Virial convergence

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

For low density (large Vm), the series (2.30) is expected to achieve useful accurary with only a few terms. Higher densities within the domain of convergence require additional terms to achieve a desired accuracy. For some densities, the virial series may not converge at all. 

This expansion in principle also includes terms proportional to (n/ V)2 and all higher powers of (n/V). However, when the density n/V is much smaller than the density of a solid or liquid, so that most of the container is empty space, this expansion converges rapidly and the higher terms can be ignored. B(T) is called the second virial coefficient, and is a function of temperature. 

Attempts to improve molecular wavefunctions so as to be able to calculate properties more accurately continue to be made, particularly via the constrained variational procedure. Two-particle hypervirial constraints were considered by Bjoma within the SCF formation,282 and he presented a perturbational approach to their solution.233 Using Scherr s wavefunction, and constraining p to satisfy the molecular virial theorem, a calculation on N2 led to rapid convergence.234-235 The constrained SCF orbitals are believed to be a closer approximation to the true tfi nearer the nucleus than further out. A later paper discussed the electron-density maps in comparison to the SCF derived maps, which confirm the conclusion that the wavefunction near the nucleus is improved.236... 

Before comparing theory and experiment let us discuss the convergence of the semiclassical expansion of the dielectric second virial coefficient. In Table 1-15 the classical dielectric virial coefficient the first and second quantum corrections, and the full quantum result are reported. An inspection of this table shows that the quantum effects are small for temperatures larger than 100 K, and /it(/) can be approximated by the classical expression with an error smaller than 2.5%. At lower temperatures the dielectric virial coefficient of 4He starts to deviate from the classical value. Still, for T > 50 K the quantum effects can be efficiently accounted for by the sum of the first and second quantum corrections. Indeed, for T = 50, 75, and 100 K the series (7) + lli 1 (7) + (7) reproduces the exact results with errors... 

The low temperature refractive properties of the He gas have not been studied extensively. However, the second virial Kerr coefficient can be related to the zeroth moment of the polarized Raman spectrum, and thus deduced from the Raman experiment. For the helium gas at the liquid nitrogen temperature the experiment gives 1.46 a.u.416, the full quantum calculation 1.45328, while the classical result computed according to Eq. (1-260) gives 1.63 328. This shows that also for the Kerr effect the quantum corrections are important. A systematic study of these corrections and of the convergence of the semiclassical expansion has not been reported thus far, even though all necessary expressions are derived328. 

The two forms of the virial expansion given by Eqs. (3.10) and (3.11) are infinite series. For engineering purposes their use is practical only where convergence is very rapid, that is, where no more than two or three terms are required to yield reasonably close approximations to the values of the series. This is realized for gases and vapors at low to moderate pressures. 

The virial expansion of Equation (4B-4) is inappropriate as the concentration increases since the series no longer converges at sufficiently high solute volume fractions. In this case,... 

An approach based on the virial expansion suffers from the difficulty of evaluating higher coefficients for highly asymmetric particles and from the non-convergence of the virial series at the concentrations required for formation of a stable nematic phase Lattice methods therefore take precedence over the virial expansion as a basis for quantitative treatment of the liquid crystalline state. 

Using eqs. (l)-(9), along with empirical pure-electrolyte parameters 3 ), 3 > 3 and and binary mixture parameters 0, one can reproduce experimental activity-coefficient data typically to a few percent and in all cases to + 20%. Of course, as noted above, the most accurate work on complex, concentrated mixtures requires that one include further mixing parameters and also for calculations at temperatures other than 25°C, include the temperature dependencies of the parameters. However, for FGD applications, a more important point is that Pitzer1s formulation appears to be a convergent series. The third virial coefficients... 

In the calculation of 0, . a mixture virial B is calculated using mixing rules described in Chap. 3 of the monograph. If B is substituted into Eqs. (11) and (12), the result is the virtual value of the partial molar enthalpy, ft Rigorous application of the Almost Band Algorithm requires that the derivative of Q be calculated with respect to temperature for use in the convergence procedure. This was not done in the solution of Examples 5-1 and 5-2. 

Examination of the terms to O(k ) in the SL expansion for the free energy show that the convergence is extremely slow for a RPM 2-2 electrolyte in aqueous solution at room temperature. Nevertheless, the series can be summed using a Pade approximant similar to that for dipolar fluids which gives results that are comparable in accuracy to the MS approximation as shown in figure A2.3.19(a). However, unlike the DHLL + i 2 approximation, neither of these approximations produces the negative deviations in the osmotic and activity coefficients from the DHLL observed for higher valence electrolytes at low concentrations. This can be traced to the absence of the complete renormalized second virial coefficient in these theories it is present... 

It is easily seen that this set of equations is in agreement with Eq. (19) in the limit of low densities, where only the first term in Eq. (39) is retained. But, of course, the equations are of interest only in the critical region. One remark on Eq. (39) should be made the virial expansion of is quite different from the virial expansion of G r), and probably converges to zero more quickly than G r). However, does not appear in every term of the expansion and it is not certain that the second moment of is finite at the critical point, which is a necessary condition for the validity of the Ornstein-Zernike theory. 

The virial expansion converges in the gas phase. Thus, the equation of state of gases can be adequately represented by the virial expansion over the entire range of density and pressure. However, in practice the virial expansion is used only when the first few terms need be kept. At... 

At 17.4°, 20.4° and 21.8°K, there appear to be no critical points for the helium-hydrogen mixtures this is indicated by phase equilibrium data calculated at these temperatures at pressures up to 7000 psia. If a critical point were reached, thepylx curves for helium and hydrogen could converge to the same value at the critical point, but there appears to be no tendency toward convergence even at pressures of 7000 psia. Unfortunately, no experimental data exist to verify this calculation. Virial coefficients calculated from the correlation are in good accord with the experimental data of Varekamp and Beenakker [ ], as shown in Table V. 

Finally, we emphasize that, even if we had several virial coefficients for a substance, the virial equations still only apply to gases and gas mixtures— both the density expansion and the pressure expansion fail to converge for liquids. Moreover, in practice we can find data or correlations for, at most, B and C, so the expansions should only be used for gases at low to moderate densities. 

The last restriction in this equation is needed to satisfy the conditions of the theorem which allows the topological reduction. In this series, there is one term with no yo 8f bonds and only one term with one yo 8f bond. This latter term has only two field points and just one bond. There are an infinite number of graphs with two yo 8f bonds, and even the series for these terms looks like a low-density virial series which might not be expected to be convergent or meaningful at high densities. 

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