Basic Approximations

Many problems with MNDO involve cases where the NDO approximation electron-electron repulsion is most important. AMI is an improvement over MNDO, even though it uses the same basic approximation. It is generally the most accurate semi-empirical method in HyperChem and is the method of choice for most problems. Altering part of the theoretical framework (the function describing repulsion between atomic cores) and assigning new parameters improves the performance of AMI. It deals with hydrogen bonds properly, produces accurate predictions of activation barriers for many reactions, and predicts heats of formation of molecules with an error that is about 40 percent smaller than with MNDO. 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

The second basic approximation is the neglect of diatomic differential overlap... 

To this point, the basic approximation is that the total wave function is a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ab initio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ab initio calculation. However, there are two main th ings to be con sidered in the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222), will require the fewest possible terms for an accurate representation of a molecular orbital. The second one is the speed of two-electron integral calculation. 

The form of the CNDO/2 equations for the Fock matrix comes from the basic approximation =5 y(]) j (]) j. This reduces the... 

The basic approximate equation for the separation process gives the water flux, m" (kg/m /s) across an RO membrane, in the absence of fouling,... 

The MNDO, AMI and PM3 methods are parameterizations of the NDDO model, where the parameterization is in terms of atomic variables, i.e. referring only to the nature of a single atom. MNDO, AMI and PM3 are derived from the same basic approximations (NDDO), and differ only in the way the core-core repulsion is treated, and how the parameters are assigned. Each method considers only the valence s- and p-functions, which are taken as Slater type orbitals with corresponding exponents, (s and... 

The Taitel and Dukler model used two basic approximations ... 

Stratified flow. A separated flow model for stratified flow was presented by Taitel and Dukler (1976a) in which the holdup and the dimensionless pressure drop, = (dpldz)TPl(dpldz)GS is calculated as a function of the Lockhart-Martinelli parameter only. (The results, however, differ from those of Martinelli and compare better with experimental data.) This model uses two basic approximations ... 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

The single-reference coupled cluster (CC) theory [1-5] has become a standard computational tool for studying ground-state molecular properties [6-10]. The basic approximations, such as CCSD (coupled cluster singles and doubles approach) [11-15], and the noniterative CCSD[T] [16,17] and CCSD(T) [18] methods, in which the cleverly designed corrections due to... 

Early theoretical treatments of liquid crystals were not surprisingly based on the molecular field approximation. However, it is neccessary to make assumptions about the pair potential employed in the calculation and it is impossible to know whether the predictions of a particular model really arise from the pair potential employed or whether they arise, at least in part, from the deficiencies of the basic approximation employed. The general problem is so complex that a better mathematical treatment of the molecular interactions in a liquid crystal is out of the question. However, with the introduction of ever more powerful computers, it has become possible to carry out meaningful numerical simulations of model liquid crystals. 

However, in some sets of data, the spacing between the points is not necessarily constant, for which we can use the most basic approximation of integration written as,... 

The London-Eyring-Polanyi-Sato (LEPS) method is a semi-empirical method.8 It is based on the London equation, but the calculated Coulombic and exchange integrals are replaced by experimental data. That is, some experimental input is used in the construction of the potential energy surface. The LEPS approach can, partly, be justified for H + H2 and other reactions involving three atoms, as long as the basic approximations behind the London equation are reasonable. 

Note that in this context the word complex does not imply an entity that has a chemically significant lifetime. The basic approximations in the theory are only valid for direct reactions. 

This last expression may be viewed as the basic approximation in order to treat more easily the quantum indirect damping. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

Many approximate self-consistent field molecular orbital schemes have been proposed, and the complete or partial neglect of differential overlap simplification has become widely accepted as a useful, but not too severe approximation to full ab initio methods. The advantages and justifications of the basic approximations in the CNDO and INDO method have been detailed by Pople ). Of course other considerations of time necessitate the use of empirical type calculations rather than full ab initio treatments when large numbers of calculations are required for series of molecules. 

Recently, Cottin and Monson [101] have shown how the LJD theory can be applied to solid phase mixtures. The basic approximation they make (in addition to single-cell occupancy) is to write the configurational partition function for an n-component system as a product of cell partition functions expressed as... 

The original derivation of Eq. (9) is given by Fleck et al. (7). However, since the split-operator scheme is closely related to the interaction representation, it can be derived rather straightforwardly in IR (3). The basic approximation is the expansion of the interaction V,(t) around the middle point in the time interval [r0, to + A],... 

Following this scheme and exploiting the same set of basic approximations characterizing continuum models (uniform charge distribution of the solvent, imiform response function), it is possible to derive the following... 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

The basic approximation used in the above discussion of the ER signal is the linearity of response to the potential perturbation (see Section 2.5). This approximation is not always appHcable. Especially when the optical signal is very weak, one should use a value much greater than RT/n ppF to obtain the signal with a reasonably high S/N ratio. Simulation of the ER signal for such a case was reported, although Rs was presumed to be zero [68]. 

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