Vector Coupling Coefficients Research Articles

Restricted Open-Shell Hartree-Fock Method for a General Configuration State Function Featuring Arbitrarily Complex Spin-Couplings.

In this work, we present a general spin restricted open-shell Hartree-Fock (ROHF) implementation that is able to generate self-consistent field (SCF) wave functions for an arbitrary configuration state function (CSF). These CSFs can contain an arbitrary number of unpaired electrons in arbitrary spin-couplings. The resulting method is named CSF-ROHF. We demonstrate that starting from the ROHF energy expression, for example, the one given by Edwards and Zerner, it is possible to obtain the values of the ROHF vector-coupling coefficients by setting up an open-shell for each group of consecutive parallel-coupled spins dictated by the unique spin-coupling pattern of any given CSF. To achieve this important and nontrivial goal, we employ the machinery of the iterative configuration expansion configuration interaction (ICE-CI) method, which is able to tackle general CI problems on the basis of spin-adapted CSFs. This development allows for the efficient generation of SCF spin-eigenfunctions for systems with complex spin-coupling patterns, such as polymetallic chains and metal clusters, while maintaining SCF scaling with system size (quadratic or less, depending on the specific algorithm and approximations chosen).

Jj-Coupling-based atomic self-consistent-field calculations with relativistic effective core potentials and two-component spinors

A self-consistent-field (SCF) program for the calculation of atomic energies and wave functions defined in jj-coupling using two-component atomic spinors and relativistic effective core potentials (RECPs) is described. The code is based on the linear combination of atomic orbitals SCF algorithm for atomic states defined in LS-coupling developed by Roothaan and Bagus. Hamiltonian matrix elements with respect to one- and two-electron operators, including RECPs, are calculated for two-component atomic spinor basis functions of either Gaussian-type orbitals (GTOs) or Slater-type orbitals (STOs). Electronic states are defined as eigenfunctions of the total angular momentum squared operator and tables of the required vector coupling coefficients that define such pure states are provided. In addition, one or more GTO expansions of large- and small-core RECPs and their corresponding GTO basis sets are supplied for all elements Z=3 through Z=118. Optimized two-component basis sets of STOs or GTOs can be calculated for use in molecular structure codes based on RECPs and relativistic electronic structure theory. Atomic asymptotic state energies at the SCF level of theory for analysis of molecular dissociation limits can be studied. Program summaryProgram title: jjatomProgram Files doi:http://dx.doi.org/10.17632/zs4twp8r67.1Licensing provisions: GPLv3Programming language: Fortran 90Nature of problem: Relativistic atomic energies and wave functions; optimization of two component spinor basis sets.Solution method: Atomic spinors defined in jj-coupling and expansions in two-component spinor basis functions of Gaussian- or Slater-type functions. Self-consistent field optimization of the total energy. Used for optimization of two-component atomic spinor basis sets and calculations of valence spectra of heavy elements.

Computation of vector coupling coefficients for atoms with one and two open shells in the Hartree–Fock approximation

UDC 539.182 A new way to introduce and calculate vector coupling coefficients (VCCs) is proposed for computations in a Hartree–Fock approximation of atoms with one and two open shells. In the case of atoms with one open shell (Roothaan method), the obtained formulas for VCCs are applicable to the calculation of all terms of such atoms without exception. The VCCs are also calculated for a number of configurations of atoms with two open shells (Huzinaga method). The correctness of the obtained VCCs is confirmed by numerous calculations of energies and dipole polarizabilities for atoms with one and two open shells.

Is Mo Involved in Hydride Binding by the Four-Electron Reduced (E4) Intermediate of the Nitrogenase MoFe Protein?

We here report the first direct evidence addressing the possible involvement of Mo in substrate interactions during catalytic turnover. When the alpha-70(Ile) MoFe protein is freeze-trapped during H(+) reduction under Ar, the majority of the resting state EPR signal from the molybdenum-iron cofactor (FeMo-co) disappears and is replaced by the S = 1/2 signal of an intermediate that has been shown to be the E(4) MoFe state, which is activated for N(2) binding and reduction through the accumulation of 4 electrons/protons by FeMo-co. ENDOR studies of E(4) showed that it contains two hydrides bound to FeMo-co. We calculate that Mo involvement in hydride binding would require a vector-coupling coefficient for Mo of |K(Mo)| approximately > 0.2 and determine K(Mo) for the E(4) intermediate state through 35 GHz ENDOR measurements of a (95)Mo enriched MoFe protein, further comparing the results with those for the E(0) resting state. The experiments show that Mo of the resting-state FeMo-co is perturbed by the alpha-70(Ile) substitution and that the isotropic (95)Mo hyperfine coupling in E(4) is a(iso) approximately 4 MHz, less than that for the resting state. The decrease in a(iso) for (95)Mo of E(4) from the already small value in the resting state MoFe protein strongly suggests that the resting Mo(IV) is not one-electron reduced during the accumulation of the four electrons of E(4). In any case, the effective K for Mo is very small; |K(Mo)| approximately < 0.04, at least 5-fold less than the lower bound required for Mo to be involved in forming a Mo-H-Fe, hydride. As the hydride couplings also are both far too small and of the wrong symmetry to be associated with a terminal hydride on Mo, we may thus conclude that Mo does not participate in binding a hydride of the catalytically central E(4) intermediate and that only Fe ions are involved. Nonetheless, the response of the Mo coupling to subtle conformational changes in E(0) and to the formation of E(4) suggests that Mo is intimately involved in tuning the geometric and electronic properties of FeMo-co in these states.

Some classes of ‘nontrivial zeroes’ of angular momentum addition coefficients

Angular momentum coupling in quantum physics obeys obvious symmetries of rotation and reflection so that the Clebsch–Gordan (vector coupling) coefficients or Wigner 3j-symbols describing the coupling vanish unless these symmetries are satisfied. However, it has long been observed that there are ‘accidental’ or ‘nontrivial’ zeroes of some coefficients even when the obvious symmetries are satisfied. Partial explanations and conjectures on the systematics of some of these zeroes have been advanced. We provide some more and propose as well ‘near zeroes’ which, while not exactly vanishing, are extremely small in magnitude. Connections are made to zeroes of Legendre and hypergeometric polynomials and to classical and semi-classical pictures for the addition of angular momenta. A convenient ordering scheme for 3j's that incorporates Regge symmetries also emerges. Further aspects of our analysis concern radial matrix elements of powers of r in a Coulomb potential that have analogous expressions to 3j's as a result of a non-compact O(2, 1) counterpart of the O(3) group symmetry of rotations. Some remarks are made on possible realization in actual physical systems.

On the construction of Kramers paired double group symmetry functions

AbstractRelativistic quantum chemical calculations on molecular systems can greatly be expedited by making use of the full molecular symmetry, including both double point group and time‐reversal symmetry. A simple yet general scheme is proposed here to make double point group symmetry functions also form Kramers pairs such that the resulting operator matrices have the desired structure (i.e., quaternion, complex, or real). The key idea is to insert appropriate phase factors into the symmetry functions or make a suitable unitary transformation of them. Arbitrary finite double point groups can be handled in the same way. As for a product basis, the same symmetrization can be achieved by simple modifications of the vector coupling (Clebsch–Gordan) coefficients. This is especially of interest to, e.g., time‐dependent relativistic density functional linear response theory or relativistic two‐electron repulsion integral evaluations in general. Several examples are provided to illustrate the algorithm and its efficiency. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Contribution of the polarizability anisotropy to Rayleigh scattering

A calculation from first principles of the differential cross section for Rayleigh scattering shows that the contribution of polarizability anisotropy found in the literature is valid only in the high rotational quantum number limit of the vector coupling coefficients, that is, in the high‐temperature or classical limit. The additional term in the total Rayleigh scattering cross section arising from the polarizability anisotropy is about four times smaller than what is called the King factor in the literature. These errors, resulting from ignoring the constraints imposed by the conservation of angular momentum, can lead to significant corrections in estimating populations from stratospheric and mesospheric measurements.

Hartree-fock equations for atoms with two open shells

We have formulated the Hartree-Fock equations for multielectron systems with two open shells (the Huzinaga method) in terms of the density matrix in the LCAO approximation. In order to solve the Hartree-Fock equations, in the algebraic approximation we have obtained expressions for the derivatives of the energy with respect to the density matrix elements and the nonlinear atomic orbital parameters (the orbital exponents). We discuss the question of calculating the open shell parameters (the vector coupling coefficients) in the configurations s1pN and s1dN within the Huzinaga method. We have calculated the energy for a series of atoms with two open shells in these configurations. Using rather narrow basis sets of Slater-type atomic orbitals), we have obtained energy values close to the results of a numerical solution of the Hartree-Fock equations with sufficiently high accuracy of the virial ratio.

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, Lμ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), \(L_n^{l+(1/2)}(\alpha r^2)r^l Y_{lm} ({\hat {\bf r}}) e^{-\alpha r^2}\). The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator

Recursive calculation of nonprimitive coupling and recoupling brakets

The concept of coupling coefficients of angular momentum for the rotation group chain [Formula: see text] can be extended to representations of any group-chain factorisation by defining the generalised notion of a braket. We give a unified approach to recoupling coefficients (rccs), vector-coupling coefficients (vccs), and 3j phases for all group-chain transformations. Category theory is the appropriate tool for studying the representations of groups and algebras. The explicit use of category theory leads to a new recursion scheme for the calculation of brakets. We derive specialisations for calculating nonprimitive rccs and nonprimitive vccs from primitive rccs, primitive vccs, and 3j phases. This new recursion scheme forms the algorithmic core of Racah v4. Racah v4 is a software package developed at the University of Canterbury to calculate group representation coefficients (brakets). PACS Nos.: 02.20.Mp, 03.65.Fd, 31.15.–p, 31.15.Hz, 02.10.Ws

3 nj-symbols and harmonic superposition coefficients: an icosahedral abacus

Angular momentum recoupling coefficients of angular momentum theory and matrix elements for basis set transformation of hyperspherical harmonics enjoy properties and sum rules crucial for applications but complicated without the guidance of graphical techniques. These coefficients being related to Racah's polynomials, the graphs also apply to polynomials of the hypergeometric family, their q-analogues and their `elliptic' extensions. A useful `abacus' exploiting the connections with presentations of icosahedral and related symmetries is introduced. Particular and limiting cases, such as those of the semiclassical type, allow a unified view of properties of angular and hyperangular momentum algebra, including relationships among vector coupling coefficients and rotation matrix elements.

Symmetry properties of the direct nonparabolic L-excitons in semiconductors of the T2d ( [formula omitted]) space group and vector-coupling coefficients for CdTe

In most III–V and II–VI compounds with the F 4 ̄ 3 m zinc-blende structure, such as GaAs, CdTe, InP, and also in fcc alkali halides and solid Xenon, the experimental reflectivity peaks near the first Λ and L-transitions are enhanced and probably caused by excitons. In CdTe the reflectivity drops sharply at 3.46 and 4.03 eV and this is to be associated with the electron-hole bound state at the L point and the near by Λ symmetry line. The total symmetry of L and Λ excitons are fully determined by the irreducible representations contained in L 6× L 4,5 and Λ 6× Λ 4,5, and these follow from momentum conservation and selection rules. We have determined the allowed L and Λ exciton momenta. The proper exciton wave function symmetries needed in the variational method for exciton binding energies have been determined by Vector-Coupling Coefficients (VCCs), (known also as the Clebsch–Gordan Coefficients (CGCs)), for the direct non parabolic excitons of the L 6× L 4,5 symmetry. The CGCs can be helpful in the evaluation of intensity of optical transitions, Raman scattering tensors and effective Hamiltonians.

Anharmonic force field and vibrational frequencies of tetrafluoromethane (CF4) and tetrafluorosilane (SiF4)

Accurate quartic anharmonic force fields for CF4 and SiF4 have been calculated using the CCSD(T) method and basis sets of spdf quality. Based on the ab initio force field with a minor empirical adjustment, the vibrational energy levels of these two molecules and their isotopomers are calculated by means of high order Canonical Van Vleck Perturbation Theory (CVPT) based on curvilinear coordinates. The calculated energies agree very well with the experimental data. The full quadratic force field of CF4 is further refined to the experimental data. The symmetrization of the Cartesian basis for arbitrary combination bands of Td group molecules is discussed using the circular promotion operator for the doubly degenerate modes, together with tabulated vector coupling coefficients. The extraction of the spectroscopic constants from our second order transformed Hamiltonian in curvilinear coordinates is discussed, and compared to a similar procedure in rectilinear coordinates.

Representation of potential energy surfaces by discrete polynomials: proton transfer in malonaldehyde

A new method for the expansion of potential energy surfaces has been developed exploiting the peculiar properties of Hahn polynomials, a class of orthogonal polynomials of a discrete variable which generalize 3j vector coupling coefficients of angular momentum algebra. The method has been tested for the Hénon–Heiles potential, a typical model for coupled oscillators, and applied to the representation of the potential energy surface of malonaldehyde, a prototype system for intramolecular proton transfer in polyatomic molecules. The representation is obtained by fitting the polynomial expansion to a set of points calculated by the density functional theory method on a hyperspherical effective three-center coordinate system, in view of perspective quantum dynamical calculations of the proton transfer process.

Graphical analysis of angular momentum for collision products

The complexity of atomic and nuclear structures and their collision processes involves conservation laws, bearing mainly on angular momenta; indeed angular momentum treatments prove most laborious. The analytic treatments, preferably carried out in Racah’s calculus, combine initially independent elements stepwise into structures branching out into resulting products. Graphical procedures that ensure phase and amplitude control of their manifold elements, illustrate these sequential steps and provide their results. The present report should familiarize readers with these procedures through examples of reactions of increasing complexity, bearing of course on structure calculations as well. The report has thus two aims: (i) computing correlation functions for reactions yielding several emitted particles (hence of arbitrary complexity) in terms of a novel method of computation, and (ii), describing the mathematical techniques relevant to solve high-complexity angular momentum problems, including the computation of many-body systems’ bound states. The complexity reflects the symmetries of the reaction products, and, more generally, of many-body system. The basic mathematical tool for such treatments is the Racah calculus which employs recoupling transformations, thus avoiding the many summations required by expansions in terms of vector coupling coefficients. The application of the Racah calculus is greatly aided by appropriate definitions and graphical procedures ensuring phase and amplitude control of their manifold elements, as well as illustrating the physical content. Beginning with photon absorption by discrete states, our examples progress to an Auger process yielding a correlation function with seven direction and polarization parameters.

Hyperspherical Symmetry of Hydrogenic Orbitals and Recoupling Coefficients among Alternative Bases

Fock's representation of momentum space hydrogenic orbitals in terms of harmonics on the hypersphere ${S}^{3}$ of a four-dimensional space is extended to classify alternative bases. These orbitals are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. Because of the correspondence between the ${S}^{3}$ manifold and the $\mathrm{SU}(2)$ group, new sum rules are established which are of relevance for the connection, not only among hydrogen atom orbitals in different bases, but also among the usual vector coupling coefficients and rotation matrix elements.

Multicenter integrals of spherical Laguerre Gaussian orbitals by generalized spherical gradient operators

Multicenter molecular integrals over the spherical Laguerre Gaussian-type functions (LGTFs), Lnl+1/2(ar2)rlYlm(r̂)e−ar2, are evaluated analytically by using the generalized spherical gradient operator method. Addition theorem to expand the generalized spherical gradient operator, Ynlm(∇), is developed. Integrals are evaluated by transforming the product of the gradient operators instead of the product of LGTFs. The transformation G coefficients for the gradient operators are explicitly given in terms of the vector-coupling coefficients, and they are much simpler to evaluate than the Talmi coefficients which transform the product of LGTFs. The integral formulas obtained are compact and general for LGTFs of unrestricted quantum numbers n, l, and m. They are four-center (as well as three-center and two-center) integrals of two-electron irregular solid harmonic operator, Ylm(r̂12)/r12l+1, where l=0, l=1, or l=2 corresponds to electron repulsion, spin-other-orbit, or spin–spin interaction, respectively. The two-center exchange-type and Coulomb-type integrals are also evaluated. In the case of two-center and three-center, integral formulas are also obtained for the one-electron irregular solid harmonic operator, Ylm(r̂)/rl+1, where l=0, l=1, or l=2 corresponds to nuclear attraction, spin-orbit interaction or electron-spin nuclear-spin interaction, respectively. Integrals of multicenter overlap as well as transition multipole moment operator, rlYlm(r̂), have also been derived. All of the integral formulas are explicitly in terms of the vector-coupling coefficients and LGTFs of the internuclear coordinates, where the analytical derivatives of these integrals with respect to the geometrical variables can be easily obtained.

Line-Mixing Effects inQBranches of CO2

A theoretical model based on the energy corrected sudden (ECS) approximation is used to account for line-mixing effects in infraredQbranches of parallel bands of CO2. Measurements have been made using a tunable difference frequency spectrometer system in a Π–Π transition: spectra of the 1111II← 0110IQbranch near 3580.3 cm−1have been recorded at room temperature for CO2–He and CO2–N2mixtures in the 0.1–1 atm total pressure range. Comparisons of experimental and computed spectra prove the quality of the ECS approach. It is shown that the influence of line mixing on the shape of parallelQbranches is very weak and much smaller than in perpendicular transitions. This result is analyzed in detail and explained in terms of the effects of the vibrational angular momenta on the dipole transition moments and vector coupling coefficients involved in the relaxation operator elements. The effects of collisions of CO2with N2and He are compared and discussed.

Molecular integrals over spherical Gaussian-type orbitals: I

A novel derivation, involving the Fourier transform and the addition theorem of harmonic polynomials, is presented for multi-centre molecular integrals over spherical Gaussian-type orbitals. Compact closed-form formulae, consisting of vector-coupling coefficients and well known functions only, are obtained for all multi-centre molecular integrals. The resulting formulae manifest the angular and geometric dependence in vector-coupling coefficients and spherical harmonic functions, respectively, and require half as many summations as those for their counterparts, Cartesian Gaussian orbitals. An efficient computational method for molecular integrals over contracted Gaussian orbitals is suggested based on the present formulae for multi-centre molecular integrals.