Energy third-order terms

With this choice, several third-order terms that appeared with the usual metric are eliminated. The new self-energy matrix in third order is asymmetric and is expressed by... 

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

Second-order and third-order results often bracket the true correction to pF - Three schemes that scale the third-order terms in various ways are known as the Outer Valence Green s Function (OVGF) [8], In OVGF calculations, one of these three recipes is chosen as the recommended one according to rules based on numerical criteria. These criteria involve quantities that are derived from ratios of various constituent terms of the self-energy matrix elements. Average absolute errors for closed-shell molecules are somewhat larger than for P3 [31]. 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

The operator H-mt in (15.4) contains, in general, terms of third and of fourth order with respect to the operators Ps and Pj. Terms of third order with respect to the operators Ps and Pj always lead to a weak exciton-exciton interaction. Since in such crystals the exciton bandwidth is much smaller than the energy required for the exciton formation, the third-order terms, which do not preserve the number of excitons, contribute to the exciton-exciton interation energy only in even orders of perturbation theory. 

Truncating of the Taylor series at the second-order terms means that the second derivatives of the Gibbs free energy difference (ACp, Ak, Ad) do not change significantly with tempierature and pressure. If this assumption is not valid, an extended analysis is necessary, where the third order terms proportional to 7 , T p, T and p are involved. As a consequence, the form of the ellipse remains but it gets distorted [114], in particular at high temperatures and pressures. 

We stress that it is specifically the second derivative that extracts the relevant information about the bond-stretching term. As noted above, this is modulated only by derivatives of the nonbonded term and the third-order terms. If, for example, we tried to parameterize v, by, say, changing just one C —H bond length in ethane and fitting the energy profile along this coordinate, then we will find that this profile contains contributions from other coordinates that actually change with the increase in the bond stretch. 

The real usefulness of expanding energy in terms of symmetry coordinates becomes apparent when we consider third-order terms [12b]. The condition for third-order terms to occur is similar to that for second-order terms, namely that Pijk is non-zero only if the product Dj ri)Dj(r )Dh(r ) transforms as the totally symmetric representation. This condition reduces the number of allowable Bijk s substantially. 

For the I3 example, the energy expansion in symmetry coordinates contains the following third-order terms. 

Hence the Leibler theory [43] indeed predicts a second order transition for f = 1/2, while for f + 1/2 where the third order term is present, a first order transition is predicted, S-1(q ) = 0 then only yields the limit of metastability of the disordered phase ( spinodal curve ). Thus using the higher order terms in Eq. (184) to actually compute the free energies of various candidates for the ordered structure, one finds which phase has the lowest free energy, and in this way the phase diagram shown in Fig. 42 (left part) has resulted [43]. 

Table I presents the results of EOM calculations of the three lowest IPs of nitrogen. Comparison of the first two columns of Table I demonstrates that there is a difference of 0.2 to 0.3 eV in the IPs when the EOM A matrix is symmetrized as by Simons, 21-order method I, and when the symmetrized form of the EOM equations, (21), 2j-order method II, is employed. The lack of symmetry in <0 (0,[//,0 ]) 0) in a 2 -order calculation arises from the inclusion of certain second-order A and terms, which contain the products of electron-electron interaction matrix elements with first-order double excitation correlation coefficients, and the neglect of other second-order A and A - terms, which involve second-order single excitation correlation coefficients multiplied by linear combinations of orbital energies. The discrepancies between the EOM 2 -order methods I and II are a measure of the importance of the terms due to single excitations in the ground-state wave function. In Section III.C, we consider the third-order terms not included in this primitive 2 -order EOM theory. The calculations imply although these terms are small, they are certainly not negligible. ...

Let us first look a little further into how higher-order terms in the pseudopotential affect properties. The counterpart of the energy wave-number characteristic given in Eq. 17-5, for third-order terms, has been evaluated (Lloyd and Sholl, IflOS Brovman, Kagan, and Holas, 1971). It contains three powers of the form factor, and a double sum over a q and a q is required in the evaluation. An interesting test for the importance of such terms was noted several years ago by... 

It is generally more convenient to transform to normal mode coordinates and into second quantized form. Then the third order term in the elastic potential energy expansion becomes... 

---