Perturbation theory transition frequencies

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

A simple method for predicting electronic state crossing transitions is Fermi s golden rule. It is based on the electromagnetic interaction between states and is derived from perturbation theory. Fermi s golden rule states that the reaction rate can be computed from the first-order transition matrix and the density of states at the transition frequency p as follows ... 

A transition linearly coupled to the phonon field gradient will experience, from the perturbation theory perspective, a frequency shift and a drag force owing to phonon emission/absorption. Here we resort to the simplest way to model these effects by assuming that our degree of freedom behaves like a localized boson with frequency (s>i. The corresponding Hamiltonian reads... 

Time-dependent perturbation theory shows that the linewidth of an n-quantum transition, generated by a single pumping frequency, should be 1/n of the corresponding... 

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

Transition probabilities. The interaction of quantum systems with light may be studied with the help of Schrodinger s time-dependent perturbation theory. A molecular complex may be in an initial state i), an eigenstate of the unperturbed Hamiltonian, Jfo I ) = E 10- If the system is irradiated by electromagnetic radiation of frequency v = co/2nc, transitions to other quantum states /) of the complex occur if the frequency is sufficiently close to Bohr s frequency condition,... 

Specifically, the collision-induced absorption and emission coefficients for electric-dipole forbidden atomic transitions were calculated for weak radiation fields and photon energies Ha> near the atomic transition frequencies, utilizing the concepts and methods of the traditional theory of line shapes for dipole-allowed transitions. The example of the S-D transition induced by a spherically symmetric perturber (e.g., a rare gas atom) is treated in detail and compared with measurements. The case of the radiative collision, i.e., a collision in which both colliding atoms change their state, was also considered. 

In order to analyze these data, the frequency shift of geffectivecan be calculated by averaging over all orientations the anisotropic shift derived from a static spin Hamiltonian [67]. This treatment is based on the assumptions that molecular motion neither changes the spin precession rate nor perturbs the states and, thus, that the center of gravity of the spectrum is invariant even in presence of some motional averaging. For the allowed 11/2) <-> <-l/2 transition under perturbation theory, with expressions valid up to the third order, this shift is given by [47] ... 

Coherent light sources are characterized by a spectral intensity distribution E(oj) and a frequency-dependent phase 0(w). According to first-order perturbation theory, linear absorption probabilities are given by the overlap between the spectrum of the light source E(w) and the optical transition, and are independent of the phase function In non-linear processes (2nd... 

The d(Ej + 2Tvd — E) term clearly identifies this second-order perturbation theory term as a process in which the molecule undergoes a transition from level Et to level. E by absorbing two photons of frequency ox [The second and third terms of Eq. (3.39), surviving when E — Eh represent (elastic) light scattering. The fourth term , surviving when E < Eh represents two-photon stimulated emission.]... 

The analogous quantities for the dissociation laser, Ed(z), ed(o>), and ed(a>) are defined similarly, with the parameters td and cod replacing tx and mx, and so forth. %The pump pulse Ex(t) induces a transition to a linear combination of the eigenstates p of the excited electronic state. The pump pulse may be chosen to encompass any. timber of states. Here we choose the pump pulse sufficiently narrow in frequency to incite only two of these states, E2) and IE3). The dump pulse Erf(r) dissociates the Sfjnolecuie by further exciting it to the continuous part of the spectrum. Both fields are chosen sufficiently weak for perturbation theory to be valid. 

The off-diagonal elements depend on the x and y components of the local field, which contain many fluctuating components oscillating at different frequencies. The parts which oscillate at the ESR frequency ( ) induce transition between the a- and P-states. By making a Fourier analysis of V(f) and using time-dependent perturbation theory [22], the transition probability between the a- and p-states (P< ) is given by... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

An early paper by Sun and Rice considered the relaxation rate of a diatomic molecule in a one-dimensional monatomic chain. An analysis similar to the Slater theoiy of unimolecular reaction was used to obtain the frequency of hard repulsive core-core collisions, and then (in the spirit of the IBC model) this was multiplied by the transition probability per collision from perturbation theory and averaged over the velocity distribution to obtain the population relaxation rate. This was apparently the first prediction that condensed-phase relaxation could occur on a time scale as long as seconds. 

This completes the solution of the problem for the evaluation of the transition frequencies the (eigenvalue, coefficient) pairs can be distinguished by a superscript (g), say, so that the first-order time-dependent perturbation theory approximation to the transition frequencies are the and the composition of the transition in terms of excitations between molecular orbitals occupied in the SCF single determinant and the virtuals of that SCF calculation are given by the elements of X and Y ... 

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