Topologically invariant reaction coordinates for simulating multistate chemical reactions.

Evaluating free energy profiles of chemical reactions in complex environments such as solvents and enzymes requires extensive sampling, which is usually performed by potential of mean force (PMF) techniques. The reliability of the sampling depends not only on the applied PMF method but also the reaction coordinate space within the dynamics is biased. In contrast to simple geometrical collective variables that depend only on the positions of the atomic coordinates of the reactants, the E(gap) reaction coordinate (the energy difference obtained by evaluating a suitable force field using reactant and product state topologies) has the unique property that it is able to take environmental effects into account leading to better convergence, a more faithful description of the transition state ensemble and therefore more accurate free energy profiles. However, E(gap) requires predefined topologies and is therefore inapplicable for multistate reactions, in which the barrier between the chemically equivalent topologies is comparable to the reaction activation barrier, because undesired "side reactions" occur. In this article, we introduce a new energy-based collective variable by generalizing the E(gap) reaction coordinate such that it becomes invariant to equivalent topologies and show that it yields more well behaved free energy profiles than simpler geometrical reaction coordinates.

Parallel coordinates in computational engineering design

Modern Engineering Design involves the deployment of many computational tools. Re- search on challenging real-world design problems is focused on developing improvements for the engineering design process through the integration and application of advanced com- putational search/optimization and analysis tools. Successful application of these methods generates vast quantities of data on potential optimum designs. To gain maximum value from the optimization process, designers need to visualise and interpret this information leading to better understanding of the complex and multimodal relations between param- eters, objectives and decision-making of multiple and strongly conflicting criteria. Initial work by the authors has identified that the Parallel Coordinates interactive visualisation method has considerable potential in this regard. This methodology involves significant levels of user-interaction, making the engineering designer central to the process, rather than the passive recipient of a deluge of pre-formatted information. In the present work we have applied and demonstrated this methodology in two differ- ent aerodynamic turbomachinery design cases; a detailed 3D shape design for compressor blades, and a preliminary mean-line design for the whole compressor core. The first case comprises 26 design parameters for the parameterisation of the blade geometry, and we analysed the data produced from a three-objective optimization study, thus describing a design space with 29 dimensions. The latter case comprises 45 design parameters and two objective functions, hence developing a design space with 47 dimensions. In both cases the dimensionality can be managed quite easily in Parallel Coordinates space, and most importantly, we are able to identify interesting and crucial aspects of the relationships between the design parameters and optimum level of the objective functions under con- sideration. These findings guide the human designer to find answers to questions that could not even be addressed before. In this way, understanding the design leads to more intelligent decision-making and design space exploration. © 2012 AIAA.

General local and rectilinear vibrational coordinates consistent with Eckart's conditions

We present a general method to construct a set of local rectilinear vibrational coordinates for a nonlinear molecule whose reference structure does not necessarily correspond to a stationary point of the potential-energy surface. We show both analytically and with a numerical example that the vibrational coordinates satisfy Eckart's conditions. In addition, we find that the Watson Hamiltonian provides a fairly robust description even of highly excited vibrational states of triatomic molecules, except for a few states of large amplitude motion sampling the singular region of the Hamiltonian. These states can be identified through slow convergence.

A curvilinear approach to the kinetic analysis of linoleate peroxidation in aqueous liposomes by 2,2'azobis(2-amidoinopropane) dihydrochloride

Lipid peroxidation is a common feature of many chemical and biological processes, and is governed by a complex kinetic scheme. A fundamental stage in kinetic investigations of lipid peroxidation is the accurate determination of the rate of peroxidation, which in many instances is heavily reliant on the method of finite differences. Such numerical approximations of the first derivative are commonly employed in commercially available software, despite suffering from considerable inaccuracy due to rounding and truncation errors. As a simple solution to this, we applied three empirical sigmoid functions (viz. the Prout-Tompkins, Richards & Gompertz functions) to data obtained from the AAPH-mediated peroxidation of aqueous linoleate liposomes in the presence of increasing concentrations of Trolox, evaluating the curve fitting parameters using the widely available Microsoft Excel Solver add-in. We have demonstrated that the five-parameter Richards' function provides an excellent model for this peroxidation, and when applied to the determination of fundamental rate constants, produces results in keeping with those available in the literature. Overall, we present a series of equations, derived from the Richards' function, which enables direct evaluation of the kinetic measures of peroxidation. This procedure has applicability not only to investigations of lipid peroxidation, but to any system exhibiting sigmoid kinetics.

Using sketch-map coordinates to analyze and bias molecular dynamics simulations

When examining complex problems, such as the folding of proteins, coarse grained descriptions of the system drive our investigation and help us to rationalize the results. Oftentimes collective variables (CVs), derived through some chemical intuition about the process of interest, serve this purpose. Because finding these CVs is the most difficult part of any investigation, we recently developed a dimensionality reduction algorithm, sketch-map, that can be used to build a low-dimensional map of a phase space of high-dimensionality. In this paper we discuss how these machine-generated CVs can be used to accelerate the exploration of phase space and to reconstruct free-energy landscapes. To do so, we develop a formalism in which high-dimensional configurations are no longer represented by low-dimensional position vectors. Instead, for each configuration we calculate a probability distribution, which has a domain that encompasses the entirety of the low-dimensional space. To construct a biasing potential, we exploit an analogy with metadynamics and use the trajectory to adaptively construct a repulsive, history-dependent bias from the distributions that correspond to the previously visited configurations. This potential forces the system to explore more of phase space by making it desirable to adopt configurations whose distributions do not overlap with the bias. We apply this algorithm to a small model protein and succeed in reproducing the free-energy surface that we obtain from a parallel tempering calculation.

Statistical modeling on coordinates

This paper is a first draft of the principle of statistical modelling on coordinates. Several causes —which would be long to detail—have led to this situation close to the deadline for submitting papers to CODAWORK’03. The main of them is the fast development of the approach along the last months, which let appear previous drafts as obsolete. The present paper contains the essential parts of the state of the art of this approach from my point of view. I would like to acknowledge many clarifying discussions with the group of people working in this field in Girona, Barcelona, Carrick Castle, Firenze, Berlin, G¨ottingen, and Freiberg. They have given a lot of suggestions and ideas. Nevertheless, there might be still errors or unclear aspects which are exclusively my fault. I hope this contribution serves as a basis for further discussions and new developments

Kriging coordinates: what does that mean?

Kriging is an interpolation technique whose optimality criteria are based on normality assumptions either for observed or for transformed data. This is the case of normal, lognormal and multigaussian kriging. When kriging is applied to transformed scores, optimality of obtained estimators becomes a cumbersome concept: back-transformed optimal interpolations in transformed scores are not optimal in the original sample space, and vice-versa. This lack of compatible criteria of optimality induces a variety of problems in both point and block estimates. For instance, lognormal kriging, widely used to interpolate positive variables, has no straightforward way to build consistent and optimal confidence intervals for estimates. These problems are ultimately linked to the assumed space structure of the data support: for instance, positive values, when modelled with lognormal distributions, are assumed to be embedded in the whole real space, with the usual real space structure and Lebesgue measure

Automatic generation of active coordinates for quantum dynamics calculations: application to the dynamics of benzene photochemistry

A new practical method to generate a subspace of active coordinates for quantum dynamics calculations is presented. These reduced coordinates are obtained as the normal modes of an analytical quadratic representation of the energy difference between excited and ground states within the complete active space self-consistent field method. At the Franck-Condon point, the largest negative eigenvalues of this Hessian correspond to the photoactive modes: those that reduce the energy difference and lead to the conical intersection; eigenvalues close to 0 correspond to bath modes, while modes with large positive eigenvalues are photoinactive vibrations, which increase the energy difference. The efficacy of quantum dynamics run in the subspace of the photoactive modes is illustrated with the photochemistry of benzene, where theoretical simulations are designed to assist optimal control experiments

Determination of vibrational polarizabilities and hyperpolarizabilities using field-induced coordinates

An analytical set of field-induced coordinates is defined and is used to show that the vibrational degrees of freedom required to completely describe nuclear relaxation polarizabilities and hyperpolarizabilities is reduced from 3N-6 to a relatively small number. As this number does not depend upon the size of the molecule, the process provides computational advantages. A method is provided to separate anharmonic contributions from harmonic contributions as well as effective mechanical from electrical anharmonicity. The procedures are illustrated by Hartree-Fock calculations, indicating that anharmonicity can be very important

Field-induced coordinates for the determination of dynamic vibrational nonlinear optical properties

Three conjugated organic molecules that span a range of polarity and valence-bond/charge transfer characteristics were studied. It was found that dispersion can be insignificant, and that adequate treatment can be achieved with frequency-dependent field-induced vibrational coordinates (FD-FICs)

Anharmonic force constant calculations

The relationship of the anharmonic force constants in curvilinear internal coordinates to the observed vibration-rotation spectrum of a molecule is reviewed. A simplified method of setting up the required non-linear coordinate transformations is described: this makes use of an / tensor, which is a straightforward generalization of the / matrix used in the customary description of harmonic force constant calculations. General formulae for the / tensor elements, in terms of the familiar L matrix elements, are presented. The use of non-linear symmetry coordinates and redundancies are described. Sample calculations on the water and ammonia molecules are reported.