Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations.

In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

Quantum computing in pharma: A multilayer embedding approach for near future applications.

Quantum computers are special purpose machines that are expected to be particularly useful in simulating strongly correlated chemical systems. The quantum computer excels at treating a moderate number of orbitals within an active space in a fully quantum mechanical manner. We present a quantum phase estimation calculation on F2 in a (2,2) active space on Rigetti's Aspen-11 QPU. While this is a promising start, it also underlines the need for carefully selecting the orbital spaces treated by the quantum computer. In this work, a scheme for selecting such an active space automatically is described and simulated results obtained using both the quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms are presented and combined with a subtractive method to enable accurate description of the environment. The active occupied space is selected from orbitals localized on the chemically relevant fragment of the molecule, while the corresponding virtual space is chosen based on the magnitude of interactions with the occupied space calculated from perturbation theory. This protocol is then applied to two chemical systems of pharmaceutical relevance: the enzyme [Fe] hydrogenase and the photosenzitizer temoporfin. While the sizes of the active spaces currently amenable to a quantum computational treatment are not enough to demonstrate quantum advantage, the procedure outlined here is applicable to any active space size, including those that are outside the reach of classical computation.

Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications.

Computational chemistry is an essential tool in the pharmaceutical industry. Quantum computing is a fast evolving technology that promises to completely shift the computational capabilities in many areas of chemical research by bringing into reach currently impossible calculations. This perspective illustrates the near-future applicability of quantum computation of molecules to pharmaceutical problems. We briefly summarize and compare the scaling properties of state-of-the-art quantum algorithms and provide novel estimates of the quantum computational cost of simulating progressively larger embedding regions of a pharmaceutically relevant covalent protein-drug complex involving the drug Ibrutinib. Carrying out these calculations requires an error-corrected quantum architecture that we describe. Our estimates showcase that recent developments on quantum phase estimation algorithms have dramatically reduced the quantum resources needed to run fully quantum calculations in active spaces of around 50 orbitals and electrons, from estimated over 1000 years using the Trotterization approach to just a few days with sparse qubitization, painting a picture of fast and exciting progress in this nascent field.

An improved chain of spheres for exchange algorithm.

In the present work, we describe a more accurate and efficient variant of the chain-of-spheres algorithm (COSX) for exchange matrix computations. Higher accuracy for the numerical integration is obtained with new grids that were developed using global optimization techniques. With our new default grids, the average absolute energy errors are much lower than 0.1 kcal/mol, which is desirable to achieve "chemical accuracy." Although the size of the new grids is increased by roughly a factor of 2.5, the excellent efficiency of the original COSX implementation is still further improved in most cases. The evaluation of the analytic electrostatic potential integrals was significantly accelerated by a new implementation of rolled-out versions of the Dupuis-Rys-King and Head-Gordon-Pople algorithms. Compared to our earlier implementation, a twofold speedup is obtained for the frequently used triple-ζ basis sets, while up to a 16-fold speedup is observed for quadruple-ζ basis sets. These large gains are a consequence of both the more efficient integral evaluation and the intermediate exchange matrix computation in a partially contracted basis when generally contracted shells occur. With our new RIJCOSX implementation, we facilitate accurate self-consistent field (SCF) binding energy calculations on a large supra-molecular complex composed of 320 atoms. The binding-energy errors with respect to the fully analytic results are well below 0.1 kcal/mol for the cc-pV(T/Q)Z basis sets and even smaller than for RIJ with fully analytic exchange. At the same time, our RIJCOSX SCF calculation even with the cc-pVQZ basis and the finest grid is 21 times faster than the fully analytic calculation.

Redesigning donor-acceptor Stenhouse adduct photoswitches through a joint experimental and computational study.

Many studies have recently explored a new class of reversible photoswitching compounds named Donor-Acceptor Stenhouse Adducts (DASAs). Upon light irradiation, these systems evolve from a coloured open-chain to a colourless closed-ring form, while the thermal back-reaction occurs at room temperature. In order to fulfill the requirements for different applications, new molecules with specific properties need to be designed. For instance, shifting the activation wavelength towards the red part of the visible spectrum is of relevance to biological applications. By using accurate computational calculations, we have designed new DASAs and predicted some of their photophysical properties. Starting from well-studied donor and acceptor parts, we have shown that small chemical modifications can lead to substantial changes in both photophysical and photoswitching properties of the resulting DASAs. Furthermore, we have also analysed how these substitutions impact the electronic structure of the systems. Finally, some pertinent candidates have been successfully synthesized and their photoswitching properties have been characterized experimentally.

Fragment-Based Local Coupled Cluster Embedding Approach for the Quantification and Analysis of Noncovalent Interactions: Exploring the Many-Body Expansion of the Local Coupled Cluster Energy.

Herein, we introduce a fragment-based local coupled cluster embedding approach for the accurate quantification and analysis of noncovalent interactions in molecular aggregates. Our scheme combines two different expansions of the domain-based local pair natural orbital coupled cluster (DLPNO-CCSD(T)) energy: the many-body expansion (MBE) and the local energy decomposition (LED). The low-order terms in the MBE are initially computed in the presence of an environment that is treated at a low level of theory. Then, LED is used to decompose the energy of each term in the embedded MBE into additive fragment and fragment-pairwise contributions. This information is used to quantify the total energy of the system while providing at the same time in-depth insights into the nature and cooperativity of noncovalent interactions. Two different approaches are introduced and tested, in which the environment is treated at different levels of theory: the local coupled cluster in the Hartree-Fock (LCC-in-HF) method, in which the environment is treated at the HF level; and the electrostatically embedded local coupled cluster method (LCC-in-EE), in which the environment is replaced by point charges. Both schemes are designed to preserve as much as possible the accuracy of the parent local coupled cluster method for total energies, while being embarrassingly parallel and less memory intensive. These schemes appear to be particularly promising for the study of large and complex molecular aggregates at the coupled cluster level, such as condensed phase systems and protein-ligand interactions.

Accurate Computation of the Absorption Spectrum of Chlorophyll a with Pair Natural Orbital Coupled Cluster Methods.

The ability to accurately compute low-energy excited states of chlorophylls is critically important for understanding the vital roles they play in light harvesting, energy transfer, and photosynthetic charge separation. The challenge for quantum chemical methods arises both from the intrinsic complexity of the electronic structure problem and, in the case of biological models, from the need to account for protein-pigment interactions. In this work, we report electronic structure calculations of unprecedented accuracy for the low-energy excited states in the Q and B bands of chlorophyll a. This is achieved by using the newly developed domain-based local pair natural orbital (DLPNO) implementation of the similarity transformed equation of motion coupled cluster theory with single and double excitations (STEOM-CCSD) in combination with sufficiently large and flexible basis sets. The results of our DLPNO-STEOM-CCSD calculations are compared with more approximate approaches. The results demonstrate that, in contrast to time-dependent density functional theory, the DLPNO-STEOM-CCSD method provides a balanced performance for both absorption bands. In addition to vertical excitation energies, we have calculated the vibronic spectrum for the Q and B bands through a combination of DLPNO-STEOM-CCSD and ground-state density functional theory frequency calculations. These results serve as a basis for comparison with gas-phase experiments.

A New Benchmark Set for Excitation Energy of Charge Transfer States: Systematic Investigation of Coupled Cluster Type Methods.

The numerous existing publications on benchmarking quantum chemistry methods for excited states rarely include Charge Transfer (CT) states, although many interesting phenomena in, e.g., biochemistry and material physics involve the transfer of electrons between fragments of the system. Therefore, it is timely to test the accuracy of quantum chemical methods for CT states, as well. In this study we first propose a new benchmark set consisting of dimers having low-energy CT states. On this set, the vertical excitation energy has been calculated with Coupled Cluster methods including triple excitations (CC3, CCSDT-3, CCSD(T)(a)*), as well as with methods including full or approximate doubles (CCSD, STEOM-CCSD, CC2, ADC(2), EOM-CCSD(2)). The results show that the popular CC2 and ADC(2) methods are much less accurate for CT states than for valence states. On the other hand, EOM-CCSD seems to have similar systematic overestimation of the excitation energies for both types of states. Among the triples methods the novel EOM-CCSD(T)(a)* method including noniterative triple excitations is found to stand out with its consistently good performance for all types of states, delivering essentially EOM-CCSDT quality results.

Unveiling the Photophysical Properties of Boron-dipyrromethene Dyes Using a New Accurate Excited State Coupled Cluster Method.

Boron-dipyrromethene (BODIPY) molecules form a class of fluorescent dyes known for their exceptional photoluminescence properties. Today, they are used extensively in various applications from fluorescent imaging to optoelectronics. The ease of altering the BODIPY core has allowed scientists to synthesize dozens of analogues by exploring chemical substitutions of various kinds or by increasing the length of conjugated groups. However, predicting the impact of any chemical change accurately is still a challenge, especially as most computational methods fail to describe correctly the photophysical properties of BODIPY derivatives. In this study, the recently developed coupled cluster method called "domain-based local pair natural orbital similarity transformed equation of motion-coupled cluster singles and doubles" (DLPNO-STEOM-CCSD) is employed to compute the lowest vertical excitation energies of more than 50 BODIPY molecules. The method performs remarkably well yielding an accuracy of about 0.06 eV compared to the experimental absorption maxima. We also provide an estimate to the error made by neglecting vibronic effects in the computed spectra. The dyes selected for investigation here span a large range of molecular sizes and chemical functionalities and are embedded in solvents with different polarities. We have also investigated if the method is able to correctly reproduce the impact of a single chemical modification on the absorption energy. To characterize the method in more specific terms, we have studied four large BODIPY analogues used in real-life applications due to their interesting chemical properties. These examples should illustrate the capacity of the DLPNO-STEOM-CCSD procedure to become a method of choice for the study of photophysical properties of medium to large organic compounds.

Computational Design of Near-Infrared Fluorescent Organic Dyes Using an Accurate New Wave Function Approach.

The extensive research focusing on fluorescent organic dyes for bioimaging has made this in vivo method available for a diverse range of applications. One way to enhance this method is to tune the absorption and emission wavelengths of dyes to the near-infrared region where better light penetration and imaging resolution can be achieved. For this purpose, the well-known BODIPY dyes and their derivatives called aza-BODIPY have been the subject of extensive synthetic efforts. The interest in these systems stems from their excellent photophysical properties. Despite numerous studies, the rational design of near-infrared active dyes with desirable properties remains difficult. Here, we present a new wave function-based method for modeling excited states of large molecules, which has numerous theoretical advantages over the most commonly used electronic structure methods. This method is employed to suggest candidates for new dyes with the desired properties and to predict the absorption and fluorescence maxima and luminescence spectra of aza-BODIPY dyes with possible applications in fluorescence imaging.

Accurate Band Gap Predictions of Semiconductors in the Framework of the Similarity Transformed Equation of Motion Coupled Cluster Theory.

In this work, we present a detailed comparison between wave-function-based and particle/hole techniques for the prediction of band gap energies of semiconductors. We focus on the comparison of the back-transformed Pair Natural Orbital Similarity Transformed Equation of Motion Coupled-Cluster (bt-PNO-STEOM-CCSD) method with Time Dependent Density Functional Theory (TD-DFT) and Delta Self Consistent Field/DFT (Δ-SCF/DFT) that are employed to calculate the band gap energies in a test set of organic and inorganic semiconductors. Throughout, we have used cluster models for the calculations that were calibrated by comparing the results of the cluster calculations to periodic DFT calculations with the same functional. These calibrations were run with cluster models of increasing size until the results agreed closely with the periodic calculation. It is demonstrated that bt-PNO-STEOM-CC yields accurate results that are in better than 0.2 eV agreement with the experiment. This holds for both organic and inorganic semiconductors. The efficiency of the employed computational protocols is thoroughly discussed. Overall, we believe that this study is an important contribution that can aid future developments and applications of excited state coupled cluster methods in the field of solid-state chemistry and heterogeneous catalysis.

Efficient simulation of overtones and combination bands in resonant Raman spectra.

In this work, we describe how Resonant Raman (RR) spectra can be predicted with good accuracy using the path integral formulation for the harmonic oscillator, including the Herzberg-Teller and Duschinsky rotation effects. In particular, we will focus on approximations to make these calculations more efficient and expand the current methodology to include overtones and combination bands in the final prediction. After outlining the theory, we explain how it can be made into a black-box method and discuss the approximations that rely on its intrinsic parameters. It is shown that the latter can improve the efficiency by orders of magnitude, without substantial loss in accuracy. We will also show that the predicted RR spectra are in good agreement with experiment for both the intensity and position of spectral peaks, even for higher order vibronic excitations. Thus, the method proposed here can be used as a computational aid to experiments to improve the quality of analysis and theoretical understanding of RR.

A domain-based local pair natural orbital implementation of the equation of motion coupled cluster method for electron attached states.

This work describes a domain-based local pair natural orbital (DLPNO) implementation of the equation of motion coupled cluster method for the computation of electron affinities (EAs) including single and double excitations. Similar to our earlier work on ionization potentials (IPs), the method reported in this study uses the ground state DLPNO framework and extends it to the electron attachment problem. While full linear scaling could not be achieved as in the IP case, leaving the Fock/Koopmans' contributions in the canonical basis and using a tighter threshold for singles PNOs allows us to compute accurate EAs and retain most of the efficiency of the DLPNO technique. Thus as in the IP case, the ground state truncation parameters are sufficient to control the accuracy of the computed EA values, although a new set of integrals for singles PNOs must be generated at the DLPNO integral transformation step. Using standard settings, our method reproduces the canonical results with a maximum absolute deviation of 49 meV for bound states of a test set of 24 molecules. Using the same settings, a calculation involving more than 4500 basis functions, including diffuse functions, takes four days on four cores, with only 48 min spent in the EA module itself.

Multilayer Approach to the IP-EOM-DLPNO-CCSD Method: Theory, Implementation, and Application.

We present a multilayer implementation of the EOM-CCSD for the computation of ionization potentials of atoms and molecules in the presence of their environment. The method uses local orbitals to partition the system into a number of hypothetical fragments and treat different fragments of the system at different levels of theory. This approach significantly reduces the computational cost with a systematically controllable accuracy and is equally applicable to describe the environmental effect of both bonded and nonbonded nature. An accurate description of the interfragment interaction has been found to be crucial in determining the accuracy of the calculated IP values.

Predicting Phosphorescence Rates of Light Organic Molecules Using Time-Dependent Density Functional Theory and the Path Integral Approach to Dynamics.

In this work, we present a general method for predicting phosphorescence rates and spectra for molecules using time-dependent density functional theory (TD-DFT) and a path integral approach for the dynamics that relies on the harmonic oscillator approximation for the nuclear movement. We first discuss the theory involved in including spin-orbit coupling (SOC) among singlet and triplet excited states and then how to compute the corrected transition dipole moments and phosphorescence rates. We investigate the dependence of these rates on some TD-DFT parameters, such as the nature of the functional, the number of roots, and the Tamm-Dancoff approximation. After that, we evaluate the effect of different SOC integral schemes and show that our best method is applicable to a large number of systems with different excited state characters.

Systematic High-Accuracy Prediction of Electron Affinities for Biological Quinones.

Quinones play vital roles as electron carriers in fundamental biological processes; therefore, the ability to accurately predict their electron affinities is crucial for understanding their properties and function. The increasing availability of cost-effective implementations of correlated wave function methods for both closed-shell and open-shell systems offers an alternative to density functional theory approaches that have traditionally dominated the field despite their shortcomings. Here, we define a benchmark set of quinones with experimentally available electron affinities and evaluate a range of electronic structure methods, setting a target accuracy of 0.1 eV. Among wave function methods, we test various implementations of coupled cluster (CC) theory, including local pair natural orbital (LPNO) approaches to canonical and parameterized CCSD, the domain-based DLPNO approximation, and the equations-of-motion approach for electron affinities, EA-EOM-CCSD. In addition, several variants of canonical, spin-component-scaled, orbital-optimized, and explicitly correlated (F12) Møller-Plesset perturbation theory are benchmarked. Achieving systematically the target level of accuracy is challenging and a composite scheme that combines canonical CCSD(T) with large basis set LPNO-based extrapolation of correlation energy proves to be the most accurate approach. Methods that offer comparable performance are the parameterized LPNO-pCCSD, the DLPNO-CCSD(T0 ), and the orbital optimized OO-SCS-MP2. Among DFT methods, viable practical alternatives are only the M06 and the double hybrids, but the latter should be employed with caution because of significant basis set sensitivity. A highly accurate yet cost-effective DLPNO-based coupled cluster approach is used to investigate the methoxy conformation effect on the electron affinities of ubiquinones found in photosynthetic bacterial reaction centers. © 2018 Wiley Periodicals, Inc.

An efficient pair natural orbital based configuration interaction scheme for the calculation of open-shell ionization potentials.

A spin adapted configuration interaction scheme is proposed for the evaluation of ionization potentials in α high spin open shell reference functions. There are three different ways to remove an electron from such a reference, including the removal of an alpha or a beta electron from doubly occupied or an alpha electron from singly occupied molecular orbitals. Ionization operators are constructed for each of these cases, and the resulting second quantized expressions are implemented using an automated code generator environment. To achieve greater computational efficiency, the virtual space is reduced using an averaged pair natural orbital machinery developed earlier and applied with great success in the calculation of X-ray absorption spectra [D. Manganas et al., J. Chem. Phys. A 122, 1215 (2018)]. Various approximate integral evaluation schemes including the resolution of identity and seminumerical techniques are also invoked to further enhance the computational efficiency. Although the resulting method is not particularly accurate in terms of predicting absolute energy values, with a simple shift in the ionization potentials, it is still possible to use it for the qualitative characterization of the basic features of X-ray photoionization spectra. While satellite intensities cannot be computed with the current method, the inclusion of vibrational effects using a path integral technique allows for the computation of vibrational transitions corresponding to main peaks.

A near-linear scaling equation of motion coupled cluster method for ionized states.

In this work, a domain-based local pair natural orbital (DLPNO) version of the equation of motion coupled cluster theory with single and double excitations for ionization potentials (IP-EOM-CCSD) equations has been formulated and implemented. The method uses ground state localized occupied and pair natural virtual orbitals and applies the DLPNO machinery to arrive at a linear scaling implementation of the IP-EOM-CCSD method. The accuracy of the method is controllable using ground state truncation parameters. Using default thresholds, the method predicts ionization potential (IP) values with good accuracy (mean absolute error of 0.08 eV). We demonstrate that our code can be used to compute IP values for systems with more than 1000 atoms and 10 000 basis functions.

Self-Consistent Field Calculation of Nuclear Magnetic Resonance Chemical Shielding Constants Using Gauge-Including Atomic Orbitals and Approximate Two-Electron Integrals.

The chain-of-spheres method (COS) for approximating two-electron integrals is applied to Hartree-Fock and density functional theory calculations of nuclear magnetic resonance chemical shielding tensors, based on gauge-including atomic orbitals. The accuracy of the approximation is compared to that of the resolution of the identity (RI) approach, using a benchmark test set of 15 small molecules. Reasonable auxiliary basis sets and grid sizes are selected on the basis of a careful investigation of how approximating each of the two-electron terms in the self-consistent field (SCF) and coupled perturbed SCF equations affects the calculated shielding constants. It is found that the errors are linearly additive but can have either sign. The mean absolute relative error due to applying the RI/COS approximations with the chosen settings to all two-electron terms is on the order of 0.01% and therefore negligible compared to the errors due to basis set incompleteness (∼1%) and the method used (10-50%). Several larger organic systems are used to assess the efficiency of the RI approximation for both Coulomb- and exchange-type integrals (RIJK) as well as a combination of RI for Coulomb and COS for exchange contributions (RIJCOSX). The RIJK approximation is more efficient for small molecules, while for systems of over 100 electrons and 1000 basis functions, the RIJCOSX approximation is superior.

On the theoretical prediction of fluorescence rates from first principles using the path integral approach.

In this work, we present and implement the theory for calculating fluorescence rates and absorption and emission spectra from first principles, using the path integral approach. We discuss some approximations and modifications to the full set of equations that improve speed and numerical stability for the case when a large number of modes are considered. New methods to approximate the excited state potential energy surface are also discussed and it is shown that for most purposes, these can be used instead of a full geometry optimization to obtain the rates mentioned above. A few examples are presented and the overall performance of the method is discussed. It is shown that the rates and spectra computed in this way are well within the acceptable range of errors and can be used in future predictions, particularly for screening purposes, with the only limitation on size being that of the electronic structure calculation itself.