Volume 136, Issue 20, 28 May 2012

Complexes of the benzenium ion () with N_{2} or CO_{2} have been studied by explicitly correlated coupled clustertheory at the CCSD(T)F12x (x = a, b) level [T. B. Adler et al., J. Chem. Phys.127, 221106 (2007)10.1063/1.2817618] and the doublehybrid density functional B2PLYPD [T. Schwabe and S. Grimme, Phys. Chem. Chem. Phys.9, 3397 (2007)10.1039/b704725h]. Improved harmonic vibrational wavenumbers for have been obtained by CCSD(T*)F12a calculations with the VTZF12 basis set. Combining them with previous B2PLYPD anharmonic contributions we arrive at anharmonic wavenumbers which are in excellent agreement with recent experimental data from pH_{2} matrix isolation IR spectroscopy[M. Bahou et al., J. Chem. Phys.136, 154304 (2012)10.1063/1.3703502]. The energetically most favourable conformer of ·N_{2} shows a πbonded structure similar to ·Rg (Rg = Ne, Ar) [P. Botschwina and R. Oswald, J. Phys. Chem. A115, 13664 (2011)10.1021/jp207905t] with D_{e} ≈ 870 cm^{−1}. For ·CO_{2}, a slightly lower energy is calculated for a conformer with the CO_{2} ligand lying in the ringplane of the moiety (D_{e} ≈ 1508 cm^{−1}). It may be discriminated from other conformers through a strong band predicted at 1218 cm^{−1}, redshifted by 21 cm^{−1} from the corresponding band of free .
 ARTICLES

 Theoretical Methods and Algorithms

Manybody calculations of lowenergy eigenstates in magnetic and periodic systems with selfhealing diffusion Monte Carlo: Steps beyond the fixed phase
View Description Hide DescriptionThe selfhealing diffusion Monte Carlo algorithm (SHDMC) [F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B79, 195117 (2009);10.1103/PhysRevB.79.195117F. A. Reboredo, Phys. Rev. B80, 125110 (2009)10.1103/PhysRevB.80.125110] is extended to study the ground and excited states of magnetic and periodic systems. The method converges to exact eigenstates as the statistical data collected increase if the wave function is sufficiently flexible. It is shown that the dimensionality of the nodal surface is dependent on whether phase is a scalar function or not. A recursive optimization algorithm is derived from the time evolution of the mixed probability density, which is given by an ensemble of electronic configurations (walkers) with complex weight. This complex weight allows the phase of the fixednode wave function to move away from the trial wave function phase. This novel approach is both a generalization of SHDMC and the fixedphase approximation [G. Ortiz, D. M. Ceperley, and R. M. Martin, Phys Rev. Lett.71, 2777 (1993)10.1103/PhysRevLett.71.2777]. When used recursively it simultaneously improves the node and the phase. The algorithm is demonstrated to converge to nearly exact solutions of model systems with periodic boundary conditions or applied magnetic fields. The computational cost is proportional to the number of independent degrees of freedom of the phase. The method is applied to obtain lowenergy excitations of Hamiltonians with magnetic field. Periodic boundary conditions are also considered optimizing wave functions with twisted boundary conditions which are included in a manybody Bloch phase. The potential applications of this new method to study periodic, magnetic, and complex Hamiltonians are discussed.

Phase diagram and universality of the LennardJones gasliquid system
View Description Hide DescriptionThe gasliquidphase transition of the threedimensional LennardJones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gasliquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.

General formulation of spinflip timedependent density functional theory using noncollinear kernels: Theory, implementation, and benchmarks
View Description Hide DescriptionWe report an implementation of the spinflip (SF) variant of timedependent density functional theory (TDDFT) within the TammDancoff approximation and noncollinear (NC) formalism for local, generalized gradient approximation, hybrid, and rangeseparated functionals. The performance of different functionals is evaluated by extensive benchmark calculations of energy gaps in a variety of diradicals and openshell atoms. The benchmark set consists of 41 energy gaps. A consistently good performance is observed for the PerdewBurkeErnzerhof (PBE) family, in particular PBE0 and PBE50, which yield mean average deviations of 0.126 and 0.090 eV, respectively. In most cases, the performance of original (collinear) SFTDDFT with 5050 functional is also satisfactory (as compared to noncollinear variants), except for the samecenter diradicals where both collinear and noncollinear SF variants that use LYP or B97 exhibit large errors. The accuracy of NCSFTDDFT and collinear SFTDDFT with 5050 and BHHLYP is very similar. Using PBE50 within collinear formalism does not improve the accuracy.

Sensitivity of the properties of ruthenium “blue dimer” to method, basis set, and continuum model
View Description Hide DescriptionThe ruthenium “blue dimer” [(bpy)_{2}Ru^{III}OH_{2}]_{2}O^{4+} is best known as the first welldefined molecular catalyst for water oxidation. It has been subject to numerous computational studies primarily employing density functional theory. However, those studies have been limited in the functionals, basis sets, and continuum models employed. The controversy in the calculated electronic structure and the reaction energetics of this catalyst highlights the necessity of benchmark calculations that explore the role of density functionals,basis sets, and continuum models upon the essential features of bluedimer reactivity. In this paper, we report KohnSham complete basis set (KSCBS) limit extrapolations of the electronic structure of “blue dimer” using GGA (BPW91 and BP86), hybridGGA (B3LYP), and metaGGA (M06L) density functionals. The dependence of solvation free energy corrections on the different cavity types (UFF, UA0, UAHF, UAKS, Bondi, and Pauling) within polarizable and conductorlike polarizable continuum model has also been investigated. The most common basis sets of doublezeta quality are shown to yield results close to the KSCBS limit; however, large variations are observed in the reaction energetics as a function of density functional and continuum cavity model employed.

Local explicitly correlated second and thirdorder Møller–Plesset perturbation theory with pair natural orbitals
View Description Hide DescriptionWe present an algorithm for computing explicitly correlated second and thirdorder Møller–Plesset energies near the basis set limit for large molecules with a cost that scales formally as with system size . This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The scaling does not rely upon the a prioridefinition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an augccpVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs.

Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized FickJacobs equation
View Description Hide DescriptionThe generalized FickJacobs equation is widely used to study diffusion of Brownian particles in threedimensional tubes and quasitwodimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized FickJacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both RegueraRubi and KalinayPercus versions of the generalized FickJacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.

Perturbative treatment of triple excitations in internally contracted multireference coupled cluster theory
View Description Hide DescriptionInternally contracted multireference coupled cluster (icMRCC) methods with perturbative treatment of triple excitations are formulated based on Dyall's definition of a zerothorder Hamiltonian. The iterative models icMRCCSDT1, icMRCC3, and their variants icMRCCSD(T), icMRCC(3) which determine the energy correction from triples by a noniterative step are consistent in the singlereference limit with CCSDT1a, CC3, CCSD(T), and CC(3), respectively. Numerical tests on the potential energy surfaces of BeH_{2}, H_{2}O, and N_{2} as well as on the structure and harmonic vibrational frequencies of the ozone molecule show that these methods account very well for higher order correlation effects. The icMRCCSD(T) method is further applied to the geometry optimization and harmonic frequencies of the symmetric vibrational modes of the binuclear transition metal oxide Ni_{2}O_{2}, to the singlettriplet splittings of o, m, and pbenzyne and to a ringopening reaction of an azirine compound with the molecular formula C_{6}H_{7}NO. The size of the active spaces used in this study ranges from CAS(2,2) to CAS(8,8). Comparisons of results based on differently sized active spaces indicate that the icMRCCSD(T) method provides a highly accurate and efficient treatment of both static and dynamic electron correlation in connection with minimal active spaces.

A sequential transformation approach to the internally contracted multireference coupled cluster method
View Description Hide DescriptionThe internally contracted multireference coupled cluster (icMRCC) approach is formulated using a new wave function ansatz based on a sequential transformation of the reference function (sqicMRCC). This alternative wave function simplifies the formulation of computationally viable methods while preserving the accuracy of the icMRCC approach. The structure of the sqicMRCC wave function allows folding the effect of the single excitations into a similaritytransformed Hamiltonian whose particle rank is equal to the one of the Hamiltonian. Consequently, we formulate an approximation to the sqicMRCC method with singles and doubles (included respectively up to fourfold and twofold commutators, sqicMRCCSD[2]) that contains all terms present in the corresponding singlereference coupled cluster scheme. Computations of the potential energy curves for the dissociation of BeH_{2} show that the untruncated sqicMRCCSD scheme yields results that are almost indistinguishable from the ordinary icMRCCSD method. The energy obtained from the computationally less expensive sqicMRCCSD[2] approximation is found to deviate from the full icMRCCSD method by less than 0.2 mE _{ h } for BeH_{2}, while, in the case of water, the harmonic vibrational frequencies of ozone, the singlettriplet splitting of pbenzyne, and the dissociation curve of N_{2}, sqicMRCCSD[2] faithfully reproduces the results obtained via the icMRCCSD scheme truncated to two commutators. A formal proof is given of the equivalence of the icMRCC and sqicMRCC methods with the internally contracted and full configuration interaction approaches.

Dispersionfree component of noncovalent interaction via mutual polarization of fragment densities
View Description Hide DescriptionComprehensive tests within a diverse set of noncovalently bonded systems are carried out to assess the performance of the recentlydeveloped dispersionfree approach in the framework of density functional theory[Ł. Rajchel, P. Żuchowski, M. Szczęśniak, and G. Chałasiński, Phys. Rev. Lett.104, 163001 (2010)]10.1103/PhysRevLett.104.163001. A numerical algorithm which cures the convergence problems of the previous implementation is presented.

Explicitly correlated secondorder MøllerPlesset perturbation theory employing pseudospectral numerical quadratures
View Description Hide DescriptionWe implemented explicitly correlated secondorder MøllerPlesset perturbation theory with numerical quadratures using pseudospectral construction of grids. Introduction of pseudospectral approach for the calculation of manyelectron integrals gives a possibility to use coarse grids without significant loss of precision in correlation energies, while the number of points in the grid is reduced about nine times. The use of complementary auxiliary basis sets as the sets of dealiasing functions is justified at both theoretical and computational levels. Benchmark calculations for a set of 16 molecules have shown the possibility to keep an error of secondorder correlation energies within 1 milihartree (mH) with respect to MP2F12 method with dense grids. Numerical tests for a set of 13 isogyric reactions are also performed.

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory
View Description Hide DescriptionThe conventional analysis of Perdew and Levy, and Sham and Schlüter shows that the functional derivative discontinuity of the exchangecorrelation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchangecorrelation functionals have been derived without invoking the concept of exchangecorrelation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, , at integers. (3) Consistent with the PerdewLevyShamSchlüter conclusion that the exact KohnSham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchangecorrelation energy, we show that the exchangecorrelation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchangecorrelation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spinup and spindown number of electrons.

Analytic gradient for second order MøllerPlesset perturbation theory with the polarizable continuum model based on the fragment molecular orbital method
View Description Hide DescriptionA new energy expression is proposed for the fragment molecular orbital method interfaced with the polarizable continuum model (FMO/PCM). The solvation free energy is shown to be more accurate on a set of representative polypeptides with neutral and charged residues, in comparison to the original formulation at the same level of the manybody expansion of the electrostatic potential determining the apparent surface charges. The analytic first derivative of the energy with respect to nuclear coordinates is formulated at the secondorder MøllerPlesset (MP2) perturbation theory level combined with PCM, for which we derived coupled perturbed HartreeFock equations. The accuracy of the analytic gradient is demonstrated on test calculations in comparison to numeric gradient. Geometry optimization of the small Trpcage protein (PDB: 1L2Y) is performed with FMO/PCM/631(+)G(d) at the MP2 and restricted HartreeFock with empirical dispersion (RHF/D). The root mean square deviations between the FMO optimized and NMR experimental structure are found to be 0.414 and 0.426 Å for RHF/D and MP2, respectively. The details of the hydrogen bond network in the Trpcage protein are revealed.

Estimating statistical distributions using an integral identity
View Description Hide DescriptionWe present an identity for an unbiased estimate of a general statistical distribution. The identity computes the distribution density from dividing a histogram sum over a local window by a correction factor from a meanforce integral, and the mean force can be evaluated as a configuration average. We show that the optimal window size is roughly the inverse of the local meanforce fluctuation. The new identity offers a more robust and precise estimate than a previous one by Adib and Jarzynski [J. Chem. Phys.122, 014114 (2005)]10.1063/1.1829631. It also allows a straightforward generalization to an arbitrary ensemble and a joint distribution of multiple variables. Particularly we derive a meanforce enhanced version of the weighted histogram analysis method. The method can be used to improve distributions computed from molecular simulations. We illustrate the use in computing a potential energy distribution, a volume distribution in a constantpressure ensemble, a radial distribution function, and a joint distribution of amino acid backbone dihedral angles.

Symmetric and asymmetric triple excitation corrections for the orbitaloptimized coupledcluster doubles method: Improving upon CCSD(T) and CCSD(T)_{Λ}: Preliminary application
View Description Hide DescriptionSymmetric and asymmetric triple excitation corrections for the orbitaloptimized coupledcluster doubles (OOCCD or simply “OD” for short) method are investigated. The conventional symmetric and asymmetric perturbative triples corrections [(T) and (T)_{Λ}] are implemented, the latter one for the first time. Additionally, two new triples corrections, denoted as OD(Λ) and OD(Λ)_{ T }, are introduced. We applied the new methods to potential energy surfaces of the BH, HF, C_{2}, N_{2}, and CH_{4} molecules, and compare the errors in total energies, with respect to full configuration interaction, with those from the standard coupledcluster singles and doubles (CCSD), with perturbative triples [CCSD(T)], and asymmetric triples correction (CCSD(T)_{Λ}) methods. The CCSD(T) method fails badly at stretched geometries, the corresponding nonparallelity error is 7–281 kcal mol^{−1}, although it gives reliable results near equilibrium geometries. The new symmetric triples correction, CCSD(Λ), noticeably improves upon CCSD(T) (by 4–14 kcal mol^{−1}) for BH, HF, and CH_{4}; however, its performance is worse than CCSD(T) (by 1.6–4.2 kcal mol^{−1}) for C_{2} and N_{2}. The asymmetric triples corrections, CCSD(T)_{Λ} and CCSD(Λ)_{T}, perform remarkably better than CCSD(T) (by 5–18 kcal mol^{−1}) for the BH, HF, and CH_{4} molecules, while for C_{2} and N_{2} their results are similar to those of CCSD(T). Although the performance of CCSD and OD is similar, the situation is significantly different in the case of triples corrections, especially at stretched geometries. The OD(T) method improves upon CCSD(T) by 1–279 kcal mol^{−1}. The new symmetric triples correction, OD(Λ), enhances the OD(T) results (by 0.01–2.0 kcal mol^{−1}) for BH, HF, and CH_{4}; however, its performance is worse than OD(T) (by 1.9–2.3 kcal mol^{−1}) for C_{2} and N_{2}. The asymmetric triples corrections, OD(T)_{Λ} and OD(Λ)_{T}, perform better than OD(T) (by 2.0–6.2 kcal mol^{−1}). The latter method is slightly better for the BH, HF, and CH_{4} molecules. However, for C_{2} and N_{2} the new results are similar to those of OD(T). For the BH, HF, and CH_{4} molecules, OD(Λ)_{T} provides the best potential energy curves among the considered methods, while for C_{2} and N_{2} the OD(T) method prevails. Hence, for singlebond breaking the OD(Λ)_{T} method appears to be superior, whereas for multiplebond breaking the OD(T) method is better.

Quantum continuum mechanics made simple
View Description Hide DescriptionIn this paper we further explore and develop the quantum continuum mechanics (QCM) of Tao et al. [Phys. Rev. Lett.103, 086401 (2009)] with the aim of making it simpler to use in practice. Our simplifications relate to the noninteracting part of the QCM equations, and primarily refer to practical implementations in which the groundstate stress tensor is approximated by its KohnSham (KS) version. We use the simplified approach to directly prove the exactness of QCM for oneelectron systems via an orthonormal formulation. This proof sheds light on certain physical considerations contained in the QCM theory and their implication on QCMbased approximations. The oneelectron proof then motivates an approximation to the QCM (exact under certain conditions) expanded on the wavefunctions of the KS equations. Particular attention is paid to the relationships between transitions from occupied to unoccupied KS orbitals and their approximations under the QCM. We also demonstrate the simplified QCM semianalytically on an example system.

First principle nonlinear quantum dynamics using a correlationbased von Neumann entropy
View Description Hide DescriptionA new concept to describe the quantum dynamics in complex systems is suggested. It extends established schemes based on the DiracFrenkel variation principle, e.g., the multiconfigurational timedependent Hartree (MCTDH) approach. The concept is based on a correlationbased von Neumann entropy (CvNentropy) definition measuring the complexity of the wavefunction.Equations of motion are derived using a CvNentropy constraint in the variational principle and result in a generally applicable effective Hamiltonian. It consists of the standard Hamilton operator and an additional nonlinear operator which limits the complexity of the wavefunction. Effectively, this nonlinear operator absorbs complex structures which are emerging in the wavefunction and allows one to introduce nonnorm conserving equations of motion. Important aspects of the new concept are outlined studying the wave packet propagation on the diabatic B _{2} potential energy surfaces of NO_{2}. First, it is demonstrated that during standard wave packet propagation the CvNentropy increases strongly with time roughly independent of the coordinate systems employed. Second, one finds that employing CvNentropy constrained MCTDH propagation yields improved wave function accuracy on longer time scales while compromising on the short time accuracy. Third, the loss of the wavefunction's norm is directly related to the overlap with the exact wavefunction. This provides an error estimate available without knowing an exact reference.

Analysis of the HeydScuseriaErnzerhof density functional parameter space
View Description Hide DescriptionThe HeydScuseriaErnzerhof (HSE) density functionals are popular for their ability to improve upon the accuracy of standard semilocal functionals such as PerdewBurkeErnzerhof (PBE), particularly for semiconductorband gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small interelectron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this twoparameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: “sXPBE” is an approximation to the sXLDA screened exchange density functionals based on the local density approximation (LDA); “HSE12” minimizes the overall error over all tests performed; and “HSE12s” is a rangeminimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.

Approximate inclusion of fourmode couplings in vibrational coupledcluster theory
View Description Hide DescriptionThe vibrational coupled cluster (VCC) equations are analyzed in terms of vibrational MøllerPlesset perturbation theory aiming specifically at the importance of fourmode couplings. Based on this analysis, new VCC methods are derived for the calculation of anharmonic vibrational energies and vibrational spectra using vibrational coupled cluster response theory. It is shown how the effect of fourmode coupling and excitations can be efficiently and accurately described using approximations for their inclusion. Two closely related approaches are suggested. The computational scaling of the socalled VCC[3pt4F] method is not higher than the fifth power in the number of vibrational degrees of freedom when up to fourmode coupling terms are present in the Hamiltonian and only fourth order when only up to threemode couplings are present. With a further approximation, one obtains the VCC[3pt4] model which is shown to scale with at most the fourth power in the number of vibrational degrees of freedom for Hamiltonians with both three and fourmode coupling levels, while sharing the most important characteristics with VCC[3pt4F]. Sample calculations reported for selected tetraatomic molecules as well as the larger dioxirane and ethylene oxide molecules support that the new models are accurate and useful.

Theoretical study of the nuclear spinmolecular rotation coupling for relativistic electrons and nonrelativistic nuclei
View Description Hide DescriptionA theoretical study of the relation between the relativistic formulation of the nuclear magnetic shielding and spinrotation tensors is presented. To this end a theoretical expression of the relativistic spinrotation tensor is formulated, considering a molecular Hamiltonian of relativistic electrons and nonrelativistic nuclei. Molecular rotation effects are introduced considering the terms of the BornOppenheimer decomposition, which couple the electrons and nuclei dynamics. The loss of the simple relation linking both spectral parameters in the nonrelativistic formulation is further analyzed carrying out a perturbative expansion of relativistic effects by means of the linear response within the elimination of the small component approach. It is concluded that relativistic effects on the spinrotation tensor are less important than those of the nuclear magnetic shieldingtensor.

Accuracy of second order perturbation theory in the polaron and variational polaron frames
View Description Hide DescriptionIn the study of open quantum systems, the polarontransformation has recently attracted a renewed interest as it offers the possibility to explore the strong systembath coupling regime. Despite this interest, a clear and unambiguous analysis of the regimes of validity of the polarontransformation is still lacking. Here we provide such a benchmark, comparing second order perturbation theory results in the original untransformed frame, the polaron frame, and the variational extension with numerically exact path integral calculations of the equilibrium reduced density matrix. Equilibrium quantities allow a direct comparison of the three methods without invoking any further approximations as is usually required in deriving master equations. It is found that the second order results in the original frame are accurate for weak systembath coupling; the results deteriorate when the bath cutoff frequency decreases. The full polaron results are accurate for the entire range of coupling for a fast bath but only in the strong coupling regime for a slow bath. The variational method is capable of interpolating between these two methods and is valid over a much broader range of parameters.