Pseudo-Perturbation Theory

Ground state weakly coupled with excited states 1 00 R(l) Ukk = Ek + X mVH(lV(E° - E ) j eSb consistent with the second-order perturbation theory pseudo Jahn-Teller effect... 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

Abstract. We present a quantum-classieal determination of stable isomers of Na Arii clusters with an electronically excited sodium atom in 3p P states. The excited states of Na perturbed by the argon atoms are obtained as the eigenfunctions of a single-electron operator describing the electron in the field of a Na Arn core, the Na and Ar atoms being substituted by pseudo-potentials. These pseudo-potentials include core-polarization operators to account for polarization and correlation of the inert part with the excited electron . The geometry optimization of the excited states is carried out via the basin-hopping method of Wales et al. The present study confirms the trend for small Na Arn clusters in 3p states to form planar structures, as proposed earlier by Tutein and Mayne within the framework of a first order perturbation theory on a "Diatomics in Molecules" type model. 

The method of epikernel principle seems to be incomplete due to its restriction to the 1st order perturbation theory and linear extension of the perturbation potential. Using more complete perturbation may produce the results comparable with the other method on account of higher elaborateness. The JT caused loss of planarity or of symmetry center in JT systems can be explained by pseudo-JT mechanisms only. Another problem is the applicability to the groups with complex characters (C , S , and C h for n > 2, T and Th). 

This is a pair of nonlinear equations and no simple solution can be written down, but they can be reduced to a single equation by making a so-called pseudo-steady state hypothesis. This is the assumption that, when Cq is much smaller than Uo, the concentration c is never very large and varies very slowly. We can then set dc/dt = 0. This hypothesis appears to be very pseudo indeed at first sight, but it can in fact be justified by what is known as the singular perturbation theory of differential equations. Setting dcjdt = 0 in Eq. (4.6.7) we can solve for c in terms of a ... 

It was Fermi who realized that it was possible to invoke an equivalent potential, which can be used to calculate the changes in the wavefunction outside the interaction by perturbation theory [13]. The unknown form of the strong nuclear interaction can be replaced by a new potential, which gives the same scattered wavefunction as the square well potential. In the derivation of Fermi s equivalent or pseudo potential [14] it is seen that the magnitude of the scattering potential depends on the scattering length of the nucleus and the mass of the neutron, m ... 

AEband can also be calculated from perturbation theory via the pseudo-potential matrix elements " ). The pseudo-potential approach, however, is only justified if the parent metals have the same valency and the same Fermi vector, and if furthermore no charge transfer takes place and the AB compounds have the same atomic volumes as A and B ). 

In principle, surface atomic and electronic structures are both available from self-consistent calculations of the electronic energy and surface potential. Until recently, however, such calculations were rather unrealistic, being based on a one-dimensional model using a square well crystal potential, with a semi-infinite lattice of pseudo-ions added by first-order perturbation theory. This treatment could not adequately describe dangling bond surface bands. Fortunately, the situation has improved enormously as the result of an approach due to Appelbaum and Hamann (see ref. 70 and references cited therein), which is based on the following concepts. 

The pseudo parameters of the HSE theory are derived from an equation of state expanded in powers of 1/kT about a hard-sphere fluid, as is developed by the perturbation theory. Consequently, it is reasonable to expect that procedures for defining optimal diameters for the perturbation theory should work well with the HSE procedure. The first portion of this chapter shows that this is indeed correct. The Verlet-Weis (VW) (5) modification of the Weeks, Chandler, and Anderson (WCA) (6) procedure was used here to determine diameters in a mixture of Len-nard-Jones (LJ) (12-6) fluids. These diameters then were used in the HSE procedure to predict the mixture properties. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

In pseudo-perturbation theory (Christiansen et al, 1996) one builds on this fact... 

As an approximate alternative one does not directly include the doubles corrections in the principal propagator but corrects the RPA excitation energies, obtained by solving the RPA eigenvalue problem, with a non-iterative doubles correction. This approach is called doubles corrected random-phase approximation-RPA(D) (Christiansen et al, 1998a) and is based on pseudo-perturbation theory that was described in Section 3.13. 

If the transition considered is the HOMO LUMO transition of an alternant hydrocarbon, then first-order theory predicts that inductive perturbation will have no effect at all, because for = fo as a consequence of the pairing theorem. Small red shifts are in fact observed that can be attributed to hyper conjugation with the pseudo-7t MO of the saturated alkyl chain.290 On the other hand, alkyl substitution gives rise to large shifts in the absorption spectra of radical ions of alternant hydrocarbons whose charge distribution is equal to the square of the coefficients of the MO from which an electron was removed (radical cations) or to which an electron was added (radical anions), and these shifts are accurately predicted by HMO theory.291... 

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