On-top Pair Density Research Articles

Self-Adapting Short-Range Correlation Functional for Complete Active Space-Based Approximations.

We propose a short-range correlation energy correction tailored for active space wave function models. The correction relies on a short-range multideterminant correlation functional computed with a local range-separation parameter that self-adapts to the underlying wave function. This approach is analogous to that of Giner et al. [J. Chem. Phys. 2018, 149, 194301] which addresses the basis set incompleteness error, with the vital distinction that in our protocol the range-separation parameter remains finite in the complete basis set limit, ensuring nonzero short-range correlation. The proposed correlation functional compensates for the missing short-range correlation via two mechanisms: (i) an automatically adapting short-range parameter, which gauges the missing correlation in the electron vicinity, and (ii) the functional's explicit dependence on the on-top pair density, which reduces short-range correlation in regions where electron correlation is mainly static. We integrate our method into the multireference adiabatic connection theory for CASSCF wave functions. The performance of the introduced CAS-AC0-(c,md) model is verified by calculating potential energy curves for alkaline-earth metal dimers (Be2, Mg2, Ca2) and for the chromium dimer, in all cases obtaining promising results.

A Merger of the Spin-Flip ORMAS Approach and the MC-PDFT Method.

The SF-ORMAS-PDFT (spin-flip occupation restricted multiple active space-pair density functional theory) approach combines the SF-ORMAS-CI method with the MC-PDFT method to treat both static and dynamic correlation in multiconfigurational systems. The static correlation description is generated via the spin-flip approach, which uses a high-spin single reference determinant to treat excited states with multiconfigurational characters. The on-top pair density functional theory uses a translation scheme applied to GGA density functionals. The SF-ORMAS-PDFT scheme has also been combined with virtual valence orbitals (VVO), a well-defined subspace of the virtual molecular orbitals, giving rise to significant speedups relative to the use of the full virtual space. The accuracy of the SF-ORMAS-PDFT method is tested by calculating 65 vertical excitation energies of 12 small- and medium-sized organic molecules. The SF-ORMAS-PDFT vertical excitation energies calculated with VVOs are comparable to those calculated with the full virtual space. The SF-ORMAS-PDFT/6-31G(d) level of theory predicts the rotational barrier of ethylene to be 65.5 and 65.9 kcal/mol, with full virtual space and VVOs, respectively. These predicted barrier heights compare well with the experimental value of 65 kcal/mol.

Variational Pair-Density Functional Theory: Dealing with Strong Correlation at the Protein Scale.

Multiconfigurational pair-density functional theory (MC-PDFT) offers a promising solution to the challenges faced by traditional density functional theory (DFT) in addressing molecular systems containing transition metals, open-shells, or strong correlations in general. By utilizing both the density and on-top pair-density, MC-PDFT can make use of a more flexible multiconfigurational wave function to capture the necessary static correlation, while the pair-density functional also includes the effect of dynamic correlation. So far, MC-PDFT has been used after a multiconfigurational self-consistent field (MCSCF) step, using the orbitals and configuration interaction coefficients from the converged MCSCF wave function to compute PDFT energies and properties. Here, instead, we propose to perform a direct optimization of the wave function using the pair-density functionals, resulting in a variational formulation of MC-PDFT. We derive the expressions for the wave function gradient and illustrate their similarity to standard MCSCF equations. Furthermore, we illustrate the accuracy on a set of singlet-triplet gaps as well as dissociation curves. Our findings highlight one of MC-PDFT's standout features: a reduced dependency on the active space size compared to conventional multiconfigurational wave function methodologies. Additionally, we show that the computational cost of MC-PDFT is potentially lower than MCSCF and often on-par with standard Kohn-Sham DFT, which is demonstrated by performing a MC-PDFT calculation of the entire ferredoxin protein with 1447 atoms and nearly 12000 basis functions.

Application of Multiconfiguration Pair-Density Functional Theory to the Diels-Alder Reaction.

Transition states for Diels-Alder reactions are strongly correlated, as evidenced by high-to-very-high M diagnostics, and therefore they require treatment by multireference methods. Multiconfiguration pair-density functional theory (MC-PDFT) combines a multiconfiguration wave function with a functional of the electron density and the on-top pair density to calculate the electronic energy for strongly correlated systems at a much lower cost than wave function methods that do not employ density functionals. Here we apply MC-PDFT to the Diels-Alder cycloaddition reaction of 1,3-butadiene with ethylene, where two kinds of reaction paths have been widely studied: concerted synchronous paths and diradical stepwise paths. The lowest-energy reaction path is now known to be a concerted synchronous one, and a method's ability to predict this is an important test. By comparison to the best available theoretical results in the literature, we test the accuracy of MC-PDFT with several choices of on-top functional for geometries and enthalpies of stable structures along both paths and for the transition state geometries. We also calculate the Arrhenius activation energies for both paths and compare these to experiment. We also compare to Kohn-Sham density functional theory (KS-DFT) with selected exchange-correlation functionals. CAS-PDFT gives consistently good energies and geometries for both the concerted and stepwise mechanisms, but none of the KS-DFT functionals gives accurate activation energies for both. The stepwise transition state is very strongly correlated, and MC-PDFT can treat it, but KS-DFT (which involves a single-configuration treatment) has larger errors. The results confirm that using a multiconfigurational reference function for strongly correlated transition states can significantly improve the reliability and that MC-PDFT can provide good accuracy at a much lower computational cost than competing multireference methods.

Double excitations in molecules from ensemble density functionals: Theory and approximations

Double excitations, which are dominated by a Slater determinant with both electrons in the highest occupied molecular orbital promoted to the lowest unoccupied orbital(s), pose significant challenges for low-cost electronic structure calculations based on density-functional theory (DFT). Here, we demonstrate that recent advances in ensemble DFT [Gould et al., Phys. Rev. Lett. 125, 233001 (2020)], which extend concepts of ground-state DFT to excited states via a rigorous physical framework based on the ensemble fluctuation-dissipation theorem, can be used to shed light on the double-excitation problem. We find that the exchange physics of double excitations is reproducible by standard DFT approximations using a linear combination formula, but correlations are more complex. In passing, to analyze correlation, we extend the random-phase approximation to ensembles. We then show, using selected test systems, that standard DFT approximations may be adapted to tackle double excitations based on theoretically motivated simple formulas that employ ensemble extensions of expressions that use the on-top pair density.

Multiconfiguration Pair-Density Functional Theory for Transition Metal Silicide Bond Dissociation Energies, Bond Lengths, and State Orderings.

Transition metal silicides are promising materials for improved electronic devices, and this motivates achieving a better understanding of transition metal bonds to silicon. Here we model the ground and excited state bond dissociations of VSi, NbSi, and TaSi using a complete active space (CAS) wave function and a separated-pair (SP) wave function combined with two post-self-consistent field techniques: complete active space with perturbation theory at second order and multiconfiguration pair-density functional theory. The SP approximation is a multiconfiguration self-consistent field method with a selection of configurations based on generalized valence bond theory without the perfect pairing approximation. For both CAS and SP, the active-space composition corresponds to the nominal correlated-participating-orbital scheme. The ground state and low-lying excited states are explored to predict the state ordering for each molecule, and potential energy curves are calculated for the ground state to compare to experiment. The experimental bond dissociation energies of the three diatomic molecules are predicted with eight on-top pair-density functionals with a typical error of 0.2 eV for a CAS wave function and a typical error of 0.3 eV for the SP approximation. We also provide a survey of the accuracy achieved by the SP and extended separated-pair approximations for a broader set of 25 transition metal–ligand bond dissociation energies.

Localized Active Space Pair-Density Functional Theory

Accurate quantum chemical methods for the prediction of spin-state energy gaps for strongly correlated systems are computationally expensive and scale poorly with the size of the system. This makes calculations for many experimentally interesting molecules impractical even with abundant computational resources. Previous work has shown that the localized active space (LAS) self-consistent field (SCF) method can be an efficient way to obtain multiconfiguration SCF wave functions of comparable quality to the corresponding complete active space (CAS) ones. To obtain quantitative results, a post-SCF method is needed to estimate the complete correlation energy. One such method is multiconfiguration pair-density functional theory (PDFT), which calculates the energy based on the density and on-top pair density obtained from a multiconfiguration wave function. In this work, we introduce localized-active-space PDFT, which uses a LAS wave function for subsequent PDFT calculations. The method is tested by computing spin-state energies and gaps in conjugated organic molecules and a bimetallic compound and comparing to the corresponding CAS-PDFT values.

Multiconfiguration Pair-Density Functional Theory.

Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

Spin-Flip Pair-Density Functional Theory: A Practical Approach To Treat Static and Dynamical Correlations in Large Molecules

We present a practical approach to treat static and dynamical correlation accurately in large multiconfigurational systems. The static correlation is taken into account by using the spin-flip approach, which is well-known for capturing static correlation accurately at low-computational expense. Unlike previous approaches to add dynamical correlation to spin-flip models which use perturbation theory or coupled-cluster theory, we explore the ability to use the on-top pair-density functional theory approaches recently developed by Gagliardi and co-workers (J. Comput. Theor. Chem., 2014, 10, 3669). External relaxations are performed in the spin-flip calculations through a restricted active space framework for which a truncation scheme for the orbitals used in the external excitation is presented. The performance of the approach is demonstrated by computing energy gaps between ground and excited states for diradicals, triradicals, and linear polyacene chains ranging from naphthalene to dodecacene. Accurate results are obtained using the new approach for these challenging open-shell molecular systems.

Learning on-top: Regressing the on-top pair density for real-space visualization of electron correlation.

The on-top pair density [Πr] is a local quantum-chemical property that reflects the probability of two electrons of any spin to occupy the same position in space. Being the simplest quantity related to the two-particle density matrix, the on-top pair density is a powerful indicator of electron correlation effects, and as such, it has been extensively used to combine density functional theory and multireference wavefunction theory. The widespread application of Π(r) is currently hindered by the need for post-Hartree-Fock or multireference computations for its accurate evaluation. In this work, we propose the construction of a machine learning model capable of predicting the complete active space self-consistent field (CASSCF)-quality on-top pair density of a molecule only from its structure and composition. Our model, trained on the GDB11-AD-3165 database, is able to predict with minimal error the on-top pair density of organic molecules, bypassing completely the need for ab initio computations. The accuracy of the regression is demonstrated using the on-top ratio as a visual metric of electron correlation effects and bond-breaking in real-space. In addition, we report the construction of a specialized basis set, built to fit the on-top pair density in a single atom-centered expansion. This basis, cornerstone of the regression, could be potentially used also in the same spirit of the resolution-of-the-identity approximation for the electron density.

Combining density-based dynamical correlation with a reduced-density-matrix strong-correlation description

A combined density and first-order reduced-density-matrix (1RDM) functional method is proposed for the calculation of potential energy curves (PECs) of molecular multibond dissociation. Its 1RDM functional part, a pair density functional, efficiently approximates the ab initio pair density of the complete active space self-consistent-field (CASSCF) method. The corresponding approximate on top pair density {\Pi} is employed to correct for double counting a correlation functional of density functional theory (DFT). The proposed ELS-DM{\Pi}DFT method with the extended L\"owdin-Shull (ELS) 1RDM functional with dispersion and multibond (DM) corrections augmented with the {\Pi}DFT functional closely reproduces PECs of multibond dissociation in the paradigmatic N_2 , H_2O, and H_2CO molecules calculated with the recently proposed CAS{\Pi}DFT (CASSCF augmented with a {\Pi} based scaled DFT correlation correction) method. Furthermore, with the additional M-correction, ELS-DM{\Pi}DFT+M reproduces well the benchmark PEC of the N_2 molecule by Lie and Clementi.

Local Enhancement of Dynamic Correlation in Excited States: Fresh Perspective on Ionicity and Development of Correlation Density Functional Approximation Based on the On-Top Pair Density.

We discuss the interplay between the nondynamic and dynamic electron correlation in excited states from the perspective of the suppression of dynamic correlation (SDC) and enhancement of dynamic correlation (EDC) effects. We reveal that there exists a connection between the ionic character of a wave function and EDC. Following this finding we introduce a quantitative measure of ionicity based solely on local functions without referring to valence bond models. The ability to recognize both the SDC and EDC regions underlies the presented method, named CASΠDFT, combining complete active space (CAS) wave function and density functional theory (DFT) via the on-top pair density (Π) function. We extend this approach to excited states by devising an improved representation of the EDC effect in the correlation functional. The generalized CASΠDFT uses different DFT functionals for ground and excited states. Numerical demonstration for singlet π → π* excitations shows that CASΠDFT offers satisfactory accuracy at a fraction of the cost of the ab initio approaches.

Relativistic short-range exchange energy functionals beyond the local-density approximation.

We develop relativistic short-range exchange energy functionals for four-component relativistic range-separated density-functional theory using a Dirac-Coulomb Hamiltonian in the no-pair approximation. We show how to improve the short-range local-density approximation exchange functional for large range-separation parameters by using the on-top exchange pair density as a new variable. We also develop a relativistic short-range generalized-gradient approximation exchange functional that further increases the accuracy for small range-separation parameters. Tests on the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges show that the latter functional gives exchange energies with a maximal relative percentage error of 3%. The development of this exchange functional represents a step forward for the application of four-component relativistic range-separated density-functional theory to chemical compounds with heavy elements.

A basis-set error correction based on density-functional theory for strongly correlated molecular systems

We extend to strongly correlated molecular systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner et al., J. Chem. Phys. 149, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the electron-electron Coulomb interaction projected in the finite basis set. This enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set. To study both weak and strong correlation regimes, we consider the potential energy curves of the H10, N2, O2, and F2 molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (i.e., independence of the energy with respect to the spin projection Sz) and size consistency. Specifically, we investigate the dependence of the functional on different types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-ζ quality basis sets for most of the systems studied here. In addition, the present basis-set incompleteness correction provides smooth potential energy curves along the whole range of internuclear distances.

Long-range-corrected multiconfiguration density functional with the on-top pair density.

We propose a multiconfiguration density functional combining a short-range density functional approximation with a novel long-range correction for dynamic correlation effects. The correction is derived from the adiabatic connection formalism so that the resulting functional requires access only to one- and two-electron reduced density matrices of the system. In practice, the functional is formulated for wavefunctions of the complete active space (CAS) type and the short-range density functional part is made dependent on the on-top pair density via auxiliary spin densities. The latter allows for reducing the self-interaction and the static correlation errors without breaking the spin symmetry. We study the properties and the performance of the non-self-consistent variant of the method, termed lrAC0-postCAS. Numerical demonstration on a set of dissociation energy curves and excitation energies shows that lrAC0-postCAS provides accuracy comparable with more computationally expensive ab initio rivals.

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.

Scaling exchange and correlation in the on-top density functional of multiconfiguration pair-density functional theory: effect on electronic excitation energies and bond energies

Multiconfiguration pair-density functional (MC-PDFT) theory provides an economical way to calculate the ground-state and excited-state energetics of strongly correlated systems. The energy is calculated from the kinetic energy, density, and on-top pair-density of a multiconfiguration wave function as the sum of kinetic energy, classical Coulomb energy, and on-top density functional energy. We have usually found good results with the translated Perdew–Burke–Ernzerhof (tPBE) on-top density functional, and in this article, we examine whether the results can be systematically improved by introducing scaling constants into the exchange and correlation terms. We find that only a small improvement is possible for electronic excitation energies and that no improvement is possible for bond energies.

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.

Calculation of Chemical Reaction Barrier Heights by Multiconfiguration Pair-Density Functional Theory with Correlated Participating Orbitals.

The accurate description of reaction barrier heights is challenging for quantum mechanical methods due to the need for a balanced treatment of dynamic and static correlation energies because their importance varies during the course of a chemical reaction. While some regions of potential energy surfaces are well-described by a single-reference wave function or by Kohn-Sham density functional theory, in other cases a multireference treatment is needed. For systems with many active electrons, most accurate multireference methods have prohibitive computational scalings with system size. Multiconfiguration pair-density functional theory, MC-PDFT, is a more affordable multireference approach that computes the total electron correlation energy in a single step by using the multiconfiguration kinetic energy, density, and on-top pair density and an on-top density functional. In this work, we apply MC-PDFT to a benchmark database (DBH24/18) of 24 diverse reaction barrier heights. We explore the role of active space and basis set selection on the performance of MC-PDFT. We find that MC-PDFT is able to calculate reaction barrier heights with a similar accuracy to complete active space second order perturbation theory, CASPT2, but at a lower computational cost, and we find that MC-PDFT is less dependent on basis set selection than CASPT2.