Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ω _k and normal modes of the isotopomer B are expressed in terms of the vibrational frequencies ν _i and normal modes of the parent molecule A. In those relations complete specification of the normal modes is not required. Only amplitudes at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies with respect to the DFT frequencies is on average . This error is due to the uncertainty of the input data (± 0.5 cm⁻¹) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment, one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better () than scaled DFT frequencies () which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation.     

Eigenvalues of the real generalized eigenvalue equation perturbed by a low-rank perturbation

The low-rank perturbation (LRP) method solves the perturbed eigenvalue equation (B +V) _k = _k (C +P) _k , where the eigenvalues and the eigenstates of the related unperturbed eigenvalue equationB _i = _i C _i are known. The method is designed for arbitraryn-by-n matricesB, V, C, andP, with the only restriction that the eigenstates _i of the unperturbed equation should form a complete set. We consider here a real LRP problem where all matrices are Hermitian, and where in addition matricesC and (C +P) are positive definite. These conditions guarantee reality of the eigenvalues _k and _i . In the original formulation of the LRP method, each eigenvalue _k is obtained iteratively, starting from some approximate eigenvalue _k . If this approximate eigenvalue is not well chosen, the iteration may sometimes diverge. It is shown that in the case of a real LRP problem, this danger can be completely eliminated. If the rank of the generalized perturbation {V, P} is small with respect ton, then one can easily bracket and hence locate to any desirable accuracy the eigenvalues _k (k = 1, ...,n) of the perturbed equation. The calculation of alln eigenvalues requiresO(² n ²) operations. In addition, if the perturbation (V, P) is local with the localizabilityl p, then onlyO(² n) operations are required for a derivation of a single eigenvalue.     

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the low rank perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at atomic sites is considered. Relations which express vibrational frequencies _k and normal modes _k of the perturbed isotopic molecule B in terms of the vibrational frequencies _i and normal modes _i of the unperturbed molecule A are derived. In these relations complete specification of the unperturbed normal modes _i is not required. Only amplitudes | _i of normal modes _i at sites affected by the isotopic substitution are needed.     

Solution of the perturbed eigenvalue equation by the low-rank perturbation method

In its simplest form, the low-rank perturbation (LRP) method solves the perturbed matrix eigenvalue equationsA (B + V) =, whereA, B andV are nth-order Hermitian matrices, and where the eigenstates and the eigenvalues of the unperturbed matrixB are known. The method can be applied to arbitrary perturbations V, but it is numerically most efficient if the rank ofV is small. A special case of low-rank perturbations are localized perturbations (e.g. replacement of one atom with another, creation and destruction of a chemical bond, local interaction of large molecules, etc.). In the case of local perturbations with a fixed localizability I, the operation count for the calculation of a single eigenvalue and/or a single eigenstate isO(l ² n). In the more general case of a delocalized perturbation with a fixed rank p, the operation count for the derivation of all eigenvalues and/or all eigenstates is O( ² n ²). For largen, the performance of the LRP method is hence at least one order of magnitude better than the performance of other methods. The obtained numerical results demonstrate that the LRP method is numerically reliable, and that the performance of the method is in accord with predicted operation counts.Research supported by the Robert A. Welch Foundation (Houston, Texas), and by the Yugoslav Ministry for Development (Grant P-339).     

The line tension and the generalized Young equation: the choice of dividing surface

Using Gibbs method of dividing surfaces, the condition of equilibrium of a sessile drop on a flat non-deformable solid substrate is investigated. The dependence of the line tension on the curvature radius of the dividing three-phase contact line is found. It has been derived a relationship between the partial derivative of the line tension with respect to the curvature radius of the three-phase contact line (which stands in the generalized Young equation) and the total derivative of the line tension with respect to the same radius along the equilibrium states. Various approximated formulas of the generalized Young equation used in the literature are analyzed.     

Setting local rank constraints by orthogonal projections for image resolution analysis: Application to the determination of a low dose pharmaceutical compound

Raman chemical imaging provides chemical and spatial information about pharmaceutical drug product. By using resolution methods on acquired spectra, the objective is to calculate pure spectra and distribution maps of image compounds. With multivariate curve resolution-alternating least squares, constraints are used to improve the performance of the resolution and to decrease the ambiguity linked to the final solution. Non negativity and spatial local rank constraints have been identified as the most powerful constraints to be used.     

Estimating the chemical rank of three-way data arrays by a simple linear transform incorporating Monte Carlo simulation

Estimating an appropriate chemical rank of a three-way data array is very important to second-order calibration. In this paper, a simple linear transform incorporating Monte Carlo simulation approach (LTMC) to estimate the chemical rank of a three-way data array was suggested. The new method determines the chemical rank through performing a simple linear transform procedure on the original cube matrix to produce two subspaces by singular value decomposition. One of two subspaces is derived from the original three-way data array itself and the other is derived from a new three-way data array produced by the linear transformation of the original one. Projection technique incorporating the Monte Carlo approach acts as distinguishing criterion to choose the appropriate component number of the system. Simulated three-way trilinear data arrays with different noise types (homoscedastic and heteroscedastic), various noise level as well as high collinearity are used to illustrate the feasibility of the new method. The results have shown that the new method could yield accurate results with different conditions appended. The feasibility of the new method is also confirmed by two real arrays, HPLC-DAD data and excitation-emission fluorescent data. All the results are compared with the other three factor-determining methods: factor indicator function (IND), core consistency diagnostic (CORCONDIA) and two-mode subspace comparison (TMSC) approach. It shows that the newly proposed algorithm can objectively and quickly determine the chemical rank to fit the trilinear model.     

Determination of rank by median absolute deviation (DRMAD): a simple method for determining the number of principal factors responsible for a data matrix

Median absolute deviation (MAD) is a well‐established statistical method for determining outliers. This simple statistic can be used to determine the number of principal factors responsible for a data matrix by direct application to the residual standard deviation (RSD) obtained from principal component analysis (PCA). Unlike many other popular methods the proposed method, called determination of rank by MAD (DRMAD), does not involve the use of pseudo degrees of freedom, pseudo F‐tests, extensive calibration tables, time‐consuming iterations, nor empirical procedures. The method does not require strict adherence to normal distributions of experimental uncertainties. The computations are direct, simple to use and extremely fast, ideally suitable for online data processing. The results obtained using various sets of chemical data previously reported in the chemical literature agree with the early work. Limitations of the method, determined from model data, are discussed. An algorithm, written in MATLAB format, is presented in the Appendix. Copyright © 2008 John Wiley & Sons, Ltd.     

Prediction of the Conductance of Strong Electrolytes and the Calculation of the Ionization Constant of Weak Electrolytes in a Dilute Solution by a New Equation

In order to predict the conductance for dilute 1-1 valent electrolyte solutions,a new conductance equation was proposed based on the Onsager and Onsagar-Fuoss-Chen conductance equation.It has only one parameter A,which can be obtained directly from the data of ionic limiting molar conductivity Λ∞m,and its expression is very simple.The new equation has been verified by the experimental molar conductivities of some single strong electrolyte and mixed electrolyte solutions at 298.15 K reported in literatures.The results are in good agreement with the experimental data.Meanwhile the ionization constants of some weak electrolyte solutions were calculated by a modified equation of this new equation,and it was also found that the calculation results are in good agreement with the data in the literature.     

Modeling phase equilibria of alkanols with the simplified PC-SAFT equation of state and generalized pure compound parameters

The simplified PC-SAFT equation of state has been applied to liquid–liquid, vapor–liquid and solid–liquid equilibria for mixtures containing 1- or 2-alkanols with alkanes, aromatic hydrocarbons, CO₂ and water. For the alkanols we use generalized pure compound parameters. This means that two of the physical pure compound parameters, m (segment number) and σ (segment diameter), are obtained from linear extrapolations, since m and mσ³, increase linearly with respect to the molar mass, and moreover, the two association parameters (association energy and association volume) were assumed to be constant for all alkanols. Only the dispersion energy is fitted to experimental data. Thus it is possible to estimate parameters for several 1- and 2-alkanols. The final aim is to develop a group contribution approach for PC-SAFT which is suitable for complex compounds, considering that the motivation of this project is to obtain a thermodynamic model which can be used in the development of sophisticated products such as pharmaceuticals, polymers, detergents or food ingredients. One of the severe limitations in applying SAFT-type equations of state to these compounds is that the procedure for obtaining the pure compound parameters is usually based on fitting to saturated vapor pressure and liquid density data over an extended temperature range. However, such data are rarely available for complex compounds. To verify the new pure compound parameters, comparisons to ordinary optimized alkanol parameters, where all five pure compound parameters were fitted to experimental liquid density and vapor pressure data, were made. The results show that the new generalized alkanol parameters from this work perform at least as well as other alkanol parameter sets.     

Determination of total phosphorus and nitrogen in turbid waters by oxidation with alkaline potassium peroxodisulfate and low pressure microwave digestion, autoclave heating or the use of closed vessels in a hot water bath: comparison with Kjeldahl digestion

The evaluation of the use of alkaline peroxodisulfate digestion with low pressure microwave, autoclave or hot water bath heating for the determination of total phosphorus and nitrogen in turbid lake and river waters is described. The efficiency of these digestion procedures were compared to a Kjeldahl digestion procedure with sulphuric acid-potassium sulfate and copper sulfate. The final solution before digestion was 0.045 M in potassium peroxodisulfate and 0.04 M in sodium hydroxide. Procedures were evaluated by the analysis of suspensions of two reference materials, National Institute of Environmental Science, Japan, no. 3 Chlorella and no. 2 pond sediment and natural turbid waters. Best recoveries of phosphorus and nitrogen by microwave heating were obtained when solutions were digested at 95 °C for 40 min. Quantitative recoveries of phosphorus from Chlorella suspensions up to 1000 mg/l were obtained by all three heating procedures, but incomplete recoveries of nitrogen occurred above 20 mg N/l in the digested sample. Good recoveries of phosphorus and nitrogen from suspended sediment suspensions were obtained only from solutions containing <150 mg/l of suspended sediments. Recoveries of phosphorus from phosphorus compounds containing COP and CP bonds added to distilled water were quantitative (94-113%) except for polyphosphates (microwave, 34±8; autoclave, 114±6; water bath, 96±4) and aluminium phosphate (8-23%). Recoveries of nitrogen compounds containing CN bonds added to distilled water were quantitative (94-96%). The analysis of a range of natural turbid water samples by alkaline peroxodisulfate and microwave, autoclave and water bath heating gave similar total phosphorus and nitrogen results. All procedures using alkaline peroxodisulfate underestimate phosphorus concentrations at high suspended sediment concentrations (>150 mg/l) and are only suitable for the analysis of very turbid samples when the turbidity is due to organic matter (algal cells, plant detritus). Underestimation of nitrogen occurs when samples contain more than 20 mg N/l.