Energy quadratic form

This calculation shows explicitly the correction to the MFA. With the quadratic form considered for and the local approximation, the calculation of k T n det 0A) can be performed exactly [39]. The change in the free energy by unit of volume, A.Fcharging when we switch on the charge is given by... 

Finally, we mention an approach based on using a different energy (or difference) function. The quadratic form used above (equation 10.58) is simple to use but it is not the only form possible. In fact, we could go through the same derivation steps as above by using any function /(Of, Sf) of the net s output Of and actual output... 

A simple quadratic form of Eq. (34.10) is due to an identical parabolic form of the free-energy surfaces f/, and U. Since the dependence of the activation free energy on AF is nonhnear, the symmetry factor a may be introduced by a differential relationship,... 

The first energy moment of the isolated system is not conserved and it fluctuates about zero. According to the general analysis of Section IIB, the entropy of the isolated system may be written as a quadratic form,... 

Equation 2 is not a poor representation of the energy function because it can be shown that a Morse potential in real distances assumes a simple quadratic form if one uses a dimensionless bond order coordinate.1263 In any event, minimization of this function leads to the conclusion that the minimum energy is attained when all bonds have the same length. Furthermore, a bond alternating distortion that lengthens and shortens a pair of adjacent bonds by Ar can be shown to raise the cr-energy as in eq 3a. 

The kinetic co-energy. t is assumed to be a quadratic form in i/t, as is customary for immiscible fluid mixtures or for materials with affine microstructure (see, e.g., [4, 5]) then Xi coincides with the kinetic energy and it is... 

The concept of adiabacity in e.t. processes has gained importance in recent years, and the question does arise to what extent it may influence the observation of the M.I.R. In principle, the occurence of the M.I.R. is related only to the quadratic form of the activation energy, not to the form of the pre-exponential factor. The M.I.R. should therefore be observed for both adiabatic and nonadiabatic reactions. However, if the observable rate of an adiabatic process is controlled by the solvent relaxation time, the influence of the exponential factor may be negligible [18]. 

In the above expression Z stands for the configuration integral which differs from the partition function Q by being calculated only over the coordinates of the molecules and not over their momenta. This is possible because the coordinate and momentum parts of the whole phase space are run over independently throughout the integration and to a simple quadratic form of the kinetic energy, which thus can be integrated immediately. 

Second order corrections to the energy of sp3 carbon atom. In order to construct the required mechanistic picture, the estimate of the restoring force which opposes both the quasi- and pseudorotation (deformation) of the hybridization tetrahedra is necessary. That can be obtained by a linear response procedure. For the sp3 carbon atom in the symmetric tetrahedral environment, the related resonance energy is a diagonal quadratic form with respect to small quasi- and pseudorotations together with triply degenerate eigenvalues [44,45] ... 

By eliminating the variables describing the motion of the center of mass and the overall rotation, the kinetic energy T of the nuclei assumes a homogeneous quadratic form in the generalized velocities Qj = dQJ/dt (J),... 

When making such a decision, some considerations should be borne in mind. First, if one is using a refinement scheme that produces a known distribution of structures, then one can calculate the likely deviation that the pseudo-energy terms will permit. For example, if an MD refinement is used, structures will be able to cross barriers of about BT. Then, if one is using a quadratic form to enforce distance restraints, one could recast Eq. [9] as follows to get an idea of the violations that would be permitted for a given force constant and temperature ... 

The meaning of 0, the angle of misalignment, is seen in Fig. 8-3. Expanding Eq. (8-7) for small 0 and small we obtain 0 = 2fi /3 the same result applies for every bond. We write the change in hybrid covalent energy, quadratic in 9, in the form... 

The normal coordinates thus transform both kinetic and potential energies into quadratic forms and are orthonormal. Each of them can therefore be treated as an harmonic oscillator. As a result of using a Cartesian system, six of the eigenvalues are zero, because they correspond to overall motions of the undeformed molecule in the absence of restoring forces. 

We next want to arrive at a form of the energy balance for the case where heat capacities are strong functions of temperature over a wide temperature range. Under these conditions the mean values used in Equation (8-30) may not be adequate for the relationship between conversion and temperature. Combining Equation (8-23) with the quadratic form of the heat capacity, Equation (8-20),... 

The second-order change in the free energy AA . thus appears to be a quadratic form in these coefficients. If A A has to be positive for arbitrary variation coefficients ca P, the Hessian of this form has to be positive definite. By Fourier transforming the coefficients, the Hessian can be block-diagonalized with blocks... 

The quadratic form of the free energy implies a linear relationship between force and the end-to-end vector, that is valid for small extensions ... 

The quadratic form, however, rests on the very general assumption that whatever structural features characterize the environment, conformational, polarization, density, or other physical properties respond linearly to the field changes, that is, changes in electric field, pressure, or other forces. The quadratic form of the activation free energy is thus broadly vahd independently of the nature of the solvation and reorganization. 

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