Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

High‐precision Hy–CI variational calculations for the ground state of neutral helium and helium‐like ions

Hylleraas–configuration interaction (Hy–CI) method variational calculations with up to 4648 expansion terms are reported for the ground ¹S state of neutral helium. Convergence arguments are presented to obtain estimates for the exact nonrelativistic energy of this state. The nonrelativistic energy is calculated to be ?2.9037 2437 7034 1195 9829 99 a.u. Comparisons with other calculations and an energy extrapolation give an estimated nonrelativistic energy of ?2.9037 2437 7034 1195 9830(2) a.u., which agrees well with the best previous variational energy, ?2.9037 2437 7034 1195 9829 55 a.u., of Korobov (Phys Rev A 2000, 61, 64503), obtained using the universal (exponential) variational expansion method with complex exponents (Frolov, A. M.; Smith, V. H. Jr. J Phys B Atom Mol Opt Phys 1995, 28, L449). In addition to He, results are also included for the ground ¹S states of H^?, Li⁺, Be⁺⁺, and B⁺³. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Calculations of Isoelectronic Series of He Using Noninteger n‐Slater Type Orbitals in Single and Double Zeta Approximations

Using integer and noninteger n-Slater type orbitals in single- and double-zeta approximations, the Hartree-Fock-Roothaan calculations were performed for the ground states of first ten cationic members of the isoelectronic series of He atom. All the noninteger parameters and orbital exponents were fully optimized. In the case of noninteger n-Slater type orbitals in double zeta basis sets, the results of calculations obtained are more close to the numerical Hatree-Fock values and the average deviations of our ground state energies do not exceed 2×10^－6 hartrees of their numerical results.     

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post‐processing analysis of quantum‐chemical electron densities is described. The code can work with Slater‐ and Gaussian‐type basis functions of arbitrary angular momentum. It has been applied to explore the basis‐set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple‐zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis‐set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple‐zeta/higher level with Gaussian/Slater functions. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Long- and intermediate-range interaction in three lowest sigma states of the HeH+ ion

Accurate variational energies have been calculated for three lowest sigma states of the HeH⁺ ion. This includes the ground state (5 ≤ R ≤ 9 a.u.) which dissociates into He + H⁺, as well as the A ¹Σ⁺ state (4 ≤ R ≤ 10) and the a ³Σ⁺ state (3 ≤ R ≤ 10) which both dissociate into He⁺ + H. The variational results are compared with those obtained using a perturbation theory expansion.     

One‐particle density matrix functional for correlation in molecular systems

Based on the analysis of the general properties for the one‐ and two‐particle reduced density matrices, a new natural orbital functional is obtained. It is shown that by partitioning the two‐particle reduced density matrix in an antisymmeterized product of one‐particle reduced density matrices and a correction Γ^c we can derive a corrected Hartree–Fock theory. The spin structure of the correction term from the improved Bardeen–Cooper–Schrieffer theory is considered to take into account the correlation between pairs of electrons with antiparallel spins. The analysis affords a nonidempotent condition for the one‐particle reduced density matrix. Test calculations of the correlation energy and the dipole moment of several molecules in the ground state demonstrate the reliability of the formalism. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 317–323, 2003     

Configuration interaction calculations on the 2P ground state of boron atom and C+ using Slater orbitals

Configuration Interaction (CI) calculations on the ground ²P state of boron atom are presented using a wave function expansion constructed with L‐S eigenfunction configurations of s‐, p‐, and d‐Slater orbitals. Two procedures of optimization of the orbital exponents have been investigated. First, CI(SD) calculations including few types of configurations and full optimization of the orbital exponents led to the energy ?24.63704575 a.u. Second, full‐CI (FCI) calculations including a large number of configuration types using a fixed set of orbital exponents for all configurations gave ?24.63405222 a.u. using the basis [4s3p2d] and 2157 configurations, and to an improved result of ?24.64013999 a.u. for 3957 configurations and a [5s4p3d] basis. This last result is better than earlier calculations of Schaefer and Harris (Phys Rev 1968, 167, 67), and compares well with the recent ones from Froese Fischer and Bunge (personal communication). In addition, using the same wave functions, CI calculations of the boron isoelectronic ion C⁺ have been performed obtaining an energy of ?37.41027598 a.u. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Radial limit of lithium revisited

Configuration interaction (CI) calculations are carried out for the ground state of lithium using a thoroughly optimized basis set of s-type Slater functions. They establish that the radial limit of the nonrelativistic energy of the ground ²S state of lithium is no higher than −7.448666443E_h. Thus, radial correlation accounts for 35.2% of the total correlation energy. The radial CI wave function predicts a significantly more accurate Fermi contact parameter than the Hartree-Fock wave function. However, the imbalanced treatment of electron correlation in the radial CI wave function leads to an excessively diffuse electron density that is worse than that of the Hartree-Fock wave function. © 1997 John Wiley & Sons, Inc.     

Convergence of the partial wave expansion of the He ground state

The configuration interaction (CI) method, using a very large Laguerre orbital basis, is applied to the calculation of the He ground state. The largest calculations included a minimum of 35 radial orbitals for each ? ranging from 0 to 12, resulting in basis sets in excess of 400 orbitals. The convergence of the energy and electron–electron δ‐function with respect to J (the maximum angular momenta of the orbitals included in the CI expansion) were investigated in detail. Extrapolations to the limit of infinite angular momentum using expansions of the type ΔX_J = A_X[J + 1/2]^?p + B_X[J + 1/2]^?p?1 + …, gave an energy accurate to 10^?7 Hartree and a value of 〈δ〉 accurate to about 0.5%. Improved estimates of 〈E〉 and 〈δ〉, accurate to 10^?8 Hartree and 0.01%, respectively, were obtained when extrapolations to an infinite radial basis were done prior to the determination of the J → ∞ limit. Round‐off errors were the main impediment to achieving even higher precision, since determination of the radial and angular limits required the manipulation of very small energy and 〈δ〉 differences. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Pair density related to one‐electron information for the ground state of spin‐compensated two‐electron systems

The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr², with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Variational functions of the padé type—application to the ground state of the helium atom

It is shown that trial functions involving Padé approximants yield satisfactory results for the ground state of the helium atom. In particular, the five-parameter form reproduces the best variational function of the type Ψ = e^–zs?(u) obtained numerically, to a remarkable extent.     

Variational calculations with a hyperspherical basis on atomic helium

We have recently formulated an expansion of the N electron wavefunction in an appropriate set of harmonics on the 3 N-dimensional hypersphere. Angular correlation appears in the usual way, while radial correlation appears as a generalized angular correlation. Calculations on¹S helium have been performed to explore the convergence of this expansion. Energies for various angular approximations have been compared with Bunge's angular limits and show a fractional error <3.5× 10^–4. A theoretical contraction procedure is shown to usefully reduce basis size without forfeiting accuracy.Supported in part by a research grant to the Johns Hopkins University from the National Science Foundation.     

Application of Löwdin's canonical orthogonalization method to the Slater‐type orbital configuration‐interaction basis set

We apply Löwdin's canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater‐type orbital configuration‐interaction (STO‐CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO‐CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that Löwdin's canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S‐wave model He atom, and then the real two‐electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two‐electron systems are obtained in reasonably high accuracy using over‐redundant STO‐CI bases, however, without using extended high‐precision technique. © 2015 Wiley Periodicals, Inc.     

Calculation of Negative Ions of B,C, N,O and F Using Noninteger n Slater Type Orbitals

Combined Hartree‐Fock‐Roothaan calculations have been performed using noninteger n Slater type orbitals for the ground states of the lowest electron configurations 1s²2s²2pⁿ (2 ≤ n ≤ 6) for negative ions of B, C, N, O and F. These results are compared with the corresponding results obtained from the use of integer n Slater type orbitals. All of the nonlinear parameters are fully optimized. The results of calculation of coupling‐projection coefficients, orbital and total energies and virial ratios are presented. It is shown that the noninteger n Slater type orbitals, in general, improve the orbital energies.     

Angular aspects of exchange correlation and the fermi hole

The complete (nonreduced) αα probability density functions evaluated from the Hartree–Fock and simple Hartree product wavefunctions have been used to elucidate the angular features of spin correlation and the Fermi hole in the 2³S state of helium and the ground state of beryllium. This approach shows that the local Fermi holes in these two cases are very similar and that the Fermi hole is essentially spherically symmetric when the reference electron is close to the nucleus. As the reference electron is removed to larger radial distances, appreciable polarization of the Fermi hole is observed. The polarization is greater in the direction of the nucleus than away from the nucleus, contrary to the situation in the Coulomb hole of the helium ground state where the polarization is greater away from the nucleus than toward the nucleus. Several other differences between the He 2³S Fermi hole and the He 1¹S Coulomb hole are noted.     

Hulthén and slater type 2d functions in some excited configurations of sulphur and phosphorus

It is shown that a substantial energy improvement is gained by the variational use of Hulthén orbitals, instead of single Slater orbitals, in the 3d shells of some excited configurations of sulphur and phosphorus. The energies obtained are close to those attained with two-term Slater functions. In some cases the radial distribution functions from Hulthén orbitals are as good an approximation of SCF radial distributions as those from two-term Slater orbitals. Single term 2d functions with only one parameter are found to give almost identical energies and radial distribution functions as those obtained from two-parameter Hulthén orbitals. It is shown that the relationship between one-term 2d orbitals and Hulthén orbitals gives a method of enforcing nuclear cusp conditions on the former with little effect on the energy.     

A note on atomic density

In atomic systems, electron density has a simple finite expansion in spherical harmonics times radial factors. The difficulties in the calculation of some radial factors are illustrated in the low‐lying states of the carbon atom. Single‐particle methods such as Hartree–Fock and approximate density functional theory cannot ensure the correct expansion of the density in spherical harmonics. Wave‐function methods are appropriate but, as some expansion terms are entirely due to correlation, these methods only will give correct results for high‐quality variational functions. Using full‐configuration integration (CI), all the terms predicted by the theory appear and are not negligible but the convergence of the term due to correlation toward its correct value is uncertain even for very large CI spaces. © 2012 Wiley Periodicals, Inc.     

Approximate lower bounds of the Weinstein and Temple variety

By using the Weinstein interval or coupling the Temple lower bound to a variational upper bound one can in principle construct an error bar about the ground‐state energy of an electronic system. Unfortunately there are theoretical and calculational issues which complicate this endeavor so that at best only an upper bound to the electronic energy has been practical in systems with more than a few electrons. The calculational issue is the complexity of 〈H²〉 which is necessary in the Temple or Weinstein approach. In this work we provide a way to approximate the 〈H²〉 to any desired accuracy using much simpler 〈H〉‐like information so that the lower bound calculations are more practical. The helium atom is used as a testing ground in which we obtain approximate error bars for the ground‐state energy of [?2.904230, ?2.903721] hartree using the variational energy with the Temple lower bound and [?2.919098, ?2.888344] hartree for the Weinstein interval. For comparison, the slightly larger error bars using the exact value of 〈H²〉 are: [?2.904358, ?2.903721] hartree and [?2.919765, ?2.887677] hartree, respectively. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Thomas–Fermi approximation for the quasi‐two‐dimensional electron gas

To take into account static correlation effects in the quasi‐two‐dimensional electron gas a screened Coulombic interaction between particles is studied. The Thomas–Fermi approximation is used and the potential screening appears as a function of the Wigner–Seitz density parameter r_s and the effective width t of the system. With the self‐consistent field theory applied to the modified deformable jellium, the ground‐state energy per particle and the conditions for electron localization are obtained in terms of the interparticle distance and the screening parameter μ. A critical minimum characteristic width t_c is obtained; below t_c no long‐range order is obtained. For larger widths a stable localized state is predicted at finite densities. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 269–276, 2001