A Monte Carlo Study of the Elastic Behavior of Polymethylene Chains

The rubber elastic behaviors of long polymethylene (PM) chains are investigated using Monte Carlo simulations and considering the rotational‐isomeric‐state model. Through the Monte Carlo method we can generate many PM chains in the equilibrium states, and obtain the average Helmholtz free energy 〈A〉 and average energy 〈U〉. Chain dimensions and thermodynamic statistical properties of long PM chains under various elongation ratios λ are also calculated. We find that the elastic force f increases with elongation ratio λ, and that energy contribution f_u to the elastic force is negative and significant. The ratio f_u /f ranges from –0.37 to –0.32 at T = 300 K and from –0.53 to –0.40 at T = 413 K, and decreases with increasing temperature, which agrees with the experimental data. Our calculations may provide some insights into the macroscopic phenomena of rubber elasticity.     

Subduction coefficients for 〈2N/2–S, 12S〉 ↓ 〈2, 1〉 ⊗ 〈2, 1〉 of U(n) ↓ U(n1) ⊗ U(n2)

The determination of the subduction coefficients for states of the unitary group U(n) under the restrictions U(n) ↓ U(n₁) ? U(n₂) have been considered for the spin free states of many electron systems. Using the transformation properties of the tensor basis spanning the irreducible representation 〈2^N/2–S, 1^2S〉 of U(n) under the permutations of electron coordinates, a simple programmable procedure has been developed for the determination of these coefficients. The procedure has been illustrated using a simple example.     

On the Escape Transition of a Tethered Gaussian Chain; Exact Results in Two Conjugate Ensembles

Summary: Upon compression between two pistons an end-tethered polymer chain undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation that is partially escaped from the gap. In the thermodynamic limit the system demonstrates a first-order phase transition. A rigorous analytical theory of this phenomenon for a Gaussian chain is presented in two ensembles: a) the H-ensemble, in which the distance H between pistons plays the role of the control parameter, and b) the conjugate f-ensemble in which the external compression force f is the independent parameter. A loop region for 〈f(H)〉 with negative compressibility exists in the H-ensemble, while in the f-ensemble 〈H(f)〉 is strictly monotonic. The average lateral forces taken as functions of H (or 〈H〉, respectively) have distinctly different behavior in the two ensembles. This result is a clear counterexample of the main principles of statistical mechanics stating that all ensembles are equivalent in the thermodynamic limit. Another theorem states that the thermodynamic potential as a function of volume must be concave everywhere. We demonstrated that the exact free energy in the H-ensemble contradicts this statement. Inapplicability of these fundamental theorems to a macromolecule undergoing the escape transition is clearly related to the fact that phase coexistence in the present system is strictly impossible. This is a direct consequence of the tethering and the absence of global translational degrees of freedom of the polymer chain.     

Partitioning technique,perturbation theory,and rational approximations

Instead of the Schródinger equation ??Ψ = EΨ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (?? ? ?)Ψ_? = aφ, where ? is a complex variable, with the same boundary condition, which gives a = 〈φ|??|Ψ_?〉 ? ? = ?₁ ? ?. Introducing the projector P = 1 ? |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψ_? = (1 ? P??/?)^?1φ = (1 + T_???)φ, where T_? = (? ? P??)^?1P = P(? ? P??P)^?1P is the reduced resolvent associated with the auxiliary Hamiltonian H? = P??P. The existence of these operators is discussed. It is shown that, if the parameter ? is real in the “discrete part” of the spectrum to ??, then ? and ?₁ = 〈φ|??|Ψ_?〉 = 〈φ|?? + ??T_???|Φ〉 ≡f(?) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(?) = ? ? f(?) = 0 is related to the variation principle. Putting ?? = ??₀ + V and expanding the inverse (? ? P??₀ ? PV)^?1 in terms of powers of V or (V ? α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h₁,h₂,…,h_n} is a linearly independent set, the inner projection Á_n = | h 〉 〈 h |A^?1| h 〉^?1〈 h | is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.     

Spin-free quantum chemistry. XXIII. The generator-state approach

In the unitary-group formulation of quantum chemistry, the spin-projected, configuration-state spaces of quantum chemistry are realized by the irreducible representation spaces (IRS ) of the freeon unitary group U_(n), where n is the number of freeon orbitals. The Pauli-allowed IRS are labeled by the partitions [λ] = [2^(N/2)?s, 1^2S], where N and S are the particle number and the spin, respectively. The generator-state approach (GSA ) to the unitary-group formulation consists of (1) the construction of the overcomplete, nonorthonormal generator basis for each IRS ; (2) the Lie-algebraic computation of matrix elements over generator states; (3) the Moshinsky–Nagel construction of the complete, orthonormal Gel'fand basis in terms of the generator basis; and (4) the computation of matrix elements over Gel'fand states in terms of matrix elements over generator states.     

A variation principle for energy differences between states of two different Hamiltonians

A modification of a variation principle due to Delves, is derived which permits the direct calculation of energy differences between states of two different Hamiltonians: [Δ ??] = 〈X₀| ??_x – W_x|X₁〉 – 〈Y₀|??_y – W_y|y₁〉 + 〈X₀| Δ ??|Y₀〉 · 〈X₀| Y₀〉^?1. Δ ?? = ??_y – ??_x, |X₀〉 and |Y₀〉 are the wave functions for the X and Y states and |X₁〉 and |Y₁〉 are functions defined in the text. The principle is applied to a few simple examples.     

Methods for the calculation of spherically averaged Compton profiles with GTOs

For the first time, an analytical and efficient algorithm for the evaluation of spherically averaged reciprocal form factors B(s)=〈B( s )〉_Ω using Gauss-type basis functions is presented. The spherically averaged Compton profile is available by Fourier transformation of the reciprocal form factor. The algorithm has been successfully implemented in connection with the quantum chemistry codes GAMESS and CRYSTAL92, which perform Hartree–Fock calculations for molecules and solids. In addition, an analytical algorithm for the direct evaluation of spherically averaged Compton profiles and the moments 〈p^ν〉 (ν≥−1) via the momentum density is proposed for Gauss-type basis functions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 213–223, 1997     

Bemerkung zur Rayleigh-Schrödinger-Störungstheorie

The time-independent Hamiltonians ? ₀ and ?=? ₀ + V have a discrete spectrum, eigenvalues, and eigenvectors E _s ^(o) , ¦s〉^(o) resp. E _s, ¦s〉. If the RS perturbation theory can be applied here then an operator \(\mathfrak{p}\) with the property $$\left| s \right\rangle ^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}\left| s \right\rangle ^{(n)} , E_s^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}E_s^{(n)} $$ exists where ¦s〉⁽ⁿ⁾ and E _s ⁽ⁿ⁾ denote the n-th order corrections of perturbation theory if E _s ^(o) is nondegenerate. In the case of degeneracy the operation \(\mathfrak{p}\) remains defined and can always be used todetermine perturbation corrections of quantum mechanical expressions which are invariant in zerothorder under transformations of the basis in degenerate subspaces of ? ₀. The equations $$\left| s \right\rangle = \sum\limits_n^{0,\infty } {\left| s \right\rangle ^{(n)} = e^\mathfrak{p} \left| s \right\rangle ^{(0)} } , E_s = \sum\limits_n^{0,\infty } {E_s^{(n)} } = e^\mathfrak{p} E_s^{(0)} $$ correspond to a basis transformation where nondegenerate eigenvectors ¦s∝> ^(o) and eigenvalues E _s ^(o) of ? ₀ transform into eigenvectors ¦s∝> and eigenvalues E _s of ?. Examples show the usefulness of this formulation.     

A new algorithm to evaluate bielectronic integrals with 1s Slater‐type orbitals obtained by the integral transforms

This article presents a variation of the integral transform method to evaluate multicenter bielectronic integrals (12|34), with 1s Slater‐type orbitals. It is proved that it is possible to define, out of the expression of (12|34) given by the integral transform method, a function F(q) that has the property of having a unique Q, such that F(Q) = (12|34). Therefore, F(q) may be used to calculate (12|34). It is shown that the evaluation of F(Q) turns out to be simpler than the three‐dimensional integral involved in the calculation of (12|34), and an algorithm is presented to calculate Q. The results show that relative errors on the order of 10^?3 or lower are obtained very efficiently. In addition, it is shown that the proposed algorithm is very stable. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Determination of Erbium by Two-Color Three-Photon Resonance Ionization Mass Spectrometry

Excited states and autoionization states of the erbium atom were investigated by the use of multicolor resonance ionization mass spectrometry. Among the observed first excited states, a level [4f¹²(³H₆)6s6p(³P₁)] located 17,348 cm⁻¹from the ground state is regarded as the most efficient state for excitation within the wavelength range investigated (560–600 nm), while a level located 17,080 cm⁻¹from this first excited state (E= 34,458 cm⁻¹) is identified as the best second excited state for the optimal photoionization scheme. Many ionization schemes adopting an autoionization state are also investigated, and the most efficient scheme is identified as 4f¹²6s²(³H₆) → 4f¹²(³H₆)6s6p(³P₁⁰), 17,348 cm⁻¹→ 34,458 cm⁻¹→ continuum state, which corresponds to the two-color (ω₁+ ω₂+ ω_1,2) scheme. Various concentrations of standard solutions for erbium are determined and the minimum amount detectable by two-color three-photon ionization was determined to be 20 pg.     

Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

A second-quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one-dimensional harmonic oscillator matrix elements. The most general case of 〈m|f(â, â⁺)|n〉 is considered, and the recurrence relations forf(â, â) = X^k, exp(?βX), and exp(?X²) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum-rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two-center integrals.     

A relativistic DFT study of magnetic exchange coupling in ketimide bimetallic uranium(IV) complexes

Magnetic exchange couplings in bis(ketimide) binuclear U^IV/U^IV complexes [Cp′₂UCl]₂(μ-ketimide) diuranium(IV) and [(C₅H₅)₂(Cl)An]₂(μ-ketimide) (Cp′ = C₅Me₄Et; ketimide = N=CMe-(C₆H₄)-MeC=N) have been investigated computationally using relativistic density functional theory (DFT) combined with the broken symmetry (BS) approach. Using the B3LYP hybrid functional, the BS ground state of these U^IV/U^IV 5f ²–5f ² complexes has been found of lower energy than the high spin (HS) quintet state, indicating an antiferromagnetic character (estimated coupling constant |J| < 5 cm⁻¹) which has not yet been evidenced unambiguously experimentally. On the contrary, the BP86 GGA functional overestimates greatly the antiferromagnetic character of the complexes (|J| > 100 cm⁻¹). As recently reported for para-bis(imido) [(C₅H₅)₃U]₂(μ-imido) uranium(V) complex, spin polarization is mainly responsible for the antiferromagnetic coupling through the π-network orbital pathway within the bis(ketimide) bridge. Furthermore, spin polarization is exalted by the combined roles of the 5f metal orbitals and of the π-conjugated ketimide bridging ligand which permit electronic communication between the two uranium atoms albeit separated by a distance of the order of 10 ?. The MO analysis clarifies which MOs contribute to the antiferromagnetic coupling in the binuclear complexes under consideration and brings to light the 5f orbitals driving contribution.     

Linewidth dependence of quasi-stationary states in large molecules

We study the quasi-stationary states, or eigenstates, of the decay matrix for a system described by one single vibronic state |s〉 coupled to a manifold {l}, both |s〉 and {l} levels being coupled to separate continua which endow them with finite linewidths γ_s, γ_l. When all the zero-order states have the same linewidth (not necessarily zero) the quasi-stationary states are identical to the molecular eigenstates; when the difference between the linewidths γ_s ? γ_l is large enough, the zero-order sets (s, {l}) are completely decoupled, a result which has been established for the two-level system. We indicate some consequences of collisional broadening involving pressure effects in the gas phase.     

Elastic behavior of uniform star polymer chains

Elastic behaviors of uniform star polymer chains with two to seven branches (namely, functionality f = 2-7) are investigated using Monte Carlo simulation and the bond fluctuation model. Here chain dimensions and thermodynamic properties of uniform star polymer chains during the process of tensile elongation are studied, and comparisons with linear chain are also made. Static properties of chains such as chain sizes and asphericities of chains are calculated, and g-value of g = 〈S²〉_star/〈S²〉_linear decreases with elongation ratio increasing for different functionality f. Thermodynamic properties such as average energy 〈U〉, free energy per bond 〈A′〉 and elastic force F are also investigated during the process of tensile elongation. In the meantime, scatting functions P(q) are calculated for star polymer chains with different functionality f. Additionally, we also discuss the influence of elongation ratio on scattering form factor. The impenetrability of the star cores is known to cause a discontinuity in the osmotic pressure showed through a peak in the scattering functions, and some different behaviors in the tensile process for uniform star chain are obtained.     

A Route to Stabilize Uranium(II) and Uranium(I) Synthons in Multimetallic Complexes

Herein, we report the redox reactivity of a multimetallic uranium complex supported by triphenylsiloxide (−OSiPh₃) ligands, where we show that low valent synthons can be stabilized via an unprecedented mechanism involving intramolecular ligand migration. The two- and three-electron reduction of the oxo-bridged diuranium(IV) complex [{(Ph₃SiO)₃(DME)U}₂(μ-O)], 4 , yields the formal “U^II/U^IV”, 5 , and “U^I/U^IV”, 6 , complexes via ligand migration and formation of uranium-arene δ-bond interactions. Remarkably, complex 5 effects the two-electron reductive coupling of pyridine affording complex 7 , which demonstrates that the electron-transfer is accompanied by ligand migration, restoring the original ligand arrangement found in 4 . This work provides a new method for controlling the redox reactivity in molecular complexes of unstable, low-valent metal centers, and can lead to the further development of f-elements redox reactivity.     

Scalar nonlinearities in physics and chemistry

Scalar nonlinearities are operators of the form N(u) = 〈Au, u〉Bu, where A, B are linear operators, and 〈·,·〉 is the inner product in a Hilbert space ??. This paper reviews applications of scalar nonlinearities. We show that operators of the form N(u) are found in equations that describe phenomena of classical mechanics, open systems of quantum mechanics, and chemical physics.     

Expansion and completeness theorems for operator manifolds

Some expansion and completeness theorems for operator manifolds, which are currently being employed in propagator theory, are derived. It is shown that excitation or ionization operators satisfying the conditions Q|0〉 = |Λ〉 and Q_Λ|0〉 = 0 for general excited states |Λ〉 and reference state |0〉 may be expanded uniquely in particular sets of basis operators. These results are then used to discuss rigorous expressions for fermion propagators.     

On the evaluation of the integral 〈pu|∂/∂ν|s〉 occurring in covalency reduction factor

Explicit expressions for integrals of the form 〈p_u|?/?ν|s〉 occurring in expression of covalence reduction factors, introduced by Stevens and Tinkham, have been derived for various types of atomic functions. Numerical values of the integral for some common ligands are given. Misconceptions regarding the sign and the magnitude of this integral existing in the literature have been clarified.     

Dilute solution properties of a polybenzimidazole

Light scattering and viscometric studies have been carried out on dilute solutions of a polybenzimidazole in N,N-dimethylacetamide. The data, which span the molecular weight range 2.9 ≦ 10^?4M_w ≦ 23.3, and the temperature range 290 ≦ T/K ≦343, yield the dependence of the mean-square radius of gyration 〈s²〉_LS, the second virial coefficient A₂, and the intrinsic viscosity [η] on molecular weight M_w and temperature. The unperturbed mean-square radius 〈s〉_LS was calculated using experimental values of 〈s²〉_LS and A₂. It was found that excluded volume effects on 〈s²〉_LS are very small. The unperturbed hydrodynamic chain dimension 〈s〉_η was estimated by considering draining effects. A small value of the draining parameter was obtained. Analysis of the temperature dependence of A₂ and [eta;] leads to the conclusion that this system approaches a lower theta temperature with increasing temperature. The steric factor σ = 〈s〉/〈s〉f, based on the value of 〈s〉_f calculated for the polymer chain with free rotation, is nearly unity. Most of these properties can be interpreted in terms of long rotational units within the main chain.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).