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.

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.

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.

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.

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.

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.

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.

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.

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 simple scheme for calculating approximate transition moments within the equation of motion expectation value formalism.

A simple scheme for calculating approximate transition moments within the framework of the equation of motion coupled cluster method is proposed. It relies on a matrix inversion technique to calculate the excited state left eigenvectors and requires no additional cost over that of the excitation energy calculation. The new approximation gives almost identical UV-Vis spectra to that obtained using the standard equation of motion coupled cluster method with single and double excitations for molecules in a standard test set.

Automatic active space selection for the similarity transformed equations of motion coupled cluster method.

An efficient scheme for the automatic selection of an active space for similarity transformed equations of motion (STEOM) coupled cluster method is proposed. It relies on state averaged configuration interaction singles (CIS) natural orbitals and makes it possible to use STEOM as a black box method. The performance of the new scheme is tested for singlet and triplet valence, charge transfer, and Rydberg excited states.

Similarity transformed equation of motion coupled-cluster theory based on an unrestricted Hartree-Fock reference for applications to high-spin open-shell systems.

The similarity transformed equation of motion coupled-cluster approach is extended for applications to high-spin open-shell systems, within the unrestricted Hartree-Fock (UHF) formalism. An automatic active space selection scheme has also been implemented such that calculations can be performed in a black-box fashion. It is observed that both the canonical and automatic active space selecting similarity transformed equation of motion (STEOM) approaches perform about as well as the more expensive equation of motion coupled-cluster singles doubles (EOM-CCSD) method for the calculation of the excitation energies of doublet radicals. The automatic active space selecting UHF STEOM approach can therefore be employed as a viable, lower scaling alternative to UHF EOM-CCSD for the calculation of excited states in high-spin open-shell systems.

Towards a pair natural orbital coupled cluster method for excited states.

The use of back-transformed pair natural orbitals in the calculation of excited state energies, ionization potentials, and electron affinities is investigated within the framework of equation of motion coupled cluster theory and its similarity transformed variant. Possible approaches to a more optimal use of pair natural orbitals in these methods are indicated.

Speeding up equation of motion coupled cluster theory with the chain of spheres approximation.

In the present paper, the chain of spheres exchange (COSX) approximation is applied to the highest scaling terms in the equation of motion (EOM) coupled cluster equations with single and double excitations, in particular, the terms involving integrals with four virtual labels. It is found that even the acceleration of this single term yields significant computational gains without compromising the desired accuracy of the method. For an excitation energy calculation on a cluster of five water molecules using 585 basis functions, the four virtual term is 9.4 times faster using COSX with a loose grid than using the canonical implementation, which yields a 2.6 fold acceleration for the whole of the EOM calculation. For electron attachment calculations, the four virtual term is 15 times and the total EOM calculation is 10 times faster than the canonical calculation for the same system. The accuracy of the new method was tested using Thiel's test set for excited states using the same settings and the maximum absolute deviation over the whole test set was found to be 12.945 cm(-1) (59 µHartree) for excitation energies and 6.799 cm(-1) (31 µHartree) for electron attachments. Using MP2 amplitudes for the ground state in combination with the parallel evaluation of the full EOM equations in the manner discussed in this paper enabled us to perform calculations for large systems. Electron affinity values for the two lowest states of a Zn protoporphyrine model compound (224 correlated electrons and 1120 basis functions) were obtained in 3 days 19 h using 4 cores of a Xeon E5-2670 processor allocating 10 GB memory per core. Calculating the lowest two excitation energies for trans-retinal (114 correlated electrons and 539 basis functions) took 1 day 21 h using eight cores of the same processor and identical memory allocation per core.

Electronic structure of the unique [4Fe-3S] cluster in O2-tolerant hydrogenases characterized by 57Fe Mossbauer and EPR spectroscopy.

Iron-sulfur clusters are ubiquitous electron transfer cofactors in hydrogenases. Their types and redox properties are important for H(2) catalysis, but, recently, their role in a protection mechanism against oxidative inactivation has also been recognized for a [4Fe-3S] cluster in O(2)-tolerant group 1 [NiFe] hydrogenases. This cluster, which is uniquely coordinated by six cysteines, is situated in the proximity of the catalytic [NiFe] site and exhibits unusual redox versatility. The [4Fe-3S] cluster in hydrogenase (Hase) I from Aquifex aeolicus performs two redox transitions within a very small potential range, forming a superoxidized state above +200 mV vs. standard hydrogen electrode (SHE). Crystallographic data has revealed that this state is stabilized by the coordination of one of the iron atoms to a backbone nitrogen. Thus, the proximal [4Fe-3S] cluster undergoes redox-dependent changes to serve multiple purposes beyond classical electron transfer. In this paper, we present field-dependent (57)Fe-Mössbauer and EPR data for Hase I, which, in conjunction with spectroscopically calibrated density functional theory (DFT) calculations, reveal the distribution of Fe valences and spin-coupling schemes for the iron-sulfur clusters. The data demonstrate that the electronic structure of the [4Fe-3S] core in its three oxidation states closely resembles that of corresponding conventional [4Fe-4S] cubanes, albeit with distinct differences for some individual iron sites. The medial and distal iron-sulfur clusters have similar electronic properties as the corresponding cofactors in standard hydrogenases, although their redox potentials are higher.

Geometry of the magic number H(+)(H2O)21 water cluster by proxy.

Abundance mass spectra, obtained upon carefully electrospraying solutions of tert-butanol (TB) in water into a mass spectrometer, display a systematic series of peaks due to mixed H(+)(TB)m(H2O)n clusters. Clusters with m + n = 21 have higher abundance (magic number peaks) than their neighbours when m ≤ 9, while for m > 9 they have lower abundance. This indicates that the mixed TB-H2O clusters retain a core hydrogen bonded network analogous to that in the famous all-water H(+)(H2O)21 cluster up to the limit m = 9. The limiting value corresponds to the number of dangling O-H groups pointing out from the surface of the degenerated pentagonal dodecahedral, considered to be the lowest energy form of H(+)(H2O)21; the experimental findings therefore support this geometry. The experimental findings are supported by ab initio quantum chemical calculations to provide a consistent framework for the interpretation of these kinds of experiments.

Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution.

Three new variants of the auxiliary-density-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and performance in all-electron B3LYP calculations with several commonly used basis sets. The effect of the choice of the exchange functional for the ADMM exchange-correction term is also investigated.

Robust fitting techniques in the chain of spheres approximation to the Fock exchange: The role of the complementary space.

The chain of spheres seminumerical integration algorithm is reexamined from the point of view of various numerical error cancellation schemes. The two sources of ideas are the robust fitting technique of Dunlap and the dealiasing concept of the pseudospectral methods of Friesner. The most efficient scheme is identified and is found to be a generalization of the overlap fitting procedure discussed earlier. It will be shown that due to the conceptual similarity between complementary auxiliary basis functions of explicitly correlated approaches and the dealiasing functions of pseudospectral methods, the former can be used in a similar manner as the latter to cancel numerical errors. In contrast to the dealiasing functions, the complementary auxiliary basis functions are not specific to the integration grid. The scheme obtained can dramatically improve reaction energies even using small grid setups at a moderate additional cost.

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.

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.