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.     

Generalized k-particle brillouin conditions and their use for the construction of correlated electronic wavefunctions

The variational form of the Schrödinger equation is shown to be equivalent to a set of generalized Brillouin conditions in terms of arbitrary antihermitean operators R. For a special choice of these R in second-quantization language, k-particle Brillouin conditions are derived that are a generalization of the “generalized one-particle Brillouin conditions” of Levy and Berthier. The application of these conditions to one-particle and two-particle hamiltonians is discussed. A two-particle generalization of the Fock operator is derived and an iterative variational pair-cluster scheme is derived. It is shown that CEPA and SCEP methods satisfy an approximate rather than an exact set of Brilouin conditions.     

N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

Shell model calculations: An efficient new algorithm

We describe an efficient new algorithm which extends the range of feasible shell model calculations. This algorithm is applicable to single shell and multiple shell configurations, where two or more quantum numbers (e.g., L and S) are required to label the states within each shell. The algorithm proceeds by factoring the shell model Hilbert space into a product of subspaces, one for each angular momentum. N-particle wave functions are built up recursively from N – 1 particle wave functions. Three kinds of N – 1- to N-particle coefficients are required to carry out the construction of N-particle electron (or fermion) states from N – 1 particle states. These are (1) coefficients of fractional parentage (CFP s) within a single shell, (2) outerproduct isoscalar factors (OISF s) within a single angular momentum subspace, and (3) innerproduct isoscalar factors (IISF s) which describe how multishell states within the complementary angular momentum subspaces are combined to form totally antisymmetric wave functions. All three types of N – 1- to N-particle coefficients are generated recursively using a single powerful and efficient matrix diagonalization algorithm. Matrix elements of single particle creation and annihilation operators are expressed in terms of single particle CFP s, OISF s, and IISF s. We also describe an efficient algorithm for computing matrix elements of products of creation and anihilation operators by inserting and summing over complete sets of intermediate states. This is the Feynman-like sum over path overlaps procedure. Timing benchmarks are presented comparing the new Drexel University shell model (DUSM ) code with a state of the art shell model code.     

Traces of spin-adapted reduced density matrices

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ² and Ŝ_z operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.     

Partitioning technique,perturbation theory,and rational approximations

Instead of the Schródinger equation ??Ψ = EΨ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (?? ? ?)Ψ_? = aφ, where ? is a complex variable, with the same boundary condition, which gives a = 〈φ|??|Ψ_?〉 ? ? = ?₁ ? ?. Introducing the projector P = 1 ? |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψ_? = (1 ? P??/?)^?1φ = (1 + T_???)φ, where T_? = (? ? P??)^?1P = P(? ? P??P)^?1P is the reduced resolvent associated with the auxiliary Hamiltonian H? = P??P. The existence of these operators is discussed. It is shown that, if the parameter ? is real in the “discrete part” of the spectrum to ??, then ? and ?₁ = 〈φ|??|Ψ_?〉 = 〈φ|?? + ??T_???|Φ〉 ≡f(?) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(?) = ? ? f(?) = 0 is related to the variation principle. Putting ?? = ??₀ + V and expanding the inverse (? ? P??₀ ? PV)^?1 in terms of powers of V or (V ? α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h₁,h₂,…,h_n} is a linearly independent set, the inner projection Á_n = | h 〉 〈 h |A^?1| h 〉^?1〈 h | is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.     

About the relationship between some necessary conditions for N-representability

The relationship between well known necessary conditions for N-representability of the reduced two-density matrix is investigated. It is shown that the G-condition implies two conditions of the operator endomorphism type: the C- and the B-condition.     

Configuration interaction matrix elements for atoms using permutation group algebra

A procedure is described for the efficient evaluation of the energy matrix elements necessary for atomic configuration-interaction calculations. With the orbital configurations of an N electron system in spin state S written as the irreducible representations [2^1/2N?S, 1^2S] of the permutation group S( N ), it is possible to evaluate readily the energy matrix elements of a spin-free Hamiltonian expressed in terms of the generators of the unitary group. We show how the use of angular momentum ladder operators permits the effective generation of a basis of eigenstates of ??², ??_z as well as ??² and ??_z, for which the energy matrix elements may be evaluated with ease.     

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

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.     

New derivation and a k-particle generalization of SCF-type theories

A hierarchy of necessary conditions that an exact density matrix of a pure state or an ensemble has to satisfy is derived, namely the hermiticity of certain operators F^(k). For k = 1 this reduces to the well-known Hartree-Fock condition. It is then shown that the kth set of conditions is equivalent to stationarity of the energy with respect to unitary k-particle transformations. k-Particle generalizations of Hartree-Fock theory are then discussed both in the spirit of k-particle pseudoeigenvalue equations and in the framework of a Newton–Raphson-type constructive scheme.     

Integral–geometrical consideration of density matrices

The ensemble N-representability problem for the k-th order reduced density matrix (k-RDM ) as well as the problem of reconstruction of the N-particle system density matrices (N-DM ) from a given k-RDM are studied. The spatial parts of the k-RDM expansion in terms of spin tensorial operators Θ are represented using particular values (at specially chosen ) of the Radon transform of the N-DM spatial parts (or their sums) ??_Nλ(x′ | x″) (here, is a d-plane in the n-space ?ⁿ of x = (x′, x″)), with n = 6N, d = 3 (N ? k), x′ ≡ (r′₁, ?, r′_N), x′ ≡ (r₁″, ?, r″_N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM , it is proved that are bounded functions. The properties of implied by the N-DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N-DM corresponding to a given k-RDM is proposed. © 1995 John Wiley & Sons, Inc.     

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.     

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.     

Dye-sensitized photooxidation of chlorpromazine

Chlorpromazine efficently quenches singlet oxygen (¹O₂) with a k_q = 3.5 × 10⁷ M^?1 s^?1. The major result of the chemical interaction between these two species is the cleavage of the N-side chain.     

Quasielastic scattering of neutrons from freely jointed polymer chains in dilute solutions

Starting with Kirkwood's Fokker–Planck equation for the polymer configuration-space distribution function and using the Zwanzig–Mori projection operator technique we have calculated the scattering law S_(q,w) for a freely jointed model polymer chain in a dilute solution. When memory effects are neglected, the theory predicts a Lorentzian for S_(q,w) with a halfwidth Ω_(q), which we have determined as a function of the momentum transfer q for all values of q. The results are compared with recent neutron scattering experiments on deuterated polytetrahydrofuran and polystyrene in dilute solution in CS₂. It is found that the observed q dependence of Ω(q) is represented satisfactorily by the present theory with a bond length b of about 6.3 Å for polystyrene and 3.8 Å for polytetrahydrofuran, and a friction coefficient ζ = 4πη0b where η₀ is the viscosity of the solvent.     

Aliphatic Organoimido Derivatives of Polyoxometalates Containing a Bioactive Ligand

A series of aliphatic organoimido derivatives of hexamolybdate based on amantadine, namely (nBu₄N)₂[Mo₆O₁₈(?NC₁₀H₁₅)] ( 1 ), (nBu₄N)₂ {cis‐[Mo₆O₁₇(?NC₁₀H₁₅)₂]} ( 2 ), (nBu₄N)₂{trans‐[Mo₆O₁₇(?NC₁₀H₁₅)₂]} ( 3 ), and (nBu₄N)₂[Mo₆O₁₆(?NC₁₀H₁₅)₃] ( 4 ), was synthesized in reasonable yield by dehydration with N,N′‐dicyclohexylcarbodiimide (DCC). They were characterized by IR and UV/Vis spectroscopy, elemental analysis, ESI mass spectrometry, and single‐crystal X‐ray structure analysis. The spectral and structural similarities and differences between monosubstituted, cis‐disubstituted, and trans‐disubstituted organoimido derivatives were elucidated and may provide guidance for related work on organoimido‐functionalized Lindqvist‐type polyoxometalates. In addition, trans‐disubstituted and polysubstituted derivatives containing aliphatic organoimido ligands have not yet been reported, and the crystal structure of the trans isomer may lead us to a deeper understanding of disubstituted derivatives. Furthermore, proliferation and morphology of MCF‐7 cells were studied with compound 1 . The present results show that the DCC‐dehydrating protocol could be an efficient approach to covalently graft bioactive ligands such as amantadine onto POMs and enhance their application in clinical cancer treatment.     

Constraints for hierarchies of many electron distribution functions

Some inequalities that constrain the reconstruction of k-electron distribution functions from lower-order distribution functions are presented. These inequalities are related to the N-representability conditions on electron distributions functions and they have two basic types: (1) general N-representability inequalities, which are very powerful but difficult to apply and (2) generalized “Davidson” inequalities, which are less powerful but which may be more facile in computational implementations. A constraint on the exchange-correlation hole is also presented.