Orbital functional theory of linear response and excitation

Orbital functional theory (OFT) is based on a rule that determines a single‐determinant reference state Φ for any exact N‐electron eigenstate Ψ. An OFT model postulates an explicit correlation energy functional E_c of occupied orbital functions {?_i} and occupation numbers {n_i}. The orbital Euler–Lagrange equations are analogous to Kohn–Sham equations, but do not in general contain local potential functions. Time‐dependent Hartree–Fock theory is generalized in OFT to a formally exact linear response theory that includes electronic correlation. In the exchange‐only limit, the theory reduces to the random‐phase approximation of many‐body theory. The formalism determines excitation energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Statistical mechanical model of diffusion of complex penetrants in polymers. II. Applications

The theory of Part I is applied to the diffusion of several aromatic diffusants in two “smooth chained” polymers: poly(ethylene terephthalate) (PET) and natural rubber. Modifications of the theory necessary to accommodate vinyl polymers are discussed and applied to benzene in PMA and PEA. In all cases the theory agrees satisfactorily with the experimental D(0,T) and D(c,T)/D(0,T) data, and the values of the disposable r_g and Δ parameters are of the expected order. The limiting Arrhenius behavior of benzene in natural rubber appears to be correctly predicted. The cell model is definitely more appropriate than the free volume model for the calculation of enfolding chain effects in highly crystalline PET. For the three amorphous polymers the two models give comparable results, the cell model being somewhat superior for natural rubber and PMA.     

The volume structural relaxation theory for amorphous polymers

An interpolation model of density fluctuations in amorphous bulk polymers is suggested for the explanation of some peculiarities of structural relaxation of amorphous polymers volume. The main peculiarities of the model are (i) its account of the influence of fluctuations in the region where the WLF theory is true (the region of large fluctuations); (ii) its account of small fluctuations in the region where the dissipation mechanism appears to be considerably different from that in the region of large fluctuations; and (iii) its joining of the solutions of the resulting kinetic equations in the crossover region. On the basis of the suggested model we have obtained an equation of the Fokker-Plank type for the distribution function for density fluctuations with nontrivial boundary conditions for large fluctuations, and a quasistatic solution for small ones. These equations are analyzed and numerically solved in the isothermal regime. The results show that the theory corresponds to the WLF theory at temperatures much higher than T_g and that memory effects are found in the T_g region. In addition, preliminary estimates show that near T_g an extreme form of temperature dependence on activation energy is observed. All these facts agree qualitatively with experiment.     

Effects of chain length distribution on viscoelastic properties of entangled linear polymers: Blending laws for binary blends

The reptation idea of de Gennes and the tube model theory of Doi and Edwards are extended to explain the terminal viscoelastic properties of binary blends in the highly entangled state of two linear monodisperse polymers with different molecular weights M₁ and M₂. A modified tube model is proposed that considers the significance of the constraint release by local tube renewal in accounting for the relaxation process of the higher molecular weight chain. Its relaxation by both reptation and the constraint release is remodeled as the disengagement by pure reptation of an equivalent primitive chain. From knowledge of the longest relaxation times of the blend components, the stress equation is formulated from which blending laws of viscoelastic properties for the binary blends are derived. To force better agreement between theory and experiment at the pure monodisperse limits of the blends, a crude treatment to include the effect of contour-length fluctuation in the equivalent-chain model is also discussed. Theoretical predictions of the zero-shear viscosity and steady-state shear compliance are shown to be in good agreement with literature data on undiluted polystyrenes and polybutadienes over a wide range of the blend composition and M₂/M₁ ratio. The asymptotic of the laws for blends with M₂/M₁ → 1 and 0 are comparable to those from the relaxation spectrum proposed by others earlier on the basis of the tube model.     

Free Energy of a Non‐Gaussian Polymer Brush

An analytical theory describing layers of polymer chains grafted to a planar surface (i.e. polymer brush) is developed. We consider a brush of chains with finite extensibility (or non‐Gaussian brush) within the framework of molecular field theory. An analytical solution for free energy of the brush and a few other brush characteristics are obtained and studied. Comparison with other known models of a brush is also made.

Chain extensibility E(x, y) for Gaussian model (dashed lines) and BCC model (solid lines) for a few chain end positions y (numbers near curves).     

On the connections between Brueckner–coupled-cluster,density-dependent Hartree–Fock,and density functional theory

The existence of an effective one-particle Hamiltonian in the Brueckner coupled cluster model naturally leads to the definition of an effective interaction G, which is a function of the T₂ amplitudes. Two types of approximations to G are proposed: One is purely phenomenological, while the other is based on approximations to the Brueckner T₂ equation. In both cases, the resulting effective interaction may be viewed as electron-density-dependent. Generalizing Hartree–Fock theory to accommodate density-dependent interactions (DDHF ), a method is obtained that is capable of accounting for correlation effects in an independent particle framework. The heuristic Skyrme force, successfully used in nuclear physics to model nucleon–nucleon interactions, is presented here as an example of an effective electron–electron correlation interaction. Due to the δ-function character of the Skyrme force, it is possible to express the energy in this model by an integral over an energy density, thus formally providing a connection between DDHF and density functional theory for this particular case. An approximation to the Brueckner T₂ equation is also proposed in the coordinate representation. In this model, the density-matrix dependence of T₂ is reduced to a nonlocal electron density dependence by means of an expansion which introduces terms that depend on the gradient of the density. The first term in this expansion amounts to a “local density approximation” to Brueckner coupled cluster theory. © 1995 John Wiley & Sons, Inc.     

Description of oxygen permeability in various high polymers using a graph theoretical approach

Graph theory methods are shown to complement group additivity methods of predicting oxygen permeability in certain types of polymers. Graph theory is a topological approach that assigns a set of indices to a molecule to describe its structure. Since many physical properties of molecules depend upon their structure, graph theory indices can be used to describe important properties of molecules. In this work a set of graph theory indices are used to describe the property of a polymer based on a modified representation of the monomer unit. More specifically, Randic indices are used to describe the log of the oxygen permeability with 3.2% average relative error. Polymers comprising the basis set contain backbones of sp², sp³, or aromatic carbons, oxygen, or silicon and have substituents that contain chloride, fluoride, alkyl groups, hydrogen, oxygen, aromatic carbons, or chloro and/or fluoro substituted alkyl groups. The correlation coefficient (R²) (0 ≤ R² ≤ 1) of a nonlinear model is 0.91. The graph theory method for describing the oxygen permeability of these selected groups of polymers is in good agreement with that predicted by the permachor model. The permachor method makes oxygen permeability predictions based upon group additivity and distinguishes the degree of crystallinity of a polymer by empirically assigning different permachor (π) values to identical groups based upon the polymer crystallinity. The inability of graph theory to explain the remaining 9% of the scatter in the data is probably due to failure to incorporate into the graph theory model terms which quantify crystallinity.     

Streaming birefringence. VII. Influence of the intermolecular hydrodynamic interaction on the viscosity and streaming birefringence of polymer solutions

The cavity model used in the theory of dielectrics was applied to hydrodynamics to calculate the force exerted by a system of soft dumbbells on a reference dumbbell in a hydrodynamic field. The influence of this force on the viscosity and flow birefringence and its dependence on both the concentration and velocity gradient were calculated. The system of equations has a real solution only for values of β = M[η]η₀γ/RT which are smaller than a critical value rapidly decreasing with increasing concentration. At zero concentration the results obtained agree with the theory of a single isolated dumbbell model. The calculated Huggins constant is k′ = 0.4. The extinction angle is connected with the relative viscosity very nearly as derived from experiments. However, the theory fails at higher concentrations and gradients yielding an increase in viscosity with the gradient and infinite zero-shear viscosity for the concentration c = 2.5/[η].     

Cages versus Sheets: A Critical Comparison in the Size Range Expected for Methylaluminoxane (MAO)

New cage models (MeAlO)_n(Me₃Al)_m (n=16, m=6 or 7) isomeric with previously reported sheet models for the principle activator found in hydrolytic MAO (h-MAO) are compared at M06-2X and MN15 levels of theory using density functional theory with respect to their thermodynamic stability. Reactivity of the neutrals or corresponding anions with formula [(MeAlO)₁₆(Me₃Al)₆Me]⁻ towards chlorination, and loss of Me₃Al is explored while reactivity of the neutrals towards formation of contact- and outer-sphere ion pairs from Cp₂ZrMe₂ and Cp₂ZrMeCl is examined. The results suggest on balance that a cage model for this activator is less consistent with experiment than an isomeric sheet model, although the latter are more stable based on free energy.     

Bond length alternation in cyclic polyenes. V. Local finite-order perturbation theory approach

The finite-order many-body perturbation theory using the localized Wannier orbital basis is applied to the problem of bond length alternation in the Pariser–Parr–Pople model of cyclic polyenes C_N H_N, N = 4v + 2, which may be regarded as a simplified model of polyacetylene. Both the Møller–Plesset and the Epstein–Nesbet-type partitionings of the model Hamiltonian are employed. The localized orbital basis enables an efficient truncation of the perturbation theory summations over the intermediate states as well as an elimination of energetically unimportant diagrams, thus enabling one to obtain the fourth-order Møller–Plesset-perturbation energies with a relatively small computational effort even for large polyenes. The results obtained with the second-, third-, and fourth-order Møller–Plesset and with the third-order Epstein–Nesbet perturbation theories yield very similar bond length distortions (about 0.05 Å) and stabilization energies per site (about 0.04 eV) as obtained earlier with the RHF , one-parameter AMO , and delocalized orbital perturbation theories. The effects of truncation and diagram elimination in the fourth-order Møller–Plesset perturbation theory and the abnormal behavior of the second-order Epstein–Nesbet perturbation theory results in the localized Wannier basis near the instability threshold of the RHF solutions are discussed.     

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.     

The elasticity of nematic networks

Nematic rubbers are composed of crosslinked polymer chains with stiff rods either incorporated into their backbones or pendant as side chains. When nematic effects axe strong, such rubbers exhibit discontinuous stress-strain relationships and spontaneous shape changes. We model such a rubber using Gaussian elasticity theory, including the nematic interaction via a mean field. Results are presented for the cases of uniaxial extension and compression. Under uniaxial extension the rubber can undergo a first order phase transition to a uniaxial nematic phase. Under uniaxial compression first or second order transitions are possible to genuinely biaxial nematic states with biaxial strains. When nematic effects are very small (i.e. T >> T_c where T_c is the nematic-isotropic phase transition temperature of the rubber) we postulate that the model is a good approximation to a conventional, non-nematic elastomer, and fit our model to data from an isoprene rubber.     

Nonadditivity of orientational interactions and perturbation theory for dipole hard spheres

The procedure of approximate summation is applied to a series of many-body perturbation theory for the internal energy of a dipolar hard sphere (DHS) fluid to produce an expression similar to the mean field approximation in the Widom-Rowlinson penetrable sphere model, which is well consistent with the experiment at moderate or high densities. Similar results are obtained from the hindered rotation model generalized for arbitrary density. The critical parameters ρ_c, P _c and T _c of both models are consistent with the data of machine experiments and are close to the parameters of the percolation transition in the DHS system.     

Gelation in Multiple Link System

The critical point theory is generalized to include gelation in multilink system with f functional units and J junction points. The equations derived include, as special cases, the cyclotrimerization model of J = 3, and the R–A_f model of J = 2. The theory is applied to the recent observation of the cyclotrimerization of bisphenol‐A dicyanate. The theoretical prediction agrees exactly with the Stutz‐Simak observation, D_c = 0.504, and accords with the Georjon‐Galy‐Pascault observation, giving a confirmation of the physical soundness of the theory. Under the smoothness assumption, we derive post‐gelation relationships with loop formation, the result suggesting the formation of permanent sol molecules that resist being absorbed into gel phase throughout an entire reaction process.

Representation of binary branching.     

Statistical thermodynamics of ABA block copolymers. III. Microstructural transitions and model verification

The predictions of a model presented previously are tested with data on five styrenebutadiene–styrene copolymers. Differential scanning calorimetry and laser light transmission are the primary tools, spanning 180–630°K, with some evidence supplied also by electron microscopy and mechanical properties. The existence of a first-order phase transition, characterized by a separation temperature T_s, is verified and found to be predictable by the model. Details of morphology, including transitions in microstructure in certain temperature ranges, are also reported and shown to be consistent with the theory.     

An average fock operator technique of approximate open-shell LCAO-MO-SCF calculation

The exchange part of the usual Hartree-Fock potential in the unrestricted Hartree-Fock (UHF) theory is suitablyaveraged to construct an, average one-electron model Hamiltonian which generates a set of spin-restricted one-electron orbitals in a self-consistent manner. These orbitals are then used to calculate the electronic energy of the open-shell system by using the proper functional form for the energy which handles the exchange terms correctly. The eigenvalues ofF ^av can be used for calculating either the spin-polarised or spin-averaged ionisation potentials of different orbitals at theKoopmans’ theorem level of approximation. Comparison ofE ^ac with the UHF-energy shows thatE _UHF<E ^ac in each case revealing some kind of an upper bound nature ofE ^ac. An approximate variational argument is given. Relationship of our model with the hyper-Hartree-Fock theory of slater is explored and the general problem of eliminating ‘self-interaction’ terms in average Fock-operator based theories is discussed.     

Theory of Monte Carlo simulations of the magnetic circular dichroism spectra of alkali metal/rare gas systems

The history of magnetic circular dichroism (MCD) spectroscopy in the study of alkali metal/rare gas (M/Rg) cryogenic systems is reviewed in the context of developing a better understanding of alkali metal/hydrogen systems of current interest to the U.S. Air Force as enhanced‐performance cryogenic rocket propellants. A new theory for simulating the MCD spectra of M/Rg systems is presented together with a careful discussion of the theory's implicit and explicit approximations and their implications. This theory uses a classical Monte Carlo (MC) simulation scheme to model the perturbing effects of the Rg environment on the ²S → ²P MCD‐active transition of the M atom. The theory sets up the MC–MCD simulation as a 6 × 6 matrix eigenvalue/eigenvector problem in the ²P manifold in which are included the effects of M–Rg interactions, metal atom spin‐orbit coupling in the ²P manifold, magnetic Zeeman perturbations of the ²S and ²P manifolds, Boltzmann temperature factors, and electric dipole transition moment integrals for left circularly polarized (LCP) and right circularly polarized (RCP) light. The theory may be applied to any type of trapping site of the host M in the guest Rg matrix; a single atom substitutional metal atom trapping site (one host Rg atom is replaced by one guest M atom) is modeled in this study for M = Na and Rg = Ar. Two temperature factors are used in these simulations; a lattice temperature to model the mobility of the Rg lattice and a magnetic temperature to model Boltzmann factors in the ²S ground manifold. The 6 × 6 eigenvalue/eigenvector problem is solved for a number of randomly generated and suitably averaged Rg configurations to yield the simulated MC‐MCD spectrum for the single substitutional Na/Ar system. The MC–MCD simulations of Na/Ar give the characteristic triplet MCD spectrum with the correct Boltzmann temperature dependence. The simulated MC–MCD spectrum correctly inverts when the direction of the applied magnetic field is reversed. Addition of the LCP and RCP absorbances gives rise to a characteristic ²S → ²P triplet absorption feature. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Enthalpy recovery in semicrystalline polymers

Constitutive equations are derived for enthalpy recovery in glassy polymers after thermal jumps. The model is based on the theory of cooperative relaxation in a version of the trapping concept. It is demonstrated that a critical temperature T_cr and a critical degree of crystallinity f_cr exist in a semicrystalline polymer above which structural relaxation vanishes.