Volume 138, Issue 9, 07 March 2013
 COMMUNICATIONS


Communication: Importance sampling including path correlation in semiclassical initial value representation calculations for time correlation functions
View Description Hide DescriptionFull semiclassical (SC) initial value representation (IVR) for time correlation functions involves a double phase space average over a set of two phase points, each of which evolves along a classical path. Conventionally, the two initial phase points are sampled independently for all degrees of freedom (DOF) in the Monte Carlo procedure. Here, we present an efficient importance sampling scheme by including the path correlation between the two initial phase points for the bath DOF, which greatly improves the performance of the SCIVR calculations for large molecular systems. Satisfactory convergence in the study of quantum coherence in vibrational relaxation has been achieved for a benchmark systembath model with up to 21 DOF.
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 ARTICLES

 Theoretical Methods and Algorithms

Localized saddlepoint search and application to temperatureaccelerated dynamics
View Description Hide DescriptionWe present a method for speeding up temperatureaccelerated dynamics (TAD) simulations by carrying out a localized saddlepoint (LSAD) search. In this method, instead of using the entire system to determine the energy barriers of activated processes, the calculation is localized by only including a small chunk of atoms around the atoms directly involved in the transition. Using this method, we have obtained Nindependent scaling for the computational cost of the saddlepoint search as a function of system size N. The error arising from localization is analyzed using a variety of model systems, including a variety of activated processes on Ag(100) and Cu(100) surfaces, as well as multiatom moves in Cu radiation damage and metal heteroepitaxial growth. Our results show significantly improved performance of TAD with the LSAD method, for the case of Ag/Ag(100) annealing and Cu/Cu(100) growth, while maintaining a negligibly small error in energy barriers.

Applicability of the wideband limit in DFTbased molecular transport calculations
View Description Hide DescriptionTransport properties of molecular junctions are notoriously expensive to calculate with ab initio methods, primarily due to the semiinfinite electrodes. This has led to the introduction of different approximation schemes for the electrodes. For the most popular metals used in experiments, such as gold, the wideband limit (WBL) is a particularly efficient choice. In this paper, we investigate the performance of different WBL schemes relative to more sophisticated approaches including the fully selfconsistent nonequilibrium Green's function method. We find reasonably good agreement between all schemes for systems in which the molecule (and not the metalmolecule interface) dominates the transport properties. Moreover, our implementation of the WBL requires negligible computational effort compared to the groundstate densityfunctional theory calculation of a molecular junction. We also present a new approximate but efficient scheme for calculating transport with a finite bias. Provided the voltage drop occurs primarily inside the molecule, this method provides results in reasonable agreement with fully selfconsistent calculations.

Recycling random numbers in the stochastic simulation algorithm
View Description Hide DescriptionThe stochastic simulation algorithm (SSA) was introduced by Gillespie and in a different form by Kurtz. Since its original formulation there have been several attempts at improving the efficiency and hence the speed of the algorithm. We briefly discuss some of these methods before outlining our own simple improvement, the recycling direct method (RDM), and demonstrating that it is capable of increasing the speed of most stochastic simulations. The RDM involves the statistically acceptable recycling of random numbers in order to reduce the computational cost associated with their generation and is compatible with several of the preexisting improvements on the original SSA. Our improvement is also sufficiently simple (one additional line of code) that we hope will be adopted by both trained mathematical modelers and experimentalists wishing to simulate their model systems.

Extension of the KLI approximation toward the exact optimized effective potential
View Description Hide DescriptionThe integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spinunrestricted exchangecorrelation potential, V _{ xcσ}, is obtained for any assumed orbitaldependent exchangecorrelation energy functional. The method extends beyond the KriegerLiIafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent firstorder perturbation theory wavefunction often referred to as the “orbital shift” function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with “sum over states” numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchangecorrelation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP V _{ xcσ}, thus allowing for the reduction of the OEP equation to a selfconsistent integral equation for the exact exchangecorrelation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchangecorrelation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbitaldependent exchangecorrelation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

Spin filtering in molecular junction: Magnetoresistance evaluation from wavefunction calculations
View Description Hide DescriptionThe conductance of magnetic molecules opens new ways to probe the electronic structure of correlated systems. Based on a 2electron/2molecular orbital prototype system, the currentpotential characteristics is inspected as a function of the differential magnetization of the electrodes sandwiching the molecule within a multideterminantal framework. The biasdependent magnetoresistance effect along the junction reflects the nature and energetics of the different multiplets, obtained within the multiconfigurational wavefunction approach. From the wavefunction description, a modulation of the magnetoresistance ratio is anticipated and both direct and inverse regimes are observed depending on the electronic structure of the junction.

Types of single particle symmetry breaking in transition metal oxides due to electron correlation
View Description Hide DescriptionVery accurate wave functions are calculated for small transition metal oxide molecules. These wave functions are decomposed using reduced density matrices to study the underlying correlation of electrons. The correlation is primarily of leftright type between the transition metals and the oxygen atoms, which is mediated by excitations from the nominal single Slater ground state into antibonding and dtype orbitals. In a localized representation, this correlation manifests itself in a 2electron hopping term that is offdiagonal. This term is of similar magnitude to the commonly considered Hubbardtype onsite interaction.

A general nuclear motion Hamiltonian and noninternal curvilinear coordinates
View Description Hide DescriptionAn exact Hamiltonian for nuclear motions in general curvilinear coordinates is derived. It is demonstrated how this Hamiltonian transforms into wellestablished expressions, such as the Wilson Howard Hamiltonian or the Meyer Günthard Hamiltonian, if the general coordinates are restricted to be rectilinear or internal. Furthermore, a compact expression for the Hamiltonian expressed in noninternal curvilinear coordinates is provided, which makes this coordinate class available for applications in a simple way, since only the Jacobian matrix with respect to the rotating frame coordinates must be calculated. An example, employing a water model potential, exemplifies how different coordinate systems from all three coordinate classes (rectilinear, internal, and noninternal) lead to vibrational spectra, which are in excellent agreement. Thereby, the applicability of the general Hamiltonian is demonstrated and also its correctness is confirmed.

Electronic transition dipole moments and dipole oscillator strengths within Fockspace multireference coupled cluster framework: An efficient and novel approach
View Description Hide DescriptionWithin the Fockspace multireference coupled cluster framework, we have evaluated the electronic transition dipole moments, which determine absorption intensities. These depend on matrix elements between two different wave functions (e.g., ground state to the excited state). We present two different ways, to calculate these transition moments. In the first method, we construct the ground and excited state wave functions with the normal exponential ansatz of Fockspace coupled cluster method and then calculate the relevant offdiagonal matrix elements. In the second approach, we linearize the exponential form of the wave operator which will generate the left vector, by use of Lagrangian formulation. The right vector is obtained from the exponential ansatz. In order to relate the transition moments to oscillator strengths, excitation energies need to be evaluated. The excitation energies are obtained from the Fockspace multireference framework. The transition dipole moments of the ground to a few excited states, together with the oscillator strengths of a few molecules, are presented.

A secondorder unconstrained optimization method for canonicalensemble densityfunctional methods
View Description Hide DescriptionA second order converging method of ensemble optimization (SOEO) in the framework of KohnSham DensityFunctional Theory is presented, where the energy is minimized with respect to an ensemble density matrix. It is general in the sense that the number of fractionally occupied orbitals is not predefined, but rather it is optimized by the algorithm. SOEO is a second order NewtonRaphson method of optimization, where both the form of the orbitals and the occupation numbers are optimized simultaneously. To keep the occupation numbers between zero and two, a set of occupation angles is defined, from which the occupation numbers are expressed as trigonometric functions. The total number of electrons is controlled by a builtin second order restriction of the NewtonRaphson equations, which can be deactivated in the case of a grandcanonical ensemble (where the total number of electrons is allowed to change). To test the optimization method, dissociation curves for diatomic carbon are produced using different functionals for the exchangecorrelation energy. These curves show that SOEO favors symmetry broken purestate solutions when using functionals with exact exchange such as HartreeFock and Becke threeparameter LeeYangParr. This is explained by an unphysical contribution to the exact exchange energy from interactions between fractional occupations. For functionals without exact exchange, such as local density approximation or Becke LeeYangParr, ensemble solutions are favored at interatomic distances larger than the equilibrium distance. Calculations on the chromium dimer are also discussed. They show that SOEO is able to converge to ensemble solutions for systems that are more complicated than diatomic carbon.

Efficient softest mode finding in transition states calculations
View Description Hide DescriptionTransition states are fundamental to understanding the reaction dynamics qualitatively in chemical physics. To date various methods of first principle location of the transition states have been developed. In the absence of the knowledge of the final structure, the softestmode following method climbs up to a transition state without calculating the Hessian matrix. One weakness of this kind of approaches is that the number of rotations to determine the softest mode is usually unpredictable. In this paper, we propose a locally optimal search direction finding algorithm, namely LOR, which is an extension of the traditional conjugate gradient method without additional calculations of the forces. We also show that the translation of forces improves the numerical stability. Experiments for the Baker test system show that the proposed algorithm is much faster than the original dimer conjugate gradient method.

Minimizing memory as an objective for coarsegraining
View Description Hide DescriptionCoarsegraining a molecular model is the process of integrating over degrees of freedom to obtain a reduced representation. This process typically involves two separate but related steps, selection of the coordinates comprising the reduced system and modeling their interactions. Both the coordinate selection and the modeling procedure present challenges. Here, we focus on the former. Typically, one seeks to integrate over the fast degrees of freedom and retain the slow degrees of freedom. Failure to separate timescales results in memory. With this motivation, we introduce a heuristic measure of memory and show that it can be used to compare competing coordinate selections for a given modeling procedure. We numerically explore the utility of this heuristic for three systems of increasing complexity. The first example is a fourparticle linear model, which is exactly solvable. The second example is a sixteenparticle nonlinear model; this system has interactions that are characteristic of molecular force fields but is still sufficiently simple to permit exhaustive numerical treatment. The third example is an atomicresolution representation of a protein, the class of models most often treated by relevant coarsegraining approaches; we specifically study an actin monomer. In all three cases, we find that the heuristic suggests coordinate selections that are physically intuitive and reflect molecular structure. The memory heuristic can thus serve as an objective codification of expert knowledge and a guide to sites within a model that requires further attention.

Molecular dynamics simulations with replicaaveraged structural restraints generate structural ensembles according to the maximum entropy principle
View Description Hide DescriptionIn order to characterise the dynamics of proteins, a wellestablished method is to incorporate experimental parameters as replicaaveraged structural restraints into molecular dynamics simulations. Here, we justify this approach in the case of interproton distance information provided by nuclear Overhauser effects by showing that it generates ensembles of conformations according to the maximum entropy principle. These results indicate that the use of replicaaveraged structural restraints in molecular dynamics simulations, given a force field and a set of experimental data, can provide an accurate approximation of the unknown Boltzmann distribution of a system.

Bond energy decomposition analysis for subsystem density functional theory
View Description Hide DescriptionWe employed an explicit expression for the dispersion (D) energy in conjunction with KohnSham (KS) density functional theory and frozendensity embedding (FDE) to calculate interaction energies between DNA base pairs and a selected set of amino acid pairs in the hydrophobic core of a small protein Rubredoxin. We use this data to assess the accuracy of an FDED approach for the calculation of intermolecular interactions. To better analyze the calculated interaction energies we furthermore propose a new energy decomposition scheme that is similar to the wellknown KS bond formation analysis [F. M. Bickelhaupt and E. J. Baerends, Rev. Comput. Chem.15, 1 (Year: 2000)10.1002/9780470125922.ch1], but differs in the electron densities used to define the bond energy. The individual subsystem electron densities of the FDE approach sum to the total electron density which makes it possible to define bond energies in terms of promotion energies and an explicit interaction energy. We show that for the systems considered only a few freezeandthaw cycles suffice to reach convergence in these individual bond energy components, illustrating the potential of FDED as an efficient method to calculate intermolecular interactions.

Oscillator strengths of electronic excitations with response theory using phase including natural orbital functionals
View Description Hide DescriptionThe key characteristics of electronic excitations of manyelectron systems, the excitation energies ω_{α} and the oscillator strengths f _{α}, can be obtained from linear response theory. In oneelectron models and within the adiabatic approximation, the zeros of the inverse response matrix, which occur at the excitation energies, can be obtained from a simple diagonalization. Particular cases are the eigenvalue equations of timedependent density functional theory (TDDFT), timedependent density matrix functional theory, and the recently developed phaseincluding natural orbital (PINO) functional theory. In this paper, an expression for the oscillator strengths f _{α} of the electronic excitations is derived within adiabatic response PINO theory. The f _{α} are expressed through the eigenvectors of the PINO inverse response matrix and the dipole integrals. They are calculated with the phaseincluding natural orbital functional for twoelectron systems adapted from the work of L wdin and Shull on twoelectron systems (the phaseincluding LöwdinShull functional). The PINO calculations reproduce the reference f _{α} values for all considered excitations and bond distances R of the prototype molecules H_{2} and HeH^{+} very well (perfectly, if the correct choice of the phases in the functional is made). Remarkably, the quality is still very good when the response matrices are severely restricted to almost TDDFT size, i.e., involving in addition to the occupiedvirtual orbital pairs just (HOMO+1)virtual pairs (R1) and possibly (HOMO+2)virtual pairs (R2). The shape of the curves f _{α}(R) is rationalized with a decomposition analysis of the transition dipole moments.
 Advanced Experimental Techniques

Comblinked, cavity ringdown spectroscopy for measurements of molecular transition frequencies at the kHzlevel
View Description Hide DescriptionWe present a low uncertainty measurement technique for determining molecular transition frequencies. This approach is complementary to subDoppler saturation spectroscopies and is expected to enable new frequency measurements for a wide variety of molecular species with uncertainties at the kHzlevel. The technique involves measurements of Doppler broadened lines using cavity ringdown spectroscopy whereby the probe laser is actively locked to the ringdown cavity and the spectrum frequencies are linked directly to an optical frequency comb that is referenced to an atomic frequency standard. As a demonstration we have measured the transition frequency of the (30012) ← (00001) P14e line of CO_{2} near 1.57 μm with a combined standard uncertainty of ∼9 kHz. This technique exhibits exceptional promise for measurements of transition frequencies and pressure shifting parameters of many weak absorbers, and indicates the potential for substantially improved measurements when compared to those obtained with conventional spectroscopic methods.

Restricted diffusion in annular geometrical pores
View Description Hide DescriptionNuclear magnetic resonance (NMR) diffusion (including diffusion MRI) experiments are only as powerful as the models used to analyse the NMR diffusion data. A major problem, especially with measurements on biological systems, is that the existing models are only very poor approximations of cellular shape. Here, diffusion propagators and pulsed gradient spinecho attenuation equations are derived in the short gradient pulse limit for diffusion within the annular region of a concentric cylinder of finite length and, similarly, within the annular region of a concentric sphere. The models include the possibility of relaxation at the boundaries and, in the case of the concentric cylinder, having the cylinder arbitrarily oriented with respect to the direction of the applied field gradient. The two models are also of interest due to their direct analogy to optical double slit diffraction. Also expressions for the mean square displacements, which are very useful information for determining the diffusion coefficient within these complex geometries, are obtained as well as for the limiting cases of diffusion on cylindrical and spherical shells and in a ring.

VUV photoionization of gas phase adenine and cytosine: A comparison between oven and aerosol vaporization
View Description Hide DescriptionWe studied the single photon ionization of gas phase adenine and cytosine by means of vacuum ultraviolet synchrotron radiation coupled to a velocity map imaging electron/ion coincidence spectrometer. Both invacuum temperaturecontrolled oven and aerosol thermodesorption were successfully applied to promote the intact neutral biological species into the gas phase. The photoion yields are consistent with previous measurements. In addition, we deduced the threshold photoelectron spectra and the slow photoelectron spectra for both species, where the close to zero kinetic energy photoelectrons and the corresponding photoions are measured in coincidence. The photoionization close and above the ionization energies are found to occur mainly via direct processes. Both vaporization techniques lead to similar electronic spectra for the two molecules, which consist of broadbands due to the complex electronic structure of the cationic species and to the possible contribution of several neutral tautomers for cytosine prior to ionization. Accurate ionization energies are measured for adenine and cytosine at, respectively, 8.267 ± 0.005 eV and 8.66 ± 0.01 eV, and we deduce precise thermochemical data for the adenine radical cation. Finally, we performed an evaluation and a comparison of the two vaporization techniques addressing the following criteria: measurement precision, thermal fragmentation, sensitivity, and sample consumption. The aerosol thermodesorption technique appears as a promising alternative to vaporize large thermolabile biological compounds, where extended thermal decomposition or low sensitivity could be encountered when using a simple oven vaporization technique.
 Atoms, Molecules, and Clusters

Rovibronically selected and resolved twocolor laser photoionization and photoelectron study of cobalt carbide cation
View Description Hide DescriptionWe have conducted a twocolor visibleultraviolet (VISUV) resonanceenhanced laser photoionization efficiency and pulsed field ionizationphotoelectron (PFIPE) study of gaseous cobalt carbide (CoC) near its ionization onset in the total energy range of 61 200–64 510 cm^{−1}. The cold gaseous CoC sample was prepared by a laser ablation supersonically cooled beam source. By exciting CoC molecules thus generated to single N′ rotational levels of the intermediate CoC*(^{2}Σ^{+}; v′) state using a VIS dye laser prior to UV laser photoionization, we have obtained N ^{+} rotationally resolved PFIPE spectra for the CoC^{+}(X ^{1}Σ^{+}; v ^{+} = 0 and 1) ion vibrational bands free from interference by impurity species except Co atoms produced in the ablation source. The rotationally selected and resolved PFIPE spectra have made possible unambiguous rotational assignments, yielding accurate values for the adiabatic ionization energy of CoC(X ^{2}Σ^{+}), IE(CoC) = 62 384.3 ± 0.6 cm^{−1} (7.73467 ± 0.00007 eV), the vibrational frequency ω_{e} ^{+} = 985.6 ± 0.6 cm^{−1}, the anharmonicity constant ω_{e} ^{+}χ_{e} ^{+} = 6.3 ± 0.6 cm^{−1}, the rotational constants (B _{ e } ^{+} = 0.7196 ± 0.0005 cm^{−1}, α _{ e } ^{+} = 0.0056 ± 0.0008 cm^{−1}), and the equilibrium bond length r _{ e } ^{+} = 1.534 Å for CoC^{+}(X ^{1}Σ^{+}). The observation of the N ^{+} = 0 level in the PFIPE measurement indicates that the CoC^{+} ground state is of ^{1}Σ^{+} symmetry. Large ΔN ^{+} = N ^{+} − N′ changes up to 6 are observed for the photoionization transitions CoC^{+}(X ^{1}Σ^{+}; v ^{+} = 0–2; N ^{+}) ← CoC*(^{2}Σ^{+}; v′; N′ = 6, 7, 8, and 9). The highly precise energetic and spectroscopic data obtained in the present study have served as a benchmark for testing theoretical predictions based on stateoftheart ab initio quantum calculations at the CCSDTQ/CBS level of theory as presented in the companion article.

Highlevel ab initio predictions for the ionization energy, bond dissociation energies, and heats of formation of cobalt carbide (CoC) and its cation (CoC^{+})
View Description Hide DescriptionThe ionization energy (IE) of CoC and the 0 K bond dissociation energies (D_{0}) and the heats of formation at 0 K (ΔH°_{f0}) and 298 K (ΔH°_{f298}) for CoC and CoC^{+} are predicted by the wavefunction based coupledcluster theory with single, double, triple and quadruple excitations (CCSDTQ) and complete basis set (CBS) approach. The CCSDTQ/CBS calculations presented here involve the approximation to the CBS limit at the coupled cluster level up to full quadruple excitations along with the zeropoint vibrational energy, highorder correlation, corevalence (CV) electronic, spinorbit coupling, and scalar relativistic effect corrections. The present calculations provide the correct symmetry, ^{1}Σ^{+}, for the ground state of CoC^{+}. The CCSDTQ/CBS IE(CoC) = 7.740 eV is found in good agreement with the experimental IE value of 7.73467 ± 0.00007 eV, determined in a twocolor laser photoion and pulsed field ionizationphotoelectron study. This work together with the previous experimental and theoretical investigations support the conclusion that the CCSDTQ/CBS method is capable of providing reliable IE predictions for 3dtransition metal carbides, such as FeC, CoC, and NiC. Among the singlereference based coupledcluster methods and multireference configuration interaction (MRCI) approach, the CCSDTQ and MRCI methods give the best predictions to the harmonic frequencies ω _{e} (ω _{e} ^{+}) = 956 (992) and 976 (1004) cm^{−1} and the bond lengths r _{e} (r _{e} ^{+}) = 1.560 (1.528) and 1.550 (1.522) Å, respectively, for CoC (CoC^{+}) in comparison with the experimental values. The CCSDTQ/CBS calculations give the prediction of D_{0}(Co^{+}–C) − D_{0}(Co–C) = 0.175 eV, which is also consistent with the experimental determination of 0.14630 ± 0.00014 eV. The theoretical results show that the CV and valencevalence electronic correlations beyond CCSD(T) wavefunction and the relativistic effect make significant contributions to the calculated thermochemical properties of CoC/CoC^{+}. For the experimental D_{0} and ΔH^{o} _{f0} values of CoC/CoC^{+}, which are not known experimentally, we recommend the following CCSDTQ/CBS predictions: ΔH^{o} _{f0}(CoC) = 775.7 kJ/mol and ΔH^{o} _{f0}(CoC^{+}) = 1522.5 kJ/mol, ΔH^{o} _{f298}(CoC) = 779.2 kJ/mol and ΔH^{o} _{298}(CoC^{+}) = 1526.0 kJ/mol.