Nature of the Approximation

Consider the simplest type of clathrate having only one kind of cavity so that we can drop the i subscripts. If a gaseous molecule K is encaged in one of the cavities this corresponds to the process 

In other words the empty cavity binds the molecule K to form an occupied cavity. If we denote an empty cavity by X and an occupied one by Y this can be described as a reversible chemical reaction 

If we may neglect interaction between neighboring if molecules the reaction will have an equilibrium constant 

Assuming that K is the only solute present one further has 

On substituting these values into the previous relation one obtains (when expressing Cr in the proper units) 

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These simplified relationships offer a clearer insight into the dependence of the equilibrium swelling ratio qm on the quality of the solvent as expressed by Xh on the extent of cross-linking. Because of the nature of the approximations introduced to obtain Eqs. (40) and (40 ), their use as quantitative expressions must be limited to networks of very low degrees of cross-linking in good solvents. 

K r,f,r) = e r e " ° r) which neglects all quantum effects arising from the noncommutativity of the operators and v. In order to appreciate the nature of the approximation, let us consider the case where the energy potential v(r) = v + A r, with v and A constant quantities. Although the QMP for a particle subjected to a constant force is one of the few cases explicitly known [32], for our convenience we adopt the following exact alternative representation of the propagator for a particle moving in a linear potential [see Appendix A, eq. (A.8)]... 

In view of the result just found, it is interesting to contrast exact and approximate behaviour of the density n(x F) [eqs. p.5) and (2.17), respectively]. Some insight into the nature of the approximations contained in our treatment is gained through the inspection of Table 1, which collects Fermi-level energy values calculated for several electron occupation numbers and two different electric field amplitudes. The entries have... 

If this were not an approximation, we could evaluate the components of the Z-matrix, the angles a and /i, and solve for y. The nature of the approximation causes this approach to fail, but we can still look for the value of y that brings the function closest to a solution. Differentiating with respect to y, we have ... 

Herzberg (Nobel prize for Chemistry, 1971) commented on the two distinct photoionizations from methane that this observation illustrates the rather drastic nature of the approximation made in the valence bond treatment of CH4, in which the 2s and 2p electrons of the carbon atom are considered as degenerate and where this degeneracy is used to form tetrahedral orbitals representing mixtures of 2s and 2p atomic orbitals. The molecular orbital treatment does not have this difficulty". 

The nature of the approximation involved here is described by S. Glasstone, Textbook of Physical Chemistry, 2nd ed. (Princeton, NJ Van Nostrand), p. 454. 

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further... 

At present it is difficult to estimate which approximate solution would be better for a particular problem, so that each case requires an independent estimate of possible errors introduced by the nature of the approximation. 

In this section, we outline a procedure for obtaining a Hamiltonian for the treatment of low-frequency vibrations in molecules. We do this, in particular, to point out the justification for some of the Hamiltonians used in the past and to make clear the nature of the approximations involved in arriving at a specific Hamiltonian. Since there is danger of overinterpreting the results obtained from approximate Hamiltonians, we indicate some of the pitfalls in doing so. 

Diabatic states are obtained from a similar approach, except that additional term (or terms) in the Hamiltonian are disregarded in order to adopt a specific physical picture. For example, suppose we want to describe a process where an electron e is transferred between two centers of attraction, A and B, of a molecular systems. We may choose to work in a basis of vibronic states obtained for the e-A system in the absence of e-B attraction, and for the e-B system in the absence of the e-A attraction. To get these vibronic states we again use a Born-Oppenheimer procedure as described above. The potential surfaces for the nuclear motion obtained in this approximation are the corresponding diabatic potentials. By the nature of the approximation made, these potentials will correspond to electronic states that describe an electron localized on A or on B, and electron transfer between centers A and B implies that the system has crossed from one diabatic potential surface to the other. 

Next consider the probability distribution itself. The solutions to the approximate Eqs (7.8) and (7.5) are the probability densities in Eqs (7.9) and (7.10), respectively, which are Gaussian functions. To gain insight on the nature of the approximation involved we consider, for simplicity, a model slightly different from that considered above, where jumps to the left or the right occur in every time-step, so that pr + Pl = 1. Let the total number of steps taken by the particle be N. The probability for a particular walk with exactly tir steps to the right (i.e. ni = N — n,- steps to the left, so that the final position relative to the origin is n Ax n = nr — iii = 2nr — N) is... 

To use the concept of chemical potential in contexts where material is being driven from point to point, i.e., in nonequilibrium situations, involves an approximation or assumption. As a procedure, it has proved successful in 30 or 40 years of use, but we should look a little more closely in the next chapter at the nature of the approximation made. 

The answers, fortunately, are Yes and Yes. For an example where the nature of the approximation we make can be quite clearly seen, consider the long bar in Figure 4.1. 

The basic ingredients of the SPT and the nature of the approximation involved are quite simple. We shall present here only a brief outline of the theory, skipping some of the more complicated details. 

Thus, Flory s approach appears to be fundamentally incorrect however, it is useful as an approximation the agreement with experiments does not seem accidental but the exact nature of the approximation remains mysterious, in spite of many attempts made to explain it. 

The convergence of the operator mapping = Ovji is determined by the wave packet nature of the wave function . The quantum mechanical nature of the approximation depends on the ability to represent the position momentum commutation relation [X, P] = ih. A close examination reveals that the function f(q) = q is not band limited on the interval [0, L] because it is discontinuous at the end of the interval q = L. Now if M = q is replaced by a periodic function f(q + Li) = f(q), then... 

Absorption models of this type have also provided important information about the nature of the zeolite framework. Calculations carried out by Furuyama and co-workers on gas absorption in mordenite and ZSM-5 zeolites have shown that the charge distributions within the channels may be quite different and that the nature of the approximation used in the calculation may have some bearing upon the reliability of the results. In these calculations the interaction of molecular dipole and quadrupole moments with the electric field have been considered, in addition to the usual repulsion dispersion and polarisation terms. 

Propagator or Green s function methods are employed in this chapter to analyze the many-electron problem in planar unsaturated molecules as treated within the Pariser-Parr-Pople (PPP) model. A derivation of the model in many-electron theory serves to demonstrate the nature of the approximations involved. Applications are presented for the case of weakly interacting atoms. A decoupling procedure for Green s functions proposed by the authors is shown capable of yielding a correct description of this case. 

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