Perturbation lowest order

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

We restrict ourselves again to symmetric tetraatomic molecules (ABBA) with linear eqnilibrium geometi7. After integrating over electronic spatial and spin coordinates we obtain for A elecbonic states in the lowest order (quartic) approximation the effective model Hamiltonian H — Hq+ H, which zeroth-order part is given by Eq. (A.4) and the perturbative part of it of the form... 

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... 

Let us next calculate JP2(0) to lowest order in perturbation theory. Quite generally, if p p ... 

The results of the Debye theory reproduced in the lowest order of perturbation theory are universal. Only higher order corrections are peculiar to the specific models of molecular motion. We have shown in conclusion how to discriminate the models by comparing deviations from Debye theory with available experimental data. 

To illustrate the accuracy of the perturbation theory these results are worth comparing with the well-known values of h and I4 for t = 1 rigorously found from first principles in [8]. It turns out that the second moment in Eq. (2.65a) is exact. The evaluation of I4, however, is inaccurate its first component is half as large as the true one. The cause of this discrepancy is easily revealed. Since M = / and (/) = J/xj, the second component in Ux) is linear in e. Hence, it is as exact in this order as perturbation theory itself. In contrast, the first component in IqXj is quadratic in A and its value in the lowest order of perturbation theory is not guaranteed. Generally speaking... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

In calculating the transition probability for the nonadiabatic reactions, it is sufficient to use the lowest order of quantum mechanical perturbation theory in the operator V d. For the adiabatic reactions, we must perform the summation of the whole series of the perturbation theory.5 (It is insufficient to retain only the first term of the series that appeared in the quantum mechanical perturbation theory.) Correct calculations in both adiabatic and diabatic approaches lead to the same results, which is evidence of the equivalence of the two approaches. 

Abstract For the case of small matter effects V perturbation theory using e = 2V E/ Am2 as the expansion parameter. We derive simple and physically transparent formulas for the oscillation probabilities in the lowest order in e which are valid for an arbitrary density profile. They can be applied for the solar and supernova neutrinos propagating in matter of the Earth. Using these formulas we study features of averaging of the oscillation effects over the neutrino energy. Sensitivity of these effects to remote (from a detector), d > PE/AE, structures of the density profile is suppressed. 

Being based on lowest-order perturbation theory, Bethe theory approximates equation (3) by T—2 Zie lp) lmv so that... 

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

Even though the total kernel in (1.23) is unambiguously defined, we still have freedom to choose the zero-order kernel Kq at our convenience, in order to obtain a solvable lowest order approximation. It is not difficult to obtain a regular perturbation theory series for the corrections to the zero-order ap>-proximation corresponding to the difference between the zero-order kernel Kq and the exact kernel Kq + 6K... 

Suppose that the system is initially uniform with an unstable disordered structure (i.e., r] = 0). For instance, the system may have been quenched from a high-temperature, disordered state, 77 = 1 represents the two equivalent equilibrium ordered variants. If the system is perturbed a small amount by a one-dimensional perturbation in the 2-direction, 77(f) = 6(t) sin(/32). Substituting this ordering perturbation into Eq. 18.26 and keeping the lowest-order terms in the amplification factor, 6(t),... 

As o) increases further above 1 In3 a single photon drives the initially populated state closer and closer to the ionization limit, and ionization occurs with the absorption of fewer photons. Few photon processes are well described by lowest order perturbation theory, which shows that the rates are proportional to E2N, where N is the number of photons absorbed. For small N such processes are not well described by a threshold field, and it is not meaningful to discuss ionization threshold fields in this case. 

In the lowest order of perturbation theory, the energy levels of the three-dimensional anharmonic oscillator are... 

The m — 6 system will again be used as an example. The guest molecules cause the mixing of the lowest (r = 0) wave function with three other wave functions derived from p = 1, p = 2, and p — 3, as described in the secular equation (14). If cx,. . ., c5 are the coefficients of the basis functions in order of increasing energy in the perturbed lowest state, we have, by perturbation theory for small a,... 

PTE and PTD describe, respectively, the effects of the solvation on the electron correlation on the solvent polarization and vice versa the PTED scheme leads instead to a comprehensive description of these two separate effects, revealing coupling between them. However, the PTDE scheme is not suitable for the calculation of analytical derivatives, even at the lowest order of the MP perturbation theory. 

Table 1. Lowest order of perturbation theory (LOPT) in which various linked and unlinked clusters first appear51)...

---