Unperturbed moment operators

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. 

The electronic magnetic multipoles (25)-(27) are unperturbed, or permanent, moment operators. In the presence of a vector potential A(r, t) (we simplify the notation, omitting the index), the canonical momentum is replaced by the mechanical momentum... 

The relative intensities of the stress-induced components have been obtained by calculating the matrix elements of the dipole moment operator for the different components of the multiplets. This has been performed for the three categories of zero stress transitions and for the stress-induced components of the r8 —s- r8 transitions, the intensities depend on parameters defined in terms of ratios of the magnitudes of matrix elements appropriate to the different unperturbed multiplets ([123], and references therein). 

Ho is die Hamiltonian for the unperturbed system and p is the dipole moment operator. From the perturbed wave functions evaluated at SCF or Cl level the dipole moment value p(f) is obtained. From the Taylor series expansion [ q. (10.1)] the coefficients a, p and y can be derived from calculations at a number of distortions along particular coordinate and field direcdons. Molecular polarizability is then obtained from the expression... 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

The dipole moment mo for an unperturbed system depends on the total angular momentum, which may be written in terms of the orbital angular momentum operator Lg and the total electron spin S. 

The first-order property with respect to an external field is the magnetic dipole moment m (eq. (10.10)). When field-independent basis functions are used, the HF magnetic dipole moment is given as the expectation value of the V2LG and S (total electron spin) operators over the unperturbed wave function, eqs (10.21) and (10.24). Since the Lg operator is imaginary it can only yield a non-zero result for spatially degenerate wave functions and the expectation value of S is only non-zero for non-singlet states. 

All the calculations reported in this work were done on a DEC 20-60 using a modified version of the GAUSSIAN 80 series of program (6). Standard ST0-3G minimal basis set (7) was considered. Polarizabilities were calculated by the finite-field SCF method of Cohen and Roothaan (8) which is virtually equivalent to the analytic Coupled Hartree-Fock scheme. A term yf, describing the interaction between the electric field, E, and the molecule is added to the unperturbed molecular Hamiltonian, H y is the total dipole moment of the molecule. At the Hartree-Fock level, the electric field appears explicitly in the one-electron part of the modified Fock operator, F( ),... 

Here, g is the degree of degeneracy of the ground state, Ej is the energy of the state y, and lii and are the operators of the electric and magnetic dipole moments, respectively. The wave functions of the ground state of the final state as well as that of the intermediate state %, over which the summation runs, are unperturbed wave functions defined in the absence of the magnetic field. 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

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