Scalar Relativistic All-Electron and Pseudopotential Ab Initio Study of a Minimal Nitrogenase [Fe(SH)4H]- Model Employing Coupled-Cluster and Auxiliary-Field Quantum Monte Carlo Many-Body Methods.

Nitrogenase is the only enzyme that can cleave the triple bond in N2, making nitrogen available to organisms. The detailed mechanism of this enzyme is currently not known, and computational studies are complicated by the fact that different density functional theory (DFT) methods give very different energetic results for calculations involving nitrogenase models. Recently, we designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S to mimic different possible protonation states of the nitrogenase active site. We showed that the energy difference between these two isomers (ΔE) is hard to estimate with quantum-mechanical methods. Based on nonrelativistic single-reference coupled-cluster (CC) calculations, we estimated that the ΔE is 101 kJ/mol. In this study, we demonstrate that scalar relativistic effects play an important role and significantly affect ΔE. Our best revised single-reference CC estimates for ΔE are 85-91 kJ/mol, including energy corrections to account for contributions beyond triples, core-valence correlation, and basis-set incompleteness error. Among coupled-cluster approaches with approximate triples, the canonical CCSD(T) exhibits the largest error for this problem. Complementary to CC, we also used phaseless auxiliary-field quantum Monte Carlo calculations (ph-AFQMC). We show that with a Hartree-Fock (HF) trial wave function, ph-AFQMC reproduces the CC results within 5 ± 1 kJ/mol. With multi-Slater-determinant (MSD) trials, the results are 82-84 ± 2 kJ/mol, indicating that multireference effects may be rather modest. Among the DFT methods tested, τ-HCTH, r2SCAN with 10-13% HF exchange with and without dispersion, and O3LYP/O3LYP-D4, and B3LYP*/B3LYP*-D4 generally perform the best. The r2SCAN12 (with 12% HF exchange) functional mimics both the best reference MSD ph-AFQMC and CC ΔE results within 2 kJ/mol.

TREXIO: A file format and library for quantum chemistry.

TREXIO is an open-source file format and library developed for the storage and manipulation of data produced by quantum chemistry calculations. It is designed with the goal of providing a reliable and efficient method of storing and exchanging wave function parameters and matrix elements, making it an important tool for researchers in the field of quantum chemistry. In this work, we present an overview of the TREXIO file format and library. The library consists of a front-end implemented in the C programming language and two different back-ends: a text back-end and a binary back-end utilizing the hierarchical data format version 5 library, which enables fast read and write operations. It is compatible with a variety of platforms and has interfaces for Fortran, Python, and OCaml programming languages. In addition, a suite of tools have been developed to facilitate the use of the TREXIO format and library, including converters for popular quantum chemistry codes and utilities for validating and manipulating data stored in TREXIO files. The simplicity, versatility, and ease of use of TREXIO make it a valuable resource for researchers working with quantum chemistry data.

Double Excitation Energies from Quantum Monte Carlo Using State-Specific Energy Optimization.

We show that recently developed quantum Monte Carlo methods, which provide accurate vertical transition energies for single excitations, also successfully treat double excitations. We study the double excitations in medium-sized molecules, some of which are challenging for high-level coupled-cluster calculations to model accurately. Our fixed-node diffusion Monte Carlo excitation energies are in very good agreement with reliable benchmarks, when available, and provide accurate predictions for excitation energies of difficult systems where reference values are lacking.

Reference Excitation Energies of Increasingly Large Molecules: A QMC Study of Cyanine Dyes.

We revisit here the lowest vertical excitations of cyanine dyes using quantum Monte Carlo and leverage recent developments to systematically improve on previous results. In particular, we employ a protocol for the construction of compact and accurate multideterminant Jastrow-Slater wave functions for multiple states, which we have recently validated on the excited-state properties of several small prototypical molecules. Here, we obtain quantum Monte Carlo excitation energies in excellent agreement with high-level coupled cluster for all the cyanines where the coupled cluster method is applicable. Furthermore, we push our protocol to longer chains, demonstrating that quantum Monte Carlo is a viable methodology to establish reference data at system sizes which are hard to reach with other high-end approaches of similar accuracy. Finally, we determine which ingredients are key to an accurate treatment of these challenging systems and rationalize why a description of the excitation based on only active π orbitals lacks the desired accuracy for the shorter chains.

Energy Derivatives in Real-Space Diffusion Monte Carlo.

We present unbiased, finite-variance estimators of energy derivatives for real-space diffusion Monte Carlo calculations within the fixed-node approximation. The derivative dλE is fully consistent with the dependence E(λ) of the energy computed with the same time step. We address the issue of the divergent variance of derivatives related to variations of the nodes of the wave function both by using a regularization for wave function parameter gradients recently proposed in variational Monte Carlo and by introducing a regularization based on a coordinate transformation. The essence of the divergent variance problem is distilled into a particle-in-a-box toy model, where we demonstrate the algorithm.

Tailoring CIPSI Expansions for QMC Calculations of Electronic Excitations: The Case Study of Thiophene.

The perturbatively selected configuration interaction scheme (CIPSI) is particularly effective in constructing determinantal expansions for quantum Monte Carlo (QMC) simulations with Jastrow-Slater wave functions: fast and smooth convergence of ground-state properties and balanced descriptions of ground and excited states of different symmetries have been reported. In particular, accurate excitation energies have been obtained by the pivotal requirement of using CIPSI expansions with similar second-order perturbation corrections for each state, that is, a similar estimated distance to the full configuration interaction limit. Here, we elaborate on the CIPSI selection criterion for excited states of the same symmetry as the ground state, generating expansions from a common orbital set. Using these expansions in QMC as determinantal components of Jastrow-Slater wave functions, we compute the lowest, bright excited state of thiophene, which is challenging due to its significant multireference character. The resulting vertical excitation energies are within 0.05 eV of the best theoretical estimates, already with expansions of only a few thousand determinants. Furthermore, we relax the ground- and excited-state structures following the corresponding root in variational Monte Carlo and obtain bond lengths that are accurate to better than 0.01 Å. Therefore, while the full treatment at the CIPSI level of this system is quite demanding, in QMC, we can compute high-quality excitation energies and excited-state structural parameters building on affordable CIPSI expansions with relatively few, well-chosen determinants.

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization.

We investigate the use of different variational principles in quantum Monte Carlo, namely, energy and variance minimization, prompted by the interest in the robust and accurate estimation of electronic excited states. For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively. On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state. This unexpected behavior is sometimes observed even when the target is the ground state and generally prevents the robust estimation of total and excitation energies. We analyze this problem using a very simple wave function and infer that the optimization finds little or no barrier to escape from a local minimum or local plateau, eventually converging to a lower-variance state instead of the target state. For the increasingly complex systems becoming in reach of quantum Monte Carlo simulations, variance minimization with current functionals appears to be an impractical route.

The key to the yellow-to-cyan tuning in the green fluorescent protein family is polarisation.

Computational approaches have to date failed to fully capture the large (about 0.4 eV) excitation energy tuning displayed by the nearly identical anionic chromophore in different green fluorescent protein (GFP) variants. Here, we present a thorough comparative study of a set of proteins in this sub-family, including the most red- (phiYFP) and blue-shifted (mTFP0.7) ones. We employ a classical polarisable embedding through induced dipoles and combine it with time-dependent density functional theory and multireference perturbation theory in order to capture both state-specific induction contributions and the coupling of the polarisation of the protein to the chromophore transition density. The obtained results show that only upon inclusion of both these two effects generated by the mutual polarisation between the chromophore and the protein can the full spectral tuning be replicated. We finally discuss how this mutual polarisation affects the correlation between excitation energies, dipole moment variation, and molecular electrostatic field.

Excited States with Selected Configuration Interaction-Quantum Monte Carlo: Chemically Accurate Excitation Energies and Geometries.

We employ quantum Monte Carlo to obtain chemically accurate vertical and adiabatic excitation energies, and equilibrium excited-state structures for the small, yet challenging, formaldehyde and thioformaldehyde molecules. A key ingredient is a robust protocol to obtain balanced ground- and excited-state Jastrow-Slater wave functions at a given geometry, and to maintain such a balanced description as we relax the structure in the excited state. We use determinantal components generated via a selected configuration interaction scheme which targets the same second-order perturbation energy correction for all states of interest at different geometries, and fully optimize all variational parameters in the resultant Jastrow-Slater wave functions. Importantly, the excitation energies as well as the structural parameters in the ground and excited states are converged with very compact wave functions comprising few thousand determinants in a minimally augmented double-ζ basis set. These results are obtained already at the variational Monte Carlo level, the more accurate diffusion Monte Carlo method yielding only a small improvement in the adiabatic excitation energies. We find that matching Jastrow-Slater wave functions with similar variances can yield excitation energies compatible with our best estimates; however, the variance-matching procedure requires somewhat larger determinantal expansions to achieve the same accuracy, and it is less straightforward to adapt during structural optimization in the excited state.

Taming the Complexity of Donor-Acceptor Stenhouse Adducts: Infrared Motion Pictures of the Complete Switching Pathway.

Switches that can be actively steered by external stimuli along multiple pathways at the molecular level are the basis for next-generation responsive material systems. The operation of commonly employed molecular photoswitches revolves around one key structural coordinate. Photoswitches with functionalities that depend on and can be addressed along multiple coordinates would offer novel means to tailor and control their behavior and performance. The recently developed donor-acceptor Stenhouse adducts (DASAs) are versatile switches suitable for such applications. Their photochemistry is well understood, but is only responsible for part of their overall photoswitching mechanism. The remaining thermal switching pathways are to date unknown. Here, rapid-scan infrared absorption spectroscopy is used to obtain transient fingerprints of reactions occurring on the ground state potential energy surface after reaching structures generated through light absorption. The spectroscopic data are interpreted in terms of structural transformations using kinetic modeling and quantum chemical calculations. Through this combined experimental-theoretical approach, we are able to unravel the complexity of the multidimensional ground-state potential energy surface explored by the photoswitch and use this knowledge to predict, and subsequently confirm, how DASA switches can be guided along this potential energy surface. These results break new ground for developing user-geared DASA switches but also shed light on the development of novel photoswitches in general.

Photoprogramming Allostery in Human Serum Albumin.

Developing strategies to interfere with allosteric interactions in proteins not only promises to deepen our understanding of vital cellular processes but also allows their regulation using external triggers. Light is particularly attractive as a trigger being spatiotemporally selective and compatible with the physiological environment. Here, we engineered a hybrid protein in which irradiation with light opens a new allosteric communication route that is not inherent to the natural system. We select human serum albumin, a promiscuous protein responsible for transporting a variety of ligands in plasma, and show that by covalently incorporating a synthetic photoswitch to subdomain IA we achieve optical control of the ligand binding in subdomain IB. Molecular dynamics simulations confirm the allosteric nature of the interactions between IA and IB in the engineered protein. Specifically, upon illumination, photoconversion of the switch is found to correlate with a less-coordinated motion of the two subdomains and an increased flexibility of the binding pocket in subdomain IB, whose fluctuations are cooperatively enhanced by the presence of ligands, ultimately facilitating their release. Our combined experimental and computational work demonstrates how harnessing artificial molecular switches enables photoprogramming the allosteric regulation of binding activities in such a prominent protein.

Perturbatively Selected Configuration-Interaction Wave Functions for Efficient Geometry Optimization in Quantum Monte Carlo.

We investigate the performance of a class of compact and systematically improvable Jastrow-Slater wave functions for the efficient and accurate computation of structural properties, where the determinantal component is expanded with a perturbatively selected configuration interaction scheme (CIPSI). We concurrently optimize the molecular ground-state geometry and full wave function-Jastrow factor, orbitals, and configuration interaction coefficients-in variational Monte Carlo (VMC) for the prototypical case of 1,3- trans-butadiene, a small yet theoretically challenging π-conjugated system. We find that the CIPSI selection outperforms the conventional scheme of correlating orbitals within active spaces chosen by chemical intuition: it gives significantly better variational and diffusion Monte Carlo energies for all but the smallest expansions, and much smoother convergence of the geometry with the number of determinants. In particular, the optimal bond lengths and bond-length alternation of butadiene are converged to better than 1 mÅ with just a few thousand determinants, to values very close to the corresponding CCSD(T) results. The combination of CIPSI expansion and VMC optimization represents an affordable tool for the determination of accurate ground-state geometries in quantum Monte Carlo.

Optimizing the Energy with Quantum Monte Carlo: A Lower Numerical Scaling for Jastrow-Slater Expansions.

We present an improved formalism for quantum Monte Carlo calculations of energy derivatives and properties (e.g., the interatomic forces), with a multideterminant Jastrow-Slater function. As a function of the number Ne of Slater determinants, the numerical scaling of O(Ne) per derivative we have recently reported is here lowered to O(Ne) for the entire set of derivatives. As a function of the number of electrons N, the scaling to optimize the wave function and the geometry of a molecular system is lowered to O(N3) + O(NNe), the same as computing the energy alone in the sampling process. The scaling is demonstrated on linear polyenes up to C60H62 and the efficiency of the method is illustrated with the structural optimization of butadiene and octatetraene with Jastrow-Slater wave functions comprising as many as 200 000 determinants and 60 000 parameters.

Electrostatic versus Resonance Interactions in Photoreceptor Proteins: The Case of Rhodopsin.

Light sensing in photoreceptor proteins is subtly modulated by the multiple interactions between the chromophoric unit and its binding pocket. Many theoretical and experimental studies have tried to uncover the fundamental origin of these interactions but reached contradictory conclusions as to whether electrostatics, polarization, or intrinsically quantum effects prevail. Here, we select rhodopsin as a prototypical photoreceptor system to reveal the molecular mechanism underlying these interactions and regulating the spectral tuning. Combining a multireference perturbation method and density functional theory with a classical but atomistic and polarizable embedding scheme, we show that accounting for electrostatics only leads to a qualitatively wrong picture, while a responsive environment can successfully capture both the classical and quantum dominant effects. Several residues are found to tune the excitation by both differentially stabilizing ground and excited states and through nonclassical "inductive resonance" interactions. The results obtained with such a quantum-in-classical model are validated against both experimental data and fully quantum calculations.

Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo.

We present a simple and general formalism to compute efficiently the derivatives of a multi-determinant Jastrow-Slater wave function, the local energy, the interatomic forces, and similar quantities needed in quantum Monte Carlo. Through a straightforward manipulation of matrices evaluated on the occupied and virtual orbitals, we obtain an efficiency equivalent to algorithmic differentiation in the computation of the interatomic forces and the optimization of the orbital parameters. Furthermore, for a large multi-determinant expansion, the significant computational gain afforded by a recently introduced table method is here extended to the local value of any one-body operator and to its derivatives, in both all-electron and pseudopotential calculations.

Introducing QMC/MMpol: Quantum Monte Carlo in Polarizable Force Fields for Excited States.

We present for the first time a quantum mechanics/molecular mechanics scheme which combines quantum Monte Carlo with the reaction field of classical polarizable dipoles (QMC/MMpol). In our approach, the optimal dipoles are self-consistently generated at the variational Monte Carlo level and then used to include environmental effects in diffusion Monte Carlo. We investigate the performance of this hybrid model in describing the vertical excitation energies of prototypical small molecules solvated in water, namely, methylenecyclopropene and s-trans acrolein. Two polarization regimes are explored where either the dipoles are optimized with respect to the ground-state solute density (polGS) or different sets of dipoles are separately brought to equilibrium with the states involved in the electronic transition (polSS). By comparing with reference supermolecular calculations where both solute and solvent are treated quantum mechanically, we find that the inclusion of the response of the environment to the excitation of the solute leads to superior results than the use of a frozen environment (point charges or polGS), in particular, when the solute-solvent coupling is dominated by electrostatic effects which are well recovered in the polSS condition. QMC/MMpol represents therefore a robust scheme to treat important environmental effects beyond static point charges, combining the accuracy of QMC with the simplicity of a classical approach.

Multiple-Resonance Local Wave Functions for Accurate Excited States in Quantum Monte Carlo.

We introduce a novel class of local multideterminant Jastrow-Slater wave functions for the efficient and accurate treatment of excited states in quantum Monte Carlo. The wave function is expanded as a linear combination of excitations built from multiple sets of localized orbitals that correspond to the bonding patterns of the different Lewis resonance structures of the molecule. We capitalize on the concept of orbital domains of local coupled-cluster methods, which is here applied to the active space to select the orbitals to correlate and construct the important transitions. The excitations are further grouped into classes, which are ordered in importance and can be systematically included in the Jastrow-Slater wave function to ensure a balanced description of all states of interest. We assess the performance of the proposed wave function in the calculation of vertical excitation energies and excited-state geometry optimization of retinal models whose π → π* state has a strong intramolecular charge-transfer character. We find that our multiresonance wave functions recover the reference values of the total energies of the ground and excited states with only a small number of excitations and that the same expansion can be flexibly used at very different geometries. Furthermore, significant computational saving can also be gained in the orbital optimization step by selectively mixing occupied and virtual orbitals based on spatial considerations without loss of accuracy on the excitation energy. Our multiresonance wave functions are therefore compact, accurate, and very promising for the calculation of multiple excited states of different character in large molecules.

Chromophore-protein coupling beyond nonpolarizable models: understanding absorption in green fluorescent protein.

The nature of the coupling of the photoexcited chromophore with the environment in a prototypical system like green fluorescent protein (GFP) is to date not understood, and its description still defies state-of-the-art multiscale approaches. To identify which theoretical framework of the chromophore-protein complex can realistically capture its essence, we employ here a variety of electronic-structure methods, namely, time-dependent density functional theory (TD-DFT), multireference perturbation theory (NEVPT2 and CASPT2), and quantum Monte Carlo (QMC) in combination with static point charges (QM/MM), DFT embedding (QM/DFT), and classical polarizable embedding through induced dipoles (QM/MMpol). Since structural modifications can significantly affect the photophysics of GFP, we also account for thermal fluctuations through extensive molecular dynamics simulations. We find that a treatment of the protein through static point charges leads to significantly blue-shifted excitation energies and that including thermal fluctuations does not cure the coarseness of the MM description. While TDDFT calculations on large cluster models indicate the need of a responsive protein, this response is not simply electrostatic: An improved description of the protein in the ground state or in response to the excitation of the chromophore via ground-state or state-specific DFT and MMpol embedding does not significantly modify the results obtained with static point charges. Through the use of QM/MMpol in a linear response formulation, a different picture in fact emerges in which the main environment response to the chromophore excitation is the one coupling the transition density and the corresponding induced dipoles. Such interaction leads to significant red-shifts and a satisfactory agreement with full QM cluster calculations at the same level of theory. Our findings demonstrate that, ultimately, faithfully capturing the effects of the environment in GFP requires a quantum treatment of large photoexcited regions but that a QM/classical model can be a useful approximation when extended beyond the electrostatic-only formulation.

Regarding the use and misuse of retinal protonated Schiff base photochemistry as a test case for time-dependent density-functional theory.

The excited-state relaxation of retinal protonated Schiff bases (PSBs) is an important test case for biological applications of time-dependent (TD) density-functional theory (DFT). While well-known shortcomings of approximate TD-DFT might seem discouraging for application to PSB relaxation, progress continues to be made in the development of new functionals and of criteria allowing problematic excitations to be identified within the framework of TD-DFT itself. Furthermore, experimental and theoretical ab initio advances have recently lead to a revised understanding of retinal PSB photochemistry, calling for a reappraisal of the performance of TD-DFT in describing this prototypical photoactive system. Here, we re-investigate the performance of functionals in (TD-)DFT calculations in light of these new benchmark results, which we extend to larger PSB models. We focus on the ability of the functionals to describe primarily the early skeletal relaxation of the chromophore and investigate how far along the out-of-plane pathways these functionals are able to describe the subsequent rotation around formal single and double bonds. Conventional global hybrid and range-separated hybrid functionals are investigated as the presence of Hartree-Fock exchange reduces problems with charge-transfer excitations as determined by the Peach-Benfield-Helgaker-Tozer Λ criterion and by comparison with multi-reference perturbation theory results. While we confirm that most functionals cannot render the complex photobehavior of the retinal PSB, do we also observe that LC-BLYP gives the best description of the initial part of the photoreaction.

Wavefunction in density functional theory embedding for excited states: which wavefunctions, which densities?

We present a detailed analysis of our recently proposed wavefunction in density functional theory method to include differential polarization effects through state-specific embedding potentials. We study methylenecyclopropene and acrolein in water by using several wavefunction approaches to validate the supermolecular reference and to assess their response to embedding. We find that quantum Monte Carlo, complete-active space second-order perturbation theory, and coupled cluster methods give very consistent solvatochromic shifts and a similar response to embedding. Our scheme corrects the excitation energies produced with a frozen environment, but the values are often overshot. To ameliorate the problem, one needs to use wavefunction densities to polarize the environment. The choice of the exchange-correlation functional in the construction of the potential has little effect on the excitation, whereas the approximate kinetic-energy functional appears to be the largest source of error.