Generalized Racah method for the evaluation of CFP for mixed electronic configurations

The Racah method, proposed for the evaluation of coefficients of fractional parentage (CFP ) for single shell configurations, is generalized for the cases of two and more than two shells. On the basis of the generalized Racah method, we have evaluated one-particle CFP for all states of s^vp^μ electronic configurations. Two-particle CFP are expressed through one-particle CFP and Racah coefficients.     

Explicit representation of Gel'fand–Tsetlin states in clifford algebra unitary group approach

A explicit expression for the unitary group Clebsch–Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand–Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ? U(n – 1) ? …? ? U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand–Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.     

Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

On the decomposition of the unitary group and the multiple coupling of angular momenta

The decomposition of the direct product group, S_n ⊗ U_N, for a system of n identical particles having access to N one-particle states is considered from the point of view of its reduction to invariant subspaces. A double-factor matrix approach is developed in terms of the frequency of occurrence of the invariant subspaces and the frequency of configurations arising from the set of one-particle states. Simple formulas are obtained for both these frequencies. It is shown that this approach is especially useful when treating the problem of multiple coupling of the quantum numbers of the states. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62 : 331–341, 1997     

On the determination of the exact number of bound states of a given potential. I. General method

A general method is developed for determining the exact number of bound states for a given potential in a space of N dimensions (N ≧ 2). This is applied to a central potential of the form where λ, r₀, α are arbitrary positive numbers, and for which analytical solutions are obtained. Some properties of the degeneracy in the energy levels, with respect to the orbital angular momentum l, are also discussed.     

All-pair wave function and reduced variational equation for electronic systems

A new type of wave function is proposed for atomic and molecular systems. This all-pair function is constructed of N(N – 1)/2 identical geminals for N electrons. For systems with the highest multiplicity this is the full space part of the wave function. For closed shell systems it has to be multiplied by a Slater determinant according to the antisymmetry condition. In the case of maximal multiplicity a reduced variational equation is derived for the geminal. This equation is independent of the dimensionality of the system and contains the particle number as a multiplicative factor only. The method is extended to the closed shell case where a restriction has to be fulfilled. The reduction of the variational equation can be done only approximately. The use of identical geminals can be treated as a first approximation. An extension of the method, called the pair interdependent configuration interaction (PICI), is proposed. The special features of the method are discussed briefly.     

Microscopic Motion of Atoms in Simple Liquids at Equilibrium and with Shear Flow

Abstract

We investigate the microscopic mechanism of atomic motion and local stress relaxation in Lennard-Jones, LJ liquids using a new class of correlation functions that emphasise the interplay between an abitrary atom in the fluid and its surrounding shells of atoms. We use the linear momenta and stress tensor to characterise the time dependence of this interaction. We consider a series of correlation functions that give complementary information and build a picture of the single particle and small cluster motion. The central particle and first shell undergo a reversal in momentum at different times after the ‘collision’ of the central particle and its first shell of neighbours. This ‘phase difference’ becomes manifest in the subsequent dynamics probed by the new correlation functions. We also consider the effect of a non-newtonian shear flow on this local dynamical relaxation, using profile biased laminar flow equations of motion. In non-newtonian shear flow we find the momentum transfer between particle and cage to be less pronounced and occur over a wider time range.     

Resonances in helium atoms associated with theN=4 andN=5 He+ thresholds

Some doubly excited autoionising states of helium atoms converging on theN=4 andN=5 He⁺ thresholds are calculated by use of a method of complex-coordinates. States withL≧2 and with parities of (?1) ^L and (?1) ^L+1 are calculated by using products of Slater-orbital type wave functions with expansion lengths up to 319 terms. Resonance parameters (both resonance energy and autoinoisation width) are calculated for states with angular momentum up toL=7 forN=4 resonances, andL=8 forN=5 resonances.     

Coulomb energy averaged over the nlN-atomic states with a definite spin

A purely group-theoretical approach (for which the symmetric group plays a central role), based upon the use of properties of fractional-parentage coefficients and isoscalar factors, is developed for the derivation of the Coulomb energy averaged over the states, with a definite spin, arising from an atomic configuration nl^N. © 1995 John Wiley & Sons, Inc.     

Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

We construct bases for the irreducible representations of the rotation group O(3) which are symmetry adapted to a Crystallographic point group. We obtain explicit expressions for the cubic groups, which are valid for arbitrary values of the angular momentum quantum number l. Our method yields an efficient algorithm for both analytical and numerical work. An explicit formula for the multiplicities of an irreducible representation for the cubic groups in an arbitrary angular momentum term l is also derived.     

The extended Koopmans' theorem Fock operator

By expanding the wave function of a system of N particles in terms of products of functions of one and (N-1) particles, the one-particle, nonlocal operator F?_EKT (extended Koopmans' theorem) is determined. It is shown that although this operator is nonhermitian, its eigenvalues and eigenfunctions represent the ionization energies and occupied orbitals, respectively. The eigenfunctions of F?_EKT are the one-particle functions that enter into the expansion of the wave function of the system as partners of the (N-1)-particle wave functions. The eingenvalues are also one-particle energies that, multipled by the orbital occupancy probalities, enter the expression for the total N-particle energy of the system.     

Information theory of D‐dimensional hydrogenic systems: Application to circular and Rydberg states

The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power‐like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D‐dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s‐states, circular states (i.e., single‐electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

The dynamics of equally spaced multilevel model systems

Coherent dynamics of multilevel systems is of interest for the elucidation of energy disposal and multiphoton chemistry of polyatomic molecules. We compare the analytical solutions for coherent multiphoton excitation of two sparse multilevel model systems: (1) The spin-J system consisting of N = (2J + 1) equally spaced levels with the radiative coupling being determined by the xth component of the angular momentum. (2) The equally coupled N level system. The transition amplitudes for system (1) are always periodic, which is in contrast to the nonperiodic behavior exhibited by system (2) for N > 3.     

Complex‐scaling calculations for resonance states of he with screened coulomb potentials

The doubly excited ¹S^e, ¹P^o, and ¹D^e resonance states of helium atom with screened Coulomb potentials are investigated. The complex scaling method with configuration interaction type basis functions are employed to extract the resonances associated with the He⁺(N = 2, 3, and 4) thresholds. 18 resonances (six below each of the He⁺ threshold) for each angular‐momentum state are calculated. The results lying below the He⁺(N = 2) threshold are in good agreement with previous calculations by the stabilization method with correlated basis wave functions. The ¹P^o and ¹D^e resonance states lying below the He⁺(N = 3 and 4) thresholds in the screening environment are reported for the first time. © 2013 Wiley Periodicals, Inc.     

Dimensionality control of coupled scattering equations using partitioning techniques

A method of variable reduction of the dimensionality of the coupled equations for inelastic scattering is presented, based upon a projection operator P with a restricted range of orbital angular momentum states. For rotational states in the range O?j ?j^* and total angular momentum large, the coupled equations have dimensionality (j^* + 1) ? N ?(j^* + 1)², where the value of N is controlled by the choice of P. This is in contrast to conventional partitioning techniques which utilize further restrictions on the important molecular rotational states. The equations for the P subspace and its complementary Q subspace are decoupled by an approximation on the equation of motion of Qψ_scat. Information about scattering into the Q subspace is retained, within this degree of approximation, and is reintroduced at the end of the computation with little additional labor. The theory is developed in terms of atom-rigid-rotor scattering, although addition of vibrational modes would not in any way interfere with the basic techniques used.     

Photochemical Reactions of N‐(2‐Halogenoalkanoyl) Derivatives of Anilines

The photochemical reactions of 2‐substituted N‐(2‐halogenoalkanoyl) derivatives 1 of anilines and 5 of cyclic amines are described. Under irradiation, 2‐bromo‐2‐methylpropananilides 1a – e undergo exclusively dehydrobromination to give N‐aryl‐2‐methylprop‐2‐enamides (=methacrylanilides) 3a – e (Scheme 1 and Table 1). On irradiation of N‐alkyl‐ and N‐phenyl‐substituted 2‐bromo‐2‐methylpropananilides 1f – m , cyclization products, i.e. 1,3‐dihydro‐2H‐indol‐2‐ones (=oxindoles) 2f – m and 3,4‐dihydroquinolin‐2(1H)‐ones (=dihydrocarbostyrils) 4f – m , are obtained, besides 3f – m . On the other hand, irradiation of N‐methyl‐substituted 2‐chloro‐2‐phenylacetanilides 1o – q and 2‐chloroacetanilide 1r gives oxindoles 2o – r as the sole product, but in low yields (Scheme 3 and Table 2). The photocyclization of the corresponding N‐phenyl derivatives 1s – v to oxindoles 2s – v proceeds smoothly. A plausible mechanism for the formation of the photoproducts is proposed (Scheme 4). Irradiation of N‐(2‐halogenoalkanoyl) derivatives of cyclic amines 5a – c yields the cyclization products, i.e. five‐membered lactams 6a , b , and/or dehydrohalogenation products 7a , c and their cyclization products 8a , c , depending on the ring size of the amines (Scheme 5 and Table 3).     

The antisymmetrized geminal power approximation to the excitation propagator

A consistent propagator approximation, denoted as the excitation propagator, is introduced. This propagator describes excitations between N-particle states and its approximation has properties required of consistent random phase approximation schemes. Several properties of this propagator are explored when based on a generalized antisymmetrized geminal power wavefunction. How singularities in the metric occur and how to remove them is discussed in detail. The excitation propagator is also contrasted with the principal (polarization) propagator.     

A density matrix variational calculation for atomic Be

The ground-state energy of the beryllium atom is calculated using a variational procedure in which the elements of the two-body reduced density matrix (particle–particle matrix) are the variational parameters. It is shown that, for this problem and with the limited number of spin-orbitals used, the trace condition and the simultaneous nonnegativity conditions on the particle–particle, the particle–hole, and the hole–hole matrices form a complete solution to the N-representability problem. The energy obtained is – 14.61425 a.u., practically identical to the value given by a configuration interaction calculation which uses the same states. The effects of weakening the nonnegativity conditions on each of the matrices in turn were also explored.