Volume 122, Issue 22, 08 June 2005
 ARTICLES

 Theoretical Methods and Algorithms

A practical treatment for the threebody interactions in the transcorrelated variational Monte Carlo method: Application to atoms from lithium to neon
View Description Hide DescriptionWe suggest a practical solution to dealing with the threebody interactions in the transcorrelated variational Monte Carlo method (TCVMC). In the TCVMC method, which was suggested in our previous paper [N. Umezawa and S. Tsuneyuki, J. Chem. Phys.119, 10015 (2003)], the Jastrow–Slatertype wave function is efficiently optimized through a selfconsistent procedure by minimizing the variance of the local energy. The threebody terms in the transcorrelated selfconsistentfield equation, which have been simply ignored in our previous works, are efficiently calculated by the Monte Carlo numerical integration. We found that our treatment for the threebody interactions is successful for atoms from Li to Ne.

Critical comparison of various connected quadruple excitation approximations in the coupledcluster treatment of bond breaking
View Description Hide DescriptionTo assess the limits of singlereference coupledcluster (CC) methods for potentialenergysurfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution [J. Chem. Phys.108, 9221 (1998)]. We compare the methods for the treatment of potentialenergy curves for small molecules. These include CCSD, where the initial contributions of triple (T) and factorized quadruple excitations are added to coupledcluster singles (S) and doubles (D), its generalization to , where instead of measuring their first contribution from orders in H, it is measured from orders in ; renormalized approximations of both, and CCSD(2) defined in [J. Chem. Phys.115, 2014 (2001)]. We also consider CCSDT, , CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for , the double bond breaking in water, and are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restrictedHartree–Fock reference function, some of these higherorder approximations have a role to play in developing new, more robust procedures.

Quantifying the effects of the selfinteraction error in DFT: When do the delocalized states appear?
View Description Hide DescriptionThe selfinteraction error in densityfunctional theory leads to artificial stabilization of delocalized states, most evident in systems with an odd number of electrons. Clear examples are dissociations of carbocation radicals that often give delocalized states at long distances and large errors in computed binding energies. On the other hand, many mixedvalence transitionmetal dimers known to exhibit valence trapping are correctly predicted to be localized. To understand the effects of the selfinteraction error on these different systems, energy differences between delocalized and localized states are calculated with B3LYP. In the dissociation of radicals into symmetric fragments at infinite distance, this energy difference equals the error of the densityfunctional treatment. The energy difference decreases with increasing size of the system, from in to for . Solvent corrections stabilize the localized state and result in smaller errors. Most reactions are asymmetric and this decreases the effect of the selfinteraction error. In many systems, delocalization will not occur if the cost to move the electron from one fragment to the other is . This estimate refers to a situation where the distance between the fragments is infinite. The limit decreases with decreasing fragment distance. B3LYP calculations on the ferromagnetic state of a Mn(III,IV) dimer predict that the correct localized state is more stable than the incorrect delocalized state. At short metal–metal distances the effect of the selfinteraction error is predicted to be small. However, as the distance between the two manganese centers is increased to , the dimer starts to delocalize and the energy artificially decreases. In the dissociation limit, the error is . This is interpreted as an artifact originating from the selfinteraction error. Delocalization is not encountered in many systems due to relatively short metal–metal distances and asymmetric ligand environments. However, some chargetransfer complexes cannot be properly calculated and delocalized states may become a problem in large models of enzyme systems with multiple transitionmetal complexes.

Timedependent densityfunctional theory calculations of triplettriplet absorption
View Description Hide DescriptionWe present densityfunctional theory calculations of triplettriplet absorption by three different approaches based on timedependent densityfunctional theory(DFT): unrestricted DFT linear response, openshell restricted DFT linear response applied to the triplet state, and quadratic response with triplet excitations applied to the ground state. Comparison is also made with corresponding results obtained by Hartree–Fock and multiconfiguration selfconsistentfield response theory. Two main conclusions concerning triplettriplet transitions are drawn in this study: First, the very good agreement between unrestricted and restricted DFT results indicates that spin contamination of the triplet state is not a serious problem when computing triplettriplet spectra of common organic molecules. Second, DFT response calculations of triplettriplet transitions can be affected by triplet instability problems, especially for the combination of DFT quadratic response with functionals containing fractional exact Hartree–Fock exchange.

Eckart axis conditions and the minimization of the rootmeansquare deviation: Two closely related problems
View Description Hide DescriptionWe highlight the fact that the rotation matrix minimizing the rootmeansquare deviation between two molecular conformations [W. Kabsch, Acta Cryst.A32, 922 (1976)] also satisfies the Eckart axis conditions [C. Eckart, Phys. Rev.47, 552 (1935)].

Analytic gradients for the spinconserving and spinflipping equationofmotion coupledcluster models with single and double substitutions
View Description Hide DescriptionAnalytic gradient expressions for the spinconserving and spinflipping equationofmotion coupledcluster models with single and double substitutions are derived using a Lagrangian approach for the restricted and unrestricted Hartree–Fock references, both for the case of all orbitals being active in correlated calculations and for the frozen core and/or virtual orbitals. Details of the implementation within the QCHEM electronic structure package are discussed. The capabilities of the new code are demonstrated by application to cyclobutadiene.

Compact multipolar representation of the electrostatic potential for flexible molecules
View Description Hide DescriptionA new method for generating a compact multipolar representation of the electrostatic potential (EP) for flexible molecules is presented. The method is based on a constrained minimization of the difference between the quantum mechanical and the classical EP. The fitting procedure used adopts the least absolute shrinkage and selection operator technique [R. Tibshirani, J. Roy. Stat. Soc. B58, 267 (1996)] which can be seen as penalized ordinary least squares. The penalty function optimized for the particular molecule of interest effectively removes redundant multipoles. It is shown that the use of multiple conformations is crucial for the predictive ability of the EP model for flexible molecules. The multipole local coordinate systems are chosen in a way that best reflects the key conformational changes. It was demonstrated that such an approach improves the predictive ability of EP models. It also allows to exploit equivalence of atoms in the calculation of multipoles components. In the case of polar flexible molecules, the augmentation of the EP model based on charges by higher multipoles decreases the relative root mean square error by a factor of 1.5–5. The corresponding effect of enlargement of the set of multipoles was significantly reduced.

Electronic energy levels with the help of trajectoryguided random grid of coupled wave packets. I. Sixdimensional simulation of
View Description Hide DescriptionAs a preliminary to future work on the behavior of atoms and molecules in strong timedependent fields, we apply the coupled coherentstates (CCS) technique of multidimensional phasespace quantum dynamics to obtain Born–Oppenheimer energy levels of electrons in molecules. Unlike traditional approaches based on atomic and molecularorbital basis sets and timeindependent Schrödinger equation the CCS method exploits the solution of the timedependent Schrödinger equation in the basis of Monte Carloselected trajectoryguided coherent states, which treat classical electron correlations exactly. In addition the CCS trajectories move over averaged potentials, which remove the Coulombic singularities.

A version of diffusion Monte Carlo method based on random grids of coherent states. II. Sixdimensional simulation of electronic states of
View Description Hide DescriptionWe report a new version of the diffusionMonte Carlo (DMC) method, based on coherentstate quantum mechanics. Randomly selected grids of coherent states in phase space are used to obtain numerical imaginary time solutions of the Schrödinger equation, with an iterative refinement technique to improve the quality of the Monte Carlo grid. Accurate results were obtained, for the appropriately symmetrized two lowest states of the hydrogen molecule, by Monte Carlo sampling and sixdimensional propagation in the full phase space.

Molecular applications of the intermediate Hamiltonian Fockspace coupledcluster method for calculation of excitation energies
View Description Hide DescriptionThe intermediate Hamiltonian Fockspace coupledcluster (FSCC) method with singles and doubles is applied to calculate vertical excitation energies (EEs) for some molecular systems. The calculations are performed for several small molecules, such as , , and CO, and for larger systems, such as , , and . Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FSCC equations, and the efficient factorization strategy, relatively large basis sets and model spaces are employed permitting a comparison of the calculated vertical EEs with the experimental data.

The microcanonical thermodynamics of finite systems: The microscopic origin of condensation and phase separations, and the conditions for heat flow from lower to higher temperatures
View Description Hide DescriptionMicrocanonical thermodynamics [D. H. E. Gross, Microcanonical Thermodynamics, Phase Transitions in “Small” Systems (World Scientific, Singapore, 2001)] allows the application of statistical mechanics both to finite and even small systems and also to the largest, selfgravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy is necessarily convex to make bimodal in . Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered; the addition of any other external constraint [A. Wehrl, Rev. Mod. Phys.50, 221 (1978)], such as a dividing surface, or the enforcement of gradients of the energy or particle profile, reduce the entropy. As will be shown here, when removing such constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the Clausius formulation of the second law: “Heat always flows from hot to cold,” can be violated. Temperature is not a necessary or fundamental control parameter of thermostatistics. However, the second law is still satisfied and the total Boltzmannentropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or nonexistence) of a critical end point of the phase separation are discussed. This is explained for the liquidgas and the solidliquidtransition.

Extended quantization condition for constructive and destructive interferences and trajectories dominating molecular vibrational eigenstates
View Description Hide DescriptionThe role of destructive quantum interference in semiclassical quantization of molecular vibrational states is studied. This aspect is crucial for correct quantization, since failure in the appropriate treatment of destructive interference quite often results in many spurious peaks and broad background to hide the true peaks. We first study the timeFourier transform of the autocorrelation function without performing summation over the trajectories. The resultant quantity, the prespectrum which is a function of individual classical trajectories, provides a clear view about how destructive interference among the trajectories should function. It turns out that the prespectrum is oscillatory but never a random noise. On the contrary, it bears a systematic and regular structure, which is sometimes characterized in terms of very sharp and high peaks in the energy space of the sampled classical trajectories. We have found an extended quantization condition that is responsible for generating these peaks in the prespectrum, which we call the prior quantization condition. Integration of the prespectrum over the trajectory space is supposed to give “zero” (practically a small value of the order of the Planck constant) at a noneigenvalue energy, which is actually a materialization of the destructive interference. Besides, certain finite peaks in the prespectrum survive after the integration to form the true spikes (eigenvalues) in the final spectrum, if they satisfy an additional resonance condition. For these resonance components, the prior quantization condition is reduced to the Einstein–Brillouin–Keller quantization condition. Based on these analyses, we propose a rather conventional filtering technique to efficiently handle tedious computation for destructive interference, and numerically verify that it works well even for multidimensional chaotic systems. This filtering technique is further utilized to extract a few trajectories that dominate an eigenstate of molecular vibration.

Extrapolated intermediate Hamiltonian coupledcluster approach: Theory and pilot application to electron affinities of alkali atoms
View Description Hide DescriptionThe intermediate Hamiltonian (IH) coupledcluster method makes possible the use of very large model spaces in coupledcluster calculations without running into intruder states. This is achieved at the cost of approximating some of the IH matrix elements, which are not taken at their rigorous effective Hamiltonian (EH) value. The extrapolated intermediate Hamiltonian (XIH) approach proposed here uses a parametrized IH and extrapolates it to the full EH, with model spaces larger by several orders of magnitude than those possible in EH coupledcluster methods. The flexibility and resistance to intruders of the IH approach are thus combined with the accuracy of full EH. Various extrapolation schemes are described. A pilot application to the electron affinities (EAs) of alkali atoms is presented, where converged EH results are obtained by XIH for model spaces of determinants; direct EH calculations converge only for a onedimensional model space. Including quantum electrodynamic effects, the average XIH error for the EAs is and the largest error is . A new reference estimate for the EA of Fr is proposed at .

Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules
View Description Hide DescriptionMolecular dynamics integrators are presented for translational and rotational motion of rigid molecules in microcanonical, canonical, and isothermalisobaric ensembles. The integrators are all time reversible and are also, in some approaches, symplectic for the microcanonical ensembles. They are developed utilizing the quaternion representation on the basis of the Trotter factorization scheme using a Hamiltonian formalism. The structure is similar to that of the velocity Verlet algorithm. Comparison is made with standard integrators in terms of stability and it is found that a larger time step is stable with the new integrators. The canonical and isothermalisobaric molecular dynamics simulations are defined by using a chain thermostat approach according to generalized Nosé–Hoover and Andersen methods.

Finite lifetime effects on the polarizability within timedependent densityfunctional theory
View Description Hide DescriptionWe present an implementation for considering finite lifetime of the electronic excited states into linearresponse theory within timedependent densityfunctional theory. The lifetime of the excited states is introduced by a common phenomenological damping factor. The real and imaginary frequencydependent polarizabilities can thus be calculated over a broad range of frequencies. This allows for the study of linearresponse properties both in the resonance and nonresonance cases. The method is complementary to the standard approach of calculating the excitation energies from the poles of the polarizability. The real and imaginary polarizabilities can then be calculated in any specific energy range of interest, in contrast to the excitation energies which are usually solved only for the lowest electronic states. We have verified the method by investigating the photoabsorptionproperties of small alkali clusters. For these systems, we have calculated the real and imaginary polarizabilities in the energy range of and compared these with excitation energy calculations. The results showed good agreement with both previous theoretical and experimental results.

Nuclearmagneticresonance shielding constants calculated by pseudospectral methods
View Description Hide DescriptionWe have developed an algorithm based upon pseudospectral (PS) ab initio electronic structure methods for evaluating nuclear magnetic shielding constants using gaugeincluding atomic orbitals (GIAOs) in the spinrestricted and spinunrestricted formalisms of Hartree–Fock (HF) theory and densityfunctional theory(DFT). The nuclear magnetic shielding constants for both and calculated using PS methodology for 21 small molecules have absolute mean errors of less than in comparison with analytic integral results. CPU timing comparisons between PS methods and conventional methods carried out for seven large molecules ranging from 510 to 1285 basis functions demonstrate that the PS methods are an order of magnitude more efficient than the conventional methods. PSHF was between 9 and 26 times faster than conventional integral technology, and PSDFT (Becke threeparameter Lee–Yang–Parr) was between 6 and 21 times faster.

Robust and accurate method for freeenergy calculation of charged molecular systems
View Description Hide DescriptionA new approach is presented to eliminate the problem of creation and/or annihilation of atoms in freeenergy calculations of charged molecular systems. The method employs a damping potential in the Ewald summation scheme, which is an exact solution of the electrostatics for threedimensional periodic systems. The proposed method enables entire molecules to be mutated from a noninteracting (ideal) state in an efficient and robust way, thus providing a means by which accurate absolute free energies of structurally complex molecules can be determined. This methodology will enable chemical and phase equilibria to be determined for large molecular species with significant charge distributions, e.g., biomolecules and drugs.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Supersonically cooled hydronium ions in a slitjet discharge: Highresolution infrared spectroscopy and tunneling dynamics of
View Description Hide DescriptionJetcooled highresolution infrared spectra of partially deuterated hydronium ion in the O–H stretch region ( band) are obtained for the first time, exploiting the high ion densities, long absorption path lengths, and concentration modulation capabilities of the slitjet discharge spectrometer. Leastsquares analysis with a Watson asymmetric top Hamiltonian yields rovibrational constants and provides high level tests of ab initio molecular structure predictions. Transitions out of both the lower and the upper tunneling levels, as well as transitions across the tunneling gap are observed. The transitions in acquire oscillator strength by loss of symmetry, and permit both groundstate and excitedstatetunneling splittings to be determined to spectroscopic precision from a single rovibrational band. The splittings and band origins calculated with recent high level ab initio sixdimensional potential surface predictions for and isotopomers [X. C. Huang, S. Carter, and J. M. Bowman, J. Chem. Phys.118, 5431 (2003); T. Rajamaki, A. Miani, and L. Halonen, J. Chem. Phys.118, 10929 (2003)] are in very good agreement with the current experimental results.

Investigation of valence orbitals of propene by electron momentum spectroscopy
View Description Hide DescriptionThe binding energy spectra and momentum distributions of all valence orbitals of propene were studied by electron momentum spectroscopy (EMS) as well as Hartree–Fock and density functionaltheoretical calculations. The experiment was carried out at impact energies of and on the stateoftheart EMS spectrometer developed at Tsinghua University recently. The experimental momentum profiles of the valence orbitals were obtained and compared with the various theoretical calculations. Moreover, the experiment with a new analysis method presents a strong support for the correct ordering of the orbital and , i.e., .

Electronimpact excitation of in carbon monoxide
View Description Hide DescriptionThe dipoleforbidden transition of of CO has been observed by a fast electronenergylossspectrometer at a large scattering angle of 7° and with an energy resolution of . The energy levels and the relative intensity distribution of , have been determined, most of the results are reported for the first time.