Closed-shell Atoms Research Articles

A spatial restriction method in the theory of many-electron systems. II. The ground state of dielectric crystals

The method of spatial restriction, developed in Part I of this work, is used to study crystals consisting of closed-shell atoms and ions. The spatial restriction imposed by permutation relations on the motion of outer shell electrons has been earlier reduced to a zero boundary condition on a spherical surface of radius r0. The r0 values obtained in Part I of this work from the ionization potentials of isolated atoms are used to study 12 alkali halide crystals, with calculated lattice constants differing from the experimental values by less than 2%. Similar calculations are performed for rare-gas crystals, with a pressure dependence of compressibilities in good agreement with experimental data in a wide range of pressures.

Semi-empirical Auger electron energies. I. General method and K-LL line energies

A general semi-empirical method has been developed to calculate Auger electron energies within an intermediate-coupling framework. The required Slater integral values and an adiabatic relaxation correction term are obtained from polynomial expressions determined by least-squares fitting to relativistic atomic Hartree-Fock data calculated for closed-shell atoms. This information is combined with experimental electron subshell binding energy data to determine the line energies. Unlike all previous semi-empirical approaches the model proposed does not require any fitting to experimental Auger energies. Furthermore the method is applicable to all classes of Auger transitions. The results for K-LL energies determined within an intermediate-coupling framework, with and without configuration-interaction effects being included, are presented. The predictions are in good agreement with known experimental values.

Z−1-expansion calculation of two-particle correlations in atoms

A calculation of the expectation value of the mass-polarization operator $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ for the two-electron atom is made using the ${Z}^{\ensuremath{-}1}$ perturbation expansion in order to determine the usefulness of this approach in calculating two-particle correlations in atoms. Evaluation of the first-order contribution to $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ yields a value in good agreement both with an earlier value determined by another method and with the value which may be deduced from calculations using very accurate wave functions. An interesting feature of the calculation is the finding that the summations are dominated by contributions from matrix elements which involve continuum states, with more than 85% of the total coming from these contributions. The second-order contribution is then obtained by demonstrating that both first- and second-order contributions are included in the first-order formula if accurate uncorrelated wave functions for the ground state are used in the matrix elements, rather than the bare hydrogenic orbitals. After observing that most of the greater accuracy of such wave functions arises from inclusion of electronic screening, the first-order calculation is repeated using simple screened hydrogenic orbitals in the matrix elements. The values of $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ obtained by means of this simple calculation for various $Z$ values agree quite well with those calculated by Hart and Herzberg using accurate 20-term wave functions. The conclusion which is drawn from this calculation is that this method appears to be relatively simple and quite promising for calculating certain types of correlation effects, such as those resulting from double excitation or ionization of valence or inner-shell electrons. As examples of this, qualitative estimates are made of the behavior of the double photoionization cross sections of the remainder of the He isoelectronic sequence ($Z\ensuremath{\ge}3$) and of other closed-shell atoms.

Charge-transfer energy in closed-shell ion–atom interactions

The importance of charge-transfer energy in the interactions between closed-shell ions and atoms is investigated. Ab initio calculations on H+–He and Li+–He are used as a guide for the construction of approximate methods for the estimation of the charge-transfer energy for more complicated systems. For many alkali ion–rare gas systems the charge-transfer energy is comparable to the induction energy in the region of the potential minimum, although for doubly charged alkaline-earth ions in rare gases the induction energy always dominates. Surprisingly, an empirical combination of repulsion energy plus asymptotic induction energy plus asymptotic dispersion energy seems to give a fair representation of the total interaction, especially if the repulsion energy is parameterized, despite the omission of any explicit charge-transfer contribution. More refined interaction models should consider the charge-transfer energy contribution.

Relation between charge and force parameters of closed-shell atoms and ions

Combining rules for parameters which characterize the short-range repulsive forces between closed-shell atoms and ions are used to determine the radius and softness for a number of alkali and halide ions and rare-gas atoms from scattering and spectroscopic data. A corresponding set of charge radii and softnesses are determined from atomic charge densities calculated in both the relativistic and nonrelativistic Hartree–Fock approximations. These characteristic atomic parameters are compared, and it is found that, for species with the same charge belonging to the same column of the periodic table, the relations between charge and force radii and between charge and force softnesses are very nearly linear. The radial distance dependence of the softness is examined, and some areas where further work is needed to follow up this empirical study are indicated.

Hartree–Fock and Gordon–Kim interaction potentials for scattering of closed-shell molecules by atoms: (H2CO,He) and (H2,Li+)

The Gordon–Kim (GK) electron gas model for calculating the forces between closed-shell atoms and molecules is applied to the He–H2CO and Li+–H2 systems. GK interaction energies are computed following the original theory and also including the self-energy correction suggested by Rae. GK interaction energies, neglecting correlation, are found to be in qualitative accord with Hartree–Fock (HF) interaction energies for the two systems. However, quantitative discrepancies are noted which are possible sources of error if GK potential energy surfaces are used to compute accurate scattering cross sections.

Dispersion energies in systems of two closed shell atoms and/or ions

A semiempirical device for the calculation of the coefficients in the dispersion energy expansion for interaction between two closed shell atoms or ions is described. Summation is over all electrons. Averages of powers of electron radial coordinates are calculated with Slater-type functions. Effective quantum numbers and screening constants were determined so as to reproduce a set of like-atom rare gas dipole–dipole coefficients. These are used to extend the calculations to coefficients for unlike rare gas and ion–ion interactions, to higher multipole coefficients for those interactions, and to polarizabilities and diamagnetic susceptibilities. The effective one level energies, numbers of polarizable electrons, and accuracy of combination rules, all as determined by these calculations, are examined. Since the principal interest here is extension to ionic crystals, discussion of a questionable lattice sum for dipole–dipole interactions is appended.

Density functional theory of interacting closed shell systems. II. The determination of charge densities and the Hellman–Feynman theorem

The density functional formalism and the additive density approximation are used to obtain the nonadditive contribution to the electron density of a set of closed shell atoms in the absence of external fields. It is shown that due to the additive density approximation, the Hellman–Feynman theorem is violated when the force on the nucleus is examined. The reason for this violation is discussed.

Dispersion forces at arbitrary distances

The formalism of Boehm and Yaris is used to evaluate explicitly the leading term of the London dispersion force between closed-shell atoms. Instead of using the usual multipole expansion, which breaks down at intermediate internuclear distances, an analytic representation of the Born amplitude together with a general angular momentum analysis is used. As a result expressions are obtained which reduce to the usual dispersion forces at large distances and are finite at all distances.

Extension of Koopmans’ theorem. II. Accurate ionization energies from correlated wavefunctions for closed-shell atoms

Extended Koopmans’ theorem ionization energies are presented for MC−SCF wavefunctions of the 1s2 2s2 (1S) ground state isoelectronic series, Li− through O4+, and also for some larger CI wavefunctions of Be and B+. Correlation in these reference states reduces the error in the Koopmans’ ionization energy for the 2s electron to approximately 0.01−0.08 eV (1/100−1/20 the SCF Koopmans’ error) in all states except the negative ion, for which the MC−SCF extended Koopmans’ error of 0.25 eV was of the same magnitude but of opposite sign to the SCF error. Our extended Koopmans’ energies for 1s ionization were only slightly better than the corresponding SCF values. Koopmans’ theorem does not yield a 2p ionization energy (1s2 2s2 → 1s2 2p), but the extended Koopmans’ theorem yields an ionization energy whose error is about one−third the SCF 2s error.

Note on the symmetry of perturbed Hartree-Fock and X -α wavefunctions

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Unified theory for the intermolecular forces between closed shell atoms and ions

A simple theoretical model for intermolecular forces is developed in such a way that it includes simultaneously the effects of short-range and long-range interactions. At long distances the calculated intermolecular forces reduce to the usual dispersion and induction effects. At short distances, the results approach those of the short-range interaction, which in the present calculation followed our previous work [J. Chem. Phys. 56, 3122 (1972)]. The results of the method are in good agreement with available experimental results for closed shell atom-atom and ion-atom interactions, over the whole range of internuclear distances.

Radial correlation in atoms

A method is described to calculate correlation energy in atoms. The total wavefunction of ann-electron system is expressed as a linear combination of products ofn one-electron basis orbitals. This function gives a correlated description of the system. Under suitable restrictions it reduces to DODS and to splitshell description of closed-shell atoms or molecules. Energies of He atom, He-like ions and also of H− ion have been calculated including radial correlation only. The calculated electron affinity of H is better than earlier split-shell calculations. The result for He shows that the energy limit for radial correlation has been attained. For the other 2-electron ions radial correlation alone explains about one-third of the total correlation.

Simplified version of the random-phase-approximation-with-exchange method

By comparing the random-phase-approximation-with-exchange (RPAE) method of Amusia et al. for the calcuation of photoionization cross sections from closed-shell atoms with the reaction-matrix method, we obtain a simplified version of the RPAE by which we calculate the dipole-transition amplitude much as in the reaction-matrix method, yet with ground-state correlation effects included simultaneously. Calculation of photoionization cross sections for the $3p\ensuremath{\rightarrow}\ensuremath{\epsilon}d$ transition of Ar and $4d\ensuremath{\rightarrow}\ensuremath{\epsilon}f$ transition of Xe by this method yields a substantial improvement on the calculation by the reaction-matrix method and good agreement with RPAE and with experiment.

Study of the potential curves of xenon with other rare gas atoms

The model for the forces between closed shell atoms recently developed by Gordon and Kim is applied to the ground state potential energy curves of Xe–Xe, Xe–He, Xe–Ne, Xe–Ar, and Xe–Kr. Agreement between experimentally determined values of the distance between the nuclei at the potential minimum and the depth of the potential well is quite good considering the simplicity of the calculation.

Local Exchange Potential for the Relativistic Hartree-Fock Equations

An exact local representation of the relativistic Hartree-Fock exchange potential for electrostatic interactions is presented for closed-shell atoms. This single-particle potential depends upon all relativistic single-particle quantum numbers except the magnetic quantum number $m$. This exchange potential should be useful for relativistic atomic Hartree-Fock calculations.

Direct calculation of ionization potentials of closed-shell atoms and molecules

Vertical ionization potentials, electron affinities and information about quasi-particles can be obtained by using the technique of the single-particle propagator. The expansion of the self-energy part up to third order perturbation theory can be evaluated numerically, but does not lead, in most cases, to satisfying results. A theoretical and numerical analysis of the diagrammatic expansion of the self-energy part requires the introduction of a renormalized interaction and renormalized hole and particle lines.

Effective van der Waals interaction between rare gas atoms in a dense classical fluid

The authors examine the theory of the effective long range interaction between two closed shell atoms in a dense liquid. An approximate theory is developed in which the effective van der Waals constant Aeff, relative to that in free space, say A, is expressed in terms of the three-atom correlation function g3 in the liquid. Results are worked out using a Kirkwood form for g3. It appears from their work that the main dependence is on the fluid density.

Z-expansion coefficients and the number of electrons

It is shown that in a closed shell atom with a large number of electrons, N, the zero- and first-order Z-expansion coefficients are given approximately by E0(N) approximately=- beta N1/3+1/2-1/18 beta 2N-1/3 and by E1(N) approximately=1/2 beta N4/3, where beta =(3/2)1/3.

Many-Body Green's Functions for Finite, Nonuniform Systems: Applications to Closed Shell Atoms

Ab initio calculations of natural orbitals, ionization potentials, and total ground state energies for the closed shell systems helium and beryllium are given using the technique of many-body Green's functions. The necessary formalism for application of the Green's function theory to finite, nonuniform, many-body systems is developed following the work of Layzer; connections with standard perturbation theory are made. The natural orbitals obtained were of high quality. Analysis of the diagrammatic expansion of the Green's function led to the surprising result that the main effect of the infinite-order summations implicit in the solution of Dyson's equation was to ``renormalize'' the second-order perturbation corrections.