A local one-electron operator for the generalized-overlap amplitudes

An approach to the N-electron behavior is presented which emphasizes the dynamics of an individual electron. The generalized overlap amplitudes (GOAS ), although formally defined by an integration over the coordinates of N ? 1 electrons, are, instead, resolved as a column vector, eigenfunction to a local one-electron differential operator. These amplitudes have no restrictions of linear independence between them, but each satisfies the one-electron boundary conditions at the nuclei and at large distances. The one-electron (or charge) density is the sum of the squares of the elements of the column. The energy density, a constant times the one-electron density, maintains this one-to-one relationship throughout modifications in total number of electrons or external potential, although the constant of proportionality, the total energy of the system, may change in the process. Indistinguishability of electrons and antisymmetry is always observed by the dynamics of each electron. A numerical example, the ground state of helium, is presented. © 1995 John Wiley & Sons, Inc.     

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.     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

A simple algorithm for the construction of genealogical spin functions

A simple algorithm is given for the construction of spin eigenfunctions according to the genealogical scheme. The method can deal directly with the N-electron problem without any knowledge of the (N-1)-electron spin eigenfunctions. It uses the representation matrices corresponding to the transpositions (k,k + 1), the latter can be written down from the knowledge of the Young tableaux.     

Aufbau principle and singlet-triplet gap in spherical Hooke atoms

Singlet and triplet spin state energies for three-dimensional Hooke atoms, that is, electrons in a quadratic confinement, with even number of electrons (2, 4, 6, 8, 10) is discussed using Full-CI and CASSCF type wavefunctions with a variety of basis sets and considering perturbative corrections up to second order. The effect of the screening of the electron–electron interaction is also discussed by using a Yukawa-type potential with different values of the Yukawa screening parameter (λ_ee = 0.2, 0.4, 0.6, 0.8, 1.0). Our results show that the singlet state is the ground state for two and eight electron Hooke atoms, whereas the triplet is the ground spin state for 4-, 6-, and 10-electron systems. This suggests the following Aufbau structure 1s < 1p < 1d with singlet ground spin states for systems in which the generation of the triplet implies an inter-shell one-electron promotion, and triplet ground states in cases when there is a partial filling of electrons of a given shell. It is also observed that the screening of electron–electron interactions has a sizable quantitative effect on the relative energies of both spin states, specially in the case of two- and eight-electron systems, favoring the singlet state over the triplet. However, the screening of the electron–electron interaction does not provoke a change in the nature of the ground spin state of these systems. By analyzing the different components of the energy, we have gained a deeper understanding of the effects of the kinetic, confinement and electron–electron interaction components of the energy.     

A graphical approach to permutation group representations for many-electron systems

A simple procedure is presented for obtaining the standard Young tableaux for the representation [(N/2) + S,(N/2) − S] of the permutation group ℒ_N for an N-electron system in spin state S directly from the spin branching diagram. We redefine the coordinate axes of the branching diagram to obtain a graph in terms of the partitions of the two-rowed Young diagram and define walks in this graph which yield directly the first rows of the allowed standard Young tableaux spanning a given representation when suitable weights have been assigned to the nodes in the graph. The allowed states are in a lexically ordered form and permit going easily from an index to an array and vice versa. © 1996 John Wiley & Sons, Inc.     

Analysis of approximations and errors in equations of motion method calculations

We analyze a number of fundamental questions associated with the use of a finite one-particle orbital basis in equations of motion (EOM) method calculations of excitation energies etc., of atomic and molecular systems. This approximation yields an approximate n_e-electron ground state and say, N excited states, while there are (N + 1)² different possible basis operators for EOM calculations. We show that sets of at most 2N basis operators can contribute to the EOM calculations. Any set of 2N basis operators, satisfying certain conditions, provides the exact EOM energies which are equivalent to complete configuration interaction results within the same orbital basis. We investigate the use of particle-particle shifting operators which are not employed in EOM calculations in model calculations on He with operator bases smaller than the complete 2V to consider the convergence of the expansion. The dependence of EOM calculations on the quality of the approximate ground state wavefunction is studied through calculations for Be where additional support is provided for the frequent need for multiconfigurational zeroth order reference functions (as corrected perturbatively). Excited state EOM wavefunctions from EOM calculations are shown to not necessarily be orthogonal to either the exact or approximate ground state wavefunction, suggesting implications in the use of EOM methods to evaluate excited state properties. The He and Be examples and a simple two-level problem are also utilized to illustrate questions concerning the use of the EOM equations to obtain an iteratively improved ground state wavefunction.     

Matrix elements of multi-electron operator in restricted Hartree-Fock theory

A theorem has been established for the matrix elements of a general t-electron operator between N-dimensional determinantal wave functions arising in the solution of the atomic and molecular multi-electron problem by the restricted Hartree-Fock (RHF) method (1 ≤ t ≤ N). The required matrix elements of this operator are sums of matrix elements over t-dimensional basic determinantal wave functions. The final results are especially useful in the determination of multi-electron properties for atoms and molecules when the Hylleraas approach in RHF theory is employed.     

Size consistency of an algebraic propagator approach

The size consistency property of a general algebraic propagator method referred to as intermediate-state representation (ISR ) is discussed. In this method intermediate states |Ψ_j constructed by a specific orthonormalization procedure from the set of “correlated excited states” Ĉ_j|Ψ_n^N are used to represent the Hamiltonian Ĥ. Here Ĉ_j denotes a physical excitation operator and |Ψ_n^N is the N-electron ground state. The ISR secular equations are shown to be separable, that is, they decouple into independent (local) sets of equations for a system consisting of noninteracting (separate) fragments. This result follows from a general factorization theorem for the intermediate states. Separability is a sufficient condition for size consistency. © 1996 John Wiley & Sons, Inc.     

Classical kinetic energy,quantum fluctuation terms and kinetic-energy functionals

We employ a recently formulated dequantization procedure to obtain an exact expression for the kinetic energy which is applicable to all kinetic-energy functionals. We express the kinetic energy of an N-electron system as the sum of an N-electron classical kinetic energy and an N-electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum. This leads to an interesting analogy with Nelson’s stochastic approach to quantum mechanics, which we use to conceptually clarify the physical nature of part of the kinetic-energy functional in terms of statistical fluctuations and in direct correspondence with Fisher Information Theory. We show that the N-electron purely quantum kinetic energy can be written as the sum of the (one-electron) Weizsäcker term and an (N?1)-electron kinetic correlation term. We further show that the Weizsäcker term results from local fluctuations while the kinetic correlation term results from the nonlocal fluctuations. We then write the N-electron classical kinetic energy as the sum of the (one-electron) classical kinetic energy and another (N?1)-electron kinetic correlation term. For one-electron orbitals (where kinetic correlation is neglected) we obtain an exact (albeit impractical) expression for the noninteracting kinetic energy as the sum of the classical kinetic energy and the Weizsäcker term. The classical kinetic energy is seen to be explicitly dependent on the electron phase, and this has implications for the development of accurate orbital-free kinetic-energy functionals. Also, there is a direct connection between the classical kinetic energy and the angular momentum and, across a row of the periodic table, the classical kinetic energy component of the noninteracting kinetic energy generally increases as Z increases. Finally, we underline that, although our aim in this paper is conceptual rather than practical, our results are potentially useful for the construction of improved kinetic-energy functionals.     

Localization measure and maximum delocalization in molecular systems

A mathematically well-defined measure of localization is presented based on Mulliken's orbital populations. It is shown that this quantity equals 1 for core- and lone-pair orbitals, 2 for two-atomic bonds, 6 for benzene rings, etc., and it is applicable for delocalized canonical HF orbitals as well. The definition of this quantity is general in the sense that ab initio MOS with overlapping AO expansion, and semiempirical wave functions using the ZDO approximation as well, can be treated. The localization quantity is essentially “intrinsic,” i.e., no subdivision of the molecule is required. For N-electron wave functions, mean delocalization can be defined. This measure is not invariant to unitary transformations of the one-electron orbitals, characterizing in this way the localized or extended representation of the N-electron wave function. It can be proven, however, that for unitary transformed wave functions a maximum delocalization exists which depends only on the physical (N-electron) properties of the molecule. It is shown that inhomogeneous charge distribution can cause strong electron localization in molecular systems. The delocalization of the canonical Hartree–Fock orbitals, the Parr–Chen circulant orbitals, and the optimum delocalized orbitals is studied by numerical calculations in extended systems.     

Generalized overlap amplitudes for the lithium atom

Generalized overlap amplitudes (GOAS) are calculated between the lithium atom and several states of Li⁺. An examination of the long-range behavior of the GOAS indicates that they are coupled, appearing to have the same exponential decay at large r. At intermediate distances from the nucleus, the GOAS decay with their unique exponential rate and the decay rates only merge at large r. Although many of the GOAS appear to be similar, their distinctness indicates that they may, in fact, be linearly independent. © 1996 John Wiley & Sons, Inc.     

N-Representability conditions generated by a one-particle grassmann factor of an antisymmetric function

N-representability conditions for a two-particle density operator implied by positive-semidefiniteness of the projection operator P^N+1(?¹ Λ Ψ^N) are derived and discussed. The operator P^N+1(?¹ Λ Ψ^N) projects onto an (N + 1)-particle antisymmetric function ?¹ Λ Ψ^N, the Grassmann product of a one-particle factor ?¹ and an N-particle factor Ψ^N. The polar subcone ??²_N(g, q) to the set of N-representable two-particle density operators ??²_N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one-particle Hilbert space. The facial structure of the convex set ??²_N exposed by elements of ??²_N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N-representability is found. It consists of the decomposition of the N-particle antisymmetric identity operator onto the mutually orthogonal projection operators.     

A new operator formulation of the many-electron problem for molecules

An n-electron operatorX _n , called a wave operator, is associated with a 2n-electron molecular wave function. Electron densities and energy are written in terms ofX _n . An equation defining an exact wave operator is found. Thus, a 2n-electron vector problem (for the wave function) is rigorously reduced to an n-electron operator problem. Conditions are formulated which guarantee thatX _n corresponds to a state with a given spin. The configuration-interaction problem is considered and methods of approximate construction ofX _n are discussed. In particular, a matrix algorithm is proposed for calculations in the two-body approximation. A generalizaton of the approach to the case of systems with an odd number of electrons is given. The waveoperator model developed forms a general basis for construction of covariant electron models of molecules.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 1, pp. 1–12, January–February, 1989.     

Symmetric group approach to relativistic CI. IV. Representations of one-electron spin operators and their products in a symmetric group-adapted basis of N-electron spin functions

A detailed study on representations of one-electron spin operators and of their products in an N-electron S_N-adapted spin space is presented. Some conclusions relevant for their evaluation and implementation in relativistic two-component SGA-CI calculations are derived. © 1997 John Wiley & Sons, Inc.     

Carbon-13 nuclear magnetic resonance spectroscopy—VII:para-substituted fluorobenzenes

C-13 and F-19 NMR spectra of seventeen para-substituted fluorobenzenes were measured and the chemical shifts as well as coupling constants with respect to substituents were analysed. The chemical shifts of the fluorine, the C₁ and the C₂ atoms were found to depend on the total electron densities. In the case of the C₃ atom, the chemical shifts seem to depend on π-electron densities rather than the total electron densities. The present calculations also indicate that the chemical shift of the C₄ atom depends mainly on σ-electron densities due to the inductive effects of substituents. The strongest factor influencing the coupling constant, ⁿJ(C? F), is also considered to be the π-electron densities on the carbon atoms. In the case of the direct couplings, ¹J(C? F), the π-bond orders are important.     

Quadrupolar Ultrafast Charge Transfer in Diaminoazobenzene-Bridged Perylenediimide Triads

Strong push-pull interactions between electron donor, diaminoazobenzene (azo), and an electron acceptor, perylenediimide (PDI), entities in the newly synthesized A−D−A type triads (A=electron acceptor and D=electron donor) and the corresponding A−D dyads are shown to reveal wide-band absorption covering the entire visible spectrum. Electrochemical studies revealed the facile reduction of PDI and relatively easier oxidation of diaminoazobenzene in the dyads and triads. Charge transfer reversal using fluorescence-spectroelectrochemistry wherein the PDI fluorescence recovery upon one-electron oxidation, deterring the charge-transfer interactions, was possible to accomplish. The charge transfer state density difference and the frontier orbitals from the DFT calculations established the electron-deficient PDI to be an electron acceptor and diaminoazobenzene to be an electron donor resulting in energetically closely positioned PDI ^δ− -Azo ^δ+ -PDI ^δ− quadrupolar charge-transfer states in the case of triads and Azo ^δ+ -PDI ^δ− dipolar charge-transfer states in the case of dyads. Subsequent femtosecond transient absorption spectral studies unequivocally proved the occurrence of excited-state charge transfer in these dyads and triads in benzonitrile wherein the calculated forward charge transfer rate constants, k_f, were limited to instrument response factor, meaning >10¹² s⁻¹ revealing the occurrence of ultrafast photo-events. The charge recombination rate constant, k_r, was found to depend on the type of donor-acceptor conjugates, that is, it was possible to establish faster k_r in the case of triads (∼10¹¹ s⁻¹) compared to dyads (∼10¹⁰ s⁻¹). Modulating both ground and excited-state properties of PDI with the help of strong quadrupolar and dipolar charge transfer and witnessing ultrafast charge transfer events in the studied triads and dyads is borne out from the present study.     

Synchrotron radiation spectroscopy of quasi-one-dimensional materials

The optical reflectivity spectra of quasi-one-dimensional polymers (SN)_x, trans-(CH)_x, and polydiacetylene are investigated in an energy range from 1 to 25 eV by use of synchrotron radiation. Observed structures associated with π-electron states are interpreted in terms of the quasi-one-dimensional one-electron bands calculated by the LCAO -type model.     

Shielding of Electron Acceptors Coordinated to Rhenium(I) by Carboxylato Groups. Intraligand and Charge‐Transfer Excited‐State Properties of fac‐(‘Anthraquinone‐2‐carboxylato')(2,2′‐bipyridine)tricarbonylrhenium (fac‐[ReI(aq‐2‐CO2)(2,2′‐bipy)(CO)3])

The photophysical and photochemical properties of (OC‐6‐33)‐(2,2′‐bipyridine‐κN¹,κN¹′)tricarbonyl(9,10‐dihydro‐9,10‐dioxoanthracene‐2‐carboxylato‐κO)rhenium (fac‐[Re^I(aq‐2‐CO₂)(2,2′‐bipy)(CO)₃]) were investigated and compared to those of the free ligand 9,10‐dihydro‐9,10‐dioxoanthracene‐2‐carboxylate (=anthraquinone‐2‐carboxylate) and other carboxylato complexes containing the (2,2′‐bipyridine)tricarbonylrhenium ([Re(2,2′‐bipy)(CO)₃]) moiety. Flash and steady‐state irradiations of the anthraquinone‐derived ligand (λ_exc 337 or 351 nm) and of its complex reveal that the photophysics of the latter is dominated by processes initiated in the Re‐to‐(2,2′‐bipyridine) charge‐transfer excited state and 2,2′‐bipyridine‐ and (anthraquinone‐2‐carboxylato)‐centered intraligand excited states. In the reductive quenching by N,N‐diethylethanamine (TEA) or 2,2′,2″‐nitrilotris[ethanol] TEOA, the reactive states are the 2,2′‐bipyridine‐centered and/or the charge‐transfer excited states. The species with a reduced anthraquinone moiety is formed by the following intramolecular electron transfer, after the redox quenching of the excited state: [Re^I(aq−2−CO₂)(2,2′‐bipy^.)(CO)₃]⁻⇌[Re^I(aq−2−CO₂^.)(2,2′‐bipy)(CO)₃]⁻ The photophysics, particularly the absence of a Re^I‐to‐anthraquinone charge‐transfer excited state photochemistry, is discussed in terms of the electrochemical and photochemical results.