Semiclassical initial value representation treatment of a hydrogen bonded complex of rigid water molecules from a single trajectory in Cartesian coordinates

A semiclassical initial value representation approach for molecular systems in Cartesian coordinates is combined with a recently proposed time averaging technique [J. Chem. Phys. 118, 7174 (2003)]. It is shown that a single trajectory can yield the zero-point energy of the water dimer with good accuracy for the model chosen when compared to fully constrained Cartesian semiclassical calculations. The convergence with respect to the number of averaging time origins is discussed.     

Geometric constraints in semiclassical initial value representation calculations in Cartesian coordinates: excited states

The authors show that a recently proposed approach [J. Chem. Phys. 123, 084103 (2005)] for the inclusion of geometric constraints in semiclassical initial value representation calculations can be used to obtain excited states of weakly bound complexes. Sample calculations are performed for free and constrained rare gas clusters. The results show that the proposed approach allows the evaluation of excited states with reasonable accuracy when compared to exact basis set calculations.     

Quantum molecular dynamics of hydrogen bonded complexes of rigid molecules using the semiclassical initial value representation in Cartesian coordinates

Semiclassical initial value representation calculations are performed for the constrained water dimer in Cartesian coordinates. The study represents the first application of a previously reported method [Issak and Roy, J. Chem. Phys. 123, 084103 (2005); 126, 024111 (2007)] to a molecular cluster. Bound state energies are calculated for a dimer of rigid water molecules (H2O)2 as well as its deuterated form (D2O)2. The results show that the approach fares well with respect to accuracy in capturing quantum effects in intermolecular interactions.     

Initial conformations of macrocyclic compounds with rotational symmetry generated from a molecular fragment

An algorithm is described for generating atomic Cartesian coordinates of conformations of macrocyclic molecules possessing exact rotational or rotation-reflection symmetries. A fragment representing the asymmetric unit of the molecule is suitably oriented in space, and then a symmetry operator is applied to generate the initial coordinates of the molecule. An empirical force field of interatomic interactions is used to generate equilibrium conformations. Results of calculations performed on two cyclic polylactones and one crown ether using this approach are given. They reveal that symmetric conformations of these molecules are often preferred. Since the latter conformations are probably responsible for the specialized properties of these molecules, this method should facilitate doing theoretical studies on these kinds of compounds.     

Geometry optimization in Cartesian coordinates: Constrained optimization

An efficient algorithm for constrained geometry optimization in Cartesian coordinates is presented. It incorporates mode-following techniques within both the classical method of Lagrange multipliers and the penalty function method. Both constrained minima and transition states can be located and, unlike the standard Z-matrix using internal coordinates, the desired constraints do not have to be satisfied in the initial structure. The algorithm is as efficient as a Z-matrix optimization while presenting several additional advantages.     

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.     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

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.     

Geometry optimization of periodic systems using internal coordinates

An algorithm is proposed for the structural optimization of periodic systems in internal (chemical) coordinates. Internal coordinates may include in addition to the usual bond lengths, bond angles, out-of-plane and dihedral angles, various "lattice internal coordinates" such as cell edge lengths, cell angles, cell volume, etc. The coordinate transformations between Cartesian (or fractional) and internal coordinates are performed by a generalized Wilson B-matrix, which in contrast to the previous formulation by Kudin et al. [J. Chem. Phys. 114, 2919 (2001)] includes the explicit dependence of the lattice parameters on the positions of all unit cell atoms. The performance of the method, including constrained optimizations, is demonstrated on several examples, such as layered and microporous materials (gibbsite and chabazite) as well as the urea molecular crystal. The calculations used energies and forces from the ab initio density functional theory plane wave method in the projector-augmented wave formalism.     

On the efficient path integral evaluation of thermal rate constants within the quantum instanton approximation

We present an efficient path integral approach for evaluating thermal rate constants within the quantum instanton (QI) approximation that was recently introduced to overcome the quantitative deficiencies of the earlier semiclassical instanton approach [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)]. Since the QI rate constant is determined solely by properties of the (quantum) Boltzmann operator (specifically, by the zero time properties of the flux-flux and delta-delta correlation functions), it can be evaluated by well-established techniques of imaginary time path integrals even for quite complex chemical reactions. Here we present a series of statistical estimators for relevant quantities which can be evaluated straightforwardly with any nonlinear reaction coordinates and general Hamiltonians in Cartesian space. To facilitate the search for the optimal dividing surfaces required by the QI approximation, we introduce a two-dimensional quantum free energy surface associated with the delta-delta correlation function and describe how an adaptive umbrella sampling can be used effectively to construct such a free energy surface. The overall computational procedure is illustrated by the application to a hydrogen exchange reaction in gas phase, which shows excellent agreement of the QI rates with those obtained from quantum scattering calculations.     

Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

Direct quantum dynamics using variational multi-configuration Gaussian wavepackets. Implementation details and test case

We present here a direct quantum dynamics method using variational multi-configuration Gaussian wavepackets. Based on the efficient multi-configuration time-dependent Hartree wavepacket propagation algorithm, it uses on-the-fly quantum chemical calculation of the potential energy and its derivatives rather than fitted surfaces. Intermediate results are stored in a database so that expensive quantum chemical computations can be recycled. This method is intended to treat quantum effects in the photochemistry of large molecules and the use of Cartesian coordinates to perform direct dynamics is discussed with a comparison between Cartesian coordinates of Jacobi vectors and Cartesian coordinates of the nuclei, using various free and constrained approaches depending on the way the rotation is treated. As a test calculation to be compared to full quantum dynamics it is applied here to the computation of the photodissociation spectrum of nitrosyl chloride (NOCl).     

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.     

Coarse-graining entropy, forces, and structures

Coarse-grained (CG) models enable highly efficient simulations of complex processes that cannot be effectively studied with more detailed models. CG models are often parameterized using either force- or structure-motivated approaches. The present work investigates parallels between these seemingly divergent approaches by examining the relative entropy and multiscale coarse-graining (MS-CG) methods. We demonstrate that both approaches can be expressed in terms of an information function that discriminates between the ensembles generated by atomistic and CG models. While it is well known that the relative entropy approach minimizes the average of this information function, the present work demonstrates that the MS-CG method minimizes the average of its gradient squared. We generalize previous results by establishing conditions for the uniqueness of structure-based potentials and identify similarities with corresponding conditions for the uniqueness of MS-CG potentials. We analyze the mapping entropy and extend the MS-CG and generalized-Yvon-Born-Green formalisms for more complex potentials. Finally, we present numerical calculations that highlight similarities and differences between structure- and force-based approaches. We demonstrate that both methods obtain identical results, not only for a complete basis set, but also for an incomplete harmonic basis set in Cartesian coordinates. However, the two methods differ when the incomplete basis set includes higher order polynomials of Cartesian coordinates or is expressed as functions of curvilinear coordinates.     

Geometry optimization with QM/MM,ONIOM, and other combined methods. I. Microiterations and constraints

Hybrid energy methods such as QM/MM and ONIOM, that combine different levels of theory into one calculation, have been very successful in describing large systems. Geometry optimization methods can take advantage of the partitioning of these calculations into a region treated at a quantum mechanical (QM) level of theory and the larger, remaining region treated by an inexpensive method such as molecular mechanics (MM). A series of microiterations can be employed to fully optimize the MM region for each optimization step in the QM region. Cartesian coordinates are used for the MM region and are chosen so that the internal coordinates of the QM region remain constant during the microiterations. The coordinates of the MM region are augmented to permit rigid body translation and rotation of the QM region. This is essential if any atoms in the MM region are constrained, but it also improves the efficiency of unconstrained optimizations. Because of the microiterations, special care is needed for the optimization step in the QM region so that the system remains in the same local valley during the course of the optimization. The optimization methodology with microiterations, constraints, and step-size control are illustrated by calculations on bacteriorhodopsin and other systems.     

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.     

Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83