Electric Hamiltonian

The quadrupolar term is more complicated, but after judicious application of the Wigner-Eckhart theorem,37 the electric Hamiltonian, for cylindrical symmetry, may be written as... 

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

This expression may be interpreted in a very similar spirit to tliat given above for one-photon processes. Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian H. In general, there is also a second-order interaction with the electric field through which returns a portion of the excited-state amplitude to surface a, with subsequent evolution on surface a. The Feymnan diagram for this second-order interaction is shown in figure Al.6.9. 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

In order to evaluate equation B1.2.6, we will consider the electric field to be in the z-direction, and express the interaction Hamiltonian as... 

Nomially the amplitude of the total incident field (or intensity of the incident light) is such that the light/matter coupling energies are sufficiently weak not to compete seriously with the dark matter Hamiltonian. As already noted, when this is tire case, tlie induced polarization, P is treated perturbatively in orders of the total electric field. Thus one writes... 

Since atomic nuclei are not perfectly spherical their spin leads to an electric quadnipole moment if I>1 which interacts with the gradient of the electric field due to all surrounding electrons. The Hamiltonian of the nuclear quadnipole interactions can be written as tensorial coupling of the nuclear spin with itself... 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

The first basic ingredient in our description of the electric double layer is the coulombic interaction. It seems quite natural to assume that the fields are coupled according to a coulombic Hamiltonian of the same... 

An analytical gradient calculation is invariably faster than a numerical one. To repeat the argument from Chapter 14, with real wavefunction Hamiltonian H (including the field terms) and parameter a (where a is a component of the external electric field)... 

I appealed to your good sense when I asked you to believe that the Hamiltonian for a particle in an external electric field could be written... 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

Eckart, criteria, 264, 298 procedure, 267 Effective charge, 274, 276 Effective Hamiltonian, 226 Elastic model, excess entropy calculation from, 141 of a solid solution, 140 Electric correlation, 248 Electric field gradient, 188, 189 Electron (s), 200... 

A physically acceptable theory of electrical resistance, or of heat conductivity, must contain a discussion of the explicitly time-dependent hamiltonian needed to supply the current at one boundary and remove it at another boundary of the macrosystem. Lacking this feature, recent theories of such transport phenomena contain no mechanism for irreversible entropy increase, and can be of little more than heuristic value. 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

Treating the radiation in the semiclassical dipole approximation, the Hamiltonian operator in the presence of an electric field e(t) may be written in atomic units as ... 

This treatment differs from the usual approach to molecule-radiation interaction through the inclusion of the contribution from the electric field from the beginning and by not treating it as a perturbation to the field free situation. The notation 7/ei(r R, e(f)) makes the parametric dependence of the electronic Hamiltonian on the nuclear coordinates and on the electric field explicit. 

Let us now examine what happens when we assume that the Hamiltonian for the interaction of the system with the electric field of the laser takes on different forms. The most common analytic form for the Hamiltonian is... 

Let us suppose that the system of interest does not possess a dipole moment as in the case of a homonuclear diatomic molecule. In this case, the leading term in the electric field-molecule interaction involves the polarizability, a, and the Hamiltonian is of the form ... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

The theoretical method, as developed before, concerns a molecule whose nuclei are fixed in a given geometry and whose wavefimctions are the eigenfunctions of the electronic Hamiltonian. Actually, the molecular structure is vibrating and rotating and the electric field is acting on the vibration itself. Thus, in a companion work, we have evaluated the vibronic corrections (5) in order to correct and to compare our results with experimental values. 

Moreover, for the observables depending on external electric field, its specific effect has to be investigated the electric field induces new terms in the nuclear Hamiltonian, due to the change of equilibrium geometry and the nuclear motion perturbation. Pandey and Santry (14) has brought to the fore this effect and calculated the correction which only concerns the parallel component. It is represented by the following expression ... 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

Orientational disorder and packing irregularities in terms of a modified Anderson-Hubbard Hamiltonian [63,64] will lead to a distribution of the on-site Coulomb interaction as well as of the interaction of electrons on different (at least neighboring) sites as it was explicitly pointed out by Cuevas et al. [65]. Compared to the Coulomb-gap model of Efros and Sklovskii [66], they took into account three different states of charge of the mesoscopic particles, i.e. neutral, positively and negatively charged. The VRH behavior, which dominates the electrical properties at low temperatures, can conclusively be explained with this model. 

---