Volume 124, Issue 5, 07 February 2006
 ARTICLES

 Theoretical Methods and Algorithms

Generalizations of the HohenbergKohn theorem: I. Legendre Transform Constructions of Variational Principles for Density Matrices and Electron Distribution Functions
View Description Hide DescriptionGiven a general, particle Hamiltonian operator, analogs of the HohenbergKohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb’s Legendre transform ansatz to the generalized HohenbergKohn functional not only solves the representability problem for these entities, but, more importantly, also solves the representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A followup paper will present a useful alternative approach to the  and representability problems based on the constrained search formalism.

Using simultaneous diagonalization and trace minimization to make an efficient and simple multidimensional basis for solving the vibrational Schrödinger equation
View Description Hide DescriptionIn this paper we improve the product simultaneous diagonalization (SD) basis method we previously proposed [J. Chem. Phys.122, 134101 (2005)] and applied to solve the Schrödinger equation for the motion of nuclei on a potential surface. The improved method is tested using coupled complicated Hamiltonians with as many as 16 coordinates for which we can easily find numerically exact solutions. In a basis of sorted products of onedimensional (1D) SD functions the Hamiltonian matrix is nearly diagonal. The localization of the 1D SD functions for coordinate depends on a parameter we denote . In this paper we present a trace minimization scheme for choosing to nearly block diagonalize the Hamiltonian matrix. Nearblock diagonality makes it possible to truncate the matrix without degrading the accuracy of the lowest energy levels. We show that in the sorted product SD basis perturbation theory works extremely well. The trace minimization scheme is general and easy to implement.

Formal relations connecting different approaches to calculate relativistic effects on molecular magnetic properties
View Description Hide DescriptionIn the present work a set of formal relations connecting different approaches to calculate relativistic effects on magnetic molecular properties are proven. The linear response (LR) within the elimination of the small component (ESC), Breit Pauli, and minimalcoupling approaches are compared. To this end, the leading order ESC reduction of operators within the minimalcoupling fourcomponent approach is carried out. The equivalence of all three approaches within the ESC approximation is proven. It is numerically verified for the NMR nuclearmagnetic shielding tensor taking and as model compounds. Formal relations proving the gauge origin invariance of the full relativistic effect on the NMR nuclearmagnetic shielding tensor within the LRESC approach are presented.

Oscillator strengths of helium computed using Monte Carlo methods
View Description Hide DescriptionWe have optimized trial wave functions for the three lowest states of the helium atom with symmetry , , , , , and using variational Monte Carlo methods. With these wave functions we then computed dipole oscillator strengths for the , , , and transitions using the length, velocity, and acceleration forms. Our values are in good agreement with the best results found in the literature.

Interpolating moving leastsquares methods for fitting potentialenergy surfaces: Further improvement of efficiency via cutoff strategies
View Description Hide DescriptionIn standard applications of interpolating moving least squares (IMLS) for fitting a potentialenergysurface (PES), all available ab initio points are used. Because remote ab initio points negligibly influence IMLS accuracy and increase IMLS timetosolution, we present two methods to locally restrict the number of points included in a particular fit. The fixed radius cutoff (FRC) method includes ab initio points within a hypersphere of fixed radius. The density adaptive cutoff (DAC) method includes points within a hypersphere of variable radius depending on the point density. We test these methods by fitting a sixdimensional analytical PES for hydrogen peroxide. Both methods reduce the IMLS timetosolution by about an order of magnitude relative to that when no cutoff method is used. The DAC method is more robust and efficient than the FRC method.

Gradient symplectic algorithms for solving the radial Schrödinger equation
View Description Hide DescriptionThe radial Schrödinger equation for a spherically symmetric potential can be regarded as a onedimensional classical harmonic oscillator with a timedependent spring constant. For solving classical dynamics problems, symplectic integrators are well known for their excellent conservation properties. The class of gradient symplectic algorithms is particularly suited for solving harmonicoscillator dynamics. By use of Suzuki’s rule [Proc. Jpn. Acad., Ser. B: Phys. Biol. Sci.69, 161 (1993)] for decomposing timeordered operators, these algorithms can be easily applied to the Schrödinger equation. We demonstrate the power of this class of gradient algorithms by solving the spectrum of highly singular radial potentials using Killingbeck’s method [J. Phys. A18, 245 (1985)] of backward NewtonRalphson iterations.

Sources of error in electronic structure calculations on small chemical systems
View Description Hide DescriptionThe sources of error in electronic structure calculations arising from the truncation of the oneparticle and particle expansions are examined with very large correlation consistent basis sets, in some cases up through valence quality, and coupled cluster methods, up through connected quadruple excitations. A limited number of full configuration interaction corrections are also considered. For cases where full configuration interaction calculations were unavailable or prohibitively expensive, a continued fraction approximation was used. In addition, errors arising from core∕valence and relativistic corrections are also probed for a number of small chemical systems. The accuracies of several formulas for estimating total energies and atomization energies in the complete basis set limit are compared in light of the present large basis set findings. In agreement with previous work, the CCSD(T) method is found to provide results that are closer to the CCSDTQ and full configurationinteraction results than the less approximate CCSDT method.

Thirdorder multireference perturbation theory: The electron valence state perturbationtheory approach
View Description Hide DescriptionA formulation of the electron valence state perturbation theory (NEVPT) at the third order of perturbation is presented. The present implementation concerns the socalled strongly contracted variant of NEVPT, where only a subspace of the firstorder interacting space is taken into account. The resulting strongly contracted NEVPT3 approach is discussed in three test cases: (a) the energy difference between the and states of the methylene molecule, (b) the potentialenergy curve of the molecule ground state, and (c) the chromium dimer groundstate potentialenergy profile. Particular attention is devoted to the last case where large basis sets comprising also orbitals are adopted and where remarkable differences between the second and thirdorder results show up.

Quadratic string method for determining the minimumenergy path based on multiobjective optimization
View Description Hide DescriptionBased on a multiobjective optimization framework, we develop a new quadratic string method for finding the minimumenergy path. In the method, each point on the minimumenergy path is minimized by integration in the descent direction perpendicular to path. Each local integration is done on a quadratic surface approximated by a damped BroydenFletcherGoldfarbShanno updated Hessian, allowing the algorithm to take many steps between energy and gradient calls. The integration is performed with an adaptive stepsize solver, which is restricted in length to the trust radius of the approximate Hessian. The full algorithm is shown to be capable of practical superlinear convergence, in contrast to the linear convergence of other methods. The method also eliminates the need for predetermining such parameters as step size and spring constants, and is applicable to reactions with multiple barriers. The effectiveness of this method is demonstrated for the MüllerBrown potential, a sevenatom LennardJones cluster, and the enolation of acetaldehyde to vinyl alcohol.

Innershell ionizations and satellites studied by the openshell reference symmetryadapted cluster/symmetryadapted cluster configurationinteraction method
View Description Hide DescriptionOpenshell reference version of the symmetryadapted cluster (SAC) and SACconfigurationinteraction (CI) methods, termed openshell reference (OR)SAC and ORSACCI methods, are developed and applied to innershell ionizations of , and HF. The innershell ionization potentials and spectra calculated by the ORSAC and ORSACCI methods are in excellent agreement with the experimental data. Including both of the electron correlation and orbital relaxation is important for quantitative agreements. Timing comparisons with the SACCI general calculations that give similar high accuracies show an efficiency of the present ORSAC and ORSACCI methods.

Diffusion and velocity relaxation of a Brownian particle immersed in a viscous compressible fluid confined between two parallel plane walls
View Description Hide DescriptionThe diffusiontensor and velocitycorrelation function of a Brownian particle immersed in a viscous compressible fluid confined between two parallel plane walls are calculated in point approximation. The fluid is assumed to satisfy stick boundary conditions at the walls. It is found that the velocitycorrelation function decays asymptotically with a negative longtime tail due to coupling to overdamped sound waves. The coefficient of the longtime tail is calculated and shown to be independent of fluid viscosity.

Efficient computation of the first passage time distribution of the generalized master equation by steadystate relaxation
View Description Hide DescriptionThe generalized master equation or the equivalent continuous time random walkequations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from shortterm microscopic simulation data. The computation of the mean first passage time and additional loworder FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steadystate relaxation, an extension of steadystate flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.

Local stress and heat flux in atomistic systems involving threebody forces
View Description Hide DescriptionLocal densities of fundamental physical quantities, including stress and heat flux fields, are formulated for atomistic systems involving threebody forces. The obtained formulas are calculable within an atomistic simulation, in consistent with the conservation equations of thermodynamics of continuum, and can be applied to systems with general two and threebody interaction forces. It is hoped that this work may correct some misuse of inappropriate formulas of stress and heat flux in the literature, may clarify the definition of site energy of manybody potentials, and may serve as an analytical link between an atomistic model and a continuum theory. Physical meanings of the obtained formulas, their relation with virial theorem and heat theorem, and the applicability are discussed.

Explicitly correlated secondorder perturbation theory using density fitting and local approximations
View Description Hide DescriptionThree major obstacles in electronic structuretheory are the steep scalings of computer time with respect to system size and basis size and the slow convergence of correlation energies in orbital basis sets. Three solutions to these are, respectively, local methods, density fitting, and explicit correlation; in this work, we combine all three to produce a loworder scaling method that can achieve accurate MP2 energies for large systems. The errors introduced by the local approximations into the R12 treatment are analyzed for 16 chemical reactions involving . Weak pair approximations, as well as local resolution of the identity approximations, are tested for molecules with up to 49 atoms, over 100 correlated electrons, and over 1000 basis functions.

A simple natural orbital mechanism of “pure” van der Waals interaction in the lowest excited triplet state of the hydrogen molecule
View Description Hide DescriptionA treatment of van der Waals (vdW) interaction by densitymatrix functionaltheory requires a description of this interaction in terms of natural orbitals (NOs) and their occupation numbers. From an analysis of the configurationinteraction (CI) wave function of the state of and the exact NO expansion of the twoelectron triplet wave function, we demonstrate that the construction of such a functional is straightforward in this case. A quantitative description of the vdW interaction is already obtained with, in addition to the standard part arising from the HartreeFock determinant , only two additional terms in the twoelectron density, one from the first “excited” determinant and one from the state of symmetry belonging to the configuration. The potentialenergy curve of the state calculated around the vdW minimum with the exact densitymatrix functional employing only these eight NOs and NO occupations is in excellent agreement with the full CI one and reproduces well the benchmark potential curve of Kolos and Wolniewicz [J. Chem. Phys.43, 2429 (1965)]. The corresponding terms in the twoelectron density , containing specific products of NOs combined with prefactors that depend on the occupation numbers, can be shown to produce exchangecorrelation holes that correspond precisely to the wellknown intuitive picture of the dispersion interaction as an instantaneous dipoleinduced dipole (higher multipole) effect. Indeed, (induced) higher multipoles account for almost 50% of the total vdW bondenergy. These results serve as a basis for both a densitymatrix functionaltheory of van der Waals bonding and for the construction of orbitaldependent functionals in densityfunctional theory that could be used for this type of bonding.

A general framework for nonBoltzmann Monte Carlo sampling
View Description Hide DescriptionNonBoltzmann sampling (NBS) methods have been extensively employed in recent years, mainly due to their ability to enhance ergodicity in simulations of complex systems. In addition, they make possible reliable computation of equilibrium properties (ensemble averages, freeenergy differences, and potentials of mean force) over continuous ranges of thermodynamic conditions. In this work, we put forward a general and systematic framework for NBS methods that allows a single set of equations and procedures to be applied to diverse systems. Moreover, we show how to exploit simulation data most effectively by obtaining continuous profiles of any mechanical properties, including structural quantities not directly related to the ensemble parameters. Finally, we demonstrate the usefulness of the developed formulation by applying it to spin systems, LennardJones fluids, and a model protein molecule (both in isolation and in the proximity of a flat wall).

Ordering of limits in the Jarzynski equality
View Description Hide DescriptionWe consider the sampling problems encountered in computing freeenergy differences using Jarzynski’s nonequilibrium work relation [Phys. Rev. Lett.56, 2690 (1997)]. This relation expresses the freeenergy change of a system, on which finitetime work is done, as an average over all possible trajectories of the system. This average can then be expressed as a cumulant expansion of the work. We study the scaling of these cumulants with an appropriately defined measure of phasespace accessibility and particle number for two simple changes in state. We find that the cumulant expansion is slowly convergent for predominantly entropic processes, those where is considerably altered during the course of the process. An accurate determination of the freeenergy change requires some knowledge of the behavior of the tails of the work distribution associated with the process. Jarzynski’s irreversible work relation is only valid with the correct ordering of the infinite limits of and , clarifying the regime of its applicability.

Validation of Markov state models using Shannon’s entropy
View Description Hide DescriptionMarkov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon’s entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Generation and characterization of highly vibrationally excited molecular beam
View Description Hide DescriptionA simple method to generate and characterize a pure highly vibrationally excited azulene molecular beam is demonstrated. Azulene molecules initially excited to the state by 266nm UVphotons reach high vibrationally excited levels of the ground electronic state upon rapid internal conversion from the electronically excited state. VUVlaser beams at 157 and , respectively, are used to characterize the relative concentrations of the highly vibrationally excited azulene and the rotationally and vibrationally cooled azulene in the molecular beam. With a laser intensity of , 75% of azulene molecules absorb a single 266nm photon and become highly vibrationally excited molecules. The remaining groundstate azulene molecules absorb two or more UVphotons, ending up either as molecular cations, which are repelled out of the beam by an electric field, or as dissociation fragments, which veer off the molecularbeam axis. No azulene without absorption of UVphotons is left in the molecular beam. The molecular beam that contains only highly vibrationally excited molecules and carrier gas is useful in various experiments related to the studies of highly vibrationally excited molecules.

Energy transfer of highly vibrationally excited azulene: Collisions between azulene and krypton
View Description Hide DescriptionThe energytransfer dynamics between highly vibrationally excited azulene molecules and Kr atoms in a series of collisionenergies (i.e., relative translational energies 170, 410, and ) was studied using a crossedbeam apparatus along with timesliced velocity map ion imaging techniques. “Hot” azulene ( internal energy) was formed via the rapid internal conversion of azulene initially excited to the state by 266nm photons. The shapes of the collisional energytransfer probability density functions were measured directly from the scattering results of highly vibrationally excited or hot azulene. At low enough collisionenergies an azuleneKr complex was observed, resulting from small amounts of translational to vibrationalrotational (TVR) energy transfer. TVR energy transfer was found to be quite efficient. In some instances, nearly all of the translational energy is transferred to vibrationalrotational energy. On the other hand, only a small fraction of vibrational energy is converted to translational energy (VT). The shapes of VT energytransfer probability density functions were best fit by multiexponential functions. We find that substantial amounts of energy are transferred in the backward scattering direction due to supercollisions at high collisionenergies. The probability for supercollisions, defined arbitrarily as the scattered azulene in the region and is 1% and 0.3% of all other collisions at collisionenergies 410 and , respectively.