Pair-density Functional Theory Research Articles

Reconciling Accuracy and Feasibility for Barrierless Reaction Steps by the PCS/DDCI/MC-PDFT Protocol: Methane and Ethylene Dissociations as Case Studies.

Several enhancements have been introduced into state-of-the-art computational protocols for the treatment of barrierless reaction steps in the framework of variable reaction coordinate variational transition state theory. The first step is the synergistic integration of the Iterative Difference Dedicated Configuration Interaction (I-DDCI) and Pisa Composite Scheme, which defines a reduced cost, yet very accurate, computational workflow. This approach provides a near black box tool for obtaining 1D reference potentials. Then, a general strategy has been devised for tuning the level of theory used in Monte Carlo (MC) sampling, employing Multiconfiguration Pair Density Functional Theory (MC-PDFT) with dynamically adjusted Hartree-Fock exchange. Concurrently, partial geometry optimizations during the MC simulations account for the coupling between the reaction coordinates and conserved modes. The protocol closely approaches full size consistency and yields highly accurate results, with several test computations suggesting rapid convergence of the I-DDCI correction with the basis set dimensions. The capabilities of the new platform are illustrated by two case studies (the hydrogen dissociation from CH4 and C2H4), which highlight its flexibility in handling different carbon hybridizations (sp3 and sp2). The remarkable accuracy of the computed rate constants confirms the robustness of the proposed method. Together with their intrinsic interest, these results pave the way for systematic investigations of complex gas-phase reactions through a reliable, user-friendly tool accessible to specialists and nonspecialists alike.

The Use of Effective Core Potentials with Multiconfiguration Pair-Density Functional Theory.

The reliable and accurate prediction of chemical properties is a key goal in quantum chemistry. Transition-metal-containing complexes can often pose difficulties to quantum mechanical methods for multiple reasons, including many electron configurations contributing to the overall electronic description of the system and the large number of electrons significantly increasing the amount of computational resources required. Often, multiconfigurational electronic structure methods are employed for such systems, and the cost of these calculations can be reduced by the use of an effective core potential (ECP). In this work, we explore both theoretical considerations and performances of ECPs applied in the context of multiconfiguration pair-density functional theory (MC-PDFT). A mixed-basis set approach is used, using ECP basis sets for transition metals and all-electron basis sets for nonmetal atoms. We illustrate the effects that an ECP has on the key parameters used in the computation of MC-PDFT energies, and we explore how ECPs affect the prediction of physical observables for chemical systems. The dissociation curve for a metal dimer was explored, and ionization energies for transition metal-containing diatomic systems were computed and compared to experimental values. In general, we find that ECP approaches employed with MC-PDFT are able to predict ionization energies with improved accuracy compared to traditional Kohn-Sham density functional theory approaches.

Electronic Structure Methods for Simulating Flavin's Spectroscopy and Photophysics: Comparison of Multi-reference, TD-DFT, and Single-Reference Wave Function Methods.

The use of flavins and flavoproteins in photocatalytic, sensing, and biotechnological applications has led to a growing interest in computationally modeling the excited-state electronic structure and photophysics of flavin. However, there is limited consensus regarding which computational methods are appropriate for modeling flavin's photophysics. We compare the energies of low-lying excited states of flavin computed with time-dependent density functional theory (TD-DFT), equation-of-motion coupled cluster (EOM-EE-CCSD), scaled opposite-spin configuration interaction [SOS-CIS(D)], multiconfiguration pair-density functional theory (MC-PDFT), and several multireference perturbation theory (MR-PT2) methods. In the first part, we focus on excitation energies of the first singlet excited state (S1) of five different redox and protonation states of flavin, with the goal of finding a suitable active space for MR-PT2 calculations. In the second part, we construct two sets of one-dimensional potential energy surfaces connecting the S0 and S1 equilibrium geometries (S0-S1 path) and the S1 (π,π*) and S2 (n,π*) equilibrium geometries (S1-S2 path). The first path therefore follows a Franck-Condon active mode of flavin while the second path maps crossings points between low-lying singlet and triplet states in flavin. We discuss the similarities and differences in the TD-DFT, EOM-EE-CCSD, SOS-CIS(D), MC-PDFT and MR-PT2 energy profiles along these paths. We find that (TD-)DFT methods are suitable for applications such as simulating the spectra of flavins but are inconsistent with several other methods when used for some geometry optimizations and when describing the energetics of dark (n,π*) states. MR-PT2 methods show promise for the simulation of flavin's low-lying excited states, but the selection of orbitals for the active space and the number of roots used for state averaging must be done carefully to avoid artifacts. Some properties, such as the intersystem crossing geometry and energy between the S1 (π,π*) and T2 (n,π*) states, may require additional benchmarking before they can be determined quantitatively.

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.

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

Global Potential Energy Surfaces by Compressed-State Multistate Pair-Density Functional Theory for Hyperthermal Collisions in the O2+O2 System.

Interactions between oxygen molecules play an important role in atmospheric chemistry and hypersonic flow chemistry in atmospheric entries. Recently, high-quality ab initio potential energy surface (PES) of the quintet O4 was reported by Paukku et al. [J. Chem. Phys. 147, 034301 (2017)]. 10543 configurations were sampled and calculated at the level of MS-CASPT2/maug-cc-pVTZ with scaled external correlation. The PES was fitted to a many-body (MB) form with the many-body part described by the permutationally invariant polynomial approach (MB-PIP). In this work, the PIP-Neural Network (PIP-NN) and MB-PIP-NN methods were used to refit the PES based on the same data by Paukku et al. Three PESs were compared. It was found that the performances differ significantly in the O+O3 region as well as in the long-range region. Therefore, additional 1300 points were sampled, and the efficient compressed-state multistate pair-density functional theory (CMS-PDFT) was used to calculate the electronic structure of these 1300 points and 10543 points by Paukku et al. Then, a completely new quintet PES was fitted using the MB-PIP-NN method. Based on this PES, the quasi-classical trajectory (QCT) approach was used to reveal all possible reaction channels for hyperthermal O2-O2 collisions.

Analytic Nuclear Gradients for Complete Active Space Linearized Pair-Density Functional Theory.

Accurately modeling photochemical reactions is difficult due to the presence of conical intersections and locally avoided crossings, as well as the inherently multiconfigurational character of excited states. As such, one needs a multistate method that incorporates state interaction in order to accurately model the potential energy surface at all nuclear coordinates. The recently developed linearized pair-density functional theory (L-PDFT) is a multistate extension of multiconfiguration PDFT, and it has been shown to be a cost-effective post-MCSCF method (as compared to more traditional and expensive multireference many-body perturbation methods or multireference configuration interaction methods) that can accurately model potential energy surfaces in regions of strong nuclear-electronic coupling in addition to accurately predicting Franck-Condon vertical excitations. In this paper, we report the derivation of analytic gradients for L-PDFT and their implementation in the PySCF-forge software, and we illustrate the utility of these gradients for predicting ground- and excited-state equilibrium geometries and adiabatic excitation energies for formaldehyde, s-trans-butadiene, phenol, and cytosine.

Organic Reactivity Made Easy and Accurate with Automated Multireference Calculations.

In organic reactivity studies, quantum chemical calculations play a pivotal role as the foundation of understanding and machine learning model development. While prevalent black-box methods like density functional theory (DFT) and coupled-cluster theory (e.g., CCSD(T)) have significantly advanced our understanding of chemical reactivity, they frequently fall short in describing multiconfigurational transition states and intermediates. Achieving a more accurate description necessitates the use of multireference methods. However, these methods have not been used at scale due to their often-faulty predictions without expert input. Here, we overcome this deficiency with automated multiconfigurational pair-density functional theory (MC-PDFT) calculations. We apply this method to 908 automatically generated organic reactions. We find 68% of these reactions present significant multiconfigurational character in which the automated multiconfigurational approach often provides a more accurate and/or efficient description than DFT and CCSD(T). This work presents the first high-throughput application of automated multiconfigurational methods to reactivity, enabled by automated active space selection algorithms and the computation of electronic correlation with MC-PDFT on-top functionals. This approach can be used in a black-box fashion, avoiding significant active space inconsistency error in both single- and multireference cases and providing accurate multiconfigurational descriptions when needed.

Minimum-Energy Conical Intersections by Compressed Multistate Pair-Density Functional Theory.

Compressed multistate pair-density functional theory (CMS-PDFT) is a multistate version of multiconfiguration pair-density functional theory that can capture the correct topology of coupled potential energy surfaces (PESs) around conical intersections. In this work, we develop interstate coupling vectors (ISCs) for CMS-PDFT in the OpenMolcas and PySCF/mrh electronic structure packages. Yet, the main focus of this work is using ISCs to calculate minimum-energy conical intersections (MECIs) by CMS-PDFT. This is performed using the projected constrained optimization method in OpenMolcas, which uses ISCs to restrain the iterations to the conical intersection seam. We optimize the S1/S0 MECIs for ethylene, butadiene, and benzene and show that CMS-PDFT gives smooth PESs in the vicinities of the MECIs. Furthermore, the CMS-PDFT MECIs are in good agreement with the MECI calculated by the more expensive XMS-CASPT2 method.

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.

Analogies between photochemical reactions and ground-state post-transition-state bifurcations shed light on dynamical origins of selectivity.

Revealing the origins of kinetic selectivity is one of the premier tasks of applied theoretical organic chemistry, and for many reactions, doing so involves comparing competing transition states. For some reactions, however, a single transition state leads directly to multiple products, in which case non-statistical dynamic effects influence selectivity control. The selectivity of photochemical reactions-where crossing between excited-state and ground-state surfaces occurs near ground-state transition structures that interconvert competing products-also should be controlled by the momentum of the reacting molecules as they return to the ground state in addition to the shape of the potential energy surfaces involved. Now, using machine-learning-assisted non-adiabatic molecular dynamics and multiconfiguration pair-density functional theory, these factors are examined for a classic photochemical reaction-the deazetization of 2,3-diazabicyclo[2.2.2]oct-2-ene-for which we demonstrate that momentum dominates the selectivity for hexadiene versus [2.2.2] bicyclohexane products.

Variational Active Space Selection with Multiconfiguration Pair-Density Functional Theory.

The selection of an adequate set of active orbitals for modeling strongly correlated electronic states is difficult to automate because it is highly dependent on the states and molecule of interest. Although many approaches have shown some success, no single approach has worked well in all cases. In light of this, we present the "discrete variational selection" (DVS) approach to active space selection, in which one generates multiple trial wave functions from a diverse set of systematically constructed active spaces and then selects between these wave functions variationally. We apply this DVS approach to 207 vertical excitations of small-to-medium-sized organic and inorganic molecules (with 3 to 18 atoms) in the QUESTDB database by (i) constructing various sets of active space orbitals through diagonalization of parametrized operators and (ii) choosing the result with the lowest average energy among the states of interest. This approach proves ineffective when variationally selecting between wave functions using the density matrix renormalization group (DMRG) or complete active space self-consistent field (CASSCF) energy but is able to provide good results when variationally selecting between wave functions using the energy of the translated PBE (tPBE) functional from multiconfiguration pair-density functional theory (MC-PDFT). Applying this DVS-tPBE approach to selection among state-averaged DMRG wave functions, we obtain a mean unsigned error of only 0.17 eV using hybrid MC-PDFT. This result matches that of our previous benchmark without the need to filter out poor active spaces and with no further orbital optimization following active space selection of the SA-DMRG wave functions. Furthermore, we find that DVS-tPBE is able to robustly and effectively select between the new SA-DMRG wave functions and our previous SA-CASSCF results.

Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules.

Modeling actinides with electronic structure theories is challenging because these systems present a strong ligand field and metal-ligand covalency. We systematically investigate the effectiveness of pair-density functional theory (PDFT) for the calculation of vertical excitation energies in An(III), [AnIIICl6]3-, and [AnVIO2]2+ (An = U, Np, Pu, and Am). We compare the performance of PDFT, hybrid PDFT, and multistate PDFT with traditional active-space methods followed by perturbation theory, like multistate CASPT2, and with experimental data. Overall, multistate PDFT gives quantitative agreement with multistate CASPT2 at a significantly reduced computational cost.

Multiconfigurational Pair-Density Functional Theory Is More Complex than You May Think.

Multiconfigurational pair-density functional theory (MC-PDFT) is a promising way to describe both strong and dynamic correlations in an inexpensive way. The functionals in MC-PDFT are often "translated" from standard spin density functionals. However, these translated functionals can in principle lead to "translated spin densities" with a nonzero imaginary component. Current developments so far neglect this imaginary part by simply setting it to zero. In this work, we show how this imaginary component is actually needed to reproduce the correct physical behavior in a range of cases, especially low-spin open shells. We showcase the resulting formalism on both local density approximation and generalized gradient approximation functionals and illustrate the numerical behavior by benchmarking a number of singlet-triplet splittings (ST gaps) of organic diradicals and low-lying excited states of some common organic molecules. The results demonstrate that this scheme improves existing translated functionals and gives more accurate results, even with minimal active spaces.

Linearized Pair-Density Functional Theory for Vertical Excitation Energies.

Multiconfiguration pair-density functional theory (MC-PDFT) is a computationally efficient method that computes the energies of electronic states in a state specific or state average framework via an on-top functional. However, MC-PDFT does not include state interaction among these states since the final energies do not come from the diagonalization of an effective model-space Hamiltonian. Recently, multistate extensions such as linearized PDFT (L-PDFT) have been developed to accurately model the potentials near conical intersections and avoided crossings. However, there has not been any systematic study evaluating their performance for predicting vertical excitations at the equilibrium geometry of a molecule, when the excited states are generally well separated. In this paper, we report the performance of L-PDFT on the extensive QUESTDB data set of vertical excitations using a database of automatically selected active spaces. We show that L-PDFT performs well on all these excitations and successfully reproduces the performance of MC-PDFT. These results further demonstrate the potential of L-PDFT, as its scaling is constant with the number of states included in the state-average manifold, whereas MC-PDFT scales linearly in this regard.

Understanding Antiferromagnetic and Ligand Field Effects on Spin Crossover in a Triple-Decker Dimeric Cr(II) Complex.

Two possible explanations for the temperature dependence of spin-crossover (SCO) behavior in the dimeric triple-decker Cr(II) complex ([(η5-C5Me5)Cr(μ2:η5-P5)Cr(η5-C5Me5)]+) have been offered. One invokes variations in antiferromagnetic interactions between the two Cr(II) ions, whereas the other posits the development of a strong ligand-field effect favoring the low-spin ground state. We perform multireference electronic structure calculations based on the multiconfiguration pair-density functional theory to resolve these effects. We find quintet, triplet, and singlet electronic ground states, respectively, for the experimental geometries at high, intermediate, and low temperatures. The ground-state transition from quintet to triplet at an intermediate temperature derives from increased antiferromagnetic interactions between the two Cr(II) ions. By contrast, the ground-state transition from triplet to singlet at low temperature can be attributed to increased ligand-field effects, which dominate with continued variations in antiferromagnetic coupling. This study provides quantitative detail for the degree to which these two effects can act in concert for the observed SCO behavior in this complex and others subject to temperature-dependent variations in geometry.

An Accurate Density Coherence Functional for Hybrid Multiconfiguration Density Coherence Functional Theory.

We present hybrid multiconfiguration density coherence functional theory (HMC-DCFT), and we optimize a density coherence functional by parametrization against a diverse data set of 59 bond energies and 60 barrier heights. We compare the results to calculations on the same data set by CASSCF, CASPT2, six Kohn-Sham and hybrid Kohn-Sham exchange-correlation functionals, and three on-top functionals for pair-density functional theory (PDFT) and hybrid PDFT. The new functional has better accuracy than all compared methods.

Pair density functional theory for excited states of Coulomb systems

Pair density functional theory is extended to excited states of Coulomb systems. It is shown that the pair density determines the Hamiltonian of the Coulomb system. A universal kinetic energy functional appropriate for the ground and all bound excited states is defined. The Euler equation can be rewritten as a two-particle auxiliary equation in which the unknown Pauli-like potential should be approximated.

Density Matrix Embedding Using Multiconfiguration Pair-Density Functional Theory.

We present a quantum embedding method for ground and excited states of extended systems that uses multiconfiguration pair-density functional theory (MC-PDFT) with densities provided by periodic density matrix embedding theory (pDMET). We compute local excitations in oxygen mono- and divacancies on a magnesium oxide (100) surface and find absolute deviations within 0.05 eV between pDMET using the MC-PDFT, denoted as pDME-PDFT, and the more expensive, nonembedded MC-PDFT approach. We further use pDME-PDFT to calculate local excitations in larger supercells for the monovacancy defect, for which the use of nonembedded MC-PDFT is prohibitively costly.

A Combined Wave Function and Density Functional Approach for K-Edge X-ray Absorption Near-Edge Spectroscopy: A Case Study of Hydrated First-Row Transition Metal Ions.

The prediction of X-ray absorption spectra (XAS) of transition metal complexes has important and broad application areas in chemistry and biology. In this letter, we have investigated the predictive ability of multiconfiguration pair-density functional theory (MC-PDFT) for X-ray absorption spectra by calculating the metal K pre-edge features of aquated 3d transition metal ions in common oxidation states. MC-PDFT results were compared with experimentally measured spectra as well as analyzed against results from restricted active-space second-order perturbation theory (RASPT2) and time-dependent density functional theory (TDDFT). As expected, TDDFT performs well for excited states that can be accurately represented by singly excited configurations but fails for excited states where higher order excitations become important. On the other hand, both RASPT2 and MC-PDFT provide quantitatively accurate results for all excited states irrespective of their character. While core-level spectroscopy with RASPT2 is accurate, it is computationally expensive. Our results show that MC-PDFT performs equally well with significantly lower computational cost and is an encouraging alternate approach for X-ray spectroscopies.