Critical analysis of the series expansion of the potential energy function of CO2

We present in this paper new values of the coefficients involved in the series expansion with respect to internal coordinates of the potential energy function of the CO₂ molecule. The relative importance of each potential coefficient and the cancellation of some potential coefficients nonsignificantly different from zero, are discussed.     

Mechanization of operations within an algebraic structure: Application to the transformation of the potential energy function of a linear triatomic molecule from internal coordinates to normal coordinates

In this article a program is described for the automatic computation of the coefficients involved in the expansion, with respect to normal coordinates, of the potential energy function of a linear triatomic molecule, starting from the coefficients of the expansion of the same function with respect to internal coordinates. The two expansions are limited to the terms of degree 6 with respect to the coordinates. This program, which is available upon request to the authors, uses a new method presented in a previous article.     

Pb2，PdPb2分子的势能函数

采用Gaussian 98程序,运用B3LYP方法,对Pd和Pb原子采用收缩价基组LANL2DZ,对Pb₂和PdPb₂分子的微观结构进行了理论计算. 由于Pb₂分子离解后一个Pb原子处于基态,另一个Pb原子处于激发态,采用最小二乘法拟合Pb₂分子的势能函数,选用的函数形式为Murrell-Sorbie势能函数加上开关函数. 使用多体展式理论导出了势函数中的参数进而给出PdPb₂分子基态势函数的解析表达式,其势能面准确地复现了PdPb₂分子的两个稳定构型(C_2V和C_∞v)及其能量关系. 关键词： 2')" href="#">Pb₂ 2')" href="#">PdPb₂ 势能函数     

N2O异构体结构与解析势能函数

在B3P86/cc-PVTZ水平上,对N₂O异构体进行优化计算,得出N₂O基态的单重态能量最低,其稳定构型为C_∞v构型,平衡核间距R₁=0.1121nm,R₂=0.1177nm,α＝180°,能量为-185.1188a.u.同时计算出基态的简正振动频率ω₁(Π)=601.5010 cm^{关键词： 异构体 多体项展式理论 解析势能函数}     

PdCO分子结构与势能函数

用密度泛函理论的B3LYP方法,对钯原子采用LANL2DZ收缩价基函数,碳原子和氧原子采用AUG-cc-pVTZ基组,对PdC,PdO和PdCO体系的结构进行优化,计算表明：PdC分子基态为¹Σ⁺态,键长为R_e=0.17285nm,离解能为4.919eV.PdO分子基态的平衡核间距为0.18546nm,其电子态为³Π,离解能为2.455eV,并拟合得到Murrell-Sorbie势能函数;PdCO分子有两 关键词： PdCO 分子结构 势能函数     

Variation in free energy with interface area: the second derivative

Using the method of rescaling particle coordinates, the derivatives of the free energy with respect to volume and area are calculated. In particular, an explicit expression for the second derivative (? ² F/?A²)_v is given together with its derivation. This derivative ought to be identically nil for ordinary interfaces such as the liquid-vapour planar interface and positive for a positive surface area for such objects as a bilayer immersed in a solvent.     

Determination of a Potential Energy Surface for CO2 Using Generalized Internal Vibrational Coordinates

A potential energy surface for CO2 is determined from experimental data using generalized internal vibrational coordinates. These coordinates are defined as two arbitrary distances and the angle between them and depend on two external parameters, which can be properly optimized. An optimal generalized internal coordinate system is obtained for CO2 by minimizing unconverged vibrational energies with respect to the external parameters. The optimal coordinates are shown to be superior to previously derived normal hyperspherical coordinates for this molecule. A nonlinear least-squares fit of the potential energy surface of CO2 to observed vibrational frequencies is made by using the optimal internal coordinates and fully variational calculations. The potential function is represented by a fourth-order Morse-cosine expansion and its quality is checked by computing highly excited vibrational transition frequencies which were not included in the fit. Copyright 1999 Academic Press.     

On the quantization of a kinetic energy with variable inertia

It is shown that fundamental properties of kinetic energy determine its operator in wave mechanics almost completely: The part containing differential operators is unequivocally given by the Laplacian calculated for the metric which is defined by the classical kinetic energy. The only remaining ambiguity is an additive scalar function proportional to ?². Invariance properties with respect to infinitesimal transformations may reduce the number of coordinates on which this function depends. In certain cases it must be constant.     

Dynamic polarizability,dispersion coefficient C6 and dispersion energy in the effective fragment potential method

The development of a fragment–fragment dispersion energy expression, for the general effective fragment potential (EFP2) method is presented. C₆ dispersion coefficients, expressed in terms of the dynamic polarizabilties over the imaginary frequency range (α(iν)), were calculated for a set of homo and hetero dimers. Using these coefficients the dispersion energy has been calculated. The dispersion energy is expressed using a simple London series expansion terminated after the n=6 term and implemented using distributed localized molecular orbitals (LMOs). The EFP2 dispersion energy is compared to symmetry adapted perturbation theory (SAPT) values. From this comparison, it is apparent that one needs to include higher order terms in the dispersion energy. Adding an estimated C₈ term to the C₆ energy greatly improves the agreement with the benchmark SAPT energies.     

氢同位素氚水T2O(X1A1)的解析势能函数

获得T₂O(X¹A₁)解析势能函数的主要困难在于Born-Oppenheimer近似下T₂O(X¹A₁)与H₂O(X¹A₁)势能函数的不可区分性.然而,在Born-Oppenheimer近似下,分子势能函数实际上是键长、键角这些 关键词： 2O(X¹A₁)')" href="#">T₂O(X¹A₁) 同位素效应 解析势能函数     

Further Dunham energy coefficients of diatomic molecules

Expressions are given for the Dunham energy coefficients Y₃₃ and Y₄₂ in terms of the potential energy coefficients a_i, for diatomic molecules. Their use is briefly discussed.     

On the use of the finite difference method in a calculation of vibration-rotation energies

The use of the finite difference method to obtain a Taylor series expansion of a potential energy function for a subsequent calculation of the rovibration energies of molecules is considered. A method is proposed that allows the stability of a finite-difference scheme to be increased against the computational inaccuracy upon numerical expansion of a multidimensional potential energy function into a high-order Taylor series. The method is based on the successive elimination of calculated expansion coefficients of a higher order in calculating the lower-order coefficients by the finite difference method. The approach is illustrated for the example of the CO and H₂S molecules.     

The vibrational energy levels of ammonia

A variational 6-dimensional method is used to determine the low lying vibrational energy levels of ammonia. The six internal coordinates were chosen to be appropriate for the symmetry and inversion motion of the molecule; they were the three NH bond lengths, r1, r2, r3, the unique angle β which each bond makes with the trisector of them, and two (of the three) angles, θ2 and θ3, between the bonds when projected on to a plane perpendicular to the trisector. The Wilson G matrix was determined for these internal coordinates both by computer algebra and by hand. An appropriate Jacobian for the motion was determined and the full Hermitian kinetic energy operator was obtained using the Podolsky transformation. Expansion functions were in the usual product form. Special attention was given to the θ2, θ3 expansion functions so that appropriate A1, A2 and E symmetry vibrational modes were obtained explicitly. Matrix elements of the kinetic energy operator were expressed in terms of one-dimensional integrals.Variational calculations have been performed with two six-dimensional surfaces: (i) that due to Martin, Lee and Taylor; and (ii) that due to Spirko and Kraemer. Although some of the vibrational levels for both surfaces are accurate, both have inadequacies: (a) because it is a Taylor expansion about an equilibrium, based on ab initio calculations, with no attention paid to planarity; and (b) because the non-inversion part of the surface was treated perturbatively in its derivation, and in fact some of the quartic displacement powers have negative coefficients. Therefore, neither surface gave good results overall, and there is a need for a refined 6 dimensional NH3 surface.     

SiO2分子的基态(X1A1)结构与分析势能函数

应用群论及原子分子反应静力学方法推导了SiO₂分子的电子态及其离解极限，采用B3P86方法，在6-311G^**水平上,优化出SiO₂基态分子稳定构型为单重态的C_2V构型，其平衡核间距R_e=R_Si—O=0.1587 nm,∠OSiO＝111.2°,能量为-440.4392 a.u..同时计算出基态的简正振动频率:对称伸缩振动频率ν(B₂)=945.4cm^-1，弯曲振动频率ν(A₁)=273.5 cm^-1和反对称伸缩振动频率ν(A₁)=1362.9cm^-1.在此基础上，使用多体项展式理论方法，导出了基态SiO₂分子的全空间解析势能函数，该势能函数准确再现了SiO₂(C_2V)平衡结构.     

The direct determination of pair potential energy functions for mixed interactions: Ar-Kr

Measurements of the viscosity coefficients for argon-krypton gas mixtures over the temperature range 120–1600 K are inverted to give the pair potential energy function for Ar-Kr. The inversion procedure requires an estimate of the well-depth parameter ε/k and a method for its determination using second virial coefficient data is described. The resulting Ar-Kr potential function has characteristic parameters ε/k=165 K, σ=0·348 nm and r _m=0·3902 nm. The conformality of this potential with those for the like-molecule interactions, and the value of parametric combining rules for this system are also investigated.     

Wigner's function and other distribution functions in mock phase spaces

This review deals with the methods of associating functions with quantum mechanical operators in such a manner that these functions should furnish conveniently semiclassical approximations. We present a unified treatment of methods and results which usually appear under expressions such as Wigner's function, Weyl's association, Kirkwood's expansion, Glauber's coherent state representation, etc.; we also construct some new associations.Section 1 gives the motivation by discussing the Thomas-Fermi theory of an atom with this end in view.Section 2 introduces new operators which resemble Dirac delta functions with operator arguments, the operators being the momenta and coordinates. Reasons are given as to why this should be useful. Next we introduce the notion of an operator basis, and discuss the possibility and usefulness of writing an operator as a linear combination of the basis operators. The coefficients in the linear combination are c-numbers and the c-numbers are associated with the operator (in that particular basis). The delta function type operators introduced before can be used as a basis for the dynamical operators, and the c-numbers obtained in this manner turn out to be the c-number functions used by Wigner, Weyl, Kirkwood, Glauber, etc. New bases and associations can now be invented at will. One such new basis is presented and discussed. The reasons and motivations for choosing different bases is then explained.The copious and seemingly random mathematical relations between these functions are then nothing else but the relations between the expansion coefficients engendered by the relations between the different bases. These are shown and discussed in this light. A brief discussion is then given to possible transformation of the p, q labels.Section 3 gives examples of how the semiclassical expansions are generated for these functions and exhibits their equivalence.The mathematical paraphernalia are collected in the appendices.     

Anharmonic vibrational wave functions, infrared band intensities, and dipole moments of water

A new and simple method to expand the anharmonic vibrational wave functions with respect to the harmonic oscillator wave functions is proposed. The coefficients of the expansion are given as matrix elements of the S function of the contact transformation in the perturbation theory and the explicit expressions of these coefficients are given within the approximation to the second order in λ. As an example of the expansion, the wave functions of water molecules were calculated and applied to the calculation of infrared band intensities and average values of dipole moments in several states.     

Investigation of a new mean temperature-dependent potential energy function for methane and its use for the prediction of transport properties

In this work an improved mean potential energy function for the interaction of an isolated pair of methane is obtained, from which the non-equilibrium properties of methane at zero pressure limit are calculated, accurately. The potential energy function of 21 different fixed orientations of (CH₄)₂ dimer has been obtained via the coupled cluster method. In order to obtain a mean potential energy function, the Boltzmann-average of the obtained potentials of the selected fixed orientations has been used. Unlike the full potential energy surface with the angle-dependent, the parameters of the mean potential are found to be temperature-dependent. The mean potential energy function is fitted well by an analytical expression at different temperatures. The mean absolute percentage deviation of analytical expression compare to the calculated value is about 0.6%. In studying a system with a great number of configurations, calculation of the potential energy function for all configurations is an impossible task; in the proposed model, 21 important fixed orientations have been selected. Introduction of a new approach for calculating the potential energy in methane and investigation of the temperature dependence of mean potential energy are the most important claims of this work. The mean potential function is used to calculate the viscosity, self-diffusion coefficient, and thermal conductivity at the zero pressure limit. The mean absolute percentage deviation in the calculated viscosity, self-diffusion coefficient and thermal conductivity is 3.1, 2.4, and 5.3%, respectively. Also, the second virial coefficient has been calculated for some temperatures.     

Exchange-Coulomb potential energy surfaces and related physical properties for Ne-N2

An exchange-Coulomb (XC) potential energy model is developed for the Ne-N₂ interaction. The construction of this new potential energy surface is based on recent results for the Heitler-London interaction energy, the long range dispersion energies, and the microwave spectra of the dimer. The adjustable parameters in the final XC1 potential energy surface have been determined by fitting the frequencies of three representative lines of the microwave spectrum for the two isotopomers ²⁰Ne-¹⁴N₂ and ²⁰Ne-¹⁵N₂ while simultaneously maintaining agreement with the experimental second virial coefficient data obtained with the initial (unadjusted XC0 potential. With no further adjustment of parameters, the final XC potential reproduces, within 0.005%, the frequencies of the 34 microwave transitions studied experimentally for four isotopomers of Ne-N₂. Excellent agreement with experiment is obtained also for the binary diffusion, the interaction viscosity, and the mixture viscosity (for all compositions) coefficients for all temperatures. Agreement with experiment for the relaxation cross-sections associated with viscomagnetic effects, and with the pressure broadening of depolarized Rayleigh light scattering, is very good given the level of computation used for the calculations and the accuracy with which some of these cross-sections currently can be determined from experimental data. Comparisons are made with predictions calculated from the three best literature potential energy surfaces or Ne-N₂. The final XC1 potential energy surface is overall the most reliable potential energy for this van der Waals complex to date. The flexibility still inherent in this potential can be exploited, if required, in future studies of the Ne-N₂ system.