Approximation exact

A basis set is a mathematical representation of the molecular orbitals within a molecule. The basis set can be interpreted as restricting each electron to a particular region of space. Larger basis sets impose fewer constraints on electrons and more accurately approximate exact molecular orbitals. They require correspondingly more computational resources. Available basis sets and their characteristics are discussed in Chapter 5. 

So far we have considered the first kind boundary conditions approximated exactly on grids. In the case of the third kind boundary conditions the question of their approximation needs investigation. In the next section we will say a little more about this. 

In this case the condition u(a ,0) = Ug x) and the boundary conditions are approximated exactly. For instance, one of the schemes arising in Section 1.2 is good enough for the difference approximation of the initial equation. No doubt, we preassumed not only the existence and continuity of the derivatives involved in the equation on the boundary of the domain in view (at. r = 0 or f = 0), but also the existence and boundedness of the third derivatives of a solution for raising the order of approximation of boundary and initial conditions. 

Note Sizes in mm are approximate. Exact values should be calculated by ... 

Here p iaa occ, L() (respectively p iaa unocc, L()) represents the probability of the atomic configuration of site i, where the orbital a with spin a is occupied (resp. unoccupied) and where L[ is a configuration of the remaining orbitals of this site. This result is similar to the expression obtained by Biinemann et al. [22], but it is obtained more directly by the density matrix renormalization (5). To obtain the expression of the qiaa factors, an additional approximation to the density matrix of the uncorrelated state was necessary. This approximation can be viewed as the multiband generalization of the Gutzwiller approximation, exact in infinite dimension [23]... 

Equation [139] is exact for a two-state solute, but differs from the traditionally used connection between the transition dipole and the emission intensity by the factor Vo/Vav." The commonly used combination miiVo/Vav appears as a result of neglect of the frequency dependence of the transition dipole mi2(v) entering Eq. [129]. It can be associated with the condensed-phase transition dipole in the two-state approximation." Exact solution for a two-state solute makes the transition dipole between the adiabatic free energy surfaces inversely proportional to the energy gap between them. This dependence, however, is eliminated when the emission intensity is integrated with the factor... 

We note that it is often possible to approximate exact lineshapes fairly closely if suitable analytical model profiles are selected whose lowest two or three spectral moments are matched to those of the measurement [231], Various suitable model profiles are known, but certain three-parameter models approximate the exact shapes so closely that lineshape calculations may be dispensible for some applications. Other analytical profiles are, however, less than convincing for the purpose [314],... 

Note Sizes in mm are approximate. Exact values should be calculated by converting from mils. For example, 2225 is actually 220mils x 250mils, and 0402 is 40mils x 20mils. 

When we compare exact and numerical solutions of (5.22) for n = 0.5, 0.8 and 1.0 we see minor quantitative but no qualitative changes in the profiles of Cb z). Similarly, for r 0.1, there is little difference between the exact profiles and the small r approximation. Exact solutions will be compared with these approximations in a forthcoming publication. 

Just before the equivalence point, the solution contains relatively high concentrations of NaCH3COO and relatively low concentrations of CH3COOH. Just after the equivalence point, the solution contains relatively high concentrations of NaCH3COO and relatively low concentrations of NaOH, both basic components. In both regions our calculations are only approximations. Exact calculations of pH in these regions are beyond the scope of this text. 

Figure 4.11 A series of Pearson VII profiles with equal peak position, peak height, and half width, but with different slope shapes (given by the exponent m), and area. For m= Lorentzian (L), m=1.5 intermediate Lorentzian (IL), m = 2 modified Lorentzian (ML). Already with m = 10 a Gaussian is approximated (exact at oo). X ray peaks mostly exhibit m values between 1.5 and 2. (After Howard and Preston, 1989. )...

The manipulations with the first relation in (246) are very simple, because the operator ii (2n) = 0. In the approximation exact up to the second order, we obtain that... 

Fig. 3. It is evident from symmetry that the effeetive force between the two particles, under the Ewald approximation, exactly vanishes for this configuration of the two particles within the central box.

Once potential parameters have been determined, we can start calculation downward following arrow in the figure. The first key quantity is radial distribution function g(r) which can be calculated by the use of theoretical relation such as Percus-Yevick (PY) or Hypemetted chain (HNC) integral equation. However, these equations are an approximations. Exact values can be obtained by molecular simulation. Ifg(r) is obtained accurately as functions of temperature and pressure, then all the equilibrium properties of fluids and fluid mixtures can be calculated. Moreover, information on fluid structure is contained in g(r) itself. 

The virial theorems derived above are exact (i.e., exact functionals satisfy these equations). However, these equations do not generally hold for approximate functionals. Therefore, they can be used to judge the quality of the approximation. Exact theorems or equations can be very useful in constructing approximate functionals, too. 

Note i2/1000 ft is approximate. Exact value depends on whether the conductor is solid or stranded, and if stranded, the type of stranding. Consult the manufacturers data sheets for exact values. 

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