Perturbation term

Results from fully relativistic calculations are scarce, and there is no clear consensus on which effects are the most important. The Breit (Gaunt) term is believed to be small, and many relativistie calculations neglect this term, or include it as a perturbational term... 

To the lowest order of approximation, which is used to evaluate the collision integrals for the perturbation terms (vzfx and we take m/M — 0. There is, thus, no interchange of energy between the electrons and neutral atoms, so that... 

Friedly (F4) expanded the theoretical analysis of Hart and McClure and included second-order perturbation terms. His analysis shows that the linear response of the combustion zone (i.e., the acoustic admittance) is not sign-ficantly altered by the incorporation of second-order perturbation terms. However, the second-order perturbation terms predict changes in the propellant burning rate (i.e., transition from the linear to nonlinear behavior) consistent with experimental observation. The analysis including second-order terms also shows that second-harmonic frequency oscillations of the combustion chamber can become important. 

The two perturbation terms are specific to the given interface and are experimentally inseparable. They measure the contact potential difference at the M/S contact. However, since no cpd is measured in this case <5/M + S%s are grouped into a single quantity denoted by X, called the interfacial... 

Figure 5 shows a sketch of the plot of Eam0 vs. 0 according to Eq. (28). If a metal is taken as a reference surface, a straight line of unit slope through its point would gather all metals with AX = O, i.e., those whose sum of perturbation terms is exactly the same. For these metals the difference in pzc is governed only by the difference in 0. 

With reference to Eq. (26), an effect of temperature is expected since both 0M and the perturbation terms depend on temperature. In particular, the effect can be written as a temperature coefficient ... 

It is worthy of mention that the perturbation terms are actually... 

If the perturbation function shows cubic symmetry, and in certain other special cases, the first-order perturbation energy is not effective in destroying the orbital magnetic moment, for the eigenfunction px = = i py leads to the same first-order perturbation terms as pi or pv or any other combinations of them. In such cases the higher order perturbation energies are to be compared with the multiplet separation in the above criterion. 

Finally, the E-state index S of atom i is the sum of the intrinsic state and of the perturbation term ... 

Perturbation terms in the Hamiltonian operator up to still lead to the uncoupling of the nuclear and electronic motions, but change the form of the electronic potential energy funetion in the equation for the nuclear motion. Rather than present the details of the Bom-Oppenheimer perturbation expansion, we follow instead the equivalent, but more elegant procedure of M. Bom and K. Huang (1954). 

Since the perturbation corrections due to b q and b q vanish in first order, we must evaluate the second-order corrections in order to find the influence of these perturbation terms on the nuclear energy levels. According to equation (9.34), this second-order correction is... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Ho and a linear perturbative term. The expansion is made around a nuclear configuration Qx, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

Finally, we discuss the effect of nonlinear coupling on domain growth, decoherence, and thermalization. As the wave functionals l/o of Ho are easily found, Eq. (16) leads to the wave functional beyond the Hartree approximation. Putting the perturbation terms (19) into Eq. (16), we first find the wave functional of the form... 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

The perturbation method is a unique method to determine the correlation energy of the system. Here the Hamiltonian operator consists of two parts, //0 and H, where //0 is the unperturbed Hamiltonian and // is the perturbation term. The perturbation method always gives corrections to the solutions to various orders. The Hamiltonian for the perturbed system is... 

The GW method seems very limited, given the very large number of perturbative terms that are omitted. However, experience has shown great improvements, and predictive precision over Hartree-Fock calculations. The reason is that, with the Coulomb potential... 

As was discussed earlier, the two intermolecular orbital perturbation terms,... 

When a molecular system is placed in static and/or dynamic external electric fields, a perturbation term has to be added to the unperturbed time independent Hamiltonian, Hg ... 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

Nakatsuji and Yasuda [56, 57] derived the 3- and 4-RDM expansions, in analogy with the Green function perturbation expansion. In their treatment the error played the role of the perturbation term. The algorithm that they obtained for the 3-RDM was analogous to the VCP one, but the matrix was decomposed into two terms one where two A elements are coupled and a higher-order one. Neither of these two terms can be evaluated exactly thus, in a sense, the difference with the VCP is just formal. However, the structure of the linked term suggested a procedure to approximate the A error, as will be seen later on. 

---