Study of correlation holes. I. Number–sum rules and infinite system

The correlation holes in a finite system of any size are known to satisfy integral number–sum rules directly related to the N-representability of the reduced density matrices. In regard to infinite systems, the same rules are known to be satisfied in the electron gas. It is shown that the number–sum rules are not generally satisfied in any infinite system, and that this happens independently of the kind of boundary conditions assumed in the N → ∞ limit. A violation is explicitly found within the alternant molecular orbital formalism. The apparent paradox is explained in terms of surface effects (end effects in a linear system), which are present in a large, but finite system. In other words the N-representability does not imply the number–sum rules. In the N → ∞ limit the rules are satisfied only in physical systems having short-range correlation.     

Generalized density-functional theory: Conquering the N -representability problem with exact functionals for the electron pair density and the second-order reduced density matrix

Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.     

The non-N-representability of the Colle–Salvetti second-order reduced density matrix

The Colle–Salvetti second-order reduced density matrix (2-matrix) is an approximation to the 2-matrix obtained from a wave function that is a product of a reference wave function containing little or no correlation times a product of correlation factors that are functions of the coordinates of pairs of electrons. A formal proof is given for the non-N-representability for the Colle–Salvetti 2-matrix using the nonnegativity condition of the 2-matrix. The nonnegativity condition of the particle-hole overlap matrix (G matrix) is also not satisfied. The proof is valid for Colle–Salvetti 2-matrices obtained from both the Hartree–Fock and small multiconfigurational-self-consistent-field wave functions. Even though the Colle–Salvetti 2-matrix is not N-representable, it does satisfy the Pauli principle component of the G-matrix condition because it reduces to an N-representable first-order reduced density matrix. © 1993 John Wiley & Sons, Inc.     

Utilization of deformations in molecular quantum chemistry and application to density functional theory

The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28 , 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998     

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.     

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.     

Two-body Reduced Density Matrix Reconstruction for Van der Waals Systems

A method for the calculation of the electronic energy of a correlated system is presented. This approach is based on the reconstruction of the total two-body reduced density matrix by doing separate configurations interaction calculations on fragments. The method has been tested on Van der Waals systems and has been implemented by considering restrictive N-representability conditions. It is shown that the computational strategy presented in this work can describe with good accuracy weak dispersion interactions, and considerably lowers the size-consistency error of a classical configuration interaction calculation.     

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.     

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.     

Molecular dynamics simulation in aqueous solution of N-methylazetidinone as a model of β-lactam antibiotics

In this article, we analyze the results of a molecular dynamics simulation in aqueous solution of the N-methylazetidinone molecule, often used to model β-lactam antibiotics. The radial distribution functions (RDFs) corresponding to the most interesting atoms, in terms of reactivity, are presented. We focus our study on the effect of a polar environment on the molecule. The solvent structure around the system is compared to the structure of β-lactam-water complexes, as obtained in a previous study of reaction mechanisms for the neutral and alkaline hydrolyses of N-methylazetidinone. Two types of complexes have been considered which are related to different hydrolysis mechanisms having similar energy barriers at the rate-limiting step of the reaction path. In the first type, the β-lactam-water interaction takes place through the oxygen carbonyl atom and there is agreement between the maxima of the RDFs obtained here and the ab initio structure of the complexes previously reported. In the second type, the interaction takes place through the nitrogen atom and we do not predict a coordination layer around the β-lactam nitrogen atom. The results suggest that in aqueous solution hydrolysis of the carbonyl group is the most probable starting point for the overall hydrolysis reaction. Some discussion on the use of cluster models to represent the solvent effect is included. Received: 29 July 1998 / Accepted: 13 October 1998 / Published online: 1 February 1999     

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.     

New N-representability results with symmetry in molecular quantum chemistry

We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $\rho \Psi $ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Görling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.     

A formulation of statistical mechanics of ordered systems

Expressions are derived for the thermodynamic functions (Gibbs free energy, Helmholtz free energy, etc.) of an ordered system in terms of the single-particle distribution function,p(x), and correlation functions. The thermodynamic functions are treated as functionals of the single-particle distribution function. By minimizing the Helmholtz free energy with respect top(x) under constraints of constantT, V andN, an integral equation is obtained from whichp(x) can be determined. The correlation function of the ordered state in the region near the coexistence surface between ordered and disordered state is expanded about the correlation function of the disordered state, and the series is truncated. Methods for calculating the thermodynamic functions and the single-particle distribution function are presented, and our result is discussed in relation to other treatments of phase coexistence in the literature.     

Novel water-swellable beads based on an acryloylated polyaspartamide

 Spherical polymeric microparticles have been prepared by a reverse-phase suspension polymerization technique. The starting polymer was α,β-poly (N-2-hydroxyethyl)-dl-aspartamide (PHEA) partially functionalised with glycidylmethacrylate (GMA) in order to introduce reactive vinyl groups in the side chain. The PHEA–GMA copolymer obtained (PHG) was cross-linked in a mixture of water/hexane–carbon tetrachloride in the presence of sorbitan trioleate (Span 85) as surfactant and ammonium persulfate/N,N,N′,N′-tetramethylethylenediamine as initiator system. The reaction was also carried out in the presence of N,N′-dimethylacrylamide as comonomer or N,N′-ethylenebisacrylamide as a cross-linking agent. The beads obtained were characterized by Fourier transformIR spectrophotometry, particle size distribution analysis and scanning electron microscopy, which revealed their microporous structure. X-ray diffractometry and differential scanning calorimetry showed the amorphous state and the dependence of the glass-transition temperature on the chemical structure of the samples prepared, respectively. Finally, in order to have information on the water affinity of the networks obtained, swelling measurements were performed. The water regain values and the high rate of swelling demonstrate a remarkable ability of the samples investigated to entrap water molecules. The dependence of aqueous swelling on the degree of cross-linking and the chemical structure of the samples is also shown. Received: 8 August 2000 Received in revised form: 6 November 2000 Accepted: 14 November 2000     

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     

An integral direct, distributed-data, parallel MP2 algorithm

Summary A scalable integral direct, distributed-data parallel algorithm for four-index transformation is presented. The algorithm was implemented in the context of the second-order M?ller-Plesset (MP2) energy evaluation, yet it is easily adopted for other electron correlation methods, where only MO integrals with two indices in the virtual orbitals space are required. The major computational steps of the MP2 energy are the two-electron integral evaluationO(N ⁴) and transformation into the MO basisO(ON ⁴), whereN is the number of basis functions, andO the number of occupied orbitals, respectively. The associated maximal communication costs scale asO(n _Σ O ² V N), whereV andn _Σ denote the number of virtual orbitals, and the number of symmetry-unique shells. The largest local and global memory requirements areO(N ²) for the MO coefficients andO(OV N) for the three-quarter transformed integrals, respectively. Several aspects of the implementation such as symmetry-treatment, integral prescreening, and the distribution of data and computational tasks are discussed. The parallel efficiency of the algorithm is demonstrated by calculations on the phenanthrene molecule, with 762 primitive Gaussians, contracted to 412 basis functions. The calculations were performed on an IBM SP2 with 48 nodes. The measured wall clock time on 48 nodes is less than 15 min for this calculation, and the speedup relative to single-node execution is estimated to 527. This superlinear speedup is a result of exploiting both the compute power and the aggregate memory of the parallel computer. The latter reduces the number of passes through the AO integral list, and hence the operation count of the calculation. The test calculations also show that the evaluation of the two-electron integrals dominates the calculation, despite the higher scaling of the transformation step.     

Diaminomethylidene derivatives of β-oxo sulfones as potential reagents in heterocyclic synthesis and chelating ligands

Active methylene β-oxo sulfones add to the C≡N bond of benzoylcyanamide in the presence of catalytic amounts of Ni(acac)₂. Debenzoylation of the reaction products under the action of MeONa in MeOH gives N,N′-unsubstituted diaminomethylidene derivatives of β-oxo sulfones (acyl(R-sulfonyl)ketene aminals), which can be used as reagents for heterocyclic synthesis and as chelating ligands. The syntheses of 2-amino-3-arylsulfonylpyridin-4(1H)-ones and 5-sulfonylcytosine derivatives are presented as examples.     

Studies on the radiolytic stability of N,N,N′,N′-tetra-2-ethylhexyl diglycolamide in n-dodecane solution containing different phase modifiers

The radiolytic stability of a branched diglycolamide extractant, namely N,N,N′,N′-tetra-2-ethylhexyl diglycolamide (T2EHDGA) dissolved in n-dodecane containing several phase modifiers, viz. N,N-dihexyloctanamide (DHOA), tri-n-butyl phosphate (TBP), 1-decanol and iso-decanol has been investigated. The distribution ratio of Am(III) decreased with increased radiation dose studied up to 1000 kGy. Nevertheless, all the composition of extractants showed satisfactory results up to 500 kGy, beyond which the extractants degraded drastically. The stripping behaviour of Am(III) with 0.2 M HNO₃ was found to be unaffected even with the ligand solution irradiated up to 1000 kGy. Extraction of fission product and structural elements was also investigated using the irradiated solvents and was found to be not significantly affected with increasing absorbed dose with the exception of Mo which showed sharp rise in the distribution coefficient values. Loading of Nd in the organic phase decreased with the irradiated solvent due to degradation of the carrier. The effect of the absorbed dose on physical parameters such as density, viscosity and interfacial tension of the solvents has also been investigated.