Second order of perturbation theory correction to the radial distribution function of a liquid with the interaction potential of hard spheres plus a square-well

A liquid with the interaction potential of hard spheres plus a square-well is analyzed using the Monte-Carlo technique. Numerical results for the perturbation theory series over a square-well potential are obtained in the form of the Barker and Henderson discrete representation. Approximating expressions for the correction to a liquid radial distribution function in the second order of perturbation theory are presented. The obtained results allow us to define this correction with a root-mean-square deviation of about 0.007. It is shown that the given approach provides a complete calculation in the second order of perturbation theory, and also the determination of the third order correction to the free energy for a liquid interacting with the potential of the Lennard-Jones type.     

Thermodynamic perturbation theory of simple liquids. A hierarchy of equations for expansion coefficients

The properties of an expansion of the statistical sum of a simple liquid with respect to the potential in thermodynamic perturbation theory are analyzed. The coefficients of this expansion are determined by the unperturbed potential, depend on temperature and density, and can be calculated by means of mathematical modeling. It is shown here that the derivatives of these coefficients with respect to temperature and density are expressed through the higher expansion coefficient (these relations are usually called a hierarchy of equations). These coefficients determine the expansion of the Helmholtz free energy and RDF with respect to the perturbation potential. The thermodynamic characteristics of the system (entropy, internal energy, pressure) are expressed through both the differential relations for the Helmholtz free energy and the integral expressions containing RDF. It is found that the hierarchy of equations obtained in this work makes these different methods equivalent. This is important for the application of thermodynamic perturbation theory because it becomes unnecessary to model any other equilibrium properties of the system apart from the expansion coefficients.     

Effect of critical fluctuations on adsorption of van der Waals fluid in a spherical cavity

A recently proposed non-uniform fifth-order thermodynamic perturbation theory (TPT) is employed to investigate the adsorption of a hard core attractive Yukawa (HCAY) fluid in a spherical cavity. Extensive comparison with available simulation data indicate that the non-uniform fifth-order TPT is sufficiently reliable in calculating the density profiles of the HCAY fluid in the highly confining geometry, and generally is more accurate than a previous third-order?+?second-order perturbation density functional theory. The non-uniform fifth-order TPT is free from numerically solving an Ornstein–Zernike integral equation, and also free of any adjustable parameter; consequently, it can be applied to both supercritical and subcritical temperature regions. The non-uniform fifth-order TPT is employed to investigate critical adsorption of the HCYA fluid in a single spherical cavity – it is disclosed that the critical fluctuations near the critical point induce depletion adsorption – quantitative theoretical calculation on relationship between the critical depletion adsorption, parameters of coexistence bulk phase and the responsible external field is in agreement with qualitative physical analysis.     

Application of SCF perturbation theory to molecular calculations

A method for solving Roothaan's molecular orbital equations by means of SCF perturbation theory is presented. An estimate of the accuracy of the third order expansion is made for the CNDO/2 approximation from a comparison of the results from direct calculations. It is found that the third order theory is sufficiently accurate for quantitative studies.     

Thermodynamic properties of pure liquids within a generalized version of the hole theory

A generalized hole theory for computing the thermodynamic and acoustic properties of liquids and liquid mixtures has been developed. Ultrasonic velocity, isothermal compressibility and thermal expansivity of different pure liquids at 293.15 and 298.15?K are evaluated by using hole theory. The calculated values of ultrasonic velocities, isothermal compressibilities, and thermal expansivities are compared with the experimental findings. A fairly good agreement between experimental and calculated values is observed.     

Relativistic perturbation theory of molecular structure

Summary The recently developed relativistic double perturbation theory is extended to handle relativistic changes of molecular structure more easily. This is achieved by simple coordinate scalings. Accurate higher order mixed perturbation energies for H ₂ ⁺ are calculated. The relativistic changes of bond energy,DE, of bond length,R _e , and especially of force constant,k, and of anharmonicity,a, are large, up to 100%·(Z/c)². The dominant contributions tok anda are due to the indirect change of the nonrelativistick anda connected with the relativistic change of bond length. Accordingly the relativistic changes obey Badger's and Gordy's rules (–RDEk).Dedicated to Prof. Klaus Ruedenberg in appreciation of his fundamental contributions to both formal theory and physical explanations in quantum chemistry     

A constant denominator perturbation theory for molecular energy

A constant denominator perturbation theory is developed based on a zeroth order Hamiltonian characterized by degenerate subsets of orbitals. Such a formulation allows for a decoupling of the numerators of the perturbation sequence, allowing for much more rapid evaluation of the resultant sums. For example, the full fourth order theory can be evaluated as an N ⁶ step rather than N ⁷, where N is proportional to the basis set.Although the theory is general, a constant denominator is chosen for this study as the difference between the average occupied and average virtual orbital energies scaled so that the first order wavefunction yields the lowest possible variational bound. The third order correction then appears naturally as a scaled Langhoff-Davidson correction. The full fourth order with this partitioning is developed. Results are presented within the localized bond model utilizing both the Pariser-Parr-Pople and CNDO/2 model Hamiltonians. The second order theory presents a useful bound, usually containing a good deal of the basis set correlation. In all cases examined the fourth order theory shows remarkable stability, even in those cases in which the Nesbet-Epstein partitioning seems poorly convergent, and the Moller-Plesset theory uncertain.     

Configuration-based multi-reference second order perturbation theory

Various configuration-based multi-reference second order perturbation approaches were investigated and a new scheme averting intruder states was suggested. The codes based on these schemes were tested by example calculations.     

Multi-reference-state perturbation theory for computation of potential-energy surfaces

The implementation of multi-reference-state Rayleigh-Schrödinger perturbation theory for the evaluation of potential-energy surfaces is reviewed. The organization of the computation and the basic algorithms are outlined. A major difficulty is the quadratic increase in computing sources as the reference space is expanded. Truncating the space of nonreference determinants is suggested as a solution for the problem. The method applies a numerical cutoff criterion for the weights of the first-order wave function. The truncation reduces the computing time significantly with negligible sacrifice in the quality of the results.This paper was presented at the international conference on The Impact of Supercomputers on Chemistry, held at the University of London, London, UK, 13–16 April 1987     

A modified multi-reference second order perturbation theory

A new scheme with extended model space is proposed to improve the calculation of multi-reference second order perturbation theory (MRPT2). The new scheme preserves the concise code structure of the original program, and avoids intruder states in constructions of the potential energy surface, which is confirmed by a series of comparable calculations. The new MRPT2 program is an available tool for the research of molecular excited states and electronic spectrum.     

Calculation of correlation energy by many-body diagrammatic perturbation theory

The many-body diagrammatic perturbation theory is used for calculation of the correlation energy of closed-shell molecular systems. We apply Brueckner's concept of the two-particle renormalized interaction defined by a non-linear diagrammatic expression containing all possible (diagonal and/or non-diagonal) particle-particle, hole-hole and particle-hole intermediate elementary processes. Then, a “second-order” simple diagrammatic expression for the correlation energy can be formed, where the correlation energy is approximated by all the diagrams with biexcited intermediate states. An illustrative numerical application for the LiH molecule is presented. This article is dedicated to the memory of our friends and colleagues Dr. Jarka Surá and Dr. Marta Černayová, who tragically died in July 1976.     

Generalized perturbation theory: Quality of the first-order wave function

A recently proposed version of generalized perturbation theory, in which the whole energy correction is taken care of in second order, is investigated with respect to the quality of its first-order wave function. It is demonstrated that the overlap of the wave function generated in this procedure with the exact solution is in most cases much closer to unity than those of the Rayleigh-Schroedinger or Brillouin-Wigner perturbation theories. Certain approximations, by means of which realistic systems become amenable to investigation within the presently discussed framework, are studied.Based on a section of a thesis to be submitted by N. M. to the senate of the Technion — Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree, and presented in the Second International Congress of Quantum Chemistry, New Orleans, April 1976.     

On a formal treatment of the Rayleigh-Schrödinger perturbation theory

The formal treatment of the Rayleigh-Schrödinger perturbation theory based on a first-order iteration procedure as described in a previous publication is discussed with reference to the properties of the terms of a Taylor series. The formalism is generalized to allow for multiple perturbation.     

Semiclassical self-consistent field perturbation theory for the hydrogen atom in a magnetic field

A recently developed perturbation theory for solving self-consistent field equations is applied to the hydrogen atom in a strong magnetic field. This system has been extensively studied using other methods and is therefore a good test case for the new method. The perturbation theory yields summable large-order expansions. The accuracy of the self-consistent field approximation varies according to field strength and quantum state but is often higher than the accuracy from adiabatic approximations. A new derivation is presented for the asymptotic adiabatic approximation, the most useful of the adiabatic approaches. This derivation uses semiclassical perturbation theory without invoking an adiabatic hypothesis. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 183–192, 1998     

Many-body perturbation theory for open-shell systems. Expansion through fourth-order

All of the diagrams which arise in the many-body perturbation theory of open-shell systems using a restricted Hartree-Fock reference function are given through fourth-order in the energy. New effects which arise in fourthorder are discussed.S.E.R.C. Advanced Fellow     

On the use of the variational method for interpreting the solubilities of gases in liquids

The Variational theory of mixtures due to Mansoori and Leland is modified to account for translational quantum effects in solution, and this modified form of the theory is applied to the problem of gas solubilities in liquids. The theory is used to derive expressions for the Henry's law constant, the molar heat of solution at infinite dilution, and the partial molar volume at infinite dilution for a solute in a liquid solution. These expressions are applied, over a range of temperatures, to the following systems; H₂ in each of Ar, N₂, and CH₄; He in each of Ar, N₂, and CH₄; and Ne in each of Ar and N₂. Lennard-Jones 6–12 pair potentials are used for these calculations. The Lennard-Jones parameters are taken from gas-phase second virial coefficient data. The results obtained are compared with experimental data and with previous calculations on these systems based on the Leonard-Henderson-Barker theory. The variational results appear to be in better agreement with experiment for the He-containing systems, while the Leonard-Henderson-Barker theory seems better for the other systems. An explanation for this is suggested.     

On the performance of the open-shell perturbation theory

A few open-shell molecules are taken as examples in order to examine the performance of the open-shell perturbation theory for electron correlation(J Chem Theory Comput,2009,5:931–936).The convergence of the perturbation series is shown to be stable for the doublet state of NH2 at both the equilibrium and stretched geometries.The equilibrium bond lengths,and harmonic and anharmonic vibrational frequencies are calculated for NO(X2),OH(X2),CH(X2)and NH(X2)with different second-order perturbation theories at t...