Open- and closed-shell states in few-particle quantum mechanics. II. Classification of atomic states

The system developed in the first paper of this series for the classification of states as open- or closed-shed type is applied to atomic states. These may be classified in the isoelectronic limit (Z → ∞) from knowledge of the true wave function in this limit. One-matrix occupation numbers are tabulated for a number of states of the first-row atoms in the limit Z → ∞ and the states classified. A classification for finite Z is possible in the framework of the Z-dependent perturbation theory by use of a thoerem for the stability of a closed-shed with respect to small perturbations. Such a stability does not hold in general for open-shel states.     

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.     

Distributed Gaussian discrete variable representation

A discrete variable representation (DVR) made from distributed Gaussians g_n(x) = e, (n = ?∞, …, ∞) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sinc function DVR of Colbert and Miller in the limit c → 0. The numerical performance of both finite and infinite grid DGDVRs and the sinc function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sinc function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sinc function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sinc function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d → 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sinc function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d → 0 to the incompleteness of this basis in this limit. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Phase transitions in polymer brushes

Liquid crystalline ordering in planar polymer brushes is investigated theoretically by numerical calculations within a self-consistent field approximation. The brushes are formed by macromolecules with mesogenic groups in main chain and immersed in a solvent. Existence of a microphase segregated brush (MSB) regime with a collapsed orientationally ordered intrinsic sublayer and a swollen external sublayer is shown. At small grafting density, the transition from a conventional brush state to the microphase segregated state is a jump-wise first order phase transition for a finite chain length (N). The magnitudes of the jumps in the average characteristics of the brush tend to zero in the limit N → ∞ since this transition occurs only in a vanishingly small part (∝ N^−1/2) of the brush. High compressibility of MSB brush is demonstrated. The origin of phase transition in planar brushes is discussed.     

The electron density distribution in CN−, LiCN and LiNC. The use of minimal and extended basis set SCF calculations

Electron density maps are reported for the CN^?ion and the LiCN and LiNC molecules, calculated from molecular wave-functions near the Hartree-Fock limit. The electron density distribution derived from CNDO/ 2 wavefunctions does not resemble the ab initio results. The ultimate ability of a minimal basis set to represent the electron density near the Hartree-Fock limit, has been tested. The requirement of N-representability of the trial electron density has been satisfied. It is found that the molecular valence density cannot be reproduced to a satisfactory extent by a minimal set of Slater orbitals, even when the exponents of the basis orbitals are optimized.     

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     

Dimensional scaling and renormalization

Chief features of dimensional scaling methods are exemplified by briefly reviewing prototypical applications and recent developments. The pseudoclassical large-D limit usually can be evaluated exactly regardless of the magnitude, nature, and number of strong, nonseparable dynamical interactions. Often, this limit can be accurately linked to D = 3 by perturbation or interpolation methods. This is because the dimension dependence of many-body effects tends to be smooth and mild when calibrated by appropriate one- or few-body problems. A simple renormalization procedure applied to atoms with up N ∼ 100 electrons yields a major part of the correlation energy. From Hartree-Fock input, a renormalized nuclear charge is determined which renders the dimensionally scaled energy at D → ∞ a good approximation to that for D = 3 with the actual Z. Prospects are discussed for other means to exploit dimensional scaling, including an analogous renormalization procedure for molecules. © 1996 John Wiley & Sons, Inc.     

Numerical solution for the ground-state energy of the anisotropic Heisenberg model

An analysis of the anisotropic Heisenberg model is carried out by solving the Bethe ansatz solution of the model numerically as a function of the anisotropy parameter for finite N. A brief introduction to the limit of the infinite chain is presented. The energy for a few special limiting cases of the anisotropy parameter in the Hamiltonian are worked out. Numerical results for finite cycles as well as for the infinite chain are given. Comparison can then be made with the case of finite increasing N. © 1997 John Wiley & Sons, Inc.     

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.     

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.     

Postgel properties in the statistical crosslinking of heterochains. II. Free‐radical crosslinking copolymerization

The heterochain crosslinking theory is applied to postgel behavior in the free‐radical crosslinking copolymerization of vinyl and divinyl monomers. In this context, the crosslinked polymer formation can be viewed as a system in which the primary chains formed at different times are combined in accordance with the statistical chain‐connection rule governed by the chemical reaction kinetics. Because the primary chains are formed consecutively, the number of chain types N must be extrapolated to infinity, N → ∞. Practically, such extrapolation can be conducted with the calculated values for only three different N values. The analytical expressions for the weight fraction and average molecular weights of the sol fraction are derived for the general primary chain length distribution function in free‐radical polymerization. Illustrative calculations show that the obtained results agree with those from the Monte Carlo method, and that the postgel properties in free‐radical crosslinking copolymerization systems could be significantly different from those in randomly crosslinked systems. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2342–2350, 2000     

Electronic structure of long cumulene chains

The analysis of the equations of the unrestricted Hartree–Fock (UHF) method for polyenes C_NH_N+2 with even and odd N » 1 is carried out. The equations of the UHF method are shown to be the same in both cases. The comparison of the UHF method with the extended Hartree–Fock (EHF) method applied to large systems is performed. The ground state and π-electron spectra of long cumulene chains C_NH₄ are treated by the EHF Method. The end effects are taken into consideration. It is shown that the EHF method gives a finite value of the first optical transition frequency and, at the same time, zero value of torsion barrier of end CH₂–groups in long cumulene chains (N → ) in contrast to previous calculations of cumulenes by the Huckel method and the restricted Hartree–Fock method.     

Extrapolation of real-space quantum chemical calculations from finite-size super cells to the ideal infinite system. II. Application to one-dimensional polymers

Calculations of finite atomic clusters based on the Hartree–Fock self-consistent field theory are modified to model more closely the ideal behavior of the infinite system. The density matrix of the standard finite supercell calculation is extrapolated to an infinite supercell so that it contains information from a continuum of k points in the first Brillouin zone. This modification is incorporated into the self-consistency loop of the MOPAC quantum chemistry program and leads to improved results compared to a standard finite supercell calculation. Heats of formation, bond lenghts, and electronic properties converge more quickly to the correct ground-state values. For polyacetylene, we obtain a reduced bond-length alternation of Δr = 0.084 Å, which is in agreement with more sophisticated calculations containing electron correlation effects. © 1994 John Wiley & Sons, Inc.     

Extrapolation of real-space quantum chemical calculations from finite-size super cells to the ideal infinite system. I. Theory

A method is proposed how to calculate the correct density matrix of an infinite polymeric chain from that of a standard finite supercell calculation. The density matrix of the finite supercell is transformed into k-space for all k-values allowed by the periodic boundary conditions. The k-dependent matrices are then unitarily transformed, with each unitary matrix being represented by a set of complex rotation matrices. It is shown that the corresponding angles can be interpolated and extrapolated toward the zone boundaries in a straghtforward manner and that this extrapolation can be done from any finite supercell with reasonable accuracy. This gives rise to an infinite system density matrix for which all fundamental properties are guaranteed by construction. This infinite system density matrix may be used to construct a corrected density matrix for the finite supercell calculation. © 1994 John Wiley & Sons, Inc.     

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.     

Molecular modeling of polymers: I. Correct and efficient enumeration of intrachain conformational energetics

Polymer conformational analyses can require being able to model the intramolecular energetics of a very long (infinite) chain employing calculations carried out on a relatively short chain sequence. A method to meet this need, based upon symmetry considerations and molecular mechanics energetics, has been developed. Given N equivalent degrees of freedom in a linear polymer chain, N unique molecular groups are determined within the chain. A molecular unit is defined as a group of atoms containing backbone rotational degrees of conformational freedom on each of its ends. The interaction of these N molecular groups, each with a finite number of nearest neighbors, properly describe the intramolecular energetics of a long (infinite) polymer chain. Thus, conformational energetics arising from arbitrarily distant neighbor interactions can be included in the estimation of statistical and thermodynamic properties of a linear polymeric system. This approach is called the polymer reduced interaction matrix method (PRIMM) and the results of applying it to isotactic polystyrene (I-PS) are presented by way of example.     

N–H(N)…S Bonding in Tetrahedral [Zn(NCS)2L]0 Complexes (L = MexH2–xN(CH2)2NH2–yMey,x, y = 0 – 2)

The crystal structures of uncharged tetrahedral dithiocyanato zinc complexes with N-methylated ethylenediamines have been determined with a view to a study of intermolecular hydrogen-bonding interactions in these compounds. It is found that the H(N) hydrogen atoms are exhaustively engaged in N–H(N) … S bonds. The majority of these bonds are branched (bifurcated or trifurcated), and the hydrogen-bond systems they form all contain one of the two characteristic primitive core motifs: either a discrete centrosymmetric […S…H…]₂ dimer or an infinite […S…H…]_∞ helix about a 2₁ or pseudo-2₁ axis. The hydrogen bonding is analyzed in detail, with particular attention to the existence of correlations between the N–H(N)–S angles and the H(N) … S distances as well as between the corresponding N–H(N)–S/H(N)…S pairs in the bifurcated N–H(N)…2 S bonds.     

Numerical calculations of the viscoelastic properties of solutions of branched polymers based on the Zimm-Kilb theory

Numerical calculations were performed for the viscoelastic properties of dilute solutions of branched star polymers with equal branch lengths as formulated in terms of a bead-spring model by Zimm and Kilb without using the integrodifferential equation approximation method to calculate the eigenvalues. The complex modulus and complex viscosity were calculated as functions of frequency for various combinations of the number of branches f (4, 8, and 13), the number of beads in one branch N_b (= N/f; 20 to 100, where N + 1 is the total number of beads, N the number of springs in the molecule) and the reduced hydrodynamic interaction parameter h^* (= h/N^1/2 0.05 to 0.3, where h is the hydrodynamic interaction parameter of Zimm and Kilb). The frequency dependence of the complex modulus in the low-frequency range depends mainly on h^* and not on N_b if N_b is large enough, and it is very close to that calculated from the eigenvalues for h→∞ obtained by Zimm and Kilb, if h^* is about 0.25. As h^* decreases from 0.25, the frequency dependence gradually approaches that of the free-draining cash (h→0). Calculations may be carried out for h^* values somewhat larger than 0.25 and result in a frequency dependence that is not intermediate to the h → 0 and h → ∞ cases as evaluated by Zimm and Kilb. The physical meaning of such “super-non-free-draining” values of h^* is uncertain, however. The intrinsic viscosity ratio g′ = [η]_f/[η]_lin is an increasing function of h^* and changes very slowly with N. For h^* = 0.25, g′ is close to the non-free-draining limit for any value of N.     

Finite chain approximation for linear and nonlinear polarizabilities of polydiacetylene and polybutatriene

Static longitudinal polarizabilities α_zz and cubic hyperpolarizabilities γ_zzzz are obtained by the FPT-INDO method for finite chain C_4N+2H_2N+4 (N = 1–15) models of polydiacetylene and polybutatriene. For the acetylenic structure the onset of saturation first becomes evident at about C₁₄? C₁₈ for the linear polarizability and C₂₂? C₂₆ for the cubic term. The values per C₄H₂ unit are accurately extrapolated to the infinite chain limit. In the case of the butatrienic structure, no saturation is observed through C₅₄.