Hamiltonian electric properties

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. 

The organization is transparent to the type of parameter provided that the derivative Fock operators are suitably constructed. The separation of terms in Eqn. (69) mentioned here corresponds to the idea of Takada, Dupuis, and King for skeleton Fock matrices [64]. A geometric derivative can be obtained with the same code as an electrical property, as can mixed derivatives. Furthermore, magnetic properties, which are unique because second derivatives of the Hamiltonian operator are not necessarily zero, fit into the DHF structure without modification. Open-shell wavefunctions can be em-... 

The non-relativistic PolMe (9) and quasirelativistic NpPolMe (10) basis sets were used in calculations reported in this paper. The size of the [uncontractd/contracted] sets for B, Cu, Ag, and Au is [10.6.4./5.3.2], [16.12.6.4/9.7.3.2], [19.15.9.4/11.9.5.2], and [21.17.11.9/13.11.7.4], respectively. The PolMe basis sets were systematically generated for use in non-relativistic SCF and correlated calculations of electric properties (10, 21). They also proved to be successful in calculations of IP s and EA s (8, 22). Nonrelativistic PolMe basis sets can be used in quasirelativistic calculations in which the Mass-Velocity and Darwin (MVD) terms are considered (23). This follows from the fact that in the MVD approximation one uses the approximate relativistic hamiltonian as an external perturbation with the nonrelativistic wave function as a reference. At the SCF and CASSCF levels one can obtain the MVD quasi-relativistic correction as an expectation value of the MVD operator. In perturbative CASPT2 and CC methods one needs to use the MVD operator as an external perturbation either within the finite field approach or by the analytical derivative schems. The first approach leads to certain numerical accuracy problems. 

Unusual magnetic and electrical properties might also arise from quasi one-dimensional crystal structures of these compounds. The acceptor stacks /especially TCNQ/ may be either regular, i, e. with equally spaced molecules, or alternating when composed of diads, triads or tetrads. In the latter case some substances exhibit EPR spectrum characteristic of mobile, thermally activated triplet states /triplet excitons/, The spectrum may result from the excitation of two or more coupled TCNQ entities /6, 7/. The triplet character of the paramagnetic excitation is shown by the anisotropic two-lines EPR spectrum which results from a zero field splitting of the triplet levels being described by the spin Hamiltonian /8/j... 

First-order electrical properties can conveniently be determined from expressions derived by using gradient theory. The Hamiltonian is augmented with an operator representing the studied property. By calculating the first derivative of the total energy with respect to the strength parameter of the property operator, one obtains an expression for the calculation of the specified first-order property. 

From the Tables one can see that both, the type of Hamiltonian (relativity) and the wave function (correlation) are important for the accurate calculations of the electric properties of molecular species. 

For purely electric properties (Xo = 0) the perturbed potential V(A) is strictly even, and the resulting DKH expressions are significantly simplified. The familiar DKH expressions for the unperturbed Dirac Hamiltonian can be directly transferred to the full system just by replacing the electron-nucleus interaction V by the even perturbed potential... 

This becomes an increasingly better approximation as the size of 8 is diminished. Of course, the numerical accuracy of the small energy difference in the numerator limits how small 8 should get. Another example is the finite field approach to finding molecular electrical properties. The energy of a molecule can be calculated in the absence of a field and then recomputed with a particular field, perhaps 0.0005 a.u. (2.6 X 10 V/cm) in the z-dirertion, entered into the Hamiltonian. From the energy difference and the size of the field used, an approximate value can be obtained for the first derivative of the energy with... 

The first step in any quantum mechanical evaluation of properties is the construaion of appropriate operators for the properties. Of course, even if not done explicitly, this means developing the Hamiltonian with the various embedded parameters. Electrical properties illustrate what is to be done in general. Using V, the rank-one polytensor ( q. [1], a molecular Hamiltonian for an applied electrical potential is... 

Our discussion of electric properties showed that at least for time-independent fields the operators derived for the relativistic case were just the four-component analogs of the nonrelativistic operators. For magnetic relativistic property operators we do not expect the connection to be so simple due to the fact that the field appears in different forms in the two versions of the Hamiltonian. For the relativistic case, we again write the Hamiltonian as... 

If applying a transformation derived from the decoupling of the unperturbed Dirac Hamiltonian was potentially unreliable for electric properties, for magnetic properties it simply will not work. The reason is that magnetic perturbations enter through the vector potential A and thus are odd operators. 

From a computational point of view FEM calculations are potentially more expensive than BEM ones. In fact, in both cases one has first to compute an electric property in some M representative points (and this is a calculation which roughly scales as MN /2 in ab initio calculations, where N is the number of basis functions), and then to introduce the coulombic potential of the charges placed at these M points in the Hamiltonian this extra computation still scales as MN /2. 

X2C ( eXact 2-Component ) is an umbrella acronym [56] for a variety of methods that arrive at an exactly decoupled two-component Hamiltonian, with X2C referring to one-step approaches [65]. Related methods to arrive at formally exact two-component relativistic operators are, for example, infinite-order methods by Barysz and coworkers (BSS = Barysz Sadlej Snijders, lOTC = infinite-order two-component) [66-69] and normalized elimination of the small component (NESC) methods [70-77]. We discuss here an X2C approach as it has been implemented in a full two-component form with spin-orbit (SO) coupling and transformation of electric property operators to account for picture-change (PC) corrections [14],... 

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. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

An averaged solvent electrostatic potential, obtained by averaging over the solvent confignrations, is included in the solnte Hamiltonian and electric and energy properties are obtained. The method provides resnlts for the dipole moment and solnte-solvent interaction which agree with the experimental valnes and with the resnlts obtained by other workers (Mendoza et al., 1998). 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

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