On the evaluation of spin–orbit coupling matrix elements in a spin‐adapted basis

In the present article, we outline a simple scheme for generating configuration interaction matrix elements for spin–orbit interactions in molecules. The procedure leads to a close parallelism with spin‐free permutation‐group approaches. Unitary shift operators were successfully used on the orbital space to generate the matching permutations necessary to evaluate the required matrix elements. The procedure is adequately illustrated using examples. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 23–27, 1999     

Efficient generation of configuration interaction matrix elements

A method for the efficient generation of CI matrix elements over spin eigenfunctions is presented. Practical application of the approach is limited to configurations with about 10 open shells, but the algorithm results in the generation of more than 2200 matrix elements/second on an IBM 370/158 computer including all overhead for a large matrix which contains 50% non-zero elements.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Calculation of molecular systems via coordinate wave functions

The system of charges is in a state with a given total spin S, which is described by a configuration of one-electron orbitals with arbitrary filling (subject to the Pauli principle). Expressions are derived for the matrix elements of operators F and G that are independent of the spin. The energy of the interaction between the completely filled orbitals and the singly filled ones is found to be independent of the spin of the latter. The formulas may be used with the tables of [2] to derive directly the expressions for the matrix elements of a configuration having an arbitrary number of completely filled orbitals and up to six singly filled ones.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Application of graphical methods of spin algebras to limited CI approaches. I. Closed shell case

The time independent diagrammatic technique based on the mathematical methods of quantum electrodynamics (second quantization, Wick's theorem, Feynman-like diagrams) is combined with graphical techniques of spin algebras to derive general expressions for the matrix elements of spin independent one- and two-particle operators between spin symmetry adapted ground, mono- and bi-excited configurations of a closed shell system. Two coupling schemes are considered for bi-excited states and their relative merits are discussed. Finally, the results are used to derive compact expressions for the coupling coefficients of the direct configuration interaction from molecular integrals (CIMI ) method.     

A method of calculation of the matrix elements between the spin-projected nonorthogonal Slater determinants

In the resonating Hartree–Fock calculation, we have to calculate the matrix element between spin-projected nonorthogonal Slater determinants (S dets). The matrix element between coherently spin-rotated S dets are given by a linear transformation of the spin-projected ones. Using the inverse transformation, we get the projected matrix elements from the coherently spin-rotated ones. By appropriately choosing the angles of the spin rotation, the required computational time is considerably reduced. © 1995 John Wiley & Sons, Inc.     

Excited states in the multireference state-specific coupled-cluster theory with the complete active space reference

The recently proposed multireference state-specific coupled-cluster theory with the complete active space reference has been used to study electronically excited states with different spatial and spin symmetries. The algorithm for the method has been obtained using the computerized approach for automatic generation of coupled-cluster diagrams with an arbitrary level of the electronic excitation from a formal reference determinant. The formal reference is also used to generate the genuine reference state in the form of a linear combination of determinants contracted to a configuration with the spin and spatial symmetries of the target state. The natural-orbital expansions of the one-electron configuration inferaction density matrix allowed us to obtain the most compact orbital space for the expansion of the reference function. We applied our approach in the calculations of singlet and triplet states of different spatial symmetries of the water molecule. The comparisons of the results with values obtained using other many-particle methods and with the full configuration interaction results demonstrate good ability of the approach to deal with electronic excited states.     

The boundary element method: applications to steady-state voltammetric simulations within domains extending to infinity

Application of the boundary element method (BEM) to the simulation of diffusion limited electrolysis reactions occurring within an infinite domain is outlined. This article focuses on the development of procedures that permit electrolysis simulations to be performed, where only an element mesh over the electrode region is required. This approach provides significant benefits over the traditional application of BEM simulations for electrochemical-based problems. In particular the reduction in mesh points to regions only over the electrode results in simpler grid generation procedures and significantly reduced computational times. The paper describes the theory and numerical details required to develop steady-state two-dimensional diffusional models for voltammetric simulations. The accuracy and versatility of the numerical procedures are tested by examining the current density at a range of well-defined electrode geometries.     

Elimination of some spectral interferences and matrix effects in inductively coupled plasma-mass spectrometry using direct sample insertion techniques

Two key problem areas in inductively coupled plasma-mass spectrometry (ICP-MS) are spectral interferences from analyte and matrix based molecular ions, particularly oxide species; and non-spectroscopic matrix effects where an excess of an element, particularly heavier elements, suppresses the signal of lighter elements. It is shown that direct sample insertion (DSI) techniques can be used to eliminate some of these problems. With the absence of water in a DSI system, oxide species are reduced to a very low level ( .1%). Examples are shown for BaO⁺, CeO⁺ and ClO⁺. In addition, by relying on differential thermal volatilization, the suppressive matrix effect of U on Zn can be eliminated. Chemical modification with NaF is a key operational factor in achieving these benefits.     

Configuration space truncation scheme for Hubbard Hamiltonian

In the present note we outline a truncation scheme for configuration space ofN-electron systems in definite spin statesS for Hubbard Hamiltonian with first neighbour transfer terms. For this sparse matrix we find that the present truncation scheme yields reasonably good ground and first excited states with very limited space being used.     

Numerical instabilities in the computation of pseudopotential matrix elements

Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion.     

Frequency-Selective Manipulations of Spins allow Effective and Robust Transfer of Spin Order from Parahydrogen to Heteronuclei in Weakly-Coupled Spin Systems

We present a selectively pulsed (SP) generation of sequences to transfer the spin order of parahydrogen (pH₂) to heteronuclei in weakly coupled spin systems. We analyze and discuss the mechanism and efficiency of SP spin order transfer (SOT) and derive sequence parameters. These new sequences are most promising for the hyperpolarization of molecules at high magnetic fields. SP-SOT is effective and robust despite the symmetry of the ¹H-¹³C J-couplings even when precursor molecules are not completely labeled with deuterium. As only one broadband ¹H pulse is needed per sequence, which can be replaced for instance by a frequency-modulated pulse, lower radiofrequency (RF) power is required. This development will be useful to hyperpolarize (new) agents and to perform the hyperpolarization within the bore of an MRI system, where the limited RF power has been a persistent problem.     

Challenges for INAA in Studies of Materials from Advanced Material Research Including Rare Earth Concentrates and Carbon Based Ceramics

Rare-earth elements are increasingly applied in advanced materials to be used, e.g., in electronic industry, automobile catalysts, or lamps and optical devices. Trace element analysis of these materials might be an interesting niche for NAA because of the intrinsic high accuracy of this technique, and the shortage of matrix matching reference materials with other methods for elemental analysis. The carbon composite materials form another category of advanced materials, where sometimes a very high degree of purity is required. Also for these materials, NAA has favorable analytical characteristics. Examples are given of the use of NAA in the analysis of both categories of materials.     

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.     

A semi-empirical model for computing radial matrix elements in one-electron atoms and ions

An easily tractable model is proposed to compute radial matrix elements between discrete states of one-electron atoms or ions. Assuming a closed-shell core, polarization effects are included in the model, and core penetration is accounted for empirically. The required inputs for this algorithm are the energies of the involved levels, and the core size and polarisabilities. Ignoring core polarization, the derived Coulomb matrix elements (possibly between levels with non-zero quantum defects) agree with those tabulated elsewhere in the literature. The method is then applied to dipolar and quadrupolar transitions in alkali atoms and singly-ionized alkaline-earth elements; it proves to be in fair agreement with available experimental data. An accuracy test of the method is proposed.     

Superposition of nonorthogonal Slater determinants towards electron correlation problems

We propose variational and nonvariational methods based on the superposition of nonorthogonal Slater determinants. Properties of the reference functions are discussed. In the nonorthogonal configuration interaction method, all the excited configurations of multiple determinants are integrated into a variational space. An efficient way to manipulate matrix elements over determinants of distinct vacuums is presented by introducing similarity transformed operator and bracket transformations. The method enables us to map a matrix multiplication in the nonorthogonal problem to an orthogonal one, and thus maintains a fundamental scaling property along with the amount of data processed in the corresponding orthogonal configuration interaction method. Furthermore, we discuss a coupled-cluster theory employing a vacuum-dependent wave operator, which is entirely size consistent as well as core extensive. These methods are applied to H₂O + nHe(n = 0−2) and a single-bond dissociation of the HF molecule, compared with conventional methods including full and multireference configuration interaction methods. Received: 7 July 1997 / Accepted: 12 September 1997     

The matrix theory of electron spin multiplets for atoms and molecules

In this article a matrix method for the construction of spin multiplets (spinconfigurations) is suggested in order to solve the multielectron problem for atoms and mulecules by means of configuration interaction.A simple graphical way is given to enumerate configurations and to break their set into subsets of configurations related to the given projection of the total spin of a system S _z . It is found that all matrices in the theory of spin multiplets are convex and in cases of two, three, and four electrons are broken into blocks of an order no higher than 3.The model of the solution of the multielectron Schrödinger equation, in which the total spin of core electrons is zero, is considered. In this model the construction of linear combinations of configurations is reduced to the construction of those for but valence electrons.     

Zero field splitting of the chalcogen diatomics using relativistic correlated wave-function methods

The spectrum arising from the (π*)(2) configuration of the chalcogen dimers, namely, the X(2)1, a2, and b0(+) states, is calculated using wave-function theory based methods. Two-component (2c) and four-component (4c) multireference configuration interaction (MRCI) and Fock-space coupled cluster (FSCC) methods are used as well as two-step methods spin-orbit complete active space perturbation theory at 2nd order (SO-CASPT2) and spin-orbit difference dedicated configuration interaction (SO-DDCI). The energy of the X(2)1 state corresponds to the zero-field splitting of the ground state spin triplet. It is described with high accuracy by the 2- and 4-component methods in comparison with experiment, whereas the two-step methods give about 80% of the experimental values. The b0(+) state is well described by 4c-MRCI, SO-CASPT2, and SO-DDCI, but FSCC fails to describe this state and an intermediate Hamiltonian FSCC ansatz is required. The results are readily rationalized by a two-parameter model; Δε, the π* spinor splitting by spin-orbit coupling and K, the exchange integral between the π(1)* and the π(-1)* spinors with, respectively, angular momenta 1 and -1. This model holds for all systems under study with the exception of Po(2).     

Cholesky decomposition of the two-electron integral matrix in electronic structure calculations

A standard Cholesky decomposition of the two-electron integral matrix leads to integral tables which have a huge number of very small elements. By neglecting these small elements, it is demonstrated that the recursive part of the Cholesky algorithm is no longer a bottleneck in the procedure. It is shown that a very efficient algorithm can be constructed when family type basis sets are adopted. For subsequent calculations, it is argued that two-electron integrals represented by Cholesky integral tables have the same potential for simplifications as density fitting. Compared to density fitting, a Cholesky decomposition of the two-electron matrix is not subjected to the problem of defining an auxiliary basis for obtaining a fixed accuracy in a calculation since the accuracy simply derives from the choice of a threshold for the decomposition procedure. A particularly robust algorithm for solving the restricted Hartree-Fock (RHF) equations can be speeded up if one has access to an ordered set of integral tables. In a test calculation on a linear chain of beryllium atoms, the advocated RHF algorithm nicely converged, but where the standard direct inversion in iterative space method converged very slowly to an excited state.