Volume 142, Issue 24, 28 June 2015
Index of content:

Many people in the materials science and solidstate community are familiar with the acronym “DFT+U.” For those less familiar, this technique uses ideas from model Hamiltonians that permit the description of both metals and insulators to address problems of electron overdelocalization in practical implementations of density functional theory (DFT). Exchangecorrelation functionals in DFT are often described as belonging to a hierarchical “Jacob’s ladder” of increasing accuracy in moving from local to nonlocal descriptions of exchange and correlation. DFT+U is not on this “ladder” but rather acts as an “elevator” because it systematically tunes relative energetics, typically on a localized subshell (e.g., d or f electrons), regardless of the underlying functional employed. However, this tuning is based on a metric of the local electron density of the subshells being addressed, thus necessitating physical or chemical or intuition about the system of interest. I will provide a brief overview of the history of how DFT+U came to be starting from the origin of the Hubbard and Anderson model Hamiltonians. This history lesson is necessary because it permits us to make the connections between the “Hubbard U” and fundamental outstanding challenges in electronic structure theory, and it helps to explain why this method is so widely applied to transitionmetal oxides and organometallic complexes alike.
 PERSPECTIVES


Perspective: Treating electron overdelocalization with the DFT+U method
View Description Hide DescriptionMany people in the materials science and solidstate community are familiar with the acronym “DFT+U.” For those less familiar, this technique uses ideas from model Hamiltonians that permit the description of both metals and insulators to address problems of electron overdelocalization in practical implementations of density functional theory (DFT). Exchangecorrelation functionals in DFT are often described as belonging to a hierarchical “Jacob’s ladder” of increasing accuracy in moving from local to nonlocal descriptions of exchange and correlation. DFT+U is not on this “ladder” but rather acts as an “elevator” because it systematically tunes relative energetics, typically on a localized subshell (e.g., d or f electrons), regardless of the underlying functional employed. However, this tuning is based on a metric of the local electron density of the subshells being addressed, thus necessitating physical or chemical or intuition about the system of interest. I will provide a brief overview of the history of how DFT+U came to be starting from the origin of the Hubbard and Anderson model Hamiltonians. This history lesson is necessary because it permits us to make the connections between the “Hubbard U” and fundamental outstanding challenges in electronic structure theory, and it helps to explain why this method is so widely applied to transitionmetal oxides and organometallic complexes alike.

 COMMUNICATIONS


Communication: Statetostate dynamics of the Cl + H2O → HCl + OH reaction: Energy flow into reaction coordinate and transitionstate control of product energy disposal
View Description Hide DescriptionQuantum statetostate dynamics of a prototypical fouratom reaction, namely, Cl + H2O → HCl + OH, is investigated for the first time in full dimensionality using a transitionstate wave packet method. The statetostate reactivity and its dependence on the reactant internal excitations are analyzed and found to share many similarities both energetically and dynamically with the H + H2O → H2 + OH reaction. The strong enhancement of reactivity by the H2O stretching vibrational excitations in both reactions is attributed to the favorable energy flow into the reaction coordinate near the transition state. On the other hand, the insensitivity of the product state distributions with regard to reactant internal excitation stems apparently from the transitionstate control of product energy disposal.

Communication: Rotational excitation of HCl by H: Rigid rotor vs. reactive approaches
View Description Hide DescriptionWe report fully quantum timeindependent calculations of cross sections for the collisional excitation of HCl by H, an astrophysically relevant process. Our calculations are based on the BianWerner ClH2 potential energy surface and include the possibility of HCl destruction through reactive collisions. The strongest collisioninduced rotational HCl transitions are those with Δj = 1, and the magnitude of the HClH inelastic cross sections is of the same order of magnitude as the HClH2 ones. Results of exact calculations, i.e., including the reactive channels, are compared to pure inelastic calculations based on the rigid rotor approximation. A very good agreement is found between the two approaches over the whole energy range 10–3000 cm^{−1}. At the highest collisional energies, where the reaction takes place, the rigid rotor approach slightly overestimates the cross sections, as expected. Hence, the rigid rotor approach is found to be reliable at interstellar temperatures.

Communication: Direct tests of singleparameter aging
View Description Hide DescriptionThis paper presents accurate data for the physical aging of organic glasses just below the glass transition probed by monitoring the following quantities after temperature up and down jumps: the shearmechanical resonance frequency (∼360 kHz), the dielectric loss at 1 Hz, the real part of the dielectric constant at 10 kHz, and the losspeak frequency of the dielectric beta process (∼10 kHz). The setup used allows for keeping temperature constant within 100 μK and for thermal equilibration within a few seconds after a temperature jump. The data conform to a new simplified version of the classical ToolNarayanaswamy aging formalism, which makes it possible to calculate one relaxation curve directly from another without any fitting to analytical functions.

 ARTICLES

 Theoretical Methods and Algorithms

A systematic benchmark of the ab initio BetheSalpeter equation approach for lowlying optical excitations of small organic molecules
View Description Hide DescriptionThe predictive power of the ab initio BetheSalpeter equation (BSE) approach, rigorously based on manybody Green’s function theory but incorporating information from density functional theory, has already been demonstrated for the optical gaps and spectra of solidstate systems. Interest in photoactive hybrid organic/inorganic systems has recently increased and so has the use of the BSE for computing neutral excitations of organic molecules. However, no systematic benchmarks of the BSE for neutral electronic excitations of organic molecules exist. Here, we study the performance of the BSE for the 28 small molecules in Thiel’s widely used timedependent density functional theory benchmark set [Schreiber et al., J. Chem. Phys. 128, 134110 (2008)]. We observe that the BSE produces results that depend critically on the meanfield starting point employed in the perturbative approach. We find that this starting point dependence is mainly introduced through the quasiparticle energies obtained at the intermediate GW step and that with a judicious choice of starting meanfield, singlet excitation energies obtained from BSE are in excellent quantitative agreement with higherlevel wavefunction methods. The quality of the triplet excitations is slightly less satisfactory.

A generalized Derjaguin approximation for electricaldoublelayer interactions at arbitrary separations
View Description Hide DescriptionDerjaguin’s approximation provides the electricaldoublelayer interaction force between two arbitrary convex surfaces as the product of the corresponding onedimensional parallelplate interaction potential and an effective radius R (function of the radii of curvature and relative orientation of the two surfaces at minimum separation). The approximation holds when both the Debye length 1/κ and minimum separation h are small compared to R. We show here that a simple transformation, yields an approximation uniformly valid for arbitrary separations h; here, Ki is the Gaussian curvature of particle i at minimum separation, and [ ⋅ ] is an operator which adds h/2 to all radii of curvature present in the expression on which it acts. We derive this result in two steps. First, we extend the twodimensional raytheory analysis of Schnitzer [Phys. Rev. E 91, 022307 (2015)], valid for κh, κR ≫ 1, to three dimensions. We thereby obtain a general closed form expression for the force by matching nonlinear diffusecharge boundary layers with a WKBJtype expansion describing the bulk potential, and subsequent integration via Laplace’s method of the traction over the medial surface generated by all spheres maximally inscribed between the two surfaces. Second, we exploit the existence of an overlap domain, 1 ≪ κh ≪ κR, where both the raytheory and the Derjaguin approximations hold, to systematically form the generalized mapping. The validity of the result is demonstrated by comparison with numerical computations.

Transition matrices and orbitals from reduced density matrix theory
View Description Hide DescriptionIn this contribution, we report two different methodologies for characterizing the electronic structure reorganization occurring when a chromophore undergoes an electronic transition. For the first method, we start by setting the theoretical background necessary to the reinterpretation through simple tensor analysis of (i) the transition density matrix and (ii) the natural transition orbitals in the scope of reduced density matrix theory. This novel interpretation is made more clear thanks to a short compendium of the oneparticle reduced density matrix theory in a Fock space. The formalism is further applied to two different classes of excited states calculation methods, both requiring a singledeterminant reference, that express an excited state as a holeparticle monoexcited configurations expansion, to which particlehole correlation is coupled (timedependent HartreeFock/timedependent density functional theory) or not (configuration interaction single/TammDancoff approximation). For the second methodology presented in this paper, we introduce a novel and complementary concept related to electronic transitions with the canonical transition density matrix and the canonical transition orbitals. Their expression actually reflects the electronic cloud polarisation in the orbital space with a decomposition based on the actual contribution of oneparticle excitations from occupied canonical orbitals to virtual ones. This approach validates our novel interpretation of the transition density matrix elements in terms of the Euclidean norm of elementary transition vectors in a linear tensor space. A proper use of these new concepts leads to the conclusion that despite the different principles underlying their construction, they provide two equivalent excited states topological analyses. This connexion is evidenced through simple illustrations of (in)organic dyes electronic transitions analysis.

Selection of active spaces for multiconfigurational wavefunctions
View Description Hide DescriptionThe efficient and accurate description of the electronic structure of strongly correlated systems is still a largely unsolved problem. The usual procedures start with a multiconfigurational (usually a Complete Active Space, CAS) wavefunction which accounts for static correlation and add dynamical correlation by perturbation theory, configuration interaction, or coupled cluster expansion. This procedure requires the correct selection of the active space. Intuitive methods are unreliable for complex systems. The inexpensive blackbox unrestricted natural orbital (UNO) criterion postulates that the Unrestricted HartreeFock (UHF) charge natural orbitals with fractional occupancy (e.g., between 0.02 and 1.98) constitute the active space. UNOs generally approximate the CAS orbitals so well that the orbital optimization in CAS SelfConsistent Field (CASSCF) may be omitted, resulting in the inexpensive UNOCAS method. A rigorous testing of the UNO criterion requires comparison with approximate full configuration interaction wavefunctions. This became feasible with the advent of Density Matrix Renormalization Group (DMRG) methods which can approximate highly correlated wavefunctions at affordable cost. We have compared active orbital occupancies in UNOCAS and CASSCF calculations with DMRG in a number of strongly correlated molecules: compounds of electronegative atoms (F2, ozone, and NO2), polyenes, aromatic molecules (naphthalene, azulene, anthracene, and nitrobenzene), radicals (phenoxy and benzyl), diradicals (o, m, and pbenzyne), and transition metal compounds (nickelacetylene and Cr2). The UNO criterion works well in these cases. Other symmetry breaking solutions, with the possible exception of spatial symmetry, do not appear to be essential to generate the correct active space. In the case of multiple UHF solutions, the natural orbitals of the average UHF density should be used. The problems of the UNO criterion and their potential solutions are discussed: finding the UHF solutions, discontinuities on potential energy surfaces, and inclusion of dynamical electron correlation and generalization to excited states.

Photoelectron circular dichroism in the multiphoton ionization by short laser pulses. I. Propagation of singleactiveelectron wave packets in chiral pseudopotentials
View Description Hide DescriptionA theoretical method to study the angleresolved multiphoton ionization of polyatomic molecules is developed. It is based on the timedependent formulation of the Single Center (TDSC) method and consists in the propagation of singleactiveelectron wave packets in the effective molecular potentials in the presence of intense laser pulses. For this purpose, the timedependent Schrödinger equation for one electron, moving in a molecular field and interacting with an arbitrary laser pulse, is solved in spherical coordinates by an efficient numerical approach. As a test, the method is applied to the one and twophoton ionizations of a model methanelike chiral system by circularly polarized short intense highfrequency laser pulses. Thereby, we analyze the photoelectron circular dichroism (PECD) in the momentum distribution. The considered model application illustrates the capability of the TDSC method to study multiphoton PECD in fixedinspace and randomly oriented chiral molecules.

On the rejectionbased algorithm for simulation and analysis of largescale reaction networks
View Description Hide DescriptionStochastic simulation for in silico studies of large biochemical networks requires a great amount of computational time. We recently proposed a new exact simulation algorithm, called the rejectionbased stochastic simulation algorithm (RSSA) [Thanh et al., J. Chem. Phys. 141(13), 134116 (2014)], to improve simulation performance by postponing and collapsing as much as possible the propensity updates. In this paper, we analyze the performance of this algorithm in detail, and improve it for simulating largescale biochemical reaction networks. We also present a new algorithm, called simultaneous RSSA (SRSSA), which generates many independent trajectories simultaneously for the analysis of the biochemical behavior. SRSSA improves simulation performance by utilizing a single data structure across simulations to select reaction firings and forming trajectories. The memory requirement for building and storing the data structure is thus independent of the number of trajectories. The updating of the data structure when needed is performed collectively in a single operation across the simulations. The trajectories generated by SRSSA are exact and independent of each other by exploiting the rejectionbased mechanism. We test our new improvement on real biological systems with a wide range of reaction networks to demonstrate its applicability and efficiency.

Linear transformation of anharmonic molecular force constants between normal and Cartesian coordinates
View Description Hide DescriptionA full derivation of the analytic transformation of the quadratic, cubic, and quartic force constants from normal coordinates to Cartesian coordinates is given. Previous attempts at this transformation have resulted in nonlinear transformations; however, for the first time, a simple linear transformation is presented here. Two different approaches have been formulated and implemented, one of which does not require prior knowledge of the translationrotation eigenvectors from diagonalization of the Hessian matrix. The validity of this method is tested using two molecules H2O and cC3H2D^{+}.

Selfconsistent KohnSham method based on the adiabaticconnection fluctuationdissipation theorem and the exactexchange kernel
View Description Hide DescriptionA selfconsistent KohnSham method based on the adiabaticconnection fluctuationdissipation (ACFD) theorem, employing the frequencydependent exact exchange kernel f x is presented. The resulting SCexactexchangeonly (EXX)ACFD method leads to even more accurate correlation potentials than those obtained within the direct random phase approximation (dRPA). In contrast to dRPA methods, not only the Coulomb kernel but also the exact exchange kernel f x is taken into account in the EXXACFD correlation which results in a method that, unlike dRPA methods, is free of selfcorrelations, i.e., a method that treats exactly all oneelectron systems, like, e.g., the hydrogen atom. The selfconsistent evaluation of EXXACFD total energies improves the accuracy compared to EXXACFD total energies evaluated nonselfconsistently with EXX or dRPA orbitals and eigenvalues. Reaction energies of a set of small molecules, for which highly accurate experimental reference data are available, are calculated and compared to quantum chemistry methods like MøllerPlesset perturbation theory of second order (MP2) or coupled cluster methods [CCSD, coupled cluster singles, doubles, and perturbative triples (CCSD(T))]. Moreover, we compare our methods to other ACFD variants like dRPA combined with perturbative corrections such as the second order screened exchange corrections or a renormalized singles correction. Similarly, the performance of our EXXACFD methods is investigated for the noncovalently bonded dimers of the S22 reference set and for potential energy curves of noble gas, water, and benzene dimers. The computational effort of the SCEXXACFD method exhibits the same scaling of N ^{5} with respect to the system size N as the nonselfconsistent evaluation of only the EXXACFD correlation energy; however, the prefactor increases significantly. Reaction energies from the SCEXXACFD method deviate quite little from EXXACFD energies obtained nonselfconsistently with dRPA orbitals and eigenvalues, and the deviation reduces even further if the Coulomb kernel is scaled by a factor of 0.75 in the dRPA to reduce selfcorrelations in the dRPA correlation potential. For larger systems, such a nonselfconsistent EXXACFD method is a competitive alternative to highlevel wavefunctionbased methods, yielding higher accuracy than MP2 and CCSD methods while exhibiting a better scaling of the computational effort than CCSD or CCSD(T) methods. Moreover, EXXACFD methods were shown to be applicable in situation characterized by static correlation.

The multiconfigurational timedependent Hartree approach revisited
View Description Hide DescriptionThe multiconfigurational timedependent Hartree (MCTDH) approach facilitates accurate highdimensional quantum dynamics simulations. In the approach, the wavefunction is expanded in a direct product of selfadapting timedependent singleparticle functions (SPFs). The equations of motion for the expansion coefficients and the SPFs are obtained via the DiracFrenkel variational principle. While this derivation yields welldefined differential equations for the motion of occupied SPFs, singularities in the working equations resulting from unoccupied SPFs have to be removed by a regularization procedure. Here, an alternative derivation of the MCTDH equations of motion is presented. It employs an analysis of the timedependence of the singleparticle density matrices up to second order. While the analysis of the first order terms yields the known equations of motion for the occupied SPFs, the analysis of the second order terms provides new equations which allow one to identify optimal choices for the unoccupied SPFs. The effect of the optimal choice of the unoccupied SPFs on the structure of the MCTDH equations of motion and their regularization is discussed. Generalized equations applicable in the multilayer MCTDH framework are presented. Finally, the effects resulting from the initial choice of the unoccupied SPFs are illustrated by a simple numerical example.

Parametrizing linear generalized Langevin dynamics from explicit molecular dynamics simulations
View Description Hide DescriptionFundamental understanding of complex dynamics in manyparticle systems on the atomistic level is of utmost importance. Often the systems of interest are of macroscopic size but can be partitioned into a few important degrees of freedom which are treated most accurately and others which constitute a thermal bath. Particular attention in this respect attracts the linear generalized Langevin equation, which can be rigorously derived by means of a linear projection technique. Within this framework, a complicated interaction with the bath can be reduced to a single memory kernel. This memory kernel in turn is parametrized for a particular system studied, usually by means of timedomain methods based on explicit molecular dynamics data. Here, we discuss that this task is more naturally achieved in frequency domain and develop a Fourierbased parametrization method that outperforms its timedomain analogues. Very surprisingly, the widely used rigid bond method turns out to be inappropriate in general. Importantly, we show that the rigid bond approach leads to a systematic overestimation of relaxation times, unless the system under study consists of a harmonic bath bilinearly coupled to the relevant degrees of freedom.

Beyond the electricdipole approximation: A formulation and implementation of molecular response theory for the description of absorption of electromagnetic field radiation
View Description Hide DescriptionWe present a formulation of molecular response theory for the description of a quantum mechanical molecular system in the presence of a weak, monochromatic, linearly polarized electromagnetic field without introducing truncated multipolar expansions. The presentation focuses on a description of linear absorption by adopting the energyloss approach in combination with the complex polarization propagator formulation of response theory. Going beyond the electricdipole approximation is essential whenever studying electricdipoleforbidden transitions, and in general, nondipolar effects become increasingly important when addressing spectroscopies involving higherenergy photons. These two aspects are examined by our study of the near Kedge Xray absorption fine structure of the alkaline earth metals (Mg, Ca, Sr, Ba, and Ra) as well as the transpolyenes. In following the series of alkaline earth metals, the sizes of nondipolar effects are probed with respect to increasing photon energies and a detailed assessment of results is made in terms of studying the pertinent transition electron densities and in particular their spatial extension in comparison with the photon wavelength. Along the series of transpolyenes, the sizes of nondipolar effects are probed for Xray spectroscopies on organic molecules with respect to the spatial extension of the chromophore.

A new class of ensemble conserving algorithms for approximate quantum dynamics: Theoretical formulation and model problems
View Description Hide DescriptionWe develop two classes of quasiclassical dynamics that are shown to conserve the initial quantum ensemble when used in combination with the FeynmanKleinert approximation of the density operator. These dynamics are used to improve the FeynmanKleinert implementation of the classical Wigner approximation for the evaluation of quantum time correlation functions known as FeynmanKleinert linearized pathintegral. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics is made with the exact results for the challenging model problems of the quartic and doublewell potentials. It is found that these dynamics provide a great improvement over the classical Wigner approximation, in which purely classical dynamics are used. In a special case, our first method becomes identical to centroid molecular dynamics.

Application of a new ensemble conserving quantum dynamics simulation algorithm to liquid parahydrogen and orthodeuterium
View Description Hide DescriptionWe apply the FeynmanKleinert QuasiClassical Wigner (FKQCW) method developed in our previous work [Smith et al., J. Chem. Phys. 142, 244112 (2015)] for the determination of the dynamic structure factor of liquid parahydrogen and orthodeuterium at state points of (T = 20.0 K, n = 21.24 nm^{−3}) and (T = 23.0 K, n = 24.61 nm^{−3}), respectively. When applied to this challenging system, it is shown that this new FKQCW method consistently reproduces the experimental dynamic structure factor reported by Smith et al. [J. Chem. Phys. 140, 034501 (2014)] for all momentum transfers considered. This shows that FKQCW provides a substantial improvement over the FeynmanKleinert linearized pathintegral method, in which purely classical dynamics are used. Furthermore, for small momentum transfers, it is shown that FKQCW provides nearly the same results as ringpolymer molecular dynamics (RPMD), thus suggesting that FKQCW provides a potentially more appealing algorithm than RPMD since it is not formally limited to correlation functions involving linear operators.

Computing intramolecular charge and energy transfer rates using optimal modes
View Description Hide DescriptionIn our recent work [X. Yang and E. R. Bittner, J. Phys. Chem. A 118, 5196 (2014)], we showed how to construct a reduced set of nuclear motions that capture the coupling between electronic and nuclear degrees of freedom over the course of an electronic transition. We construct these modes, referred to as “Lanczos modes,” by applying a search algorithm to find linear combinations of vibrational normal modes that optimize the electronic/nuclear coupling operator. Here, we analyze the irreducible representations of the dominant contributions of these modes and find that for the cases considered here, these belong to totally symmetric irreducible representations of the donor and acceptor moieties. Upon investigating the molecular geometry changes following the transition, we propose that the electronic transition process can be broken into two steps, in the agreement of BornOppenheimer approximation: a fast excitation transfer occurs, facilitated by the “primary Lanczos mode,” followed by slow nuclear relaxation on the final electronic diabatic surface.

Variance decomposition in stochastic simulators
View Description Hide DescriptionThis work aims at the development of a mathematical and computational approach that enables quantification of the inherent sources of stochasticity and of the corresponding sensitivities in stochastic simulations of chemical reaction networks. The approach is based on reformulating the system dynamics as being generated by independent standardized Poisson processes. This reformulation affords a straightforward identification of individual realizations for the stochastic dynamics of each reaction channel, and consequently a quantitative characterization of the inherent sources of stochasticity in the system. By relying on the SobolHoeffding decomposition, the reformulation enables us to perform an orthogonal decomposition of the solution variance. Thus, by judiciously exploiting the inherent stochasticity of the system, one is able to quantify the variancebased sensitivities associated with individual reaction channels, as well as the importance of channel interactions. Implementation of the algorithms is illustrated in light of simulations of simplified systems, including the birthdeath, Schlögl, and MichaelisMenten models.

An EQTcDFT approach to determine thermodynamic properties of confined fluids
View Description Hide DescriptionWe present a continuumbased approach to predict the structure and thermodynamic properties of confined fluids at multiple lengthscales, ranging from a few angstroms to macrometers. The continuum approach is based on the empirical potentialbased quasicontinuum theory (EQT) and classical density functional theory (cDFT). EQT is a simple and fast approach to predict inhomogeneous density and potential profiles of confined fluids. We use EQT potentials to construct a grand potential functional for cDFT. The EQTcDFTbased grand potential can be used to predict various thermodynamic properties of confined fluids. In this work, we demonstrate the EQTcDFT approach by simulating LennardJones fluids, namely, methane and argon, confined inside slitlike channels of graphene. We show that the EQTcDFT can accurately predict the structure and thermodynamic properties, such as density profiles, adsorption, local pressure tensor, surface tension, and solvation force, of confined fluids as compared to the molecular dynamics simulation results.