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 supercells to the ideal infinite system. III. Application to two-dimensional systems

A previously presented extrapolation method of the density matrix of a standard finite supercell calculation to an infinite supercell is extended to two-dimensional systems. The density matrix of the finite supercell is transformed into q space where it is interpolated and extrapolated in such a way that all its fundamental properties are guaranteed by construction. The resulting modified density matrix contains information from a continuum of q points in the first Brillouin zone which allows for a more realistic calculation of properties like the total energy E_tot per atom and the band structure ϵ₁(q). It is shown that this more realistic calculation takes better care of the crystal symmetries and is essential in reproducing both important degeneracies in the band structure and rotationally symmetric results. In the special case of π electrons only, an exact analytical solution for the density matrix of the infinite supercell is presented. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 895–911, 1997     

Efficient method for computation of representation matrices of the unitary group generators

The representation matrices of the unitary group generators E_k1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ? 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.     

Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single-particle states is defined as the (reduced) k-state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one- and two-particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density-matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off-diagonal elements of the density matrices.     

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.     

Geometry of density matrices. III. Spin components

A definition is given for the spin-tensorial components of a p-electron density matrix for arbitrary p. Certain of these are defined to be standard components, and it is shown that all others can be obtained from them by linear combinations of permutations. When the density matrix is reduced, the standard components either map into standard components of the fewer-electron density matrix or into the origin. One- and two-electron density matrices are treated in greater detail. Reducing bases are given for the spaces in which the spatial coefficients of the spin components are elements. For spin eigenstates traces of all components are determined, and the π = 1 parts of all components are determined in terms of two independent components, which are themselves determined by the two-electron charge density matrix. Geometric consequences are investigated.     

Formulation of a phonon band calculation for molecular crystals using a coarse-grained coordinate approach under periodic boundary conditions

A phonon band calculation scheme based on our previously proposed coarse-graining theory under periodic boundary conditions was formulated. Starting with a simple one-dimensional, one-body periodic system, we introduced the basis set of a phase-shift coordinate system that can easily afford the discrete Fourier transformation of vectors and matrices with infinite dimensions. When the unit cell contains two or more bodies, the basis set of the phase-shift coordinate system is represented with tensors. By choosing an appropriate tensor basis set of a coarse-grained space, we can approximately block-diagonalize the dynamical matrix. Then, we can obtain the inertia and stiffness matrices represented by the given coarse-grained coordinate system, upon which the application of the mass-weighted Hessian equation affords a set of angular frequencies (ω) as functions of the wavenumber (k). Thus, the phonon band structure (k–ω plot) is obtained based on the coarse-graining approximation. When this approximation is applied to molecular assemblies comprising hydrogen bonding, the computational error resulting from this scheme is expected to be a maximum of a few cm^?1.

Graphical abstract

     

Surface Green's function calculations: a nonrecursive scheme with an infinite number of principal layers

A novel computational method for a surface Green's function matrix is introduced for the calculation of electrical current in molecular wires. The proposed nonrecursive approach includes an infinite number of principal layers and yields the second-order matrix equation for the transformed Green's function matrix. The solution is found by the direct diagonalization of the auxiliary matrix without any iteration process. As soon as complex roots of the auxiliary matrix (approximately GS) are calculated, the gaps and the bands in the surface electronic structure are found. It is shown that the solution of a second-order matrix equation determines the spectral density matrix, that is, the density of states for noninteracting electrons. Single and double principal layer models are studied both analytically and numerically. The energy interval for nonvanishing spectral matrices is determined. This method is applicable to matrices of any rank.     

Ab initio SCF calculations of the linear infinite chain of LiH according to the pseudo-lattice method

The pseudo-lattice (PL) method has been reformulated for ab initio self-consistent-field (SCF) calculations. The translational symmetries of infinite systems have been applied to the finite model chain by manipulating all the intramolecular and intermolecular Fock matrices. The nuclear repulsion energy has been corrected accordingly. The method has been tested for the linear chain of lithium hydride under the constraint of equidistance between all neighboring lithium and hydrogen atoms. The calculated results of the infinite chain have been compared with those of finite chains of lithium hydride under the same geometric constraint. The equilibrium geometries, band structures, intermolecular stabilization energies and potential curves have been studied. It is found that the infinite systems cannot be described by considering only first nearest neighbor interactions, and the intermolecular interactions must be considered at least up to third nearest neighbors in order to obtain accurate value of force constant of infinite systems. We can conclude from band structures of infinite chains that the boundary effect of the finite model chain is effectively removed by the PL method.     

Structure and spin-purity conditions for reduced density matrices of arbitrary order

For arbitrary k, the separation of spin variables is performed in the reduced density matrix of the kth order (RDM -k) on the basis of the Fock coordinate function method. The independent spatial components of RDM -k are analyzed. For RDM -k of the total spin eigenstate, their number is proved never to exceed its spin multiplicity 2s + 1. Integral and other nontrivial interrelations between spatial components are established which turn out to be the necessary and sufficient conditions of spin purity of a wavefunction corresponding to a given RDM -k. It is shown that the r-rank k-particle spin distribution matrix F, defined as a spatial coefficient at the spin-tensorial operator of rank r in the RDM -k expansion, can be obtained by reduction of the (k + r)-particle charge density matrix F. In particular, all spatial components of RDM -2 are explicitly expressed in terms of the four-electron charge density matrix only. This allows us to purpose some approximative formulas for the McWeeny-Mizuno spin–orbit and spin–spin coupling functions in the case of the weak spin contamination.     

Pseudomerohedrally twinned crystal structure of 2,3‐diphenyl­buta‐1,3‐diene

The title compound, C₁₆H₁₄, is twinned by reticular pseudomerohedry of twin index 2. The primitive monoclinic cell of the single crystal can be transformed into a B‐centred pseudo‐orthorhombic supercell with a fourfold volume. The twofold twin operation about the reciprocal a* axis of the primitive monoclinic cell is co‐directional with the approximate C₂ axis of the mol­ecule and a symmetry element of the orthorhombic supercell. A tentative twin domain model is proposed.     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

Lattice dynamics with second neighbor interactions. II. Green's formula

A satisfactory definition of spectral density for the normal modes of lattice dynamics problems requires the study of singular recurrence relations which is carried out in detail for one-dimensional chains with pth neighbor interactions. The relationship of transfer matrices to the dynamical matrix is explored in order to obtain Green's formula. By using Green's formula, a mapping is defined between V_n, whose basis is formed from the normal modes of vibration of an n-particle chain, and V_2p, which is the space of boundary conditions for the recurrsion equations. Most of the properties of this mapping may be deduced from a symplectic bilinear form in V_2p which is associated with the Hermitean inner product in V_n. This symplectic form defines a geometry which is invariant under the recursion relation, as well as canonical initial and boundary conditions, and a maximal isotropic subspace which may be used to determine square summability of the normal modes and the spectral density in the limit as the number of particles becomes infinite.     

Some basic properties of the correlation matrices

On the basis of the properties of correlation matrices, it is shown here that the set of all the first‐order transition reduced density matrices of a system provide complete information about that system. Also, the interrelation between the properties of the correlation matrix and the 2‐RDM N‐representability conditions is studied. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

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.     

Modified tight-binding equations for electron wave functions in crystals with point defects

A system of tight-binding equations for the electron wave function in infinite, semi-infinite, or interface crystals with point defects is transformed to a finite system for the coefficients of specially constructed converging and diverging waves. The solution of this finite system allows the construction of the electron wave function over all space. This makes it possible to calculate the electronic structure of a defect and its “images” in the various experimental methods used for studying defects in the bulk, on the surface, and at the interface of crystals, without conventional structural simplifications. The proposed method is applied to the calculation of point defects in a two-dimensional square lattice. G. K. Boreskov Institute of Catalysis, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 37, No. 6, pp. 985–993, November–December, 1996. Translated by I. Izvekova     

Improvement to the hypervirial theorem and Kramer's rule for the calculation of central potential matrix elements

Ameliorated recurrence relations for the calculation of matrix elements of central potential wavefunctions are presented. These were obtained by using the hypervirial theorem with V(r) three-dimensional potentials and f (r) arbitrary functions. This procedure leads to a generalization of the usual l = l′ diagonal three-dimensional and l == 0 one-dimensional hypervirial relations of first and second class. The use of this kind of generalization to the calculation of r^k integrals, allows one to obtain all off-diagonal recursion formulas for any V(r). Besides, for hydrogenic wavefunctions one gets to equations that reduce to the usual Kramer's rule as a particular diagonal case. The proposed approach can be straightforwardly extended to obtain recurrence relations for the calculation of two center integrals.     

Sweeping with an Enhanced Electric Field of Neutral Analyte Zones in Electrokinetic Chromatography

The concept of sweeping has been used in electrokinetic chromatography to explain the concentrating mechanism in micellar electrokinetic chromatography when the sample matrix is a high resistivity non-micellar aqueous solution. Theoretical and experimental studies were undertaken. It was found that the total focusing effect is brought about by the cumulative effect of sweeping and sample stacking. For better analytical results, compounds showing low to moderate retention factors (k) and compounds showing high values of k must be dissolved in low and high conductivity matrices, respectively.     

Direct calculation of lattice sums. A method to account for the crystal field effects

A method of direct calculation of lattice sums in three-dimensional crystals is reported. The method is based on annihilation of some lowest multipole moments of the unit cell by a redefinition of the unit cell content. As a result, properties of the infinite crystal can be calculated as usual by taking a finite cluster of unit cells, but surrounded by an additional surface layer of a charge density (e.g., a layer of point charges). This charge density distribution produces the electric field approximating that one of the rest of the infinite crystal. The method proposed is easily applicable in the SCFLCAO procedure as well as in any method using a cluster representation for an infinite crystal. The validity of the infinite crystal model for a finite crystal is also discussed.