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.     

Sum of ranking differences for method discrimination and its validation: comparison of ranks with random numbers

This paper describes the theoretical background, algorithm and validation of a recently developed novel method of ranking based on the sum of ranking differences [TrAC Trends Anal. Chem. 2010; 29 : 101–109]. The ranking is intended to compare models, methods, analytical techniques, panel members, etc. and it is entirely general. First, the objects to be ranked are arranged in the rows and the variables (for example model results) in the columns of an input matrix. Then, the results of each model for each object are ranked in the order of increasing magnitude. The difference between the rank of the model results and the rank of the known, reference or standard results is then computed. (If the golden standard ranking is known the rank differences can be completed easily.) In the end, the absolute values of the differences are summed together for all models to be compared. The sum of ranking differences (SRD) arranges the models in a unique and unambiguous way. The closer the SRD value to zero (i.e. the closer the ranking to the golden standard), the better is the model. The proximity of SRD values shows similarity of the models, whereas large variation will imply dissimilarity. Generally, the average can be accepted as the golden standard in the absence of known or reference results, even if bias is also present in the model results in addition to random error. Validation of the SRD method can be carried out by using simulated random numbers for comparison (permutation test). A recursive algorithm calculates the discrete distribution for a small number of objects (n < 14), whereas the normal distribution is used as a reasonable approximation if the number of objects is large. The theoretical distribution is visualized for random numbers and can be used to identify SRD values for models that are far from being random. The ranking and validation procedures are called Sum of Ranking differences (SRD) and Comparison of Ranks by Random Numbers (CRNN), respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Relation between the Slater–Kohn–Sham orbitals generated by the one-body potential V(r) of DFT and the Löwdin natural orbitals for inhomogeneous electron liquids

The Löwdin natural orbitals (NO) are defined as those orbitals which bring the first-order density matrix of a correlated electron assembly into diagonal form. Another one-particle density matrix with the same diagonal elements is the single-particle idempotent Dirac density matrix generated by the one-body potential of density functional theory. Here, we compare the off diagonal form of γ expanded in terms of the Slater–Kohn–Sham (SKS) orbitals generated by V(r) with the NO expansion of Löwdin for general inhomogeneous electron liquids. In particular, the equation of motion of the correlated γ is corrected from that of γ _s , both now containing the one-body potential V(r). To illustrate the theory presented here, we first construct an approximate, albeit accurate, correlated 1DM for the ground state of the He atom and display connections between the resulting NOs and the SKS orbitals. The second example we discuss, but now quite briefly, is that of the inhomogeneous electron liquid in crystalline Si, where the NO expansion is available from the literature.     

Normalized irreducible tensorial matrix expansions applied to an effective sp-type Hamiltonian for the PES of water

Any matrix can be expanded on a basis of SU(2) normalized irreducible tensorial matrices, NITM , defined in terms of 3-j symbols or coupling coefficients of SU (2). The NITM transform under rotations according to Wigner's matrices. If one dimension of an NITM is odd and the other even, the tensor has half-integer rank. A simple NITM basis consists of all NITM having the same numbers of rows and columns as the expanded matrix. A compound NITM basis consists of two or more simple bases, each spanning a corresponding block in the expanded matrix. The choice of NITM basis for expanding an effective Hamiltonian matrix is a crucial step in formulating a model. To illustrate the use of a compound NITM basis, including nonsquare NITM , an effective sp-type overlap-free superposition Hamiltonian is constructed and applied to the photoelectron ionization potential spectrum of water.     

Order theoretical tools for the evaluation of complex regional pollution patterns

The mathematical and statistical evaluation of environmental data gains an increasing importance in environmental chemistry as the data sets become more complex. It is inarguable that different mathematical and statistical methods should be applied in order to compare results and to enhance the possible interpretation of the data. Very often several aspects have to be considered simultaneously, for example, several chemicals entailing a data matrix with objects (rows) and variables (columns). In this paper a data set is given concerning the pollution of 58 regions in the state of Baden-Württemberg, Germany, which are polluted with metals lead, cadmium, zinc, and with sulfur. For pragmatic reasons the evaluation is performed with the dichotomized data matrix. First this dichotomized 58 x 13 data matrix is evaluated by the Hasse diagram technique, a multicriteria evaluation method which has its scientific origin in Discrete Mathematics. Then the Partially Ordered Scalogram Analysis with Coordinates (POSAC) method is applied. It reduces the data matrix in plotting it in a two-dimensional space. A small given percentage of information is lost in this method. Important priority objects, like maximal and minimal objects (high and low polluted regions), can easily be detected by Hasse diagram technique and POSAC. Two variables attained exceptional importance by the data analysis shown here: TLS, Sulfur found in Tree Layer, is difficult to interpret and needs further investigations, whereas LRPB, Lead in Lumbricus Rubellus, seems to be a satisfying result because the earthworm is commonly discussed in the ecotoxicological literature as a specific and highly sensitive bioindicator.     

A coupled-cluster study of linear and rhombic boron nitride dimers: a revisit

The potential energy surfaces of both singlet and triplet B₂N₂ have been investigated computationally at the coupled-cluster level with a polarized triple zeta basis set augmented with diffuse functions. Calculated vibrational frequencies and intensities are also reported. The triplet species are consistently more stable than their singlet analogs and the stabilities of the linear B₂N₂ isomers increase with increasing number of B–N bonds. The most stable isomer is the linear triplet BNBN isomer with a rhombic form with a short diagonal BB distance close in energy. Our results are consistent with the results of the matrix IR studies of Andrews et al. nucleus-independent chemical shift (NICS) values were calculated for the singlet D_2h rhombic form and its C_2v dication, and these were compared to those of the D_2h cyclobutadiene and its D_2d dication, respectively. Electron density plots for the linear and rhombic B₂N₂ minima showed similar distributions for the singlet and triplet states. These plots confirmed weak BB bonding interactions in both rhombic forms but larger BN bond orders.     

A method for the construction of orthogonal spin eigenfunctions

A method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group S_n is proposed. For the irreducible representation [λ] = [λ₁, λ₂] characterising a spin state S of an n-electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7-electron system in the spin state S = M_S = 1/2 and the results are listed in the Appendix.     

A Closed-Form Relation for Dimension-Dependent Two-Electron Matrix Elements of the Interelectronic Distance

The evaluation of matrix elements of two electron atoms is fundamental for the study of the electronic properties of those systems. We add to this knowledge by presenting an explicit expression for the matrix elements of the inverse of the interelectronic distance of two-electron atoms in any spatial dimension D. The basis functions used are the D-dependent hydrogenic wavefunctions {1s ²,2p ²,3d ²,4f ²,5g ²,...,21y ²,...}, extending and including, in this way, the results of the previous basis set {1s ²,2p ²,3d ²,4f ²}. The methodology used does not employ Fourier integral transforms as in previous works but hypergeometric transformation formulas.     

New optimization method for conformational energy calculations on polypeptides: Conformational space annealing

A new optimization method is presented to search for the global minimum-energy conformations of polypeptides. The method combines essential aspects of the build-up procedure and the genetic algorithm, and it introduces the important concept of “conformational space annealing.” Instead of considering a single conformation, attention is focused on a population of conformations while new conformations are obtained by modifying a “seed conformation.” The annealing is carried out by introducing a distance cutoff, D_cut, which is defined in the conformational space; D_cut effectively divides the whole conformational space of local minima into subdivisions. The value of D_cut is set to a large number at the beginning of the algorithm to cover the whole conformational space, and annealing is achieved by slowly reducing it. Many distinct local minima designed to be distributed as far apart as possible in conformational space are investigated simultaneously. Therefore, the new method finds not only the global minimum-energy conformation but also many other distinct local minima as by-products. The method is tested on Met-enkephalin, a 24-dihedral angle problem. For all 100 independent runs, the accepted global minimum-energy conformation was obtained after about 2600 minimizations on average. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1222–1232     

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.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Obtaining the key set of typical vectors by factor analysis and subsequent isolation of component spectra

The key set of typical rows (or typical columns), used to reproduce a data matrix, are obtained by finding the set of rows (or set of columns) most orthogonal to each other. If the data matrix consists of the spectra of a series of related mixtures, and there exists at least one data point unique to each of the components, the spectra of the pure components can be isolated. The method is applied to problems in nuclear magnetic resonance, gas—liquid chromatography and mass spectrometry.     

Spin-independent three-body effective valence-shell operators: Application to molecular oxygen

A mathematical construction is presented that uniquely defines a set of spin-independent effective valence-shell Hamiltonian (H^v) three-body matrix elements. These spin-independent H^v matrix elements separate direct and exchange portions of the three-body H^v matrix elements and therefore provide the most natural form for comparisons with parameterization schemes of semiempirical electronic structure methods in which the three-body matrix elements are incorporated into semiempirical one- and two-body Hamiltonian matrix elements in an averaged manner. Ab initio H^v three-body matrix elements of O₂ are computed through third order of quasidegenerate perturbation theory and are analyzed as a function of internuclear distance and atomic orbital overlap to aid in understanding how these three-body matrix elements may be averaged into semiempirical one- and two-body matrix elements. © 1992 John Wiley & Sons, Inc.     

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.     

Orbitally resolved charge sensitivity analysis: Basic concepts and relations

Following the recent developments of the charge sensitivity analysis (CSA ) in the atoms-in-molecules (AIM ) resolution, the corresponding CSA quantities in the orbital (or shell) resolution (OR ) are defined. The OR electron population variables, in the ordinary closed-shell SCF problem, are the elements of the bond-order matrix P , and their conjugates, “chemical potentials,” F ^T = ?E/? P , are the respective Fock matrix elements, appropriate for the representation in question; here E is the SCF energy. The second derivatives ?²E/? P ? P define the OR hardness tensor from which all related OR CS s, e.g., the hardness, softnesses, Fukui function (FF ) indices, etc., can be determined. The rigid potentials and hardness tensor, corresponding to the “frozen” orbital approximation, are examined in more detail, and the decoupled representation of the normal orbitals (N oO ) is introduced, in which the rigid hardness tensor becomes diagonal. Illustrative valence-shell N oO contours for the water molecule are given and discussed. The new approximation for the OR FF indices, as the orbital occupation probabilities, is proposed on the basis of the density matrix functional development of Donnely and Parr for natural orbitals, and the relevant expressions for the molecular fragment (collection of orbitals) quantities are summarized.     

Discrimination of isomeric structures using information theoretic topological indices

Three newly defined information theoretic topological indices, namely “degree complexity (I^d),” “graph vertex complexity (H^V),” and “graph distance complexity (H^D)” along with three other information indices have been used to study their discriminating power of 45 trees and 19 monocyclic graphs. It is found that the newly defined indices have satisfactory discriminating power while H^D has been found to be the only index to discriminate all the graphs studied.