A quantum mechanical Hamiltonian for unhindered macromolecular chains: consistency with Gaussian models

A model for an open unhindered three-dimensional macromolecular chain, based upon quantum mechanics and proposed in previous works, is studied in order to investigate its physical properties and consistency. The chain is formed by N particles interacting through harmonic-like vibrational potentials in the high-frequency limit (in which all successive bond lengths become fixed). This formulation leads to a specific Hamiltonian for the chain: it constitutes an improvement in comparison with standard Gaussian models, which do not. The classical partition function Z_c resulting from the quantum formulation is represented through an integral, which exhibits explicitly rotational invariance in the integrand and provides the basis for further approximations for large N. Approximate formulae are obtained for correlations between pairs of bond vectors, the distribution function for the end-to-end vector, distribution functions for individual bond vectors, “rubber eleasticity” (when stretching forces act) and the structure factor for small wave vector. In all cases, the results which have arisen from the quantum mechanical formulation coincide with those obtained for the standard Gaussian chain. This agreement appears to confirm the physical consistency of the quantum Hamiltonian characterizing the model.     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

A steered molecular dynamics simulation on the elastic behavior of adsorbed star polymer chains

<正>Elastic behavior of 4-branched star polymer chain with different chain length N adsorbed on attractive surface is investigated using steered molecular dynamics(SMD) simulation method based on the united-atom(UA) model for branched alkanes.The simulation is realized by pulling up the chain via a linear spring with a constant velocity v = 0.005 nm/ps.At the beginning,the chain lies extensionally on adsorbed surface and suffers continuous deformations during the tensile process.Statistical parameters as mean-square radii of gyration S~2_(xy),S~2_z,shape factor δ,describing the conformational changes,sectional density den which gives the states of the chain,and average surface attractive energy U_a,average total energy U,average force f probed by the spring,which characterize the thermodynamic properties, are calculated in the stimulant process.Remarkably,distinguishing from the case in linear chains that there only exists one long plateau in the curve of f,the force plateau in our study for star chains is multiple,denoting different steps of desorption,and this agrees well with the experimental results in essence.We find during the tensile process,there are three characteristic distances Z_c,Z_t and Z_0 from the attractive surface,and these values vary with N.When Z=Z_c,the chain is stripped from the surface,but due to the form of wall-monomer interaction,the surface retains weak influence on the chain till Z = Z_c.From Z=Z_t,parameters U_a,U and f respectively reach a stable value,while the shape and the size of the chain still need adjustments after Z_t till Z_0 to reach their equilibrium states.Specifically,for short chain of N= 41,Z_t and Z_0 are incorporated.These results may help us to deepen the knowledge about the elastic behavior of adsorbed star polymer chains.     

New Method to Calculate Thermodynamic and Transport Properties of a Multi-Temperature Plasma: Application to N2 Plasma

Multi-temperature thermal plasmas have often to be considered to account for the nonequilibrium effects. Recently André et al. have developed the calculation of concentrations in a multi-temperature plasma by artificially separating the partition functions into a product by assuming that the excitation energies are those of the lower levels (electronic, vibration, and rotation). However, at equilibrium, differences, increasing with temperature, can be observed between partition functions calculated rigorously and with their method. This paper presents a modified method where it has been assumed that the preponderant rotational energy is that of the vibrational level v=0 of the ground electronic state and the preponderant vibrational energy is that of the ground electronic state. The internal partition function can then be expressed as a product of series expressions. At equilibrium for N ₂ and N ₂ ⁺ partition functions the values calculated with our method differ by less than 0.1% from those calculated rigorously. The calculation has been limited to three temperatures: heavy species T_h , electrons T_e , and vibrational T _v temperatures. The plasma composition has been calculated by minimizing the Gibbs free enthalpy with the steepest descent numerical technique. The nonequilibrium properties have been calculated using the method of Devoto, modified by Bonnefoi and Aubreton. The ratio =T_e/T_h was varied between 1 and 2 as well as the ratio _v =T _v /T _h for a nitrogen plasma. At equilibrium the corresponding equilibrium transport properties of Ar and N ₂ are in good agreement with those of Devoto and Murphy except for T>10,000 K where we used a different interaction potential for N–N ⁺ . The effects of _v and _e on thermodynamic and transport properties of N ₂ are then discussed.     

Computer simulation of brownian motion of a polyelectrolyte molecule (charged bead-spring model)

A Langevin equation of motion for a charged bead-spring statistical chain is written in difference form and the relaxation and equilibrium behavior of the chain is studied by computer simulation. Results are presented for the behavior of end-to-end length h, principal axes of the polymer ellipsoid L₁, L₂, L₃, and chain contour length c in terms of their averages, root mean square values, root mean square fluctuations, orientations, and relaxation strengths and times. The simulation was made with various sets of parameters, bead number N, charge on the bead q, and radius of ion atmosphere around the bead k^?1. It is found that 〈h²〉^1/2 and 〈L₁²〉^1/2 increase more strongly with increasing q and decreasing κ than 〈L₂²〉^1/2, 〈L₃²〉^1/2, and 〈c₁²〉^1/2, indicating that the chain is expanded in three dimensions and at the same time is extended along the end-to-end direction. The relaxation time τ_rot of rotation of the end-to-end vector, which is proportional to N² at q = 0, increases with increasing q and tends to be proportional to N³ for an extended chain, while the relaxation time τ_conf of the magnitude of h is almost independent of q and is always proportional to N². It is concluded that the extended chain possesses a well-defined end-to-end axis and the chain rotates as a whole with a relaxation time τ_rot which is much longer than τ_conf. The complex viscosity of the chain is calculated from the Fourier transform of the time–correlation function of momentum flux and is found to have a frequency spectrum similar to that observed for aqueous solutions of poly(acrylic acid). The dominant mode appearing in the low-frequency range is evidenced to arise from the rotation of the extended chain.     

Molecular dynamics simulation of local motion of polystyrene chain end—comparison with the fluorescence depolarization study

Molecular dynamics (MD) simulation of the local motion of a polystyrene (PS) chain with anthryl group at the chain end surrounded by benzene molecules was performed and the results were compared with those obtained experimentally by the fluorescence depolarization method. The molecular weight dependence of the relaxation time of the probe obtained by the MD simulation was qualitatively in agreement with the results obtained by the fluorescence depolarization method. We also estimated the molecular weight dependence of the relaxation time for the end-to-end vector. Below the degree of polymerization (DP)≤3, the mean relaxation time T_m for the end-to-end vector was similar to that for the vector corresponding to the transition moment of the probe. With the increase of DP, the T_m for the probe tended to reach an asymptotic value unlike that for the end-to-end vector, which monotonically increased with DP. This indicates that the entire motion of a polymer coil contributes to the local motion to a lesser extent as the molecular weight increases. The MD simulations using artificial restraints showed that the rotational relaxation of the probe at the chain end for a dynamically stiff PS chain is realized by the cooperative rotation of the main chain bonds. The internal modes which takes place below 5 monomer units mainly led to the rotational relaxation of the probe at the PS chain end. Finally, the change of T_m with the position along the PS main chain was examined.     

A one-dimensional azido-bridged manganese(III) complex with bidentate Schiff base: Crystal structure and magnetic properties

The synthesis, structural characterization, and magnetic behavior of a novel one-dimensional azido-bridged manganese(III) complex of formula [Mn(L)₂N₃] (1) is reported, where HL is the bidentate Schiff base obtained from the condensation of salicylaldehyde with 4-methoxy aniline. Complex 1 crystallizes in the monoclinic system, space group P2₁/n, with a=11.743(4) Å, b=24.986(9) Å, c=13.081(5) Å, β=95.387(7)° and Z=2. The complex is of one-dimensional chain structure with single end-to-end azido bridges and the manganese(III) ion has an elongated octahedral geometry. Magnetic studies show that the weak antiferromagnetic interaction is mediated by the single end-to-end azido bridge with the exchange parameter J=−5.84 cm⁻¹.     

CONFIGURATION-DEPENDENT PROPERTIES OF POLY(DIMETHYLSILOXANE) CHAIN*

The intermolecular rotational potential energies for poly(dimethylsiloxane) (PDMS) chains aredirectly obtained from a priori probability P_(αβ). Here the differing statistical weight matrices for the Si-Oand O-Si bonds are considered in calculating the configuration partition function. In the Bahar's model, asthe same statistical weight matrices for the Si -O and O- Si bonds are adopted, there exists a large deviationof αpriori probability P_(αβ) between the theory and the molecular dynamics (MD) simulation. Our model givessatisfactory agreement with experiment on the mean-square unperturbed end-to-end distance, the mean-square dipole moment and its temperature dependence, and the molar cyclization equilibrium constants fordimethylsiloxane oligomers. This new rotational isomeric state approach can be widely applied to otherchains, such as -CH_2-C[(CH_2)_mH]_2- and -O-Si[(CH_2)_mH]_2 for arbitrary m.     

Studies of solution properties of copolymers. II. Copolymer of maleic anhydride and styrene

Styrene and maleic anhydride were copolymerized in benzene. The whole polymer thus obtained was fractionated with acetone and petroleum ether as the solvent and precipitant, respectively. The viscosities and osmotic pressures of the fractions were determined in tetrahydrofuran. The relation between the intrinsic viscosity and the molecular weight, [η] = 5.07 × 10^?5 M?_n^0.81, was obtained in tetrahydrofuran. The unperturbed mean square end-to-end distance was estimated by the Stockmayer-Fixman equation. A theoretical equation for the mean square end-to-end distance for a chain of repeating units of different bond lengths a and b with a fixed valence angle θ and without restriction of internal rotation was presented and applied to this copolymer. In addition, the equation of the mean-square end-to-end distance derived by Wall for trans-polyisoprene without rotational restriction was modified for application to this copolymer. The result evaluated with our equation was about 26% smaller than that from the modified Wall equation. A steric parameter for the present copolymer is defined and discussed in comparison with those of polystyrenesulfone and polystyrene.     

Statistical representation of atomic systems structure. I. Total binding energy as a function of the relative information content

A nonorbital representation of the many-electron atomic systems is proposed. It is obtained by considering a certain equivalence class of mappings ?: ? → Π from the set ? of N electrons into the set Π of Z protons. Total binding energy of the systems (Z = 3,4,…, 18; Z ? N = k = 0, 1,…, 8) arranged according to the Periodic Table criterion, turns out to be the linear function of ZI_r, where I_r is an information functional related to our representation.     

Rotation-Inversion Isomerization of Tertiary Carbamates: Potential Energy Surface Analysis of Multi-Paths Isomerization Using Boltzmann Statistics

Potential energy surface (PES) analyses at the SMD[MP2/6–311++G(d,p)] level and higher-level energies up to MP4(fc,SDTQ) are reported for the fluorinated tertiary carbamate N-ethyl-N-(2,2,2-trifluoroethyl) methyl carbamate ( VII ) and its parent system N,N-dimethyl methyl carbamate ( VI ). Emphasis is placed on the analysis of the rotational barrier about the CN carbamate bond and its interplay with the hybridization of the N-lone pair (NLP). All rotational transition state (TS) structures were found by computation of 1D relaxed rotational profiles but only 2D PES scans revealed the rotation-inversion paths in a compelling fashion. We found four unique chiral minima of VII , one pair each of E- and Z-rotamers, and we determined the eight unique rotational TS structures associated with every possible E/Z-isomerization path. It is a significant finding that all TS structures feature N-pyramidalization whereas the minima essentially contain sp²-hybridized nitrogen. We will show that the TS stabilities are affected by the synergetic interplay between NLP/CO₂ repulsion minimization, NLP→σ^*(CO) negative hyperconjugation, and two modes of intramolecular through-space electrostatic stabilization. We demonstrate how Boltzmann statistics must be applied to determine the predicted experimental rotational barrier based on the energetics of all eight rotamerization pathways. The computed barrier for VII is in complete agreement with the experimentally measured barrier of the very similar fluorinated carbamate N-Boc-N-(2,2,2-trifluoroethyl)-4-aminobutan-1-ol II . NMR properties of VII were calculated with a variety of density functional/basis set combinations and Boltzmann averaging over the E- and Z-rotamers at our best theoretical level results in good agreement with experimental chemical shifts δ(¹³C) and J(¹³C,¹⁹F) coupling constants of II (within 6 %).     

Large-Z and -N dependence of atomic energies from renormalization of the large-dimension limit

By combining Hartree–Fock results for nonrelativistic ground-state energies of N-electron atoms with analytic expressions for the large-dimension limit, we have obtained a simple renormalization procedure. For neutral atoms, this yields energies typically threefold more accurate than the Hartree–Fock approximation. Here, we examine the dependence on Z and N of the renormalized energies E(N, Z) for atoms and cations over the range Z, N = 2 → 290. We find that this gives for large Z = N an expansion of the same form as the Thomas–Fermi statistical model, E → Z^7/2(C₀ + C₁Z^?1/3 + C₂Z^?2/3 + C₃Z^?3/3 + ?), with similar values of the coefficients for the three leading terms. Use of the renormalized large-D limit enables us to derive three further terms. This provides an analogous expansion for the correlation energy of the form δE δZ^4/3(δC₃ + δC₅Z^?2/3 + δC₆Z^?3/3 + ?); comparison with accurate values of δE available for the range Z ? 36 indicates the mean error is only about 10%. Oscillatory terms in E and δE are also evaluated. © 1994 John Wiley & Sons, Inc.     

Symmetry assignment in the distributed Gaussian functions method to study homonuclear rotating trimers

An approximate method based on the use of distributed Gaussian functions (DGF) to describe the interparticle distances is employed to study the rovibrational spectrum of trimers. Rotational energy levels are obtained by assuming that vibration and rotation are separated. Thus, eigenstates of the Hamiltonian for the zero total angular momentum, J = 0, are used as basis set to solve the rotational Hamiltonian. A procedure to identify the corresponding symmetry character for the rovibrational bound states is proposed. The DGF approach is applied to the case of the rotating Ar₃ trimer. The reliability of the method is tested by comparison with results from an exact hyperspherical coordinate calculation for J = 0, 1 and 6.     

1,3‐Ninhydrin dihydrazone featuring an R44(12) motif with symmetry and comprising only N and H atoms

In the title compound [systematic name: (1Z,3Z)‐1,3‐dihydrazinylidene‐1H‐inden‐2(3H)‐one], C₉H₈N₄O, isolated molecules possess approximate noncrystallographic C_2v symmetry and their cis conformation and planarity are assisted by a pair of short intramolecular N—H...O hydrogen bonds. Each molecule is asymmetrically involved in an extensive three‐dimensional network of N—H...O and N—H...N hydrogen bonds, and the structure also exhibits weaker π–π and C=O...C interactions. The structure features an R₄⁴(12) motif consisting solely of N and H atoms and possessing crystallographic symmetry.     

Hindered internal rotation in molecular systems: quantum statistics of equilibrium and rate constants in the Wigner function formalism

Summary The quantum statistics of a symmetric hindered internal rotator in a molecule or molecular complex is developed within the Wigner function formalism. Different shapes of the rotational barrier are considered. The partition function and the thermodynamic functions are given as Wigner-Kirkwood series expansions in terms of powers of Planck's constant squared. One gets simple closed expressions containing the modified Bessel functionsJ ₀ andJ ₁ of the argumentiV ₀/2kT whereV ₀ is the barrier height. Some problems concerning the evaluation of equilibrium and rate constants of chemical reactions are discussed.Supported by the Alexander von Humboldt-Stiftung     

Approximate sizes of monoatomic negative ions with fractional nuclear charge using electrostatic potentials

Summary Using the criterion that for negative monoatomic ions withN electrons, the location of minimum in the electrostatic potentialV(r) gives an approximate estimate of ionic radii,r _m. Calculations of the latter are reported for quark atoms with fractional nuclear chargesZ=N– andN– , respectively. Quark atoms withN=1–10, 18 and 36 have also been considered.     

Computer simulation of the system N2 in fluid argon-correlation functions and relaxation times

The correlation functions of the dipole moment, P₂[u(0) · u(t)], angular and linear velocity, and bond forces have been calculated from computer simulated data for four different density-temperature states of N₂ in fluid argon. From these functions infrared and Raman line shapes, NMR relaxation times, and rotational and classical vibrational relaxation times have been computed.     

Conformational statistics of polymer chain terminally attached to wall (III) —NRW model loop chain

When the two end groups of a linear polymer chain are absorbed on a solid surface, the polymer chain forms the “loop” conformation. Investigation has been made on the conformational statistics of a model loop chain by the normal random walk (NRW) on a lattice confined in the half-infinite space. Based on the conformational distribution function of the NRW model tail chain, it is easy to deduce an analytical formula expressing the conformational number of the model loop chain. It was found that the ratio of the conformational number of the model loop chain to that of the free chain varies with the power functionN ^-2/3 when the chain lengthN→οο. The same result -was obtained by means of the recursion equation. The ratio of the mean square end-to-end distanceh ² for the model loop chain to its mean square bond lengthl ² is 2N/3. Compared with the free chain with the same lengthN, the mean square end-to-end distance of the model loop chain contracts to a certain extent. The basic relationships deduced were supported by the exact enumeration and Monte Carlo simulations. Project supported by the National Natural Science Foundation of China.     

N.m.r. studies of the keto–enol tautomerism of acylthioacetamides

Acetylthioacetamides exist as different keto and enol isomers in chloroform solutions. The keto form with intramolecular hydrogen bonding between the NH and the carbonyl group is the dominant keto isomer. On the other hand the enol forms with intramolecular hydrogen bonding between the OH and the thioketo group are the dominant enol isomers in the temperature range 60°C to ?60°C. The thermodynamic data of the keto-enol equilibria were obtained by measuring the intensities of appropriate high resolution proton signals as a function of temperature. At low temperatures all lines characteristic of the enol forms are doubled in the N-phenyl-substituted derivatives because the rotation of the NH? C₆H₅ group around the C? N bond becomes slow and the chemical shifts characteristic of the E and Z isomers are different. We estimated approximate thermodynamic data of the E/Z equilibrium in some of the compounds. The changes of the line shape as well as the chemical shifts as a function of temperature indicate the presence of various additional exchange processes. In order to obtain further information we performed curve fittings of the chemical shifts of one acetylthioacetanilide and of a series of monothio-β-diketones (studied in another paper) assuming a fast two site exchange process. On the basis of the results obtained a reaction scheme for N-substituted acylthioacetanilides in solution is proposed.     

Application of Lie algebraic method to the calculation of rotational spectra for linear triatomic molecules

The Hamiltonian describing rotational spectra of linear triatomic molecules has been derived by using the dynamical Lie algebra of symmetry group U1(4) U2(4). After rovibrational interactions being considered, the eigenvalue expression of the Hamiltonian has the form of term value equation commonly used in spectrum analysis. The molecular rotational constants can be obtained by using the expression and fitting it to the observed lines. As an example, the rotational levels of v2 band for transition (02°0-0110) of molecules N2O and HCN have been fitted and the fitting root-mean-square errors (RMS) are 0.00001 and 0.0014 cm-1, respectively.