Cross-interaction terms

In the original molecular mechanics work, a steric energy, E, for a molecule was defined as the sum of the potentials for bond stretch, angle bend, Ee, torsional strain, E, nonbonded interactions, Ey f, and other terms, such as Urey-Bradley terms, cross-interaction terms, and electrostatic terms, (8). 

The subtraction scheme can be easily understood in the following way the whole system is described at the low level of theory (Efhigh level (E high w - E low lmel). The additive scheme is simply sum of low and high level descriptions of outer and inner parts, respectively, augmented by a cross-interaction term (E JJ ). 

The cross interaction terms Vi ° and (t//k)i with i j that appear in the mixing rules are related to the pure fluid quantities by... 

We need a better model, and the next step, naturally, is to include proper quadratic terms, in addition to the cross (interaction) terms already in the model. To have a sufficient number of degrees of freedom, we must include levels other than +1 in the training set runs. Using all 52 runs in the table to fit a full quadratic model, we arrive at the equation... 

In Equation 4, M, and Mj are the molar masses of components 1 and 2, respectively. The two cross interaction terms and Vj, are adjustable parameters of the fit they... 

All applications of the lattice-gas model to liquid-liquid interfaces have been based upon a three-dimensional, typically simple cubic lattice. Each lattice site is occupied by one of a variety of particles. In the simplest case the system contains two kinds of solvent molecules, and the interactions are restricted to nearest neighbors. If we label the two types of solvents molecules S and Sj, the interaction is specified by a symmetrical 2x2 matrix w, where each element specifies the interaction between two neighboring molecules of type 5, and Sj. Whether the system separates into two phases or forms a homogeneous mixture, depends on the relative strength of the cross-interaction W]2 with respect to the self-inter-action terms w, and W22, which can be expressed through the combination ... 

The background is substantial under the edges to be measured the cross-section term, a, in equation (1) contains contributions from other interactions, together with the required edges that combine to produce a signal ... 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

The cross-bonded terms cxCy/ ax and caCaFay result from interaction of X and Y with the wrong bonding hybrids on A. The magnitudes of these terms can usually be judged from simple overlap considerations. Unless X and Y are of quite dissimilar electronic character, the two cross-bonded terms are inherently of similar magnitude and therefore tend to cancel one another out. Thus, cross-bonded terms tend to make only minor contributions to geminal delocalization. 

It is assumed that most of the electron spin density resides on the metal, but that a certain small part of it, given by the quantity p , is delocalized to the ligand heteroatom L. The first term is the point-dipole interaction term, the second corresponds to the dipolar interaction between the nuclear spin under consideration and the spin-density on the atom L and the last term describes the cross-correlation of the two dipolar interactions (we discuss the issue of cross-correlation phenomena in more general terms in Section II. D and III.B). The quantity is the effective distance from the nuclear spin... 

For several covariates we simply introduce a cross-product term for each covariate with corresponding coefficients d, 2 and dj. The presence of treatment-by-covariate interactions can then be investigated through these coefficients. 

We can also investigate the presence of treatment-by-covariate interactions by including cross-product terms ... 

In earlier chapters we examined systems with one or two types of diffusing chemical species. For binary solutions, a single interdiffusivity, D, suffices to describe composition evolution. In this chapter we treat diffusion in ternary and larger multicomponent systems that have two or more independent composition variables. Analysis of such diffusion is complex because multiple cross terms and particle-particle chemical interaction terms appear. The cross terms result in TV2 independent interdiffusivities for a solution with TV independent components. The increased complexity of multicomponent diffusion produces a wide variety of diffusional phenomena. 

Now, the non-adiabatic electron transitions is examined only when electron matrix element Fif is small (see the criterion (10) and (10a)). It is the criterion of applicability of the perturbation theory on F f, but it is not the criterion of applicability of the concept of non-adiabatic transition between two crossing diabatic terms. As it is known (see, for example, ref. [5]) the true image of terms is changed on taking into account the interaction V. Denote two terms without inter-term interaction as E[(R) and E (R), where R is the generalized nuclear coordinate. If the crystal phonons (or the outer-sphere variables in a polar medium) only participate in the transition, then E[(R) and E (R) are the parabolic terms independent of the value of shift of... 

Fig. 2.3. The adiabatic terms of a system as functions of the generalized coordinate R. The dotted lines denote the crossing diabatic terms in the region of strong influence of the inter-term interaction. The indexes i and f denote the regions of the initial and final states.

Bell et al. [33] proposed an analytical formula, widely known in the literature as the Belfast ionization (BELI) formula [34] that contains the dipole interaction term for the electron-impact ionization of atoms and ions. It has been applied to light atomic and ionic targets with species-dependent parameters. Godunov and Ivanov [34] applied the BELI formula to the El ionization of Ne 1 ions. Here also no generality as to parameters of the formula was provided regarding the species-dependent parameters. Moreover, the BELI formula does not make any allowance for relativistic effects. Haque et al. [35-38] have proposed a modification of this BELI model for evaluating the El K-, L-, and M-shell ionization cross sections of atoms. The relativistic and ionic effects are also incorporated in their modified BELI (MBELL) [35-38] model in addition to generalizing the species-independent... 

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