Systematic Perturbation Expansion

This section is devoted to a reconsideration of system (3.2.2), to obtain the frequency change as a power series in e. A slightly careful examination of Method I reveals that a major obstacle to systematic perturbation expansions lies in the fact that the surfaces of constant phase are generally curved in state space. Since the definition of the surfaces of constant phase is entirely at our disposal. 

Imagine a tubular region G as in Sect. 3.2 (Fig. 3.1). It is supposed that G is so thin that no intersections between different r(0) occur within G. Assume further that the perturbation ep is sufficiently weak so that no state points can get out of G for all t possibly except for initial transients. Imagine a state point X(t) which obeys the perturbed equation (3.2.2). It will move from one T to another T, and we choose 0(f) in such a way that X t) is found on T (p t)) for any t. Note that 0(t) represents the phase of A (t) in the new deHnition of phase. For any t, X t) can then be split uniquely as 

The formulation becomes a little more transparent by working with h(0), defined by 

It is easy to confirm from the general expression for B that By, 0) contains only the lower-order unknowns 2v (0) and C/y/(0) (v v), which says that (4.1.17, 18) represent a system of linear inhomogeneous equations. 

Since we are only interested in the long-time behavior, we take the limit 0 00 (equivalent to / oo). Then the first term on the right-hand side in (4.1.19) vanishes by virtue of the assumed property Re A/ 0 (/ 4= 0). Furthermore, it can be proved that Qv(0) for lU v, approach T-periodic functions of 0 as 

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An alternative approach is to work in a frozen basis and to calculate excitations by systematic perturbation expansions, which are most elegantly written out using Feynman graphs. An excellent introduction to Feynman graphs in this context is given by Kelly [241]. 

Apart from the necessity to project onto the physical subspace Fj, the form of the slave boson Hamiltonian H b has the advantage that usual many-body techniques, like the generalization of Wick s theorem to finite temperatures, can be used to produce a systematic perturbation expansion in the mixing term. Coleman performs such an expansion neglecting the restriction on Q and carries out the projection onto the 2 = 1 subspace in the end of the calculation. For that purpose he extended Abrikosov s scheme (Abrikosov 1965, Barnes 1976) of weighting the 2 > 1 states by... 

Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

One way to proceed would be to device a procedure for obtaining systematic local approximations to S x.x -.E). This could possibly be obtained from known perturbation expansions... 

In the experiments that followed, the pre- and post-expansion temperature, pressure and composition were systematically perturbed around their values in this base case and the results were compared to identify the effects of each variable on the morphology and size of the precipitated Naphthalene particles. The pre-expansion temperature, while varied, was kept high enough in all experiments so that the jet of expanded fluid downstream of the nozzle was not visible, i.e. no condensation of carbon dioxide occurred upon expansion. Tbe base case post-expansion pressure corresponds roughly to the solubility minimum of Naphthalene in CO2 at 45 C (2). All of the experiments performed in this study are listed in Table I (Run 1 is the above mentioned base case). 

The convergence of the quantum chemical calculations can be studied in terms of two types of hierarchies. First, the quality of the calculations depends on the flexibility of the MO space the AOs that are used to expand the MOs may be extended in a well-defined and systematic manner, thereby establishing a one-electron hierarchy. Second, we can increase the excitation level in coupled-cluster theory or the order of perturbation expansion, thus setting up an n-electron hierarchy of approximate electronic wave functions. In Fig. 5, the roles of the one-electron and the ra-electron hierarchies are illustrated. 

Many-body theory starts out from the principle that all wavefunctions (for both ground and excited states) should be calculated in the same atomic field, i.e. from the same Hamiltonian. The perturbative expansion then allows the higher-order corrections to be calculated systematically. It can then be shown [250] that in the pure RPAE, the dipole length and dipole velocity forms of the cross section are precisely equal, by construction. For this reason, the pure RPAE is often referred to as exact, which means simply that it satisfies equation (5.31) exactly, and not that one should necessarily expect it to agree exactly with experiment. 

Early non-relativistic many-body perturbation theory studies of correlation energies in molecules established that the error associated with truncation of the finite basis set is most often much more significant than that resulting from truncation of the perturbation expansion.15 The chosen basis set is required to support not only an accurate description of the occupied Hartree-Fock orbitals but also a representation of the virtual spectrum. Over the past twenty years significant progress has been reported on the systematic design of basis sets for electron correlation studies in general and many-body perturbation theory calculations in particular.18... 

E)q. (3) shows that the Z— and D— oo limits are complementary to one another the D— oo limit is to the kinetic terms what the Z— oo limit is to the potential terms. More precisely, the D— co limit amounts to ignoring the harder set of kinetic terms (the second derivatives T ), while the Z— oo amounts to ignoring the harder set of potential terms (the interelectron repulsions V). Moreover, one can systematically develop perturbation expansions /D and 1/Z expansions) about each of these limits. It will be shown below that this complementarity is very useful, since it makes it possible to use one approach to correct the shortcomings of the other. 

Numerical calculations of the ionic spectra rely on a systematic skeleton expansion, defined via perturbation theory, with respect to the hybridization based on band states and ionic many-body states. This perturbation technique results in integral equations of the type which are usually solved... 

Hence, for this model it is possible to investigate the dependence of each diagrammatic component of the perturbation expansion on the basis set employed. The results of such an investigation using a universal systematic sequence of even-tempered basis sets of exponential-type functions is presented in Table XII, where it can be seen that each component does indeed converge to the corresponding exact value. It is also of interest to compare the accuracy obtain in the basis set study with the achieved in numerical methods, and this is done in Table XIII, where it can be seen that... 

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