Approximate versions

Equation 16 is the correct material balance expression for calculating the chlorine efficiency of diaphragm cells. Whereas many approximate versions are used (8), the one closest to equation 16 is the "six equation" ... 

It has later been shown that the resulting equations are identical to CCSD where some of the terms have been omitted. The omitted terms are computationally inexpensive, and there appears to be no reason for using the less complete QCISD over CCSD (or QCISD(T) in place of CCSD(T)), although in practice they normally give very similar results. There are a few other methods which may be considered either as CISD with addition of extra terms to make them approximately size extensive, or as approximate versions of CCSD. Some of the methods falling into this category are Averaged... 

So far everything is exact. A complete manifold of excitation operators, however, means that all excited states are considered, i.e. a full Cl approach. Approximate versions of propagator methods may be generated by restricting the excitation level, i.e. tmncating h. A complete specification furthermore requires a selection of the reference, normally taken as either an HF or MCSCF wave function. 

In classic options analysis, one flrst creates an approximate version of the decisions and risks to be considered and then tries to exactly optimize the decision policies. 

In this chapter, we review the elements of G3 theory and related techniques of computational thermochemistry. This review is restricted almost exclusively to the techniques that we have developed and the reader is referred to the remaining chapters in this volume for other complementary approaches. An important part of the development of such quantum-chemical methods is their critical assessment on test sets of accurate experimental data. Section 3.2 provides a brief description of the comprehensive G3/99 test set [26] of experimental data that we have collected. Section 3.3 discusses the components of G3 theory as well as the approximate versions such as G3(MP3) [22] and G3(MP2) [23], and their performance for the G3/99 test set. The G3S method [29] that includes multiplicative scale factors is presented in section 3.4 along with other related variants. Section 3.5 discusses the recently developed G3X method [30] that corrects for most of the deficiencies of G3 theory for larger molecules. The performance of these methods is compared to... 

The correlation methods in G3 theory are still computationally demanding and it is of interest to find modifications to reduce the computational requirements. Two approximate versions of G3 theory have been proposed to make the methods more widely applicable. The first is G3(MP3) [22] that eliminates the expensive MP4/2df,p calculation by evaluating the larger basis set effects at the MP3 level. It also eliminates the MP4/plus calculation,... 

Even with the progress that has been made in rigorous quantum approaches, it is nevertheless possible to carry out such calculations only for relatively simple chemical systems. For example, the largest molecular system for which such calculations have been carried out is for the reaction H2 + OH — H2O + H. There is clearly interest, therefore, in the development of approximate versions of the approach that can be applied to more complex systems. Section III describes a semiclassical approximation for doing this, and Section IV concludes. 

Table 4.3 shows that even this approximate version of transition state theory gives a satisfactory interpretation of the experimental p factors. As the complexity of the reactants increases so p decreases. 

The HF method tends to overestimate the barriers, making unstable molecules seem stabler than they really are. Geometries are discussed further in Section 5.5.1. Approximate versions of the MP2 method that speed up the process with little loss of accuracy are available in some program suites LMP2, localized MP2, and RI-MP2, resolution of identity MP2. LMP2 starts with a Slater determinant which has been altered so that its MOs are localized, corresponding to our ideas of bonds and lone pairs (Section 5.2.3.1), and permits only excitations into spatially nearby virtual orbitals [93]. RI-MP2 approximates four-center integrals (Section 5.3.2) by three-center ones [94]. 

Only if there were a transfer of, say, an H+, H, or H in a reaction, AH + B - A + HB (charges not indicated), at a fairly large AB separation distance, would the situation be rather analogous to that of ETs. The H transfer would occur at an approximately fixed position of A and B, fixed because of the substantially larger masses of A and B compared with that of H. That is, an approximate version of the Franck-Condon principle would apply. Under such conditions of an H transfer, the description of the reaction via two intersecting approximate parabolas would be a reasonable first approximation. 

The binary-encounter-dipole (BED) model of Kim and Rudd [31] couples the modified form of Mott cross section [32] with the Bom-Bethe theory [27]. BED requires the differential continuum oscillator strength (DOS) which is rather difficult to obtain. The simplest approximate version of BED is the binary-encounter-Bethe (BEB) [31] model, which does not need the knowledge of DOS for calculating the EISICS. 

According to the approximate version of (5.18b) it follows from (5.32) that... 

We shall next give an explicit formula for the half-width T on the energy scale of a not too broad Stark level. To this purpose we write (5.17b) with the use of the approximate version of (5.18a) as u +1... 

If one compares Eq. (3.II9) of the fmite-ion-size model with Eq. (3.90) of the point-charge approximation, it is clear that the only difference between the two expressions is that the former contains a term 1/(1 + ku) in the denominator. Now, one of the tests of a more general version of a theory is the correspondence principle, i.e., the general version of a theory must reduce to the approximate version under the conditions of applicability of the latter. Does Eq. (3.119) from the finite-ion-size model reduce to Eq. (3.90) from the point-charge model ... 

In the two final sections an approximate version of the theory, based on a lattice distribution, is used to discuss the thermodynamic properties of liquids and liquid mixtures. 

Characteristics of the biphasic equilibria calculated from the lattice theory are summarized in Table 1. Results obtained for the volume fractions Vp and v at coexistence, for their ratio Vp/Vp, for xVp and for y/x according to the 1956 approximate version of the theory are given in columns 2, 3 and 4 for the neat fluid, for rods with X = 20 and in the limit x = oo, respectively. Corresponding calculations from the exact version of the theory are given in the last three columns. The latter calculations yield somewhat lower volume fractions for the coexisting phases. The ratios Vp/Vp are smaller in the limit x -> oo this ratio is 1.465 compared with 1.592 in the earlier approximation The differences are comparatively small, however. Hence, use of the 1956 treatment with its advantages of greater simplicity is vindicated for most purposes. 

Owing to the discontinuity in the p(r) versus N plot, three different approximate versions but well suited for different varieties of chemical reactions have been proposed as follows [36],... 

Approximate versions of the translational EPR state, wherein the -function correlations are replaced by finite-width (Gaussian) distributions, have been shown to characterize the quadratures of the two optical-field outputs of parametric down-conversion, or of a fiber interferometer with Kerr nonlinearity. Such states allow for various schemes of continuous-variable quantum information processing such as quantum teleportation [Braunstein 1998 (b) Furu-sawa 1998] or quantum cryptography [Silberhorn 2002], A similar state has also been predicted and realized using collective spins of large atomic samples [Polzik 1999 Julsgaard 2001]. It has been shown that if suitable interaction schemes can be realized, continuous-variable quantum states of the original EPR type could even serve for quantum computation. 

The orbital optimization was, in the first implementation of the CASSCF method, performed using an approximate version of the super-CI method, which avoided the calculation of the third-order density matrix s. This was later developed as a procedure entirely based on the average MCSCF Fock operator ... 

As already pointed out, the super-CI procedure can be regarded as an approximate version of the augmented Hessian approach. This is of course also true for the method described above, the only difference being that the approximations made are more severe. 

Many different isotherms have been derived for molecular adsorption at interfaces, including the liquid gas, liquid liquid, and liquid solid interfaces. Some of these isotherms can be shown to be approximate versions of the general isotherm derived here. 

The basic idea of the algorithm stems from the observation that the requirement knum = k is too strict to fulfill for all frequencies and angles of propagation. To circumvent this problem, an approximate version of (5.39), based on Taylor expansion, is utilized and then knum = k is applied in the modified equation [28], The difference between the two sides of the expression, so extracted, is defined as the error function 2D ... 

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