Normally ordered Franck–Condon operator: A new approach for the calculation of Frank–Condon factors

The idea of a Franck–Condon (FC ) operator is introduced, and its normally ordered form is obtained through the newly developed technique of “integration within an ordered product of operators (IWOP ).” It is shown that the FC operator leads to a new approach for the calculation of FC factors. The results of existing theories are viewed, and the connection between the FC operator and the “squeeze-operator” is pointed out.     

Falloff curves for the recombination reaction Cl + FC(O)O + M --> FC(O)OCl + M

The pressure dependence of the recombination reaction Cl + FC(O)O + M --> FC(O)OCl + M has been investigated at 296 K. FC(O)O radicals and Cl atoms were generated by laser flash photodissociation of FC(O)OO(O)CF at 193 nm in mixtures with Cl2 and He or SF6 over the total pressure range 8-645 Torr. The measured FC(O)O radical and F atom yields in the photolysis are 0.33 +/- 0.06 and 0.67 +/- 0.06. The reaction lies in the falloff range approaching the high-pressure limit. The extrapolations toward the limiting low- and high-pressure ranges were carried out using a reduced falloff curves formalism, which includes a recent implementation for the strong-collision broadening factors. The resulting values for the low-pressure rate coefficients are (2.2 +/- 0.4) x 10(-28)[He], (4.9 +/- 0.9) x 10(-28)[SF6], (1.9 +/- 0.3) x 10(-28)[Cl2] and (5.9 +/- 1.1) x 10(-28)[FC(O)OO(O)CF] cm3 molecule(-1) s(-1). The derived high-pressure rate coefficient is (4.4 +/- 0.8) x 10(-11) cm3 molecule(-1) s(-1). For the reaction Cl + FC(O)OCl --> Cl2 + FC(O)O a rate coefficient of (1.6 +/- 0.3) x 10(-11) cm3 molecule(-1) s(-1) was determined. The high-pressure rate coefficient was theoretically interpreted using SACM/CT calculations on an ab initio electronic potential computed at the G3S level of theory. Standard heat of formation values of -99.9 and -102.5 kcal mol(-1) were computed at the G3//B3LYP/6-311++G(3df,3pd) level of theory for cis-FC(O)OCl and trans-FC(O)OCl, respectively. The computed electronic barrier for the conversion between the trans and cis conformers is 8.9 kcal mol(-1). On the basis of the present results, the above reactions are expected to have a negligible impact on stratospheric ozone levels.     

Computation of Rotation Matrices Making Lined‐Up to the Local Cartesian Coordinates

Rotation matrices were expressed in terms of Gaunt coefficients and complex spherical harmonics. The rotation matrices were calculated using two different ways. In the first, Gaunt coefficients and normalized complex spherical harmonics were directly calculated using binomial coefficients; in the second, Gaunt coefficients and complex spherical harmonics were recursively calculated. The methods were compared with respect to accuracy and computation time (CPU) for low and very high quantum numbers.     

Determination of Anafranil in Pharmaceuticals by Difference and Difference-Derivative Spectrophotometry

Abstract

Difference (ΔA) and difference first- (ΔD1) and second- (ΔD2) derivative spectrophotometric methods are described for the assay and quality control of anafranil, a powerful antidepressant, in pharmaceutical formulations.

The procedures are based upon the measurement of ΔA, ΔD1 and ΔD2 of anafranil in alkaline solutions against their acidic solutions as blanks.

Interferences of the excipients and diluents or irrelevant absorptions are nullified. Calibration graphs of ΔA, ΔD1 and ΔD2 versus the concentration of the drug showed linear relationships up to 10 μg/ml, with correlation coefficients ranging from 0.9998 to 0.9999.

Detection limits at p = 0.01 level of significance were calculated to be 0.060 (ΔA), 0.056 (ΔD1) and 0.063 (ΔD2) μg/ml. The limits of quantification were 0.45 (ΔA), 0.36 (ΔD1) and 0.81 (ΔD2) μg/ml.

The procedures have been successfully applied to the determination of anafranil in synthetic samples and in commercial pills and injections for this drug with high reliability and repeatability.     

Polynomial coefficients. Application to spin–spin splitting by N equivalent nuclei of spin I > 1/2

The NMR intensity pattern of a nucleus split by N identical nuclei of spin 1/2 is given by the binomial coefficients. These are conveniently obtained from Pascal's triangle, equivalent to the chemist's branching diagram. Much less well‐known is the pattern from splitting by N identical nuclei of spin I > 1/2. This was originally presented in terms of multinomial coefficients, but polynomial coefficients are more convenient. These describe the number of ways that N objects can be distributed to 2I + 1 numbered boxes. They arise in the polynomial expansion and are conveniently obtained from generalizations of Pascal's triangle. Examples and predictions are given.     

Use of Filter-Steinborn B and Guseinov $${Q_{ns}^q}$$ auxiliary functions in evaluation of two-center overlap integrals over Slater type orbitals

An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov \({Q_{ns}^q}\) auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those obtained by other method for arbitrary principal quantum numbers and different screening constants.     

A comparison of diffusion coefficients for ternary mixed micelle solutions measured by macroscopic gradient and dynamic light scattering techniques

Taylor dispersion is used to measure ternary mutual diffusion coefficients (D(ik)) for aqueous solutions of decylsulfobetaine (SB10) (1) + dodecylsulfobetaine (SB12) (2), SB10 (1) + SB14 (2), and SB12 (1) + SB14 (2) mixed zwitterionic micelles. Cross-coefficient D(21) for the coupled flow of surfactant 1 produced by a concentration gradient in surfactant 2 is relatively small for these solutions, but D(12) reaches values as large as the main D(ii) coefficients. The results are interpreted by using the equation D(ik) = partial differential(C(i)D(i))/ partial differentialC(k) to relate the ternary mutual diffusion coefficients to the concentration-weighted average diffusion coefficients D(i) of the micellar and free-monomer forms of the surfactants. The macroscopic-gradient Taylor measurements are compared with diffusion coefficients measured by dynamic light scattering (DLS), which monitors microscopic concentration fluctuations. At most compositions, the intensity autocorrelation function G(tau) is a single exponential decay in D((2)), the smaller eigenvalue of the mutual diffusion coefficient matrix. A contribution from D((1)) is identified at high solute fractions of surfactant 1. The DLS results are consistent with contributions to G(tau) from uncoupled fluctuations in the concentrations of eigencomponents defined as the linear combinations of surfactants 1 and 2 that diagonalize the D(ik) matrix. A procedure for the rapid and convenient DLS measurement of ternary mutual diffusion coefficients, including the cross-coefficients for coupled diffusion, is suggested, using the Onsager reciprocal relation together with the eigenvalues and pre-exponential factors from G(tau).     

Development and testing of an algorithm for efficient MP2/CCSD(T) energy estimation of molecular clusters with the 2–body approach

This work reports the development and testing of an automated algorithm for estimating the energies of weakly bound molecular clusters employing correlated theory. Firstly, the monomers and dimers of (homo/hetero) clusters are identified, and the sum of one-body and two-body contributions to correlation energy is calculated. The addition of this contribution to the Hartree-Fock full calculation (FC) energies provides a good estimate of the total energies at Møller–Plesset second-order perturbation theory (MP2)/coupled-cluster method with singles and doubles (CCSD) (T)-level theory using augmented Dunning basis sets. The estimated energies for several test clusters show an excellent agreement with their FC counterparts, with a substantial wall-clock time saving employing off-the-shelf hardware. Furthermore, the complete basis set (CBS) limit for MP2 energy computed using the two-body approach also agrees with its CBS energy with its FC counterpart.     

Clebsch–Gordan coefficients for the group chains O ⊃ T ⊃ D2 and Td ⊃ T ⊃D2 by the method of projective representations

The spinor representations for the groups forming the chains O ? T ? D ₂ and T _d ? T ? D ₂ are constructed as projective representations. The Clebsch–Gordan coefficients are then calculated using a standard method. Projective factor systems, irreducible representations, canonical bases, and tables of Clebsch–Gordan coefficients are given. The subduction from O to D ₃ is discussed.     

Dramatic changes in geometry after ionization: experimental and theoretical studies on the electronic properties of fluorocarbonyl (mono-, di-, and tri-) sulfur compounds

In this work, we present a complete study on He I photoelectron spectroscopy (PES) for the fluorocarbonyl mono-, di-, and trisulfur compounds FC(O)SCl, FC(O)SSCH(3), and FC(O)SSSC(O)F. After optimizations of the structure for stable conformers at different levels of theory, a complete theoretical study involving the calculation of the ionization energies using orbital valence Green's functional (OVGF) was performed. Calculations of radical-cationic forms were carried out in order to compare their properties with those of the neutral molecules. The first IP values are 10.7, 9.0, and 10.5 eV for FC(O)SCl, FC(O)SSCH(3), and FC(O)SSSC(O)F, respectively. The groups bonded to the S atom mainly influence the ionizations originating from the sulfur lone pairs. A wide electronic delocalization in the FC(O)S moiety can be deduced from experimental and theoretical results, which leads to a strong energetic stabilization of the n' '(S) (sulfur lone pair pi orbital). Other conclusions relate to effects on the substituents attached to the S atom and the importance of the molecular planarity in the orbital stabilization of the FC(O)S moiety for the neutral molecules. It is worthwhile mentioning that FC(O)SCl retains its planar structure after ionization, but drastic changes occur in the geometry of both FC(O)SSCH(3) and FC(O)SSSC(O)F. The FC(O)SSCH(3) molecule adopts a heavy atom planar structure after ionization. The FC(O)SSS moiety becomes a planar form after the ionization of the FC(O)SSSC(O)F molecule, whereas the second C(O)F group maintains its original conformation with respect to the SSS group.     

Electron density distribution in stacked benzene dimers: a new approach towards the estimation of stacking interaction energies

The potential energy surface for the benzene dimer in stacked conformations (84 points calculated) was computed at the MP2(FC)6-31+G(2d,2p) level of theory. Electron density (ED) distribution computed using the MP2(FC), B3LYP, and Hartree-Fock methods with the same basis set is studied in the frame of topological analysis. It is found that ED topology does not depend on the method of calculation. The values of the ED and its Laplacian in the cage critical point calculated using different methods are determined to be linearly dependent with the slope depending on basis set. Correlation equations based on these properties allow the interaction energy between benzene rings to be predicted with 8% mean relative error in the energy for the given region of the potential energy surface. This provides a new method for the estimation of stacking interaction energy using ED properties calculated with low level quantum-chemical methods.     

Analytical interaction potentials of the long range alkali-metal dimers

Using almost degenerate perturbation theory the calculation of long range limits of alkali-metal dimer molecular states which dissociate to the atom pairs S_1/2-P_{1/2, 3/2} and S_1/2-D_{3/2, 5/2} is presented. Hund's coupling casec is adopted. The calculation includes correlations up to the second order of perturbation theory and exchange energy contributions. The results are expressed in a simple analytical form in terms of dispersion coefficients.     

Calculation of the tau dependence of the vibration-internal rotation-overall rotation interactions in CH3OH from molecular structure and molecular dynamics

The Guan and Quade theory for vibration-large-amplitude internal-motion-rotation interactions has been applied to the internal rotation problem in CH(3)OH. Through the molecular dynamics, the cos 3tau and sin 3tau dependence of the torsional-rotational coefficients in the effective Hamiltonian have been calculated from molecular structure. The internal rotation coordinate tau(') for the vibrationally distorted molecule is shown to have the necessary threefold symmetry for all values of tau('). For the methyl deformation modes, the vibrational dependence of the internal rotation potential energy is shown to have a threefold symmetry. The S(t) and S(t)S(t) dependence of the inertia tensor and Coriolis coupling coefficients has been developed in terms of curvilinear internal coordinates. The T transformation separating rotation from vibrations in zeroth order is then applied, the kinetic-energy tensor inverted to momentum space, and finally the effective torsion-rotation coefficients are calculated by Van Vleck perturbation theory. When compared to the empirical results, the kinetic-energy contributions to the cos 3tau and sin 3tau dependence of the coefficients are as follows: 54% of P(a)(2) is accounted for, 28% of P(a)P(b), 16% of P(a)P(c), and 91% of the asymmetry. The calculation is inadequate to account for the P(b)(2),P(c)(2), and P(b)P(c) coefficients, ranging from factors of 20-70, even with the incorrect sign for some of the terms. Anharmonic force contributions from the vibrations have not been used in the calculation since these forces are not known at this time.     

Electronic transitions of aluminum monoxide

In view of the renewed interest in the electronic spectra of the AlO molecule, which is of both astrophysical and aeronomical interest, a systematic analysis of the available spectral data is carried out. A valid and consistent set of vibrational constants are derived for the A and D states of the molecule. Potential curves (Morse and RKRV) are obtained for the X, A, C and D states. Franck-Condon (FC) intensity factors and r-centroids are reported for the CX, DX and DA transitions. The calculated FC-factors are used to explain the intensity distribution of the bands in the various systems.     

Use of binomial coefficients in fast and accurate calculation of Löwdin-<Emphasis Type="Italic">α</Emphasis> radial functions

In the present paper, we introduce a new algorithm for the development of efficient numerical methods for the analytical solution of the Löwdin-α radial function. The final results are expressed through the binomial coefficients, which enable fast and accurate calculation of the Löwdin-α radial function. The comparisons of the obtained results with the literature have shown that the presented approach can be used for the fast calculation of multicenter multielectron integrals over Slater-type orbitals (STOs). The efficiency of the algorithm is discussed and its performance with several examples is demonstrated.     

Use of Filter-Steinborn B and Guseinov $${Q_{ns}^q }$$ auxiliary functions in evaluation of two-center overlap integrals over Slater type orbitals

An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those obtained by other method for arbitrary principal quantum numbers and different screening constants. An erratum to this article can be found at     

Effects of modified starch and fat on the rheological characteristics of newly formulated processed cheese: Use of experimental design method

This study aims to investigate the impact of the incorporation of starch (modified and non-modified) on the texture of the processed cheese in order to appreciate a new recipe on one hand and to try to reduce fat content (FC) on the other. The modeling of the experimental results has been performed. An experimental design was used (the variable factors are the amount of starch and FC). The dry extract (DE) of cheese and parameters derived from the modeling of rheological results are the selected responses. The cheese with a concentration of modified starch that is greater than 2% has a high viscosity at rest, while the cheese with non-modified starch has a lower viscosity. When the FC exceeds 15%, the viscosity increases as the concentration of the starch exceeds 2%. Below this concentration, the FC effect is negligible except when the starch concentration is 1%, in this case the viscosity decreases.     

Sampling hazelnuts for aflatoxin: effect of sample size and accept/reject limit on reducing the risk of misclassifying lots

About 100 countries have established regulatory limits for aflatoxin in food and feeds. Because these limits vary widely among regulating countries, the Codex Committee on Food Additives and Contaminants began work in 2004 to harmonize aflatoxin limits and sampling plans for aflatoxin in almonds, pistachios, hazelnuts, and Brazil nuts. Studies were developed to measure the uncertainty and distribution among replicated sample aflatoxin test results taken from aflatoxin-contaminated treenut lots. The uncertainty and distribution information is used to develop a model that can evaluate the performance (risk of misclassifying lots) of aflatoxin sampling plan designs for treenuts. Once the performance of aflatoxin sampling plans can be predicted, they can be designed to reduce the risks of misclassifying lots traded in either the domestic or export markets. A method was developed to evaluate the performance of sampling plans designed to detect aflatoxin in hazelnuts lots. Twenty hazelnut lots with varying levels of contamination were sampled according to an experimental protocol where 16 test samples were taken from each lot. The observed aflatoxin distribution among the 16 aflatoxin sample test results was compared to lognormal, compound gamma, and negative binomial distributions. The negative binomial distribution was selected to model aflatoxin distribution among sample test results because it gave acceptable fits to observed distributions among sample test results taken from a wide range of lot concentrations. Using the negative binomial distribution, computer models were developed to calculate operating characteristic curves for specific aflatoxin sampling plan designs. The effect of sample size and accept/reject limits on the chances of rejecting good lots (sellers' risk) and accepting bad lots (buyers' risk) was demonstrated for various sampling plan designs.     

The infrared fundamental intensities and polar tensor of allene.

The polar tensor of allene was calculated from the infrared fundamental band intensities of C3H4 and C3D4. The ambiguities in the signs of the dipole moment derivatives with respect to their normal coordinates were resolved by comparison of tensor elements with ab initio calculations at the B3LYP, MP2(FC) and CCD(FC) levels with a 6/311 + + G(3d,3p) basis set. The results are similar to those previously obtained by Koga and co-workers except for the choice of an average of two sign combinations for the E symmetry elements. The values of the mean dipole moment derivatives for the sp and sp2 carbon atoms obtained in this work, 0.032 and -0.133 e, respectively, are in good agreement with the CCD(FC)/6-311 + + G(3d,3p), 0.061 and -0.128 e, and MP2(FC)/6-311 + + G(3d,3p), 0.072 and -0.153 e, theoretical results. The mean dipole moment derivatives are shown to be consistent with potential models relating 1s electron ionization energies and atomic charges.