Nondynamical Correlation Effects Research Articles

Toward a correct treatment of core properties with local hybrid functionals.

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23ptfunctional relates to the previous LH20t functional but specifically improves upon the core region.

Reassessing the Composition of Hybrid Orbitals in Contemporary VB Calculations.

Large variations in the ratios between the p and s components of individual hybrid orbitals that have been observed in contemporary ab initio VB calculations are reassessed, and links are established to specific energy terms that drive bond formation. It is demonstrated that the ratios between the p and s components for individual hybrid orbitals are not indicative of the overall hybridization status of the relevant atom, which exhibits only relatively small variations with the level of theory, irrespective of whether or not non-dynamical and dynamical electron correlation effects are accounted for. An alternative orbital representation that turns out to be far more consistent with the overall hybridization of the relevant atom is examined. The chosen test cases, which can be compared with the classical sp3, sp2, and sp hybridization models for a central carbon atom, are CH4 (Td), trigonal CH3 (D3h), and triplet CH2 distorted from its ground state geometry so as to be linear (D∞h).

Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals.

We develop a new family of electronic structure methods for capturing at the same time the dynamic and nondynamic correlation effects. We combine the natural orbital functional theory (NOFT) and many-body perturbation theory (MBPT) through a canonicalization procedure applied to the natural orbitals to gain access to any MBPT approximation. We study three different scenarios: corrections based on second-order Møller-Plesset (MP2), random-phase approximation (RPA), and coupled-cluster singles doubles (CCSD). Several chemical problems involving different types of electron correlation in singlet and multiplet spin states have been considered. Our numerical tests reveal that RPA-based and CCSD-based corrections provide similar relative errors in molecular dissociation energies (De) to the results obtained using a MP2 correction. With respect to the MP2 case, the CCSD-based correction improves the prediction, while the RPA-based correction reduces the computational cost.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions.

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Spin-Coupled Generalized Valence Bond Theory: New Perspectives on the Electronic Structure of Molecules and Chemical Bonds

Spin-Coupled Generalized Valence Bond (SCGVB) theory provides the foundation for a comprehensive theory of the electronic structure of molecules. SCGVB theory offers a compelling orbital description of the electronic structure of molecules as well as an efficient and effective zero-order wave function for calculations striving for quantitative predictions of molecular structures, energetics, and other properties. The orbitals in the SCGVB wave function are usually semilocalized, and for most molecules, they can be interpreted using concepts familiar to all chemists (hybrid orbitals, localized bond pairs, lone pairs, etc.). SCGVB theory also provides new perspectives on the nature of the bonds in molecules such as C2, Be2 and SF4/SF6. SCGVB theory contributes unparalleled insights into the underlying cause of the first-row anomaly in inorganic chemistry as well as the electronic structure of organic molecules and the electronic mechanisms of organic reactions. The SCGVB wave function accounts for nondynamical correlation effects and, thus, corrects the most serious deficiency in molecular orbital (RHF) wave functions. Dynamical correlation effects, which are critical for quantitative predictions, can be taken into account using the SCGVB wave function as the zero-order wave function for multireference configuration interaction or coupled cluster calculations.

Extension of the Fock-space coupled-cluster method with singles and doubles to the three-valence sector.

The single-reference coupled-cluster method has proven very effective in the ab initio description of atomic and molecular systems, but its successful application is limited to states dominated by a single Slater determinant, which is used as the reference. In cases where several determinants are important in the wave function expansion, i.e., we have to deal with nondynamic correlation effects, a multi-reference version of the coupled-cluster method is required. The multi-reference coupled-cluster approaches are based on the effective Hamiltonian formulation providing a two-step procedure, in which dynamic correlation effects can be efficiently evaluated by the wave operator, while nondynamic correlation contributions are given by diagonalization of the effective Hamiltonian in the final step. There are two classical multi-reference coupled-cluster formulations. In this paper, the focus is on the so-called Fock-space coupled-cluster method in its basic version with one- and two-particle operators in the exponent. Computational schemes using this truncation of the cluster operator have been successfully applied in calculations in one- and two-valence sectors of the Fock space. In this paper, we show that the approach can be easily extended and effectively employed in the three-valence sector calculations.

Excitation energies from thermally assisted-occupation density functional theory: Theory and computational implementation.

The time-dependent density functional theory (TDDFT) has been broadly used to investigate the excited-state properties of various molecular systems. However, the current TDDFT heavily relies on outcomes from the corresponding ground-state DFT calculations, which may be prone to errors due to the lack of proper treatment in the non-dynamical correlation effects. Recently, thermally assisted-occupation DFT (TAO-DFT) [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)], a DFT with fractional orbital occupations, was proposed, explicitly incorporating the non-dynamical correlation effects in the ground-state calculations with low computational complexity. In this work, we develop TDTAO-DFT, which is a TD, linear-response theory for excited states within the framework of TAO-DFT. With tests on the excited states of H2, the first triplet excited state (13Σu +) was described well, with non-imaginary excitation energies. TDTAO-DFT also yields zero singlet-triplet gap in the dissociation limit for the ground singlet (11Σg +) and the first triplet state (13Σu +). In addition, as compared to traditional TDDFT, the overall excited-state potential energy surfaces obtained from TDTAO-DFT are generally improved and better agree with results from the equation-of-motion coupled-cluster singles and doubles.

Taming the excited states of butadiene, hexatriene, and octatetraene using state specific multireference perturbation theory with density functional theory orbitals.

To compute the electronic excitation energies, a state-specific multireference Møller-Plesset perturbation theory (SSMRPT) with a complete active space configuration interaction reference function constructed using the orbitals obtained by the density functional theory (DFT) is presented as an accurate, as well as computationally affordable, and efficient protocol at the level of second order. The global hybrid B3LYP (Becke, 3-parameter, Lee-Yang-Parr) functional has been used to generate orbitals. The present method, called DFT-SSMRPT, uses perturbers that are individual Slater determinants and accounts for the coupling between the nondynamical and dynamical correlation effects. We have applied the new method to compute excitation energies in conjugated systems of π-electrons such as trans-1,3-butadiene, trans,trans-1,3,5-hexatriene, and all-trans-1,3,5,7-octatetraene. The ordering of the excited states is correctly reproduced by the DFT-SSMRPT calculations. The relative ordering of low-lying excited 1Bu and 1Ag states alters when the length of the polyene changes. The results match reasonably well with the literature including experimental and best theoretical findings. The accuracy of the method is sufficient to discern the energy gap between the close low-lying singlet and triplet states. The DFT-SSMRPT appears as an affordable computational ab initio avenue for a qualitatively correct description of excitation energies.

An efficient method for strongly correlated electrons in one dimension

The one-particle reduced density matrix functional theory in its natural orbital functional (NOF) version is used to study strongly correlated electrons. We show the ability of the Piris NOF 7 (PNOF7) to describe non-dynamic correlation effects in one-dimensional (1D) systems. An extensive study of 1D systems that includes Hydrogen (H) chains and the 1D Hubbard model with periodic boundary conditions is provided. Different filling situations and large sizes with up to 122 electrons are considered. Compared to quasi-exact results, PNOF7 is accurate in different correlation regimes for the 1D Hubbard model even away from the half-filling, and maintains its accuracy when the system size increases. The symmetric and asymmetric dissociations of the linear H chain composed of 50 atoms are described to remark the importance of long-range interactions in presence of strong correlation effects. Our results compare remarkably well with those obtained at the density-matrix renormalization group level of theory.

Multireference perturbation theory with improved virtual orbitals for radicals: More degeneracies, more problems

AbstractIVO‐SSMRPT is an affordable and accurate type of state‐specific multireference perturbation (SSMRPT) theory that adds dynamic correlation energy to improved virtual orbital (IVO) complete active space configuration interaction (CASCI) wave functions using a single‐root parametrization of multi‐root Hilbert‐space ansatz. We applied it to many chemically important di‐ and tri‐radicals to analyze the geometries and electronic properties of spectroscopic interest for both closed‐ and open‐shell singlet‐ and nonsinglet ground as well as excited states. We observed that IVO‐SSMRPT identifies optimized geometries, splitting between multiplets and frequencies for several radicals that are similar to those displayed by current generation state‐of‐the‐art methods but with admiringly decreased computational effort. This study illustrates the importance of having an accurate treatment of both nondynamical and dynamical correlation effects when examining multiradical species. Chemically and spectroscopically relevant answers can be obtained using our computationally tractable method. Our method will be a serviceable avenue for portraying open‐shell interactions in other radicals.

On the conversion XCN ⟷ XNC via an efficient and economic perturbative wave function approach

ABSTRACTThe energetics, structures, reaction paths and mechanisms for the conversion XCN ⟷ XNC, with X = H, Li, CH3 and H2B have been investigated through the use of improved virtual orbitals (IVO) in a second-order multireference perturbation theory (MRPT) treatment within the framework of state-specific (SS) parametrisation of the wave function for the target state (IVO-SSMRPT). Due to its flexibility, IVO-SSMRPT formulation allows us to take into account the major part of the coupling between the dynamic and nondynamic correlation effects in an accurate manner. The results obtained by our calculations agree well with the state-of-the-art electronic structure methods and reference values which illustrates that IVO-SSMRPT, when applied judiciously, can be very accurate for reaction surfaces of XCN ⟷ XNC isomerisation, barrier and bond-parameters.

W4-17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods.

Atomization reactions are among the most challenging tests for electronic structure methods. We use the first-principles Weizmann-4 (W4) computational thermochemistry protocol to generate the W4-17 dataset of 200 total atomization energies (TAEs) with 3σ confidence intervals of 1 kJmol-1 . W4-17 is an extension of the earlier W4-11 dataset; it includes first- and second-row molecules and radicals with up to eight non-hydrogen atoms. These cover a broad spectrum of bonding situations and multireference character, and as such are an excellent benchmark for the parameterization and validation of highly accurate ab initio methods (e.g., CCSD(T) composite procedures) and double-hybrid density functional theory (DHDFT) methods. The W4-17 dataset contains two subsets (i) a non-multireference subset of 183 systems characterized by dynamical or moderate nondynamical correlation effects (denoted W4-17-nonMR) and (ii) a highly multireference subset of 17 systems (W4-17-MR). We use these databases to evaluate the performance of a wide range of CCSD(T) composite procedures (e.g., G4, G4(MP2), G4(MP2)-6X, ROG4(MP2)-6X, CBS-QB3, ROCBS-QB3, CBS-APNO, ccCA-PS3, W1, W2, W1-F12, W2-F12, W1X-1, and W2X) and DHDFT methods (e.g., B2-PLYP, B2GP-PLYP, B2K-PLYP, DSD-BLYP, DSD-PBEP86, PWPB95, ωB97X-2(LP), and ωB97X-2(TQZ)). © 2017 Wiley Periodicals, Inc.

Analytic energy gradient of projected Hartree-Fock within projection after variation.

We develop a geometrical optimization technique for the projection-after-variation (PAV) scheme of the recently refined projected Hartree-Fock (PHF) as a fast alternative to the variation-after-projection (VAP) approach for optimizing the structures of molecules/clusters in symmetry-adapted electronic states at the mean-field computational cost. PHF handles the nondynamic correlation effects by restoring the symmetry of a broken-symmetry single reference wavefunction and moreover enables a black-box treatment of orbital selections. Using HF orbitals instead of PHF orbitals, our approach saves the computational cost for the orbital optimization, avoiding the convergence problem that sometimes emerges in the VAP scheme. We show that PAV-PHF provides geometries comparable to those of the complete active space self-consistent field and VAP-PHF for the tested systems, namely, CH2, O3, and the [Cu2O2]2+ core, where nondynamic correlation is abundant. The proposed approach is useful for large systems mainly dominated by nondynamic correlation to find stable structures in many symmetry-adapted states.

Multiconfigurational Effects in Theoretical Resonance Raman Spectra.

We analyze resonance Raman spectra of the nucleobase uracil in the short‐time approximation calculated with multiconfigurational methods. We discuss the importance of static electron correlation by means of density‐matrix renormalization group self‐consistent field (DMRG‐SCF) calculations. Our DMRG‐SCF results reveal that a minimal active orbital space that leads to a qualitatively correct description of the resonance Raman spectrum of uracil should encompass parts of the σ/σ* bonding/anti‐bonding orbitals of the pyrimidine ring. We trace these findings back to the considerable entanglement between the σ/σ* bonding/anti‐bonding as well as valence π/π* orbitals in the excited‐state electronic structure of uracil, which indicates non‐negligible non‐dynamical correlation effects that are less pronounced in the electronic ground state.

Externally and internally corrected coupled cluster approaches: an overview

The single-reference (SR) coupled-cluster (CC) approaches proved to be remarkably efficient in handling the dynamic correlation and continue to be widely exploited in quantum chemical computations of the molecular electronic structure. Yet, in the presence of quasi-degeneracy, e.g., when handling molecules away from their equilibrium geometry or when dealing with strongly correlated systems, a proper account of a non-dynamic correlation becomes essential and calls, in general, for a multi-reference (MR) type methodology. However, in view of the ambiguity, complexity, and computational demands of such MR approaches it is tempting to design SR CC type methods that are capable of accommodating both kinds of correlation effects. One avenue how to achieve this goal is to employ the complementarity of perturbative (i.e., CC) and variational (i.e., UHF, VB, CI, CAS SCF, etc.) type approaches and exploit the latter to remedy the CC methods—primarily CCSD or CCSD(T)—for the lack of the static or non-dynamic correlation effects. This leads to the so-called externally corrected (ec) ecCCSD or ecCCSD(T) approaches, which amend the standard CCSD or CCSD(T) methods by accounting for higher-than-pair clusters, extracted from various external sources. The same goal that leads simultaneously to more efficient SR CC algorithms may also be achieved by an effective implicit account of higher-order clusters via the so-called internally corrected methods (e.g., ACP-D45, ACCD, ACPQ, etc). While both types of these approaches were formulated and exploited long time ago, they recently enjoyed a certain renaissance. The objective of this work is to review, classify, and interrelate these efforts and highlight the advances made in this direction.

H4: A challenging system for natural orbital functional approximations.

The correct description of nondynamic correlation by electronic structure methods not belonging to the multireference family is a challenging issue. The transition of D(2h) to D(4h) symmetry in H4 molecule is among the most simple archetypal examples to illustrate the consequences of missing nondynamic correlation effects. The resurgence of interest in density matrix functional methods has brought several new methods including the family of Piris Natural Orbital Functionals (PNOF). In this work, we compare PNOF5 and PNOF6, which include nondynamic electron correlation effects to some extent, with other standard ab initio methods in the H4 D(4h)/D(2h) potential energy surface (PES). Thus far, the wrongful behavior of single-reference methods at the D(2h)-D(4h) transition of H4 has been attributed to wrong account of nondynamic correlation effects, whereas in geminal-based approaches, it has been assigned to a wrong coupling of spins and the localized nature of the orbitals. We will show that actually interpair nondynamic correlation is the key to a cusp-free qualitatively correct description of H4 PES. By introducing interpair nondynamic correlation, PNOF6 is shown to avoid cusps and provide the correct smooth PES features at distances close to the equilibrium, total and local spin properties along with the correct electron delocalization, as reflected by natural orbitals and multicenter delocalization indices.

Assessment of TD-DFT and LF-DFT for study of d - d transitions in first row transition metal hexaaqua complexes.

Herein, we present the systematic, comparative computational study of the d - d transitions in a series of first row transition metal hexaaqua complexes, [M(H2O)6](n+) (M(2+/3+) = V (2+/3+), Cr(2+/3+), Mn(2+/3+), Fe(2+/3+), Co(2+/3+), Ni(2+)) by the means of Time-dependent Density Functional Theory (TD-DFT) and Ligand Field Density Functional Theory (LF-DFT). Influence of various exchange-correlation (XC) approximations have been studied, and results have been compared to the experimental transition energies, as well as, to the previous high-level ab initio calculations. TD-DFT gives satisfactory results in the cases of d(2), d(4), and low-spin d(6) complexes, but fails in the cases when transitions depend only on the ligand field splitting, and for states with strong character of double excitation. LF-DFT, as a non-empirical approach to the ligand field theory, takes into account in a balanced way both dynamic and non-dynamic correlation effects and hence accurately describes the multiplets of transition metal complexes, even in difficult cases such as sextet-quartet splitting in d(5) complexes. Use of the XC functionals designed for the accurate description of the spin-state splitting, e.g., OPBE, OPBE0, or SSB-D, is found to be crucial for proper prediction of the spin-forbidden excitations by LF-DFT. It is shown that LF-DFT is a valuable alternative to both TD-DFT and ab initio methods.

11,11-dimethyl-1,6-methano[10]annulene--an annulene with an ultralong CC bond or a fluxional molecule?

Extensive quantum chemical calculations involving more than 20 different methods and including vibrational, temperature, entropic, and environmental corrections suggest that 11,11-dimethyl-1,6-methano[10]annulene (1) is characterized by a broad, asymmetric single well potential minimum in which the molecule can carry out a large-amplitude vibration. This result is obtained by using CASPT2(14,14) and CCSD(T) together with a VTZ basis set. The average R(C1C6) distance of 1 is close to 1.8 Å, in agreement with X-ray diffraction measurements. Lower level methods fail because a reliable account of the electronic structure of bridged annulenes requires a balanced description of nondynamical and dynamical electron correlation effects as well as a correct assessment of bridge-annulene interactions. An independent determination of the distance R using the mean deviation between the calculated and measured (13)C NMR chemical shifts of 1 leads to a value of 1.79 Å. By using electron density, energy density, and the local C1C6 stretching mode, it is demonstrated that the covalent bond ceases to exist at 1.695 Å and that for larger R values through-space homoaromatic interactions lead to some stabilization. The peculiar potential of 1 is shown to be a result of the interaction of the methyl groups with the perimeter CC bonds bisected by the symmetry plane of the molecule. CASPT2(14,14), CASPT2(10,10), CCSD(T), and BD(T) calculations were also used to provide for the first time reliable descriptions of the valence tautomeric potentials for the parent molecule, 1,6-methano[10]annulene (2), and the system 1,3,5-cycloheptatriene-norcardiene (3). In the latter case, calculations confirm a previous kinetic measurement of the free activation energy but correct NMR-based estimates. The methodology described can be applied to other annulenes and fullerenes.

Electronic Structure of One-Electron-Oxidized Form of the Methylcobalamin Cofactor: Spin Density Distribution and Pseudo-Jahn–Teller Effect

The electronic and structural properties of the one-electron-oxidized form of methylcobalamin (MeCbl) cofactor have been investigated using density functional theory (DFT) and CASSCF/MC-XQDPT2 calculations. We applied two types of functionals (hybrid and GGA) which produced quite different results in terms of spin density profiles: the B3LYP description was consistent with Co(III) and the π-cation corrin radical while the BP86 result was more in line with the Co(IV) oxidation state. A closer inspection of both outcomes indicates that the oxidized species have a mixed π-cation corrin radical and Co(III)/Co(IV) character. This mixed character was further supported by high-level ab initio CASSCF/MC-XQDPT2 calculations, which reveal the strong mixing of the electronic states due to nondynamical correlation effects. The near degeneracy, which takes place between the ground and first excited state, was consistent with the presence of a pseudo-Jahn-Teller (pJT) effect in the oxidized form of MeCbl. In addition, the DFT-based investigation of the structurally related porphyrin complexes gives a description consistent with corrin-based analogues and reveals that the corrin species have more Co(IV) character. The most important finding of the present study, regardless of the type of functional used, was the significant lowering of dissociation energy (∼35%), which might be due to the partial depopulation of the Co-C σ orbital upon removal of an electron.

A simple DFT-based diagnostic for nondynamical correlation

We propose a simple DFT-based diagnostic for nondynamical correlation effects, namely Aλ = (1 - TAE[XλC]/TAE[XC])/λ where TAE stands for the molecular total atomization energy, XC is a pure-DFT exchange-correlation functional, and XλC represents the corresponding hybrid with 100λ % Hartree–Fock-type exchange. The diagnostic is a good predictor for sensitivity of energetics to the level of theory, unlike most wavefunction- based diagnostics. For GGA functionals, Aλ values approaching unity indicate severe nondynamical correlation, while values between 0 and about 0.1 indicate systems where correlation is predominantly dynamical in character (or entirely absent). The diagnostic is only weakly sensitive to the basis set (beyond polarized valence double zeta) and can easily be applied to problems beyond the practical reach of wavefunction ab initio methods required for other diagnostics. We also propose a simple measure for the importance of dynamic correlation..