Polarization approximation two molecules

According to the Rayleigh-Schrddinger perturbation theory (Chapter 5) the unperturbed Hamiltonian is a sum of the isolated molecules Hamiltonians + Hg. Following quantum theory tradition in the present chapter the symbol for the perturbation operator will be changed (when compared to Chapter 5) //(i) s V. 

Despite the fact that we may also formulate the penurbation theory for excited states, we will assume that we are dealing with the ground state (and denote it by subscript 0 ). In what is called the polarization approximation. the zeroth-order wave function will be taken as a product 

We will assume that, because of the large separation of the two molecules, the electrons of molecule A are distinguishable from the electrons of molecule B. We have to stress the classical flavour of this approximation. Secondly, we assume that the exact wave functions of both isolated molecules i/r o and are at our disposal. 

Of course, function is only an approximation to the exact wave function of the total system. Intuition tells us that this approximation is probably very good, because we assume the perturbation is small and the product function o 

The chosen has a wonderful feature, namely it represents an eigenfunction of the operator, as is required by the Rayleigh-Schrodinger perturbation theory (Chapter 5). 

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Perturbation theory is a natural tool for the description of intemioleciilar forces because they are relatively weak. If the interactmg molecules (A and B) are far enough apart, then the theory becomes relatively simple because tlie overlap between the wavefiinctions of the two molecules can be neglected. This is called the polarization approximation. Such a theory was first fomuilated by London [3, 4], and then refomuilated by several others [5, 6 and 7]. 

Though limited to pressures where the two-term virial equation in pressure has approximate vahdity, this correlation is applicable to most chemical-processing conditions. As with all generalized correlations, it is least accurate tor polar and associating molecules. 

In 1947 the present writer published a paper [1] in which he studied the interaction between the polar covalent diatomic molecule A—X and the atom B which has a complete electronic shell. The atom X, in particular, may be a hydrogen atom. To simplify the problem only four electrons were considered two electrons from the bond A—H, and the unshared pair of electrons from atom B, viz. A H B. The respective problem was solved approximately by the method of valence structures. The following structures were adopted ... 

With the exception of highly polar materials, London dispersion forces account for nearly all of the van der Waals attraction which is operative. The London attractive energy between two molecules is very short-range, varying inversely with the sixth power of the intermolecular distance. For an assembly of molecules, dispersion forces are, to a first approximation, additive and the van der Waals interaction energy between two particles can be computed by summing the attractions between all interparticle molecule pairs. 

The effective correlation times for an approximately isotropic motion, tr, ranged from 40.3 ps in methanol to 100.7 ps in acetic acid for 5a, and from 61.6 ps to 180.1 ps for 5b in the same solvents. Neither solvent viscosity nor dielectric constant bore any direct relationship to the correlation times found from the overall motion, and attempts to correlate relaxation data with parameters (other than dielectric constant) that reflect solvent polarity, such as Kosover Z-values, Win-stein y-values, and the like, were unsuccessful.90 Based on the maximum allowed error of 13% in the tr values derived from the propagation of the experimental error in the measured T, values, the rate of the overall motion for either 5a or 5b in these solvents followed the order methanol N,N-dimethylformamide d2o < pyridine < dimethyl sulfoxide. This sequence appears to reflect both the solvent viscosity and the molecular weight of the solvated species. On this basis, and assuming that each hydroxyl group is hydrogen-bonded to two molecules of the solvent,137 the molecular weights of the solvated species are as follows in methanol 256, N,N-dimethylformamide 364, water 144, pyridine 496, and dimethyl sulfoxide 312. 

Fig. 6.12 Stereo views of the two forms of 6-XXII (MNP). On the right approximate orientations for the molecular dipole moments are also shown (see text). Note that in both cases the molecular dipole moments are nearly perpendicular to the polar b axis, which is perpendicular to the plane of the paper. Top, Form 2, with two molecules in the asymmetric unit bottom, Form 3, with four molecules in the asymmetric unit, designated by capital letters A, B, C, D. A and B are pseudocentrosymmetrically related, as are C and D.

Here is an intriguing idea the polarization approximation should be an extremely good approximation for the interaction of a molecule with an antimolecule (built from antimatter). Indeed, in the molecule we have electrons in the antimolecule positrons, and no antisjmunetrization (between the systems) is needed. Therefore, a product wave function should be a very good starting point. No valence repulsion will appear, and the two molecules will penetrate like ghosts. Soon after the annihilation takes place and the system disappear . 

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