Static Correlation Effects Research Articles

Exploring Avenues beyond Revised DSD Functionals: I. Range Separation, with xDSD as a Special Case.

We have explored the use of range separation as a possible avenue for further improvement on our revDSD minimally empirical double hybrid functionals. Such ωDSD functionals encompass the XYG3 type of double hybrid (i.e., xDSD) as a special case for ω → 0. As in our previous studies, the large and chemically diverse GMTKN55 benchmark suite was used for evaluation. Especially when using the D4 rather than D3BJ dispersion model, xDSD has a slight performance advantage in WTMAD2. As in previous studies, PBEP86 is the winning combination for the semilocal parts. xDSDn-PBEP86-D4 marginally outperforms the previous “best in class” ωB97M(2) Berkeley double hybrid but without range separation and using fewer than half the number of empirical parameters. Range separation turns out to offer only marginal further improvements on GMTKN55 itself. While ωB97M(2) still yields better performance for small-molecule thermochemistry, this is compensated in WTMAD2 by the superior performance of the new functionals for conformer equilibria. Results for two external test sets with pronounced static correlation effects may indicate that range-separated double hybrids are more resilient to such effects.

Gas-Phase Thermochemistry of MX3 and M2X6 (M = Sc, Y; X = F, Cl, Br, I) from a Composite Reaction-Based Approach: Homolytic versus Heterolytic Cleavage.

A domain-based local-pair natural-orbital coupled-cluster approach with single, double, and improved linear-scaling perturbative triple correction via an iterative algorithm, DLPNO-CCSD(T1), was applied within the framework of the Feller-Peterson-Dixon approach to derive gas-phase heats of formation of scandium and yttrium trihalides and their dimers via a set of homolytic and heterolytic dissociation reactions. All predicted heats of formation moderately depend on the reaction type with the most and least negative values obtained for homolytic and heterolytic dissociation, respectively. The basis set size dependence, as well as the influence of static correlation effects not covered by the standard (DLPNO-)CCSD(T) approach, suggests that exploitation of the heterolytic dissociation reactions with the formation of M3+ and X- ions leads to the most robust heats of formation. The gas-phase formation enthalpies ΔHf°(0 K)/ΔHf°(298.15 K) and absolute entropies S°(298.15 K) were obtained for the first time for the Sc2F6, Sc2Br6, and Sc2I6 species. For ScBr3, ScI3, Sc2Cl6, and Y2Cl6, we suggest a reexamination of the experimental heats of formation available in the literature. For other compounds, the predicted values were found to be in good agreement with the experimental estimates. Extracted MX3 (M = Sc, Y; X = F, Cl, Br, and I) 0 K atomization enthalpies indicate weaker bonding when moving from fluorine to iodine and from yttrium to scandium. Likewise, the stability of yttrium trihalide dimers degrades when going from fluorine to iodine. Respective scandium trihalide dimers are less stable, with 0 K dimer dissociation energy decreasing in the row fluorine - chlorine - bromine ≈ iodine. Correlation of the (n - 1)s2p6 electrons on bromine and iodine, inclusion of zero-point energy, relativistic effects, and the effective-core-potential correction as well as amelioration of the DLPNO localization inaccuracy are shown to be of similar magnitude, which is critical if accurate heats of formation are a goal.

Hole-hole Tamm-Dancoff-approximated density functional theory: A highly efficient electronic structure method incorporating dynamic and static correlation.

The study of photochemical reaction dynamics requires accurate as well as computationally efficient electronic structure methods for the ground and excited states. While time-dependent density functional theory (TDDFT) is not able to capture static correlation, complete active space self-consistent field methods neglect much of the dynamic correlation. Hence, inexpensive methods that encompass both static and dynamic electron correlation effects are of high interest. Here, we revisit hole-hole Tamm-Dancoff approximated (hh-TDA) density functional theory for this purpose. The hh-TDA method is the hole-hole counterpart to the more established particle-particle TDA (pp-TDA) method, both of which are derived from the particle-particle random phase approximation (pp-RPA). In hh-TDA, the N-electron electronic states are obtained through double annihilations starting from a doubly anionic (N+2 electron) reference state. In this way, hh-TDA treats ground and excited states on equal footing, thus allowing for conical intersections to be correctly described. The treatment of dynamic correlation is introduced through the use of commonly employed density functional approximations to the exchange-correlation potential. We show that hh-TDA is a promising candidate to efficiently treat the photochemistry of organic and biochemical systems that involve several low-lying excited states-particularly those with both low-lying ππ* and nπ* states where inclusion of dynamic correlation is essential to describe the relative energetics. In contrast to the existing literature on pp-TDA and pp-RPA, we employ a functional-dependent choice for the response kernel in pp- and hh-TDA, which closely resembles the response kernels occurring in linear response and collinear spin-flip TDDFT.

Global Hybrid Multiconfiguration Pair-Density Functional Theory.

A global hybrid extension of multiconfiguration pair-density functional theory (MC-PDFT) is developed. Using a linear decomposition of the electron-electron repulsion term, a fraction λ of the nonlocal exchange interaction, obtained from variational two-electron reduced-density matrix (v2RDM)-driven complete active-space self-consistent field (CASSCF) theory, is combined with its local counterpart, obtained from an on-top pair-density functional. The resulting scheme (called λ-MC-PDFT) inherits the benefits of MC-PDFT (e.g., its simplicity and the resolution of the symmetry dilemma) and, when combined with the v2RDM approach to CASSCF, requires only polynomially scaling computational effort. As a result, λ-MC-PDFT can efficiently describe static and dynamical correlation effects in strongly correlated systems. The efficacy of the approach is assessed for several challenging multiconfigurational problems, including the dissociation of molecular nitrogen, the double dissociation of a water molecule, and the 1,3-dipolar cycloadditions of ozone to ethylene and ozone to acetylene in the O3ADD6 benchmark set.

Parametrisation of the optimised effective potential method based on the Coulson–Fischer wave function for excited states

ABSTRACT This article further develops the Coulson–Fischer formalism in conjunction with the optimised effective potential method [V.N. Glushkov and S. Wilson, Molec. Phys. 110, 149 (2012)] to incorporate both exchange and static correlation effects for excited states having the same symmetry as the ground state. By using this formalism, we compute complete potential energy curves for the ground and first excited states of the molecule and the isoelectronic heteronuclear ion. Effective potentials are generated in according to a novel density functional theory [A.K. Theophilou, J. Chem. Phys. 149, 074104 (2018)] in which the Kohn–Sham (KS) like potential has spherical symmetry around each nucleus. In this way, the inconsistencies of standard density functional theory concerning the asymmetry of the KS potential are remedied. In our implementation, an effective potential which satisfies this requirement is expressed as a direct mapping of the external potential. Different approximations for the effective potentials are investigated. It is shown that applications of the proposed formalism support good agreement with the exact Coulson–Fischer theory for both the ground and first excited state potential energy curves and thus a qualitatively correct description of dissociation.

Static correlation in vibrational frequencies studied using thermally-assisted-occupation density functional theory

Thermally-assisted-occupation density functional theory is a method that incorporates approximate static correlation at a relatively low computational cost and allows static correlation effects to be more easily incorporated into vibrational frequency calculations. Here the effects of thermally-assisted-occupation are tested on a wide range of frequencies, indicating that static correlation plays an important role in the vibrations of several molecules that make up popular test sets of experimental vibrational frequency data. Small but persistent effects are also noted for vibrations involving significant hydrogen stretching character and mode-specific differences are seen for several molecules that are known to have multi-reference character.

Correlation effects in B1s core-excited states of boronic-acid derivatives: An experimental and computational study.

We performed a theoretical investigation on the influence of electronic correlation effects on the B1s NEXAFS spectrum of boronic acid derivatives, namely, boric acid [B(OH)3], phenyl boronic acid (PBA), and 1,4-phenyl diboronic acid (PDBA), employing different computational schemes of increasing complexity, ranging from the purely one-electron scheme based on the transition potential method of density functional theory (DFT-TP), time-dependent DFT (TDDFT), and multiconfigurational self-consistent field (MCSCF). We also report experimental measurements of the B1s NEXAFS spectra of the aforementioned molecules together with the high-resolution C1s NEXAFS spectrum of PBA. We demonstrate that due to the shallow B1s core energy levels compared to C, O, and N, the inclusion of static correlation effects, which can be incorporated by using multireference approaches to excited states, assumes a decisive role in reconciling experiment and theory on B1s core-electron excitation energies and oscillator strengths to valence states. This claim is corroborated by the good agreement that we find between the DFT-TP calculated C1s NEXAFS spectrum and that experimentally measured for PBA and by the failure of both DFT-TP and TDDFT approaches with a selection of xc functionals kernels to properly describe the B1s NEXAFS spectrum of PBA and PDBA, at variance with the good agreement with the experiment that is found by employing the MCSCF wave function approach.

Frozen-density embedding employing configuration interaction as a subsystem method

ABSTRACT We report the implementation of a general configuration interaction (CI) implementation combined with frozen-density embedding (FDE). The implemented direct CI algorithm for ground and excited states covers all truncated CI schemes ranging from CI singles (CIS) to full CI (FCI) and also the complete active-space (CAS) ansatz to treat molecules in which dynamic and static correlation effects dominate, respectively. Combining this general CI ansatz with FDE enables to study environment effects upon molecular properties, required to obtain a balanced description of correlation and environment effects. In the framework of the uncoupled FDE (FDEu) scheme, the implementation allows to combine truncated CI schemes, e.g. CI singles doubles triples quadruples (CISDTQ), embedded in water molecules computed using second-order Møller–Plesset perturbation theory (MP2) and density-functional theory (DFT), also denoted CISDTQ-in-MP2-in-DFT.

Electronic Properties of Linear and Cyclic Boron Nanoribbons from Thermally-Assisted-Occupation Density Functional Theory

It remains rather difficult for traditional computational methods to reliably predict the properties of nanosystems, especially for those possessing pronounced radical character. Accordingly, in this work, we adopt the recently formulated thermally-assisted-occupation density functional theory (TAO-DFT) to study two-atom-wide linear boron nanoribbons l-BNR[2,n] and two-atom-wide cyclic boron nanoribbons c-BNR[2,n], which exhibit polyradical character when the n value (i.e., the number of boron atoms along the length of l-BNR[2,n] or the circumference of c-BNR[2,n]) is considerably large. We calculate various electronic properties associated with l-BNR[2,n] and c-BNR[2,n], with n ranging from 6 to 100. Our results show that l-BNR[2,n] and c-BNR[2,n] have singlet ground states for all the n values examined. The electronic properties of c-BNR[2,n] exhibit more pronounced oscillatory patterns than those of l-BNR[2,n] when n is small, and converge to the respective properties of l-BNR[2,n] when n is sufficiently large. The larger the n values, the stronger the static correlation effects that originate from the polyradical nature of these ribbons. Besides, the active orbitals are found to be delocalized along the length of l-BNR[2,n] or the circumference of c-BNR[2,n]. The analysis of the size-dependent electronic properties indicates that l-BNR[2,n] and c-BNR[2,n] can be promising for nanoelectronic devices.

Benchmarking the Accuracy of Seniority-Zero Wave Function Methods for Noncovalent Interactions.

In this paper, we scrutinize the ability of seniority-zero wave function-based methods to model different types of noncovalent interactions, such as hydrogen bonds, dispersion, and mixed noncovalent interactions as well as prototypical model systems with various contributions of dynamic and static electron correlation effects. Specifically, we focus on the pair Coupled Cluster Doubles (pCCD) ansatz combined with two different flavors of dynamic energy corrections, (i) based on a perturbation theory correction and (ii) on a linearized coupled cluster ansatz on top of pCCD. We benchmark these approaches against the A24 data set [ Řezáč and Hobza J. Chem. Theory Comput. 2013 , 9 , 2151 - 2155 .] extrapolated to the basis set limit and some model noncovalent complexes that feature covalent bond breaking. By dissecting different types of interactions in the A24 data set within the Symmetry-Adapted Perturbation Theory (SAPT) framework, we demonstrate that pCCD can be classified as a dispersion-free method. Furthermore, we found that both flavors of post-pCCD approaches represent encouraging and computationally more efficient alternatives to standard electronic structure methods to model weakly bound systems, resulting in small statistical errors. Finally, a linearized coupled cluster correction on top of pCCD proved to be most reliable for the majority of investigated systems, featuring smaller nonparallelity errors compared to perturbation-theory-based approaches.

Spin-dependent properties in the framework of the dynamic correlation dressed complete active space method.

We report an extension of the recently proposed 2nd order dynamic correlation dressed complete active space method [S. Pathak et al., J. Chem. Phys. 147, 234109 (2017)] to incorporate spin-dependent relativistic effects into the Hamiltonian. The result is an effective Hamiltonian that contains the effects of static correlation, dynamic correlation, and relativistic effects on an equal footing. All contributions necessary for the description of magnetic phenomena and electron paramagnetic resonance (EPR) spectroscopy, namely, spin-orbit coupling, magnetic hyperfine coupling, Zeeman interaction, and direct electronic spin-spin coupling, are incorporated. We also suggest a novel analysis of g-matrices and A-matrices based on the singular value decomposition, which can provide not only the magnitude but also the sign of the principal components and allows for a transparent decomposition into different physical contributions. The new method was tested for excitation energies of first-row transition metal ions as well as D-tensors and g-shifts of first-row transition metal complexes using minimal active spaces. It was observed that state-mixing effects are usually small in these cases and that the results are comparable to nondegenerate N-electron valence state perturbation theory (NEVPT2) in conjunction with quasi-degenerate perturbation theory (QDPT). Results on EPR parameters of pseudo-square-planar Cu(ii) complexes show that state-mixing with a ligand-to-metal-charge-transfer configuration greatly improves results compared with NEVPT2/QDPT but also demonstrate that future modifications of the 0th order Hamiltonian or more elaborate electron correlation treatments will be necessary in order to achieve better agreement with the experiment.

Relativistic theory of two-photon photoemission from ferromagnetic materials

A fully relativistic theory of angle-resolved two-photon photoemission for ferromagnetic materials is derived from the general theory of pump-probe photoemission that is based on the Keldysh formalism. Two-photon photoemission spectroscopy is a widely used analytical tool to study nonequilibrium phenomena in nonmagnetic as well as in magnetic solids, as for example ultrafast demagnetization processes. Our theoretical approach aims at a quantitative description of the time-dependent spectroscopic properties of specific solid systems under consideration, and it allows for the inclusion of static correlation effects via the local spin-density and dynamical mean-field theory electronic structure approach. To this end we follow Pendry's one-step theory of the photoemission process as closely as possible and make extensive use of concepts of relativistic multiple - scattering theory within the layer-Korringa-Kohn-Rostoker method in order to guarantee angular resolution in the spectroscopic calculations. As usual the final state of the two-photon photoemission process is represented by a time-reversed low-energy electron diffraction state. The formalism allows for a quantitative calculation of the time-dependent photocurrent for moderately correlated systems like simple ferromagnetic metals. A first application to the Fe(100) surface is discussed in detail.

Combining Pair-Density Functional Theory and Variational Two-Electron Reduced-Density Matrix Methods.

Complete active space self-consistent field (CASSCF) computations can be realized at polynomial cost via the variational optimization of the active-space two-electron reduced-density matrix (2-RDM). Like conventional approaches to CASSCF, variational 2-RDM (v2RDM) driven CASSCF captures nondynamical electron correlation in the active space, but it lacks a description of the remaining dynamical correlation effects. Such effects can be modeled through a combination of v2RDM-CASSCF and on-top pair-density functional theory (PDFT). The resulting v2RDM-CASSCF-PDFT approach provides a computationally inexpensive framework for describing both static and dynamical correlation effects in multiconfigurational and strongly correlated systems. On-top pair-density functionals can be derived from familiar Kohn-Sham exchange-correlation (XC) density functionals through the translation of the v2RDM-CASSCF reference densities [ Li Manni et al. J. Chem. Theory Comput. 2014 , 10 , 3669 - 3680 ]. Translated and fully translated on-top PDFT versions of several common XC functionals are applied to the potential energy curves of N2, H2O, and CN-, as well as to the singlet/triplet energy splittings in the linear polyacene series. Using v2RDM-CASSCF-PDFT and the translated PBE functional, the singlet/triplet energy splitting of an infinitely long acene molecule is estimated to be 4.87 kcal mol-1.

Orbital-invariant spin-extended approximate coupled-cluster for multi-reference systems.

We present an approximate treatment of spin-extended coupled-cluster (ECC) based on the spin-projection of the broken-symmetry coupled-cluster (CC) ansatz. ECC completely eliminates the spin-contamination of unrestricted CC and is therefore expected to provide better descriptions of dynamical and static correlation effects, but introduces two distinct problems. The first issue is the emergence of non-terminating amplitude equations, which are caused by the de-excitation effects inherent in symmetry projection operators. In this study, we take a minimalist approach and truncate the Taylor series of the exponential ansatz at a certain order such that the approximation safely recovers the traditional CC without spin-projection. The second issue is that the nonlinear equations of ECC become underdetermined, although consistent, yielding an infinitude of solutions. This problem arises because of the redundancies in the excitation manifold, as is common in other multi-reference approaches. We remove the linear dependencies in ECC by employing an orthogonal projection manifold. We also propose an efficient solver for our method, in which the components are usually sparse but not diagonal-dominant. It is shown that our approach is rigorously orbital-invariant and provides more accurate results than its configuration interaction and linearized CC analogues for chemical systems.

Using reduced density matrix techniques to capture static and dynamic correlation in the energy landscape for the decomposition of the CH2CH2ONO radical and support a non-IRC pathway.

The unexpected abundance of HNO in the photodecomposition of the radical 2-nitrosooxy ethyl (CH2CH2ONO) is investigated through calculations of the potential energy surface by the anti-Hermitian contracted Schrödinger equation (ACSE) method, which directly generates the 2-electron reduced density matrix. The ACSE, which is able to balance single-reference (dynamic) and multi-reference (static) correlation effects, reveals some subtle correlation effects along the intrinsic reaction coordinate (IRC) en route to NO + oxirane, an IRC which offers a potential bifurcation to the HNO + vinoxy product channel. These effects were not fully captured by either single-reference techniques, such as coupled cluster, or multi-reference techniques, such as second-order multi-reference perturbation theory. These correlation effects reveal small to moderate energy changes in key transition states, which have implications for the reaction mechanism as related to the production of HNO.

Relative stability between the manganese hydroxide- and oxo-models for water oxidation by CCSD, DMRG CASCI, CASSCF, CASPT2 and CASDFT methods; Importance of static and dynamical electron correlation effects for OEC of PSII

Relative stability between manganese hydroxide (H2O)2(OH)4Mn(IV)OH (1) and manganese oxo (H2O)3(OH)3Mn(IV)O (2) species were investigated by coupled cluster (CC) SD and SD(T), DMRG CASCI, CASSCF, CASPT2 and CASDFT methods. The DMRG CASCI and DMRG CASSCF based on thirteen active orbitals and thirteen active electrons {13, 13} CAS space indicted the greater stability of 2 than 1. On the other hand, the relative stability between 1 and 2 was reversed by UCC SD, DLPNO-UCC SD, CASPT2 and CAS-DFT, indicating an important role of dynamical electron correlation corrections. Implications of present results are discussed to elucidate scope and applicability of hybrid DFT (HDFT) methods for 1 and 2.

Excited state non-adiabatic dynamics of the smallest polyene, trans 1,3-butadiene. II. Ab initio multiple spawning simulations.

The excited state non-adiabatic dynamics of the smallest polyene, trans 1,3-butadiene (BD), has long been the subject of controversy due to its strong coupling, ultrafast time scales and the difficulties that theory faces in describing the relevant electronic states in a balanced fashion. Here we apply Ab Initio Multiple Spawning (AIMS) using state-averaged complete active space multistate second order perturbation theory [SA-3-CAS(4/4)-MSPT2] which describes both static and dynamic electron correlation effects, providing a balanced description of both the initially prepared bright 11Bu (ππ*) state and non-adiabatically coupled dark 21Ag state of BD. Importantly, AIMS allows for on-the-fly calculations of experimental observables. We validate our approach by directly simulating the time resolved photoelectron-photoion coincidence spectroscopy results presented in Paper I [A. E. Boguslavskiy et al., J. Chem. Phys. 148, 164302 (2018)], demonstrating excellent agreement with experiment. Our simulations reveal that the initial excitation to the 11Bu state rapidly evolves via wavepacket dynamics that follow both bright- and dark-state pathways as well as mixtures of these. In order to test the sensitivity of the AIMS results to the relative ordering of states, we considered two hypothetical scenarios biased toward either the bright 1Bu or the dark 21Ag state. In contrast with AIMS/SA-3-CAS(4/4)-MSPT2 simulations, neither of these scenarios yields favorable agreement with experiment. Thus, we conclude that the excited state non-adiabatic dynamics in BD involves both of these ultrafast pathways.

Properties of coupled-cluster equations originating in excitation sub-algebras

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

The role of topology in organic molecules: origin and comparison of the radical character in linear and cyclic oligoacenes and related oligomers.

We discuss the nature of electron-correlation effects in carbon nanorings and nanobelts using an analysis tool known as fractional occupation number weighted electron density (ρFOD) and the RAS-SF method, revealing for the first time significant differences in static correlation effects depending on how the rings (i.e. chemical units) are fused and/or connected until closing the loop. We choose to study in detail linear and cyclic oligoacene molecules of increasing size, and relate the emerging differences with the difficulties for the synthesis of the latter due to their radicaloid character. We finally explore how minor structural modifications of the cyclic forms can alter these results, showing the potential use of these systems as molecular templates for the growth of well-shaped carbon nanotubes as well as the usefulness of theoretical tools for molecular design.

Effects of Static Correlation between Spin Centers in Multicenter Transition Metal Complexes.

Multicenter transition metal complexes are the key moieties of many processes in chemistry, biochemistry, and materials science such as in the active sites of enzymes, molecular catalysts, and biological electron carriers. Their electronic structure, often characterized by high-spin-polarized metal sites, is a challenge for theoretical chemists because of their high degree of dynamical and static correlation. Static correlation is necessary both for the appropriate description of the metal-ligand bonding and for a correct description of the multideterminant character arising from the magnetic interactions between spin centers. Density functional theory (DFT) is usually applied using a single-determinant broken-symmetry state that is lacking the correct spin symmetry when the ground state has total low-spin character. To alleviate this drawback, we use the extended broken-symmetry (EBS) approach to derive approximate ground-state energies and, for the first time, forces for the correctly symmetric ground state of an arbitrary number of spin centers within the framework of the Heisenberg-Dirac-van Vleck Hamiltonian. Remarkably, the proposed procedure supplies relaxed geometries that are fully consistent with the calculated J-coupling constants. We apply the method to investigate the relaxed geometrical structure of the low-spin ground state of iron-sulfur clusters with two, three, and four iron centers. We observed significant differences in both geometrical parameters and coupling constant J between the symmetrized ground state, the high-spin, and the broken-symmetry optimized structures. These changes are often comparable with the differences observed by using different functionals, and the use of EBS always improves the description of the studied systems. It will be therefore important to include it in any DFT attempt to quantitatively describe multicenter transition metal complexes in the future.