Molecular-orbital-free algorithm for excited states in time-dependent perturbation theory

A nonlinear conjugate gradient optimization scheme is used to obtain excitation energies within the random phase approximation (RPA). The solutions to the RPA eigenvalue equation are located through a variational characterization using a modified Thouless functional, which is based upon an asymmetric Rayleigh quotient, in an orthogonalized atomic orbital representation. In this way, the computational bottleneck of calculating molecular orbitals is avoided. The variational space is reduced to the physically-relevant transitions by projections. The feasibility of an RPA implementation scaling linearly with system size N is investigated by monitoring convergence behavior with respect to the quality of initial guess and sensitivity to noise under thresholding, both for well- and ill-conditioned problems. The molecular-orbital-free algorithm is found to be robust and computationally efficient, providing a first step toward large-scale, reduced complexity calculations of time-dependent optical properties and linear response. The algorithm is extensible to other forms of time-dependent perturbation theory including, but not limited to, time-dependent density functional theory.     

Diagonalization of a real-symmetric Hamiltonian by genetic algorithm: A recipe based on minimization of Rayleigh quotient

A genetic algorithm-based recipe involving minimization of the Rayleigh quotient is proposed for the sequential extraction of eigenvalues and eigenvectors of a real symmetric matrix with and without basis optimization. Important features of the method are analysed, and possible directions of development suggested     

Development and validation of an improved probabilistic quotient normalization method for LC/MS- and NMR-based metabonomic analysis

Robust normalization is a prerequisite for reliable metabonomic analysis especially when intervention treatments cause drastic metabolomic changes or when spot urinary samples are employed without knowing the drinking water quantity. With the simulated and real datasets, here, we report a probabilistic quotient normalization method based on the mode-of-quotients (mPQN) which is suitable for metabonomic analysis of both NMR and LC–MS data with little and/or drastic metabolite changes. When applied to metabonomic analysis of both animal plasma samples and human urinary samples, this newly proposed method has clearly shown better robustness than all classical normalization methods especially when drastic changes of some metabolites occur.     

Normalization and Fermi–Coulomb and Coulomb hole sum rules for approximate wave functions

For approximate wave functions, we prove the theorem that there is a one‐to‐one correspondence between the constraints of normalization and of the Fermi–Coulomb and Coulomb hole charge sum rules at each electron position. This correspondence is surprising in light of the fact that normalization depends on the probability of finding an electron at some position. In contrast, the Fermi–Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion, while the Coulomb hole sum rule depends on Coulomb repulsion. We demonstrate the theorem for the ground state of the He atom by the use of two different approximate wave functions that are functionals rather than functions. The first of these wave function functionals is constructed to satisfy the constraint of normalization, and the second that of the Coulomb hole sum rule for each electron position. Each is then shown to satisfy the other corresponding sum rule. The significance of the theorem for the construction of approximate “exchange‐correlation” and “correlation” energy functionals of density functional theory is also discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Evaluation of standard and advanced preprocessing methods for the univariate analysis of blood serum 1H-NMR spectra

Proton nuclear magnetic resonance (¹H-NMR)-based metabolomics enables the high-resolution and high-throughput assessment of a broad spectrum of metabolites in biofluids. Despite the straightforward character of the experimental methodology, the analysis of spectral profiles is rather complex, particularly due to the requirement of numerous data preprocessing steps. Here, we evaluate how several of the most common preprocessing procedures affect the subsequent univariate analyses of blood serum spectra, with a particular focus on how the standard methods perform compared to more advanced examples. Carr–Purcell–Meiboom–Gill 1D ¹H spectra were obtained for 240 serum samples from healthy subjects of the Asklepios study. We studied the impact of different preprocessing steps—integral (standard method) and probabilistic quotient normalization; no, equidistant (standard), and adaptive-intelligent binning; mean (standard) and maximum bin intensity data summation—on the resonance intensities of three different types of metabolites: triglycerides, glucose, and creatinine. The effects were evaluated by correlating the differently preprocessed NMR data with the independently measured metabolite concentrations. The analyses revealed that the standard methods performed inferiorly and that a combination of probabilistic quotient normalization after adaptive-intelligent binning and maximum intensity variable definition yielded the best overall results (triglycerides, R = 0.98; glucose, R = 0.76; creatinine, R = 0.70). Therefore, at least in the case of serum metabolomics, these or equivalent methods should be preferred above the standard preprocessing methods, particularly for univariate analyses. Additional optimization of the normalization procedure might further improve the analyses.     

Finite barrier corrections for the Kramers rate problem in the spatial diffusion regime

Finite barrier height corrections for the rate of thermally activated escape of a particle over a barrier out of a metastable well are determined within the framework of Fokker—Planck processes in the spatial diffusion regime. Different existing concepts to obtain the rate are extended and compared to each other. The central role that is played by Kramers' stationary current carrying probability density is demonstrated. An improvement of this function by means of a perturbation theory allows one to calculate corrections to the rate by means fo the flux over population expression, the Rayleigh quotient and the mean first passage time to the stochastic separatrix. The Rayleigh quotient is in a sense superior as it allows one to obtain from the same stationary current carrying Kramers function higher order corrections than by means of other methods. Explicit results are obtained for the case of a cubic and a quartic potential and compared with existing results.     

Third-order many-body perturbation theory for the ground state of the carbon monoxide molecule

Many-body perturbation calculations have been performed for the ground state of the carbon monoxide molecule at its equilibrium internuclear separation. The calculations are complete through third order within the algebraic approximation; i.e., the state functions are parameterized by expansion in a finite basis set. All two-, three-, and four-body terms are rigorously determined, and many-body effects are found to be very important. A detailed comparison is made with a previously reported configuration interaction study. Padé approximants to the energy expansion are constructed. The many-body perturbative wave function is used in the Rayleigh quotient to produce upper bounds to the electronic energy.     

Related upper and lower bounds to atomic binding energies

The method proposed by Singh for the calculations of lower bounds to atomic binding energies has been generalized to encompass upper bounds as well. The result is a pair of related matrix eigenvalue problems, constructed from similar sets of basic matrix elements, with the solution of one yielding the tower bound, and of the other, the upper bound. The upper bounds are identical to those calculated by the Rayleigh–Ritz method, which can be useful when the inversion of the normalization matrix is ill-conditioned. The lower bounds are comparable with the best available in the literature. © 1993 John Wiley & Sons, Inc.     

Fundamental importance of the Coulomb hole sum rule to the understanding of the Colle-Salvetti wave function functional

In this paper we consider the general form of the correlated-determinantal wave function functional of Colle and Salvetti (CS) for the He atom. The specific form employed by CS is the basis for the widely used CS correlation energy formula and the Lee-Yang-Parr correlation energy density functional of Kohn-Sham density functional theory. We show the following: (i) The key assumption of CS for the determination of this wave function functional, viz., that the resulting single-particle density matrix and the Hartree-Fock theory Dirac density matrix are the same, is equivalent to the satisfaction of the Coulomb hole sum rule for each electron position. The specific wave function functional derived by CS does not satisfy this sum rule for any electron position. (ii) Application of the theorem on the one-to-one correspondence between the Coulomb hole sum rule for each electron position and the constraint of normalization for approximate wave functions then proves that the wave function derived by CS violates charge conservation. (iii) Finally, employing the general form of the CS wave function functional, the exact satisfaction of the Coulomb hole sum rule at each electron position then leads to a wave function that is normalized. The structure of the resulting approximate Coulomb holes is reasonably accurate, reproducing both the short- and the long-range behavior of the hole for this atom. Thus, the satisfaction of the Coulomb hole sum rule by an approximate wave function is a necessary condition for constructing wave functions in which electron-electron repulsion is represented reasonably accurately.     

Functional analysis concepts and Fartree-Fock instability conditions

Concepts of functional analysis, namely, regular points, tangent subspaces, constraint surfaces, Lagrangian matrix restricted to the tangent subspace of a constraint surface, are presented in connection with the Hartree-Fock (HF) problem. The energy functional in LCAO approximation is considered to be a polynomial function of several variables subject to subsidiary conditions. General HF equations and instability conditions for the unrestricted Hartree-Fock (UHF) solutions are derived from this standpoint.     

Normalized Lithium Growth from the Nucleation Stage for Dendrite‐Free Lithium Metal Anodes

Inducing uniform deposition of lithium from the stage of metal crystallization nucleation is of vital importance to achieve dendrite‐free lithium anodes. Herein, using experiments and simulation, homogenization of Li nucleation and normalization of Li growth can be achieved on PNIPAM polymer brushes with lithiophilic functional groups modified Cu substrates. The lithiophilic functional groups of amide O can homogenize ion mass transfer and induce the uniform distribution of Li nucleation sites. What is more, the ultra‐small space between each brush can act as the channels for Li transportation and normalization growth. Owing to the synergistic effect of homogenization and normalization of electrodeposited Li, the obtained planar columnar Li anode exhibits excellent cycle stability at an ultra‐high current density of 20 mA cm^?2.     

Physical basis for constrained lattice density functional theory

To study nucleation phenomena in an open system, a constrained lattice density functional theory (LDFT) method has been developed before to identify the unstable directions of grand potential functional and to stabilize nuclei by imposing a suitable constraint. In this work, we answer several questions about the method on a fundamental level, and give a firmer basis for the constrained LDFT method. First, we demonstrate that the nucleus structure and free energy barrier from a volume constraint method are equivalent to those from a surface constraint method. Then, we show that for the critical nucleus, the constrained LDFT method in fact produces a bias-free solution for both the nucleus structure and nucleation barrier. Finally, we give a physical interpretation of the Lagrange multiplier in the constraint method, which provides the generalized force to stabilize a nucleus in an open system. The Lagrange multiplier is found to consist of two parts: part I of the constraint produces an effective pressure, and part II imposes a constraint to counteract the supersaturation.     

Some properties of the Lagrange multiplier μ in density functional theory

Some new properties of the Lagrange multiplier μ introduced through the normalization constraint on ρ in the variations of energy density functionals are determined. Through arguments concerning the homogeneity properties of these functionals with respect to μ, it is demonstrated that at the point of variation μ = μ₀ = E₀/N, where E₀ is the ground state energy and N is the total particle number. It is also shown that the value of μ₀ is independent of the normalization imposed on ρ. The interpretation of μ₀ as a chemical potential is discussed in the light of these findings.     

Algorithmen zum automatischen Vergleich von Spektren in der HPLC/UV-Kopplung

Summary For normalization of spectra in spectral libraries in UV/VIS range the maximum of absorbance, the area under the spectrum or a linear regression technique may be used. Using the error propagation law the best method should be the linear regression followed by area normalization. In practical work the area normalization has no advantages compared to a modified point normalization, where the mean of the maximum and the two next data points are taken. Therefore, in an automated comparison of UV spectra the modified point normalization is used for searching for similar spectra. The final comparison is done after linear regression normalization. This has advantages in calculation of error propagation and significance levels of two spectra. The reference spectrum is used as independent, the spectrum of the sample as dependent variable.

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Herrn Prof. Dr. W. Fresenius zu seinem 75. Geburtstag gewidmet

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1. Mitteilung siehe [12]     

Powered partial least squares discriminant analysis

From the fundamental parts of PLS‐DA, Fisher's canonical discriminant analysis (FCDA) and Powered PLS (PPLS), we develop the concept of powered PLS for classification problems (PPLS‐DA). By taking advantage of a sequence of data reducing linear transformations (consistent with the computation of ordinary PLS‐DA components), PPLS‐DA computes each component from the transformed data by maximization of a parameterized Rayleigh quotient associated with FCDA. Models found by the powered PLS methodology can contribute to reveal the relevance of particular predictors and often requires fewer and simpler components than their ordinary PLS counterparts. From the possibility of imposing restrictions on the powers available for optimization we obtain an explorative approach to predictive modeling not available to the traditional PLS methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Investigation of the conformation of hyperbranched poly(arylene oxindole)s using hyper‐Rayleigh scattering

A hyperbranched poly(arylene oxindole), a poly(methacrylate), and a dendrimer, to which the same nonlinear optical chromophore was attached via a small, rigid spacer, were prepared. The difference in hyper‐Rayleigh scattering intensities was measured and compared. From this study, it was concluded that the chromophores, and hence the functional groups in the macromolecule before functionalization, are orientationally correlated in the dendrimer, whereas they are not in the linear and hyperbranched polymer. More in particular, the chromophores in the dendrimer are fixed in a centrosymmetric way because of the globular structure, whereas there is no orientational correlation between the chromophores in linear and hyperbranched polymer. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 3740–3747, 2009     

A size extensive energy functional derived from a double configuration interaction approach: the role of N representability conditions

Starting from a configuration interaction (CI) approach including only doubly excited configurations, the corresponding energy functional is modified by introduction of a topological factor in the normalization condition of the CI coefficients in such a way that it gets inherently size extensive. Constraints imposed by N representability conditions on the modified functional are discussed and lead to a specific choice of the topological factor. The basic variable in the modified energy functional is the second-order reduced density matrix parametrized in terms of CI-like coefficients. Test calculations for a variety of small molecules show that the numerical results obtained with the new functional are in very good agreement with those obtained from coupled cluster singles doubles calculations.     

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.     

Rayleigh–Ritz variation method and connected‐moments polynomial approach

We show that the connected‐moments polynomial approach proposed recently is equivalent to the well known Rayleigh–Ritz variation method in the Krylov space. We compare the latter with one of the original connected‐moments methods by means of a numerical test on an anharmonic oscillator. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009