Operators dipole moment

Because of difficulties in calculating the non-adiabatic conpling terms, this method did not become very popular. Nevertheless, this approach, was employed extensively in particular to simulate spectroscopic measurements, with a modification introduced by Macias and Riera [47,48]. They suggested looking for a symmetric operator that behaves violently at the vicinity of the conical intersection and use it, instead of the non-adiabatic coupling term, as the integrand to calculate the adiabatic-to-diabatic transformation. Consequently, a series of operators such as the electronic dipole moment operator, the transition dipole moment operator, the quadrupole moment operator, and so on, were employed for this purpose [49,52,53,105]. However, it has to be emphasized that immaterial to the success of this approach, it is still an ad hoc procedure. 

In Ih e quail tiiin mechanical description of dipole moment, the charge is a continuous distribution that is a I linction of r. and the dipole moment man average over the wave function of the dipole moment operator, p ... 

The dipole moment operator is a sum of one-electron operators r , and as such the electronic conlribution to the dipole moment can be written as a sum of one-electron contributions. The eleclronic contribution can also be written in terms of the density matrix, P, as follows ... 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

The integrations over the eleetronie eoordinates eontained in I)f p , as well as the integrations over vibrational degrees of freedom yield "expeetation values" of the eleetrie dipole moment operator beeause the eleetronie and vibrational eomponents of i and f are identieal ... 

Here r is the vector giving, together with e, the unit charge, the quantum mechanical dipole moment operator... 

The electric dipole moment operator jx has components along the cartesian axes ... 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

In order to compare calculated and observed dipole moments, we should replace the classical expression of the dipole moment by its quantum analogue jx = f F x1Ir dr where /I is the dipole moment operator (given by jx = —eri + . eZ-Blj with i and j running over the electronic and nuclear coordinates, respectively, and — e the electron charge). The actual calculation of a VB dipole moment is described below. 

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... 

In order to solve this problem of unboundedness, Argyres and Sfiat [17] decomposed the dipole moment operator into a periodic sawtooth function and its non periodic stair-case complement. The stair-case component is responsible of the localization of the electronic wavefunction whereas the sawtooth potential is associated with the periodic character of the polarization. Otto and Ladik [18-19] have proposed an alternative decomposition of the dipole moment operator. 

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 operator M (dipole moment operator) is composed of x, y, and z components ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

Excited states formed by light absorption are governed by (dipole) selection rules. Two selection rules derive from parity and spin considerations. Atoms and molecules with a center of symmetry must have wavefunctions that are either symmetric (g) or antisymmetric (u). Since the dipole moment operator is of odd parity, allowed transitions must relate states of different parity thus, u—g is allowed, but not u—u or g—g. Similarly, allowed transitions must connect states of the same multiplicity—that is, singlet—singlet, triplet-triplet, and so on. The parity selection rule is strictly obeyed for atoms and molecules of high symmetry. In molecules of low symmetry, it tends to break down gradually however,... 

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 ... 

Now, let us look at the autocorrelation function of the dipole moment operator within the adiabatic approach. In the representation I it is... 

Langevin equation. RR [58] have obtained for the dipole moment operator the following autocorrelation function which may be written after a correction (corresponding to a zero-point-energy they neglected) [8] ... 

This integral of the electronic dipole moment operator is a function of a nuclear coordinate Q. The integral may be expanded in a Taylor series with respect to Q (equation 4) and... 

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