Energy quadratic term

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. 

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... 

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. 

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. 

An example that is closely related to organic photochemistry is the E x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic term in Eq. (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat. The polar coordinates p and [Pg.462]

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

Energies of reorganization are typically of the order of 0.5 - 1.5 eV applied overpotentials are often not higher than 0.1 - 0.2 V. For small overpotentials, when A 2> eo, the quadratic term in the energy of activation may be expanded to first order in eo this gives the following expression for the rate constant of the oxidation reaction ... 

Burdett (35-38) has extended the AOM by the introduction of a quartic term in the expansion of the perturbation determinant as a power series in the overlap integral Sx. In the conventional AOM, only the quadratic term (proportional to Sx) is considered. In closed-shell systems, the sum of the energies of the relevant orbitals is independent of angular variations in the molecular geometry if only the quadratic term is used. This is no longer true if the quartic term is included, and it is possible to rationalise many stereochemical observations. 

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

If the proof in section 2 is studied carefully, it will be seen that there is nothing in the derivation of this law which could not be applied mutatis mutandis to any kind of energy representable in the form mec2. For, apart from general considerations of probability, the only condition assumed in the proof was that the total kinetic energy could be expressed as a sum of three quadratic terms— equation 1 of section 2 multiplied by I/2m expresses this condition. 

The simple exponential factor e ElRT, as we have seen, is only applicable strictly to represent the probability of a total energy E of the colliding molecules in two quadratic terms. As an approximation it may be used for any quite small number of terms. In fact its applicability with such success to bimolecular reactions might be taken to show that the activation process in such reactions was a relatively simple one, whatever its exact nature might be. 

If we regard the two colliding molecules as one system, the fraction of such systems which possess energy greater than E in n quadratic terms is... 

Several typically negligible effects have been neglected in the derivation of Eq. 3.83, including (1) interactions between the interstitials, (2) effects of the interstitials on the local elastic constants, (3) quadratic terms in the elastic energy, and (4) nonlinear stress-strain behavior. A more complete treatment, applicable to the present problem, takes into account many of these effects and has been presented by Larche and Cahn [21]. 

Again, as in the derivation of Eq. 3.82, quadratic terms in the elastic energy, which are of lower order in importance, have been neglected (see Larche and Cahn [21]). 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

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