Coupled-static approximation

Fig. 4.2. The results of various calculations of the 1 = 1 partial-wave contribution to the positronium formation cross section in positron-hydrogen scattering in the Ore gap A, Brown and Humberston (1985) B, Chan and McEachran (1976) C, Winick and Reinhardt (1978b) D, coupled-static approximation E, Born approximation.

Fig. 3.2. The s-wave phase shift for positron-hydrogen scattering A, static approximation B, result for six-term coupled state (Is, 2s, 2p, 3s, 3p, 3d of H) (McEachran and Fraser, 1965) C, exact variational result (Schwartz, 1961b Bhatia et al., 1971).

A three-dimensional, quasipotential reduction of the Bethe-Salpeter equation, based on the static approximation for each two-particle propagator, was developed which adequately approximates the full solution to the BS equation. The parameters of the model include the ir, 8, q,p,a>,e, rrNA, and pNA couplings and the NN and NA cut-off masses. 

As a simplest example, we shall again invoke the linear three atomic molecule XMX. Suppose the two states denoted by s> and > are nearly degenerate. If these states have different parity, say s> being symmetric and > antisymmetric, then they will couple via the perturbation term i odd <2odd- In the static approximation, we regard the normal coordinate Oodd as a parameter. Then we get a secular matrix of the Hamiltonian in the form... 

The feasibility of time-dependent density matrix functional theory (TDDMFT) for calculating response properties has been demonstrated by proposing an improvement over the static approximation of the coupling matrix K(co) and by applying it to the HeH prototype system. 

While with-in the mobile x-ray system, the waste in the sampler, is contained within a replaceable (and disposable) polyvinyl chloride (PVC) sleeve with a wall thickness of approximately 0.2-inches and a sealed bottom. It was anticipated that the PVC tube or sleeve would, with use, become highly contaminated with waste residues which drip of fall-off the sampler. The sleeve is coated with a conductive coating to prevent static electricity buildup . There are no sources of ignition in this sealed spare. The sampler (and waste) is coupling which includes a positive pressure gasket. This barrier is further isolated by a second barrier consisting of an epoxy coated aluminum sleeve also sealed-off from the main x-ray cabinet and PVC sleeve. There are also no potential sources of ignition in this isolated secondary space as well. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

In comparison with eqns. 3.14 and 3.15 for a static inert electrode, eqns. 3.34 and 3.35 differ only in the power of D /D as for the usual reversible diffusion-controlled redox couples Drei and Dox are approximately equal, ( red/F)ox)1/2 will even be closer to unity, so that we can simply write... 

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