Numerical techniques for investigating an excited state of the anisotropic Heisenberg model

An analysis of the anisotropic Heisenberg model is carried out by solving the Bethe ansatz solution of the model numerically as a function of finite N. A brief introduction to the infinite chain limit is presented and the energy for a few limiting cases of the anisotropy parameter are evaluated. Numerical results for the infinite chain are given which can be compared with the case of finite increasing N. It is shown that the calculation can be extended to the case of an excited state of the model.     

Molecular modeling of polymers: I. Correct and efficient enumeration of intrachain conformational energetics

Polymer conformational analyses can require being able to model the intramolecular energetics of a very long (infinite) chain employing calculations carried out on a relatively short chain sequence. A method to meet this need, based upon symmetry considerations and molecular mechanics energetics, has been developed. Given N equivalent degrees of freedom in a linear polymer chain, N unique molecular groups are determined within the chain. A molecular unit is defined as a group of atoms containing backbone rotational degrees of conformational freedom on each of its ends. The interaction of these N molecular groups, each with a finite number of nearest neighbors, properly describe the intramolecular energetics of a long (infinite) polymer chain. Thus, conformational energetics arising from arbitrarily distant neighbor interactions can be included in the estimation of statistical and thermodynamic properties of a linear polymeric system. This approach is called the polymer reduced interaction matrix method (PRIMM) and the results of applying it to isotactic polystyrene (I-PS) are presented by way of example.     

Viscoelasticity of dilute chain-molecule solutions: Evaluation of hydrodynamic interaction

The viscoelastic properties of chain molecules varying in flexibility and length have been calculated by use of the bead-spring model theory of Zimm. In the evaluation of the hydrodynamic interaction parameter, the number of springs in the bead-spring model, N, has been selected from the range in which the properties predicted by the theory are insensitive to the value of N. The results for limiting viscosity number agree with those predicted by the Yamakawa–Fujii theory of the limiting viscosity number of wormlike chains. The theory also fits the experimental data of Johnson on a sample of polystyrene of molecular weight 860,000 in theta solvents at infinite dilution. The viscoelastic properties of some moderate molecular weight deoxyribonucleic acid solutions are predicted to deviate from the non-free-draining behavior toward the free-draining behavior.     

Finite chain approximation for linear and nonlinear polarizabilities of polydiacetylene and polybutatriene

Static longitudinal polarizabilities α_zz and cubic hyperpolarizabilities γ_zzzz are obtained by the FPT-INDO method for finite chain C_4N+2H_2N+4 (N = 1–15) models of polydiacetylene and polybutatriene. For the acetylenic structure the onset of saturation first becomes evident at about C₁₄? C₁₈ for the linear polarizability and C₂₂? C₂₆ for the cubic term. The values per C₄H₂ unit are accurately extrapolated to the infinite chain limit. In the case of the butatrienic structure, no saturation is observed through C₅₄.     

Chiral Supramolecular Nanotubes of Single‐Chain Magnets

We report a single‐chain magnet (SCM) made of a terbium(III) building block and a nitronyl‐nitroxide radical (NIT) functionalized with an aliphatic chain. This substitution is targeted to induce a long‐range distortion of the polymeric chain and accordingly it gives rise to chains that are curled with almost 20 nm helical pitch. They self‐organize as a chiral tubular superstructure made of 11 chains wound around each other. The supramolecular tubes have a 4.5 nm internal diameter. Overall, this forms a porous chiral network with almost 44 % porosity. Ab initio calculations highlight that each Tb^III ion possesses high magnetic anisotropy. Indeed, notwithstanding the supramolecular arrangement each chain behaves as a SCM. Magnetic relaxation with both finite and infinite‐size regimes is observed and confirms the validity of the Ising approximation. This is associated with quite strong coercive field and magnetic remanence (H_c=2400 Oe M_R=2.09 μ_B at 0.5 K) for this class of compounds.     

Ab initio SCF calculations of the linear infinite chain of LiH according to the pseudo-lattice method

The pseudo-lattice (PL) method has been reformulated for ab initio self-consistent-field (SCF) calculations. The translational symmetries of infinite systems have been applied to the finite model chain by manipulating all the intramolecular and intermolecular Fock matrices. The nuclear repulsion energy has been corrected accordingly. The method has been tested for the linear chain of lithium hydride under the constraint of equidistance between all neighboring lithium and hydrogen atoms. The calculated results of the infinite chain have been compared with those of finite chains of lithium hydride under the same geometric constraint. The equilibrium geometries, band structures, intermolecular stabilization energies and potential curves have been studied. It is found that the infinite systems cannot be described by considering only first nearest neighbor interactions, and the intermolecular interactions must be considered at least up to third nearest neighbors in order to obtain accurate value of force constant of infinite systems. We can conclude from band structures of infinite chains that the boundary effect of the finite model chain is effectively removed by the PL method.     

Polymer chains confined into tubes with attractive walls: A Monte Carlo simulation

A bead-spring off-lattice model of a polymer chain with repulsive interactions among repeating units confined into straight tubes of various cross sections, D_T², is studied by Monte Carlo simulation. We are also varying the chain length from N = 16 to 128 and the strength of a short-range attractive interaction between the repeating units and the walls of the tube. Longitudinal and perpendicular static linear dimensions of the chains are analyzed, as well as the density profile of repeating units across the tube. These data are interpreted in terms of scaling concepts describing the crossover between three-dimensional and quasi-one-dimensional chain conformations and the adsorption transition of chains at flat infinite walls, respectively. We also study the time-dependent mean-square displacements of repeating units and obtain various relaxation times. It is shown that both relaxation times scaling proportional to N² and to N³ play a role in the reptative motion of the chain in the tubes.     

Extrapolation of real-space quantum chemical calculations from finite-size super cells to the ideal infinite system. I. Theory

A method is proposed how to calculate the correct density matrix of an infinite polymeric chain from that of a standard finite supercell calculation. The density matrix of the finite supercell is transformed into k-space for all k-values allowed by the periodic boundary conditions. The k-dependent matrices are then unitarily transformed, with each unitary matrix being represented by a set of complex rotation matrices. It is shown that the corresponding angles can be interpolated and extrapolated toward the zone boundaries in a straghtforward manner and that this extrapolation can be done from any finite supercell with reasonable accuracy. This gives rise to an infinite system density matrix for which all fundamental properties are guaranteed by construction. This infinite system density matrix may be used to construct a corrected density matrix for the finite supercell calculation. © 1994 John Wiley & Sons, Inc.     

Study of correlation holes. I. Number–sum rules and infinite system

The correlation holes in a finite system of any size are known to satisfy integral number–sum rules directly related to the N-representability of the reduced density matrices. In regard to infinite systems, the same rules are known to be satisfied in the electron gas. It is shown that the number–sum rules are not generally satisfied in any infinite system, and that this happens independently of the kind of boundary conditions assumed in the N → ∞ limit. A violation is explicitly found within the alternant molecular orbital formalism. The apparent paradox is explained in terms of surface effects (end effects in a linear system), which are present in a large, but finite system. In other words the N-representability does not imply the number–sum rules. In the N → ∞ limit the rules are satisfied only in physical systems having short-range correlation.     

Asymptotic behavior of the RHF energy of the PPP model of alternant cyclic polyenes

The large N expansion of the restricted Hartree–Fock (RHF) exchange energy per atom E(N) of the Pariser–Parr–Pople (PPP) model of cyclic polyenes (annulenes) C_NH_N is derived in detail. We explicitly derive the coefficients E₀ and E₁ of the asymptotic expansion: E(N)=E₀+E₁ ln N/N²+O(N⁻²), N→∞, in the very simple case of half-filling and no bond alternation. The exchange energy per atom in the infinite chain can be written as E_ex=(2/π²)∑^∞_j=1{[γ_∞(2j−1)]/[(2j−1)²]}, where γ_∞ is the two-electron repulsion integral in the infinite chain. On the other hand, the second coefficient E₁ giving a finite-size correction is found to be 1/2b, where b is the bond length. This value of E₁ differs slightly from that of a linear chain with periodic boundary conditions because the distance between sites depends upon the radius of the ring, i.e., upon N. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 397–407, 1998     

Predictions of a recently proposed non-markovian model for polymer melts and concentrated solutions

Several predictions for a recently proposed mesoscopic model for polymer melts and concentrated solutions is presented. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by a fractal “waiting-time” distribution function, with a single adjustable parameter β, which describes the long-time behavior of the distribution: ∼ 1/t^1+β. We find that the model predicts D ∼ 1/N² and η₀ ∼ N^3.4 for β ≈︁ 1.4, where N is the degree of polymerization. The generalized model predicts that the relaxation spectrum has a plateau regime whose height is independent of N, but whose width is strongly N dependent, in agreement with experiment. The model also predicts that rings will diffuse somewhat more slowly than linear chains of the same molecular weight (about 80% as fast), with the same scaling dependence on N as linear chains, also in agreement with preliminary data.     

Chain direction elastic modulus of PE crystal and interlamellar force constant of n-alkane crystals from RAMAN measurements

Spectroscopic data can deliver force constants only if the exact chain conformation is known. For the longitudinal acoustic modes (LAM), however, simple linear chain models can be used to yield the effective longitudinal chain modulus from spectroscopic data of oligomer crystals. The model of p-coupled linear chain molecules with N masses and only nearest neighbor interactions was used to investigate the longitudinal acoustic modes with s nodes. The frequencies plotted versus s/N fall onto different branches for different s. The intermolecular coupling and the heavier endmasses shift the LAM branches to higher and lower frequencies, respectively. There exists a value x₀ depending on the masses and force constants, where the branches cut the dispersion curve of the infinite molecule. For s/N ≥ x₀ the effect of endmasses dominates. Low-frequency RAMAN spectra of n-alkanes (N = 20, …, 40 C atoms) were recorded and analyzed. The LAM1 branch runs clearly above a smooth fit through all other LAM data and the origin. This fit approximates to first order the dispersion curve of the infinite PE molecule in an ideal crystal. Its curvature exceeds that of the dispersion curve of the simple linear chain, but is somewhat smaller than that of the dispersion curve of the planar zig-zag chain with rigid bonds. The slope at the origin yields the limiting elastic modulus E_c = 315 GPa in chain direction of crystalline polyethylene. From our measurements on n-alkanes we obtained the frequency shift of LAM1 due to the interlayer coupling and the heavier endmasses. Calculation of the intermolecular coupling constant of the model of a long row of linear chain molecules with the same frequency shift yield the mean value f_l = 2.5 N/m. This value decreases with increasing chain length. The relevance and applicability of the model is discussed. © 1997 John Wiley & Sons, Inc.     

Extrapolation of the linear and nonlinear polarizabilities from ab initio finite oligomer calculations

Different definitions of property per unit cell and different fitting functions are employed to obtain the asymptotic limit values per unit cell of the polarizability (α), the first (β), and the second (γ) hyperpolarizabilities of an infinite oligomer. A 1/n power series function is found to be suitable for the average value and logarithmic average value per unit cell definition, and an exponentially decreasing function is found to be suitable for the difference value per unit cell definition. These conclusions are derived based on an equation expressing the total energy per unit cell of a finite linear oligomer as a power series of 1/n, presented from a perturbation treatment. Several calculations of long chain systems have been carried out to reach our conclusions. An equation of p(n)/n = a + b/n + c/n² is strongly recommended for a least‐squares fitting of the properties per unit cell to achieve a stabilization behavior when the chain length is increased. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations

A wormlike chain model for rod type blocks in a rod‐coil diblock copolymer is implemented in the self‐consistent field theory (SCFT) formalism. A pseudo‐spectral method is used to solve for the single‐chain partition function of this copolymer system. Orientation degrees of freedom are discretized using Lebedev sphere rules such that orientation integrations are carried out through a Lebedev quadrature, an approach not used previously in tandem with the pseudo‐spectral method. Phase behavior in the rigid‐rod limit as a function of rod segment volume fraction, Flory–Huggins interaction parameter χ , degree of polymerization N , and rod contour length ratio β are examined in detail in one and two dimensions. Examples extending to three dimensions are included. Semiflexible behavior via the rod bending rigidity κ is explored. An approximation is used for rigid‐rods that do not need spherical harmonics leading to increased speed in finding equilibrium morphologies. The results show that standing vertical structures may be more easily produced with rigid‐rod blocks compared to coil‐coil lamellae, an important feature in nanolithographic applications. Suggestions are made for using the model in future molecular orientation studies where the model can be used with inverse search methods to measure the values of the model parameters for the real systems. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 29–39     

Chain collapse by lattice simulation

Monte Carlo simulations have been performed on a self-avoiding simple cubic lattice chain with the nearest-neighbor interactions for a range of chain lengths N from 40 to 1000 segments to investigate equilibrium properties of polymer chains from an athermal to a collapsed state. Both the fraction of segments in the clusters and the number of contacts exhibit the three stage process for the chain collapse, consistent with our previous molecular dynamics simulations of a fully atomistic chain. In the collapse region corresponding to the nearest-neighbor interaction parameter larger than 0.5 for a segment-solvent pair, polymer chains are quite spherical and both ends lie nearly randomized within the sphere. The peak height of the specific heat is proportional to N(In N)^3/11, as predicted by the renormalization group theory.     

Flow birefringence and optical anisotropy of poly(N-vinylpyrrolidone) molecules

Concentration dependences of flow birefringence and viscosity of poly(N-vinylpyrrolidone) solutions in water and benzyl alcohol are investigated. The intrinsic anisotropy for a poly(N-vinylpyrrolidone) macromolecular segment, (α₁ ? α₂) = ?(82 ± 8) × 10^?25 cm³, is determined from the results of birefringence measurements in benzyl alcohol. For aqueous solutions, a strong concentration dependence of the specific anisotropy of solution is obtained, a result that may be explained by the heterogeneity of coils. A model allowing for this heterogeneity is suggested. It makes it possible to fit the concentration dependence to a hyperbolic function, to separate contributions of heterogeneity anisotropy and form anisotropy to the birefringence of a solution, and to estimate the segment asymmetry parameter as p = 3.0 ± 0.5.     

Uniform electron gases

We show that the traditional concept of the uniform electron gas (UEG)—a homogeneous system of finite density, consisting of an infinite number of electrons in an infinite volume—is inadequate to model the UEGs that arise in finite systems. We argue that, in general, a UEG is characterized by at least two parameters, viz. the usual one-electron density parameter ρ and a new two-electron parameter η. We outline a systematic strategy to determine a new density functional E (ρ, η) across the spectrum of possible ρ and η values.     

Transition from two-to three-dimensional behavior in anisotropic polymer layers

Two types of anisotropic polymer systems were studied in the spherical approximation used in the classical theory of ferromagnetism. These were a three-dimensional system composed of weakly interacting layers with isotropic interactions between chain segments in the layer planes and thin quasi-two-dimensional polymer films possessing intra-and interchain interaction anisotropy, whose behavior is close to that of two-dimensional systems. Laws that govern a change in the temperature T _cr of phase transition from the long-range order state to a disordered state depending on the magnitude of anisotropy and the size of the layers were established. For systems of the former type in which interlayer interactions is weakened, T _cr tends to zero, being inversely proportional to lng, where g is the ratio of the interaction constant between the layers to that of inplane interaction in a layer. For systems of the latter type, the transition temperature T _cr → 0 according to the T _cr ～ √? law, where ? is the parameter that characterizes the anisotropy of intra-and interchain interactions. The number of layers required for the systems to be considered three-dimensional was estimated. Regardless of the type of boundary conditions for finite systems, the number of layers increases with enhancement of interaction anisotropy (a decrease in g and ?) and an increase in the number of chains in the layers, especially for systems of the former type. Transverse orientational correlations of chain segments with respect to the arrangement of the layers decrease according to a power law, as in the case of infinite two-dimensional systems. There are fluctuations of three-dimensional long-range orientation order in the plane of the layers, the fluctuations are enhanced with an increase in the anisotropy of interactions in the system.     

Calculation of molecular crystals within the PCILOCC framework

Stabilization energies and equilibrium distances of one-dimensional (HF)_n and (H₂O)_n chains have been calculated by means of the PCILO method for finite chain length and by the PCILOCC method for infinite chain length. Both types of calculation are compared with corresponding CNDO /2-MO and CNDO /2-CO calculations. Further we have performed an analysis of the individual contributions of the stabilization energies per monomer of the PCILO and PCILOCC calculations. The results show that the PCILOCC method is well suited for the calculation of molecular associations with translational symmetry.     

Aggregation of Poly(γ‐benzyl L‐glutamate)s Followed by Time‐Resolved Emission Anisotropy

Syntheses of poly(γ‐benzyl L ‐glutamate)s (PBLGs) labeled with various fluorophores (tryptophan, dansyl, and anthracene) having different molecular weights are reported. Association of PBLG chains was studied by time‐resolved emission anisotropy in the solvents supporting the aggregation process (1,4‐dioxane and tetrahydrofuran) and in N,N‐dimethylformamide, where the aggregates were not formed. The influence of molecular weight and polymer concentration on PBLG association was studied as well. The limiting emission anisotropy (r_∞) and rotational correlation times (ϕ) were determined. The chain relaxation dynamics were compared with the fluorescence lifetimes of the fluorophores and spectroscopically suitable labels were selected. Tryptophan was found to be an inconvenient fluorophore for the association study of PBLGs because of its short excited‐state lifetime. Dansyl and anthracene fluorophores, however, proved to be suitable labels for the chain dynamics study of PBLGs in solution. The mobilities of PBLG chains in 1,4‐dioxane were slower than those in tetrahydrofuran and N,N‐dimethylformamide because of PBLG association in this solvent.