Zeroth-order approximation calculation

Fig. 3. Schematic diagram for the variation with concentration of the partial molar heat of solution of the liquid noble metals into liquid tin, taken from reference 51. The numbers are the experimental and calculated AHt for the solutes, in cal/g atom, at the two concentrations of 0 and 0.02 mole fraction. Q-C labels the curves calculated by the quasichemical theory in first order B-W labels the curves calculated by the Bragg-Williams, or zeroth-order approximation, which assumes a random...

We have called the vibrational quantum numbers here Vj, v2, v3 in order to distinguish them from the local quantum numbers, va, vl , vc. Note that, in view of the presence of the missing label, %, the normal basis is not very convenient for calculations. The spectrum corresponding to Eq. (4.59) is shown in Figure 4.8. There are fewer examples of molecules for which the dynamical symmetry of the normal chain II, provides a realistic zeroth-order approximation. The normal behavior arises when the masses of the three atoms are comparable, as, for example in XY2 molecules with mx = mY. More examples are discussed in the following sections. 

The first order perturbation energy is calculated by the following zeroth-order approximation to the ground-state wave function P° of the two molecules ... 

Such a definition of R is rather conventional we assume it as some zeroth-order approximation. Further, the derivatives of various characteristics with respect to R are calculated. They describe the interaction of the nucleus with the outer elec-tron and this permits the recalculation of results when R varies within reasonable limits. The Coulomb potential for the spherically symmetric density p(r i ) is ... 

Further we present the results of our calculations of the Li- ike iGplasma satellite lines on the basis of QED PT with ab initio zeroth-order approximation for three-quasiparticle systems, together with the optimized Dirac-Fock results and experimental data for comparison. In Table 4 there are displayed the experimental value (A) for wavelength (in A) of the Ti-like lines dielectron satellites to the ls So-ls3p Pi line of radiation in the K plasma, and the corresponding theoretical results (B) PT on 1/Z (C) QED PT (our data) (D) calculation by the AUTOJOLS method, and (E) MCDF [12, 21],... 

The calculated modes at 1282, 1296, and 1306 cm-1 represent motions of the diazene moiety. The assignment of these vibrations in (101) as overtones of the modes at about 650 cm-1, which was confirmed by the experimental spectra of the isotopomers, was well reproduced by the calculations (103), in which we used the harmonic oscillator model as a zeroth-order approximation for the estimation of the overtone wavenumbers. 

Experience in a variety of applications of the C ASSCF method has shown it to be a valuable tool for obtaining good zeroth-order approximations to the wavefunctions. Attempts have been made to extend the treatment to include also the most important dynamical correlation effects. While this can be quite successful in some specific cases (see below for some examples), it is in general an impossible route. Dynamical correlation effects should preferably be included via multireference Cl calculations. It is then rarely necessary to perform very large CASSCF calculations. Degeneracy effects are most often described by a rather small set of active orbitals. On the other hand experience has also shown that it is important to use large basis sets including polarization functions in order to obtain reliable results. The CASSCF calculations will in such studies be dominated by the transformation step rather than by the Cl calculation. A mixture of first- and second-order procedures, as advocated above, is then probably the most economic alternative. 

The first-order perturbation theory corrections to the quasi-relativistic energies obtained with the lORA and the ERA Hamiltonian as the zeroth-order approximation to the Dirac equation are given in Table 6 and Table 7, respectively. Direct perturbation theory calculations on top of the MIORA and MERA Hamiltonians has not been studied computationally. 

The second problem centers about the use of an approximate ground-state wave function that eminates from a multiconfigurational zeroth-order approximation. The N2 calculations in Section III.C suggest that the restriction to a single configuration zeroth-order ground state imposes a fundamental limitation on the quality of the calculated EOM ionization potentials for that system. 

The above calculations, which are a zeroth order approximation, and which follow essentially the ideas of Fisher [28], can be further refined. Higher order approximations can... 

An interesting approach to the problem of calculating the local wave function of a fragment in interaction with its surrounding has been proposed by Kirtman and de Melo [88]. Their approach is based on the density matrix formulation of the HF problem. The zeroth order approximation to the density matrix of the total system is a simple direct sum of the density matrices of the fragments, provided that the AO basis is orthogonal by construction or it is properly orthogonalized. 

Here C ° and E denote a zeroth-order approximation for the quasi-particle states. In our Si calculation this zeroth-order approximation was extracted from an empirically fitted pseudopotential band-structure (see ref.4 and 35). This bandstructure is fitted in terms of a fourth-nearest neighbor (in the fcc-lattice sites) overlap model of bonding and antibond ng orbitals as described n our earlier work on optical properties and impurity screening. Also the calculation of the two-particle Green s function is based on this bandstructure and follows closely the impurity studies (for details see in particular, ref.35). 

In our previous diamond work we determined the bare HF part of E by making a Slater-Koster fit to an existing HF band calculation, while the correlation part was determined, as in the present Si work, by evaluating the matrix elements in an explicit basis set, which represents a zeroth-order approximation to the actual quasi-particle states. 

Here we look at some details of the cumulative reaction probability of the reaction of H + H2 and discuss what picture can be obtained from the analytical expressions of the current theory. Figure 7.7 shows the cumulative reaction probability N E) as a function of the energy E both in linear and log scales. The result of the harmonic approximation (dotted curve) and that of the NF theory (solid curve) are compared. The calculation uses the potential energy surface of Mielke et al. The total angular momentum is fixed to zero for simplicity. This allows us to take the zeroth order approximation as in eqn (7.11), i.e. a collection of harmonic oscillators and a parabolic barrier. Readers interested in the extension of the theory to include the rotational motions should refer to the literature. ... 

Calculate the energy to first order of He in its lowest-energy state. Use the hydrogen atom in its ground state as your zeroth-order approximation. Use atomic units. Predict the signs (plus, zero, minus) of and g. Explain your reasoning. 

Assuming certain initial values for calculate approximate path, instanton energy, and avoided crossing point Xc. Then we can compute cr(x ) and )[2f(T)], and determine the zeroth order approximate instanton path. 

The investigation of the response of macromolecules to external mechanical forces or to electromagnetic fields may basically contribute to our understanding of the structural and functional properties of these systems. The starting point of all studies of this kind is the proper description of the equilibrium (ground) state of the molecule without external fields. In our a priori calculations, the ground state energy is obtained in two steps as a zeroth order approximation the Hartree-Fock (HF) contribution is calculated by the ab initio crystal orbital method (1,2) and electronic correlation effects are included by perturbation theory afterwards. 

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