Internal‐to‐Cartesian back transformation of molecular geometry steps using high‐order geometric derivatives

In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high‐order path‐expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization. © 2013 Wiley Periodicals, Inc.     

Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms

In previous research [J. Chem. Phys. 111, 3800 (1999)] a Hessian-based integration algorithm was derived for performing direct dynamics simulations. In the work presented here, improvements to this algorithm are described. The algorithm has a predictor step based on a local second-order Taylor expansion of the potential in Cartesian coordinates, within a trust radius, and a fifth-order correction to this predicted trajectory. The current algorithm determines the predicted trajectory in Cartesian coordinates, instead of the instantaneous normal mode coordinates used previously, to ensure angular momentum conservation. For the previous algorithm the corrected step was evaluated in rotated Cartesian coordinates. Since the local potential expanded in Cartesian coordinates is not invariant to rotation, the constants of motion are not necessarily conserved during the corrector step. An approximate correction to this shortcoming was made by projecting translation and rotation out of the rotated coordinates. For the current algorithm unrotated Cartesian coordinates are used for the corrected step to assure the constants of motion are conserved. An algorithm is proposed for updating the trust radius to enhance the accuracy and efficiency of the numerical integration. This modified Hessian-based integration algorithm, with its new components, has been implemented into the VENUS/NWChem software package and compared with the velocity-Verlet algorithm for the H(2)CO-->H(2)+CO, O(3)+C(3)H(6), and F(-)+CH(3)OOH chemical reactions.     

A method for the calculation of normal-mode vibrational frequencies using symmetry coordinates. Application to the calculation of secondary deuterium isotope effects on carbocations

A method is introduced for the calculation of normal-mode vibrational frequencies of polyatomic molecules based on numerical differencing of analytical gradients in symmetry coordinates. This procedure requires a number of gradient evaluations equal to the largest number of symmetry coordinates belonging to any single irreducible representation of the molecular point group (plus a single gradient evaluation at the equilibrium configuration), which is fewer than the 3N-6 (N atoms) gradient evaluations needed for schemes based on Cartesian or internal coordinates. While the proposed method will not generally be as efficient as procedures which involve the direct calculation of energy second derivatives analytically (as are now available for single-determinant wavefunctions) it appears to be equally accurate, and it should be the method of choice for frequency calculations involving multideterminant wavefunctions for which analytical second-derivative algorithms have yet to be developed. The method is illustrated by the calculation of equilibrium secondary deuterium-isotope effects on a number of reactions involving simple carbocations.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Multimode wavelet basis calculations via the molecular self-consistent-field plus configuration-interaction method

Wavelets provide potentially useful quantum bases for coupled anharmonic vibrational modes in polyatomic molecules as well as many other problems. A single compact support wavelet family provides a flexible basis with properties of orthogonality, localization, customizable resolution, and systematic improvability for general types of one-dimensional and separable systems. While direct product wavelet bases can be used in coupled multidimensional problems, exponential scaling of basis size with dimensionality ultimately provides limits on the number of coupled modes that can be treated simultaneously in exact quantum calculations. The molecular self-consistent-field plus configuration-interaction method is used here in multimode wavelet calculations to reduce the basis size without sacrificing flexibility or the ability to systematically control errors. Both two-dimensional Cartesian coordinate and three-dimensional curvilinear coordinate systems are examined with wavelets serving as universal bases in each case. The first example uses standard Daubechies [Ten Lectures on Wavelets (SIAM, Philadelphia (1992)] wavelets for each mode and the second adapts symmlet wavelets to intervals for each of the curvilinear coordinates.     

Quantum dynamics of solid Ne upon photo-excitation of a NO impurity: A Gaussian wave packet approach

A high-dimensional quantum wave packet approach based on Gaussian wave packets in Cartesian coordinates is presented. In this method, the high-dimensional wave packet is expressed as a product of time-dependent complex Gaussian functions, which describe the motion of individual atoms. It is applied to the ultrafast geometrical rearrangement dynamics of NO doped cryogenic Ne matrices after femtosecond laser pulse excitation. The static deformation of the solid due to the impurity as well as the dynamical response after femtosecond excitation are analyzed and compared to reduced dimensionality studies. The advantages and limitations of this method are analyzed in the perspective of future applications to other quantum solids.     

The use of internal cartesian coordinates for describing molecular vibrations

An analysis of the influence of isotope substitution on the system of electronic-nuclear equations for an arbitrary molecular system was used as a basis for formulating invariance conditions with respect to isotope substitution of the potential energy surface written in the Cartesian coordinates rigidly bound with the center of mass of the molecule (internal Cartesian coordinates). This property of the potential function obviates the necessity of using curvilinear natural coordinates, which can be replaced by Cartesian coordinates, in theoretical studies of the vibrational spectra of molecules and their isotopomers and in solving the direct and inverse anharmonic problems. An equation for the quantum-mechanical Hamiltonian of a normal molecule in internal Cartesian coordinates was obtained.     

Geometry optimization of metal complexes using natural internal coordinates: Representation of skeletal degrees of freedom

The geometry optimization using natural internal coordinates was applied for transition metal complexes. The original definitions were extended here for the skeletal degrees of freedom which are related to the translational and rotational displacements of the ηⁿ-bonded ligands. We suggest definitions for skeletal coordinates of ηⁿ-bonded small unsaturated rings and chains. The performance of geometry optimizations using the suggested coordinates were tested on various conformers of 14 complexes. Consideration was given to alternative representations of the skeletal internal coordinates, and the performance of optimization is compared. Using the skeletal internal coordinates presented here, most transition metal complexes were optimized between 10 and 20 geometry optimization cycles in spite of the usually poor starting geometry and crude approximation for the Hessian. We also optimized the geometry of some complexes in Cartesian coordinates using the Hessian from a parametrized redundant force field. We found that it took between two and three times as many iterations to reach convergence in Cartesian coordinates than using natural internal coordinates. © 1997 by John Wiley & Sons, Inc.     

Exact quantum statistics for electronically nonadiabatic systems using continuous path variables

We derive an exact, continuous-variable path integral (PI) representation of the canonical partition function for electronically nonadiabatic systems. Utilizing the Stock-Thoss (ST) mapping for an N-level system, matrix elements of the Boltzmann operator are expressed in Cartesian coordinates for both the nuclear and electronic degrees of freedom. The PI discretization presented here properly constrains the electronic Cartesian coordinates to the physical subspace of the mapping. We numerically demonstrate that the resulting PI-ST representation is exact for the calculation of equilibrium properties of systems with coupled electronic and nuclear degrees of freedom. We further show that the PI-ST formulation provides a natural means to initialize semiclassical trajectories for the calculation of real-time thermal correlation functions, which is numerically demonstrated in applications to a series of nonadiabatic model systems.     

A combined ONIOM quantum chemical-molecular dynamics study of zinc-uracil bond breaking in yeast cytosine deaminase

A QM/MM method that combines ONIOM quantum chemistry and molecular dynamics is developed and applied to a step in the deamination of cytosine to uracil in yeast cytosine deaminase (yCD). A two-layer ONIOM calculation is used for the reaction complex, with an inner part treated at a high level for the chemical reaction (bond breaking) and a middle part treated at a lower level for relevant protein residues that are frozen in the quantum optimization. An outer layer (protein and solvent) is treated using MD. Configurations for the entire system are generated using MD and optimized with ONIOM. The method permits the use of high-level quantum calculations along with sufficient configurational sampling to approximate the potential of mean force for certain bond-breaking reactions. A previously proposed reaction mechanism for deamination (Sklenak, S.; Yao, L. S.; Cukier, R. I.; Yan, H. G. J. Am. Chem. Soc. 2004, 126, 14879) requires breaking the bond between a catalytic zinc and the O4 of uracil in order to permit product release. Using an ONIOM approach, direct bond cleavage was found to be energetically unfavorable. In the work presented here, the combined ONIOM MD method is used to show that the barrier for bond cleavage is small, approximately 3 kcal/mol, and, consequently, should not be the rate-limiting step in the reaction.     

Hamilton-Jacobi equation for the least-action/least-time dynamical path based on fast marching method

Classical dynamics can be described with Newton's equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethian's fast marching method. Using this method, we can--starting from an arbitrary initial conformation--find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H(2)-->H(2)+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales.     

Second derivatives in generalized Born theory

Generalized Born solvation models offer a popular method of including electrostatic aspects of solvation free energies within an analytical model that depends only upon atomic coordinates, charges, and dielectric radii. Here, we describe how second derivatives with respect to Cartesian coordinates can be computed in an efficient manner that can be distributed over multiple processors. This approach makes possible a variety of new methods of analysis for these implicit solvation models. We illustrate three of these methods here: the use of Newton-Raphson optimization to obtain precise minima in solution; normal mode analysis to compute solvation effects on the mechanical properties of DNA; and the calculation of configurational entropies in the MM/GBSA model. An implementation of these ideas, using the Amber generalized Born model, is available in the nucleic acid builder (NAB) code, and we present examples for proteins with up to 45,000 atoms. The code has been implemented for parallel computers using both the OpenMP and MPI environments, and good parallel scaling is seen with as many as 144 OpenMP processing threads or MPI processing tasks.     

Adaptive biasing force method for scalar and vector free energy calculations

In free energy calculations based on thermodynamic integration, it is necessary to compute the derivatives of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivatives as the average of a mean force acting on the order parameters, which involves first derivatives with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivatives of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, especially for complicated order parameters. It is also straightforward to implement in a molecular dynamics code, as can be seen from the pseudocode given at the end. We further discuss how the approach based on time derivatives can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calculations. Using the backbone dihedral angles Phi and Psi in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivatives, a calculation that presents some difficulties in the vector case because of the statistical errors affecting the derivatives.     

The choice of internal coordinates in complex chemical systems

This article presents several considerations for the appropriate choice of internal coordinates in various complex chemical systems. The appropriate and black box recognition of internal coordinates is of fundamental importance for the extension of internal coordinate algorithms to all fields where previously Cartesian coordinates were the preferred means of geometry manipulations. Such fields range from local and global geometry optimizations to molecular dynamics as applied to a wide variety of chemical systems. We present a robust algorithm that is capable to quickly determine the appropriate choice of internal coordinates in a wide range of atomic arrangements. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

A mathematical model of photochemical transformations: Analysis of the influence of basic characteristics of the model

Basic aspects of the choice of parameters entering a system of differential equations that simulate the sequence of chemical transformations in the case of initiation of a reaction by electromagnetic pulse excitation of reactant molecules are discussed. These parameters are the probabilities of spontaneous and stimulated dipole–dipole transitions and the frequency of quantum beats that lead to the reaction. A method for an a priori estimation of optimum values of these frequencies on the basis of fundamental characteristics of intramolecular processes has been specified. It has been noted that in theoretical prediction of the course of chemical reactions, the solution of the problem in the natural coordinate system (distance between the atoms), rather than Cartesian coordinates, can be more appropriate.     

Self-consistent treatment of the local dielectric permittivity and electrostatic potential in solution for polarizable macromolecular force fields

A self-consistent method is presented for the calculation of the local dielectric permittivity and electrostatic potential generated by a solute of arbitrary shape and charge distribution in a polar and polarizable liquid. The structure and dynamics behavior of the liquid at the solute∕liquid interface determine the spatial variations of the density and the dielectric response. Emphasis here is on the treatment of the interface. The method is an extension of conventional methods used in continuum protein electrostatics, and can be used to estimate changes in the static dielectric response of the liquid as it adapts to charge redistribution within the solute. This is most relevant in the context of polarizable force fields, during electron structure optimization in quantum chemical calculations, or upon charge transfer. The method is computationally efficient and well suited for code parallelization, and can be used for on-the-fly calculations of the local permittivity in dynamics simulations of systems with large and heterogeneous charge distributions, such as proteins, nucleic acids, and polyelectrolytes. Numerical calculation of the system free energy is discussed for the general case of a liquid with field-dependent dielectric response.     

On the use of projection operators in molecular dynamics

Symmetry coordinates are easily constructed when operating in mass-weighted Cartesian space. Such coordinates can be used to block diagonalize automatically the secular equation occurring in the harmonic approximation of dynamics of large molecules. Examples of the usefulness of the method are given.     

Path integral calculation of thermal rate constants within the quantum instanton approximation: application to the H + CH4 --> H2 + CH3 hydrogen abstraction reaction in full Cartesian space

The quantum instanton approximation for thermal rate constants of chemical reactions [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)], which is modeled after the earlier semiclassical instanton approach, is applied to the hydrogen abstraction reaction from methane by a hydrogen atom, H + CH4 --> H2 + CH3, using a modified and recalibrated version of the Jordan-Gilbert potential surface. The quantum instanton rate is evaluated using path integral Monte Carlo approaches based on the recently proposed implementation schemes [Yamamoto and Miller, J. Chem. Phys. 120, 3086 (2004)]. The calculations were carried out using the Cartesian coordinates of all the atoms (thus involving 18 degrees of freedom), thereby taking explicit account of rotational effects of the whole system and also allowing the equivalent treatment of the four methane hydrogens. To achieve such a treatment, we present extended forms of the path integral estimators for relevant quantities that may be used for general N-atom systems with any generalized reaction coordinates. The quantum instanton rates thus obtained for the temperature range T = 200-2000 K show good agreement with available experimental data, which gives support to the accuracy of the underlying potential surface used.