Dipole perturbations

To illustrate how the above developments are earried out and to demonstrate how the results express the desired quantities in terms of the original wavefunetion, let us eonsider, for an MCSCF wavefunetion, the response to an external eleetrie field. In this ease, the Hamiltonian is given as the eonventional one- and two-eleetron operators to whieh the above one-eleetron eleetrie dipole perturbation V is added. The MCSCF wavefunetion P and energy E are assumed to have been obtained via the MCSCF proeedure with H=H0+ iV, where X ean be thought of as a measure of the strength of the applied eleetrie field. 

In MCD literature (Piepho and Schatz, 1983 Schellman, 1975 Stephens, 1976) and in the paper written by Judd (Gorller-Walrand and Binnemans, 1998 Judd, 1962) the electric dipole perturbation Hamiltonian is given as... 

This simple proportionality for dipole perturbations does not carry over to other symmetries, i.e. the quadrupole term is a linear combination of two Gegenbauer polynomials. 

By adopting, as for OPA, an approximation for the interaction operator including only the electric dipole perturbation, the isotropically averaged two-photon transition strength (011,012) can be written as... 

In the third order of long-range perturbation theory for a system of tluee atoms A, B and C, the leading nonadditive dispersion temi is the Axilrod-Teller-Mutd triple-dipole interaction [58, 59]... 

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

Assuming that the system has no pennanent dipole moment, tire existence ofP(t) depends on a non-stationary j induced by an external electric field. For weak fields, we may expand the polarization in orders of the perturbation. 

Adsorbates can physisorb onto a surface into a shallow potential well, typically 0.25 eV or less [25]. In physisorption, or physical adsorption, the electronic structure of the system is barely perturbed by the interaction, and the physisorbed species are held onto a surface by weak van der Waals forces. This attractive force is due to charge fiuctuations in the surface and adsorbed molecules, such as mutually induced dipole moments. Because of the weak nature of this interaction, the equilibrium distance at which physisorbed molecules reside above a surface is relatively large, of the order of 3 A or so. Physisorbed species can be induced to remain adsorbed for a long period of time if the sample temperature is held sufficiently low. Thus, most studies of physisorption are carried out with the sample cooled by liquid nitrogen or helium. 

The expansion of the perturbation w(r 2> tenns of inultipole potentials (e.g. dipole-dipole, dipole-... 

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

The influence of an applied magnetic field, as introduced in section Bl.5.2.2. is quite different from that of an applied electric field. A magnetic field may perturb the interfacial nonlinear response (and that of the weak bulk tenns), but it does not lead to any dipole-allowed bulk nonlmear response. Thus, in the presence of magnetic fields and magnetization, SHG remains a probe that is highly specific to surfaces and interfaces. It... 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

Not only can electronic wavefiinctions tell us about the average values of all the physical properties for any particular state (i.e. above), but they also allow us to tell us how a specific perturbation (e.g. an electric field in the Stark effect, a magnetic field in the Zeeman effect and light s electromagnetic fields in spectroscopy) can alter the specific state of interest. For example, the perturbation arising from the electric field of a photon interacting with the electrons in a molecule is given within die so-called electric dipole approximation [12] by ... 

This perturbation method is claimed to be more efficient than the fluctuating dipole method, at least for certain water models [Alper and Levy 1989], but it is important to ensure that the polarisation (P) is linear in the electric field strength to avoid problems with dielectric saturation. 

Before elosing this ehapter, it is important to emphasize the eontext in whieh the transition rate expressions obtained here are most eommonly used. The perturbative approaeh used in the above development gives rise to various eontributions to the overall rate eoeffieient for transitions from an initial state i to a final state f, these eontributions inelude the eleetrie dipole, magnetie dipole, and eleetrie quadrupole first order terms as well eontributions arising from seeond (and higher) order terms in the perturbation solution. 

Essentially all experimentally measured properties can be thought of as arising through the response of the system to some externally applied perturbation or disturbance. In turn, the calculation of such properties can be formulated in terms of the response of the energy E or wavefunction P to a perturbation. For example, molecular dipole moments p are measured, via electric-field deflection, in terms of the change in energy... 

The perturbation V = H-H appropriate to the particular property is identified. For dipole moments ( i), polarizabilities (a), and hyperpolarizabilities (P), V is the interaction of the nuclei and electrons with the external electric field... 

Force Constants and Dipole-Moment Derivatives of Molecules from Perturbed Hartree-Fock Calculations I J. Gerratt and I. M. Mills Journal of Physical Chemistry 49 (1968) 1719... 

---