Pseudostate method

Fig. 8.3. Differential cross section for electron scattering to the Is, 2s and 2p states of hydrogen at 54.4 eV. Experimental data for Is are interpolated (Williams, 1975), for 2s and 2p they are taken from Williams (1981). Calculations are solid curve, convergent close coupling (Bray and Stelbovics, 1992h) long-dashed curve, coupled channels optical (Bray et al, 1991c) short-dashed curve, distorted-wave second Born (Madison et al, 1991) chain curve, intermediate-energy R matrix (Scholz et al, 1991) dotted curve, pseudostate method (van Wyngaarden and Walters, 1986).

Fig. 8.9. Integrated cross section for electron scattering to the 2s (above) and 2p (below) states of hydrogen below the n=3 threshold. The positions and quantum numbers of resonances are shown on the upper scale. Experiment, Williams (1988) solid curve, coupled channels optical (equivalent local) (McCarthy and Shang, 1992) long-dashed curve, pseudostate method (Callaway, 1982) short-dashed curve, 9-state coupled channels.

A time-dependent extension of the pseudostate method, where the true spectrum is replaced by a small number of effective states , has been proposed by Visser and Wormer [76], and applied to the TDHF calculation of dynamic polarizabilities for He, Ne, H2, N2 and the related dispersion coefficients [42]. The TDHF values (without correlation) calculated in this way for He and He2, as well as those resulting from a SDT-MBPT calculation which includes the real correlation effects in second order of the correlation potential [42], are compared in Table 5 with estremely accurate values taken from [77, 78]. 

A field-free approach to polarizabilities of excited states has been designed to overcome the difficulties of the finite-field version of the Diffusion Monte Carlo method. It has been applied to the n = 2 hydrogen atom, whose hybrid orbitals partition into two nodal regions. The pseudostate method has been applied to calculate polarizabilities of the positronium negative ion. 

The B-spline K-matrix method follows the close-coupling prescription a complete set of stationary eigenfunctions of the Hamiltonian in the continuum is approximated with a linear combination of partial wave channels (PWCs) [Pg.286]

Table (8.1) shows results of test calculations of e-He partial wave phase shifts, compared with earlier variational calculations [383], The polarization pseudostate was approximated here for He by variational scaling of the well-known hydrogen pseudostate [76]. The present method is no more difficult to implement for polarization response (SEP) than it is for static exchange (SE). 

This method simply involves the solution of the Lippmann—Schwinger equations (6.73) or (6.87) with the potential matrix elements (7.35). The states i) are not eigenstates of the target Hamiltonian. They are configuration-interaction states or pseudostates obtained by diagonalising the target Hamiltonian in a square-integrable basis as described in section 5.6. 

Pseudostate calculations have the advantage over Born and optical-potential methods that they constitute a numerically-exact solution of a problem. The problem is not identical to a scattering problem but can be made quite realistic for useful classes of scattering phenomena by an appropriate basis choice. The state vectors, or equivalently the set of half-off-shell T -matrix elements, for such a calculation contain quite realistic information about the ionisation space. 

The comparison of theory and experiment in table 8.3 is somewhat unsatisfactory. The coupled-channels-optical and pseudostate calculations agree with each other and with the convergent-close-coupling calculation within a few percent, yet there are noticeable discrepancies with the experimental estimates. The convergent-close-coupling method calculates total ionisation cross sections in complete agreement with the measurements... 

Other scattering calculations that account for the complete target space can also be tested. The method (10.55) can be used for the pseudostate... 

The excited pseudostates occurring in Equations (4.18) and (4.19) can be obtained using the extension of the Ritz method to the calculation of second-order energies introduced in Chapter 1. 

Probably the most important characteristic of military and commercial explosives and solid rocket propellants is performance as related to end use and safety. Performance can be described by a variety of conventional properties such as thermal stability, shock sensitivity, friction sensitivity, explosive power, burning, or detonation rate, and so on. Thermal analysis methods, according to Maycock (51), show great promise for providing information on both these conventional properties and other parameters of explosive and propellant systems. The thermal properties have been determined mainly by TG and DTA techniques and isothermal or adiabatic constant-volume decomposition. Physical processes in pseudostable ma-... 

An alternative method based on a discrete variational representation of the continuum in terms of pseudostates has been developed by Drake and Goldman [11]. The method is simplest to explain for the case of hydrogen. The key idea is to define a variational basis set containing a huge range of distance scales according to ... 

Equations (81)-(83) give our final result for the dispersion energy once the k-dependent polarizabilities have been calculated for a given series of molecules by any one of the standard methods at our disposal (pseudostates, RPA, TDHF, MC-TDHF), they can be used to evaluate the dispersion energies for any pair of molecules from that set, for whatever orientation Q and whatever separation R. In this sense, the generalized expansion (75) is given a separable form [58]. The method is relatively new, and has been applied so far to the accurate calculation of the... 

Expansion in a finite set of discrete pseudostates evidently converges much faster than expansion in true eigenstates of Hq (which would involve also continuum states). The same behaviour is observed for the expansion in eigenstates of Fq (the one-particle Fock Hamiltonian), so that convergence of conventional TDHF methods may prove rather slow, especially with large bases of atomic orbitals... 

The FHBS method is unitary, i.e., probability is preserved. We can accurately identify the probability of making a transition to all bound states as the probability calculated from transitions to all negative energy pseudostates. Within this approximation we can identify ionization as the missing probability, i.e.. 

Berrington et al have recently calculated this cross section using the R-matrix method. They included Is, 2s, and 2p eigenstates and also 3s, 3p, and 3d pseudostates in an expansion of the initial and final wave functions. As a check on their calculations, they find that the position, width, and shape... 

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