Computing a molecule: A mathematical viewpoint

We give here an overview of the mathematical results known to this day on the models used in Quantum Chemistry for the numerical computations of molecules. We focus on the problems related to the ground state, in the framework of Hartree–Fock type models and Thomas–Fermi type models. More precisely, we outline the most recent results on the following questions: existence and uniqueness of the minimum, and existence of an optimized geometry for the nuclei. We eventually give a list of open problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computing the energy of a water molecule using multideterminants: a simple, efficient algorithm

Quantum Monte Carlo (QMC) methods such as variational Monte Carlo and fixed node diffusion Monte Carlo depend heavily on the quality of the trial wave function. Although Slater-Jastrow wave functions are the most commonly used variational ansatz in electronic structure, more sophisticated wave functions are critical to ascertaining new physics. One such wave function is the multi-Slater-Jastrow wave function which consists of a Jastrow function multiplied by the sum of Slater determinants. In this paper we describe a method for working with these wave functions in QMC codes that is easy to implement, efficient both in computational speed as well as memory, and easily parallelized. The computational cost scales quadratically with particle number making this scaling no worse than the single determinant case and linear with the total number of excitations. Additionally, we implement this method and use it to compute the ground state energy of a water molecule.     

Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation

This paper proposes a novel mathematical model for the formation of spatio-temporal patterns in electrodeposition. At variance with classical modelling approaches that are based on systems of reaction–diffusion equations just for chemical species, this model accounts for the coupling between surface morphology and surface composition as a means of understanding the formation of morphological patterns found in electroplating. The innovative version of the model described in this work contains an original, flexible and physically straightforward electrochemical source term, able to account for charge transfer and mass transport: adsorbate-induced effects on kinetic parameters are naturally incorporated in the adopted formalism. The relevant non-linear dynamics is investigated from both the analytical and numerical points of view. Mathematical modelling work is accompanied by an extensive, critical review of the literature on spatio-temporal pattern formation in alloy electrodeposition: published morphologies have been used as a benchmark for the validation of our model. Moreover, original experimental data are presented—and simulated with our model—on the formation of broken spiral patterns in Ni–P–W–Bi electrodeposition.     

Supramolecular reactor from self-assembly of rod-coil molecule in aqueous environment

We synthesized an amphiphilic coil-rod-coil triblock molecule consisting of hexa-p-phenylene as a rod block and poly(ethylene oxide) with the number of repeating units of 17 as coil blocks and investigated aggregation behavior in aqueous environment. The rod-coil molecule was observed to aggregate into discrete micelles consisting of hydrophobic disklike rod bundles encapsulated by hydrophilic poly(ethylene oxide) coils. The aromatic bundles of the micelles were demonstrated to be used as an efficient supramolecular reactor for the room temperature Suzuki cross-coupling reaction of a wide range of aryl halides, including even aryl chlorides with phenylboronic acids in aqueous environment. These results demonstrate that self-assembly of amphiphilic rod-coil molecules can provide a useful strategy to construct an efficient supramolecular reactor for aromatic coupling reaction.     

The influence of director fluctuations on molecular reorientation of a small probe molecule in a liquid-crystalline environment

The influence of director fluctuations on a probe molecule dissolved in a nematic liquid crystal is calculated to all orders using the gaussian properties of the director field. Consequences for the N.M.R. spectrum and relaxation through quadrupolar interaction of the probe molecule are indicated. It is found that the equilibrium distribution of molecular orientations, and consequently quadrupolar spectral splittings are virtually unchanged by fluctuations in the director field. The spectral densities are also obtained and it is shown that in addition to J₀₁ also J₀₀ and J₀₂ become frequency dependent in the N.M.R. frequency range. The frequency dependence of J₀₀ and J₀₂ has a logarithmic contribution; the magnitude of the frequency dependent contribution to J₀₂ is, however, insufficient to explain experimental results.     

A mathematical model for a copolymer in an emulsion

In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus α times the number of hydrophobic monomers in oil minus β times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone \({\{(\alpha,\beta)\in \mathbb{R}^2\colon\,|\beta|\leq\alpha\}}\). We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks.     

Atomic spectral overlaps: A mathematical model and its use for studying line shape parameters

A mathematical model which describes an atomic spectral overlap is presented. The model is developed using hyperfine structure, overlap separation values, and line broadening calculations coupled with relative absorbance measurements of overlap and non-overlap situations. With good hyperfine structure data, the model can be used to determine the a-parameter values for both transitions of the overlap. Even in the absence of high quality data, the model can provide meaningful insights into line shapes, hyperfine structure, and wavelength assignments of overlapping spectral transitions.     

Enantioseparation of binaphthol and its monoderivatives by cyclodextrin-modified capillary zone electrophoresis: A mathematical approach

A simple model for the separation of atropisomers of binaphthol and its monoderivatives by means of cyclodextrin-modified capillary zone electrophoresis (CD-CZE) was used to describe the migration behavior of poly charged enantiomers in a chiral separation system. This mathematical approach allowed for the determination of the optimal cyclodextrin concentrations for the enantioseparation of binaphthols by taking into account the influence of the formed complex mobilities. Moreover, using this theoretical approach, the reversal of the enantiomers’ migration order as a function of cyclodextrin concentration was predicated. The apparent complexation constants between the cyclodextrins and the binaphthol and its monoderivatives could be calculated using a non-linear curve fitting method and three linear plotting methods (x-reciprocal, y-reciprocal and double reciprocal). Good agreements between the theoretical and experimental cyclodextrin concentrations were obtained.     

Using a small molecule inhibitor of peptide: N-glycanase to probe its role in glycoprotein turnover

Peptide:N-glycanase (PNGase) is ostensibly the sole enzyme responsible for deglycosylation of unfolded N-linked glycoproteins dislocated from the ER to the cytosol. Here we show the pan-caspase inhibitor, Z-VAD-fmk, to be an active site-directed irreversible inhibitor of yeast and mammalian PNGase at concentrations below those used to inhibit caspases in vivo. Through chemical synthesis we determined that the P1 residue, electrophile position, and leaving group are important structural parameters for PNGase inhibition. We show that Z-VAD-fmk inhibits PNGase in living cells and that degradation of class I MHC heavy chains and TCRalpha, in an identical cellular setting, is markedly different. Remarkably, proteasome-mediated turnover of class I MHC heavy chains proceeds even when PNGase is completely inhibited, suggesting that the function of PNGase may be to facilitate more efficient proteasomal proteolysis of N-linked glycoproteins through glycan removal.     

Exploration of single molecule events in a haloperoxidase and its biomimic: localization of halogenation activity

In situ observation of single oxidation/halogenation events by catalytically generated hypobromite species using single molecule fluorescence microscopy allows monitoring of the diffusion behavior of these halonium species from the catalyst into the bulk solution. The fluororgenic probe specifically reacts with hypohalites, yielding fluorescein that can be detected with single molecule sensitivity. It was found for two investigated catalysts (Curvularia verruculosa enzymes and tungstate-exchanged LDH crystals) that in steady-state conditions hypobromite is able to diffuse over 800 nm in the bulk solution before it oxidizes organic substrates.     

Temperature and velocity fields in a gas stream exiting a plasma torch. A mathematical model and its experimental verification

A mathematical model was developed to predict the velocity and temperature fields in a free plasma jet issuing from a D.C. plasma torch. It was assumed that the temperature and velocity at the torch nozzle were specified and the turbulent Navier-Stokes equations were solved in conjunction with a two-equation model of turbulence and the energy transport equation. The model was formulated in terms of the two-dimensional elliptic equations to facilitate its future extension to nonparabolic problems. The predictions of the model were compared with experimental measurements obtained from laser Doppler and spectroscopic techniques. Good overall agreement was found between the theoretical predictions and the experimental measurements for two sets of initial conditions corresponding to nitrogen/hydrogen and argon/hydrogen plasmas. Radiation was found to have a small effect on the overall heat transfer process, and it is suggested that the assumption of an optically thin radiation loss per unit volume is sufficiently accurate for most engineering applications. The significance of this work lies in the fact that, for the first time, it is possible to test the assumptions of the current model against a reliable set of experimental measurements.Notation C ₁,C ₂,C _D constants inK- turbulence model, Table III - C _P average specific heat of plasma at constant pressure - h ₁ length of integration region - h ₂ width of integration region - K turbulent kinetic energy per unit mass - P pressure - r radial coordinate - R ₀ internal radius of anode - S _K source term forK equation - S _R radiation loss per unit volume - S source term for equation - T ₀ temperature of plasma atz=0 - T temperature of plasma - T _a ambient temperature - u velocity inz direction - u ₀ velocity of plasma atz=0 - v radial velocity of plasma - z axial coordinate - dissipation rate of turbulence energy - , _eff, _t molecular, effective, and turbulent viscosities, respectively - density - _K, _T, Prandtl numbers forK, T, and, respectively     

Finding the 3D structure of a molecule in its IR spectrum

 The labile iron(II) and iron(III) species are complexed directly in the sample solution with 1,10 phenanthroline and ferron (8-hydroxy-7-iodoquinoline-5-sulfonic acid), respectively. The complexes thus formed are mutually adsorbed and separated by solid phase extraction. The direct determination of iron(III) and iron(II) species with flame atomic absorption spectrometry (FAAS) follows the elution of the iron(III)-ferron complex adsorbed by an anion-exchange and an iron(II)-phenanthroline complex adsorbed by a non-polar RP-18 phase. In the case of indirect determination, the iron(II)-phenanthroline complex that passes through the anion-exchange phase, is measured, and the content of iron(III) is calculated by the difference of the iron(II) and the total iron content. A direct determination with this method has been applied to the iron species analysis in wine samples and the results are compared with those obtained for the determination with adsorptive stripping voltammetry (ASV) as reference method. Received: 17 August 1995/Revised: 12 February 1996/Accepted: 14 February 1996     

Accurate theoretical prediction of vibrational frequencies in an inhomogeneous dynamic environment: a case study of a glutamate molecule in water solution and in a protein-bound form

We propose a hierarchical approach to model vibrational frequencies of a ligand in a strongly fluctuating inhomogeneous environment such as a liquid solution or when bound to a macromolecule, e.g., a protein. Vibrational frequencies typically measured experimentally are ensemble averaged quantities which result (in part) from the influence of the strongly fluctuating solvent. Solvent fluctuations can be sampled effectively by a classical molecular simulation, which in our model serves as the first, low level of the hierarchy. At the second high level of the hierarchy a small subset of system coordinates is used to construct a patch of the potential surface (ab initio) relevant to the vibration in question. This subset of coordinates is under the influence of an instantaneous external force exerted by the environment. The force is calculated at the lower level of the hierarchy. The proposed methodology is applied to model vibrational frequencies of a glutamate in water and when bound to the Glutamate receptor protein and its mutant. Our results are in close agreement with the experimental values and frequency shifts measured by the Jayaraman group by the Fourier transform infrared spectroscopy [Q. Cheng et al., Biochem. 41, 1602 (2002)]. Our methodology proved useful in successfully reproducing vibrational frequencies of a ligand in such a soft, flexible, and strongly inhomogeneous protein as the Glutamate receptor.     

Single molecule solvation and its effects on tautomeric equilibria in a self-assembled capsule

A self-assembled cylindrical capsule provides a nanoscale environment that affects keto-enol equilibria. The equilibrium constants for encapsulated beta-ketoesters show values that differ by an order of magnitude from that of the free tautomers in solution. For complexes with a single, large encapsulated guest, the inner surfaces of the capsule and the seam of the hydrogen bonds influence the equilibrium between the encapsulated keto and enol forms. For complexes of smaller beta-ketoesters, the coencapsulated solvent influences the equilibria. The solvent reduces the space available and affects the positioning of the ester in the capsule.     

A diastereoselective process induced in a curved aromatic molecule: oxidation of thioether-substituted subphthalocyanines

Different sulfoxide-substituted subphthalocyanines have been synthesized by oxidation reaction of the corresponding thioether precursors. We have observed for the first time that when the oxidation takes place close to the macrocyclic core, the intrinsically chiral subphthalocyanine can induce a high degree of diastereoselectivity in this process.     

Potential energy of the smoothed surface of a molecule in its interaction with a solvent

A procedure was developed for calculating the potential of the smoothed surface of a solute molecule in its interaction with the solvent. The potential map of the smoothed surface was calculated for a series of molecules, including those containing various heteroatoms. A method was suggested for calculating the averaged potential of the solute molecule surface, taking into account the dynamics of solvation. The total energy of interaction of a solute molecule with the surrounding solvent was estimated.     

Geometric structure of a molecule of Epitalon tetrapeptide: A molecular-dynamics simulation

Variation of the geometric parameters of a molecule of Epitalon tetrapeptide (Ala-Glu-Asp-Gly) over a period of 1500 ps was simulated by the method of molecular dynamics using AMBER force field. The structure of the molecule is stabilized by two salt bridges formed by the N-terminal nitrogen atom and oxygen atoms of Asp and Glu side chains. The biological effect of Epitalon was attributed to formation of salt or hydrogen bonds involving one or several ionizable functional groups of the molecules.