Linear term

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

CO, CO, co, and o, respectively. The integrals in Eqs. (E.9) and (E.IO) will then be different from zero only if the integrands are invariant under all symmetry operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Q+ and Q- vanish in and H In turn. 

For the external eleetrie field ease at hand, this result says that the field-dependenee of the state energy will have a linear term equal to... 

Keeping only the linear term, the transition dipole moment is given by... 

Can the relationship be approximated by an equation involving linear terms for the quantitative independent variables and two-factor interaction terms only or is a more complex model, involving quadratic and perhaps even multifactor interaction terms, necessary As indicated, a more sophisticated statistical model may be required to describe relationships adequately over a relatively large experimental range than over a limited range. A linear relationship may thus be appropriate over a narrow range, but not over a wide one. The more complex the assumed model, the more mns are usually required to estimate model terms. 

To translate the axes to a new origin at (h, k), substitute for x and y in the original equation x + h and y + k. Translation of the axes can always be accomplished to eliminate the linear terms in the second-degree equation in two variables having no xy term. 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

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... 

Figure 15.28 Decomposition of a reaction barrier into a parabola and a linear term...

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Substituting (14.42) into Eq. (14.19), we sec that, once more, the linear term Vs can be neglected with respect to the exponential term. In other words, most of the gate-source ohmic drop occurs in the insulator, so that Eq. (14.23) becomes simply... 

Of course, any set of experimental data can be described by selecting an appropriate empirical equation with an arbitrary set of constants. However, comparing a vast wealth of the known results of measurements of suspension viscosity, it should be admitted that a universal formula for ther (cp) dependence does not exist, and significant discrepancies may begin already from a linear term, so that physical reasons for exagerated values of the coefficient bt as compared to 2.5 should be looked for. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

It is shown that under normal conditions (and only these are generally of interest in applications) one can disregard the terms P3 and Q2 and consider only the linear terms. 

In other words, in normal cases the nature of equilibrium is determined only by the linear terms. This is also intuitively obvious since, as the trajectory approaches the singular point (at the origin), both x and y decrease indefinitely so that ultimately only the linear terms of the first order of magnitude remain. 

Stability on the Basis of Abridged Equations.—The argument used in deriving the characteristic equation (3.8) was to neglect P2(x,y) and Qz(x,y), and to proceed on the basis of linear terms. It is possible to obtain more precise information regarding the validity of this assumption. Consider the system of differential equations... 

The linear term in Cp m for metals results from the contribution to the heat capacity of the free electrons. It can become important at very low temperatures where the T3 relationship becomes very small. For example, the electronic contribution to the heat capacity of Cu metal is 1.2% at 30 K, but becomes 80% of the total at 2 K.e... 

Tor the purpose of this brief account we will provide only a notional definition of optical aberrations. In an optical system, the angular coordinates of incident rays are transformed according to sequential applications of Descarte s law from one optical surface to the next. Aberrations are essentially the non-linear terms of the transformation, the angular coordinates of emerging rays not being strictly proportional to those of the incident ones -thereby generating distorted and/or blurred images. 

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... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

While we are at it, we estimate the interaction of the domain with the higher order strain, at least due to the term (B.l), in the frequency region of interest. The next order term in the k expansion in the surface integral from Eq. (B.2) has the same structure but is scaled down from the linear term by a factor of kR. At the plateau frequencies 0C) )/3O, kR < 0.5 as immediately follows from the previous paragraph. While this is not a large number, it is not very small either. Therefore this interaction term is of potential importance. 

Assuming that at the initial instant the angular velocity dp/dt — 0, we conclude that the mass m, placed at any point around the point of equilibrium, remains at rest. Of course, it is only an approximation, because we preserved in the power series, (Equation (3.146)), only the linear term and discarded terms of higher orders. Formally, this case is characterized by infinitely large period of free vibrations... 

The advantage of the explanations of Equation (3) is its simplicity. It seems certain that the bulk (r ) and surface (r ) terms exist. The simplest way to get a minimum in AG is to include the linear term. The problem with this explanation is that it leaves us with the question why would edge sites have negative hence favorable energies when face sites have positive, disfavorable energies We do not know. We do know, however, that the system with surface-adsorbed ligands is complex. Perhaps that complexity can lead to such an unexpected consequence. 

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