Electron-pair density relaxation holes

The electron-pair density relaxation hole has been defined as the electron-pair density of the real molecule minus the electron-pair density of a reference system consisting of overlapping, spherically averaged, undeformed atoms, positioned at the molecular nuclear coordinates. We have shown how it can be calculated from one- and two-electron reduced density matrices expanded in a Gaussian type basis set. Analysis of the calculated radial electron-pair density holes, from full configuration interaction one- and two-electron reduce density matrices, for the ground states of the hydrogen molecule, the helium dimer and the lithium and beryllium hydrides reveal that the different types of bonding interactions yield distinctively visually recognizable different topological patterns of the electron-pair density relaxation hole.     

On the inner and outer radial density functions

Starting from the two-electron radial density D ₂(r ₁,r ₂), a generalized partitioning of the one-electron radial density function D(r) into two component densities D _a (r) and D _b (r) is discussed for many-electron systems. The literature partitioning (Koga and Matsuyama Theor Chem Acc 115:59, 2006) of D(r) into the inner D _<(r) and outer D _>(r) radial densities is shown to minimize the average variance of the two component density functions D _a (r) and D _b (r). It is also found that the average radial separation halved, , constitutes a lower bound to the standard deviation σ of D(r).     

Inner and outer radial density functions in many-electron atoms

When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D _<(r) and outer D _>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V _en and the diamagnetic susceptibility χ _d are the sum of the inner and outer contributions. The electron–electron repulsion energy V _ee has an approximate relation with the minus first moment of the outer density D _>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities D _<(r), D _>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components to V _en and are also discussed for these atoms.     

Approximate radial functions

The approximate 4s and 3d radial wavefunctions of Richardsonet al. [J, chem. Physics 36, 1057 (1962)] for first-row transition-metal atoms and ions have been extended to additional electronic configurations. The results suggest several improvements in the 4s wavefunction parameters. Formulas are reported for extending the double- 3d wavefunctions over the range of atomic orbitalsd ¹ throughd ¹⁰. The results are intended for use in calculations of chemical bonding.     

Electron-pair distributions and chemical bonding

The electron-pair (intracule and extracule) densities of the first-row hydrides may be understood on the basis of the different chemical-bonding schemes in this series. Thus, the LiH pair densities show a strongly ionic nature and may be fairly well described by Li⁺H^?. The CH-pair densities, on the other hand, may be approximated by the promolecular superposition of a carbon and a hydrogen atom with an accumulation of pair density in the internuclear region signifying covalency. In the case of FH, its pair densities show a predominately ionic structure and are closer to those of F^? than to those of the promolecular superposition of a fluorine and a hydrogen atom. The slight deformation of longitudinal pair densities observed in FH is largely due to the presence of the H⁺. © 1994 John Wiley & Sons, Inc.     

Software to obtain spatially localized functions from different radial functions

We have developed an algorithm that enables simplified box orbital functions (SBO) to be obtained with optimized coefficients by fitting them to functions of many types. SBOs are a linear combination of radial functions useful in quantum chemistry calculations which can be spatially restricted (defined in $$0 le r < r_{0}$$ interval, and zero for $$r ge r_{0}$$). The algorithm proposed makes it possible to obtain the optimal radius $$r_{0}$$ and the coefficients of the SBOs of any number of terms from the functions to be fitted, but also allows the user to define a particular radius r and calculate the coefficients of the combination of terms of the SBOs. SBOs have proved to be useful in the calculation of molecular properties, and can reduce the complexity of the integral calculations, especially in huge chemical systems such as atomic clusters. These types of functions are also adequate for studying confined systems such as molecules in solution or big chemical systems such as atomic clusters. In addition, while carrying out the examples presented in this study we have tested the suitability of SBO functions to calculate molecular reactivity, showing that the basis functions provide results as good as the basis sets typically used for this kind of calculations.     

Electron-pair densities of singly charged atoms

For the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground states, spherically averaged electron-pair intracule (relative-motion) density h(u), extracule (center-of-mass-motion) density d(R), and their moments uⁿ and Rⁿ are examined, where u and R are the interelectronic distance and the center-of-mass radius of a pair of electrons, respectively. The intracule and extracule densities of all the 96 ions are found to be monotonically decreasing functions, as for neutral atoms. Approximate relations d(R)8h(2R) and uⁿ/Rⁿ2ⁿ are confirmed to be valid for the charged atoms as well.     

Effect of available volumes on radial distribution functions

The traditional method of analyzing solution structuring properties of solutes using atom–atom radial distribution functions (rdfs) can give rise to misleading interpretations when the volume occupied by the solute is ignored. It is shown by using the examples of O(4) in α- and β-D-allose that a more reliable interpretation of rdfs can be obtained by normalising the rdf using the available volume, rather than the traditional volume of a spherical shell. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 363–367, 1998     

Marginal electron density and density-difference functions

Summary For visual analysis of the density reorganization and distortion, the one-dimensional cut (x, y ₀,z ₀) and the two-dimensional cut (x, y, z ₀) of the three-dimensional electron density difference function (x, y, z) are frequently employed. However, these cut functions do not satisfy any sum rules in contrast to the original difference function (x, y, z). To avoid this difficulty, the use of the marginal electron density functions _x (x) and _xy (x, y) and their difference functions _x (x) and _xy (x, y) is proposed. The marginal densities are condensation of the three-dimensional density onto a particular plane or line of our interest, and they satisfy the sum rule (i.e., the conservation of the number of electrons) exactly. Some basic properties of the marginal electron density are clarified for typical diatomic molecular orbitals. An illustrative application is given for the bonding and antibonding processes in the H₂ system.     

Continuum radial dielectric functions for ion and dipole solution systems

Radial dielectric constant (permittivity) functions for ionic solute, polar solvent systems of the type obtainable from the Lorentz-Debye continuum field formulations are reexamined. Major interest is focused on the assumptions underlying these formulations and their expression in limiting field behavior. The analysis is extended to dipolar solutes and the importance of two types of corrections are evaluated. The first draws connections with the concept of the reaction field as employed by Onsager. This correction is shown to be significant as regards range of predicted saturation effects and for dipole moment self-consistency, for the same type molecule serving as solute and solvent. The second type correction involves the phenomenon of electrostriction whose effects appear much more limited both in range and on the intensity of the fields necessary for its observation. Application of the permittivity functions developed to compute modified Born model hydration energies for a variety of ions is illustrated. Excellent asymptotic approximations for all radial permittivity equations of interest are also presented which should enhance their future utility.     

Eight-point density correlation functions in lennard-jones fluids

Molecular dynamics calculations of components of the eight-point four-time density correlation function around a probe mulecule in a Lennard-Jones host fluid have been made. This function has important consequences for the interpretation of line broadening and other collision-induced spectroscopic experiments.     

Symmetrical radial distribution functions in the potentially theory of electrolyte solutions

A symmetrical radial distribution function g_ij in electric double layer theory has been proposed by Féat and Levine. We adopt their work to the electrolyte bulk to show how a symmetrical g_ij may be obtained in the potential theory of electrolyte solutions.     

Cubic response functions in time-dependent density functional theory

We present density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively, the quasienergy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly-through the time dependence of the density itself rather than through the form of the exchange-correlation functionals-it generalizes previous time-dependent implementations in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange correlation at the generalized-gradient approximation level and fractional exact Hartree-Fock exchange. With our implementation the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. As illustration we explore the use of various combinations of functionals for applications of nonlinear optical hyperpolarizabilities of a few centrosymmetric systems; molecular nitrogen, benzene, and the C(60) fullerene. Considering that vibrational, solvent, and local field factors effects are left out, we find in general that very good experimental agreement can be obtained for the second dynamic hyperpolarizability of these systems. It is shown that a treatment of the response of the density beyond the local density approximation gives a significant effect. The use of different functional combinations are motivated and discussed, and it is concluded that the choice of higher order kernels can be of similar importance as the choice of the potential which governs the Kohn-Sham orbitals.     

New evaluation of reconstructed spatial distribution function from radial distribution functions

Although three dimensional (3D) solvation structure is much more informative than one dimensional structure, its evaluation is difficult experimentally and theoretically. In our previous Communication [Yokogawa et al., J. Chem. Phys. 123, 211102 (2005)], we proposed a new method to present reconstructed spatial distribution function (RC-SDF) from a set of radial distribution functions (RDFs). In this article, we successfully extended the method more accurately with new basis sets. This new method was applied to two liquid solvation structures, methanol and dimethyl sulfoxide, as examples. Their RC-SDFs evaluated here clearly show that the former solvation structure is well defined while the latter one is broad, which agrees well with the SDFs calculated directly from molecular dynamics simulations. These results indicate that the method can reproduce well these 3D solvation structures in reasonable computational cost.     

The effective fragment potential: small clusters and radial distribution functions

The effective fragment potential (EFP) method for treating solvent effects provides relative energies and structures that are in excellent agreement with the analogous fully quantum [i.e., Hartree-Fock (HF), density functional theory (DFT), and second order perturbation theory (MP2)] results for small water clusters. The ability of the method to predict bulk water properties with a comparable accuracy is assessed by performing EFP molecular dynamics simulations. The resulting radial distribution functions (RDF) suggest that as the underlying quantum method is improved from HF to DFT to MP2, the agreement with the experimental RDF also improves. The MP2-based EFP method yields a RDF that is in excellent agreement with experiment.     

Application of density functional theory to capillary phenomena in cylindrical mesopores with radial and longitudinal density distributions

In this paper, we applied a version of the nonlocal density functional theory (NLDFT) accounting radial and longitudinal density distributions to study the adsorption and desorption of argon in finite as well as infinite cylindrical nanopores at 87.3 K. Features that have not been observed before with one-dimensional NLDFT are observed in the analysis of an inhomogeneous fluid along the axis of a finite cylindrical pore using the two-dimensional version of the NLDFT. The phase transition in pore is not strictly vapor-liquid transition as assumed and observed in the conventional version, but rather it exhibits a much elaborated feature with phase transition being complicated by the formation of solid phase. Depending on the pore size, there are more than one phase transition in the adsorption-desorption isotherm. The solid formation in finite pore has been found to be initiated by the presence of the meniscus. Details of the analysis of the extended version of NLDFT will be discussed in the paper.     

Approximate density matrices and wigner distribution functions from density,kinetic energy density,and idempotency constraints

The Wigner distribution function and the corresponding density matrix are calculated using a form for the distribution function suggested by maximization of the entropy. Wigner functions and density matrices are determined by imposing conditions of idempotency on the density matrix. Exchange energies and Compton profiles calculated from density matrices obtained by imposing the idempotency constraints are compared with the results of calculations using the Hartree–Fock density matrix and a Gaussian approximation for the density matrix for H and the noble gases He through Xe. Compton profiles from Wigner functions with idempotency constraints show improvements over the Gaussian approximation for the lighter atoms, but do not show significant changes for the heavier atoms. Exchange energies from density matrices with idempotency constraints show improvements over the Gaussian approximation except for the heavier atoms Kr and Xe.