Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond

Density functional theory (DFT) results are mistrusted at times due to the presence of an unknown exchange correlation functional, with no practical way to guarantee convergence to the right answer. The use of a known exchange correlation functional based on wave-function theory helps to alleviate such mistrust. The exchange correlation functionals can be written exactly in terms of the density-density response function using the adiabatic-connection and fluctuation-dissipation framework. The random phase approximation (RPA) is the simplest approximation for the density-density response function. Since the correlation functional obtained from RPA is equivalent to the direct ring coupled cluster doubles (ring-CCD) correlation functional, meaning only Coulomb interactions are included, one can bracket RPA between many body perturbation theory (MBPT)-2 and CCD with the latter having all ring, ladder, and exchange contributions. Using an optimized effective potential strategy, we obtain correlation potentials corresponding to MBPT-2, RPA (ring-CCD), linear-CCD, and CCD. Using the suitable choice of the unperturbed Hamiltonian, Kohn-Sham self-consistent calculations are performed. The spatial behavior of the resulting potentials, total energies, and the HOMO eigenvalues are compared with the exact values for spherical atoms. Further, we demonstrate that the self-consistent eigenvalues obtained from these consistent potentials used in ab initio dft approximate all principal ionization potentials as demanded by ionization potential theorem.     

The exchange-correlation potential in ab initio density functional theory

From coupled-cluster theory and many-body perturbation theory we derive the local exchange-correlation potential of density functional theory in an orbital dependent form. We show the relationship between the coupled-cluster approach and density functional theory, and connections and comparisons with our previous second-order correlation potential [OEP-MBPT(2) (OEP-optimized effective potential)] [I. Grabowski, S. Hirata, S. Ivanov, and R. J. Bartlett, J. Chem. Phys. 116, 4415 (2002)]. Starting from a general theoretical framework based on the density condition in Kohn-Sham theory, we define a rigorous exchange-correlation functional, potential and orbitals. Specifying initially to second-order terms, we show that our ab initio correlation potential provides the correct shape compared to those from reference quantum Monte Carlo calculations, and we demonstrate the superiority of using Fock matrix elements or more general infinite-order semicanonical transformations. This enables us to introduce a method that is guaranteed to converge to the right answer in the correlation and basis set limit, just as does ab initio wave function theory. We also demonstrate that the energies obtained from this generalized second-order method [OEP-MBPT2-f] and [OEP-MBPT2-sc] are often of coupled-cluster accuracy and substantially better than ordinary Hartree-Fock based second-order MBPT=MP2.     

Comparing ab initio density-functional and wave function theories: the impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order M?ller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.     

Comparative study of DFT methods applied to small titanium/oxygen compounds

The performance of a number of different local and nonlocal density functional theory (DFT) methods has been investigated for some small titanium—oxygen systems. Equilibrium geometries, ionization potentials, dipole moments, atomization energies, and harmonic vibrational frequencies have been calculated for the TiO, TiO₂, and Ti₂ molecules, and the results are compared with experimental data and ab initio calculations. It is shown that most DFT methods perform much better than the ab initio Hartree—Fock (HF), second-order perturbation theory (MP2), and configuration interaction including single and double excitations (CISD) treatments. For good agreement with experimental data, gradient corrections to the exchange part of the DFT functional are needed, as well as some type of correction for the errors in the calculated energy splittings between different atomic states of titanium. Hybrid methods including a mixture of HF exchange with DFT exchange correlation do not perform as well as “pure” DFT methods for the studied systems. © 1996 John Wiley & Sons, Inc.     

Ab initio molecular dynamics of hydrogen dissociation on metal surfaces using neural networks and novelty sampling

We outline a hybrid multiscale approach for the construction of ab initio potential energy surfaces (PESs) useful for performing six-dimensional (6D) classical or quantum mechanical molecular dynamics (MD) simulations of diatomic molecules reacting at single crystal surfaces. The algorithm implements concepts from the corrugation reduction procedure, which reduces energetic variation in the PES, and uses neural networks for interpolation of smoothed ab initio data. A novelty sampling scheme is implemented and used to identify configurations that are most likely to be predicted inaccurately by the neural network. This hybrid multiscale approach, which couples PES construction at the electronic structure level to MD simulations at the atomistic scale, reduces the number of density functional theory (DFT) calculations needed to specify an accurate PES. Due to the iterative nature of the novelty sampling algorithm, it is possible to obtain a quantitative measure of the convergence of the PES with respect to the number of ab initio calculations used to train the neural network. We demonstrate the algorithm by first applying it to two analytic potentials, which model the H2/Pt(111) and H2/Cu(111) systems. These potentials are of the corrugated London-Eyring-Polanyi-Sato form, which are based on DFT calculations, but are not globally accurate. After demonstrating the convergence of the PES using these simple potentials, we use DFT calculations directly and obtain converged semiclassical trajectories for the H2/Pt(111) system at the PW91/generalized gradient approximation level. We obtain a converged PES for a 6D hydrogen-surface dissociation reaction using novelty sampling coupled directly to DFT. These results, in excellent agreement with experiments and previous theoretical work, are compared to previous simulations in order to explore the sensitivity of the PES (and therefore MD) to the choice of exchange and correlation functional. Despite having a lower energetic corrugation in our PES, we obtain a broader reaction probability curve than previous simulations, which is attributed to increased geometric corrugation in the PES and the effect of nonparallel dissociation pathways.     

Applicability of multi-reference many-body perturbation theory to the determination of potential energy surfaces: A model study

The recently written CI -based multi-reference many-body perturbation theory (MR-MBPT ) program package is exploited to study a simple ab initio minimum basis set model involving four hydrogen atoms in a rectangular configuration. This model was examined earlier by several authors using both coupled cluster (CC ) and finite-order MBPT approaches. Here we present the MR-MBPT results up to the 50th order and examine the effect of various shifting techniques on the convergence behavior of this approach. It is shown that in contrast with CC methods, both single and MR finite-order MBPT potential energy calculations are plagued with convergency and intruder state problems, which can be particularly severe when the latter approach is employed for non-degenerate situations.     

Information theoretic indices for characterization of chemical structures. By Danail Bonchev Wiley,New York, 1983

Wilson, Jankowski, and Paldus have recently applied nondegenerate many-body perturbation theory (MBPT ) to simple models, in which the degree of quasidegeneracy could be varied continuously, and concluded that the nondegenerate theory was applicable even near degeneracy. The error in their results changes, however, considerably with geometry, leading to an incorrect potential surface. An extension of their calculations shows convergence even at exact degeneracy (square planar H₄). It is shown here that the apparently good convergence is due to the suppression of the large (infinite at exact degeneracy) component of the perturbation energy in low order by the way the Hamiltonian is partitioned. This component will, however, resurface at higher orders, leading to slow convergence or even divergence. The low-order sum of the perturbation series is not very meaningful, depends strongly on details of the zero-order Hamiltonian, and yields, in general, incorrect potential surfaces. Multireference MBPT eliminates these problems.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Ab initio density functional theory: the best of both worlds?

Density 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 initio DFT is a promising route to pursue. Whereas conventional DFT cannot describe weak interactions, photoelectron spectra, or many potential energy surfaces, ab initio DFT, even in its initial, optimized effective potential, second-order many-body 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 coupled-cluster 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 Ne2,Be2,F2, and benzene.     

A pseudoatom approach to molecular truncation: application in ab initio MBPT methods

In this paper, we test the performance of the molecular truncation method of Mallik et al., which was originally applied at the semiempirical NDDO level, in ab initio MBPT methods. Pseudoatoms developed for the replacement of -OCH(3) and -OCH(2)CH(3) functional groups are used in optimizations of selected clusters, and the resulting geometries are compared to reference values taken from the full molecules. It is shown that the pseudoatoms, which consist of parametrized effective core potentials for the nearest neighbor interactions and an external charge field for long-range Coulomb effects, perform well at the MP2 and CCSD levels of theory for the suite of molecules to which they were applied. Representative timings for some of the pseudoatom-terminated clusters are presented, and it is seen that there is a significant reduction in computational time, yet the geometric configurations and deprotonation energies of the pseudoatom-terminated clusters are comparable to the more computationally expensive all-atom molecules.     

Applications of effective core potentials and density functional theory to the spin states of iron porphyrin

We investigated the performance of the B3LYP density functional in combination with ab initio effective core potentials (ECPs) that are derived from either Hartree-Fock or Dirac-Fock calculations. The transferability of ab initio ECPs is assessed on the basis of comparison with all-electron density functional calculations. For iron(II) porphyrin in particular, our assessment focused on the relative energetic ordering of five low-lying spin states, 1A1G, 3A1G, 3B2G, 5A2G, and 5B1G, and their properties, including optimized structures, charge distribution, spin density, and vibrational frequencies. Our results show that core electron correlation and core-valence electron correlation do not have significant effects on the relative energetics of the spin states of iron porphyrin. Our calculations suggest that effects of replacing the core electrons with ECPs are less significant than the choice of basis functions. We conclude that ab initio ECPs such as LANL2, RCEP, and MEFIT-R may be combined with the B3LYP density functional theory to provide consistent and accurate results.     

Investigating magnetostructural correlations in the pseudooctahedral trans-[Ni(II){(OPPh2)(EPPh2)N}2(sol)2] complexes (E = S, Se; sol = DMF, THF) by magnetometry, HFEPR, and ab initio quantum chemistry

In this work, magnetometry and high-frequency and -field electron paramagnetic resonance spectroscopy (HFEPR) have been employed in order to determine the spin Hamiltonian (SH) parameters of the non-Kramers, S = 1, pseudooctahedral trans-[Ni(II){(OPPh(2))(EPPh(2))N}(2)(sol)(2)] (E = S, Se; sol = DMF, THF) complexes. X-ray crystallographic studies on these compounds revealed a highly anisotropic NiO(4)E(2) coordination environment, as well as subtle structural differences, owing to the nature of the Ni(II)-coordinated solvent molecule or ligand E atoms. The effects of these structural characteristics on the magnetic properties of the complexes were investigated. The accurately HFEPR-determined SH zero-field-splitting (zfs) D and E parameters, along with the structural data, provided the basis for a systematic density functional theory (DFT) and multiconfigurational ab initio computational analysis, aimed at further elucidating the electronic structure of the complexes. DFT methods yielded only qualitatively useful data. However, already entry level ab initio methods yielded good results for the investigated magnetic properties, provided that the property calculations are taken beyond a second-order treatment of the spin-orbit coupling (SOC) interaction. This was achieved by quasi-degenerate perturbation theory, in conjunction with state-averaged complete active space self-consistent-field calculations. The accuracy in the calculated D parameters improves upon recovering dynamic correlation with multiconfigurational ab initio methods, such as the second-order N-electron valence perturbation theory NEVPT2, the difference dedicated configuration interaction, and the spectroscopy-oriented configuration interaction. The calculations showed that the magnitude of D (～3-7 cm(-1)) in these complexes is mainly dominated by multiple SOC contributions, the origin of which was analyzed in detail. In addition, the observed largely rhombic regime (E/D = 0.16-0.33) is attributed to the highly distorted metal coordination sphere. Of special importance is the insight by this work on the zfs effects of Se coordination to Ni(II). Overall, a combined experimental and theoretical methodology is provided, as a means to probe the electronic structure of octahedral Ni(II) complexes.     

Non-Hamiltonian molecular dynamics implementation of the Gibbs ensemble method. II. Molecular liquid-vapor results for carbon dioxide

The Gibbs ensemble molecular dynamics algorithm introduced in the preceding paper (paper I) [C. Bratschi and H. Huber, J. Chem. Phys. v126, 164104 (2007)] is applied to two recently published CO2 ab initio pair potentials, the Bock-Bich-Vogel and symmetry-adapted perturbation theory site-site potentials. The critical properties of these potentials are calculated for the first time. Critical values and points in the single and two-phase zones are compared with Monte Carlo results to demonstrate the accuracy of the molecular dynamics algorithm, and are compared with experiment to test the accuracy of the potentials. Pressure calculations in the liquid, gas, and supercritical states are carried out and are used to explain potential-related effects and systematic discrepancies. The best ab initio potential yields results in good agreement with experiment.     

Adsorption of benzene on coinage metals: a theoretical analysis using wavefunction-based methods

The interaction of benzene with a Ag(111) surface has been determined using reliable ab initio electronic structure calculations. The results are compared to a recent detailed analysis of the interaction of benzene with copper and gold surfaces, thus making it possible to derive a consistent picture for the electronic structure changes encountered when benzene is brought into contact with the densely packed coinage metal surfaces. To avoid the problems encountered when the presently most frequently employed computational approach, density functional theory (DFT), is applied to adsorbate systems where dispersion (or van der Waals) forces contribute substantially, we use a wavefunction-based approach. In this approach, the weak van der Waals interactions, which are dominated by correlation effects, are described using second-order perturbation theory. The surface dipole moment and the work function changes induced upon adsorption are also discussed.     

Structural and optical properties of a neutral Nickel bisdithiolene complex: density functional versus ab initio methods

Density functional theory (DFT) and ab initio computations are applied to examine different properties of diamagnetic, square planar neutral nickel complexes that contain two bidentate ligands derived from bis ((ethylene)-1,2-dithiolato) ligands. Geometry, vibrational spectra (IR and Raman) are well reproduced in the density functional framework whereas TD-DFT methods are clearly insufficient to reproduce absorption properties. Multiconfigurational perturbation theory based on a complete active space self-consistent field wave function, i.e. MRPT2 and MRPT4 methods, reveal the pronounced multiconfigurational character of the ground state wave function. The singlet–triplet energy gap, the energy gained from symmetry breaking and the singlet diradical character are discussed in the DFT and ab initio frameworks. The complex of interest does not display a strong singlet diradical character. This molecule having a peculiar electronic structure; strong delocalization as shown by a new electron pair localization function analysis (EPLF); exemplifies the fragility of the TD-DFT method and thus, caution should be taken in the determination of the energetic properties of such compounds.     

Density functional theory and post-Hartree-Fock studies on molecular structure and harmonic vibrational spectrum of formaldehyde

Density functional theory (DFT) with the Becke's three-parameter exchange correlation functional and the functional of Lee, Yang and Parr, gradient-corrected functionals of Perdew, and Perdew and Wang [the DFT(B3LYP), DFT(B3P86) and DFT(B3PW91) methods, respectively], and several levels of conventional ab initio post-Hartree-Fock theory (second- and fourth-order perturbation theory M?ller-Plesset MP2 and MP4(SDTQ), coupled cluster with the single and double excitations (CCSD), and CCSD with perturbative triple excitation [CCSD(T)], configuration interaction with the single and double excitations [CISD], and quadratic configuration interaction method [QCISD(T)], using several basis sets [ranging from a simple 6-31G(d,p) basis set to a 6-311+ +G(3df, 2pd) one], were applied to study of the molecular structure (geometrical parameters, rotational constants, dipole moment) and harmonized infrared (IR) spectrum of formaldehyde (CH₂O). High-level ab initio methods CCSD(T) and QCISD(T) with the 6-311+ +G(3df, 2pd) predict correctly molecular parameters, vibrational harmonic wavenumbers and the shifts of the harmonic IR spectrum of ¹²CH₂ ¹⁶O upon isotopic substitution. Received: 30 January 1997 / Accepted: 7 May 1997     

13C carbonyl chemical shielding tensors: Comparing SCF,MBPT (2), and DFT predictions to experiment

In this work, we calculate the ¹³C nuclear magnetic resonance chemical shielding tensors for 18 carbonyl-containing compounds. The many-body perturbation theory (MBPT), self-consistent field (SCF), and density functional theory (DFT) formalisms were used with gauge including atomic orbitals (GIAO) to calculate the shielding tensors. Our data suggest that shielding tensors can be efficiently estimated by performing one MBPT(2) correlated calculation (e.g., at a reference geometry) and SCF-level calculations at other geometries and taking the SCF-to-correlated tensor element differences to be geometry independent. That is, the correlation contribution to the chemical shielding seems to be relatively constant over a considerable range of distortions. Treatment of correlation using DFT methods is shown to not be as systematically reliable as with MBPT(2). Data on 18 carbonyl compounds show that the single largest influence on the shielding tensor is the presence of nearby electron-withdrawing or electron-donating groups. Finally, although good agreement with powder or single-crystal experimental data is achieved for two or three tensor eigenvalues, systematic differences remain for one element; the origins of these differences are discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 875–894, 1997     

Density-functional theory with effective potential expressed as a direct mapping of the external potential: applications to atomization energies and ionization potentials

In this paper the authors further develop and apply the direct-mapping density functional theory to calculations of the atomization energies and ionization potentials. Single-particle orbitals are determined by solving the Kohn-Sham [Phys. Rev. A. 140, 1133 (1965)] equations with a local effective potential expressed in terms of the external potential. A two-parametric form of the effective potential for molecules is proposed and equations for optimization of the parameters are derived using the exchange-only approximation. Orbital-dependent correlation functional is derived from the second-order perturbation theory in its Moller-Plesset-type zeroth-order approximation based on the Kohn-Sham orbitals and orbital energies. The total atomization energies and ionization potentials computed with the second-order perturbation theory were found to be in agreement with experimental values and benchmark results obtained with ab initio wave mechanics methods.