Volume 123, Issue 6, 08 August 2005
 COMMUNICATIONS


Decay behavior of leastsquares coefficients in auxiliary basis expansions
View Description Hide DescriptionMany quantum chemical methods, both wave function and density based, rely on an expansion of elements of the electron density in an auxiliary basis. However, little is known about the analytical behavior of the expansion coefficients and, in particular, about their rate of decay with distance. We discuss an exactly solvable model system and characterize the expansion coefficients for various fitting metrics and various dimensionalities of the auxiliary basis. In the case of Coulomb fitting, we find that the decay rate depends critically on the effective dimensionality of the auxiliary basis, varying from to to for , 2, or 3.
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 SPECIAL TOPIC: RECENT DEVELOPMENTS IN DENSITY FUNCTIONAL THEORY: ORBITAL DEPENDENT FUNCTIONALS


Preface: Recent developments in density functional theory: Orbital dependent functionals
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Prescription for the design and selection of density functional approximations: More constraint satisfaction with fewer fits
View Description Hide DescriptionWe present the case for the nonempirical construction of density functional approximations for the exchangecorrelation energy by the traditional method of “constraint satisfaction” without fitting to data sets, and present evidence that this approach has been successful on the first three rungs of “Jacob’s ladder” of density functional approximations [local spindensity approximation (LSD), generalized gradient approximation (GGA), and metaGGA]. We expect that this approach will also prove successful on the fourth and fifth rungs (hyperGGA or hybrid and generalized randomphase approximation). In particular, we argue for the theoretical and practical importance of recovering the correct uniform density limit, which many semiempirical functionals fail to do. Among the beyondLSD functionals now available to users, we recommend the nonempirical Perdew–Burke–Ernzerhof (PBE) GGA and the nonempirical Tao–Perdew–Staroverov–Scuseria (TPSS) metaGGA, and their oneparameter hybrids with exact exchange. TPSS improvement over PBE is dramatic for atomization energies of molecules and surface energies of solids, and small or moderate for other properties. TPSS is now or soon will be available in standard codes such as GAUSSIAN, TURBOMOLE, NWCHEM, ADF, WIEN, VASP, etc. We also discuss old and new ideas to eliminate the selfinteraction error that plagues the functionals on the first three rungs of the ladder, bring up other related issues, and close with a list of “do’s and don’t’s” for software developers and users.

Away from generalized gradient approximation: Orbitaldependent exchangecorrelation functionals
View Description Hide DescriptionThe localdensity approximation of density functional theory(DFT) is remarkably accurate, for instance, for geometries and frequencies, and the generalized gradient approximations have also made bondenergies quite reliable. Sometimes, however, one meets with failure in individual cases. One of the possible routes towards better functionals would be the incorporation of orbital dependence (which is an implicit density dependency) in the functionals. We discuss this approach both for energies and for response properties. One possibility is the use of the Hartree–Focktype exchange energy expression as orbitaldependent functional. We will argue that in spite of the increasing popularity of this approach, it does not offer any advantage over Hartree–Fock for energies. We will advocate not to apply the separation of exchange and correlation, which is so ingrained in quantum chemistry, but to model both simultaneously. For response properties the energies and shapes of the virtual orbitals are crucial. We will discuss the benefits that Kohn–Sham potentials can offer which are derived from either an orbitaldependent energy functional, including the exactexchange functional, or which can be obtained directly as orbitaldependent functional. We highlight the similarity of the Hartree–Fock and Kohn–Sham occupied orbitals and orbital energies, and the essentially different meanings the virtual orbitals and orbital energies have in these two models. We will show that these differences are beneficial for DFT in the case of localized excitations (in a small molecule or in a fragment), but are detrimental for chargetransfer excitations. Again, orbital dependency, in this case in the exchangecorrelation kernel, offers a solution.

Orbital and statedependent functionals in densityfunctional theory
View Description Hide DescriptionShortcomings of present densityfunctional methods are considered. Kohn–Sham and timedependent densityfunctional methods using orbital and statedependent functionals for exchangecorrelation energies, potentials, and kernels are discussed as possible remedy for some of these shortcomings. A view on the Kohn–Sham formalism is presented which differs somewhat from the one conventionally taken. The crucial step of constructing local multiplicative exchangecorrelation potentials in Kohn–Sham methods based on orbital and statedependent functionals is discussed. The description of openshell systems via a symmetrized Kohn–Sham formalism employing statedependent exchangecorrelation functionals is elucidated. The generalized adiabatic connection Kohn–Sham approach for the selfconsistent treatment of excited states within a densityfunctional framework is considered. In the latter approach orbital and statedependent exchangecorrelation functionals occur in a densityfunctional framework which is no longer based on the Hohenberg–Kohn theorem but on a more general relation between electron densities and local multiplicative potentials.

Orbitaldependent correlation energy in densityfunctional theory based on a secondorder perturbation approach: Success and failure
View Description Hide DescriptionWe have developed a secondorder perturbation theory (PT) energy functional within densityfunctional theory(DFT). Based on PT with the Kohn–Sham (KS) determinant as a reference, this new ab initioexchangecorrelation functional includes an exact exchange (EXX) energy in the first order and a correlationenergy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlationenergy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully selfconsistent calculations for any orbitaldependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For twoelectron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both secondorder Møller–Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital and eigenvaluedependent algebraic structure of the correlation functional. The failure emerges in the selfconsistent approach but not in perturbative postEXX calculations, emphasizing the crucial importance of selfconsistency in testing new orbitaldependent energy functionals.

Ab initio density functional theory: The best of both worlds?
View Description Hide DescriptionDensity functional theory(DFT), in its current local, gradient corrected, and hybrid implementations and their extensions, is approaching an impasse. To continue to progress toward the quality of results demanded by today’s ab initio quantum chemistry encourages a new direction. We believe ab initioDFT is a promising route to pursue. Whereas conventional DFT cannot describe weak interactions, photoelectron spectra, or many potential energy surfaces,ab initioDFT, even in its initial, optimized effective potential, secondorder manybody perturbation theory form [OEP (2)semi canonical], is shown to do all well. In fact, we obtain accuracy that frequently exceeds MP2, being competitive with coupledcluster theory in some cases. Furthermore, this is accomplished within a relatively fast computational procedure that scales like iterative second order. We illustrate our results with several molecular examples including , and benzene.

Timedependent density functional theory: Past, present, and future
View Description Hide DescriptionTimedependent density functional theory (TDDFT) is presently enjoying enormous popularity in quantum chemistry, as a useful tool for extracting electronic excited state energies. This article discusses how TDDFT is much broader in scope, and yields predictions for many more properties. We discuss some of the challenges involved in making accurate predictions for these properties.

Recent development of selfinteractionfree timedependent densityfunctional theory for nonperturbative treatment of atomic and molecular multiphoton processes in intense laser fields
View Description Hide DescriptionIn this paper, we present a short account of some recent developments of selfinteractionfree densityfunctional theory(DFT) and timedependent densityfunctional theory (TDDFT) for accurate and efficient treatment of the electronic structure, and timedependent quantum dynamics of manyelectron atomic and molecular systems. The conventional DFT calculations using approximate and explicit exchangecorrelation energy functional contain spurious selfinteraction energy and improper longrange asymptotic potential, preventing reliable treatment of the excited, resonance, and continuum states. We survey some recent developments of DFT/TDDFT with optimized effective potential (OEP) and selfinteraction correction (SIC) for both atomic and molecular systems for overcoming some of the above mentioned difficulties. These DFT (TDDFT)/OEPSIC approaches allow the use of orbitalindependent singleparticle local potential which is selfinteraction free. In addition we discuss several numerical techniques recently developed for efficient and highprecision treatment of the selfinteractionfree DFT/TDDFT equations. The usefulness of these procedures is illustrated by a few case studies of atomic, molecular, and condensed matter processes of current interests, including (a) autoionizing resonances, (b) relativistic OEPSIC treatment of atomic structure , (c) shellfilling electronic structure in quantum dots, (d) atomic and molecular processes in intense laser fields, including multiphoton ionization, and veryhighorder harmonic generation, etc. For the timedependent processes, an alternative Floquet formulation of TDDFT is introduced for timeindependent treatment of multiphoton processes in intense periodic or quasiperiodic fields. We conclude this paper with some open questions and perspectives of TDDFT.
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 ARTICLES

 Theoretical Methods and Algorithms

Relativistic dynamics of halfspin particles in a homogeneous magnetic field: An atom with nucleus of spin
View Description Hide DescriptionAn investigation of the relativistic dynamics of spin particles placed in an external, homogeneous magnetic field is carried out. The system can represent an atom with a fermion nucleus and electrons. Quantum electrodynamical interactions, namely, projected Briet and magnetic interactions, are chosen to formulate the relativistic Hamiltonian. The quasifreeparticle picture is retained here. The total pseudomomentum is conserved, and its components are distinct when the total charge is zero. Therefore, the centerofmass motion can be separated from the Hamiltonian for a neutral fermion system, leaving behind a unitarily transformed, effective Hamiltonian at zero total pseudomomentum. The latter operator represents the complete relativistic dynamics in relative coordinates while interaction is chosen through order . Each oneparticle part in the effective Hamiltonian can be brought to a separable form for positive and negativeenergy states by replacing the odd operator in it through two successive unitary transformations, one due to Tsai [Phys. Rev. D7, 1945 (1973)] and the other due to Weaver [J. Math. Phys.18, 306 (1977)]. Consequently, the projector changes and the interaction that involves the concerned particle also becomes free from the corresponding odd operators. When this maneuver is applied only to the nucleus, and the nonHermitian part of the transformed interaction is removed by another unitary transformation, a familiar form of the atomic relativistic Hamiltonian emerges. This operator is equivalent to . A good Hamiltonian for relativistic quantum chemical calculations, , is obtained by expanding the nuclear part of the atomic Hamiltonian through order for positiveenergy states. The operator is obviously an approximation to . When the same technique is used for all particles, and subsequently the nonHermitian terms are removed by suitable unitary transformations, one obtains a Hamiltonian that is equivalent to but is in a completely separable form. As the semidiscrete eigenvalues and eigenfunctions of the oneparticle parts are known, the completely separable Hamiltonian can be used in computation. A little more effort leads to the derivation of the correct atomic Hamiltonian in the nonrelativistic limit, . The operator is an approximation to . It not only retains the relativistic and radiative effects, but also directly exhibits the phenomena of electron paramagnetic resonance and nuclear magnetic resonance.

Decoupling and recoupling using continuouswave irradiation in magicanglespinning solidstate NMR: A unified description using bimodal Floquet theory
View Description Hide DescriptionThe application of two or more different timedependent coherent perturbations with, in general, incommensurable frequencies occurs quite commonly in NMR experiments. Here we develop a unified description of the entire class of experiments using bimodal Floquet theory and van Vleck–Primas perturbation theory. This treatment leads to a timeindependent effective Hamiltonian in Hilbert space and can be looked at as a generalization of average Hamiltonian theory to several incommensurate time dependencies. As a prototype experiment we treat the application of continuouswave (cw) radiofrequency irradiation in combination with magicangle sample spinning. Practically relevant examples of this type of experiments are heteronuclear spin decoupling and recoupling experiments using cw irradiation, e.g., rotaryresonance recoupling. Perturbations up to the third order must be taken into account to explain all experimentally observed resonance conditions.

Electronic decoherence time for nonBornOppenheimer trajectories
View Description Hide DescriptionAn expression is obtained for the electronic decoherence time of the reduced density electronic matrix in mixed quantumclassical moleculardynamics simulations. The result is obtained by assuming that decoherence is dominated by the time dependence of the overlap of minimumuncertainty packets and then maximizing the rate with respect to the parameters of the wave packets. The expression for the decay time involves quantities readily available in nonBornOppenheimer moleculardynamics simulations, and it is shown to have a reasonable form when compared with two other formulas for the decay time that have been previously proposed.

Investigation of isotope effects with the nuclearelectronic orbital approach
View Description Hide DescriptionThis paper addresses fundamental issues that arise in the application of the nuclearelectronic orbital (NEO) approach to systems with equivalent quantum nuclei. Our analysis illustrates that HartreeFock nuclear wave functions do not provide physically reasonable descriptions of systems comprised of equivalent lowspin fermions or equivalent bosons. The physical basis for this breakdown is that the ionic terms dominate due to the localized nature of the nuclear orbitals. Multiconfigurational wave functions that include only covalent terms provide physically reasonable descriptions of these types of systems. The application of the NEO approach to a variety of chemical systems is presented to elucidate the isotope effects on the geometries and electronic wave functions. Deuteration of hydrogen halides, water, ammonia, and hydronium ion decreases the bond length and the magnitude of negative partial atomic charge on the heavy atom. These results are consistent with experimental spectroscopic data. Deuteration at the beta position for formate anion and a series of amines increases the magnitude of negative partial atomic charge on the protonation site for the unprotonated species. This observation is consistent with the experimentally observed increase in basicity upon deuteration at the beta position for carboxylic acids and amines.

On the nature of the MøllerPlesset critical point
View Description Hide DescriptionIt has been suggested [F. H. Stillinger, J. Chem. Phys.112, 9711 (2000)] that the convergence or divergence of MøllerPlesset perturbation theory is determined by a critical point at a negative value of the perturbation parameter at which an electron cluster dissociates from the nuclei. This conjecture is examined using configurationinteraction computations as a function of and using a quadratic approximant analysis of the highorder perturbation series. Results are presented for the He, Ne, and Ar atoms and the hydrogen fluoride molecule. The original theoretical analysis used the true Hamiltonian without the approximation of a finite basis set. In practice, the singularity structure depends strongly on the choice of basis set. Standard basis sets cannot model dissociation to an electron cluster, but if the basis includes diffuse functions then it can model another critical point corresponding to complete dissociation of all the valence electrons. This point is farther from the origin of the plane than is the critical point for the electron cluster, but it is still close enough to cause divergence of the perturbation series. For the hydrogen fluoride molecule a critical point is present even without diffuse functions. The basis functions centered on the H atom are far enough from the F atom to model the escape of electrons away from the fluorine end of the molecule. For the Ar atom a critical point for a oneelectron ionization, which was not previously predicted, seems to be present at a positive value of the perturbation parameter. Implications of the existence of critical points for quantumchemical applications are discussed.

Multidimensional timedependent discrete variable representations in multiconfiguration Hartree calculations
View Description Hide DescriptionIn the multiconfiguration timedependent Hartree (MCTDH) approach, the wave function is expanded in timedependent basis functions, called singleparticle functions, to increase the efficiency of the wavepacket propagation. The correlation discrete variable representation (CDVR) approach, which is based on a timedependent discrete variable representation (DVR), can be employed to evaluate matrix elements of the potential energy. The efficiency of the MCTDH method can be further enhanced by using multidimensional singleparticle functions. However, up to now the CDVR approach could not be used in MCTDH calculations employing multidimensional singleparticle functions, since this would require a general multidimensional nondirectproduct DVR scheme. Recently, Dawes and Carrington presented a practical scheme to implement general nondirectproduct multidimensional DVRs [R. Dawes and T. Carrington, Jr., J. Chem. Phys.121, 726 (2004)]. The present work utilizes their scheme in the MCTDH/CDVR approach. The accuracy is tested using the photodissociation of NOCl as example. The results show that the CDVR scheme based on multidimensional timedependent DVRs allows for an accurate evaluation of the potential in MCTDH calculations with multidimensional singleparticle functions.

Systematically convergent basis sets for transition metals. I. Allelectron correlation consistent basis sets for the elements Sc–Zn
View Description Hide DescriptionSequences of basis sets that systematically converge towards the complete basis set (CBS) limit have been developed for the firstrow transition metal elements Sc–Zn. Two families of basis sets, nonrelativistic and DouglasKrollHess (DK) relativistic, are presented that range in quality from triple to quintuple. Separate sets are developed for the description of valence electron correlation ( and ; , 5) and valence plus outercore correlation ( and ; , 5), as well as these sets augmented by additional diffuse functions for the description of negative ions and weak interactions ( and ). Extensive benchmark calculations at the coupled cluster level of theory are presented for atomic excitation energies, ionization potentials, and electron affinities, as well as molecular calculations on selected hydrides (TiH, MnH, CuH) and other diatomics (TiF, ). In addition to observing systematic convergence towards the CBS limits, both electron correlation and scalar relativity are calculated to strongly impact many of the atomic and molecular properties investigated for these firstrow transition metal species.

A local correlation model that yields intrinsically smooth potentialenergy surfaces
View Description Hide DescriptionWe demonstrate an algorithm for computing local coupledcluster doubles (LCCD) energies that form rigorously smooth potentialenergysurfaces and which should be fast enough for application to large systems in the future. Like previous LCCD algorithms, our method solves iteratively for only a limited number of correlation amplitudes, treating the remaining amplitudes with secondorder perturbation theory. However, by employing bump functions, our method smoothes the transition from iteratively solved amplitude to perturbationtreated amplitude, invoking the implicit function theorem to prove that our LCCD energy is an infinitely differentiable function of nuclear coordinates. We make no explicit amplitude domains nor do we rely on the existence of atomcentered, redundant orbitals in order to get smooth potentialenergy curves. In fact, our algorithm employs only localized orthonormal occupied and virtual orbitals. Our approach should be applicable to many other electron correlation methods.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Collisional and photoinitiated reaction dynamics in the ground electronic state of Ca–HCl
View Description Hide Descriptionreactive collisions were studied for different rovibrational states of the HCl reactant using wavepacket calculations in reactant Jacobi coordinates. A recently proposed potentialenergy surface was used with a barrier of followed by a deep well. The possibility of an insertion mechanism due to this last well has been analyzed and it was found that once the wave packet passes over the barrier most of it goes directly to products, which shows that the reaction dynamics is essentially direct. It was also found that there is no significant change in the reaction efficiency as a function of the initial HCl rovibrational state, because CaHCl at the barrier has an only little elongated HCl bond. Near the threshold for reaction with , however, the reaction shows significant steric effects for . In a complementary study, the infrared excitation from the Ca–HCl van der Waals well was simulated. The spectrum thus obtained shows several series of resonances which correspond to quasibound states correlating to excited vibrations. The Ca–HCl binding energies of these quasibound states increase dramatically with , from , because the wave function spreads increasingly over larger HCl bond lengths. Thus it explores the region of the barrier saddle point and the deep insertion well. Although also the chargetransfer contribution increases with , the reaction probability for resonances of the manifold, which are well above the reaction threshold, is still negligible. This explains the relatively long lifetimes of these resonances. The reaction probability becomes significant at . Our simulations have shown that an experimental study of this type will allow a gradual spectroscopic probing of the barrier for the reaction.

FranckCondon analysis of the absorption and fluorescence spectra of all trans ,diphenylpolyenes with one to seven polyene double bonds
View Description Hide DescriptionFluorescence and absorption spectra have been measured for alltrans ,diphenylpolyenes with one to seven polyene double bonds in roomtemperature solution, along with the fluorescence spectra of those with one to six polyene double bonds in alkane matrices at 77 K. All the spectral data were fitted by sums of Gaussians to treat the FranckCondon envelopes of the measured spectra quantitatively. The FranckCondon analyses of the spectra in the harmonic limit revealed that the displacements of the CC and C–C stretching vibrational modes in the 2 Ag state relative to those in the ground state,, increase, while those in the state show a slight decrease with the increase of the polyene chain length. It is also shown that the bandwidths of the absorption and fluorescence spectra exhibit a monotonic decrease with the increase of the chain length.

The weakly bound He–HCCCN complex: Highresolution microwave spectra and intermolecular potentialenergy surface
View Description Hide DescriptionRotational spectra of the weakly bound He–HCCCN and He–DCCCN van der Waals complexes were observed using a pulsednozzle Fouriertransform microwave spectrometer in the 7–26GHz frequency region. Nuclear quadrupole hyperfine structures due to the and D nuclei (both with nuclearspin quantum number ) were resolved and assigned. Both strong and weaker type transitions were observed and the assigned transitions were used to fit the parameters of a distortable asymmetric rotor model. The dimers are floppy, near Tshaped complexes. Three intermolecular potentialenergysurfaces were calculated using the coupledcluster method with single and double excitations and noniterative inclusion of triple excitations. Boundstate rotational energy levels supported by these surfaces were determined. The quality of the potentialenergysurfaces was assessed by comparing the experimental and calculated transition frequencies and also the corresponding spectroscopic parameters. Simple scaling of the surfaces improved both the transition frequencies and spectroscopic constants. Five other recently reported surfaces [O. AkinOjo, R. Bukowski, and K. Szalewicz, J. Chem. Phys.119, 8379 (2003)], calculated using a variety of methods, and their agreement with spectroscopic properties of He–HCCCN are discussed.