Lowest order expansion

The fact that matrix elements of the fundamental band are dependent on the rotational quantum numbers j, f cannot be ignored. As a consequence, many more B coefficients must now be evaluated which, in principle, poses no special problem. The volume of data needed renders the task awkward. Molecular spectroscopists have for generations coped with similar problems which were solved with the so-called Dunham expansion in terms of j(j+1). Specifically, for our purpose, the lowest-order expansion for the fundamental band looks like [63]... 

OAI further claimed that the lowest order expansion was sufficiently good. However, one sees that the OAI tables does not actually show that such is the case. Higher order terms are definitely needed [15], indicating that a microscopic IBA is not as simple as is the phenemenological IBA. 

Where does this fundamental difference to (2.27) come from To answer this question one has to remember that in the first two approaches some arbitrary orbital-dependent xc has been assumed, i.e. the form of E c has not been specified. On the other hand, in the present approach the use of the Dyson equation for both the KS and the interacting system automatically implies the use of the exact E c- In order to make closer contact between the first two and this third derivation, one thus has to study the exact xc in more detail. This will be the subject of Sect. 2.4. In the present section the comparison of (2.45) with (2.27) will for simplicity be restricted to the x-only limit, which corresponds to a lowest order expansion of E in the coupling constant e. In this limit the right-hand side of (2.45) reduces to... 

Wc now consider the lowest-order expansion coefficients of (fc) appearing on the right side of Eq. (F.29). Using the general definition (P.19a) for the Fourier coefficients and performing the required derivatives, we obtain... 

In the multipole expansion of the two-electron integral (9.13.23), the interaction tensor (9.13.22) depends only on the relative position of the origins about the charge distributions - all information about the charge distributions is contained in the multipole moments (9.13.20) and (9.13.21). As an illustration, consider the lowest-order expansion of the two-electron integrals. The moments of the charge distributions are given as... 

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... 

Introducing these expansions into the expression for af i gives rise to terms of various powers in MX. The lowest order terms are ... 

FIG. 8 PIMC results (symbols) of the imaginary-time correlations G r) versus imaginary time for densities p = 0.1,0.2,..., 0.7 from bottom to top the temperature is T = 1. The full line shows the results for Q r) according to the lowest-order virial expansion the dashed lines give the MF values of Q r) for the densities p = 0.7, 0.6, and 0.5 from top to bottom. (Reprinted with permission from Ref. 175, Fig. 1. 1996, American Physical Society.)... 

The charge distribution of the molecule can be represented either as atom centred charges or as a multipole expansion. For a neutral molecule, the lowest-order approximation considers only the dipole moment. This may be quite a poor approximation, and fails completely for symmetric molecules which do not have a dipole moment. For obtaining converged results it is often necessarily to extend the expansion up to order 6 or more, i.e. including dipole, quadrupole, octupole, etc. moments. 

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. 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

There are other sources of nonlinearity in the system, such as the intrinsic anharmonicity of the molecular interactions present also in the corresponding crystals. While these issues are of potential importance to other problems, such as the Griineisen parameter, expression (B.l) only considers the lowest order harmonic interactions and thus does not account for this nonlinear effect. We must note that if this nonlinearity is significant, it could contribute to the nonuniversality of the plateau, in addition to the variation in Tg/(do ratio. It would thus be helpful to conduct an experiment comparing the thermal expansion of different glasses and see whether there is any correlation with the plateau s location. 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

When going beyond the lowest-order term in derivatives, we need a counting scheme. For theories with only one relevant scale (such as QCD at zero chemical potential), each derivative is suppressed by a factor of This is not the case for theories with multiple scales. In the CFL phase, we have both and the gap, A, and the general form of the chiral expansion is [31] ... 

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. 

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

If we further assume that o- (/) does not change significantly in the interval xlu - 8 t xlu + 8, and hence a- t) = cr xlu) within the integration limits, we conclude that only the first term in the expansion will contribute to lowest order in 8, since to order 8,... 

One must expect the presence of mixed terms of the form k B in the expansion. The term of lowest order a —2, d = l), contributing oczf to the stopping cross section, would indicate a difference between the Barkas-Andersen correction evaluated from the Born series and the Bohr model, respectively. While such a comparison has not been performed in general terms, a numerical evaluation for the specific case of Li in C revealed a negligible difference [24]. 

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