Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators

Non-iterative Triples for Transcorrelated Coupled Cluster Theory

We present an implementation of a perturbative triples correction for the coupled cluster ansatz including single and double excitations based on the transcorrelated Hamiltonian. Transcorrelation introduces explicit electron correlation in the electronic Hamiltonian through similarity transformation with a correlation factor. Due to this transformation, the transcorrelated Hamiltonian includes up to three-body couplings and becomes non-Hermitian. Since the conventional coupled cluster equations are solved by projection, it is well suited to harbor non-Hermitian Hamiltonians. The arising three-body operator, however, creates a huge memory bottleneck and increases the runtime scaling of the coupled cluster equations. As it has been shown that the three-body operator can be approximated, by expressing the Hamiltonian in the normal-ordered form, we investigate this approximation for the perturbative triples correction. Results are compared with a code-generation based transcorrelated coupled cluster implementation up to quadruple excitations.

Shortcut to chemically accurate quantum computing via density-based basis-set correction

Using GPU-accelerated state-vector emulation, we propose to embed a quantum computing ansatz into density-functional theory via density-based basis-set corrections to obtain quantitative quantum-chemistry results on molecules that would otherwise require brute-force quantum calculations using hundreds of logical qubits. Indeed, accessing a quantitative description of chemical systems while minimizing quantum resources is an essential challenge given the limited qubit capabilities of current quantum processors. We provide a shortcut towards chemically accurate quantum computations by approaching the complete-basis-set limit through coupling the density-based basis-set corrections approach, applied to any given variational ansatz, to an on-the-fly crafting of basis sets specifically adapted to a given system and user-defined qubit budget. The resulting approach self-consistently accelerates the basis-set convergence, improving electronic densities, ground-state energies, and first-order properties (e.g. dipole moments), but can also serve as a classical, a posteriori, energy correction to quantum hardware calculations with expected applications in drug design and materials science.

Striking the right balance of encoding electron correlation in the Hamiltonian and the wavefunction ansatz

Multi-configurational electronic structure theory delivers the most versatile approximations to many-electron wavefunctions, flexible enough to deal with all sorts of transformations, ranging from electronic excitations, to open-shell molecules and chemical reactions. Multi-configurational models are therefore essential to establish universally applicable, predictive ab initio methods for chemistry. Here, we present a discussion of explicit correlation approaches which address the nagging problem of dealing with static and dynamic electron correlation in multi-configurational active-space approaches. We review the latest developments and then point to their key obstacles. Our discussion is supported by new data obtained with tensor network methods. We argue in favor of simple electron-only correlator expressions that may allow one to define transcorrelated models in which the correlator does not bear a dependence on molecular structure.

Orbital optimisation in xTC transcorrelated methods

We present a combination of the bi-orthogonal orbital optimisation framework with the recently introduced xTC version of transcorrelation. This allows us to implement non-iterative perturbation based methods on top of the transcorrelated Hamiltonian. Additionally, the orbital optimisation influences results of other truncated methods, such as the distinguishable cluster with singles and doubles. The accuracy of these methods in comparison to standard xTC methods is demonstrated, and the advantages and disadvantages of the orbital optimisation are discussed.

Towards efficient quantum computing for quantum chemistry: reducing circuit complexity with transcorrelated and adaptive ansatz techniques

The near-term utility of quantum computers is hindered by hardware constraints in the form of noise. One path to achieving noise resilience in hybrid quantum algorithms is to decrease the required circuit depth - the number of applied gates - to solve a given problem. This work demonstrates how to reduce circuit depth by combining the transcorrelated (TC) approach with adaptive quantum ansätze and their implementations in the context of variational quantum imaginary time evolution (AVQITE). The combined TC-AVQITE method is used to calculate ground state energies across the potential energy surfaces of H4, LiH, and H2O. In particular, H4 is a notoriously difficult case where unitary coupled cluster theory, including singles and doubles excitations, fails to provide accurate results. Adding TC yields energies close to the complete basis set (CBS) limit while reducing the number of necessary operators - and thus circuit depth - in the adaptive ansätze. The reduced circuit depth furthermore makes our algorithm more noise-resilient and accelerates convergence. Our study demonstrates that combining the TC method with adaptive ansätze yields compact, noise-resilient, and easy-to-optimize quantum circuits that yield accurate quantum chemistry results close to the CBS limit.

Complete Active Space Iterative Coupled Cluster Theory

In this work, we investigate the possibility of improving multireference-driven coupled cluster (CC) approaches with an algorithm that iteratively combines complete active space (CAS) calculations with tailored CC and externally corrected CC. This is accomplished by establishing a feedback loop between the CC and CAS parts of a calculation through a similarity transformation of the Hamiltonian with those CC amplitudes that are not encompassed by the active space. We denote this approach as the complete active space iterative coupled cluster (CASiCC) ansatz. We investigate its efficiency and accuracy in the singles and doubles approximation by studying the prototypical molecules H4, H8, H2O, and N2. Our results demonstrate that CASiCC systematically improves on the single-reference CCSD and the externally corrected CCSD methods across entire potential energy curves while retaining modest computational costs. However, the tailored coupled cluster method shows superior performance in the strong correlation regime, suggesting that its accuracy is based on error compensation. We find that the iterative versions of externally corrected and tailored coupled cluster methods converge to the same results.

Compactification of determinant expansions via transcorrelation

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects.

Quantum Information Orbitals (QIO): Unveiling Intrinsic Many-Body Complexity by Compressing Single-Body Triviality

The simultaneous treatment of static and dynamic correlations in strongly correlated electron systems is a critical challenge. In particular, finding a universal scheme for identifying a single-particle orbital basis that minimizes the representational complexity of the many-body wave function is a formidable and longstanding problem. As a contribution toward its solution, we show that the total orbital correlation actually reveals and quantifies the intrinsic complexity of the wave function, once it is minimized via orbital rotations. To demonstrate the power of this concept in practice, an iterative scheme is proposed to optimize the orbitals by minimizing the total orbital correlation calculated by the tailored coupled cluster singles and doubles (TCCSD) ansatz. The optimized orbitals enable the limited TCCSD ansatz to capture more nontrivial information on the many-body wave function, indicated by the improved wave function and energy. An initial application of this scheme shows great improvement of TCCSD in predicting the singlet ground state potential energy curves of the strongly correlated C2 and Cr2 molecule.

Toward Real Chemical Accuracy on Current Quantum Hardware Through the Transcorrelated Method

Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and connectivity) hamper the implementation of most quantum algorithms and call for more noise-resilient solutions. We propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach to target these major roadblocks directly. This method transfers, without any approximation, correlations from the wave function directly into the Hamiltonian, thus reducing the resources needed to achieve accurate results with noisy quantum devices. We show that the TC approach allows for shallower circuits and improves the convergence toward the complete basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, thus, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using two and four qubits, respectively. To demonstrate our approach's current and near-term potential, we perform hardware experiments, where our results confirm that the TC method paves the way toward accurate quantum chemistry calculations already on today's quantum hardware.

Efficient Exploitation of Numerical Quadrature with Distance-Dependent Integral Screening in Explicitly Correlated F12 Theory: Linear Scaling Evaluation of the Most Expensive RI-MP2-F12 Term

We present a linear scaling atomic orbital based algorithm for the computation of the most expensive exchange-type RI-MP2-F12 term by employing numerical quadrature in combination with CABS-RI to avoid six-center-three-electron integrals. Furthermore, a robust distance-dependent integral screening scheme, based on integral partition bounds [Thompson, T. H.; Ochsenfeld, C. J. Chem. Phys. 2019, 150, 044101], is used to drastically reduce the number of the required three-center-one-electron integrals substantially. The accuracy of our numerical quadrature/CABS-RI approach and the corresponding integral screening is thoroughly assessed for interaction and isomerization energies across a variety of numerical integration grids. Our method outperforms the standard density fitting/CABS-RI approach with errors below 1 μEh even for small grid sizes and moderate screening thresholds. The choice of the grid size and screening threshold allows us to tailor our ansatz to a desired accuracy and computational efficiency. We showcase the approach's effectiveness for the chemically relevant system valinomycin, employing a triple-ζ F12 basis set combination (C54H90N6O18, 5757 AO basis functions, 10,266 CABS basis functions, 735,783 grid points). In this context, our ansatz achieves higher accuracy combined with a 135× speedup compared to the classical density fitting based variant, requiring notably less computation time than the corresponding RI-MP2 calculation. Additionally, we demonstrate near-linear scaling through calculations on linear alkanes. We achieved an 817-fold acceleration for C80H162 and an extrapolated 28,765-fold acceleration for C200H402, resulting in a substantially reduced computational time for the latter─from 229 days to just 11.5 min. Our ansatz may also be adapted to the remaining MP2-F12 terms, which will be the subject of future work.

Partial Separability of the Schrödinger Equation Combined with a Jastrow Factor

Describing the Coulomb interactions between electrons in atomic or molecular systems is an important step to help us obtain accurate results for the different observables in the system. One convenient approach is to separate the dynamic electronic correlation, i.e., Coulomb electron-electron repulsion, from the motion of the electrons in the nucleus electric field. The wave function is written as the product of two terms, one accounting for the electron-electron interactions, which is symmetric under identical particle exchange, and the other is antisymmetric and represents the dynamics and exchange of electrons within the nuclear electric field. In this work, we present a novel computational scheme based on this idea that leads to an expression of the energy as the sum of two terms. To illustrate the method, we look into few-body Coulombic systems, H2, H3+, and Li(1s2,2s), and discuss the possible extension to larger systems. A simple correlation factor, based on the Jastrow exponential term, is employed to represent the dynamics of the electron pairs, leading to simple analytical forms and accurate results. We also present and illustrate a different approach with the Li atom based on the partial separability applied to a portion of the atom.

Nonunitary projective transcorrelation theory inspired by the F12 ansatz

An alternative nonunitary transcorrelation, inspired by the F12 ansatz, is investigated. In contrast to the Jastrow transcorrelation of Boys-Handy, the effective Hamiltonian of this projective transcorrelation features: 1. a series terminating formally at four-body interactions. 2. no spin-contamination within the non-relativistic framework. 3. simultaneous satisfaction of the singlet and triplet first-order cusp conditions. 4. arbitrary choices of pairs for correlation including frozen-core approximations. We discuss the connection between the projective transcorrelation and F12 theory with applications to small molecules, to show that the cusp conditions play an important role to reduce the uncertainty arising from the nonunitary transformation.

The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

Transcorrelated selected configuration interaction in a bi-orthonormal basis and with a cheap three-body correlation factor

In this work, we develop a mathematical framework for a selected configuration interaction (SCI) algorithm within a bi-orthogonal basis for transcorrelated (TC) calculations. The bi-orthogonal basis used here serves as the equivalent of the standard Hartree-Fock (HF) orbitals. However, within the context of TC, it leads to distinct orbitals for the left and right vectors. Our findings indicate that the use of such a bi-orthogonal basis allows for a proper definition of the frozen core approximation. In contrast, the use of HF orbitals results in bad error cancellations for ionization potentials and atomization energies (AE). Compared to HF orbitals, the optimized bi-orthogonal basis significantly reduces the positive part of the second-order energy (PT2), thereby facilitating the use of standard extrapolation techniques of hermitian SCI. While we did not observe a significant improvement in the convergence of the SCI algorithm, this is largely due to the use in this work of a simple three-body correlation factor introduced in a recent study. This correlation factor, which depends only on atomic parameters, eliminates the need for re-optimization of the correlation factor for molecular systems, making its use straightforward and user-friendly. Despite the simplicity of this correlation factor, we were able to achieve accurate results on the AE of a series of 14 molecules on a triple-zeta basis. We also successfully broke a double bond until the full dissociation limit while maintaining the size consistency property. This work thus demonstrates the potential of the BiO-TC-SCI approach in handling complex molecular systems.

Studies on the Transcorrelated Method

We investigate the possibility of using a transcorrelated (TC) Hamiltonian to describe electron correlation. A method to obtain TC wavefunctions was developed based on the mathematical framework of the bi-variational principle. This involves the construction of an effective TC Hamiltonian matrix, which can be solved in a self-consistent manner. This was optimized using a method we call second-order-moment minimization and demonstrate that it is possible to obtain highly accurate energies for some closed-shell atoms and helium-like ions. The effects of certain correlator terms on the description of electron-electron and electron-nuclear cusps were also examined graphically, and some TC wavefunctions were compared against near-exact Hylleraas wavefunctions.

Biorthonormal Orbital Optimization with a Cheap Core-Electron-Free Three-Body Correlation Factor for Quantum Monte Carlo and Transcorrelation

We introduce a novel three-body correlation factor that is designed to vanish in the core region around each nucleus and approach a universal two-body correlation factor for valence electrons. The transcorrelated Hamiltonian is used to optimize the orbitals of a single Slater determinant within a biorthonormal framework. The Slater-Jastrow wave function is optimized on a set of atomic and molecular systems containing both second-row elements and 3d transition metal elements. The optimization of the correlation factor and the orbitals, along with an increase in the basis set, results in a systematic lowering of the variational Monte Carlo energy for all systems tested. Importantly, the optimal parameters of the correlation factor obtained for atomic systems can be transferred to molecules. Additionally, the present correlation factor is computationally efficient and uses a mixed analytical-numerical integration scheme that reduces the costly numerical integration from R6 to R3.

Density Matrix Renormalization Group for Transcorrelated Hamiltonians: Ground and Excited States in Molecules

We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.