Unique perturber approximation

The idea of an effective Hamiltonian for diatomic molecules was first articulated by Tinkham and Strandberg (1955) and later developed by Miller (1969) and Brown, et al., (1979). The crucial idea is that a spectrum-fitting model (for example Eq. 18 of Brown, et al., 1979) be defined in terms of the minimum number of linearly independent fit parameters. These fit parameters have no physical significance. However, if they are defined in terms of sums of matrix elements of the exact Hamiltonian (see Tables I and II of Brown, et al., 1979) or sums of parameters appropriate to a special limiting case (such as the unique perturber approximation, see Table III of Brown, et al., 1979, or pure precession, Section 5.5), then physically significant parameters suitable for comparison with the results of ab initio calculations are usually derivable from fit parameters. 

In the unique perturber approximation (Zare, et al., 1973), only one 2E electronic state is considered to be responsible for the A-doubling of the 2II state. If the interacting 2II and 2E states belong to the 7r1 and a1 configurations, then the matrix elements of the exact many-electron wavefunctions may be replaced by the one-electron matrix elements (see Section 3.5.4),... 

Finally, the pure precession approximation requires, in addition to the unique perturber, identical potential assumptions, that the interacting 2II and 2 states axe each well described by a single configuration, that these configurations are identical except for a single spin-orbital, and that this spin-orbital is a pure... 

For the OH radical, the values of p and q for the X2n (ground state can be attributed to a unique perturber interaction with the A2E+ (ow4) state. The pure precession approximation simply ignores the contribution of the atomic orbital to the per molecular orbital. For all hydrides, the oTsh orbital makes a negligible contribution to Hso and BL+ matrix elements. For OH, the H-atom contributions to a+ and b are 4 x 10 4% and 1%, respectively (Hinkley, et al., 1972). 

However, by constructing a nested sequence of successively larger discrete spaces and approximations therein we hope to end up with some approximation of a unique invariant measure, which is then implicitly defined via the constructed sequence of subspaces. An expression of this mathematical consideration is the multilevel structure of the suggested algorithm - details see below (Section 3.2). In physical terms, we hope that the perturbations introduced by discretization induce a unique and smooth invariant measure but are so weak that they do not destroy the essential physical structure of the problem. 

The number of applications of the non-Abelian Stokes theorem is not as large as in the case of the Abelian Stokes theorem nevertheless, it is the main motivation for formulating the non-Abelian Stokes theorem. It is interesting to note that in contradistinction to the Abelian Stokes theorem, whose formulation is homogenous (unique), different formulations of the non-Abelian Stokes theorem are useful for particular purposes and applications. From a purely techincal point of view, one can classify applications of the non-Abelian Stokes theorem as exact and approximate. The term exact applications means that one can perform successfully an exact calculus to obtain an interesting result, whereas the term approximate application means that a more or less controllable approximation (typically, perturbative) is involved in the calculus. Since exact applications seem to be more convincing and more illustrative for the subject, we will basically confine our discussion to presentation few of them. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

For the actinides the crystal entropies follow approximately the decreasing average radius produced by f-electron participation in metallic bonding. They are also clearly shown to be non-magne-tic, as the f s are itinerant. However, the entropy correlation itself cannot predict these values, since there is no model in terms of a like metal that can be used to compare these totally unique early actinides. There are also of course perturbations due to the high electronic specific heats, caused by high densities of states at the Fermi level. 

Another characteristic of turbulent flows is unpredictability, that is the high sensitivity of the solution to very small perturbations that are always present in real physical systems or numerical simulations. This unpredictability, also known as dynamical chaos, is a well known feature of much simpler low-dimensional nonlinear dynamical systems. Although in a strict mathematical sense a unique solution of the Navier-Stokes equation always exists for well-posed initial conditions (at least for large finite times), in practice the details of the forcing and boundary conditions are only known within some approximations and thus the solution in the turbulent regime repre-... 

The second term of the expansion, i.e., the first element of the second order contributions, corresponds to the polarization of A, due to the fixed unperturbed charge distribution of B the next term gives the polarization of B, due to the fixed charge distribution of A. The two terms, summed together, approximate IND. We have already commented that perturbation theory in a standard formulation cannot give IND with a unique term further refinements regarding mutual polarization effects have to be searched at higher order of the PT expansion. 

The High-Field Approximation In most NMR experiments the nuclear Zeeman interaction with the static external magnetic field is much stronger than all other interactions of the nuclear spins. As a result of these differences in the size, it is usually possible to treat these interactions in first order perturbation theory, i.e. use only those terms which commute with the Zeeman Hamiltonian, the so called secular terms. This approximation is called the high field approximation. While the single particle interactions like CSA or quadrupolar interaction have a unique form, for bilinear interactions, one has to distinguish between a homonuclear and a hetero-nuclear case. The secular parts of Hamiltonians discussed in the previous section are collected in Table 1. 

The first two of these results (4.141) and (4.142) are of little use unless we know or can approximate the function c(ri. t2 ap) for all values of a. They were obtained first by Stillinger and Buff by a cluster expansion and by Lebowitz and Percus by using functional integration, but we owe to Saam and Ebner the comment that since F[p] is a unique functional of p(i) then the values of F and 0 calculated in this way are independent of the path in p-space (4.136). The third result (4.146), although restricted to pair potentials, is a useful starting point for the development of perturbation theories of both bulk liquids and of the gas-liquid surface. 

The first-order approximations discussed in the next section, although historically derived from non-regular perturbation series, yield valid first-order expectation values, and thus their value should agree with the first-order expectation value of any regular expansion in 1/c, since the latter is unique as far as the series is regular. 

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