The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

On the use of multipole expansion of the Coulomb potential in quantum chemistry

Some features of the multipole expansion of the Coulomb potential V for a system of point charges are studied. It is shown that multipole expansion is convergent both locally in L₂(R³) and weakly on some classes of functions. One-particle Hamiltonians H_n = H₀ + V_n, where H₀ is the kinetic energy operator and V_n is the n-th partial sum of the multipole expansion of V, are discussed, and the convergence of their eigenvalues to those of H = H₀ + V with increasing n is proved. It is also shown that the discrete spectrum eigenfunctions of H_n converge to those of H both in L₂(R³) (together with their first and second derivatives) and uniformly on R³. © 1996 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.     

Matrix elements of one- and two-electron operators

A graphical representation of matrix elements of spin-free one- and two-electron operators is used for deriving a simple algorithm for the evaluation of their values. The method covers all the cases which may occur when wave functions are taken as mutually orthogonal antisymmetrized products of spinorbitals (which are assumed to form an orthonormal set) and are eigenfunctions of L ² and L _z operators. The resulting formulas are suitable as well for computer programming as for hand calculations.

---

Zusammenfassung Mit Hilfe einer graphischen Darstellung wird ein einfacher Algorithmus zur Bestimmung der Werte der Matrix-Elemente von spinfreien Ein- und Zweielektronenoperatoren abgeleitet. Durch diese Methode werden alle Fälle erfaßt, die auftreten, wenn die Wellenfunktionen wechselweise orthogonale, antisymmetrisierte Produkte von Spinorbitalen (von denen angenommen wird, daß sie einen orthogonalen Satz bilden) darstellen und Eigenfunktion der L ² und L _z Operatoren sind. Die erhaltenen Formeln eignen sich für Rechnungen mit und ohne Verwendung eines Computers.

---

---

Sponsored by the Mathematical Institute, Polish Academy of Sciences.     

The geometry and spectra of hyperbolic manifolds

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceH ⁿ by discrete subgroups Г, i.e. ℳ =H ⁿ/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles. Research supported in part by NSF grant DMS93-07438     

Syntheses and structures of copper complexes of tetradentate enaminones derived from condensation of benzoylacetone and ferrocenoylacetone with 1,2- bis (2-aminophenoxy)ethane

Reaction of 1,2-bis(2-aminophenoxy)ethane with benzoylacetone and ferrocenoylacetone yields tetradentate enaminones H₂L¹ and H₂L², respectively. Reaction of copper acetate with two enaminones affords the corresponding complexes 1 and 2, which are formulated as [L¹Cu] and [L²Cu], respectively. The structures of H₂L¹, H₂L² and 1 have been determined by single-crystal X-ray crystallography. Enaminone H₂L¹ crystallizes with Z = 4 in space group C2/c, the molecule of which lies on a C ₂ axis. Enaminone H₂L² crystallizes with Z = 2 in space group P2₁/c, the molecule of which possesses a symmetric center. For complex 1, which crystallizes in space group P 1, H₂L¹ is a bianionic tetradentate donor via two carbonyl oxygens and two deprotonated enamine nitrogens; the coordination geometry of copper(II) is a distorted tetrahedron.     

Some properties of the bivariational Hartree–Fock scheme for complex symmetric many-particle operators

The bivariational Hartree–Fock scheme for a general many-body operator T is discussed with particular reference to the complex symmetric case: T^? = T*. It shown that, even in the case when the complex symmetric operator T is real and hence also self-adjoint, the complex symmetric Hartree–Fock scheme does not reduce to the conventional real form, unless one introduces the constraint that the N-dimensional space spanned by the Hartree–Fock functions ? should be stable under complex conjugation, so that ?* = ?α. If one omits this constraint, one gets a complex symmetric formulation of the Hartree–Fock scheme for a real N-electron Hamiltonian having the properties H = H* = H^?, in which the effective Hamiltonian H_eff (1) may have complex eigenvalues ε_k. By using the method of complex scaling, it is indicated that these complex eigenvalues—at least for certain systems—may be related to the existence of so-called physical resonance states, and a simple example is given. Full details will be given elsewhere.     

Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Nonadiabatic effects in the nuclear dynamics of the H₂⁺ molecular ion, initiated by ionization of the H₂ molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H₂⁺ and D₂⁺ molecules are analyzed. In particular, it is shown that the position-momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.     

Self-Assembly of a Three-Dimensional [Ga6(L2)6] Metal–Ligand “Cylinder”

A near trigonal antiprism with metal–metal distances in the nanometer regime is formed by the six metal ions in the crystalline, homochiral [Ga₆(L²)₆] (see structure). This metal–ligand “cylinder” is based on a threefold symmetric, β-diketone ligand, and represents a new geometry for metal–ligand clusters.     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

Novel Trinuclear MnII/MnII/MnII Complexes – Crystal Structures and Catalytic Properties

The trinuclear manganese(II) complexes [Mn₃(L¹)₂(μ‐OAc)₄]·2Et₂O {HL¹ = (1‐hydroxy‐4‐nitrobenzyl)((2‐pyridyl)methyl)((1‐methylimidazol‐2‐yl)methyl)amine} ( 1·2EtOH ), [Mn₃(L²)₂(μ‐OAc)₄] {HL² = ((1‐methylimidazol‐2‐yl)methyl)(1‐hydroxybenzyl)amine} ( 2 ) and [Mn₃(L³)₂(μ‐OAc)₆] {L³ = bis(1‐methylimidazol‐2‐yl)methanone} ( 3 ) were synthesized. The compounds were characterized by X‐ray crystallography, mass spectrometry, IR, UV‐vis spectroscopy, and elemental analysis. The manganese atoms in 1 and 2 are bridged by phenol moieties of the ligands and acetates. In 3 they are only bridged by acetates in a mono‐ and bi‐dentate way. The resulting Mn···Mn distances are 3.233(1) Å ( 1 ), 3.364(1) Å ( 2 ) and 3.643(7) Å ( 3 ). In the present compounds different limiting cases for the phenomenon of the carboxylate shift are realized. Besides symmetric mono‐ and bi‐dentate bridging an unusual intermediate is also observed. 1·2EtOH is the first example of a trinuclear model for the OEC that shows catalase activity. Furthermore it was characterized by temperature dependent magnetic susceptibility measurements and a total spin ground state of S_t = 5/2 was found. The results for 1 reveals antiferromagnetic coupling between the central and the terminal manganese ions, with J = ?1.2 cm^?1, g = 2.00 (fixed), χ_TIP = 150×10^?6 cm³mol^?1.     

Synthesis,structure and biological activity of diorganotin derivatives with pyridyl functionalized bis(pyrazol‐1‐yl)methanes

Three pyridyl functionalized bis(pyrazol‐1‐yl)methanes, namely 2‐[(4‐pyridyl)methoxyphenyl] bis(pyrazol‐1‐yl)methane (L¹), 2‐[(4‐pyridyl)methoxyphenyl]bis(3,5‐dimethylpyrazol‐1‐yl)methane (L²) and 2‐[(3‐pyridyl)methoxyphenyl]bis(pyrazol‐1‐yl)methane (L³) have been synthesized by the reactions of (2‐hydroxyphenyl)bis(pyrazol‐1‐yl)methanes with chloromethylpyridine. Treatment of these three ligands with R₂SnCl₂ (R = Et, n‐Bu or Ph) yields a series of symmetric 2:1 adducts of (L)₂SnR₂Cl₂ (L = L¹, L² or L³), which have been confirmed by elemental analysis and NMR spectroscopy. The crystal structures of (L²)₂Sn(n‐Bu)₂Cl₂·0.5C₆H₁₄ and (L³)₂SnEt₂Cl₂ determined by X‐ray crystallography show that the functionalized bis(pyrazol‐1‐yl)methane acts as a monodentate ligand through the pyridyl nitrogen atom, and the pyrazolyl nitrogen atoms do not coordinate to the tin atom. The cytotoxic activity of these complexes for Hela cells in vitro was tested. Copyright © 2010 John Wiley & Sons, Ltd.     

Characters of two-row representations of the symmetric group

Characters of irreducible representations (irreps) of the symmetric group corresponding to the two-row Young diagrams, i.e., describing transformation properties of N-electron eigenfunctions of the total spin operators, have been expressed as explicit functions of the number of electrons N and of the total spin quantum number S. The formulas are useful in various areas of theory of many-electron systems, particularly in designing algorithms for evaluation of spectral density moments. © 1997 John Wiley & Sons, Inc.     

Synthesis of pyridyl derivatives for the future functionalization of biomolecules labeled with the fac-[<Superscript>188</Superscript>Re(CO)<Subscript>3</Subscript>(H<Subscript>2</Subscript>O)<Subscript>3</Subscript>]<Superscript>+</Superscript> precursor

In this paper, we investigated three ligand systems, symmetric and asymmetric pyridyl-containing tridentate ligands (L¹NH₂ = (bis(2-pyridylmethyl)-amino)-ethylamine, L²H = (bis(2-pyridylmethyl)-amino)-acetic acid, L³NH₂ = [(6-amino-hexyl)-pyridyl-2-methyl-amino]-acetic acid) as bifunctional chelating agents for labeling biomolecules. These ligands reacted with the precursor fac-[¹⁸⁸Re(CO)₃(H₂O)₃]⁺ and yielded the radioactive complexes fac-[¹⁸⁸Re(CO)₃L] (L = three ligands), which were identified by RP-HPLC. The corresponding stable rhenium tricarbonyl complexes (1–3) were allowed for macroscopic identification of the radiochemical compounds. ¹⁸⁸Re tricarbonyl complexes, with log P _o/w values ranging from −1.36 to −0.32, were obtained with yields of ≥90% using ligand concentrations within the 10⁻⁶−10⁻⁴M range. Challenge studies with cysteine and histidine revealed the high stability properties of these radioactive complexes, and biodistribution studies in normal mice indicated a fast rate of blood clearance and high rate of total radioactivity excretion, primarily through the renal-urinary pathway. In summary, these asymmetric and symmetric pyridyl-containing tridentate ligands are potent bifunctional chelators for the future biomolecules labeling of fac-[¹⁸⁸Re(CO)₃(H₂O)₃]⁺.     

Use of the squared dirac operator in variational relativistic calculations

The squared Dirac operator D² is bounded from below and its eigenvalues and eigenfunctions can be obtained from the variation principle. Errors of eigenvalues of D² in a matrix representation are of the same order as for the non-relativistic hamiltonian, while those from a matrix representation of D are larger by ○ (c²). Numerical results are presented for O⁷⁺ and for H₂⁺.     

Rotation energies for deuterated H+3 oscillators in zero-point states of vibration

Previous ab initio studies of deuterated H⁺₃ molecular ions are extended to include rotational modes for the zero-point states of vibration. Rotation energies are obtained using direct numerical diagonalization of vibration—rotation hamiltonian matrices, and nuclear wavefunctions as superpositions of mode-coupled anharmonic rotationless vibrators and related prolate symmetric top eigenfunctions. Relevance to recent searches for interstellar H₂D⁺ is noted.     

On the resolution of a double determinant

The resolution of a double determinant into a sum of spin-free orthogonal spin eigenfunctions is presented; its equivalent resolution of a spin function product θ^M₀ is also given. Furthermore, a more general resolution of a Slater determinant into all spin eigenfunctions with S ≥ M is also obtained. Present work has also provided a method to calculate the linear coefficients in the expression of the primitive θ^M_i in terms of the linear combination of spin eigenfunctions X? (S, M).     

Nicke(II) complexes of half unit,symmetric and unsymmetric Schiff base ligands

Summary The template reaction of isonitrosoacetylacetone (Hina) witho-phenylenediamine (o-phenen) in the presence of (MeCO₂)₂Ni·4H₂O in EtOH yielded three types of nickel(II) complexes (depending on the molar ratio of the reactants) formulated as L¹Ni(O₂CMe)·2H₂O (1), (L¹)₂Ni (2), and L²Ni·H₂O (3). HL¹ and H₂L² are the half unit and symmetric Schiff base ligands obtained from the (11) and (21) condensation of (Hina) with (o-phenen) respectively. The (11) molar ratio reaction of (1) with either acetylacetone (Hacac) or (Hina) in CHCl₃ led to the formation of mixed ligand complexes L¹Ni(acac)·H₂O (4) and L¹Ni(ina)·H₂O (5) whereas a similar reaction with salicylaldehyde (Hsal) produced L³Ni (6); H₂L³ is the unsymmetric Schiff base formed by the (11) condensation of the amino group in (1) with (Hsal). Analytical, spectral and magnetic moment evidence are compatible with the suggested structures of the metal complexes.This paper is a summary of the M.Sc. thesis of S. M. Imam.     

Synthesis and characterisation of four novel N x O y -Schiff-base macrocyclic ligands and their metal complexes

Macrocyclic Schiff-bases, L¹ and L², and bis-macrocyclic ligands, L³ and L⁴, have been prepared by [1 + 1] and [2 + 1] cyclocondensation of 1,7-bis(2-formylphenyl)-1,4,7-trioxaheptane or 2,6-bis(2-formylphenoxymethyl)pyridine with 2,6-bis(o-aminophenoxymethyl)pyridine or 1,2,4,5-tetraaminebenzene · 4HCl, respectively. The related Y^III, M^II (M=Co, Ni, Zn, Cd or Pb) and Ln^III nitrate or perchlorate complexes have been synthesised following a template procedure, demonstrating the effectiveness of these metal ions to promote the assembly of these four symmetric Schiff-base macrocycles. The complexes were characterised by elemental analyses, molar conductivity, mass spectrometry, i.r., u.v.–vis, and ¹H-n.m.r spectroscopy.