Analytical HDMR formulas for functions expressed as quadratic polynomials with a multivariate normal distribution

High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for systems with many variables. A general formulation for the HDMR component functions with independent and correlated variables was obtained previously. Since the HDMR component functions generally are coupled to one another and involve multi-dimensional integrals, explicit formulas for the component functions are not available for an arbitrary function with an arbitrary probability distribution amongst their variables. This paper presents analytical formulas for the HDMR component functions and the corresponding sensitivity indexes for the common case of a function expressed as a quadratic polynomial with a multivariate normal distribution over its variables. This advance is important for practical applications of HDMR with correlated variables.     

On calculations of correlated wave functions with heavy configurational mixing

We discuss aspects of the theory and computation of wave functions and energies of discrete states of polyelectronic atoms that are represented in zero order by configurations with holes in subshells below the valence subshell. Both in zero order and in the remaining correlation components, such wave functions have particularities stemming from the state‐specific self‐consistent field and the heavy configurational mixing associated with near‐degeneracies and hole‐filling correlations. By referring to a variety of examples from small‐ and large‐scale calculations, it is noted that appropriate penetration into the many‐body problem can provide, in an economic and physically transparent way, reliable interpretations and semi‐ and fully quantitative understanding of issues related to states with inner holes and to cases of near‐degeneracies that result in strongly correlated wave functions. Whenever hole‐filling correlations are allowed, multiple correlations (i.e., beyond single‐ and double‐orbital substitutions in the single reference configuration) acquire increased importance relative to that in ordinary electronic structures. This is demonstrated via large‐scale multiconfigurational Hartree–Fock (MCHF) plus configuration interaction (CI) calculations on the Cl KL3s3p^{6 2}S discrete state, which is the lowest of its symmetry. The calculations incorporated correlations up to selected sextuple orbital excitations from the M shell. MCHF plus CI calculations at the level of quadruple orbital substitutions were also carried out for the Cl KL3s²3p^{5 2}P^o ground state and the excitation energy at this level of calculation was found to be 85,364 cm^?1, in excellent agreement with the experimental value of the fine‐structure‐weighted average, 85,385 cm^?1 (10.59 eV). Within the approximations of the calculation, the hole‐filling triple and quadruple orbital correlations, which, of course, are absent from the ²P^o state, contribute about 1 eV, which is significant. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Explicitly correlated wave functions: summary and perspective

We summarize explicitly correlated electronic structure theory in perspective of future work in the field. Earlier stages of approaches with different Ansätze in physics and chemistry are described. We then discuss recent advances focusing on explicitly correlated wave functions using cusp conditions. Removal of Coulomb singularities in terms of the rational generator is brought out from the viewpoint of many-body perturbation theory. On the basis of decomposition schemes for many-electron integrals in R12 and F12 methods, we further discuss the possibility of increasing the accuracy of molecular numerical integration and massively parallel calculations of explicitly correlated methods.     

Three stages of optimization and simple correlated wave functions

It is advocated to carry out an optimization procedure, which is based upon the variational method, in such a way that the optimum values of the variational parameters are expressed as functions of physical constants, such as the atomic number, Z. The three stages involved in this treatment are illustrated by the optimization of nine correlated wave functions, which describe the ground states of atomic two-electron systems. An analysis of the Z-expansions of the total energies associated with these functions leads to the concept of a class of variational functions. The performances of functions belonging to the same class differ only marginally, especially at larger values of Z. Consequently, the concept of class may be used to bring some order in the plethora of variational functions. © 1994 John Wiley & Sons, Inc.     

Product of group-function expansions for correlated wave functions

A theorem is proved which demonstrates the relationship between a product of group functions describing the correlated motion of a particular group of electrons in an N-electron system and a wave function obtained from the exact wave function which describes the correlation of the same group of electrons. By considering such products of group functions as elements in a variational wave function, an expansion for correlated wave functions is suggested, which emphasizes the correlated motion of groups of electrons in the whole system.     

Cross validation and uncertainty estimates in independent component analysis

A data analysis tool, known as independent component analysis (ICA), is the main focus of this paper. The theory of ICA is briefly reviewed, and the underlying statistical assumptions and a practical algorithm are described. This paper introduces cross validation/jack-knifing and significance tests to ICA. Jack-knifing is applied to estimate uncertainties for the ICA loadings, which also serve as a basis for significance tests. These tests are shown to improve ICA performance, indicating how many components are mixed in the observed data, and also which parts of the extracted sources that contain significant information. We address the issue of stability for the ICA model through uncertainty plots. The ICA performance is compared to principal component analysis (PCA) for two selected applications, a simulated experiment and a real world application.     

General formulation of the first-order perturbation graph theory

A generalization of the perturbation graph theory of Herndon and Párkányi is developed, enabling the calculation of resonance energies of conjugated heterocyclic and inorganic systems possessing any number of π-electrons.     

Approximate density matrices and Husimi functions using the maximum entropy formulation with constraints

The entropy of an electronic system is defined in terms of the Husimi function, a nonnegative distribution function in phase space. The Husimi function is calculated by maximizing the entropy subject to the constraints that the Husimi function give a Gaussian convolution of the desity when integrating over the momentum coordinates and that its second moment with respect to momentum give a sum of Gaussian convolutions of the density and the kinetic energy density. The result is compared with the Wigner function. Equations are given for calculating the density matrix from the Husimi function. The resulting equation for the exchange energy requires a difficult numerical integration. An alternate method is used to obtain the density matrix from an approximate partially collapsed Husimi matrix that gives the maximum entropy Husimi function as its diagonal. The results are exact for the harmonic oscillator ground state. Exchange energies calculated for H and the He isoelectronic series through C⁺⁴ show slight improvements over those calculated using a maximum entropy Wigner function.     

Iterative Monte Carlo formulation of real-time correlation functions

We present an iterative Monte Carlo path integral methodology for evaluating thermally averaged real-time correlation functions. Standard path integral Monte Carlo methods are used to sample paths along the imaginary time contour. Propagation of the density matrix is performed iteratively on a grid composed of the end points of the sampled paths. Minimally oscillatory propagators are constructed using energy filtering techniques. A single propagation yields the values of the correlation function at all intermediate time points. Model calculations suggest that the method yields accurate results over several oscillation periods and the statistical error grows slowly with increasing propagation time.     

An algorithm for calculating atomic D states with explicitly correlated gaussian functions

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.     

Matrix elements of N-particle explicitly correlated Gaussian basis functions with complex exponential parameters

In this work we present analytical expressions for Hamiltonian matrix elements with spherically symmetric, explicitly correlated Gaussian basis functions with complex exponential parameters for an arbitrary number of particles. The expressions are derived using the formalism of matrix differential calculus. In addition, we present expressions for the energy gradient that includes derivatives of the Hamiltonian integrals with respect to the exponential parameters. The gradient is used in the variational optimization of the parameters. All the expressions are presented in the matrix form suitable for both numerical implementation and theoretical analysis. The energy and gradient formulas have been programmed and used to calculate ground and excited states of the He atom using an approach that does not involve the Born-Oppenheimer approximation.     

Generalized formulation of quantum and classical crystallography using Green's functions

Quantum crystallography is a methodology by which structural information about a crystalline material obtained from X‐ray crystallography is combined with quantum mechanical methods. The objective is to enhance the data obtained from the X‐ray diffraction experiment, which are related to the atomic structure of the crystal, and to predict the properties and efficacy of those chemical compounds from which the crystals are derived. One approach in quantum crystallography is to use a projector matrix with a normalized trace. In this approach, quantum mechanical parameters in the projector matrix are fit into crystallographic data. During this fitting, the properties of the projector matrix called idempotency and normalization are used. Throughout this implementation procedure, Clinton's iteration scheme has been used in addition to the least‐squares technique. The purpose of the present study is to generalize Clinton's iterative equations in quantum crystallography by means of single‐particle Green's functions with the aid of the equal atoms model in the theory of direct methods. Convergency characters of the novel iterative equations are discussed by the steepest descent procedure. Furthermore, whether the calculations are valid in nonorthogonal bases was also examined. The iteration schemes widely used in quantum crystallography have been generalized but, in addition, the generalized expressions relating to the phase determination procedure and the probabilities of the sign relations between the structure factors are obtained and discussed comprehensively. The phrase order of crystallography has been put forward as a new concept. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Multicenter and multiparticle integrals for explicitly correlated cartesian gaussian-type functions

An analytical derivation of multicenter and multiparticle integrals for explicitly correlated Cartesian Gaussian-type cluster functions is demonstrated. The evaluation method is based on the application of raising operators that transform spherical cluster Gaussian functions into Cartesian Gaussian functions.     

Vulcanization kinetics study of natural rubber compounds having different formulation variables

Vulcanization kinetics of natural rubber (NR) compounds with efficient vulcanization system was studied through phenomenological approach using the experimentally cure data obtained from a moving die rheometer. The cure kinetic parameters were defined using the proposed models by Claxton?CLiska and Deng?CIsayev with the support of curve fitting software. The effects of the amount of accelerators, sulfur and silica in the formulations on the cure characteristics and cure kinetic parameters at high cure temperatures were investigated. Kinetic data results showed that the above two models were able to describe the curing behaviour of the studied compounds satisfactorily. It showed that the fitting of the experimental data with Claxton?CLiska and Deng?CIsayev could provide a good platform to investigate the cure kinetics of the prepared NR compounds.     

On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

Explicitly correlated multireference configuration interaction with multiple reference functions: avoided crossings and conical intersections

We develop an explicitly correlated multireference configuration interaction method (MRCI-F12) with multiple reference functions. It can be routinely applied to nearly degenerate molecular electronic structures near conical intersections and avoided crossings, where the reference functions are strongly mixed in the correlated wave function. This work is a generalization of the MRCI-F12 method for electronic ground states, reported earlier by Shiozaki et al. [J. Chem. Phys. 134, 034113 (2011)]. The so-called F12b approximation is used to arrive at computationally efficient formulas. The doubly external part of the wave function is expanded in terms of internally contracted configurations generated from all the reference functions. In addition, we introduce a singles correction to the CASSCF reference energies, which is applicable to multi-state calculations. As examples, we present numerical results for the avoided crossing of LiF, excited states of ozone, and the H(2)?+?OH (A(2)Σ(+)) reaction.     

Two-electron integrations in the quantum theory of atoms in molecules with correlated wave functions

A recent method proposed to compute two-electron integrals over arbitrary regions of space [Martin Pendas, A. et al., J Chem Phys 2004, 120, 4581] is extended to deal with correlated wave functions. To that end, we use a monadic factorization of the second-order reduced density matrix originally proposed by E. R. Davidson [Chem Phys Lett 1995, 246, 209] that achieves a full separation of the interelectronic components into one-electron terms. The final computational effort is equivalent to that found in the integration of a one determinant wave function with as many orbitals as occupied functions in the correlated expansion. Similar strategies to extract the exchange and self-interaction contributions from the two-electron repulsion are also discussed, and several numerical results obtained in a few test systems are summarized.     

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.     

Theoretical modeling of the relationship between Young's modulus and formulation variables for segmented polyurethanes

We describe a new modeling approach to prediction of Young's modulus of segmented polyurethanes. This approach combines micromechanical models with thermodynamic considerations based on the theory of block copolymers. The resulting model predicts both the equilibrium morphology and the “ideal” Young's modulus of a segmented polyurethane polymer as a function of its formulation (hard segment chemical structure, hard segment weight fraction, soft segment equivalent weight) and temperature. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2123–2135, 2007     

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.