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In earlier work [A. Y. Sokolov and G. K.-L. Chan, J. Chem. Phys. 144, 064102 (2016)], we introduced a time-dependent formulation of the second-order N-electron valence perturbation theory (t-NEVPT2) which (i) had a lower computational scaling than the usual internally contracted perturbation formulation and (ii) yielded the fully uncontracted NEVPT2 energy. Here, we present a combination of t-NEVPT2 with a matrix product state (MPS) reference wavefunction (t-MPS-NEVPT2) that allows us to compute uncontracted dynamic correlation energies for large active spaces and basis sets, using the time-dependent density matrix renormalization group algorithm. In addition, we report a low-scaling MPS-based implementation of strongly contracted NEVPT2 (sc-MPS-NEVPT2) that avoids computation of the four-particle reduced density matrix. We use these new methods to compute the dissociation energy of the chromium dimer and to study the low-lying excited states in all-trans polyenes (C_4H_6 to C_(24)H_(26)), incorporating dynamic correlation for reference wavefunctionswith up to 24 active electrons and orbitals.

A statistical theory of nematic systems has been developed for semiflexible polymers on the basis of a new wormlike model of chain statistics. This wormlike chain model closely represents the short‐range statistics of real chains and allows the evaluation of the orientation‐dependent configurational partition function, which replaces the rotational partition function employed in the classical treatments of nematic systems of rodlike particles. This theory has been applied to polyethylene (PE) and polytetrafluoroethylene (PTFE) chains in the bulk, and leads to the conclusion that PE chains form a stable isotropic melt (with its first order isotropic‐nematic transition expected to occur around ∼75 K), whereas PTFE forms a stable nematic phase in the melt. Furthermore, the entropy changes accompanying the isotropic‐nematic transitions of semiflexible polymers are predicted to be quite small, and the predicted order parameters at the transitions are not very high, falling in the range 0.4–0.7.

The statistical theory of nematic systems of semiflexible polymers is extended to polymer chains of finite length in the bulk. Isotropic‐nematic transition temperature, entropy change, orientational order, and conformational order at the transition are computed as a function of chain flexibility and chain length, using the new worm‐like chain model with limiting curvature developed recently. The theory can cover a wide range of chain flexibility and its results converge to those of the theory of rod‐like particles in the limit of completely rigid chains. Absolute stability of the nematic phase at all temperatures is predicted within suitable ranges of chain flexibility and chain length.

The low frequency dynamics of a swollen network is investigated by means of a many‐body Langevin equation which treats the junctions like slowly moving structural units, as suggested by recent improvements of the equilibrium theory of rubber elasticity. Both the viscoelastic constitutive equation and the optical autocorrelation function S (k, t) of a junction‐labeled network are then calculated for a model system with very long circuits. Deviations of S (k, t) from exponential behavior and quadratic linewidth are predicted to occur if the scattering wavelength is not much larger than the average distance between adjacent crosslinks. Calculated deviations are shown to be strongly dependent on functionality and swelling ratio. In particular, we predict a remarkable difference between trifunctional and tetrafunctional networks.

The Langevin equation is applied to the calculation of the dynamics of a Gaussian ring chain in the free draining limit. An analytical expression is obtained for the correlation functions, which provides the well known t1/2 asymptotic behavior. The dynamic shear moduli are obtained in the conventional low‐frequency range, and also in the high‐frequency range, where G′ (ω) and G″ (ω) are shown to be independent of the molecular weight, with finite limits G′ (∞) and [ωG″ (ω)]ω→∞; these asymptotic results anticipate a discussion of the far‐reaching analogy existing between the high‐frequency range of the Gaussian chain and the plateau region of an entangled system.

A mean field theory for two‐ and three‐dimensional cubic lattice solutions of long semiflexible chains is developed using Di Marzio’s treatment of excluded volume. The configurational term of the partition function at a fixed level of macroscopic orientation is evaluated in close analogy with recent treatments of wormlike chain systems. In the presence of macroscopic orientation, the effective one‐chain partition function shows typical Ising features. Mesomorphic phases of intermediate orientation are predicted to become more stable than the isotropic solution below a characteristic temperature which depends on concentration. In two dimensions, a continuous set of coexisting phases of different concentration and corresponding orientation joins the isotropic solution at the transition temperature. In three dimensions our analysis, restricted to states of uniaxial symmetry, predicts a first order transition whose order parameter decreases with concentration and approaches the limiting value 1/2 for very dil...

The effect of configurational hindrances in unswollen rubberlike networks is taken into account by assuming the existence of a proper correlation between the macroscopic deformation and the rms fluctuations of the network junctions along the various directions of space. Following James and Guth’s approach, some junctions have been fixed at the surface of a proper parallelepiped in order to avoid collapse of the network, and the above correlation is expressed through the introduction of appropriate δ functions into the configurational partition function. Results identical with those previously arrived at by Flory–Hoeve–Ciferri are obtained by assuming that the rms fluctuation components transform as the macroscopic stretch ratios. This assumption cannot hold under high deformations because it implies that the junction fluctuations increase without limit with the deformation. Therefore an intermediate model is proposed where (i) the average ms fluctuations of the junctions are independent of strain, as in J...

The mean‐square distances between various points of regular Gaussian networks are calculated using a new electrical analog and solving exactly a set of difference equations both in two and in three variables through an appropriate Fourier lattice transformation. The results show that the mean‐square radius of gyration of the network is of the same order of magnitude as that of a single chain for networks having a three‐dimensional connectivity, while it increases logarithmically with the total number of chains for networks having a two‐dimensional connectivity.

Configurational properties of the worm‐like chain with limiting curvature (WCLC), developed recently to investigate nematic systems of semiflexible polymers, are analyzed in detail. Employing this WCLC model, expressions for the characteristic ratio, the distribution of the end‐to‐end chain vector, and the persistence length are derived. Specific numerical results are obtained for polymethylene chains of finite length, and compared with the corresponding results of the rotational isomeric state theory and the classical Kratky–Porod worm‐like chain. Orientational correlations of chain segments with chain vector and the temperature dependence of the internal energy are also derived and compared with the predictions of other chain models.

Elastic interactions between polymer chains in concentrated liquids are incorporated into the equation of motion by adding a memory term whose kernel is related to the relaxation modulus of the system. Below a characteristic chain length mechanical and dynamical properties are predicted to approach Rouse‐like behavior. For long chain systems, an approximate self‐consistency condition is obtained for the dynamic viscosity in the form of a nonlinear integral equation. In this limit, the mechanical spectrum is shown to split into a molecular weight independent branch separated from terminal relaxations by a wide plateau region whose elastic modulus is calculated as a function of the same parameters commonly used for the Rouse characterization of low molecular weight polymers. Concentration dependences of plateau modulus, entanglement length, and transitional relaxation time are correctly predicted. Zero shear viscosity and terminal relaxation time are shown to scale with the same power of the molecular weight. Heuristic considerations on the viscosity exponent suggest a value of three. The calculated expression of the single chain scattering function in the transition region shows a continuous change from Rouse‐like behavior to flat response with decreasing wave vector. Application to multichain scattering by concentrated solutions predicts long‐time exponential behavior only above a critical scattering wavelength.

Formal similarities between the Hamiltonian of a chain of harmonic oscillators and the effective Hamiltonian of the mean spherical model are exploited by introducing a ’’white noise’’ Langevin equation which reproduces the equilibrium properties of the 3D mean spherical model. Application to the dynamics with no magnetic field is made in three steps: (a) determination of the space–time correlation function at equilibrium; (b) spin relaxation calculations at different constant temperatures; (c) prediction of thermodynamic behavior after sudden coolings from T0?Tc to T?Tc. Dynamic transitions with a final equilibrium state below Tc have a common feature: The approach to global equilibrium requires a time proportional to νγ, where ν is the total number of lattice sites; γ=2/3 for cooling to T?Tc and spin relaxation at T=Tc, whereas γ=1 for spin relaxation at T

Biphasic equilibria of lyotropic solutions of semiflexible polymers are calculated using the worm‐like chain model with limiting curvature. The threshold concentration at incipient phase separation varies gradually with molecular weight to reach a finite value at infinite chain length. This limiting concentration is found to depend on the axial ratio of the persistence length and on the temperature. The theory is then applied specifically to poly(n‐hexyl isocyanate) (PHIC) solutions. Deriving the configurational parameters from measurements on chain dimensions in dilute solutions, theoretical calculations are found to satisfactorily match recent experimental results on the molecular weight dependence of the threshold concentration of PHIC in toluene.

The modified Rouse equation of motion recently introduced for the study of multiple elastic interactions of polymer chains in concentrated liquids is applied to investigate the onset of entanglement coupling. For undiluted systems of ring polymers we conjecture a possible dynamic transition at M=3.938 Me, preceded and induced by strong alterations of the mechanical spectrum. Extrapolation to linear polymers localizes the onset of entanglement coupling at M≂1.97 Me.