Single-term approximation

As in Section 6.3, the weighting function W(ro) is confined to a narrow region around the ground-state equilibrium re such that the sum in (6.32) reduces to a single term. Approximating N(ro) around re according to... 

Because of the single-term approximation of the series, solutions for the off-center temperatures may also be expressed in the following simple forms. 

Helsler and Grober Charts—Single-Term Approximations... 

The Heisler [36] cooling charts and the Grober [29] heat loss fraction charts for the three geometries can be calculated accurately by the single-term approximations [28,56]... 

Note that the above dispersion coefficient corresponds to a single-term approximation to the bare (non-interacting) spherically averaged dipolar dynamic polarizability associated with the localized orbital i,... 

We now examine the possibility of transcending the single-term approximation considered in the last section by admitting other terms in the expansion (14.1.2) and, in view of the close analogy with the orbital approximations used in earlier chapters, we then refer to the admission of configuration interaction , where now the excited configurations are associated with local excitations in which one, two or several groups may be described as excited-state functions 0, . . (a, b, ... 

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

Both the BO and adiabatic approximation can be based on choosing a single term in... 

Clearly then, the closed expression for the density-transformed 1-matrix in Eq. (52) (i.e., the 1-matrix constructed from transformed orbitals) is a function of the single-electron density p(r). Hence, when we express the total energy of an N-particle system (in the single-determinantal approximation) in terms of the transformed one-matrix described by Eq. (52), one can readily obtain an energy functional which depends on p. This fact, which has been exploited by several authors [59-62,85], is considered below with particular reference to the work of Ludeha [60]. 

Although in principle an exact solution to the Schrodinger equation can be expressed in the form of equation (A.13), the wave functions and coefficients da cannot to determined for an infinitely large set. In the Hartree-Fock approximation, it is assumed that the summation in equation (A.13) may be approximated by a single term, that is, that the correct wave function may be approximated by a single determinantal wave function , the first term of equation (A.13). The method of variations is used to determine the... 

Next, we examine the term i2. In a gas-like single segment approximation, this term can be replaced by 1212. The molecular conformation statistics are independent of each other. This might be due to the fact that in the absence of a three-dimensional lattice-potential, nematic shifts of neighboring segments are very likely to occur. In this approximation the configuration does not depend on which individual pair of molecules k, 1 is picked out The molecular structure factor is independent of the indexes k and L Hence 1 inter, d can be written as... 

The total moment of a complex of configurations may be presented as a sum of two terms nl(K) representing the moment of the total spectrum in single-configuration approximation, and the second, An (K), describing the correction due to superposition of configurations, i.e. 

This is called a steady-state approximation and is expressed mathematically by setting the rate of ES formation equal to the rate of ES consumption (Equations 4.6 and 4.7). After a number of rearrangements, Equation 4.7 can be solved for [ES] (Equation 4.8). The collection of three rate constants is replaced with a single term, Km, the Michaelis constant. 

Now we see clear the problem while the new dot Hamiltonian (154) is very simple and exactly solvable, the new tunneling Hamiltonian (162) is complicated. Moreover, instead of one linear electron-vibron interaction term, the exponent in (162) produces all powers of vibronic operators. Actually, we simply remove the complexity from one place to the other. This approach works well, if the tunneling can be considered as a perturbation, we consider it in the next section. In the general case the problem is quite difficult, but in the single-particle approximation it can be solved exactly [98-101]. 

This assumption is distinctly different from a commonly used approximation that the signal shape must be described by an intensity crosscorrelation of the pump and probe pulses. Let us consider a signal that is described by a single term of the type given in equation (4). The... 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

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