot) of frequency where p. denotes the dipole moment operator... 

Owing to the fact that the wavefunction perturbed to first order only contains double excitations D > and that the electric moment operators occurring in electron operators, the energy correction reduces to the Mollcr-Plessct (MP2) correction... 

Is it possible to detect excited states by exciting the ground state Well, there is a promising path showing how to do it. From Chapter 2, we know that this requires the time-dependent periodic perturbation —fi exp ( icot) of frequency co, where p denotes the dipole moment operator of the system, and is the electric field amplitude. Such a theory is valid under the assumption that the perturbation is relatively small and the electronic states of the isolated molecule are still relevant. In view of that, we consider only a linear response of the system to the perturbation. Let us focus on the dipole moment of the system as a function of a>. It turns out that at certain... 

The perturbation is now readily expressed in terms of the multipole moment operators ... 

In Chapters 4, 5 and 6 explicit forms for these perturbation Hamiltonian operators will be derived by expressing the scalar and vector potentials in terms of components of the electric field a, the electric field gradient cj3, the magnetic induction Ba, the nuclear moment and the rotation of the molecule. The resulting operators are also collected in Appendix A. 

In the second approach we will use the fact that the moments are defined as derivatives of the energy of a molecule in the presence of an inhomogeneous electric field, Eqs. (4.19), (4.20) and (4.21). In order to apply these definitions we need to find an expression for the energy of a molecule in the presence of an inhomogeneous electric field. Here, we are using perturbation theory as developed in Section 3.2. The first step is thus to define the perturbation Hamiltonian operators and to derive explicit expressions for them in terms of components of the electric field a(Ro) and field gradient tensor a/3(Ro)- The electric field and field gradient enter the molecular Hamiltonian in the form of the scalar potential From Eq. (4.15) we can see... 

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