Volume 117, Issue 20, 22 November 2002
 ARTICLES

 Theoretical Methods and Algorithms

Nuclear cusp of the virial exchange energy density for spherical atoms
View Description Hide DescriptionUsing the virial exchange energy density defined by the integrand of the Levy–Perdew exchange virial relation, it is shown that for spherical atoms with nuclear chargeZ, the nuclear cusp of exchange energy density For the local density approximation, this condition is given by Numerical results are presented for the rare gas atoms He–Xe using a variety of exchangeonly local effective potentials. For the optimized effective potential and local density approximation the above conditions are obeyed quantitatively. It is found that the Krieger–Li–Iafrate approximation closely reproduces the optimized effective potential results, whereas those derived from the popular potentials due to Becke and Perdew–Wang give rise to much larger values. The exchange energy density defined analogously as the integrand of the directly calculated exchange energy of the model potential leads to the exact cusp values of for the optimized effective potential and for the local density approximation.

A comparison of some variational formulas for the free energy as applied to hardsphere crystals
View Description Hide DescriptionWe examine several variational methods for determining bounds on the free energy of model crystalline phases, as applied to hard spheres in one and three dimensions. Cell and harmonicbased reference systems are considered. Methods that provide the tightest bounds on the free energy are similar in form to freeenergyperturbation, and are prone to inaccuracy from inadequate sampling. Gibbs–Bogoliubov formulas are reliable but weaker. For hard potentials they can give only a lower bound, indicating that their ability to provide upper bounds for other potentials is limited. Nevertheless, bounds given by Gibbs–Bogoliubov when applied with the optimal harmonic system prescribed by Morris and Ho [Phys. Rev. Lett. 74, 940 (1995)] yields impressive results; for hard spheres at higher density it is, within confidence limits, equal to the exact hardsphere free energy.

Cross correlation functions via Lanczos algorithms without diagonalization
View Description Hide DescriptionIt is shown how the quasiminimal residual algorithm (QMR), based on the Lanczos algorithm, can be modified to compute crosscorrelation functions without any diagonalization by recursively updating a small number of scalars. Only three Lanczos vectors need to be stored. Several lefthand side vectors and multiple shifts E can be considered simultaneously. The new method is termed the quasiminimal recursive residue generation method (QMRRGM) and is applied to the collinear problem to illustrate its convergence behavior. The properties of two different formulations of the Lanczos algorithm, the usual threeterm and a coupled twoterm recursion, are also discussed. The QMRRGM exhibits smooth convergence behavior, and it is shown that the stopping criteria used in the QMR algorithm can also be used for computing correlation functions.

An analytical model for vibrational nonBorn–Oppenheimer induced electron ejection in molecular anions
View Description Hide DescriptionWe introduce an analytical model designed to capture the most important features of the electronic matrix elements arising in nonBorn–Oppenheimer couplings between a bound anion state and a neutralmoleculeplusejectedelectron state. In this particleinaradialbox model, vibrations are assumed to cause modulations in the depth and length parameters of the box. The most important elements of this model are that is chosen to reproduce the proper dependence of the radial size of the anion’s orbital on electron binding energy, and is chosen to produce the correct electron affinity. Within this model, which is shown to be consistent with trends seen in ab initio calculations of associated electron ejection rates, the coupling matrix elements can be evaluated analytically to provide closedform expressions for how the rates depend upon (1) the kinetic energy of the ejected electron, (2) the energy spacing between the anion and neutral energy surfaces as a function of geometry, (3) the difference in the slopes of the anion and neutral energy surfaces, and (4) overlaps of the neutral’s vibration–rotation wave function with the spatial derivative of that of the anion.

Statistical angular correlation coefficients of atomic Hartree–Fock orbitals
View Description Hide DescriptionWhen the probe function is chosen to be the statistical correlation coefficient reduces to the angular correlation coefficient which provides an overall measure of the difference between the electronpair density and the product of singleelectron densities. For a pair of atomic Hartree–Fock spin–orbitals and the characteristics of the angular correlation coefficient are studied. The coefficient is found to be negative for two spin–orbitals with the same spin and with azimuthal quantum numbers different by unity and zero for other cases. It is shown that is expressible in terms of the generalized electronpair moments relevant to the spin–orbitals and Especially, the relation for means that is nothing but the difference between the centerofmass motion and relative motion contributions of two electrons in those spin–orbitals. A numerical examination of 102 neutral atoms shows that the angular correlation is generally largest between the outermost and subshells.

nelectron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants
View Description Hide DescriptionThe nelectron valence state perturbation theory is reformulated in a spinfree formalism, concentrating on the “strongly contracted” and “partially contracted” variants. The new formulation is based on the introduction of average values in the unperturbed state of excitation operators which bear resemblance with analogous ones occurring in the extended Koopmans’ theorem and in the equationsofmotion technique. Such auxiliary quantities, which allow the secondorder perturbation contribution to the energy to be evaluated very efficiently, can be calculated at the outset provided the unperturbed fourparticle spinless density matrix in the active orbital space is available. A noticeable inequality concerning secondorder energy contributions of the same type between the strongly and partially contracted versions is proven to hold. An example concerning the successful calculation of the potential energy curve for the molecule is discussed.

The analog of Koopmans’ theorem in spindensity functional theory
View Description Hide DescriptionFor spinunrestricted Kohn–Sham (KS) calculations on systems with an open shell ground state with total spin quantum number S, we offer the analog of the Koopmans’type relation between orbital energies and ionization energies familiar from the Hartree–Fock model. When (case I) the lowest ion state has spin (typically when the neutral molecule has a (less than) half filled open shell), the orbital energy of the highest occupied orbital belonging to the open shell with majority spin (α) electrons, is equal to the ionization energy to this lowest ion state with spin For lower (doubly occupied) orbitals the ionization leaves an unpaired electron that can couple to the open shell to states: (exact identity for reducing to a simple average in the case of a doublet ground state (single electron outside closed shells). When the lowest ion state has spin (case II; typically for more than half filled open shells): for A physical basis is thus provided for the KS orbital energies also in the spin unrestricted case and an explanation is given for the common observation in approximate Kohn–Sham calculations of more negative majority spin (α) levels for than minority spin levels

The efficient optimization of molecular geometries using redundant internal coordinates
View Description Hide DescriptionThe optimization of ab initio molecular geometries is discussed. Based on comparisons of 30 minimizations and 15 saddlepoint optimizations, the most efficient combination of coordinate system, approximate and exact Hessians, and step control is determined. Use of a proposed set of extraredundant internal coordinates is shown to reduce the number of geometry steps significantly relative to the use of redundant coordinates. Various update schemes are tested for minimum and saddlepoint optimizations, including combination formulas. The complete expressions for the first and second derivatives of the Wilson B matrix are presented, thereby avoiding the need to calculate this by finitedifference methods. The presented scheme appears to be the most efficient, robust and generally applicable scheme to date.

A transferable polarizable electrostatic force field for molecular mechanics based on the chemical potential equalization principle
View Description Hide DescriptionA polarizable electrostatic potential model for classical molecular mechanics is presented. Based on the chemical potential equalization (CPE) principle, the model is developed starting from the original formulation of Mortier, Ghosh, and Shankar [J. Am. Chem. Soc. 108, 4315 (1986)]. Following York and Yang [J. Chem. Phys. 104, 159 (1996)] we present an SPbasis CPE parametrization to describe realistically any sort of molecular system. By fitting ab initio electronic properties, such as dipole moment,polarizability and global molecular hardness of a restricted set of organic molecules, we derive atomic parameters to be applied to a more vast target set of compounds. We show, indeed, that the atomic CPE parameters calculated for the learning set of molecules give reliable values for several electronic properties of various compounds not included in the learning set. The multipole moments obtained by using the proposed CPE parametrization are compared to the results of a fixed charge parametrization like that used by a popular classical molecular mechanics force field, such as AMBER. We show that the fixed charge parametrization can well reproduce only the multipole moments of the molecular conformation or the isomer used for the fit, while being inaccurate when different molecular conformations or isomers are considered. On the contrary, the CPE model realistically reproduces the charge reorganization due to nuclear structural changes of the molecule, such as isomerization or conformational transition. The CPE model has been also tested on various molecular complexes to investigate the polarization response in the case of realistic molecule–molecule interactions. The main result of the paper is the demonstration that the construction of a general polarizable electrostatic force field for classical molecular mechanics is now a viable way.

Implementation of generalized valence bondinspired coupled cluster theories
View Description Hide DescriptionWe present an implementation of the recently proposed imperfect pairing (IP) and generalized valence bond restricted coupled cluster (GVBRCC) methods. Our algorithm centers on repeated construction of Coulomb and exchange matrices. These operations are the computational bottleneck, scaling with the third power of system size for large systems. Robust optimization of the valence orbitals is attained using a geometrically consistent form of direct minimization. Analytic gradients of the IP and GVBRCC energies are also obtained by a simple modification of the energy optimization scheme. As an illustration of the potential of these new methods, we use IP to compute the equilibrium geometry and energetics of a cluster that is a crude model for silicon dimerization on the Si(001) surface. We thus demonstrate a valuable role for IP and GVBRCC as a diagnostic for the accuracy of reduced active space calculations as compared to their full valence analogs.

Accurate method for the Brownian dynamics simulation of spherical particles with hardbody interactions
View Description Hide DescriptionIn Brownian Dynamics simulations, the diffusive motion of the particles is simulated by adding random displacements, proportional to the square root of the chosen time step. When computing average quantities, these Brownian contributions usually average out, and the overall simulation error becomes proportional to the time step. A special situation arises if the particles undergo hardbody interactions that instantaneously change their properties, as in absorption or association processes, chemical reactions, etc. The common “naı̈ve simulation method” accounts for these interactions by checking for hardbody overlaps after every time step. Due to the simplification of the diffusive motion, a substantial part of the actual hardbody interactions is not detected by this method, resulting in an overall simulation error proportional to the square root of the time step. In this paper we take the hardbody interactions during the time step interval into account, using the relative positions of the particles at the beginning and at the end of the time step, as provided by the naı̈ve method, and the analytical solution for the diffusion of a point particle around an absorbing sphere. Öttinger used a similar approach for the onedimensional case [Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996), p. 270]. We applied the “corrected simulation method” to the case of a simple, secondorder chemical reaction. The results agree with recent theoretical predictions [K. Hyojoon and Joe S. Kook, Phys. Rev. E 61, 3426 (2000)]. The obtained simulation error is proportional to the time step, instead of its square root. The new method needs substantially less simulation time to obtain the same accuracy. Finally, we briefly discuss a straightforward way to extend the method for simulations of systems with additional (deterministic) forces.

The generalized Douglas–Kroll transformation
View Description Hide DescriptionWe derive the most general parametrization of the unitary matrices in the Douglas–Kroll (DK) transformation sequence for relativistic electronic structure calculations. It is applied for a detailed analysis of the generalized DK transformation up to fifth order in the external potential. While DKH2–DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial freeparticle Foldy–Wouthuysen transformation. The freedom in the choice of this expansion coefficient vanishes consistently if the optimum unitary transformation is sought for. Since the standard protocol of the DK method is the application of unitary transformations to the oneelectron Dirac operator, we analyze the DKH procedure up to fifth order for hydrogenlike atoms. We find remarkable accuracy of the higherorder DK corrections as compared to the exact Diracground state energy. In the case of manyelectron atomic systems, we investigate the order of magnitude of the higherorder corrections in the light of the neglect of the DK transformation of the twoelectron terms of the manyparticle Hamiltonian. A careful analysis of the silver and gold atoms demonstrates that both the fourth and fifthorder oneelectron DK transformation yield a smaller contribution to the total electronic energy than the DK transformation of the twoelectron terms. In order to improve significantly on the thirdorder correction DKH3, it is thus mandatory to include the DK transformation of the twoelectron terms as well as the spindependent terms before proceeding to higher orders in the transformation of the oneelectron terms. However, an analysis of the ionization energies of these atoms indicates that already DKH3 yields a highly accurate treatment of the scalarrelativistic effects on properties.

An optimized mean first passage time approach for obtaining rates in activated processes
View Description Hide DescriptionThe mean first passage time has recently become a useful analytic and computational quantity for estimating reaction rates in manydimensional activated processes. Unfortunately, the accuracy of this association is limited by the indeterminacy of the appropriate boundary surface with respect to which the first passage times are obtained. The standard choices for this boundary result in an overestimate of the rates in stochastic models using the Langevin equation in the low friction limit. We propose a boundary surface which is a subspace of phase space that results in rates that are accurate in the entire friction regime. It is to be contrasted with equally accurate meanfirstpassagetime rates that are obtained using noninvariant subspaces of either the configuration space or phase space and hence are not amenable to nonnumerical analysis. The proposed boundary surface is also shown heuristically and numerically to result from a new kind of variational principle.

Polarization consistent basis sets. III. The importance of diffuse functions
View Description Hide DescriptionA sequence of diffuse functions to be used in connections with the previously defined polarization consistent basis sets are proposed based on energetic criteria and results for molecular properties. At the Hartree–Fock level the addition of a single set of diffuse s and pfunctions significantly improves the convergence of calculated electron affinities. A corresponding analysis at the density functional level indicates that only systems with high electron affinities have welldefined basis set limits with common exchangecorrelation functionals that have electron selfinteraction errors. The majority of reported density functional calculations of electron affinities appear to be artifacts of the limited basis set used. The good agreement with experiments for such calculations is most likely due to a reasonable modeling of the physics of the anionic species, rather than being a theoretically sound procedure. For molecular properties like dipole and quadrupole moments, and static polarizabilities, the addition of diffuse functions up to dfunctions is required to reach the basis set limit in a consistent fashion, but higher order angular momentum functions are significantly less important.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

The photodissociation of dimethylformamide: A complete active space selfconsistent field study
View Description Hide DescriptionThe and equilibrium geometries of N,Ndimethylformamide (NNDMF) and transition state structures of the and bond cleavages in the and states were determined with the complete active space selfconsistent field method using the basis set. The surface intersections among the and states were optimized with the stateaveraged complete active space selfconsistent field method. The most probable mechanisms, leading to different products for the NNDMF photodissociation at 193 nm, were provided in the present paper. The obtained mechanisms are slightly different from those proposed previously on the basis of qualitative analyses of state correlation and electronic configurations between the reactants and products. This is discussed in detail.

Dissociative photoionization of in the region of the state, studied by ion–electron velocity vector correlation
View Description Hide DescriptionImaging and timeofflightresolved coincidence techniques are combined to extend the vector correlation method to the study of dissociativephotoionization of small polyatomic molecules breaking into two heavy fragments. Dissociativephotoionization (DPI) of the linear molecule into the ionic state, induced by linearly polarized synchrotron radiation (P), is chosen as an example. The ion–electron kinetic energy correlation enables the identification of all the DPI processes producing the and fragments. The molecular frame photoelectron angular distributions (MFPADs), deduced from the spatial analysis of the vector correlations, exhibit remarkable features. When the molecule is aligned parallel to the polarization axis, a preferred electron emission from the molecular ion in the direction perpendicular to the molecular axis, as well as a strong forward–backward asymmetry that favors electron emission along the molecular axis in the same direction as the or N fragment, are demonstrated. The measured MFPADs are found in good agreement with the reported multichannel Schwinger configuration interaction calculations, when molecular rotation prior to dissociation is taken into account. This comparison provides an estimation of the lifetime of the state prior to dissociation into the dominant channels and which is found to be about 2 ps.

Dynamics of CO elimination from reactions of yttrium atoms with formaldehyde, acetaldehyde, and acetone
View Description Hide DescriptionReactions of neutral, groundstateyttrium atoms with formaldehyde, acetaldehyde, and acetone where were studied in crossed molecular beams. At collision energies greater than 24 kcal/mol, four product channels were observed corresponding to elimination of CO, H, and nonreactive scattering. For the dominant CO elimination channel, a large fraction (34%–41%) of the available energy appeared as kinetic energy of the products. RRKM modeling indicated this was a result of two factors: a large potential energy barrier for migration leading to and dissociation of this complex prior to complete energy randomization. The CM angular distributions were all forward–backward symmetric, indicating the existence of at least one longlived reaction intermediate. The angular distributions ranged from being quite forward–backward peaking for the reaction to isotropic for A simple equation is derived based on statistical complex theory that relates the shape of the CM angular distributions to the structure of the dissociating complex.

The 212.8nm photodissociation of formic acid: Degenerate fourwave mixing spectroscopy of the nascent radicals
View Description Hide DescriptionThe 212.8nm photodissociation dynamics of formic acid was investigated utilizing degenerate fourwave mixing spectroscopy. The backgroundfree rotational spectrum of the nascent OH radicals was obtained, and a cold rotational energy distribution peaking at was extracted from the DFWM spectrum. The distribution was well approximated by a Boltzmann distribution with a rotational temperature of K, which corresponds to an average rotational energy of ∼498 The observation of a nonstatistical spin–orbit state distribution, with a preference for the lowenergy manifold, implies the absence of any interactions with nearby triplet states during dissociation. Preferential population of the Λdoublet was observed, indicating that the H–O–C bending vibration in and the recoil impulse are the principal sources of the OH rotation.

Vibrational predissociation of van der Waals molecules: An internal collision, angular momentum model
View Description Hide DescriptionWe describe an “internal collision” model of vibrational predissociation (vpd) in triatomic van der Waals (vdW) molecules based on the angular momentum (AM) model of collisioninduced vibration–rotation transfer. The probability of vpd is related to the probability of disposing the vibrational energy into rotational and orbital AM. In Tshaped species, two internal collision configurations are likely to dominate namely, the turning points of excursions by the weakly bound species relative to the diatomic. These two geometries result in a bimodal distribution of final rotational states. VelocityAM diagrams demonstrate why halogen and hydride vdW molecules have very different properties and illustrate the physics of quantitative calculations that reproduce experimental distributions in a wide range of vdW molecules. We introduce an analogy between a metastable dissociative state and the optical resonator and define a quality factor that relates vpd lifetime to stored energy and to ease of generating rotational and orbital AM by dissociation. Data on vdW molecules of OH are analyzed using the concept and the accelerated dissociation on forming the vdW complex with an efficient energy acceptor is likened to the formation of a particularly lowQ molecular resonator.

Molecular dynamics simulations of structural transitions and phase coexistence in water pentamers
View Description Hide DescriptionMolecular dynamics simulations of water pentamers were carried out using the semiempirical Parameterization Method 3 method to calculate the forces. Simulations were performed in the microcanonical ensemble, at several (average) vibrational temperatures. Three distinct transitions were observed in these systems, corresponding to the onset of phase coexistence and of two hydrogen bond rearrangements that were predicted by previous structural calculations. A detailed study of the hightemperature pentamer dynamics is presented, which clarifies the distinction between liquidlike behavior and simple structural rearrangements in these systems.