Coulomb Operators Research Articles

Developments in the Theory of Complex Spectra

The use of orthogonal operators to fit atomic energy levels often requires the calculation of matrix elements of multi-electron operators defined by the irreducible representations (irreps) of Lie groups. One way to do this is to imagine the 24/ +2 states of an atomic l shell as arising from the coupling in SO(2l + 1) of four inequivalent quasi-particles, each of which is labeled by the elementary spinor irrep (½ ½... ½) (of dimension 2l) of SO(2l + 1). Two parity labels (specifying the oddness or evenness of the numbers of electrons with spins up and spins down) complete the classification scheme, which leads to N (electron number) and S (total spin) as good quantum numbers. A unique correspondence (to within a phase) is established with the classic states of Racah. The Coulomb operators e2 and e3 for f electrons are examined, and the selection rule ΔS = 0 for e3 reappears as symmetry conditions on the 9-W symbols, where W is an irrep of SO(7). Work underway (with Shaozhong Li) is outlined. In keeping with a presentation to a general EGAS audience, these matters are described broadly and through examples.

Integral processing in beyond‐Hartree‐Fock calculations

AbstractThe increasing rate at which improvements in processing capacity outstrip improvements in input/output performance of large computers has led to recent attempts to bypass generation of a disk‐based integral file. The “direct” SCF method of Almlöf and co‐workers represents a very successful implementation of this approach. The present work is concerned with the extension of this general approach to CI and MCSCF calculations. After a discussion of the particular types of MO integrals for which — at least for most current generation machines — disk‐based storage seems unavoidable, it is shown how all the necessary integrals can be obtained as matrix elements of Coulomb and exchange operators that can be calculated using a direct approach. Computational implementations of such a scheme are discussed.

The a b i n i t i o model potential method. Main group elements

In this paper we present the ab initio core model potential method, in which the Coulomb and exchange core operators Jc and Kc are represented as accurately as possible using adequate local and nonlocal potentials, and the valence basis set is optimized in atomic MP SCF calculations following the variational principle. Nonrelativistic model potential parameters and valence basis sets are presented for the main group elements from Li to Xe. The pilot SCF molecular calculations on the ground states of N2, P2, As2, Sb2, and CaO show a good agreement between all-electron and model potential results, in particular the changes observed in molecular results due to improvements of the valence part of the all-electron basis set are reproduced by the model potential calculations.

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms Rb through Cd

Following a previously described method which approximates the Coulomb and exchange integrals in valence-electron-only SCF calculations, the necessary parameters for the atoms from Rb to Cd are reported. Extension to other atoms is in progress.

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms Ga through Kr

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms Ga through Kr

Spin-quasispin interchange in atomic-shell theory

The inverted-term structure of the octet and sextets of f7 compared with that of the singlets and triplets of f2 is related to the interchange of spin and quasispin. This interchange is formalised by an operator R. The R classification of the annihilation and creation operators for the electrons in an atomic l shell leads to several new Lie groups, among which is SO(6l+3). Racah's Coulomb operators ei are broken up into their R-symmetric and R-antisymmetric parts. The limited nature of the generalisation of the term inversion to other fN configurations is related to selection rules on the irreducible representations of SO(7) and G2.

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms Li through Ar

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms Li through Ar

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms K through Zn

Approximate calculation of the matrix elements of Coulomb and exchange operators for the ?core? electrons of the atoms K through Zn

Recent computational developments with the self‐consistent electron pairs method and application to the stability of glycine conformers

AbstractSignificant improvements in calculational efficiency and capability with the self‐consistent electron pairs (SCEP) method has resulted from several new computational developments. A new procedure for constructing the important internal Coulomb and exchange operators has substantially reduced the preiteration time. A general scheme for utilizing molecular symmetry has been used to advantage in reducing the number of pair functions and external operators that must be found explicitly at each iteration. A projection operator tool has been implemented and found to be quite effective at minimizing the number of iterations required at some point on a potential energy surface when an SCEP wavefunction exists for some nearby point. These and other improvements in the program construction have yielded sizable reductions in time for some representative test cases, including water and a potential energy curve for formaldehyde. The new SCEP program also performs low‐order perturbation theory treatments and coupled electron pair approximation (CEPA) calculations using the same operator approach. The usefulness of the approach is demonstrated by very large scale calculations on the stability of the two interstellar glycine conformers. These calculations involve the variational treatment of 82,205 symmetry‐adapted singly and doubly substituted configurations involving 225 internal electron pairs.

Scattering amplitudes and integral equations for the collision of two charged composite particles

Transition operators for the collision of two clusters composed of an arbitrary number of charged and neutral particles are represented as a sum of pure Coulomb and Coulomb-modified short-range operators. Sandwiching this relation between the corresponding channel states, correct two-fragment scattering amplitudes are obtained by adapting the conventional two-body screening and renormalization procedure. Furthermore, integral equations are derived for off-shell extensions of the full screened amplitudes and of the unscreened Coulomb-modified short-range amplitudes. For three particles, the final results coincide with those derived previously in a different approach. The proposed theory is valid for pure Coulomb scattering as well as for systems containing, in addition, two-body interactions of short range.NUCLEAR REACTIONS $N$-body scattering theory for charged particles. Scattering amplitudes for two-fragment collisions defined via screening and renormalization procedure. Derived effective two-body integral equations. Formalism also applicable to atomic problems.

Relativistic pseudopotential theories and corrections to the Hartree-Fock method

The fully relativistic Dirac-Fock generalizations of non-relativistic Phillips-Kleinman and generalized Phillips-Kleinman pseudopotentials are used to derive pseudopotential theories based on the Schrodinger hamiltonian which incorporate relativistic effects. In the theory in which all the orbitals are treated to the lowest non-vanishing order (1/c 2), the relativistic corrections to the energies are obtained by taking the expectation value of a perturbing operator over the non-relativistic wavefunctions. This perturbation consists of coulomb and exchange operators built from the relativistically induced changes in the charge distributions of other orbitals and two new operators arising from the pseudopotential, in addition to the standard mass-velocity, spin-orbit coupling and Darwin terms. The leading corrections to Hartree-Fock theory are obtained by omitting the pseudopotential operators and correct two expressions reported previously. The relative contributions of the standard operators and the coulo...

Calculation of the electronic structure of transition metalhexafluoro-compounds by the MO LCAO method with averaging with respect to configuration

where H is the operator of the kinetic energy and the energy of attraction of the electron to the nuclei shielded by the core electrons, aiu is the population of the i-thatomic orbital at the center u, Jiu and Kiu are the Coulomb and exchange operators, constructed on the i-thAO of the center'u. Self-consistency was achieved for the ground (t~ge~} and excited (t52ge~) configurations. ~ae parameter 10Dq was determined as the difference in the oneelectron orbital energies 5 3 ~0~q = ~ ( e g, ~ g e g) --~ (~, t~ogj.~ (2) It may be noted that Eq. (2) is applicable only in the approximation of "frozen MO," and with allowance for the presence of open shells the effective one-electron operator used in Eq. (2) differs from the operator defining the coefficients in the MO [5]. The authors of [1-4] studied different approaches to the analysis of the populations and the influence of the choice of basis functions, but even the most successful versions of the method give d-level populations which are too high, compared with nonempirical calculations [4, 7], and overestimate the covalency of the t2g- and eg-MO. In [8] a semi-empirical version was proposed for calculating the specific features of the open shells of a complex in the approximation involving averaging of the many-electron energy with respect to the terms of the configuration. For the systematic semi-empirical calculation, assuming a self-consistent solution of different Hartree--Fock equations for open and filled MO of the same symmetry, it is necessary to have reliabIe approximate expressions for the matrix elements (ME) of the operators of their orthogonalizatio n with respect to one another. Allowance for these expressions complicates considerably the usual calculation procedure, so that in the present work it is assumed that the composition of all the MO can be obtained with one effective Hamiltonian, that is that all the MO, as in [1-4], are obtained from the solution of the secular equation with the operator (1). The 10Dq is determined using the many-electron approach with allowance for the presence of open levels, proposed in [8], where the following expression was obtained for the energy of the configuration, averaged with respect to the terms,

A vector perturbation series based upon a representation in a finite banach space

The atomic finite-difference Hartree-Fock equations are written as an operator equation in a finite Banach space, the finite-difference variables and Lagrange multipliers forming the components of a solution vector, U, in this space. The Coulomb and exchange operators are treated as Λ-dependent perturbations, and the solution vector is expressed in the form of a vector perturbation series, U(Λ) = Σ n U n Λ n . An algorithm for calculating the coefficient vectors U n is given. Through a scaling of units, one set of coefficient vectors suffices for the calculation of an isoelectronic sequence of states, starting from the neutral atom. Results are presented for the 1 s 2 1S and 1 s 22 s 2S isoelectronic sequences. The Richardson procedure is used to extrapolate to the differential limit.

Hartree–Fock equations for open shell states

Fock equations are derived for multideterminantal single configuration states with any number of open shells of any symmetry. New operators, ’’projection–transition’’ operators and ’’off-diagonal’’ Coulomb and exchange operators, are required for cases that do not fit a Roothaan-like mold.

Theory of self-consistent electron pairs. An iterative method for correlated many-electron wavefunctions

A rapidly converging iterative method is presented to solve the many-electron Schrödinger equation within a Hilbert space confined to functions with at most two electrons outside an internal space defined by the orbitals of a reference function. The wavefunction is given in terms of external two-electron clusters represented by coefficients and density matrices referring directly to the basis functions. All matrix elements are obtained from generalized Coulomb and exchange operators. Only one operator per correlated electron pair is required for each iteration cycle.

Applicability of SCF Theory to Some Open-Shell States of CO, N2, and O2

By working with the real functions π x and π y instead of π + and π −, we show how to express the SCF Hamiltonians for the Σ states of the configurations (π u)3 (π g), (π u)3(π g)3, and (1π)3 (2π) of diatomic molecules in terms of the Coulomb and exchange operators only. With these results, we have used conventional SCF programs to solve for the wavefunctions of many interesting states of N2, O2, and CO, e.g., the B 3Σu− state of O2. For many states, the SCF results are in good agreement with experiment. However, SCF theory runs into serious trouble if electron correlation is important in determining the relative locations of excited states.

Hamiltonien de contact spin-spin et interaction coulombienne

It is shown that the spin-spin contact Hamiltonian has always the same angular part as some Coulomb interaction operators.

On the Structure of Light Nuclei

Extensive calculations based on the approximation of single particle wave functions (the Hartree method) have been made for the nuclei between ${\mathrm{He}}^{6}$ and ${\mathrm{O}}^{16}$ using the general symmetrical interaction operator given by Eq. (1). The Coulomb interaction is treated as a small perturbation. Secular equations are avoided by the construction of space wave functions in the normal state configuration belonging to irreducible representations of the symmetric group. These functions yield an energy matrix which is diagonal in the ordinary and Majorana interaction energies. The contributions of the spin exchange and Coulomb operators to the energy terms are found by a first-order perturbation calculation. Although the general symmetrical operator contains several parameters as yet undetermined, only those parameters which have been fixed by consideration of the two, three and four particle problems are involved in the energy differences within the group of low lying terms belonging to the normal state configuration. These term differences are identical with those recently computed for unsymmetrical interaction operators of the saturation type. New results for mass defects, excitation energies and energy relations between isobars are compared with experimental values.