Computer simulation studies of anisotropic systems. XI. Second- and fourth-rank molecular interactions

The Maier-Saupe theory for nematic liquid crystals provides a reasonable account of their orientational order and its temperature dependence. The theory is based on second-rank anisotropic interactions and its predictions can be improved by the introduction of higher-rank terms as in the Humphries-James-Luckhurst theory. However comparison with the properties of real nematogens does not allow an unambiguous test of the theory because the form of the pair potential is unknown. This is not the case for computer simulations where the intermolecular potential is defined. We have therefore undertaken a Monte Carlo study of the influence fourth-rank interactions on nematic behaviour and report the results of our simulations here. The model nematogen used as a reference is that developed by Lebwohl and Lasher in which the particles are confined to the sites of a simple cubic lattice and interact via a second-rank anisotropic potential. The simulation gives the internal energy, the heat capacity at constant volume and the second-rank order parameter as a function of temperature, as well as the nematic-isotropic transition temperature. These results are used to provide the first unambiguous test of the Humphries-James-Luckhurst theory. We also discuss the changes in the transition temperature which are caused by the introduction of the fourth-rank term in the pair potential using thermodynamic perturbation theory for the Helmoltz free energy.     

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second‐order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties—excitation energies, spectroscopic parameters, and potential energy curves—for five molecules: ethylene, butadiene, benzene, N₂, and O₂. The calculated results are compared with those obtained with second‐ and third‐order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second‐order multireference perturbation theory using Møller‐Plesset partitioning. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1390–1400, 2003     

N.M.R. investigations and the molecular field theory of nematic mixtures

N.M.R. measurements have shown that the mixture E5 can be described by a single order parameter. Based on molecular field theory the nematic-isotropic transition and the temperature dependence of the order parameter in the nematic phase are considered for a binary mixture of nematogens. Guided by the results of the N.M.R. measurements the binary mixture is treated as an effective medium characterized by a single order parameter. Soft attractive forces are taken into account as well as the excluded volume. The coexistence of nematic and isotropic phases in the phase transition region is discussed in detail.     

Spin-free quantum chemistry. XXVI. The Ising,small-bipolaron theory of cuprate superconductivity

The Ising, small-bipolaron (ISB ) theory is a strong-coupling theory of cuprate superconductivity which is based on the negative-U, Hubbard Hamiltonian. Its ground state is composed of (small) bipolarons and (small-bipolaron) holes with a vibronically induced, bipolaron-hole exchange interaction, J_BH, between them. The energy gap, Δ(0), is taken to be equal to the dissociation energy of a small bipolaron and which, since it is defined spectroscopically, is not an order parameter. The application of the Ising mean-field theory to the highly degenerate ground-state yields a second-order phase change with kT_C/2 = J_BH and a real order parameter, Ω(T), which is valid over the entire temperature range from zero to T_C. Near T_C, the Ising free-energy functional takes the same form as does the Landau. In the presence of an electromagnetic field, the Ising functional is a generalization of the Ginzburg-Landau functional which employs a complex order parameter and which is invariant under the electromagnetic gauge transformation. The breaking of the gauge invariance yields the London theory of superconductivity. © 1996 John Wiley & Sons, Inc.     

Comment on ‘Pople versus Dunning basis sets for group IA metal hydrides and some other second row hydrides: The case against a De Facto standard’ by R.A. Klein and M.A. Zottola [Chem. Phys. Lett. 419 (2006) 254–258]

In their recent Letter, Klein and Zottola [Chem. Phys. Lett. 419 (2006) 254–258] emphasized the superior performance of density functional theory combined with the 6-311++G(2d, p) Pople-style basis set relative to conventional methods, such as second order perturbation theory and coupled cluster theory, with correlation consistent basis sets for the prediction of bond lengths. In this work, we present evidence that the former approach may be less capable of achieving uniformly accurate results for a variety of spectroscopic properties.     

Model for a three-center decomposition reaction. I. Perfluorodiazirine

A semiempirical approach has been used to evaluate rate parameters for a three-center decomposition reaction from the point of view of transition state theory, with perfluorodiazirine serving as the prototype molecule. Several activated complex models are considered in which the reaction coordinate is chosen as the ? NCN bending mode. The constraints imposed include the principle of concerted bond-order conservation in passing from the initial to the final state, and use is made of empirical bond order–bond length and bond order–force constant relationships. The geometric configuration of the transition state sought is one which conforms with the lowest energy path and is also consistent with the observed entropy of activation. The potential energy of activation is taken as the optimum difference in binding energies (based on the INDO method) between the transition and initial states, and the critical energy is obtained by applying a correction for the zero-point energy difference, derived from normal coordinate analysis. Satisfactory agreement is found in the case of the activated complex model for which the total bond order is conserved and bonds undergoing rupture are assigned a fractional bond order (FBO) of 2/3, derived from the postulate (FBO) = α/β whe re α(=2) is the number of bonds breaking, and β(=3) is the number of bonds undergoing change in the ring opening.     

Order parameters from U.V. spectra with incompletely polarized absorption bands

A new method is developed for the determination of Saupe's order parameters S* and D* from I.R. or U.V. spectra and the temperature dependence of their degree of anisotropy R. This method can be applied to molecules with one uniformly polarized band and another band of mixed polarization if all bands concerned are polarized in the same plane. For that a relation D* = f(S*) between the order parameters as a function of temperature is necessary, which can be obtained from the lines of constant entropy in the order triangle. These isentropics are calculated from the orientational distribution function of the ordered system. The resulting function D* = f(S*) is equal to that of the mean field theory given by Luckhurst et al., but can be derived here in a very easy way. The method is applied to diaminoanthraquinones with a point symmetry C_2v.     

Formulation of N- and v-representable density functional theory. IV. Non-Born–Oppenheimer approach

Local-scaling transformations are employed in order to formulate a desity functional theory where both electrons and nuclei are treated quantum mechanically. Because of the properties of these transformations, the ensuing version of density functional theory is N- and v- representable. In particular, we study the effect of these transformations on both densities and generating wave functions, and as a result, a functional for the energy expressed in terms of the electronic and nuclear densities is constructed. Plausible physical approximations for simplifying this electronic-nuclear functional are considered, and upon variation of the fuctional with respect to the electronic and nuclear densities, a system of coupled Euler–Lagrange equations is obtained.     

Localized bond model for molecular energy to fourth order in perturbation theory

The localized bond model of Malrieu, Diner, and Claverie is extended to fourth order in perturbation theory. Single, double, triple, and quadruple replacements from the doubly occupied bonding reference function are included utilizing a symmetric form of diagrammatic perturbation theory. The fourth order theory derived executes on a computer as quickly as does the third order theory. Results are examined utilizing the Pariser–Parr–Pople and CNDO/2 model Hamiltonians, and are compared with third order results and with either exact results where they are known, or with a configuration interaction of all singles and doubles. The influence of the initial hybridization, localization, and bond polarization is discussed. In general, the fourth order corrections are of comparable size to third order. Improvement in results appears to be marginal in the Nesbet–Epstein scheme in passing to fourth order because of the oscillating nature of the series; for Moller–Plesset theory errors are approximately halved. The relative energies as a function of modest geometry change about minima is about the same at third order as it is at fourth for most cases examined.     

The alpha–beta transformation in polypivalolactone. II. A simple mechanical model

A simple mechanical model based on the equilibrium rate theory of Burte and Halsey is presented in order to explain the stress–strain curve of polypivalolactone (PPL) fibers. A semiquantitative application of this model indicates that the PPL alpha–beta transformation implies the cooperative deformation of at least thrity chain units and that the free energy difference between the alpha and beta states is of the order of 2 kcal/mole of monomer.     

Uncertainty calculations in the certification of reference materials. 1. Principles of analysis of variance

 The preparation and certification of reference materials is a rapidly developing area. Many innovative reference materials have limited homogeneity and stability, and, additionally, the uncertainty estimation of the property values must be brought in agreement with the principles of the “Guide to the expression of uncertainty in measurement” (GUM). The results of the homogeneity and stability studies must be included to a certain extent in the uncertainty of the property values of the reference material, in order to comply with these requirements. The basic theory needed to accomplish this is essentially the theory of analysis of variance (ANOVA). As GUM also allows alternative evaluations other than Type A evaluations, a reinterpretation of the theory of ANOVA is necessary to establish a model for the certification of reference materials that is widely applicable. For this, analysis of variance can be used as a statistical technique to derive standard uncertainties from homogeneity, stability and characterisation data. Received: 10 May 2000 / Accepted: 29 July 2000     

Quantum-chemical Treatment of Cyclization and Recyclization Reactions: XXVI. Effects of Substituents in the Thermal Rearrangement of Methylenecyclopropane

Basing on the coupled perturbation Hartree-Fock theory an effect of substituents on the thermal rearrangement of methylenecyclopropane derivatives was considered. The effect is described in the second order of the theory. The activation energy depends both on the first and second power of the parameter characterizing the substituent, and introduction of several groups as a rule cannot be described in the framework of an additive scheme and requires accounting for the interactions between the substituents. The results of computation are in good agreement with the results of kinetic measurements.     

A constant denominator perturbation theory for molecular energy

A constant denominator perturbation theory is developed based on a zeroth order Hamiltonian characterized by degenerate subsets of orbitals. Such a formulation allows for a decoupling of the numerators of the perturbation sequence, allowing for much more rapid evaluation of the resultant sums. For example, the full fourth order theory can be evaluated as an N ⁶ step rather than N ⁷, where N is proportional to the basis set.Although the theory is general, a constant denominator is chosen for this study as the difference between the average occupied and average virtual orbital energies scaled so that the first order wavefunction yields the lowest possible variational bound. The third order correction then appears naturally as a scaled Langhoff-Davidson correction. The full fourth order with this partitioning is developed. Results are presented within the localized bond model utilizing both the Pariser-Parr-Pople and CNDO/2 model Hamiltonians. The second order theory presents a useful bound, usually containing a good deal of the basis set correlation. In all cases examined the fourth order theory shows remarkable stability, even in those cases in which the Nesbet-Epstein partitioning seems poorly convergent, and the Moller-Plesset theory uncertain.     

Diffusion-controlled vinyl polymerization. IV. Comparison of theory and experiment

Bulk polymerization data of methyl methacrylate, ethyl methacrylate, ethyl acrylate, n-propyl acrylate, vinyl acetate, and styrene were compared with the predictions of the theory proposed in the earlier parts of this series (I-III). This theory of polymerization kinetics uses the concepts of free volume and chain entanglements to describe the relationship between chain mobility and chain length dependent termination reactions. Excellent agreement was found between the predictions of the theory and the polymerization rate and molecular weight data of the six polymerization systems studied. Emphasis was placed on the ability to explain the development of higher order molecular weight averages (M?_w, M?_z, etc.) because they provide the most crucial tests for such a model. No changes were required in the model as it was applied to the different polymerization systems for a variety of reaction conditions. The theory offers a unified understanding of the diverse polymerization behavior displayed by such systems.     

C. A. coulson and the surface energy of metals: The distribution of eigenvalues as a difficult problem in number theory

An article by Coulson on the relationhship of physics and mathematics has attracted our attention to problems already discussed by Titchmarsh, namely the distribution of quantum energy levels of a particle confined in a square or a cube with infinite potential-energy walls. These problems are identical to some classical problems in the theory of lattice points, which belongs to number theory. They are seemingly simple, but in fact they are very difficult and not yet completely solved. We summarize the most relevant results from number theory. Whereas the beautiful theorems bearing directly on the error with respect to elementary estimates only provide indications concerning the order of that error; theorems concerning mean values can be used as a basis for a physicist's guess on its asymptotic mean value. This paper has also been intended to contribute to bridging the gap between quantum mechanics and number theory.     

Multiple scattering theory. II. Collision induced dissociation in first order

Computationally tractable formulae for one particle differential cross sections and for coincidence cross sections for atom— diatom collision induced dissociation (CID) are obtained within the framework of first order multiple scattering theory. The first order formulation is then used to analyze the simple “knockout circle” model for CID and to derive a more reliable multiple scattering spectator model. Finally, differential and coincidence cross sections are computed for Li⁺ + H₂ at 2O eV and are compared with experiment. The qualitative insight afforded by coincidence studies of CID is clearly demonstrated.     

Theoretical study on the sound absorption of electrolytic solutions. II. Assignments of relaxations

The theory on the ultrasonic absorption spectrum of electrolytic solutions recently proposed by us is applied to the model system that resembles to the aqueous solution of MgSO4. The charges on ions are reduced to +/-1.5e in order to obtain the equilibrium structure by the integral equation theory. The theory reproduces the existence of two relaxations around 100 kHz and 1 GHz. The physical origin of the relaxation is analyzed based on the theoretical expression. The slower relaxation is shown to originate in the formation of contact ion pair, in harmony with the conventional assignment. The amplitude of this relaxation agrees with the experimental one fairly well. The absorption cross section is a weakly increasing function of the concentration of the salt in theory, whereas it depends little on the concentration in experiment, which is ascribed to the weaker association of the pair in the theory. The deviation from the Debye relaxation is found for the faster process, and the concentration dependence is small. The analysis shows that this relaxation stems from the coupling between the pressure and the long-range concentration fluctuation, and the concentration independence and the non-Debye relaxation are explained based on the theoretical analysis. In particular, the theory demonstrates that this process has the t(-3/2) tail in the time domain, which is confirmed by numerical calculation. The deviation of the theoretical relaxation amplitude from the experimental one is elucidated in terms of the theoretical expression of the coefficient.