Intramolecular Correlation Research Articles

Molecular Configuration in Bulk Polymers

Abstract Quantitative experimental evidence on (i) the stress in stretched rubbers, (ii) temperature coefficients of the stress, (iii) cyclization equilibria, and (iv) thermodynamic activities of solvents in the Henry's law range reveal no discemible effects of dilution that could be attributed to dispersal of chain order postulated to exist in the undiluted polymer. The results in support of this statement have been obtained by diverse methods applied in many different laboratories. They appear to be confirmed by neutron scattering studies now in progress. The values of 〈r2〉0 deduced for PDMS from (iii) and of d ln〈r2〉0/dT from (ii) for several bulk polymers are in excellent agreement with measurements in dilute solutions. The body of evidence as a whole is compelling to the effect that the configurations of macromolecular chains in amorphous bulk polymers are random, and that they differ imperceptibly from the configurations occurring at infinite dilution (apart from the effect of excluded volume in dilute media). Scarcely less compelling is the evidence for local intermolecular order furnished by optical anisotropies observed by depolarized light scattering and by strain birefringence as discussed above. These observations do not directly contradict those cited in the paragraph above. We have pointed out earlier that intermolecular correlations need to be carefully distinguished from intramolecular correlations; it is the latter that are closely bound to the molecular configuration. Several aspects of the evidence for correlations obtained from optical anisotropy should be carefully noted. First, the range of intramolecular correlation of the group polarizability tensors is short. This is apparent from the fact that 〈γ2〉#x002F;n for the isolated chain increases by a factor of only ca. 1.3 for PM chains CH3—(CH2)n—H from n=1 to ∞. It is almost as if the individual group tensors αCH2 were mutually independent, in which case 〈γ2〉/n would be constant with n. In contrast, the characteristic ratio 〈r2〉0/n increases about eightfold (at ordinary temperature) from n=1 to ∞. The apparent optical anisotropy is therefore more vulnerable to intramolecular effects as judged by relative changes in this quantity. Secondly, intermolecular correlative effects manifested in the optical anisotropy do not necessarily denote parallelization of chains. In the case of n-alkane chains, for example, the same enhancement of the apparent 〈γ2〉 could be achieved by alignment of the vectors that are normal to the planes defined by the carbon atoms in trans sequences occurring in neighboring chains. Finally, the ‘bundle’, ‘micelle’, and mesomorphic models so often advocated for bulk amorphous polymers would require the optical anisotropy to be at least an order of magnitude greater than observed. The effects observed are small. They cannot be reconciled with the degree of order postulated. They are comparable in magnitude to the correlative effects identified in liquids consisting of small molecules. The concatenation of units comprising the polymer chain evidently does not of itself greatly enhance intermolecular correlations in the amorphous state.

New type of cluster theory for molecular fluids: Interaction site cluster expansion

A new cluster series, called the interaction site cluster expansion, is derived for classical molecular fluids. For a general class of molecular models, the series provides an exact formula for the equilibrium pair correlations between interaction sites on different molecules. In the models considered, each molecule contains m of these interaction sites. The total intermolecular potential between two molecules is the sum of m2 site–site potentials. A site–site potential depends on the separation between the sites only. However, because the sites are not necessarily located at the centers of molecules, the total molecular pair interaction can depend strongly on molecular orientations. Even though the interaction site cluster expansion is exact, the cluster integrals in the series involve the translational coordinates of the interaction sites only. The molecular orientational coordinates are removed by a transformation which introduces intramolecular correlation functions into the series. Thus, the new cluster series expresses the intermolecular site–site correlation functions in terms of diagrams involving site–site cluster bonds and intramolecular correlation bonds. The interaction site cluster expansions are used to derive several important results. It is shown that when the site–site interactions are hard core potentials, the pair correlation functions rigorously contain not only discontinuities but also cusps. The locations for these singularities are derived. A method is presented which shows how these singularities are smoothed when the hard core potentials are softened. The singularities as well as special limiting results derived herein are used to provide a justification for the reference interaction site model (RISM) equation.

Heitler-London renormalization for long-range many-molecule interactions

It is shown that the group function formalism of McWeeny leads in a natural way to a renormalization of the perturbation expansion of the partition function in terms of intramolecular correlation functions.

Perturbative ab initio calculations of intermolecular energies. I. Method

AbstractStarting from a knowledge of approximate wave functions of the isolated molecules (or atoms) A and B, a method is proposed to build up a zeroth‐order ground state and excited configurations for the complex AB in which the molecular orbitals keep their local significance. The standard Rayleigh‐Schrödinger perturbation in this basis provides a decomposition of the zeroth‐order interaction energy as a sum of the electrostatic and repulsion energy. In the second order, it is possible to identify the classical dispersion and polarization forces (modified by a term of order S2) and two additional contributions which are linked to the exchange possibility. The intramolecular correlation component is taken into account and compared with the correlation on the isolated molecules. It is moreover suggested that since we work in a rather limited basis set, the perturbed energy of AB must be compared with the unperturbed energies of A and B calculated in a basis including the vacant orbitals on their respective partner. Finally a possibility for going beyond the second order is described.

On a unified treatment of molecular interactions at finite temperatures

The by now standard expression for the free energy of interaction of a pair of molecules in terms of molecular susceptibilities at imaginary frequencies follows from a straightforward perturbation expansion of the statistical operator exp(− βH) and from the possibility to express the terms in the expansion in terms of intramolecular charge density correlations at imaginary times.

Interactions between Molecules in the Statistical Treatment of Real Gases. II. Derivation of van der Waals Interactions

The current perturbation theory for classical real gases, which views interaction energies between pairs, triplets, etc., of molecules, given in advance, as a perturbation to the ideal-gas behavior, is replaced by a systematic perturbation expansion in terms of the Coulomb interaction between the electrons and nuclei in the different molecules and the intramolecular correlations of these particles in the isolated molecules. The latter refer to correlations at different imaginary times t = − iτ, where 0 ≤ τ ≤ β with β = 1 / kT the reciprocal temperature. The Fourier transform of the pair correlation, expressed as a function of τ, is directly related to the susceptibility of the molecule at imaginary frequencies. In the expansion of the grand partition function the terms containing only particle pair correlations represent the contributions of molecular interactions due to the familiar additive and nonadditive van der Waals forces.

Quantum Mechanics of the H2–H2 Interaction. III. Nonorthogonal SCF—GF Calculations in the One-Configuration Approximation

Short-range intermolecular forces in the H2–H2 system have been approached by a rigorous quantum-mechanical calculation using a wavefunction consisting of one configuration of nonorthogonal group functions constructed from a minimal basis set of 1s Slater orbitals. The need of avoiding, with restricted wavefunctions, the forced orthogonalization of the atomic-orbital basis, which amounts to the relaxation of the strong orthogonality condition in a group-function approach, appears to be essential in order to give the correct angular dependence of the intermolecular energy. About 98% of the potential `barrier' resulting from the minimal-basis-set calculation with complete configuration interaction is accounted for by nonorthogonal group functions which allow only for intramolecular correlation. The even simpler one-determinant wavefunction constructed from SCF bond orbitals gives numerical results within 5% to 10% of the reference value for the intersection and within 3% to 5% for the energy difference between different relative orientations. Both inter- and intramolecular correlations seem therefore to be relatively unimportant in determining the relative orientation of the two molecules in the short-range region. The partitioning of the short-range interaction energy into a Coulomb component (slightly attractive and slowly varying with the dihedral angle) and into a correction or penetration term, which arises from electron interchange between the electron groups when they begin to overlap, shows that the largest part of the interaction and its orientation dependence arise from the repulsion due to negative overlap between the closed-shell charge clouds.

Nonadditive Interactions between Molecules in a Statistical Treatment of Dilute Gases

A dilute gas is treated by the method of diagrammatic perturbation expansion for a system of electrons and nuclei. A reinterpretation of the diagrams in terms of interacting molecules leads to their re-expression as functionals of intramolecular particle correlation functions. The first-order diagrams depend on the single-electron and electron-pair distributions in the molecule.

The scattering of light by freely flexible linear macromolecules in a hydrodynamic field: II. The interference factor

In part I such macromolecules were approximated by a ‘bead’ model and the correlation function for pairs of orientable beads calculated. The function is here used to calculate the scattering envelope. When the orientation and separation of beads is not correlated the envelope depends on an interference factor. The values of this factor are compared with those of Peterlin, Heller and Nakagaki. In the light of PHN's very different intramolecular statistics and correlation function the values of the factor show little change; but the new statistics may change that part of the scattering induced by the field in the particular direction of greatest extension of the molecule by upto 75% within the range of accessible shear rates; in the direction of greatest compression that part of the scattering induced by the field is changed even more. At still higher shear rates the scattering in almost all directions approaches the values of PHN more closely; but the difference in scattering in the particular direction of greatest compression increases to a limiting value of several hundred per cent. In the following paper III the effect of anisotropically polarisable beads on the depolarisation is considered.