Approximation zeroth-order

In the limit of low densities, A2.4.15 shows that the zeroth-order approximation forg(r) has the fonn... 

Electronic structures of SWCNT have been reviewed. It has been shown that armchair-structural tubes (a, a) could probably remain metallic after energetical stabilisation in connection with the metal-insulator transition but that zigzag (3a, 0) and helical-structural tubes (a, b) would change into semiconductive even if the condition 2a + b = 3N s satisfied. There would not be so much difference in the electronic structures between MWCNT and SWCNT and these can be regarded electronically similar at least in the zeroth order approximation. Doping to CNT with either Lewis acid or base would newly cause intriguing electronic properties including superconductivity. 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Euler s equation is thus recovered as a direct consequence of momentum conservation, but only via the zeroth-order approximation to the full solution to the Boltzman-equation. 

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...

In this section we estimate the magnitude of these quantum mixing effects. Even though the strictly semiclassical theory agrees well with experiment as is, making such estimates that go beyond it is useful for two distinct reasons. First, we must check to what extent the semiclassical picture, tacitly assumed earher, is a consistent zeroth order approximation to a more complete treatment. 

Figure 10. Low-energy vibronic levels in the X2II state of HCCS computed in various approximations [152]. Hq zeroth-order approximation (both vibronic and spin-orbit couplings neglected). Hi. vibronic coupling taken into account, spin-orbit interaction neglected. Hi + Hs0 both vibronic and spin-orbit couplings taken into account. Solid horizontal lines K = 0 vibronic levels dashed line K — 1 dash-dotted lines K = 2 dotted lines K — 3. Values of the quantum numbers V4, N of the basis functions dominating the vibronic wave function of the level in question are indicated. Approximate correlation of vibronic states computed in various approximations is indicated by thin lines. In all cases the stretching quantum numbers are assumed to be zero.

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 ... 

In the approach pursued here, the recovery of correlation is perceived as a two-stage process First, the determination of a zeroth-order approximation in form of a MCSCF wavefunction that is in some way related to the full valence space and determines the molecular orbitals then, the determination of refinements that recover the remaining dynamic correlation. Section 2 of this paper deals with the elimination of all configurational deadwood from full valence spaces. In Section 3, a simple approach for obtaining an accurate estimate of the dynamic correlation is discussed. 

A remarkable number of organic compounds luminesce when subjected to consecutive oxidation-reduction (or reduction-oxidation) in aprotic solvents1-17 under conditions where anion radicals are oxidized or cation radicals are reduced. In many instances, the emission is identical with that of the normal solution fluorescence of the compound employed. In these instances the redox process has served to produce neutral molecules in an excited electronic state. These consecutive processes which result in emission are not special examples of oxidative chemiluminescence, but are more properly classified as electron transfer luminescence in solution since the sequence oxidation-reduction can be as effective as reduction-oxidation.8,10,12 A simple molecular orbital diagram, although it is a zeroth-order approximation of what might be involved under some conditions, provides a useful starting... 

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],... 

Here, we seek to obtain wave functions - molecular orbitals - in a manner analogous to atomic orbital (AO) theory. We harbour no preconceptions about the chemical bond except that, as in VB theory, the atomic orbitals of the constituent atoms are used as a basis. A naive, zeroth-order approximation might be to regard each AO as an MO, so that the distribution of electron density in a molecule is simply obtained by superimposing the constituent atoms whose AOs remain essentially unaltered. But since there is inevitably an appreciable amount of orbital overlap between atoms in any stable molecule - without it there would be no bonding - we must find a set of orthogonal linear combinations of the constituent atomic orbitals. These are the MOs, and their number must be equal to the number of AOs being combined. 

What is the proper interpolation and extrapolation technique Experience has shown that differences in families are never quite constant, implying that linear interpolation and extrapolation is only a zeroth-order approximation. In some of the examples to follow, enough information is available to permit the use of a quadratic scheme, which may be the best approximation. For most systems, the differences between the linear and quadratic methods will be small. 

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. 

If we take the HF MOSCF occupied levels as a zeroth-order approximation, the hole levels are then influenced by a number of relaxation and correlation effects. 

The second approximation often used to truncate Cl wavefunctions is to neglect configurations formed by more than two-electron excitations. It is easily seen that they cannot mix into the Cl wavefunction in first order. The third method comes from perturbation theory. If one takes the Hartree-Fock energy as the zeroth-order approximation to the energy... 

It is clear that this zeroth-order approximation is, in fact, the classical QSSA defined in Sect. 4.5.2(a), stiff variables being free radical concentrations and non-stiff variables being concentrations of molecular species. The outer solution constitutes a good approximation to the true solution only when the inner solution becomes negligible, i.e. when r - °°, since by the definition of boundary layer solutions... 

Although the function M4>0 would seem a natural zeroth-order approximation for a perturbation expansion of the interaction energies, it unfortunately is not an eigenfunction ofy/0. J)ne could try to introduce a new partitionmg of the Hamiltonian, H = H0 + V, such that is an eigenfunction of H0, but no... 

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