N-electron Valence State Perturbation Theory Research Articles

Combined Multireference-Multiscale Approach to the Description of Photosynthetic Reaction Centers.

A first-principles description of the primary photochemical processes that drive photosynthesis and sustain life on our planet remains one of the grand challenges of modern science. Recent research established that explicit incorporation of protein electrostatics in excited-state calculations of photosynthetic pigments, achieved for example with quantum-mechanics/molecular-mechanics (QM/MM) approaches, is essential for a meaningful description of the properties and function of pigment-protein complexes. Although time-dependent density functional theory has been used productively so far in QM/MM approaches for the study of such systems, this methodology has limitations. Here we pursue for the first time a QM/MM description of the reaction center in the principal enzyme of oxygenic photosynthesis, Photosystem II, using multireference wave function theory for the high-level QM region. We identify best practices and establish guidelines regarding the rational choice of active space and appropriate state-averaging for the efficient and reliable use of complete active space self-consistent field (CASSCF) and the N-electron valence state perturbation theory (NEVPT2) in the prediction of low-lying excited states of chlorophyll and pheophytin pigments. Given that the Gouterman orbitals are inadequate as a minimal active space, we define specific minimal and extended active spaces for the NEVPT2 description of electronic states that fall within the Q and B bands. Subsequently, we apply our multireference-QM/MM protocol to the description of all pigments in the reaction center of Photosystem II. The calculations reproduce the electrochromic shifts induced by the protein matrix and the ordering of site energies consistent with the identity of the primary donor (ChlD1) and the experimentally known asymmetric and directional electron transfer. The optimized protocol sets the stage for future multireference treatments of multiple pigments, and hence for multireference studies of charge separation, while it is transferable to the study of any photoactive embedded tetrapyrrole system.

Multireference Ab Initio Calculations on Excited Electronic States of Carbazole-Based Organic Compounds for Thermally Activated Delayed Fluorescence.

The emerging technology of organic light-emitting diodes takes advantage of the thermally activated delayed fluorescence (TADF) mechanism for improved efficiency. Carbazole-based organic molecules are suitable for TADF emission because of charge transfer excitations between the electron-donor carbazole and an electron-acceptor unit. Computational design of new TADF molecules with the desired properties is challenging because charge-transfer excitations cannot be predicted accurately by time-dependent density functional theory. Four groups of carbazole-based donor-acceptor molecules have been studied using multireference ab initio methods to understand the nature of excited electronic states. The state-averaged complete active space self-consistent field (SA-CASSCF) and the N-electron valence state perturbation theory (NEVPT2) were used to calculate energies and oscillator strengths for multiple excited electronic states. The number of active electrons and orbitals and the number of excited states included in state-averaged CASSCF were selected such that the accuracy of ab initio predictions could be improved systematically. The procedure introduced here for the calculation of multiple excited electronic states of TADF candidates can be used to accelerate the computational search for efficient TADF materials.

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.

Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals.

Chemical phenomena involving near-degenerate electronic states, such as conical intersections or avoided crossing, can be properly described using quasi-degenerate perturbation theory. This study proposed a highly scalable quasi-degenerate second-order N-electron valence state perturbation theory (QD-NEVPT2) using the local pair-natural orbital (PNO) method. Our recent study showed an efficient implementation of the PNO-based state-specific NEVPT2 method using orthonormal localized virtual molecular orbitals (LVMOs) as an intermediate local basis. This study derived the state-coupling (or off-diagonal) terms to implement QD-NEVPT2 in an alternative manner to enhance efficiency based on the internally contracted basis and PNO overlap matrices between different references. To facilitate further acceleration, a local resolution-of-the-identity (RI) three-index integral generation algorithm was developed using LMOs and LVMOs. Although the NEVPT2 theory is considered to be less susceptible to the intruder-state problem (ISP), this study revealed that it can easily suffer from ISP when calculating high-lying excited states. We ameliorated this instability using the imaginary level shift technique. The PNO-QD-NEVPT2 calculations were performed on small organic molecules for the 30 lowest-lying states, as well as photoisomerization involving the conical intersection of 1,1-dimethyldibenzo[b,f] silepin with a cis-stilbene skeleton. These calculations revealed that the PNO-QD-NEVPT2 method yielded negligible errors compared to the canonical QD-NEVPT2 results. Furthermore, we tested its applicability to a large photoisomerization system using the green fluorescent protein model and the ten-state calculation of the large transition metal complex, showcasing that off-diagonal elements can be evaluated at a relatively low cost.

Ab Initio Calculation of UV-vis Absorption of Parent Mg, Fe, Co, Ni, Cu, and Zn Metalloporphyrins.

Relativistic restricted active space (RAS) second-order multireference perturbation theory (MRPT2) methods, incorporating spin-orbit (SO) coupling perturbatively via state interaction (SO-MRPT2/RASSCF), were used to reproduce the absorption spectra of parent metalloporphyrins containing the Mg2+, Zn2+, Co2+, Ni2+, Cu2+, or FeCl2+ ions in the 12,500-40,000 cm-1 region. Particular attention was paid to the interaction between the porphyrin ring and the metal 3d electrons in states of different multiplicities (we used metal 3d and double d-shell or 3d' orbitals). For this class of compounds, the N-electron valence state perturbation theory (NEVPT2) method is superior to the complete active space perturbation theory (CASPT2) and successfully reproduces the energies of all four characteristic transitions (Q, B, N, and L) of closed-shell metalloporphyrins. Inclusion of SO coupling was found to have very little effect on excitation energies and oscillator strengths. For FeCl2+ porphyrin, we treated ligand-to-metal charge-transfer (LMCT; π,d), metal ligand field (d,d), and metal-to-ligand charge-transfer (MLCT; d,π*) transitions within the same framework. The broad and intense spectral features associated with its B (Soret) band are attributed to multiconfigurational LMCT (d,π*) bands involving strong metal-ligand orbital mixing.

Spinless formulation of linearized adiabatic connection approximation and its comparison with the second order N-electron valence state perturbation theory.

The adiabatic connection (AC) approximation, along with its linearized variant AC0, was introduced as a method of obtaining dynamic correlation energy. When using a complete active space self-consistent field (CASSCF) wave function as a reference, the AC0 approximation is considered one of the most efficient multi-reference perturbation theories. It only involves the use of 1st- and 2nd-order reduced density matrices. However, some numerical results have indicated that the excitation energies predicted by AC0 are not as reliable as those from the second-order N-electron valence state perturbation theory (NEVPT2). In this study, we develop a spinless formulation of AC0 based on the Dyall Hamiltonian and provide a detailed comparison between AC0 and NEVPT2 approaches. We demonstrate the components within the correlation energy expressions that are common to both methods and those unique to either AC0 or NEVPT2. We investigate the role of the terms exclusive to NEVPT2 and explore the possibility of enhancing AC0's performance in this regard.

The trials and triumphs of modelling X-ray absorption spectra of transition metal phthalocyanines.

This study explores the electronic structure of Co, Fe and Mn phthalocyanines (TMPcs) as well as their perfluorinated counterparts through a series of electronic structure calculations utilizing multireference methods and by simulating their metal L-edge and ligand (nitrogen and fluorine) K-edge X-ray absorption spectra (XAS) in an angle-resolved manner. Simulations targeting different ground-state symmetries, where relevant, have been conducted to observe changes in the N K-edge lineshape. The applicability of the quasi-degenerate formulation of n-electron valence state perturbation theory (QD-NEVPT2) for L-edge X-ray absorption spectroscopy (XAS) is evaluated, alongside the use of a restricted active space (RAS) formalism to describe the final-state multiplets generated by L-shell X-ray processes. Our findings provide valuable insights into the electronic properties of TMPcs, in particular with respect to the effect of fluorination, and demonstrate the broad applicability of various formulations of NEVPT2 in spectral simulations. Moreover, this study highlights the utility of manual truncation of the configuration spaces in order to allow for large active orbital spaces in aforementioned calculations.

Preparation and Spectroscopic Identification of the Cyclic CO2 Dimer 1,2-Dioxetanedione.

We report the preparation and infrared spectroscopic identification of 1,2-dioxetanedione, which is one of the two possible cyclic dimers of carbon dioxide. We prepared this hitherto experimentally incompletely characterized species in a solid nitrogen matrix at 3 K from the reaction of oxalyl dichloride with the urea·hydrogen peroxide complex. Surprisingly, irradiation at 254 nm does not lead to its dissociation into carbon dioxide but rather yields cyclic carbon trioxide. We further assert our spectroscopic assignments by 18O isotopic labeling and high-level N-electron valence state perturbation theory and coupled-cluster computations. The successful isolation of 1,2-dioxetanedione supports its viability as the postulated high-energy intermediate in the well-known and ubiquitously exploited "peroxyoxalate" chemiluminescent system.

Optical Properties of Neutral F Centers in Bulk MgO with Density Matrix Embedding.

The optical spectra of neutral oxygen vacancies (F0 centers) in the bulk MgO lattice are investigated using density matrix embedding theory. The impurity Hamiltonian is solved with the complete active space self-consistent field and second-order n-electron valence state perturbation theory (NEVPT2-DMET) multireference methods. To estimate defect-localized vertical excitation energies at the nonembedding and thermodynamic limits, a double extrapolation scheme is employed. The extrapolated NEVPT2-DMET vertical excitation energy value of 5.24 eV agrees well with the experimental absorption maxima at 5.03 eV, whereas the excitation energy value of 2.89 eV at the relaxed triplet defect-localized state geometry overestimates the experimental emission at 2.4 eV by only nearly 0.5 eV, indicating the involvement of the triplet-singlet decay pathway.

Magneto-Structural Analysis of Hydroxido-Bridged CuII2 Complexes: Density Functional Theory and Other Treatments

A selection of dimeric Cu(II) complexes with bidentate N,N′ ligands with the general formula [Cu(L)(X)(μ-OH)]2·nH2O and [Cu(L)(μ-OH)]2X2·nH2O were magneto-structurally analyzed using the Density Functional Theory (DFT). A Broken Symmetry-Density Functional Theory (BS-DFT) study was undertaken for these complexes with relevant decomposition schemes that gave insight into the effect of the nature of the ligand and coordination environment on the DFT-predicted coupling constants (J). The impact of the spin population, which correlates well with the Cu-O-Cu bridging angles and the calculated coupling constant (J) values, was studied. The models were further refined using a complete active space self-consistent field (CASSCF) while expanding the active space from 2 orbitals 2 electrons (2,2) to 10 orbitals 18 electrons (18,10). These models were approximated using multireference methods (n-electron valence state perturbation theory and difference dedicated configuration interaction), and a better approximation of J values was found as expected. Orbitals involved in the superexchange pathway were also visualized.

Local N-electron valence state perturbation theory using pair-natural orbitals based on localized virtual molecular orbitals.

Second-order N-electron valence state perturbation theory (NEVPT2) is an exactly size-consistent and intruder-state-free multi-reference theory. To accelerate the NEVPT2 computation, Guo and Neese combined it with the local pair-natural orbital (PNO) method using the projected atomic orbitals (PAOs) as the underlying local basis [Guo et al., J. Chem. Phys. 144, 094111 (2016)]. In this paper, we report the further development of the PNO-NEVPT2 method using the orthonormal and non-redundant localized virtual molecular orbitals (LVMOs) instead of PAOs. The LVMOs were previously considered to perform relatively poor compared to PAOs because the resulting orbital domains were unacceptably large. Our prior work, however, showed that this drawback can be remedied by re-forming the domain construction scheme using differential overlap integrals [Saitow et al., J. Chem. Phys. 157, 084101 (2022)]. In this work, we develop further refinements to enhance the feasibility of using LVMOs. We first developed a two-level semi-local approach for screening out so-called weak-pairs to select or truncate the pairs for PNO constructions more flexibly. As a refinement specific to the Pipek-Mezey localization for LVMOs, we introduced an iterative scheme to truncate the Givens rotations using varying thresholds. We assessed the LVMO-based PNO-NEVPT2 method through benchmark calculations for linear phenyl alkanes, which demonstrate that it performs comparably well relative to the PAO-based approach. In addition, we evaluated the Co-C bond dissociation energies for the cobalamin derivatives composed of 200 or more atoms, which confirms that the LVMO-based method can recover more than 99.85% of the canonical NEVPT2 correlation energy.

SparseMaps-A systematic infrastructure for reduced-scaling electronic structure methods. VI. Linear-scaling explicitly correlated N-electron valence state perturbation theory with pair natural orbital.

In this work, a linear scaling explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12) is presented. By using the idea of a domain-based local pair natural orbital (DLPNO), computational scaling of the conventional NEVPT2-F12 is reduced to near-linear scaling. For low-lying excited states of organic molecules, the excitation energies predicted by DLPNO-NEVPT2-F12 are as accurate as the exact NEVPT2-F12 results. Some cluster models of rhodopsin are studied using the new algorithm. Our new method is able to study systems with more than 3300 basis functions and an active space containing 12 π-electrons and 12 π-orbitals. However, even larger calculations or active spaces would still be feasible.

Ab Initio Investigation of the Na3[Ln(ODA)3]·2NaClO4·6H2O (Ln = Ce-Yb; ODA = Oxydiacetate) Series.

In this work, the Na3[Ln(ODA)3]·2NaClO4·6H2O (Ln = Ce-Yb; ODA = oxydiacetate) series was analyzed with the ab initio ligand field theory (AILFT) module of the ORCA computational suite. The results were discussed within the framework of the angular overlap model (AOM) and compared to literature data. We find that the structural changes observed across the series exemplifies the effects of the lanthanide contraction also manifesting in the value of the AOM parameters. It is also shown that the complete active space self-consistent field (CASSCF) methodology is sufficient to describe the ligand field interactions in mononuclear lanthanide complexes, and the effects of dynamic correlation, through n-electron valence state perturbation theory (NEVPT2), are discussed. The calculated ligand field parameters of the present work are compared to the experimentally derived values from the literature.

Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers

An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. To perform this assessment reliably, quantum resource estimates must be coupled with classical computations attempting to answer relevant chemical questions and to define the classical algorithms simulation frontier. Herein, we explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical-quantum advantage boundary. This is accomplished by analyzing the convergence of density matrix renormalization group plus n-electron valence state perturbation theory (DMRG+NEVPT2) and coupled-cluster singles doubles with noniterative triples [CCSD(T)] calculations for spin gaps in models of the CYP catalytic cycle that indicate multireference character. The quantum resources required to perform phase estimation using qubitized quantum walks are calculated for the same systems. Compilation into the surface code provides runtime estimates to compare directly to DMRG runtimes and to evaluate potential quantum advantage. Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the potential to be a quantum advantage problem and emphasizes the important interplay between classical computations and quantum algorithms development for chemical simulation.

An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices.

A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to the one needed for the construction of the third-order reduced density matrix at the most is presented. Thus, the computational bottleneck arising from explicit evaluation of the fourth-order density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability.

Magnetic exchange coupling in Cu dimers studied with modern multireference methods and broken-symmetry coupled cluster theory

Spin-state energetics of exchange-coupled copper complexes pose a persistent challenge for applied quantum chemistry. Here, we provide a comprehensive comparison of all available theoretical approaches to the problem of exchange coupling in two antiferromagnetically coupled bis-μ-hydroxo Cu(II) dimers. The evaluated methods include multireference methods based on the density matrix renormalization group (DMRG), multireference methods that incorporate dynamic electron correlation either perturbatively, such as the N-electron valence state perturbation theory, or variationally, such as the difference-dedicated configuration interaction. In addition, we contrast the multireference results with those obtained using broken-symmetry approaches that utilize either density functional theory or, as demonstrated here for the first time in such systems, a local implementation of coupled cluster theory. The results show that the spin-state energetics of these copper dimers are dominated by dynamic electron correlation and represent an impossible challenge for multireference methods that rely on brute-force expansion of the active space to recover correlation energy. Therefore, DMRG-based methods even at the limit of their applicability cannot describe quantitatively the antiferromagnetic exchange coupling in these dimers, in contrast to dinuclear complexes of earlier transition metal ions. The convergence of the broken-symmetry coupled cluster approach is studied and shown to be a limiting factor for the practical application of the method. The advantages and disadvantages of all approaches are discussed, and recommendations are made for future developments.

Toward an Accurate Ab Initio Description of Low-Lying Singlet Excited States of Polyenes.

The low-lying excited states of carotenoids play a crucial role in many important biophysical processes such as photosynthesis. Most of these excited states are strongly correlated, which makes them both challenging for a qualitative ab initio description and an engaging model system for trying out emerging multireference methods. Among these methods, driven similarity renormalization group (DSRG) and its perturbative version (DSRG-MRPT2) are especially attractive in terms of both accuracy and moderate numerical complexity. In this paper, we applied density matrix renormalization group (DMRG) followed by DSRG-MRPT2 for the calculation of vertical and adiabatic excitation energies into the 2Ag-, 1Bu-, and 1Bu+ electronic states of polyenes containing from 8 to 13 conjugating double bonds acting as a model for natural carotenoids. It was shown that the DSRG flow parameter should be adjusted to ensure both the energy convergence with respect to it and the agreement with the experimental data. With the increased flow parameter, the proposed combination of methods provides a reasonable agreement with the experiment. The deviations of the adiabatic excitation energies are less than 1000 cm-1 for the 2Ag- and less than 3000 cm-1 for the excited states of the Bu symmetry, which in terms of accuracy significantly outperforms the N-electron valence state perturbation theory. At the same time, DSRG-MRPT2 is shown to be robust with respect to variation of quality of the DMRG reference wave function such as the orbital optimization or the number of electronic states in the averaging.

Revisiting the Electronic Structure of Cobalt Porphyrin Nitrene and Carbene Radicals with NEVPT2-CASSCF Calculations: Doublet versus Quartet Ground States.

Cobalt porphyrin complexes are established catalysts for carbene and nitrene radical group-transfer reactions. The key carbene and mono- and bisnitrene radical complexes coordinated to [Co(TPP)] (TPP = tetraphenylporphyrin) have previously been investigated with a variety of experimental techniques and supporting (single-reference) density functional theory (DFT) calculations that indicated doublet (S = 1/2) ground states for all three species. In this contribution, we revisit their electronic structures with multireference N-electron valence state perturbation theory (NEVPT2)-complete-active-space self-consistent-field (CASSCF) calculations to investigate possible multireference contributions to the ground-state wave functions. The carbene ([CoIII(TPP)(•CHCO2Et)]) and mononitrene ([CoIII(TPP)(•NNs)]) radical complexes were confirmed to have uncomplicated doublet ground states, although a higher carbene or nitrene radical character and a lower Co–C/N bond order was found in the NEVPT2-CASSCF calculations. Supported by electron paramagnetic resonance analysis and spin counting, paramagnetic molar susceptibility determination, and NEVPT2-CASSCF calculations, we report that the cobalt porphyrin bisnitrene complex ([CoIII(TPP•)(•NNs)2]) has a quartet (S = 3/2) spin ground state, with a thermally accesible multireference and multideterminant “broken-symmetry” doublet spin excited state. A spin flip on the porphyrin-centered unpaired electron allows for interconversion between the quartet and broken-symmetry doublet spin states, with an approximate 10-fold higher Boltzmann population of the quartet at room temperature.

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction.

Over the last decade, the second-order N-electron valence state perturbation theory (NEVPT2) has developed into a widely used multireference perturbation method. To apply NEVPT2 to systems with large active spaces, the computational bottleneck is the construction of the fourth-order reduced density matrix. Both its generation and storage become quickly problematic beyond the usual maximum active space of about 15 active orbitals. To reduce the computational cost of handling fourth-order density matrices, the cumulant approximation (CU) has been proposed in several studies. A more conventional strategy to address the higher-order density matrices is the pre-screening approximation (PS), which is the default one in the ORCA program package since 2010. In the present work, the performance of the CU, PS, and extended PS (EPS) approximations for the fourth-order density matrices is compared. Following a pedagogical introduction to NEVPT2, contraction schemes, as well as the approximations to density matrices, and the intruder state problem are discussed. The CU approximation, while potentially leading to large computational savings, virtually always leads to intruder states. With the PS approximation, the computational savings are more modest. However, in conjunction with conservative cutoffs, it produces stable results. The EPS approximation to the fourth-order density matrices can reproduce very accurate NEVPT2 results without any intruder states. However, its computational cost is not much lower than that of the canonical algorithm. Moreover, we found that a good indicator of intrude states problems in any approximation to high order density matrices is the eigenspectra of the Koopmans matrices.

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation.

In Paper I, the performances of pre-screening (PS), extended PS (EPS), and cumulant (CU) approximations to the fourth-order density matrix were examined in the context of second-order N-electron valence state perturbation theory (NEVPT2). It has been found that the CU, PS, and even EPS approximations with loose thresholds may introduce intruder states. In the present work, the origin of these "false intruder" states introduced by approximated density matrices is discussed. Canonical NEVPT2 implementations employ a rank reduction trick. By analyzing its residual error, we find that the omission of the rank reduction leads to a more stable multireference perturbation theory for incomplete active space reference wave functions. Such a full rank (FR)-NEVPT2 formulation is equivalent to the conventional NEVPT2 method for the complete active space self-consistent field/complete active space configuration interaction reference wave function. A major drawback of the FR-NEVPT2 formulation is the necessity of the fifth-order density matrix. To avoid the construction of the high-order density matrices, the combination of the FR-NEVPT2 with the CU approximation is studied. However, we find that the CU approximation remains problematic as it still introduces intruder states. The question of how to robustly and efficiently perform internally contracted multireference perturbation theories with approximate densities remains a challenging field of investigation.