Bound Energy for the Exponential-Cosine-Screened Coulomb Potential

An alternative approximation scheme has been used in solving the Schrödinger equation for the exponential-cosine-screened Coulomb potential. The bound state energ?es for various eigenstates and the corresponding wave functions are obtained analytically up to the second perturbation term.     

Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. II. Consideration of perturbations in the metric operator

A previous relativistic shielding calculation theory based on the regular approximation to the normalized elimination of the small component approach is improved by the inclusion of the magnetic interaction term contained in the metric operator. In order to consider effects of the metric perturbation, the self-consistent perturbation theory is used for the case of perturbation-dependent overlap integrals. The calculation results show that the second-order regular approximation results obtained for the isotropic shielding constants of halogen nuclei are well improved by the inclusion of the metric perturbation to reproduce the fully relativistic four-component Dirac-Hartree-Fock results. However, it is shown that the metric perturbation hardly or does not affect the anisotropy of the halogen shielding tensors and the proton magnetic shieldings.     

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

This contribution presents an approximate solution of the enzyme kinetics problem for the case of excess of an enzyme over the substrate. A first order perturbation approach is adopted where the perturbation parameter is the relation of the substrate concentration to the total amount of enzyme. As a generalization over existing solutions for the same problem, the presented approximation allows for nonzero initial conditions for the substrate and the enzyme concentrations as well as for nonzero initial complex concentration. Nevertheless, the approximate solution is obtained in analytical form involving only elementary functions like exponentials and logarithms. The presentation discusses all steps of the procedure, starting from amplitude and time scaling for a non-dimensional representation and for the identification of the perturbation parameter. Suitable time constants lead to the short term and long term behaviour, also known as the inner and outer solution. Special attention is paid to the matching process by the definition of a suitable intermediate layer. The results are presented in concise form as a summary of the required calculations. An extended example compares the zero order and first order perturbation approximations for the short term and long term solution as well as the uniform solution. A comparison to the numerical solution of the initial set of nonlinear ordinary differential equations demonstrates the achievable accuracy.     

Random-phase-approximation-based correlation energy functionals: benchmark results for atoms

The random phase approximation for the correlation energy functional of the density functional theory has recently attracted renewed interest. Formulated in terms of the Kohn-Sham orbitals and eigenvalues, it promises to resolve some of the fundamental limitations of the local density and generalized gradient approximations, as, for instance, their inability to account for dispersion forces. First results for atoms, however, indicate that the random phase approximation overestimates correlation effects as much as the orbital-dependent functional obtained by a second order perturbation expansion on the basis of the Kohn-Sham Hamiltonian. In this contribution, three simple extensions of the random phase approximation are examined; (a) its augmentation by a local density approximation for short-range correlation, (b) its combination with the second order exchange term, and (c) its combination with a partial resummation of the perturbation series including the second order exchange. It is found that the ground state and correlation energies as well as the ionization potentials resulting from the extensions (a) and (c) for closed subshell atoms are clearly superior to those obtained with the unmodified random phase approximation. Quite some effort is made to ensure highly converged data, so that the results may serve as benchmark data. The numerical techniques developed in this context, in particular, for the inherent frequency integration, should also be useful for applications of random phase approximation-type functionals to more complex systems.     

Calculating thermodynamic properties from perturbation theory: I. An analytic representation of square-well potential hard-sphere perturbation theory

The Barker–Henderson macroscopic compressibility approximation of the second-order perturbation term is improved by assuming that the numbers of molecules in every two neighbour shells are correlated, based upon the original assumptions. The results are better than those for the original macroscopic compressibility and local compressibility approximation, especially at high densities. A simple analytic representation of square-well potential hard-sphere perturbation theory is derived based upon this improvement. The method is tested by calculating thermodynamic properties with the four-term truncated form, and the results are in good agreement with those of Monte Carlo and Molecular Dynamics simulation.     

Theory of diamagnetic anisotropy for conjugated monocyclic systems

Deficiencies are pointed out in the theory based on the one-electron approximation. The anisotropy is calculated for a cyclic polyene with 4n electrons in the one-configuration and multiconfiguration approximations for complete axial symmetry and for perturbed axial symmetry. The relation of the paramagnetic term to the perturbation is derived for both approximations.     

On the validity range of the Born-Oppenheimer approximation: a semiclassical study for all-particle quantization of three-body Coulomb systems

The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (mu) and antiproton (p) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppmu), and (ppp) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 3/2 to the mass of negative particles, that is, m(1.5). The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m(5/4) is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m(6/4). Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m(1/4).     

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.     

Validation of dispersion-corrected density functional theory approaches for ionic liquid systems

The performance of several general gradient approximation, meta general gradient approximation, and hybrid functionals is tested against M?ller-Plesset perturbation theory second-order for ionic liquid systems. Additionally, two dispersion-corrected approaches (addition of van der Waals forces by a 1/r(6) term and employing a dispersion-corrected atom-center dispersion pseudopotential) were studied. For the 1-butyl-3-methylimidazolium cation neglecting dispersion results in different trends for structural stabilities. The two applied correction schemes for density functional theory improve the results tremendously. Investigating several 1-butyl-3-methylimidazolium dicianamide ion pairs shows a mean absolute deviation from M?ller-Plesset perturbation theory of 35.7 kJ/mol for Hartree-Fock and up to 33.2 kJ/mol for the density functional theory methods. The dispersion-corrected methods reduce the mean absolute deviation to less than 10 kJ/mol. Comparing adducts of the 1-ethyl-3-methylimidazolium dicianamide ion pair with Diels-Alder educts (cyclopentadiene and methylacrylate) shows similar energetic differences as for the ion pairs. Furthermore large deviations in geometries for the intermolecular distances were found for the Hartree-Fock approach (mean absolute deviation: 190 pm) and density functional theory (mean absolute deviation up to 178 pm) while for the dispersion-corrected methods the mean absolute deviation is less than 50 pm.     

Convergence of the perturbation theory expansion for spin-spin coupling constants

The convergence of the full perturbation theory expansion for the Fermi contact term in the spin-spin coupling constant of hydrogen fluoride has been studied. By examining the contribution of higher virtual states as the basis set is expanded to 18 atomic orbitals, it is shown that at this level of approximation the expansion does not converge. The need to also establish convergence before accepting values calculated from configuration interaction wavefunctions is discussed.The Puerto Rico Nuclear Center is operated by the University of Puerto Rico for the United States Atomic Energy Commission under Contract No. AT(40-1)-1833.     

A Thermodynamic Model for Square-well Chain Fluid: Theory and Monte Carlo Simulation

A thermodynamic model for the freely jointed square-well chain fluids was developed based on the thermodynamic perturbation theory of Barker-Henderson, Zhang and Wertheim. In this derivation Zhang's expressions for square-well monomers improved from Barker-Henderson compressibility approximation were adopted as the reference fluid, and Wertheim's polymerization method was used to obtain the free energy term due to the bond connectivity. An analytic expression for the Helmholtz free energy of the square-well chain fluids was obtained. The expression without adjustable parameters leads to the thermodynamic consistent predictions of the compressibility factors, residual internal energy and constant-volume heat capacity for dimer, 4-mer, 8-mer and 16-mer square-well fluids. The results are in good agreement with the Monte Carlo simulation. To obtain the MC data of residual internal energy and the constant-volume heat capacity needed, NVT MC simulations were performed for these square-well chain fluids.     

Equation of state based on the weeks-chandler-andersen perturbation theory

An equation of state based on the Weeks-Chandler-Andersen (WCA) perturbation theory utilizing the simplified random-phase approximation (RPA) term is p of state is applied successfully to calculate the P-V-T relationship for a spherical molecule, argon.For nonspherical molecules, the effects of anisotropic interactions are treated empirically. The calculated P-V-T relationships and saturated properties for nonspherical and nonpolar molecules agree well with experimental data. The potential parameters for nonpolar substances are well correlated with the acentric factors.     

External coupled-cluster perturbation theory: description and application to weakly interaction dimers. Corrections to the random phase approximation

The formalism for developing perturbation theory by using an arbitrary fixed (external) set of amplitudes as an initial approximation is presented in a compact form: external coupled-cluster perturbation theory (xCCPT). Nonperturbative approaches also fit into the formalism. As an illustration, the weakly interacting dimers Ne(2) and Ar(2) have been studied in the various ring-coupled-cluster doubles (CCD) approximations; ring, direct-ring, antisymmetrized ring, and antisymmetrized direct ring, and a second-order correction in the xCCPT approach is added. The direct approaches include the summation of just Coulomb terms with the intention of selectively summing the largest terms in the perturbation first. "Coulomb attenuation" is effected by taking the random phase approximation to define such amplitudes, whose results are then improved upon using perturbation theory. Interaction energies at the ring-CCD level are poor but the xCCPT correction employed predicts binding energies which are only a few percent from the coupled-cluster single double (triple) values for the direct ring-CCD variants. Using the MP2 amplitudes which neglect exchange, the initial Coulomb-only term, leads to very accurate Ne(2) and Ar(2) potentials. However, to accurately compute the Na(2) potential required a different initial wavefunction, and hence perturbation. The potential energy surfaces of Ne(2) and Ar(2) are much too shallow using linear coupled-cluster doubles. Using xCCPT(2) with these amplitudes as the initial wavefunction led to slightly worse results. These observations suggest that an optimal external set of amplitudes exists which minimizes perturbational effects and hence improve the predictability of methods.     

Energy shifts in the relative coordinates treatment of the collinear collision between an atom and an oscillator. Exponential repulsive potential

The treatment of the kinetic term which may be taken as the perturbation when the collision between an atom and an oscillator is described with relative coordinates is shown to require the use of energy shifted basis functions in the calculation of the transition probabilities. For an exponential repulsive potential, a simple analytical formula is obtained for one quantum jumps in the first order Born approximation, which has a simple relationship with the first order Born distorted wave expression, and which lead to better results than the other available analytical formulas.     

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     

Multipole interaction in high energy approximation. Calculation of the total collision cross section

For molecules of cylindrical symmetry the total collision cross section is calculated in the high energy approximation, taking into account various long range intermolecular interactions. Results are obtained for the limiting cases that the rotational motion of the molecules is fast or slow with respect to the duration of the molecular interaction. The assumption is made that either one multipole term of the intermolecular potential predominates or that all multipole terms can essentially be treated as a small perturbation of the van der Waals r^?6 interaction.     

Perturbation theory for excited states. IV

Two of the perturbation methods developed in Paper I (CIPT and SCPT) are extended for use with uncoupled Hartree-Fock perturbation theory. Corrections to the uncoupled results are discussed and it is found that it is possible to correct using functions which when applied to ground state calculations are equivalent to the geometric approximation. It is also shown that CIPT and SCPT can be applied to ground state calculations and that when coupled perturbed orbitals are used CIPT and SCPT are equivalent to the usual coupled perturbation theory.     

Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory

Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used "average of degenerate states approximation" for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases.     

Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory

Range-separated density-functional theory combines wave function theory for the long-range part of the two-electron interaction with density-functional theory for the short-range part. When describing the long-range interaction with non-variational methods, such as perturbation or coupled-cluster theories, self-consistency effects are introduced in the density functional part, which for an exact solution requires iterations. They are generally assumed to be small but no detailed study has been performed so far. Here, the authors analyze self-consistency when using M?ller-Plesset-type (MP) perturbation theory for the long range interaction. The lowest-order self-consistency corrections to the wave function and the energy, that enter the perturbation expansions at the second and fourth order, respectively, are both expressed in terms of the one-electron reduced density matrix. The computational implementation of the latter is based on a Neumann series which, interestingly, even though the effect is small, usually diverges. A convergence technique, which perhaps can be applied in other uses of Neumann series in perturbation theory, is proposed. The numerical results thus obtained show that, in weakly bound systems, self-consistency can be neglected since the long-range correlation does not affect the density significantly. Although MP is not adequate for multireference systems, it can still be used as a reliable analysis tool. Though the density change is not negligible anymore in such cases, self-consistency effects are found to be much smaller than long-range correlation effects (less than 10% for the systems considered). For that reason, a sensible approximation might be to update the short-range energy functional term while freezing its functional derivative, namely, the short-range local potential, in the wave function optimization. The accuracy of such an approximation still needs to be assessed.