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The local and regional stabilities in clusters of water, methanol, and formaldehyde up to the tetramers have been analyzed from an atomic energy perspective. We optimize structures at the MP2/6-311++G(d,p) level with some CCSD(T)/6-311++G(d,p) single point energies, and then decompose the electronic densities into atomic parts using the atoms in molecules (AIM) approach. We consider the changes in atomic energy in the clusters vs. the isolated monomer. This method of analysis allows us to reveal the variety of stabilities within these hydrogen-bonded clusters, including indications of cooperative, anti-cooperative, and non-cooperative interactions. Cooperatively interacting clusters have increasing stability at the atomic level as the cluster size grows. This is not observed in the anti- and non-cooperative arrangements of water and formaldehyde clusters. The cooperativity in methanol clusters is dominated by the OH regions, with negligible energy change in the methyl regions. Formaldehyde clusters, including the lowest minimum “bucket” cluster, do not show significant cooperativity. Atomic energy analysis is supported with bond critical point data as well as charge and geometric values. We represent the local stability in the clusters using a simple visual approach that allows areas of increased or decreased stability to be easily interpreted.

The mutual interaction between beryllium bonds and halogen bonds within H2Be···FCl···Base complexes, where Base includes a wide set of N- and O-containing Lewis bases, has been studied at the M06-2X/6-31+G(d,p) level of theory. The reliability of this theoretical model was assessed by comparison with ab initio CCSD/aug-cc-pVTZ reference calculations. Cooperative effects were investigated within the framework of the atoms in molecules theory (AIM) by analyzing the topology of the electron density and the changes in the atomic energy components. The decomposition of the total stabilization energy into atomic components is found to be a very reliable tool to describe halogen bond interactions. Both the topological analysis of the electron density and the changes in the atomic energy components of the binding energy show the existence of strong cooperative effects between beryllium and halogen bonds, which are in some cases very intense. In general, there is a correlation between the intrinsic basicity of the Lewis base participating in the halogen bond and the resulting cooperativity in the sense that the stronger the base, the larger the cooperative effects.

Atomic energies are used to describe local stability in eight low-lying water hexamers: prism, cage, boat 1, boat 2, bag, chair, book 1, and book 2. The energies are evaluated using the quantum theory of atoms in molecules (QTAIM) at MP2/aug-cc-pVTZ geometries. It is found that the simple, stabilizing cooperativity observed in linear hydrogen-bonded water systems is diminished as clusters move from nearly planar to three-dimensional structures. The prism, cage, and bag clusters can have local water stabilities differing up to 5 kcal mol(-1) as a result of mixed cooperative and anticooperative interactions. At the atomic level, in many cases a water may have a largely stabilized oxygen atom but the net water stability will be diminished due to strong destabilization of the water's hydrogen atoms. Analysis of bond critical point (BCP) electron densities shows that the reduced cooperativity results in a decrease in hydrogen bond strength and an increase in covalent bond strength, most evident in the prism. The chair, with the greatest cooperativity, has the largest average electron density at the BCP per hydrogen bond, whereas the cage has the largest total value for BCP density at all hydrogen bonds. The cage also has the second largest value (after the prism) for covalent bond critical point densities and an oxygen-oxygen BCP which may factor into the experimentally observed stability of the structure.

The molecular structure can be defined quantum mechanically thanks to the theory of atoms in molecules. Here, we report a new molecular model that reflects quantum mechanical properties of the chemical bonds. This graphical representation of molecules is based on the topology of the electron density at the critical points. The eigenvalues of the Hessian are used for depicting the critical points three-dimensionally. The bond path linking two atoms has a thickness that is proportional to the electron density at the bond critical point. The nuclei are represented according to the experimentally determined atomic radii. The resulting molecular structures are similar to the traditional ball and stick ones, with the difference that in this model each object included in the plot provides topological information about the atoms and bonding interactions. As a result, the character and intensity of any given interatomic interaction can be identified by visual inspection, including the noncovalent ones. Because similar bonding interactions have similar plots, this tool permits the visualization of chemical bond transferability, revealing the presence of functional groups in large molecules.

The local value of the single-particle momentum provides a direct three-dimensional representation of bonding interactions in molecules. It is given exclusively in terms of the electron density and its gradient, and therefore is an ideal localized electrons detector (LED). The results introduced here extend to molecular systems our study of the single-particle local momentum in atomic systems (Bohorquez and Boyd in J Chem Phys 129:024110, 2008; Chem Phys Lett 480:127, 2009). LED is able to clearly identify covalent and hydrogen bonding interactions by depicting distinctive regions around the bond critical points, emerging as a complementary tool in conventional atoms in molecules studies. The local variable we introduce here is an intuitively interpretable 3D electron-pairs locator in atoms and molecules that can be computed either from theoretical or experimentally derived electron densities.

We propose a general definition of the radius of an atom in terms of its ionization energy. A relationship between these two fundamental properties is derived from the radial distribution function and the exponential decay of the electron density. Strong correlations with well-known atomic radii lead us to conclude that we have found a universally valid definition of the atomic radius, defined exclusively in terms of an intrinsic property, the ionization energy of an atom, which is equally available from experiments and theoretical computations.

We discuss the exact Coulomb hole for the ground state of the helium atom and helium-like ions. We find that the correlated wavefunction yields a smaller probability of finding the electrons at large separations than does the Hartree–Fock wavefunction, leading to the counterintuitive conclusion that correlation brings distant electrons closer together. This effect becomes less pronounced as the nuclear charge increases.

This study investigates the effect of basis set on the DFT-derived geometries and energies of halide ion-water complexes towards determining the optimum computational scheme for the study of larger clusters. It is shown that currently available methods are unable to adequately describe the anion-water clusters, mainly due to the residual BSSE even when the largest basis sets are used. This is also reflected in the fact that a basis-set converged geometry is not obtained even at the B3LYP/aug-cc-pV5Z level. There also appears to be a significant difference in geometry when counterpoise-corrected optimizations are performed at this level, suggesting that BSSE is the major contributor to the difficulties in modelling these systems.

We report a detailed characterization of the valence shell charge concentration (VSCC) of fluorine in methyl fluoride through the atoms in molecules theory. A comparison between the VSCCs of fluorine and neighboring elements (N, O, P, S, Cl) in the periodic table is also presented. The fluorine VSCC is shown to be tetrahedral and the most compact with the highest values of ρ and −∇ 2 ρ . Our results calculated at several post Hartree–Fock levels of theory using a large basis set show a remarkable insensitivity to the level of theory.

The study of ion-solvent clusters has been suggested as a means towards understanding the process of solvation of Group I and II metal cations. This study investigates the effect of basis set on the DFT-derived geometries and energies of Group I and II metal ion-solvent complexes towards determining the optimum computational scheme for the study of larger clusters. The ideal method is one that can generate BSSE-free geometries to a high degree of accuracy. It is also required that the binding energies obtained be capable of closely reproducing experimental data. Results indicate that B3LYP/6-311 + G(3df, 3pd) should be used, as it reaches the basis set converged geometry in addition to providing good binding energies. As well, the results at this level are BSSE-free, leading to greater confidence in the geometries and energies. The binding energies differ from available experimental data by less than 5 kJ mol-1, outstripping G2 and CCSD(T) at this basis set level.

The question of why the iron displacement out of the porphyrin plane is enhanced in quintet states of singly ligated iron-porphyrin complexes compared to lower spin states and unligated iron-porphyrin is addressed. The spatial size of the Fe2+ atom is analyzed with respect to different spin states, and it is shown that the ion size decreases with increasing spin state for the d6 electronic configuration. This contradicts the common belief that the iron out-of-plane location in the quintet state of ligated Fe(II)-porphyrins is due to an increased required space of the iron within the porphyrin ring. Therefore, the singlet, triplet, and quintet ground states of imidazole-ligated iron-porphyrin have been calculated employing density functional theory, and the relevant molecular orbitals have been analyzed. Additional comparison with the unligated iron-porphyrin molecules reveals that the enhanced doming in the quintet state is the result of a combination of the weakening of the iron-ring nitrogen bonds by oc...

The radial moments of the electron density, 〈rn〉, have been calculated for N2 and H2O at the quadratic configuration interaction (with singles and doubles) levels of theory by use of a variety of basis sets. An optimal basis set was chosen from the first set of calculations and then used with several conventional ab initio and density functional methods to compute the moments for a representative set of molecules: N2, CO, CH4, NH3, H2O, HF, SiH4, PH3, H2S, HCl, CH3OH, and CH3SH. The effects of solvation on the radial moments were also studied using the Onsager model and an extension of the Tomasi polarized continuum model.

The hyperfine coupling constants (HFCCs) in the hydroxyl radical are investigated through comparison of results obtained from a variety of quantum chemical methods. The couplings obtained from the multi-reference configuration interaction (MRCI) wave function, built upon the restricted open-shell Hartree–Fock (ROHF) reference determinant, are investigated in terms of the basis set, the configuration selection energy threshold, and the size of the reference space. Overall results which converge to the experimental couplings are obtained for hydrogen, but not for oxygen. In particular, the MRCI method shows no improvement over density functional theory (the B3LYP functional), for the calculation of Aiso(17O). On the other hand, results in excellent agreement with experiment are obtained through the use of the quadratic configuration interaction (QCISD) method based on the unrestricted HF (UHF) reference determinant with the identical basis sets. Examination of UHF and ROHF based coupled-cluster methods, CCS...

By analogy with the Carbo overlaplike similarity measure, a spin-density polarization index (SPI) is defined. This index gives the Euclidean distance between the α and β electron densities for an open-shell molecule. An extensive study of the O2 molecule reveals that the SPI is highly dependent on the theoretical method, but is less sensitive to the basis set provided flexible extended basis sets are used. The radial distribution of the O2 spherically averaged electron density also reveals that near the center of the bond there is an inversion of the α and β electron densities and that there is a minimum in the β electron density near the nuclei within the internuclear region. Finally, the SPI is calculated for a variety of small molecules and the results are compared with those obtained for O2.

A new semiquantitative measure of the difference between two electron density distributions is proposed. Specifically, the natural orbitals of an accurate electron density, such as those given by a configuration interaction method, are used to expand the Kohn–Sham orbitals of an approximate density functional method. The difference between the two density distributions is then reduced to a single number, the orbital-based density difference index (DDI). With the reference densities calculated from quadratic configuration interaction calculations including single and double substitutions, DDIs were obtained for four diatomic molecules using three basis sets and seven approximate functionals. Results are also included for an additional six small molecules with a flexible extended basis set and the same set of functionals. These results show that the DDI leads to conclusions that are consistent with those obtained previously by visual comparison of density difference plots. The orbital-based density differen...

The convergence of basis set contractions is examined at the multireference configuration interaction level with respect to the 14N isotropic hyperfine coupling constant in NH2, a case that is very sensitive to the choice of the basis set and the degree of correlation recovery. The contraction schemes studied include the atomic natural orbital (ANO) approach and two Hartree–Fock-based contraction methods: Raffenetti’s general contraction method and the segmented method. Contractions of a (13s8p2d) primitive set ranging from [4s3p2d] to [8s3p2d] in the s space and from [7s3p2d] to [7s6p2d] in the p space are employed. It is found that the ANO contraction yields the smoothest and fastest convergence, although all three contractions converge to the uncontracted results. Thus, the ANO contraction starts to converge at [6s3p2d] while the other two converge at [7s4p2d]. The correlation recovery of the core electrons versus that of the valence electrons is also studied. In contrast to previous speculation, it is...

The molecular quadrupole moments of benzene and naphthalene are calculated using a range of ab initio and density functional methods, with basis sets as large as aug-CC-PVTZ for benzene and CC-PVTZ for naphthalene. The quadrupole moments calculated using Hartree-Fock wavefunctions are lower than the experimentally determined values, and all DFT values are higher than the experimental values. The use of the hybrid B3P86 and B3PW91 functionals, with a large basis set, produce quadrupole moments which are within experimental error for these two small aromatic molecules.

The electron densities of H2, Li2, N2, O2, and F2 have been calculated by density functional theory (DFT) methods with various exchange and correlation functionals such as SVWN, BLYP, B3LYP, BP86, B3P86, and B3PW91 and compared with the results from quadratic configuration interaction calculations including all single and double substitutions (QCISD) using Dunning's correlation consistent polarized valence double- and triple-ζ basis sets augmented with the corresponding diffuse functions (aug-CC-PVDZ and aug-CC-PVTZ). The DFT methods yield electron densities and Laplacians of the densities in good agreement with the QCISD results. The gradient-corrected functionals improve upon the densities calculated by use of the local spin density approximation (SVWN) relative to the corresponding QCISD densities. All gradient-corrected functionals are generally insensitive to the grids used in the study. The basis set, however, has a significant effect on the electron density and shows a strong dependence on the choi...

The energy surface of CaOH has been studied and an isomer of the form of HCaO with an electronic state of 2Σ+ symmetry has been found. It is linear with H–Ca and Ca–O bond lengths of 2.021 and 2.002 A, respectively, at the CISD level with a basis set of triple‐zeta plus double polarization quality. The overall electronic structure is largely ionic and can be described as H(1s2)Ca(4s0)O(2s22p2x2p2y2p1z) (H−Ca2+O−). The minimum on the energy surface is well defined and the harmonic vibrational frequencies have been calculated. The O–H bond length and stretching frequency and the bending frequency of CaOH have also been calculated and compared with experiment.

The thermal motion of CD+4 in the gas phase at 4 and 40 K is studied by means of a molecular dynamics method using the MP2/6‐31G(d,p) force field. The thermal motion of CD+4 in the temperature range from 4 to 40 K does not involve a deuterium permutation on the picosecond time scale. The electronic energy fluctuations of CD+4 due to thermal motion are less than 0.7 kcal/mol at 4 K and 1.6 kcal/mol at 40 K, which is less than the energy barrier (3.4 kcal/mol at MP2 level) required for the permutation between two energy minimum C2v structures. On the basis of these calculations it is suggested that the observation of four equivalent hydrogen atoms in the ESR spectrum of CH+4 can be attributed to fast quantum tunneling, with an estimated rate of 4.3×106 s−1 for CH+4 and 1.6×103 s−1 for CD+4.

The effect of using ‘geometry-adapted’ atomic charges in computer simulations is studied by carrying out molecular dynamics simulations of trans-stilbene (tS) in methanol solution. The atomic charges on tS have been calculated at the Hartree-Fock/6-31G(d, p) level according to the Merz-Singh-Kollman scheme as a function of the two torsional angles φ1 and φ2 across the phenyl groups and ethylenic carbons (Cph-Ce), fitted to analytical functions q i (φ1, φ2) and added to the force field for calculation of the Coulombic interactions. In spite of the non-polar character of tS, the quantum mechanically calculated atomic charges in the linkage region are observed to vary more than 0-1 |e| with respect to the changes of the torsional angles φ1 and φ2. Compared to reference molecular dynamics simulations, carried out simultaneously with conventional ‘fixed’ charges, there is a clear effect on the solvation structure when ‘flexible’ atomic charges on tS are used. It is expected that the effect of geometry...

The hyperfine structure of the 14NH2 radical is investigated by means of multireference single and double configuration interaction (MRCI) techniques. Particular attention is paid to the dependence of the coupling constants on the basis set, reference space, and configuration selection energy threshold. It is found that convergence can be obtained only if more than 83 reference configurations are included with an energy threshold of at least 10−7 hartree. With up to 126 reference configurations, an energy threshold smaller than 10−8 hartree and an uncontracted (13s8p2d/8s2p) basis set, the MRCI isotropic couplings (27.44 and −68.47 MHz for N and H, respectively) are in very good agreement with experimental data (27.9 and −67.2 MHz, respectively).

The interactions between an alkali metal atom (Li, Na, K) and acetylene/vinylidene are investigated by means of gradient corrected (‘PWP’) density functional theory. Particular emphasis is made on the determination of equilibrium geometries, energetics and atomic hyperfine structures of the MHCCH van der Waals addition complexes, and the MHCCH and MCCH2 charge-transfer complexes. The PWP optimized geometries are found to be similar to previous ab initio data, and the generated hyperfine structures are in excellent accord with experimental matrix isolation ESR data recorded at 4 K. In the charge transfer complex formation, the metal atoms are found to donate 0.6–0.8 electron from an excited p orbital to the antibonding CC π∗ orbital of the hydrocarbon fragment. The calculated trend in charge donation matches the order of increasing atomic s → p excitation energies, K < Li < Na. This is the first application of the present method to metal atom hyperfine structures, and the results are promising for future studies.

The electronic structures of the Rydberg and related molecules belonging to the series [Formula: see text], NH4, and [Formula: see text] and H3O+, H3O, and H3O− have been investigated by analysis of the electron densities generated by the Hartree–Fock method (HF), the quadratic configuration interaction method including single and double excitations (QCISD), the coupled-cluster method including double excitations (CCD), and the density functional method with the Becke–Perdew gradient correction functionals (DFT-BP). Adding an electron to the stable cation core ([Formula: see text] and H3O+) results in an increase in the bond length, a larger increase in the distance of the bond critical point (BCP) from the nitrogen atom than from the hydrogen atom, a decrease in the electron density at the bond critical point, and an increase of the Laplacian of the electron density at the BCP. These trends are consistent with a decrease in the electronegativity of the central atom and a weakening of the bond as electrons are added to diffuse Rydberg orbitals. There is little effect, however, when a second electron is added to form the double Rydberg molecules ([Formula: see text] and H3O−). For NH4, the preferred geometry corresponds to a high atomic energy associated with the hydrogen basin, and to a low kinetic energy and a high potential energy in the basin. The high atomic energy associated with the hydrogen basin is attributed to the dominant contribution of the electron–nuclear attraction energy. Similar effects are observed in the Jahn–Teller distorted molecules: [Formula: see text] and [Formula: see text]. The DFT-BP method yields results in qualitative agreement with the results of the QCISD and CCD methods; however, it underestimates the bond strengths of the Rydberg molecules relative to high-level ab initio methods.

The small and subtle effects of electron correlation in the Rydberg-like 3{sup 1}D and 3{sup 3}D states of the helium-like ions are studied using very accurate explicitly correlated wave functions. In particular, the effects of electron correlation on one-electron densities, probability distributions for the interelectronic distance, and their moments are examined with the help of density and Coulomb holes. Statistical electron correlation is examined with radial and angular correlation coefficients. The different methods of analysis provide complementary insight.

The variation of the charge and intracule densities of singly excited states of two‐electron ions with respect to spin multiplicity, nuclear charge, degree of excitation, and angular momentum quantum number is studied systematically. The n 1S, n 3S, n 1P, n 3P, n 1D, and n 3D states with n=3–6 are considered for all the ions from He through Ne8+ using highly accurate explicitly correlated wave functions. Special attention is paid to spin multiplicity differences, that is differences between the densities of a pair of states arising from the same electron configuration of the same ion.

Statistical electron correlation coefficients provide overall measures of the difference between the electron pair density and the product of one-electron densities. Explicitly correlated wave functions are used to compute radial and angular correlation coefficients for 29 low-lying states (n1S, n3S, n1P, n3P, n1D, and n3D with n < 7) of the two-electron ions from helium through Ne8+. This data base of 1566 correlation coefficients permits a systematic study of trends with respect to variation of nuclear charge, degree of excitation, angular momentum, and spin multiplicity. © 1993 John Wiley & Sons, Inc.

Extensive configuration‐interaction calculations with double‐zeta plus polarization and near triple‐zeta plus polarization basis sets are used to analyze the Jahn–Teller (JT) effect in the methane radical cation. Energy component analysis shows that the Jahn–Teller effect leads to a decrease in the expectation value of the electron‐nuclear attraction energy, an increase in the expectation value of the interelectronic repulsion energy, and an increase in the internuclear repulsion energy. These observations are consistent with a contraction of the electron cloud. The dominant factor in the −0.0550 hartree Jahn–Teller distortion (Td→C2v) in CH+4 is the −0.5262 hartree change in the electron–nuclear attraction energy. The differences in all energy components are large in relation to the JT distortion. Interelectronic repulsion plays a dominant role in determining the relative energies of the possible JT distorted structures, but electron correlation effects are relatively unimportant.

Multireference configuration interaction calculations for the low-lying electronic states of C2 (a 3Πu and A 1Πu) and O2 (X 3Σg−, a 1Δg) over a range of internuclear separations reveal that the singlet-triplet electron-electron repulsion energy differences tend to have the same sign as the singlet-triplet kinetic energy differences, while the corresponding electron-nuclear attraction energy differences tend to have the opposite sign at a given geometry. Adiabatic energy component differences for C2 and O2 show that the electron-electron repulsion in the triplet state is greater than in the singlet state, while there is more electron-nuclear attraction in the triplet than in the singlet state. The latter observations are consistent with the results for atomic states arising from the same configuration, but are contrary to the traditional explanation of Hund's rule.

Extensive configuration‐interaction calculations with double‐zeta plus polarization and near triple‐zeta plus polarization basis sets are used to analyze the Jahn–Teller (JT) effect in the methane radical cation. Energy component analysis shows that the Jahn–Teller effect leads to a decrease in the expectation value of the electron‐nuclear attraction energy, an increase in the expectation value of the interelectronic repulsion energy, and an increase in the internuclear repulsion energy. These observations are consistent with a contraction of the electron cloud. The dominant factor in the −0.0550 hartree Jahn–Teller distortion (Td→C2v) in CH+4 is the −0.5262 hartree change in the electron–nuclear attraction energy. The differences in all energy components are large in relation to the JT distortion. Interelectronic repulsion plays a dominant role in determining the relative energies of the possible JT distorted structures, but electron correlation effects are relatively unimportant.

Configuration interaction calculations have been carried out for low-lying singlet and triplet states of LiH({sup 1,3}{Sigma}{sup +}), BH({sup 1,3}II), NH({sup 1}{Delta}, {sup 3}{Sigma}{sup {minus}}), and FH({sup 1,3}II), with the aim of evaluating singlet-triplet energy component differences. These calculations represent the first systematic study of the quantum mechanical interpretation of Hund's rule in a series of molecules. While early molecular work on Hund's rule suggested that the electron-nuclear attraction may always be deeper in the triplet than in the singlet, exceptions are found in the present work. Also, in certain cases, where configuration mixing and molecular charge separation play a key role, the singlet electron-electron repulsion exceeds that of the triplet.

Extracule (centre of mass) and intracule (relative motion) pair densities are reported for the electronic ground states of the first-row hydrides, LiH, BeH, BH, CH, NH, OH and FH. For each type of two-particle coordinate and for each molecule, the radial, longitudinal and transverse densities are evaluated from self-consistent field molecular orbital wavefunctions constructed from 3-21G Gaussian basis sets. The radial intracule density I(r), the longitudinal intracule density L(i)(z) and the longitudinal extracule density L(e)(Z) evolve systematically from double maxima functions in LiH to single maximum functions in FH, whereas the extracule radial density E(R), the transverse extracule density T(i)(b) and the transverse extracule density T(e) (B) exhibit a single maximum in every case. These observations are explained in terms of the ionic character of the bond, the number of electron pairs, the probable orientations of the interelectronic vectors and the orientation of the vectors describing the centre of mass of an electron pair.

The aim was to investigate the relationship between the bond length and the electron density at the bond critical point in homonuclear X--X and Z--Z and heteronuclear C--Z bonds (X = Li-F, Z = Na-Cl). The d,rho(c) pairs were obtained from 472 target bonds in DFT-optimized (B3LYP/6-311+G(d,p)) small molecular species. These species were selected arbitrarily but with a view to maximize the range widths WR for each atom combination. It was found that (i) with one clear exception, the d(A - A) means (A = X or Z) correlate linearly with the bond lengths d(A(2)) of the respective diatomic molecules; (ii) the d(A - A) means correlate parabolically with n, the formal number of valence electrons in the atoms of the bond; and (iii) with increasing sample size N the ratio WR(rho(c))/WR(d) appears to converge toward a representation f [WR(rho(c))/WR(d)](N-->infinity) characteristic of A. Detailed analysis of the d,rho(c) relationship has shown that by and large simple power regression accounts best for the DFT data. The regression coefficients of d = arho(c) (-b) and rho(c) = alphad(-beta) (b, beta > 0) vary with n in a seemingly irregular manner but one that is consistent with simple chemical notions. The d(A(2)) can be approximated in terms of multilinear MO electron occupancies.

Analysis of the partial Fermi hole in the 23S state of helium and the ground states of Li and Be shows that the radius of the Fermi hole increases with the distance of the reference electron from the nucleus, and that there exists a non-zero lower limit for the radius. The limit increases in a regular manner for the two-, and three- and four-electron isoelectronic sequences. The radius of the Fermi hole in the ground states of the Li and Be isoelectronic sequences varies with the atomic number but is independent of the number of electrons. The latter observations stands in contrast to the differences in the sizes of the electron clouds of each isonuclear pair.

The pair distribution function h(r12;r1, γ) and the virial theorem are used to derive a general expression for the local contributions to the total correlation energy of an atom. A direct link between correlation effects and the correlation energy is obtained by use of G(r1, γ) and Γ(r1, y). The former is the probability associated with a given choice of r1 and γ, while the latter describes the local contribution to the correlation energy. Explicit calculations for the ground state of helium indicate that the angular dependence of the local contribution to the correlation energy is essentially independent of r1, whereas the local correlation energy shows a strong r1 dependence. The maximum contribution to the correlation energy occurs at intermediate values of γ where there is close agreement between the Hartree–Fock and exact densities.

The K–L intershell correlation effects for both singlet and triplet states arising from the 1s2s electronic configuration of the helium atom are analyzed in detail with the aid of the complete (nonreduced) interelectronic probability density and the local Coulomb hole functions. The calculations for both singlet and triplet states suggest the existence of nuclear Coulomb well channels similar to those recently described for the intrashell case of Berry and Krause. When the reference electron is close to the nucleus, the 2 1S Coulomb hole resembles the ground state Coulomb hole, whereas this feature is lost in the 2 3S state due to the effect of exchange correlation. Electron correlation is shown to reduce the nuclear shielding provided by the K‐shell distribution and, as a consequence, to lead to a relative shrinking of the outer shell density. Other differences between the ground state and excited state correlation effects are described.

Mean values of r 1 and r 12 for the ground and several excited states of the helium isoelectronic sequence are used to demonstrate that a simple scaling which superimposes the distribution function f(r 12) as a function of the atomic number leads to a similar result for the electron density distribution D(r1). On the basis of a screening interpretation of the scaling parameter δ, it is concluded that screening is greater in the singlet than the triplet state of a particular configuration, that screening is greater in the P states than the corresponding S states, and that the screening approaches the limiting value of 1 for the highly excited states. The perturbation expansions of Scherr and Knight are used to determine the limiting value of δ when Z→∞ and the relationship between the scaling parameter and the scale factor, chosen so that a trial wave function satisfies the virial theorem, is discussed. A brief discussion of the scaling of the Coulomb hole is presented.

The radius of the Coulomb hole in an atom or molecule is discussed in general terms and illustrated in detail for the ground state of the helium isoelectronic sequence. The calculations demonstrate a general tendency for the radius of the Coulomb hole to increase as a function of the distance of the reference electron from the nucleus and the existence of a nonzero lower limit (0.800 a.u. in He and 0.111 a.u. in Mg10+) for the radius of the Coulomb hole. This limit is slightly larger than the radial distance at which the radial density distribution reaches its maximum. By expressing the correlation energy of an atom in terms of certain one and two electron expectation values, the relationship between the correlation energy and the size and shape of the Coulomb hole is discussed. Detailed comparisons for the ground and 23S states of He are used to demonstrate the relationship.

An electrostatic model, according to which the electronegativity of an atom is identified with the electrostatic force between the effective nuclear charge and an electron at a distance equal to the relative radius of the atom, is used to calculate a nonempirical electronegativity scale for the first 54 elements. The relative radius is calculated from the free‐atom wave function by use of the density contour approach, while intergration of the radial density from the nucleus to the relative radius yields the effective nuclear charge. In contrast to the empirical methods, the nonempirical electrostatic method is not only applicable to all elements, but treats all elements equally. In addition to correlating well with empirical scales, the proposed model is consistent with the major conclusions of the density functional approach of Parr and his co‐workers. In particular, the model is easily extended to yield valence state electronegativities.

Analytical expansions for the radial density D(r), as calculated within the Hartree–Fock approximation, are reported for the neutral atoms from helium to xenon. As part of a brief discussion of one-electron properties which may be calculated from D(r) or the related density distribution ρ(r), expectation values for rn, n = −2, −1, 1, 2, 3, 4, and 5 are presented.

The density difference ΔD(r1), defined as the difference between the exact and the Hartree-Fock radial density functions is reported for the 2 3S and the 2 1S excited states of the helium atom. The density differences in both cases resemble the inverse of the helium ground-state density difference, but are more long-ranged due to the diffuseness of the excited states. The effect of electron correlation on one-electron distributions is also demonstrated by comparison of (r1n) expectation values and several indices of the relative sizes of atoms. Electron correlation leads to a contraction of the Hartree-Fock electron clouds.

Ab initio SCF MO calculations are reported for benzene, s -triazine, borazine and boroxine. The Laplacian of the charge density and the Mulliken population analysis procedure demonstrate that the delocalization of the π electrons decreases and the polarity of the ring bonds increases substantially as the atoms in the ring become more dissimilar. Several other properties, including distortion of the ring angles, puckering of the ring and nuclear quadrupolar coupling constants, emphasize the different chemical properties within the isoelectronic series.

The pair distribution functions evaluated for the 23S state of the helium isoelectronic sequence from the Hart and Herzberg correlated wave functions and those corresponding to the Hartree-Fock approximation are used to determine the shape of the corresponding Coulomb holes. As a consequence of a discontinuity in the Hartree-Fock solution between He and Li+, the Coulomb hole has a different shape for He than for Li+ and the other isoelectronic ions.

The effect of water-anion and water-cation interactions on the stretching and bending force constants of a water molecule was studied by ab initio calculations. The OH stretching force constant f, is decreased by water-anion and water-cation interactions, its decrease correlating linearly with the elongation of the O-H bond length. The H-O-H bending force constantj= is increased by anion attachment but appears to be almost unaffected by cation attachment. The observed near-constancy off, upon cation attachment is internreted as the result of a decrease due to C%H bond elongation and an approximately equal increase due to greater forces opposing the bending motion. and F- . HOH . F- (VI), which were optimized under C,, symmetry. The optimized geometries of V and VI correspond to potential minima only under the constraint of C,, symmetry; the two complexes are unstable with respect to dissociation into a free ion and an ion-molecule complex. METHOD OF CALCULATION

The spherically averaged intracule density I(r) is calculated with minimal and split-valence basis sets in the ground state of the hydrogen molecule. Restricted Hartree-Fock calculations indicate a high probability of interelectronic distances (r) of the order of 1 au as the internuclear separation (R) goes to infinity due to the incorrect dissociation of H2. The inadequacy of the restricted Hartree-Fock I(r) at large R is further illustrated by calculating (1/r) as a function of R. Configuration interaction calculations are shown to yield the correct qualitative form for I(r) at large R and when combined with the Hartree-Fock results yield the approximate Coulomb hole in H2 as a function of internuclear separation. Explicit formulae for the evaluation of I(r) and its moments with s-type Gaussian basis functions are included.

Potential curves for the ground and all excited singlet states which dissociate into the ground states of the atoms, Cl(2pu) and F(2Pu), or into the ionic species, Cl+(1Dg) and F−(1Sg), have been calculated by means of the multi-reference configu interaction (MRD CI) method. The Gaussian basis set consists of a total of 70 contracted functions, including Rydberg and bond functions. The calculations show that adequate treatment of Rydberg-valence mixing is essential for an understanding of the ClF electronic absorption spectrum. The calculated Do value for the ground state is 2.47 eV compared to the experimental value of 2.617 eV. Transitions of potential spectroscopic interest include the dipole allowed 1Σ+ → 1/gP transition, having a vertical transitio energy of 8.92 eV, and the allowed 1Σ+ → 1Σ+ transition, with a vertical transition energy of 10.3 eV.

The effect of electron correlation on the Politzer–Parr procedure for partitioning the electron density of an atom into core and valence regions is investigated by use of configuration interaction (CI) wave functions. Although the effect is generally small, substantial qualitative differences are observed between the 3, 4, and 6 electron cases. Analytical Hartree–Fock (HF) wave functions are used to reveal the trends in isoelectronic and isonuclear sequences. Correlated wave functions, which explicitly include the interelectronic distance, are used to study the effect of spin and electronic excitation on the populations and dimensions of the core and valence regions in the 1sns excited states of He.

The density difference ΔD (r1), defined as the difference between the exact radial density function Dex(r1), or our best approximation to it, and the best uncorrelated, that is, the Hartree–Fock, radial density function DHF(r1), is reported for the He, Li, and Be isoelectronic sequences and to a lesser accuracy for the carbon atom. By means of an analysis of the Be density difference in terms of K and L shell contributions some insight is gained for the evolution of the density difference through the He, Li, and Be sequence. In general the effect of electron correlation on the radial density function D (r1) and, of course, on the related density distribution ρ (r1) diminishes as the charge cloud becomes less diffuse. The effect of electron correlation on one‐electron distributions is also demonstrated by comparison of 〈rn1〉 expectation values. Analytical Hartree–Fock wavefunctions are reported for H− (energy =−0.4879 2973 a.u.) and Li− (energy =−7.428 2317 a.u.).