Pseudomolecular atoms: geometry of two-electron intrashell excited states

Supermultiplet patterns indicating pronounced collective motion of electrons are prominent among intrashell excited states of helium or alkaline earth atoms. Many aspects of these patterns have previously been discussed in terms of an empirical rovibrator model, e-core-e, analogous to alinear triatomic molecule. However, this has appeared incompatible with a dimensional scaling treatment that predicts pseudomolecular features and relates properties of some excited states to the ground state, but corresponds to abent equilibrium geometry. We examine both the full two-electron Hamiltonian, including angular momentum, and a prototype model for angular correlation which restricts the two particles to the surface of a sphere. By virtue of the boundary conditions imposed by the Jacobian volume element, we find that alinear equilibrium geometry is not allowed. If the Hamiltonian is transformed to reduce the Jacobian to unity, an infinite barrier appears in the effective potential when the electrons are 180° apart and equidistant from the nucleus. We find the supermultiplet patterns, including some “antimolecular” features, are consistent with a floppy butbent geometry, anasymmetric rotor model. The most probable interelectron angle ? _m varies markedly with the principal quantum number and also with the space-quantization of the angular momentum with respect to body-fixed axes.     

Perturbation calculations on the split-shell wavefunctions of two-electron atoms

In a special case of the Löwdin-Adams variant of perturbation theory, knowledge of the perturbed wavefunction through first order is sufficient for the calculation of the energy correct to the fifth order. This is applied to the split-shell wavefunctions of H^–, He, and Li^–. The results are presented, compared with other approaches, and discussed.     

Coupling of radial and angular correlations in two-electron atoms

The nature of interelectronic correlation in the ground and low lying excited states of the helium sequence is investigated. It is demonstrated that as a consequence of the coupling between radial and angular correlation the electrons are actually pulled together rather than pushed apart, in the low nuclear charge end of the isoelectronic sequence. A uniform treatment of correlation effects in both the ground and the excited states is thus shown to be feasible.     

Exploiting non-abelian point group symmetry in direct two-electron integral transformations

Summary A simple and general scheme to exploit any discrete point group symmetry in two-electron integral transformations is introduced. It has been implemented together with integral pre-screening techniques in direct two-electron integral transformations. Its application has also been extended to subsequent MO integral processing steps like MP2 or solution of the coupled-perturbed Hartree-Fock equations (CPHF). Timings for representative applications are presented.     

Proof of a conjecture by Dacre on the symmetry of two-electron integrals

A format proof is given of a statement by Dacre concerning the effect of symmetry operators on two-electron integrals.     

Electron correlation study for two-electron atoms by a simple correlated wave function

The difference in the electron correlation between H^- and other two-electron atoms is clarified by the introduction of the r₁₂ term in the wave function. By using the expansion of r₁₂, a certain modification of the usual electron correlation factor 1 + Cr₁₂ is introduced and its effectiveness is examined. Calculations are carried out for the ground state and the three lowest excited states (2³S, 2³P and 2¹P). The peculiar electron correlation in the ground state of H^? is shown by looking at the Coulomb hole for closed- and open-shelf models in comparison with those for other two-electron atoms.     

Electric-field gradient calculations for atoms with axial symmetry

Restricted open-shell Hartree-Fock and unrestricted Hartree-Fock calculations of the electric-field gradient in atoms B, N, O, Al, S, and Cl were performed by relieving the spherical symmetric constraint. The Sternheimer's core polarization effect is then automatically taken into account. The orbitals produced by the axial symmetric self-consistent field are found to have axial symmetry of s-d and p-f mixing types. However, the nonequivalence of the three p orbitals also gives rise to ambiguity. © 1996 John Wiley & Sons, Inc.     

Spatial symmetry holes in many-electron atoms and molecules

When a many-electron system has a spatial symmetry, it is shown that there exist spatial symmetry holes, which imply that two or more electrons are prohibited from being at certain spatial positions simultaneously. Inversion holes, rotation holes, and reflection holes, which result from inversion, twofold rotation, and reflection symmetries, respectively, are discussed in detail. The electron-electron counterbalance hole reported in literature is a particular case of the inversion hole. The spatial symmetry holes are illustrated for simple atoms and molecules.     

Effects of static electric fields on the photoionization spectra of two-electron atoms and ions

The simultaneous photoexcitation of two electrons is a highly correlated process, producing doubly excited states, observed as resonant structure in the ionization cross-section. This review paper explores the ability of a static electric field to affect these correlations, as evidenced by changes in the shape and energy of resonances. A comparison of field effects on the spectra of H⁻, Ps⁻, He, Ba and Cs⁻ summarizes current understanding of the processes and the capability of theory to predict the cross-sectional features.     

Electrocatalyst design for promoting two-electron oxygen reduction reaction: Isolation of active site atoms

Selective two-electron (2 e^?) pathway oxygen reduction reaction (ORR) has gained prominence for enabling small-scale, on-site electrochemical H₂O₂ production and has emerged as a promising alternative to the conventional anthraquinone process. The rational design of catalysts that can suppress the competing four-electron pathway ORR is critical. This review highlights catalyst design strategies for promoting the selective 2 e^? pathway ORR, including alloying with inert metals, partial surface poisoning, and generating atomically dispersed sites. The major results and advances, as well as unresolved challenges are summarized.     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

The principle-quantum-number (and the radial-quantum-number) expansion of the correlation energy of two-electron atoms

The second-order correlation energy of two-electron ions is studied in terms of an expansion in minimal approximations to the first-order natural orbitals (NOs). The non-linear parameters of these NOs are determined by minimization of the second-order energy. An approximation to the total second-order correlation energy is obtained as a sum of increments e(lp), depending on the angular quantum number l and the radial quantum number p. (Either l or p can be eliminated in favor of the principal quantum number n = l + p.) Closed expressions for these energy increments are derived. For fixed p the increments go as (l + 1)(-5). This is consistent with the behavior of the exact partial wave increments (that depend on the parameter l only) as (l + 1/2)(-4). While the partial wave increments correspond to a summation of e(lp) over p, other partial summations of the two-parameter increments lead to either the principal-quantum-number expansion (PQNE) with energy increments approximately n(-4), or the radial-quantum-number expansion, with a less transparent convergence pattern. Unfortunately these partial summations can neither be done in closed form nor from the asymptotic expansion, but some insight is obtained from a numerical summation. The hope to find a rigorous derivation of the PQNE has not been fulfilled.     

Effective potential model for calculating the energy shift in the ground state of two-electron atoms

An effective operator for the electron–electron interaction derived earlier has been shown to be successful for calculating the ground-state energy shift values for 10 members of the helium isoelectronic sequence in terms of a single free parameter. Some possible applications of the modified interaction operator for semiempirical calculations are suggested.     

Use of molecular symmetry in two-electron integral transformation An MP2 program compatible with HONDO 5

A computer program is described which evaluates the second-order Møller-Plesset energy using the integral list formed by HONDO 5. In this program use may be made of full molecular symmetry for most common point groups, even if they contain two-dimensional representations. The algorithm for the integral transformation may also be applied to other methods beyond Hartree-Fock. Some numerical results and timings are presented.     

Molecular symmetry and ab initio calculations: IV. Symmetry-matrix and symmetry-supermatrix in calculations of two-electron repulsion integrals

The product of two Gaussians having different centers is itself a one-center Gaussian, thus multicenter integrals with a Cartesian Gaussian basis can be reduced to one-center integrals. Recurrence relations for overlap integrals and electron repulsion integrals (ERIs) are derived at these centers. The calculations of overlap integrals and ERIs are carried out step by step from the highest symmetry case (one center) to required cases (different centers) by using the translation of Cartesian Gaussians. Full exploitation of symmetry in calculation processes can result in optimal use of these recurrence relations. Compared with the recently published algorithms, based on the recurrence relations derived by Obara and Saika [J. Chem. Phys., 84 , 3963 (1986)], the floating point operations (FLOPs) for ERI calculations (having four different centers) can be reduced by a factor of ca. 2. A significant extra saving in calculations and storage can be obtained if atoms, linear, or planar molecules are discussed. © 1997 John Wiley & Sons, Inc.     

Dynamical symmetry analysis of ionization and harmonic generation of atoms in bichromatic laser pulses

Recently the effect of the relative phase ? in a high‐intensity (～10¹⁴ W/cm²) two‐color (bichromatic) CW laser with frequencies ω and 2ω on the high‐order harmonic generation (HHG) was studied within the framework of the non‐Hermitian quantum mechanics (NHQM) [Phys Rev A 2004, 69, 043404/1]. Here we emphasize the study of symmetries in bichromatic HHG spectra within the framework of the conventional Hermitian QM, and in particular by taking the duration of the laser pulse into consideration (an effect that has not been included in the non‐Hermitian studies due to the time asymmetry problem in NHQM). The phase dependence of HHG and intense‐field ionization probability in a 1D Xe atom with symmetric field‐free potential and symmetric initial wave function were studied numerically and analytically. From simulations based on a single‐particle response it can be seen that the HHG spectra is symmetric with respect to inversion in the relative phase between the two colors ? only if ionization is forbidden in the system and the laser pulse is an adiabatic one. The HHG spectra is symmetric with respect to a π‐shift in ? whenever the laser pulse is an adiabatic one, either for bound or open (ionized) systems. The ionization probability is symmetric both to inversion or π‐shift in ?; the component probabilities (right‐ and left‐ionization probabilities) have the same ?‐dependence, up to a shift of π. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005