E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.     

Primary isotopic effect on the magnetic shielding of tin nuclei (117Sn and 119Sn)

The Kirkwood-Buff statistical mechanical theory of surface tension γ for monatomic fluids is extended to molecular fluids. A rigorous expression for γ is derived in terms of the angular pair distribution function f(z ₁ R ₁₂θ₁θ₂) of an equilibrium fluid-fluid system (liquid-gas, liquid-liquid, or gas-gas). The Fowler approximation is applied for the liquid-gas case, and a simple expression for γ is derived in terms of the bulk liquid angular pair correlation function g(R ₁₂θ₁θ₂). Thermodynamic perturbation theory for g(R ₁₂θ₁θ₂) is also used to calculate γ theoretically. The theoretical results are compared with experimental values.     

Computation and analysis of the dynamic structure factor S(k, ω) for small wave vectors

Molecular dynamics (MD) calculations have been performed for the Lennard-Jones (12-6) potential function using 2048 particles. Using conventional parameters the results may be compared with those for liquid argon.

The dynamic structure factor S(k, ω) has been determined both by Fourier inversion of the intermediate scattering function F(k, t) and from the longitudinal current-current correlation function C _∥(k, t). Particular attention was paid to the recurrence time of the system. The results for S(k, ω) by the two methods agree within 5 per cent for the whole region of small k-vectors considered. Double Fourier inversion of the van Hove function G(r, t) led to insufficiently accurate results for these small k-values. In view of the present data, the MD-results of Levesque et al. [1] for S(k, ω) have only a qualitative character. These latter data appear to contain truncation errors due to incomplete Fourier transformations.

Using a hydrodynamic assumption for F(k, t) we were able to extract the transport coefficients, the velocity of sound and the ratio of the specific heats in the limit of large wave lengths or small k. The velocity of sound was obtained by exploiting the MD generated anomalous dispersion curve of sound waves. Anomalous dispersion was found to set in for kσ ～ 0·25. A sound speed of 880 ms^-1 has been determined which is in excellent agreement with experimental values for liquid argon. The total error for the MD value amounts to about 5 per cent. In contrast, the ratio of the specific heats γ and the transport coefficients D _T and Γ (thermal diffusivity and sound attenuation) were determinable only with an accuracy of 15 per cent due to the need for larger extrapolations. Nevertheless, we found D _T, Γ and γ in agreement with experimental values within 5-10 per cent.     

Binary hard-sphere solute-solvent radial distribution function in the colloidal limit: exact calculation from an equation of state

An exact formula is derived relating the contact value of the solute-solvent radial distribution function for an additive binary hard-sphere (HS) mixture at infinite dilution, g ^∞ ₁₂(d ₁₂), to the mixture equation of state (EOS) (1 denotes the solvent and 2 denotes the solute). This result can also be considered to be a consistency condition involving approximations for g ^∞ ₁₂(d ₁₂) and for the mixture EOS. Employing three approximate HS mixture equations of state from the literature, we use our formula to derive corresponding analytical approximations for g ^∞ ₁₂(d ₁₂) In addition, new computer simulations were performed to obtain accurate results for g ^∞ ₁₂(d ₁₂) and for g ^∞ ₁₂(r ₁₂) at the solute-solvent diameter ratios {1, 3, 5, 7, 10, 20} and the reduced solvent density ρ? = 0.8. We compare our results for g ^∞ ₁₂(d ₁₂) with the simulation results and with the results of approximate analytical expressions for g ^∞ ₁₂(d ₁₂) proposed by several workers. The results obtained from our formula in conjunction with two of the EOS expressions considered are more accurate than all previously proposed approximations, with the exception of the approximation of Matyushov and Ladanyi [1997, J. chem. Phys., 107, 5815], which is of comparable accuracy.     

Heterodyne coincidence spectroscopy

A new type of light-scattering experiment, which should measure directly the triple static structure factor S ⁽³⁾ (k, q) of a fluid, is proposed. S ⁽³⁾(k, q) is the full spatial Fourier transform of the equilibrium triplet distribution function g ⁽³⁾(r ₁, r ₂, r ₃). The experiment may also be used to study dynamic correlation functions of the form <a_k (t)a_q (t′)a_k_q(t″)> (where a_k () is the kth spatial Fourier component of the density), thereby giving new information on mode-mode coupling. The method obtains its information from triple correlations in the arrival of scattered photons at three detectors. The detectors must be operated in the heterodyne mode (i.e. with a local oscillator); the scattering volume must be much larger than the volume over which molecular positions are correlated. Comparison is made with previous analyses of other multi-detector experiments.     

An approximate expression for the Wigner-Kirkwood ℏ4 quantum correction to the pair distribution function in a one-component plasma

We propose an accurate approximate expression for the exact ℏ⁴ quantum correction to the pair distribution function g₂^q(r₁₂) that we have derived recently in an OCP using Wigner-Kirkwood ℏ² expansion. Our expression, depending only the classical pair distribution function g₂^c(r₁₂), reproduces the behavior of Wigner-Kirkwood g₂^q(r₁₂) at order ℏ⁴, at small, intermediate and large r₁₂.     

Phonons,Diffusons, and the Boson Peak in Two-Dimensional Lattices with Random Bonds

Within the model of stable random matrices possessing translational invariance, a two-dimensional (on a square lattice) disordered oscillatory system with random strongly fluctuating bonds is considered. By a numerical analysis of the dynamic structure factor S(q, ω), it is shown that vibrations with frequencies below the Ioffe-Regel frequency ω_IR are ordinary phonons with a linear dispersion law ω(q) ∝ q and a reciprocal lifetime б ~ q³. Vibrations with frequencies above ω_IR, although being delocalized, cannot be described by plane waves with a definite dispersion law ω(q). They are characterized by a diffusion structure factor with a reciprocal lifetime б ~ q², which is typical of a diffusion process. In the literature, they are often referred to as diffusons. It is shown that, as in the three-dimensional model, the boson peak at the frequency ωb in the reduced density of vibrational states g(ω)/ω is on the order of the frequency ω_IR. It is located in the transition region between phonons and diffusons and is proportional to the Young’s modulus of the lattice, ω _b ≃E.

     

Motion in the complex plane of the poles of the quantum-mechanical partial amplitude for a bond

Details are given of the analytic features of this amplitudef _g(l, k) in the g plane for a very large class of potentials that satisfy conditions (1). It is shown that there is a region (l, k²) around the point g = 0 that is free from singularities inf _g(l, k), so the Mittag-Leffler method can be applied to findf _g(l, k) and hence also the total amplitude T_g(k, t) for any g to any specified degree of accuracy with reference to the information contained in the coefficients of a finite number of terms of the series given by perturbation methods forf _g(l, k).This paper was read in substance on 8th April 1963 at Uzhgood University [1] and on 29 June 1963 at the All-Union Symposium on the Theory of Elementary Particles at Novosibirsk.     

Perturbation theory for the radial distribution function

Perturbation theory is used to consider expansions for the radial distribution function, g ₂(r), of a fluid with a soft core. We consider the Lennard-Jones (12, 6) potential and divide it into repulsive and attractive regions. In the repulsive region we expand the function exp (β u(r))g ₂(r) about a hard sphere value. For the first-order contribution of the attractive region we consider a simple approximation to the exact analytical expression. The resulting g ₂(r) is accurate at densities below about ρσ ³=0·5.     

Fourier Transform Infrared Emission Spectroscopy of thebΠ–aΣSystem of BN

The emission spectrum of BN has been investigated in the 1800–9000 cm⁻¹region using a Fourier transform spectrometer. BN was formed in a microwave discharge of He with a trace of BCl₃and N₂. The bands observed in the 3000–7800 cm⁻¹interval have been assigned as theb¹Π–a¹Σ⁺transition, with the 0–0 band at 3513.99040(43) cm⁻¹. This transition is analogous to theA¹Π_u–X¹Σ⁺_g(Phillips) system of the isoelectronic C₂molecule. The rotational analysis of the 0–0, 1–1, 1–0, 2–1, 3–2, 2–0, 3–1, 4–2, and 4–1 bands has been obtained and the molecular constants for theb¹Π anda¹Σ⁺states have been determined. A local perturbation has been observed in thev= 1 vibrational level of theb¹Π state nearJ= 18 caused by the interaction with thev= 3 vibrational level of thea¹Σ⁺state. The principal equilibrium constants for thea¹Σ⁺state are: ω_e= 1705.4032(11) cm⁻¹, ω_ex_e= 10.55338(52) cm⁻¹,B_e= 1.683771(10), α_e= 0.013857(16) cm⁻¹, andr_e= 1.2745081(37) Å. Although theb¹Π–a¹Σ⁺transition has recently been seen in emission from boron nitride trapped in solid neon matrices [J. Chem. Phys.104,3143–3146 (1996)], our work represents the first observation of this transition of BN in the gas phase.     

The 2‐matrix of the spin‐polarized electron gas: contraction sum rules and spectral resolutions

The spin‐polarized homogeneous electron gas with densities ρ_↑ and ρ_↓ for electrons with spin ‘up’ (↑) and spin ‘down’ (↓), respectively, is systematically analyzed with respect to its lowest‐order reduced densities and density matrices and their mutual relations. The three 2‐body reduced density matrices γ_↑↑, γ_↓↓, γ_a are 4‐point functions for electron pairs with spins ↑↑, ↓↓, and antiparallel, respectively. From them, three functions G_↑↑(x,y), G_↓↓(x,y), G_a(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g_↑↑(r) = G_↑uarr;(0,r), g_↓↓(r) = G_↓↓(0,r), g_a(r) = G_a(0,r) with r = | r ₁ ‐ r ₂|, but also the 1‐body reduced density matrices γ_↑ and γ_↓ being 2‐point functions according to γ_s = ρ_sf_s and f_s(r) = G_ss(r, ∞) with s = ↑,↓ and r = | r ₁ ‐ r ^′₁|. The contraction properties of the 2‐body reduced density matrices lead to three sum rules to be obeyed by the three key functions G_ss, G_a. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n_↑(k) and n_↓(k), following from f_↑(r) and f_↓(r) by Fourier transform, are correctly normalized through f_s(0) = 1. In addition to the non‐negativity conditions n_s(k),g_ss(r),g_a(r) ≥ 0 [these quantities are probabilities], it holds n_s(k) ≤ 1 and g_ss(0) = 0 due to the Pauli principle and g_a(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin‐unpolarized homogeneous electron gas in terms of 2‐body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin‐polarized case and (ii) to the 2‐body reduced density matrix giving thus its spectral resolutions.     

Investigations on the g factors of the ground and excited states for Cu2+ ions in ZnO crystal

The g factors g _// and g _⊥ of the ground Γ₆(² T ₂) and excited Γ_4,5(² E), Γ₆(² E) states for trigonal Cu²⁺ centres in ZnO crystals are calculated from three theoretical methods, the complete diagonalization (of the energy matrix) method, the second-order perturbation method (PTM-I) and the simplified second-order perturbation method (PTM-II, this method was described in an earlier paper). These methods are based on the cluster approach in which the spin-orbit coupling parameters ζ, ζ′ and the orbital reduction factors k, k′ are calculated from a semi-empirical molecular orbital method. The crystal-field parameters used in the calculations are obtained from the superposition model and so the defect structure of Cu²⁺ centres in ZnO can be acquired. The calculated g factors from the three methods are in reasonable agreement with the experimental values and the defect structure of Cu²⁺ centres in ZnO is acquired. It appears that in some cases the approximate PTM can be applied in the studies of g factors of various states. The conditions that the PTM are ineffective are discussed.     

Feldgleichungen der TREDERschen Gravitationstheorie,die aus einem Variationsprinzip ableitbar sind. I

In the frame work of TREDER 's gravitational theory we consider two classes of field equations which are derivable from two classes of LAGRANGE ian densities Ω⁽¹⁾(ω₁, ω₂), Ω⁽²⁾(s?₁, s?₂). ω₁, ω₂; s?₁, s?₂ are parameters. Ω⁽²⁾(ω₁, ω₂) gives us field equations which are up to the post-NEWTON ian approximation in the sense of NORDTVEDT , THORNE and WILL equivalent to the field equations given by BRANS and DICKE . For ω₂ = ?1 ?2ω₁ field equations follow from Ω⁽¹⁾(ω₁, ?1 ?2ω₁) which are in the above mentioned sense of post-NEWTON ian approximation equivalent to EINSTEIN 's equations. The field equations following from Ω⁽¹⁾(ω₁, ω₂) have a cosmological model with the well known cosmological singularities for T → ± ∞ in case that ω₁/(1 +3ω₁ +ω₂) ? γ > 0. For ω₁/(1 +3ω₁ +ω₂) ≤ 0 cosmological models with no cosmological singularities exist. From Ω⁽²⁾(s?₁, s?₂) we obtain field equations which at the best give us perihelion rotation 7% above EINSTEIN 's value and light deflection 7% below the corresponding EINSTEIN 's value. But in that case we are able to show the existence of a cosmological model without any cosmological singularity.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

Electronic absorption spectrum of Ni2+ in lithium potassium sulphate single crystal

Single crystals of nickel-doped lithium potassium sulphate were grown by slow evaporation method at room temperature. From the nature and position of the bands observed, a successful interpretation of all the bands could be made assumingO _h symmetry for the Ni²⁺ ion in the crystal. The bands have been assigned transitions from the ground³A_2g(F) state to the excited³T_2g(F),¹E_g(D),³T_1g(F),¹T_2g(D) and³T_1g(P) states. The crystal field parameters derived areDq=910cm^–1,B=890cm^–1 andC=3560cm^–1.The authors wish to express their thanks to Prof. K. Sreerama Murthy for his constant encouragement throughout this investigation. The authors are also thankful to Prof. Mihir Chowdhury, Indian Association for the cultivation of Science, Calcutta for giving permission to take the spectra.     

Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σ _g ×S ², where Σ _g is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya–Floer cohomology of Σ _g ×S ¹. Received: 22 July 1999 / Accepted: 12 June 2000     

Kinetics of processes in the middle atmosphere during the laser excitation of O2 molecules

Kinetic processes taking place in the atomic-molecular system O-O₂-O₃ in the middle atmosphere with the participation of oxygen molecules in the excited electronic states O₂(a ¹Δ_g) and O₂(b ¹Σ _g ⁺ ) are analyzed in detail. The possibility of increased ozone production under the influence of solar radiation during the laser excitation of O₂ molecules in the a ¹Δ_g state is demonstrated on the basis of numerical modeling. Upper and lower bounds are determined for the densities of O₂(a ¹Δ_g) molecules at which the ozone concentration increases in the irradiated zone. Zh. Tekh. Fiz. 68, 15–23 (August 1998)     

On the relative intensities of the Raman active fundamentals,r0 structural parameters,and pathway of chair–boat interconversion of cyclohexane and cyclohexane‐d12

Raman spectra of liquid cyclohexane, C₆H₁₂, and deuterated cyclohexane, C₆D₁₂, were recorded with both parallel and perpendicular polarizations. The observed vibrational wavenumbers, depolarization ratios, and their intensities were measured and compared with the corresponding predicted values as well as the experimental values previously reported. The conformational energetics were obtained with the Møller–Plesset perturbation method to the second order [MP2(full)] as well as with density functional theory by the B3LYP method utilizing a variety of basis sets. The average ab initio predicted difference in energy between the more stable chair form (D_3d) and the less stable twisted‐boat form (D₂) is 2213 cm⁻¹ (26.47 kJ/mol), with a similar value of 2223 cm⁻¹ (26.59 kJ/mol) from the density function theory calculations. By using two dihedral angles as variables, we calculated the chair–boat interconversion pathway for cyclohexane at the MP2(full)/6‐31G(d) level. The harmonic force constants, Raman intensities, depolarization values, and the potential energy distribution were predicted from both MP2(full) and B3LYP calculations with the 6‐31G(d) basis set and compared with the experimental values for the chair form when available. The ‘adjusted’ r₀ structural parameters were obtained from MP2/6‐311 + G(d,p) calculations and previously reported microwave rotational constants of five isotopomers of cyclohexane: i.e. 1,1‐d₂, ¹³C‐1,1‐d₂, 1,1,2,2,3,3‐d₆, and d₁ (equatorial and axial). The determined distances in Å are: r(CC) = 1.536(3), r(CH)_ax = 1.098(1); r(CH)_eq = 1.095(1); and the angles in degrees: ∠CCH_ax = 108.8(3); ∠CCH_eq = 110.2(3); ∠CCC = 111.1(3); and ∠HCH = 107.6(3) with dihedral angle ∠CCCC = 55.7(3). These values are compared with those previously reported and it is found that the difference in the r₀ distances (0.003 Å) between the two CH values is much smaller than the difference (0.008 Å) previously reported for the r_s values. Copyright © 2008 John Wiley & Sons, Ltd.     

On the kinetics of a two-level system with a diffusion-modulated interaction

For a two-level system corresponding to a particle of spin ½ in a random field in the Z direction, the relaxation function

has been estimated, the magnitude Ω(t) being the sum of the isotropic interactions of the particle in question with particles j executing diffusional motion. Specifically Ω(t)=ω₀ + Σ ω(Mj, rj), where ω₀ = constant, M_j is a random time-independent parameter, ω(M, r) decreases with r faster than r ^-3 and r _j = r _j (t) is a diffusional, random function. From the expression for <σ⁺(t)>, we establish general features of the relaxation phenomenon for diffusional processes, and calculate the relaxation rate 1/T ₂ and relaxation shift Δω to be 1/T ₂-iΔω = 4π CAv_M λ _M , where C is the concentration of particles and λ _M is the scattering length for an equation of the Schrödinger type with an imaginary potential -iω(M, r) instead of U/?, and diffusion coefficient instead of ?/2m. We also found that for the case of ‘external’ relaxation, the Redfield approach proved valid only under the simultaneous restrictions of low concentration and weak interaction.     

An efficient method for calculating the pair distribution functions of a ternary hard-sphere system in r space within the Percus-Yevick approximation

We present a direct method which allows an accurate and—at the same time—economical calculation of the pair distribution functions (PDFs) g_ij (r) of an additive ternary hard-sphere system within the Percus—Yevick approximation. The approach is based on the fact that for this approximation the Laplace transforms [?_ij (s)] of the PDFs are known analytically, so that the inversion of the ?_ij (s) into r space can be performed exactly. The expressions presented here allow the determination of the ?_ij (r) for r values up to 8R ₁, R ₁ being the diameter of the smallest species; this range in r space should be sufficient for applications in standard algorithms of liquid state theory, such as thermodynamic perturbation theories or integral-equation approaches.