Subspace Methods for the Simulation of Molecular Response Properties on a Quantum Computer.

We explore Davidson methods for obtaining excitation energies and other linear response properties within the recently developed quantum self-consistent linear response (q-sc-LR) method. Davidson-type methods allow for obtaining only a few selected excitation energies without explicitly constructing the electronic Hessian since they only require the ability to perform Hessian-vector multiplications. We apply the Davidson method to calculate the excitation energies of hydrogen chains (up to H10) and analyze aspects of statistical noise for computing excitation energies on quantum simulators. Additionally, we apply Davidson methods for computing linear response properties such as static polarizabilities for H2, LiH, H2O, OH-, and NH3, and show that unitary coupled cluster outperforms classical projected coupled cluster for molecular systems with strong correlation. Finally, we formulate the Davidson method for damped (complex) linear response, with application to the nitrogen K-edge X-ray absorption of ammonia, and the C6 coefficients of H2, LiH, H2O, OH-, and NH3.

Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing.

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH2 in the series of STO-3G/6-31G/6-31G* basis sets and of H4 and H2O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.

Which Options Exist for NISQ-Friendly Linear Response Formulations?

Linear response (LR) theory is a powerful tool in classic quantum chemistry crucial to understanding photoinduced processes in chemistry and biology. However, performing simulations for large systems and in the case of strong electron correlation remains challenging. Quantum computers are poised to facilitate the simulation of such systems, and recently, a quantum linear response formulation (qLR) was introduced [Kumar et al., J. Chem. Theory Comput. 2023, 19, 9136-9150]. To apply qLR to near-term quantum computers beyond a minimal basis set, we here introduce a resource-efficient qLR theory, using a truncated active-space version of the multiconfigurational self-consistent field LR ansatz. Therein, we investigate eight different near-term qLR formalisms that utilize novel operator transformations that allow the qLR equations to be performed on near-term hardware. Simulating excited state potential energy curves and absorption spectra for various test cases, we identify two promising candidates, dubbed "proj LRSD" and "all-proj LRSD".

The variational quantum eigensolver self-consistent field method within a polarizable embedded framework.

We formulate and implement the Variational Quantum Eigensolver Self Consistent Field (VQE-SCF) algorithm in combination with polarizable embedding (PE), thereby extending PE to the regime of quantum computing. We test the resulting algorithm, PE-VQE-SCF, on quantum simulators and demonstrate that the computational stress on the quantum device is only slightly increased in terms of gate counts compared to regular VQE-SCF. On the other hand, no increase in shot noise was observed. We illustrate how PE-VQE-SCF may lead to the modeling of real chemical systems using a simulation of the reaction barrier of the Diels-Alder reaction between furan and ethene as an example.

Multiconfigurational short-range density functional theory for nuclear magnetic resonance shielding constants with gauge-including atomic orbitals.

In this paper, we present the theory and implementation of nuclear magnetic resonance shielding constants with gauge-including atomic orbitals for the hybrid multiconfigurational short-range density functional theory model. As a special case, this implementation also includes Hartree-Fock srDFT (HF-srDFT). Choosing a complete-active space (CAS) wave function as the multiconfigurational parameterization of the wave function, we investigate how well CAS-srDFT reproduces experimental trends of nuclear shielding constants compared to DFT and complete active space self-consistent field (CASSCF). Calculations on the nucleobases adenine and thymine show that CAS-srDFT performs on average the best of the tested methods, much better than CASSCF but only marginally better than HF-srDFT. The performance, compared to regular DFT, is similar when functionals containing exact exchange are used. We generally find that the inclusion of exact exchange is important for an accurate description of the shielding constants. In cases where no exact exchange is included, we observe that the HF- and CAS-srDFT often outperform regular DFT. For calculations on transition metal nuclei in organometallic compounds with significant static correlation, the CAS-srDFT method again outperforms CASSCF compared to experimental shielding constants, and the change from HF-srDFT is substantial. In conclusion, the static correlation posed by the metal complexes seems to be captured by CAS-srDFT, which is promising since this type of correlation is not well described by regular DFT.

Multi-configurational short-range density functional theory can describe spin-spin coupling constants of transition metal complexes.

The multi-configurational short-range (sr) density functional theory has been extended to the calculation of indirect spin-spin coupling constants (SSCCs) for nuclear magnetic resonance spectroscopy. The performance of the new method is compared to Kohn-Sham density functional theory and the ab initio complete active space self-consistent field for a selected set of molecules with good reference values. Two density functionals have been considered, the local density approximation srLDA and srPBE from the GGA class of functionals. All srDFT calculations are of Hartree-Fock-type HF-srDFT or complete active space-type CAS-srDFT. In all cases, the calculated SSCC values are of the same quality for srLDA and srPBE functionals, suggesting that one should use the computationally cost-effective srLDA functionals in applications. For all the calculated SSCCs in organic compounds, the best choice is HF-srDFT; the more expensive CAS-srDFT does not provide better values for these single-reference molecules. Fluorine is a challenge; in particular, the FF, FC, and FO couplings have much higher statistical errors than the rest. For SSCCs involving fluorine and a metal atom CAS-srDFT with singlet, generalized Tamm-Dancoff approximation is needed to get good SSCC values although the reference ground state is not a multi-reference case. For VF6 -1, all other considered models fail blatantly.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems.

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

Cost-Effective Potential for Accurate Polarizable Embedding Calculations in Protein Environments.

The fragment-based polarizable embedding (PE) model combined with an appropriate electronic structure method constitutes a highly efficient and accurate multiscale approach for computing spectroscopic properties of a central moiety including effects from its molecular environment through an embedding potential. There is, however, a comparatively high computational overhead associated with the computation of the embedding potential, which is derived from first-principles calculations on individual fragments of the environment. To reduce the computational cost associated with the calculation of embedding potential parameters, we developed a set of amino acid-specific transferable parameters tailored for large-scale PE-based calculations that include proteins. The amino acid-based parameters are obtained by simultaneously fitting to a set of reference electric potentials based on structures derived from a backbone-dependent rotamer library. The developed cost-effective polarizable protein potential (CP3) consists of atom-centered charges and isotropic dipole-dipole polarizabilities of the standard amino acids. In terms of reproduction of electric potentials, the CP3 is shown to perform consistently and with acceptable accuracy across both small tripeptide test systems and larger proteins. We show, through applications on realistic protein systems, that acceptable accuracy can be obtained by using a pure CP3 representation of the protein environment, thus altogether omitting the cost associated with the calculation of embedding potential parameters. High accuracy comparable to that of the full fragment-based approach can be achieved through a mixed description where the CP3 is used only to describe amino acids beyond a threshold distance from the central quantum part.

CPPE: An Open-Source C++ and Python Library for Polarizable Embedding.

We present a modular open-source library for polarizable embedding (PE) named CPPE. The library is implemented in C++, and it additionally provides a Python interface for rapid prototyping and experimentation in a high-level scripting language. Our library integrates seamlessly with existing quantum chemical program packages through an intuitive and minimal interface. Until now, CPPE has been interfaced to three packages, Q-Chem, Psi4, and PySCF. Furthermore, we show CPPE in action using all three program packages for a computational spectroscopy application. With CPPE, host program interfaces only require minor programming effort, paving the way for new combined methodologies and broader availability of the PE model.

Triplet excitation energies from multiconfigurational short-range density-functional theory response calculations.

Linear response theory for the multiconfigurational short-range density functional theory (MC-srDFT) model is extended to triplet response with a singlet reference wave function. The triplet linear response equations for MC-srDFT are derived for a general hybrid srGGA functional and implemented in the Dalton program. Triplet excitation energies are benchmarked against the CC3 model of coupled cluster theory and the complete-active-space second-order perturbation theory using three different short-range functionals (srLDA, srPBE, and srPBE0), both with full linear response and employing the generalized Tamm-Dancoff approximation (gTDA). We find that using gTDA is required for obtaining reliable triplet excitations; for the CAS-srPBE model, the mean absolute deviation decreases from 0.40 eV to 0.26 eV, and for the CAS-srLDA model, it decreases from 0.29 eV to 0.21 eV. As expected, the CAS-srDFT model is found to be superior to the HF-srDFT model when analyzing the calculated triplet excitations for molecules in the benchmark set where increased static correlation is expected.

Importance of Accurate Structures for Quantum Chemistry Embedding Methods: Which Strategy Is Better?

Quantum chemistry embedding methods have become a popular approach to calculate molecular properties of larger systems. In order to account for finite temperature effects, including both configurational and conformational averaging, embedding methods are often combined with molecular dynamics (MD) simulations either in a direct or sequential manner. One of the decisive factors for a successful application of embedding methods is that that the underlying structures provided by the MD simulation are accurate, if not this will result in low-quality prediction of the molecular properties in question. Here we investigate different approaches for generating a set of molecular structures to be used in subsequent embedding calculations ranging from classical MD using a standard molecular mechanics (MM) force field to combined quantum mechanics/molecular mechanics (QM/MM) MD. Overall, we find an intermediate approach relying on classical MD followed by a constrained QM/MM geometry optimization to be a fairly accurate and very cost-effective approach, although this procedure naturally leads to underestimation of, for example, spectral bandwidths.

A comparative study of binding affinities for 6,7-dimethoxy-4-pyrrolidylquinazolines as phosphodiesterase 10A inhibitors using the linear interaction energy method.

The linear interaction energy (LIE) method was used to estimate the free energies of binding for a set of 27 pyrrolidylquinazoline derivatives as phosphodiesterase 10A inhibitors. Twenty-six X-ray crystal structures of phosphodiesterase 10A and two sampling methods, minimization and Hybrid Monte Carlo, were used to assess the affinity models based on the linear interaction energies. The best model was obtained based on the parameters α=0.16 and ß=0.04, which represent non-polar and polar interactions, respectively, with a root mean square error (RMSE) of 0.42kcal/mol (R(2)=0.71) and 0.52kcal/mol (R(2)=0.86) for the training and test sets, respectively. In addition, the applicability domain of the model was investigated. After validation of the models, the best model was subsequently used in a virtual screening process, which resulted in a set of optimized compounds. The models developed in this study could be useful as filter for virtual screening and lead optimization processes for phosphodiesterase 10A drug developments.