Nondynamical Correlation Research Articles

An efficient method for strongly correlated electrons in two-dimensions.

This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen square lattices. The singlet ground-state is studied, as well as the doublet mixed quantum state obtained by extracting an electron from the system. Our method satisfies two-index necessary N-representability conditions of the two-particle RDM (2RDM) and guarantees the conservation of the total spin. We show the ability of PNOF7 to describe strong correlation effects in two-dimensional (2D) systems by comparing our results with the exact diagonalization, density matrix renormalization group (DMRG), and auxiliary-field quantum Monte Carlo calculations. PNOF7 overcomes variational 2RDM methods with two- and three-index positivity N-representability conditions, reducing computational cost to mean-field scaling. Consistent results are obtained for small and large systems up to 144 electrons, weak and strong correlation regimes, and many filling situations. Unlike other methods, there is no dependence on dimensionality in the results obtained with PNOF7 and no particular difficulties have been observed to converge PNOF7 away from half-filling. Smooth double occupancy of sites is obtained, regardless of the filling. Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic correlation dramatically affects electron detachment energies. PNOF7 compares well with DMRG along the dissociation curve.

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.

Embracing local suppression and enhancement of dynamic correlation effects in a CASΠDFT method for efficient description of excited states.

The recently proposed CASΠDFT method combines the reliable description of nondynamic electron correlation with the complete active space (CAS) wavefunction and the efficient treatment of dynamic correlation by density functional theory (DFT). This marriage is accomplished by adopting the DFT correlation energy functional modified with the local correction function of the on-top pair density (Π). The role of the correction function is to sensitize the correlation functional to local effects of suppression and enhancement of dynamic correlation and to account for an adequate amount of dynamic correlation energy. In this work we show that the presence of covalent and ionic configurations in a wavefunction gives rise to spatial regions where the effects of suppression and enhancement of correlation energy, respectively, dominate. The results obtained for the potential energy curves of the excited states of the hydrogen molecule prove that CASΠDFT is reliable for states that change their character along the dissociation curve. The method is also applied to the lowest excited states of six-membered heterocyclic nitrogen compounds such as pyridine, pyrazine, pyrimidine, and pyridazine. The obtained excitation energies for the n → π* and π → π* excitations confirm the good performance of CASΠDFT for excited states. The absolute average error of the method is 0.1 eV lower than that of the CCSD method and higher by the same amount than that of the more expansive CC3 variant. Compared with the coupled cluster methods, this encouraging performance of CASΠDFT is achieved at the negligible computational cost of obtaining the correlation energy.

Reproducing benchmark potential energy curves of molecular bond dissociation with small complete active space aided with density and density-matrix functional corrections.

Various effects of electron correlation accompany molecular bond dissociation, which makes the efficient calculation of potential energy curves a notoriously difficult problem. In an attempt to reliably reproduce both absolute energies and shapes of the benchmark dissociation curves, calculations with the combined CASΠDFT method are carried out for the prototype molecules H2, BH, F2, and N2. The complete active space (CAS) part of CASΠDFT accounts for long-range nondynamic correlation, while short-range dynamic correlation is accounted for with the corrected Lee-Yang-Parr correlation functional of density functional theory (DFT). The correction represents the suppression of dynamic correlation with nondynamic correlation, and it is a function of the ratio x(r) between the conditional and conventional densities obtained with the CAS on-top pair density Π(r). For the single-bonded molecules H2, BH, and F2, CASΠDFT succeeds in reproducing the shapes and absolute energies (for H2 and BH) of the benchmark curves, while for the triple-bonded N2 molecule, the addition to CASΠDFT of a multibond correction is required. It accounts for the middle-range dynamic correlation of the same-spin electrons in the (symmetrized) high-spin atomic electron configurations of the dissociating N2.

M11plus: A Range-Separated Hybrid Meta Functional with Both Local and Rung-3.5 Correlation Terms and High Across-the-Board Accuracy for Chemical Applications.

The way to improve Kohn-Sham density functional theory is to improve the exchange-correlation functionals, and functionals have been successively improved by adding new ingredients, especially local spin density gradients, nonlocal Hartree-Fock exchange, and local meta terms based on kinetic energy density. Here, we present a new kind of functional obtained by adding rung-3.5 terms to a functional including local gradients, local meta terms, and range-separated Hartree-Fock exchange. A rung-3.5 term has short-range nonlocality designed to account for nondynamic correlation; we add two kinds of rung-3.5 terms, one kind modeled on position-dependent Hartree-Fock exchange and another modeled on the spin density at a point interacting with the opposite-spin exchange hole at the same point. Optimization of the functional yields broad accuracy for both ground states and excited states with especially significant improvement for systems with strong correlation.

Complete active space and corrected density functional theories helping each other to describe vertical electronic π → π* excitations in prototype multiple-bonded molecules.

The CASΠDFT method, which combines the complete active space (CAS) wave function approach and density functional theory (DFT), offers an efficient description of important excitations to the lowest excited states. CASΠDFT employs a correlation DFT functional corrected with a function P[x] of the ratio xr of the conditional and conventional electron densities obtained with the CAS on-top pair density Π(r). The sectors of P[x] for x(r) ≤ 1 and x(r) > 1 represent the opposite effects of the suppression of dynamic correlation with nondynamic correlation and its enhancement due to the ionic-type excitation. The present combination of the self-consistent-field CAS and the corrected Lee-Yang-Parr correlation functional closely reproduces in the relatively small double-zeta basis the benchmark experimental lowest singlet vertical π → π* excitations in the prototype multiple-bonded molecules N2, CO, C2H2, and C2H4.

Performance of new density functionals of nondynamic correlation on chemical properties.

Becke'13 (B13) is a general density functional theory (DFT) designed for strongly correlated molecules. Kong-Proynov'16/B13 (KP16/B13) is a modification of B13 that treats the correlation kinetic term through modeling the adiabatic connection and contains fewer parameters. A question remains whether functionals designed for extremely strong correlations can handle correlation of normal strength. In this work, both functionals are benchmarked with a variety of molecular systems from the Minnesota-2015 datasets and compared with the mainstream functionals such as B3LYP and a spectrum of other types of functionals, as well as Becke'05 (B05), the predecessor of B13. Our calculations, made possible with an efficient self-consistent-field implementation of those new functionals, show that B05, B13, and KP16/B13, based on full HF exchange, recover a majority of the correlation where it is very significant (i.e., where HF has large errors). They perform very well for reaction barriers. On the other hand, their performance on multireference systems is mixed. Overall, B05, B13, and KP16/B13 are competitive to B3LYP, and B05 is even competitive to heavily parameterized functionals for most sets despite containing 100% HF exchange. KP16/B13, in particular, contains only three empirical parameters, demonstrating the possibility of treating correlation of all strength with single-determinant Kohn-Sham DFT.

Simultaneous Attenuation of Both Self-Interaction Error and Nondynamic Correlation Error in Density Functional Theory: A Spin-Pair Distinctive Adiabatic-Connection Approximation.

We present a spin-pair distinctive algorithm in the context of adiabatic-connection fluctuation-dissipation (ACFD) theorem, which enables to quantify the self-interaction error (SIE) and the nondynamic/strong correlation error (NCE) in the direct random-phase approximation (dRPA). Using this knowledge, we propose a spin-component scaled dRPA (scsRPA) correlation model with simultaneous attenuation of both the SIE and the NCE. Along with the exact exchange, scsRPA is shown to present a comprehensive improvement over dRPA, as well as the well-established PBE and PBE0 functionals, for bonding energies of pronounced multireference characters and transition-metal complexes of strongly correlated systems, while consistently providing an accurate description for reaction energies, reaction barriers, and noncovalent bond interactions of weakly correlated systems.

Natural orbital functional for spin-polarized periodic systems

Natural orbital functional theory is considered for systems with one or more unpaired electrons. An extension of the Piris natural orbital functional (PNOF) based on electron pairing approach is presented, specifically, we extend the independent pair model, PNOF5, and the interactive pair model PNOF7 to describe spin-uncompensated systems. An explicit form for the two-electron cumulant of high-spin cases is only taken into account, so that singly occupied orbitals with the same spin are solely considered. The rest of the electron pairs with opposite spins remain paired. The reconstructed two-particle reduced density matrix fulfills certain N-representability necessary conditions, as well as guarantees the conservation of the total spin. The theory is applied to model systems with strong non-dynamic (static) electron correlation, namely, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, PNOF7 compares well with exact diagonalization results so the model presented here is able to provide a correct description of the strong-correlation effects.

Electron correlation energy with a combined complete active space and corrected density-functional approach in a small basis versus the reference complete basis set limit: A close agreement

Approximate CASΠDFT scheme of the Ec calculation is considered, in which its non-dynamic Ecnd and dynamic Ecd components are calculated with the complete active space (CAS) configuration interaction (CI) for valence bonds in a small basis and density functional theory (DFT) in the same basis, respectively. The DFT contribution is corrected for the suppression of dynamic correlation (SDC) with non-dynamic correlation. The present CASΠDFT closely reproduces both complete basis set (CBS) Ec limits and their trends in the series of the prototype multiple-bonded molecules C2, HCN, N2, and CO.

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.

Efficient evaluation of electron correlation along the bond-dissociation coordinate in the ground and excited ionic states with dynamic correlation suppression and enhancement functions of the on-top pair density

Calculation of the electron correlation energy ${E}_{c}$ of ground and excited states through the partitioning and evaluation of the dynamic ${E}_{c}^{d}$ and nondynamic ${E}_{c}^{nd}$ components requires one to account for the interplay of these correlation modes as well as for the excitation effect. In this paper it is demonstrated that both local suppression of dynamic correlation (SDC) by nondynamic correlation and enhancement of dynamic correlation (EDC) due to excitation to a state of the ionic nature can be quantified with the ratio $x(\mathbit{r})$ between the correlated and uncorrelated on-top pair densities $\mathrm{\ensuremath{\Pi}}(\mathbit{r})$. A $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$ scheme is proposed, in which ${E}_{c}^{nd}$ is calculated with the complete active space approach in a small basis, while ${E}_{c}^{d}$ is calculated in the same basis with a functional of density functional theory corrected for SDC and EDC with an original correction function of $x(\mathbit{r})$. Correlation energies calculated with $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$ along the bond-dissociation coordinate for the paradigmatic ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ molecules as well as for the ${\mathrm{C}}_{2}$ molecule with strong nondynamic correlation at the equilibrium agree well with the reference data, thus providing a proof of concept for $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$.

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.

Electronic processes in NO dimerization on Ag and Cu clusters: DFT and MRMP2 studies.

Experimentally observed NO dimerization on Cu and Ag surfaces is surprising because binding energy of NO dimer is very small in gas phase. MRMP2, MP2 to MP4, CCSD(T), and DFT studies of NO dimerization on Ag2 and Cu2 clusters disclosed that the CCSD(T) method could be applied to this reaction on Ag2 and Cu2 unlike NO dimerization in gas phase which exhibits significantly large nondynamical electron correlation effect. Charge-transfer (CT) from Ag2 and Cu2 to NO moieties plays important role in NN bond formation between two NO molecules. This CT considerably decreases nondynamical correlation effect. Also, the DFT method could be applied to this NO dimerization, if appropriate DFT functional is used; all pure functionals examined here and most of the hybrid functionals underestimated the activation barrier (Ea ), while only ωB97X provided Ea similar to CCSD(T)-calculated value. NO dimerization on similar Cu2 and Cu5 needs moderately larger Ea than those on Ag2 and Ag5 , because frontier orbital participating in the CT exists at lower energy in Cu2 and Cu5 than in Ag2 and Ag5 . The Ea decreases in the order Ag2 >> Ag38 > Ag7 ∼ Ag5 and the reaction energy (ΔE) is positive (endothermic) in Ag2 but significantly negative in Ag38 , Ag7 , and Ag5 , indicating that various Ag clusters could be effective for NO dimerization except for Ag2 . The decreasing order of Ea and increasing order of exothermicity are attributed to increasing order of the frontier orbital energy of Ag2 < Ag38 < Ag7 ∼ Ag5 . © 2018 Wiley Periodicals, Inc.

Density functionals for nondynamical correlation constructed from an upper bound to the exact exchange energy density

ABSTRACTHyper-generalised-gradient approximations (hGGAs) for the exact exchange-correlation functional are increasingly popular in density functional theory. HGGAs model nondynamical correlation using a flexible local combination of exact (Hartree–Fock, HF) exchange and approximate exchange. We present a simplified ‘Rung 3.5’ upper bound to the HF exchange energy density, the essential ingredient of hGGAs. We also present a nonempirical generalised gradient approximation for this upper bound. Both upper bounds go to zero in the high-density and density tail limits, facilitating the construction of hGGAs that recover HF exchange in these limits. The ‘Rung 3.5’ construction enables facile evaluation of analytic derivatives and calculations in periodic boundary conditions. Extensive numerical tests show that the upper bounds capture the critical difference between HF and approximate exchange, showing these ingredients' promise for building simple hGGAs. The tests also indicate a need for more sophisticated semi-local upper bounds.

Practical approach for beryllium atomic clusters: TD-DFT potential energy surfaces from equilibrium to dissociation for excited states of 2s → 2p

To shed light on the spectroscopic energy-smearing mechanism of the recently developed technique for multi-element analysis (e.g., forensic tests on artworks), which is called plume laser-induced fluorescence, practically computer simulations need to be accurate and fast enough for generating wide-range excited-state potential energy surfaces (PESs) from equilibrium to dissociation of large (non-equilibrated) atomic clusters. Currently, all the very high-level post-Hartree–Fock theories (e.g., coupled-cluster theory) are computationally too expensive for this purpose (e.g., taking at least ~ 23 years). After considering the extensive applications of time-dependent density-functional theory (TD-DFT) in computing the optical properties of many-body systems, we wonder how TD-DFT performs in the wide-range-PES generations because no such TD-DFT application is found by us in the literature (perhaps due to its insufficient static/non-dynamical correlation energy). In this work, we carried out TD-DFT calculations on beryllium atomic clusters from Be1 to Be50 (in contrast to open-shell H2 or Li2 system, we think the individual closed-shell Be systems at the dissociation limit could help TD-DFT to describe the static/non-dynamical correlation. For example, it is usually considered that Be2 has zero non-dynamical correlation energy in the literature) with and without Tamm–Dancoff approximation (TDA). Electronic excited-state properties (e.g., PES, equilibrium position, dissociation energy, fork intersection) associated with 2s → 2p are systematically investigated at a total of 48 levels of TD-DFT. Our calculations are thoroughly benchmarked against experimental results and (very) high-level theoretical values. In short, we find that for accurate simulations, it is necessary to use diffuse basis sets and the TDA approximation. TD-DFT with the small basis set 3-21+G* took ~ 2 min to finish a single-point excitation energy calculation on a linear Be50 (a toy model), while the large basis set 6-311+G(2df,2p) took 30 min. By contrast, a couple-cluster theory EOM-CCSD with 6-311+G(2df,2p) took 2.5 h to do the same but merely for a smaller linear Be10. In general, we suggest that TD-DFT:TDA with 3-21+G* should be practically fast and accurate enough for emulating our desired wide-range excited-state PESs of large Be atomic clusters (including conical intersection points of close energies) in the future studies. Nevertheless, benchmarking work on (much) larger sizes of Be clusters will always be more desirable whenever its computational costs at very high level of theories becomes more tractable.

Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix.

The use of the mixed reference (MR) reduced density matrix, which combines reduced density matrices of the MS = +1 and -1 triplet-ground states, is proposed in the context of the collinear spin-flip-time-dependent density functional theory (SF-TDDFT) methodology. The time-dependent Kohn-Sham equation with the mixed state is solved by the use of spinor-like open-shell orbitals within the linear response formalism, which enables to generate additional configurations in the realm of TD-DFT. The resulting MR-SF-TDDFT computational scheme has several advantages before the conventional collinear SF-TDDFT. The spin-contamination of the response states of SF-TDDFT is nearly removed. This considerably simplifies the identification of the excited states, especially in the "black-box" type applications, such as the automatic geometry optimization, reaction path following, or molecular dynamics simulations. With the new methodology, the accuracy of the description of the excited states is improved as compared to the collinear SF-TDDFT. Several test examples, which include systems typified by strong non-dynamic correlation, orbital (near) degeneracy, and conical intersections, are given to illustrate the performance of the new method.

Post-CCSD(T) contributions to total atomization energies in multireference systems.

We examine the magnitude and the basis set convergence of post-coupled-cluster with single, double, and perturbative triple excitations (CCSD(T)) contributions (up to CCSDTQ567) for a wide and diverse set of 21 first- and second-row molecules with up to four non-hydrogen atoms. We focus on multireference systems for which post-CCSD(T) effects are particularly pronounced. The considered molecules are BN(1∑+), C2(1∑+), O2, FO, P2, S2, ClO, N2O, NO2, O3, FNO, FO2, F2O, S2O, S3, ClNO, ClOO, Cl2O, N2C2, P4, and S4. This set spans the gamut from molecules dominated by moderate nondynamical correlation (e.g., FO, ClO, NO2, S2O, N2C2, and P4) to systems dominated by strong nondynamical correlation (e.g., BN, C2, FO2, O3, ClOO, and S4). We examine the basis set convergence of the CCSDT, CCSDT(Q), CCSDTQ, CCSDTQ(5), CCSDTQ5, CCSDTQ5(6), CCSDTQ56, CCSDTQ56(7), and CCSDTQ567 methods. The largest basis sets employed in each category are cc-pV6Z (CCSDT(Q)), cc-pV5Z (CCSDTQ), cc-pVTZ (CCSDTQ5(6)), and cc-pVDZ (CCSDTQ567). Apart from examining the basis-set convergence of post-CCSD(T) contributions near the one-particle basis-set limit, this work explores cost-effective approaches for obtaining these contributions from fairly small basis sets. We consider both effective basis-set extrapolations and scaling factors. An important finding is that extrapolating the perturbative connected quadruples, (Q), from the cc-pVDZ(4s3p1d) and cc-pVTZ basis sets yields near basis-set limit results and represents a significant improvement relative to cc-pV{D,T}Z extrapolation at no additional computational cost (where cc-pVDZ(4s3p1d) is an extended version of the cc-pVDZ basis set). Combining the (Q)/cc-pV{D(4s3p1d),T}Z extrapolations with the fully iterative connected quadruples, Q-(Q), contribution calculated with the cc-pVDZ (or even the cc-pVDZ(3s2p)) basis set is a cost-effective way for obtaining the connected quadruples component close to the basis-set limit (where cc-pVDZ(3s2p) is a truncated version of the cc-pVDZ basis set). In addition, we show that the (5)/cc-pVDZ(3s2p) and (6)/cc-pVDZ(3s2p) components provide reasonable approximations for the connected quintuple and sextuple components close to the basis-set limit, respectively.

Phase dilemma in natural orbital functional theory from the N-representability perspective

Any rigorous approach to first-order reduced density (1RDM) matrix functional theory faces the phase dilemma, that is, having to deal with a large number of possible combinations of signs in terms of the electron-electron interaction energy. This problem was discovered by reducing a ground-state energy generated from an approximate N-particle wavefunction into a functional of the 1RDM, known as the top-down method. Here, we show that the phase dilemma also appears in the bottom-up method, in which the functional E[1RDM] is generated by progressive inclusion of N-representability conditions on the reconstructed two-particle reduced density matrix. It is shown that an adequate choice of signs is essential to accurately describe model systems with strong non-dynamic (static) electron correlation, specifically, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, the Piris natural orbital functional 7 (PNOF7), with phases equal to -1 for the inter-pair energy terms containing the exchange-time-inversion integrals, agrees with exact diagonalization results.

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.