Electron pair correlation in atoms and molecules

Accurate methods for computing energies and electronic properties of atoms and molecules have been derived from direct treatment of localized pairs of electrons. The conceptual development and implementation of such methods is reviewed.     

A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $N-1$ particles could be constructed from a 2DM ( $\Gamma $ ) for $N$ fermions divided by the diagonal of the 1DM, the density $n(\mathbf{r}_1)$ , as $2\Gamma (\mathbf{r}_1,\mathbf{r}^{\prime }_2;\mathbf{r}_1,\mathbf{r}_2)/n(\mathbf{r}_1)$ for any arbitrary fixed $\mathbf{r}_1$ . Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.     

Product pair correlation in bimolecular reactions

The vast majority of chemical reactions involve polyatomic species as reactants and/or products. The added degree of complexity offers opportunities to address dynamical questions other than those already encountered in a typical atom + diatom reaction. Product pair correlation is one of them. This article introduces the basic concept, outlines the experimental approach we developed and then highlights some of the applications to bimolecular reaction dynamics. Particular emphasis is placed on the information contents and unique insights gained from this type of measurements, which otherwise would have been lost by the conventional approach.     

Subshell-pair correlation coefficients of atoms in momentum space

Angular correlation coefficients τ ^{nl,n^′ l^′} [p] between linear momenta of an electron in a subshell nl and another electron in a subshell n′ l′ are studied for the 102 neutral atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. We theoretically find that electron momenta are negatively correlated or uncorrelated; τ ^{nl,n^′ l^′} [p] < 0 when |l − l′|=1, while τ ^{nl,n^′ l^′} [p]=0 when |l − l′| ≠ 1. Numerical examinations of the atoms show that except for the He–B atoms, negative correlations are largest between 1s and 2p subshells, which have the most diffuse electron distributions in momentum space.     

Role of angular electron pair correlation in stabilizing C

A pseudo‐potential that was successfully employed in an earlier study by the Compton group is used to describe the binding of a single electron to a C₆₀ molecule to form C. Then, the interaction of a second electron with the C anion is treated in two manners. First, as performed in the earlier Compton study, a mean‐field (i.e., Hartree–Fock) approach is used to estimate the C‐to‐C energy difference for the singlet state of the dianion and, much as in the earlier study, this dianion is predicted to be unstable by ∼0.4 eV. Second, for this same singlet state, a configuration interaction wave function is employed that allows for the angular correlation of the two excess electrons, allowing them to avoid one another by moving on opposite sides of the C₆₀ skeleton. The energy of the dianion is lowered by 0.3 eV when angular correlation is included, suggesting that the singlet dianion is unstable with respect to electron loss by only ∼0.1 eV. A Coulomb barrier (>1 eV high) and angular momentum barriers then combine to trap electrons of singlet C from detaching, thus producing the very long observed lifetimes. In addition, the energy of the lowest triplet state of C is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

The polarizability of a pair of hydrogen atoms at long range

The coefficients of R^?6 and R^?9 in the long-range expansion of the parallel and perpendicular components of the polarizability of a pair of hydrogen atoms at a separation R have been calculated. The results are A⁽⁶⁾₆ = 2558.59. A⁽⁶⁾_⊥ = 1268.25, A⁽⁸⁾₆ = 90639.5 and A⁽⁸⁾_⊥ = 22010.3 au. The values of A⁽⁶⁾₆ and A⁽⁶⁾₆ and A⁽⁶⁾_| are somewhat larger than previous theoretical estimates and much larger than the classical values of 729 and 182.25 au. The terms in R^?8 arise from distortion associated with the dispersion forces and from the field of the quadrupole moment induced by the field-gradient at each atom.     

Radial correlation limits of helium and heliumlike atoms

Radial correlation limits of two-electron atoms with atomic numbers Z = 1–10 are calculated by using modified Kinoshita wave functions in which all the parameters are optimized. The optimal Kinoshita functions show a rapid energy convergence with the increasing number N of constituent terms, and the radial energies convergent to 10 significant figures are obtained. The results show that both the calculated and estimated values of the radial correlation limits in the literature are insufficiently accurate. In the case of He, for example, the present calculation gives ?2.879 028 764 hartrees with N = 40, while the best literature value is ?2.879 028 6 hartrees.     

A calculation of the well depth for a pair of helium atoms

A new methods is proposed for the elimination of the basis superposition error in the calculation of small potentials. The method which involves CI is applied to the calculation of the well depth of He₂. It includes interatomic and intraatomic correlation effects simultaneously. A value for the well depth of 10.54 K at R = 3.03 au is obtained, in excellent agreement with current theoretical estimates.     

New correlation effects in the photoionization of atoms and ions

Recent advances in experimental and computational capabilities have provided the impetus for highly detailed studies of the photoionization process in atoms and ions. As a result, there have been a number of new phenomena uncovered. And these new phenomena are, in a general sense, of correlation among the atomic electrons, i.e., the breakdown of the single-particle approximation. Of particular interest are cases where correlation is not merely a small perturbation on the single-particle model, but where correlation dominates the process. In this paper, a number of illustrative experimental and theoretical examples are presented in which electron–electron correlation is seen to be a primary determinant of the process.     

Macromolecular pair correlation functions from fluorescence depolarization experiments

Depolarization of fluorescence resulting from the transport of electronic excitations in chromophore-containing polymers is investigated as a technique for detecting deviations from ideal-chain statistics. An approximate expression for the fluorescence anisotropy that depends only on the pair correlation function of chromophore labels is presented. This approximation is shown to be accurate for particular cases of long-range correlations, short-range correlations, and no correlations among chromophore positions. The formalism allows fluorescence depolarization experiments to be used as a probe of macromolecular pair correlation functions.     

Convergence of third order correlation energy in atoms and molecules

We have investigated the convergence of third order correlation energy within the hierarchies of correlation consistent basis sets for helium, neon, and water, and for three stationary points of hydrogen peroxide. This analysis confirms that singlet pair energies converge much slower than triplet pair energies. In addition, singlet pair energies with (aug)-cc-pVDZ and (aug)-cc-pVTZ basis sets do not follow a converging trend and energies with three basis sets larger than aug-cc-pVTZ are generally required for reliable extrapolations of third order correlation energies, making so the explicitly correlated R12 calculations preferable.     

Electron pair correlation energies for ZN2+

Pair energies contributing to the correlation energies of the outer-shell electrons (n = 3) as well as for the 1s² and 2s² pairs are computed for the Zn²⁺ closed-shell ion by means of the variational-perturbation method starting with the sum of one-electron Hartree–Fock operators as the zeroth-order Hamiltonian. The results allow an understanding of the electron correlation for pairs of electrons of the p and d type. For 3p3d pairs it has been found that the correlation energy for the singlet pair of ¹D symmetry is lower than for the triplet pair ³D. The 3l-3l′ correlation energies are compared with the MBPT results of Kelly and Ron for Fe. The total correlation energy of the outer shell is ?1.032 a. u.     

Breit-type three-electron equation in the Pauli approximation

We have reduced the Breit-type equation written down for 3 electrons to the 8 large components of the wave function in the (v/c)² (Pauli) approximation. This procedure required appropriate handling of the 64 scalar equations which result in this instance. According to the results obtained within the accuracy of (v/c)² all the terms are the same as those obtained by summing up the terms of the reduced 2-electron Breit's equation to 3 electrons. This statement is no longer valid, however, if during the reduction procedure we would go beyond the (v/c)² approximation. Finally there is a discussion about how to begin developing a relativistic many-electron theory valid for the inner shell electrons of heavy atoms within the accuracy of the fine structure constant.     

Efficient evaluation of one-center three-electron Gaussian integrals

 An algorithm is presented for the efficient evaluation of two types of one-center three-electron Gaussian integrals. These integrals are required to avoid the resolution-of-identity (RI) approximation in explicitly correlated linear R12 methods. Without the RI approximation, it is possible to enforce rigorously the strong orthogonality of the second-order M?ller–Plesset R12 ansatz. A test calculation is performed using atomic Gaussian-type orbitals of the neon atom. Received: 21 November 2000 / Accepted: 6 April 2001 / Published online: 9 August 2001     

Enhanced second‐order treatment of electron pair correlation

Ab initio calculations have been performed for F₂, HCCH, H₂O, HF, (HF)₂, and (H₂O)₂, comparing certain electron pair correlation methods, or methods for doubly substituted configurations. In these model systems, the reweighting of substituted configurations that occurs beyond a second‐order perturbative treatment of electron correlation can be partly built into the second‐order analysis in a computationally trivial step. Specific means for doing this are explored, and they offer improvement in certain cases or else very little change. A consistent improvement in the correlation energy when judged against treatment with double substitution coupled cluster theory for the test species is obtained through one of these schemes. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 226–236, 2000     

Semiempirical estimates of the correlation energy in small clusters of hydrogen atoms

The Pamuk EPCE-F2σ method is applied to neutral and charged clusters composed from 2–9 hydrogen atoms. The range of applicability of the method is demonstrated with H₂, H ₃ ⁺ , and H₃ by comparing the results with the reported rigorous SCF and CI calculations. Predictions of the correlation energy were made for larger hydrogen atom systems, the emphasis being laid in the discussion on H₄, H ₅ ⁺ , and H₆.     

Random-phase-approximation-based correlation energy functionals: benchmark results for atoms

The random phase approximation for the correlation energy functional of the density functional theory has recently attracted renewed interest. Formulated in terms of the Kohn-Sham orbitals and eigenvalues, it promises to resolve some of the fundamental limitations of the local density and generalized gradient approximations, as, for instance, their inability to account for dispersion forces. First results for atoms, however, indicate that the random phase approximation overestimates correlation effects as much as the orbital-dependent functional obtained by a second order perturbation expansion on the basis of the Kohn-Sham Hamiltonian. In this contribution, three simple extensions of the random phase approximation are examined; (a) its augmentation by a local density approximation for short-range correlation, (b) its combination with the second order exchange term, and (c) its combination with a partial resummation of the perturbation series including the second order exchange. It is found that the ground state and correlation energies as well as the ionization potentials resulting from the extensions (a) and (c) for closed subshell atoms are clearly superior to those obtained with the unmodified random phase approximation. Quite some effort is made to ensure highly converged data, so that the results may serve as benchmark data. The numerical techniques developed in this context, in particular, for the inherent frequency integration, should also be useful for applications of random phase approximation-type functionals to more complex systems.     

Effective dynamic correlation in multiconfigurational wave-function calculations on atoms and molecules

An intuitive understanding of dynamic correlation in terms of a regularized electron repulsion expression is outlined. Expressions for cusp kinetic energy corrected regularized electron repulsion integrals are deduced and implemented in a multiconfigurational wave-function framework. A regularized complete active space self-consistent field (reg-CASSCF) technique is suggested and tested on atomic total energies, molecular structures and binding energies. Received: 26 November 1996 / Accepted: 21 April 1997     

Electron–positron pair production and bremsstrahlung at intermediate energies in the field of heavy atoms

The Coulomb corrections (CC) to the processes of bremsstrahlung and pair production are investigated. The next-to-leading term in the high-energy asymptotics is found. This term becomes very essential in the region of intermediate energies. The influence of screening for CC is small for differential cross section, spectrum, and the total cross section of pair production. The same is true for the spectrum of bremsstrahlung, but not for the differential cross section, where the influence of screening can be very large. The corresponding screening corrections as well as the modification of the differential cross section of bremsstrahlung are found. A comparison of our results for the total cross section of pair production with the experimental data available is performed. This comparison has justified our analytical results and allowed to elaborate a simple ansatz for the next-to-leading correction. The influence of the electron beam shape on CC for bremsstrahlung is investigated. It turns out that the differential cross section is very sensitive to this shape.     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. I. Basic integrals

The modified and extended version of the Neumann expansion of the interrelectronic distance function r for u = ?1, 0, 1, 2, using the set of orthogonal polynomials normalized to unity, is presented. This expansion has been utilized to obtain analytical expressions for evaluating two-center two- and three-electron integrals in the Slater orbital basis occurring if variational correlated functions are used.