Quadratic term

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Harmonic analysis is an alternative approach to MD. The basic assumption is that the potential energy can be approximated by a sum of quadratic terms in displacements. 

To ensure that the arrangement of four atoms in a trigonal planar environment (e.g., a sp -hybridized carbon atom) remains essentially planar, a quadratic term like V(0) = (fe/2) is used to achieve the desired geometry. By calculating the angle 9 between a bond from the central atom and the plane defined by the central... 

A larger value for the bending force constant K0 leads to a greater tendency for the angle to remain at its equilibrium value 0g. There may be cubic, quartic, etc. terms as with the corresponding bond stretch term in addition to the quadratic term shown here. 

Thus, from the form of (4.7), shock pressure is given as the sum of a linear and quadratic term in particle velocity, based on the data of Table 4.1. A pressure volume relation can be obtained by combining (4.6) with (4.1) to yield... 

In the new coordinates the action, expanded up to quadratic terms, reads... 

Tijssen et al. (7) showed that the dimensionless retention time of a macromolecule, T, can be described by a modified quadratic term ... 

The left side is expanded in a binomial series, which is truncated after the quadratic term. Combination leads to... 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

Rule i 4, on the other hand, has both a linear and quadratic term, so that / (p = 0) > 0 in general, and is therefore predicted to have a second order (or continuous) phase transition. Although the mean-field predictions are, of course, dimension-independent, they are expected to become exact as the dimension d —7 oo. In practice, it is often found that there exists a critical dimension dc above which the mean-field critical exponents are recovered exactly. 

If the perturbation is small enough that the quadratic term can be ignored, the re-equilibration process is first-order. Its relaxation time is... 

Substitution of the second form of each of these equations into the rate law, imposition of the equilibrium condition i[A]"[B] = -i [P] [R] , and the dropping of quadratic terms give this expression for the relaxation time ... 

We assume that the energy E of the molecule contains a quadratic term in the charge... 

As these expressions correspond to the CC energy derivative, they must give size-extensive results. However, the price we pay is that the energy of a given order requires wave function contributions of the same order. Furthermore, these non linear terms are difficult to evaluate. The quadartic in term in second-order, requires comparable difficulty to the quadratic terms in a CCSD calculation... 

Some experimenting nught be necessary if it turns out that quadratic terms such as improve the fit between the model and the data. However, the relevant point is that such a model is only a means to refine and speed up the process of finding optimal conditions. For this purpose it is counter-productive to try for a perfect fit, it might even be advantageous to keep the model simple and throw out all but the best five to 10 experiments, choose new conditions, and then return to the work bench. 

The simplest way to model isomerization is to add a quadratic term to the spherical pendulum Hamiltonian [10, 24]. Thus... 

The volume integral in Eq. (B.2) produces a quadratic term, which is roughly equal to (Vcj)) fy (Pr (d k). We then proceed in a completely identical fashion to our earlier estimate of g. Assuming the diplacements within the droplet are random, one gets for the integral P dik, where the factor of comes... 

The selection of the number of PLS-components to be included in the model was done according to the PRESS criterion (Section 36.3). Note that the result is comparable to the one which we obtained earlier by means of the simple Hansch analysis (Section 37.1.1). Hence, in this case, there is no obvious benefit to include a quadratic term of log P in the model. 

We next consider the more general situation where the angular frequency co(k) is not proportional to k, but is instead expanded in the Taylor series (1.13) about (k — ko). Now, however, we retain the quadratic term, but still neglect the terms higher than quadratic, so that... 

The subscript y has been included in the notation y(x, t) in order to distinguish that wave packet from the one in equations (1.14) and (1.15), where the quadratic term in cD(k) is omitted. The integral over k may be evaluated using equation (A. 8), giving the result... 

Each term of this may be equated to half the corresponding one on the right-hand side of Eq. (98). From the quadratic term it follows that... 

The exchange repulsion energy in EFP2 is derived as an expansion in the intermolecular overlap. When this overlap expansion is expressed in terms of frozen LMOs on each fragment, the expansion can reliably be truncated at the quadratic term [44], This term does require that each EFP carries a basis set, and the smallest recommended basis set is 6-31-1— -G(d,p) [45] for acceptable results. Since the basis set is used only to calculate overlap integrals, the computation is very fast and quite large basis sets are realistic. 

Fig. 66. Concentration dependence of r H(c)/r s result of seperate fits o results of viscosity measurements on PDMS solutions (Mw = 7400). The result of the simultaneous fit considering the linear term in r H(c) = r 0(l + [r ]c + kH[r ]2c2) is given by the solid line the inclusion of a quadratic term leads to the dashed line. The point-dashed line indicates the macroscopic viscosity for M = 60000 g/mol. (Reprinted with permission from [40]. Copyright 1984 American Chemical Society, Washington)...

The subscripts have the same meaning as in (12.29), (12.30). Indeed, (12.29), (12.30) show that in the parabolic free energy function, the static free energy is the linear term with respect to Aq, while the relaxation free energy is the quadratic term. Thus,. A/l[A) and /L4[fxc are, respectively, the static and the relaxation free energies to insert a unit charge into the reactant state. 

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