Multireference Coupled Cluster Theory Research Articles

Code generation in ORCA: progress, efficiency and tight integration.

An improved version of ORCA's automated generator environment (ORCA-AGE II) is presented. The algorithmic improvements and the move to C++ as the programming language lead to a performance gain of up to two orders of magnitude compared to the previously developed PYTHON toolchain. Additionally, the restructured modular design allows for far more complex code engines to be implemented readily. Importantly, we have realised an extremely tight integration with the ORCA host program. This allows for a workflow in which only the wavefunction Ansatz is part of the source code repository while all actual high-level code is generated automatically, inserted at the appropriate place in the host program before it is compiled and linked together with the hand written code parts. This construction ensures longevity and uniform code quality. Furthermore the new developments allow ORCA-AGE II to generate parallelised production-level code for highly complex theories, such as fully internally contracted multireference coupled-cluster theory (fic-MRCC) with an enormous number of contributing tensor contractions. We also discuss the automated implementation of nuclear gradients for arbitrary theories. All these improvements enable the implementation of theories that are too complex for the human mind and also reduce development times by orders of magnitude. We hope that this work enables researchers to concentrate on the intellectual content of the theories they develop rather than be concerned with technical details of the implementation.

Excitation spectra of fully correlated donor-acceptor complexes by density matrix renormalisation group

In this paper, we investigate the convergence properties of density matrix renormalisation group (DMRG) calculations for vertical electronic excitations performed on three bimolecular complexes motivated by Szalay's recent benchmark results [JCTC. 2020;16:7]. Besides the high-level coupled-cluster reference spectrum, the extrapolated truncation-free DMRG results are compared against several alternative solutions based on multireference configuration interaction and coupled-cluster theories. By taking advantage of the error cancellation effects, which are inherent in the state-averaged computational scheme, we demonstrate that the extrapolated solution based on only low bond-dimensional DMRG data could already provide quantitative predictions even for large active spaces. We also show that orbital optimisation improves further the accuracy of the DMRG results which systematically approach the excitation energies of coupled cluster single-double-triple at similar computational costs.

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory.

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory.

We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving for the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of the three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.

Electronic states of NaLi molecule: Benchmark results with Fock space coupled cluster approach.

Accurate potential energy curves (PECs) are obtained for 20 lowest lying electronic states of the NaLi molecule. The computational scheme used here is based on the multireference coupled cluster theory formulated in the (2,0) sector of the Fock space. The latter sector provides the description of states obtained by attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionized NaLi+2 molecule as a Fermi vacuum. The latter has a very concrete advantage in calculations of the PECs since it dissociates into closed shell fragments (NaLi+2 → Na+ + Li+); hence, the restricted Hartree-Fock method can be used within the whole range of interatomic distances. Computed PECs and spectroscopic constants stay very close to the experimental values (if the latter are available) with the accuracy exceeding the other theoretical approaches including those based on the effective core polarization potentials. Relativistic corrections included at the infinite-order two-component level have a non-negligible effect on the accuracy of computed excitation and dissociation energies with contributions up to 50 cm-1.

Fock-Space Coupled Cluster Theory: Systematic Study of Partial Fourth Order Triples Schemes for Ionization Potential and Comparison with Bondonic Formalism.

In this paper, we have made a systematic study of partial fourth order perturbative schemes due to triples to compute the ionization potential within Fock-space multi-reference coupled-cluster theory. In particular, we have obtained computationally less expensive correlation schemes due to fourth order triples. Prototype examples have been considered to explore the efficacy of the approximate methods mentioned, while the bondonic formalism supporting the bonding phenomenology is also respectively for the first time here advanced.

A multireference coupled cluster theory based on quantum electrodynamics: Physical aspects of open-shell treatment

An electrodynamical coupled cluster (CC) formulation employing standard QED Hamiltonian that is written in Coulomb gauge and uses a multiconfiguration SCF picture of matter fields is discussed here. It employs a radiative cluster, matter clusters and their pair modifications, and considers a number state distribution of photons in thermal equilibrium. QED and electron correlation through CC are treated on the same footing. Thus an averaging over the radiation state is done to generate Lamb, Breit and hyperfine interactions from the radiative cluster. Because of the thermal distribution, it leaves a residual transverse interaction that may modify the static correlation. Dynamical correlation effects are determined next while using the exponential matter cluster in traditional ways of multireference CC. Deexcitations to negative-energy levels, included in matter clusters, generate vacuum polarization effects. The dynamical correlation energy includes relativistic corrections as well as QED contributions, namely, Lamb, Breit, hyperfine and pair energy.

Reference spaces for multireference coupled-cluster theory: the challenge of the CoH molecule

In multireference calculations on transition metal compounds, large active spaces including a second set of d orbitals are often used. However, with the increase in the system size and the complexity of the method, such calculations quickly become impractical. In this work, we looked for an inner valence active space for atomic cobalt and its hydride that correctly describes the transition energies among states with distinct configurations. Potential energy curves for the CoH molecule were obtained using a fixed reference space, where the orbitals are optimised only for the isolated atoms, avoiding undesirable sudden variations of the active orbitals. The calculations made use of the multireference configuration interaction (MRCI) and coupled-cluster (MRCC) theories. The accuracy of these calculations is carefully analysed, and we show that MRCC results are accurate, although not all electronic states are quantitatively described at the MRCI level. We conclude that the use of the fixed reference space is a viable approach to reduce the computational cost when large active spaces are prohibitive. Finally, a careful comparison of the present results with experimental values is carried out.

Improved and simplified orthogonalisation scheme and connected triples correction within the internally contracted multireference coupled-cluster method

The internally contracted multireference coupled-cluster theory (icMRCC) is reviewed, providing additional insights that we have gained over recent years. In addition, several modifications are presented that lead to an overall improvement in the method. These include a simpler and size extensive way to generate the excitation manifold and therefore determine the cluster amplitudes. Modifications and approximations are also introduced to the perturbative triples correction, icMRCCSD(T), which lead to a small improvement over the previous manifestation, at a reduced computational cost. With these reformulated methods, we can approach sub-chemical accuracy results.

Reaction Profiles and Kinetics for Radical-Radical Hydrogen Abstraction via Multireference Coupled Cluster Theory.

Radical-radical abstractions in hydrocarbon oxidation chemistry are disproportionation reactions that are generally exothermic with little or no barrier yet are underappreciated and poorly studied. Such challenging multireference electronic structure problems are tackled here using the recently developed state-specific multireference coupled cluster methods Mk-MRCCSD and Mk-MRCCSD(T), as well as the companion perturbation theory Mk-MRPT2 and the established MRCISD, MRCISD+Q, and CASPT2 approaches. Reaction paths are investigated for five prototypes involving radical-radical hydrogen abstraction: H + BeH → H2+ Be, H + NH2 → H2 + NH, CH3 + C2H5 → CH4 + C2H4, H + C2H5 → H2 + C2H4, and H + HCO → H2 + CO. Full configuration interaction (FCI) benchmark computations for the H + BeH, H + NH2, and H + HCO reactions prove that Mk-MRCCSD(T) provides superior accuracy for the interaction energies in the entrance channel, with mean absolute errors less than 0.3 kcal mol-1 and percentage deviations less than 10% over the fragment separations of relevance to kinetics. To facilitate combustion studies, energetics for the CH3 + C2H5, H + C2H5, and H + HCO reactions were computed at each level of theory with correlation-consistent basis sets (cc-pVXZ, X = T, Q, 5) and extrapolated to the complete basis set (CBS) limit. These CBS energies were coupled with CASPT2 projected vibrational frequencies along a minimum energy path to obtain rate constants for these three reactions. The rigorous Mk-MRCCSD(T)/CBS results demonstrate unequivocally that these three reactions proceed with no barrier in the entrance channel, contrary to some earlier predictions. Mk-MRCCSD(T) also reveals that the economical CASPT2 method performs well for large interfragment separations but may deteriorate substantially at shorter distances.

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation.

This study examines several techniques to improve the efficiency of the linearized multireference driven similarity renormalization group truncated to one- and two-body operators [MR-LDSRG(2)]. We propose a sequential MR-LDSRG(2) [sq-MR-LDSRG(2)] scheme, in which one-body substitutions are folded exactly into the Hamiltonian. This new approach is combined with density fitting (DF) to reduce the storage cost of two-electron integrals. To further avoid storage of large four-index intermediates, we propose a noninteracting virtual orbital (NIVO) approximation of the Baker-Campbell-Hausdorff series that neglects commutators terms with three and four virtual indices. The NIVO approximation reduces the computational prefactor of the MR-LDSRG(2), bringing it closer to that of coupled cluster with singles and doubles (CCSD). We test the effect of the DF and NIVO approximations on the MR-LDSRG(2) and sq-MR-LDSRG(2) methods by computing properties of eight diatomic molecules. The diatomic constants obtained by DF-sq-MR-LDSRG(2)+NIVO are found to be as accurate as those from the original MR-LDSRG(2) and coupled cluster theory with singles, doubles, and perturbative triples. Finally, we demonstrate that the DF-sq-MR-LDSRG(2)+NIVO scheme can be applied to chemical systems with more than 550 basis functions by computing the automerization energy of cyclobutadiene with a quintuple-ζ basis set. The predicted automerization energy is found to be similar to the value computed with Mukherjee's state-specific multireference coupled cluster theory with singles and doubles.

Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs.

Quantum chemistry is a discipline which relies heavily on very expensive numerical computations. The scaling of correlated wave function methods lies, in their standard implementation, between and , where N is proportional to the system size. Therefore, performing accurate calculations on chemically meaningful systems requires (i) approximations that can lower the computational scaling and (ii) efficient implementations that take advantage of modern massively parallel architectures. Quantum Package is an open-source programming environment for quantum chemistry specially designed for wave function methods. Its main goal is the development of determinant-driven selected configuration interaction (sCI) methods and multireference second-order perturbation theory (PT2). The determinant-driven framework allows the programmer to include any arbitrary set of determinants in the reference space, hence providing greater methodological freedom. The sCI method implemented in Quantum Package is based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm which complements the variational sCI energy with a PT2 correction. Additional external plugins have been recently added to perform calculations with multireference coupled cluster theory and range-separated density-functional theory. All the programs are developed with the IRPF90 code generator, which simplifies collaborative work and the development of new features. Quantum Package strives to allow easy implementation and experimentation of new methods, while making parallel computation as simple and efficient as possible on modern supercomputer architectures. Currently, the code enables, routinely, to realize runs on roughly 2 000 CPU cores, with tens of millions of determinants in the reference space. Moreover, we have been able to push up to 12 288 cores in order to test its parallel efficiency. In the present manuscript, we also introduce some key new developments: (i) a renormalized second-order perturbative correction for efficient extrapolation to the full CI limit and (ii) a stochastic version of the CIPSI selection performed simultaneously to the PT2 calculation at no extra cost.

All-electron calculation of ground state vibration-rotation energy levels of <sup>7</sup>Li<sub>2</sub><sup>(0, ±1)</sup> molecular systems

The investigation of spectroscopic information is important for understanding the mechanisms of molecular photochemical and photophysical reactions. As a prototype to study the electronic structures and spectra of diatomic molecular systems, the vibration-rotational spectra of alkali dimer and its ions have aroused considerable research interest in the last two decades. Single-reference and multi-reference coupled cluster theory in combination with correlation consistent Gaussian basis set are adopted to study the ground-state potential energy curves of <sup>7</sup>Li<sub>2</sub><sup>(0,± 1)</sup> molecular systems. The correlation effect and relativistic effect of all the electrons are taken into account in the calculation. And the spectroscopic constants, including the equilibrium internuclear distance <i>R</i><sub>e</sub>, the harmonic vibrational frequency <i>ω</i><sub>e</sub>, the anharmonic constant <i>ω</i><sub>e</sub><i>x</i><sub>e</sub>, the equilibrium rotational constant <i>B</i><sub>e</sub>, and the dissociation energy <i>D</i><sub>e</sub> of the molecular system and vibration-rotational energy level information of the ground states are obtained by solving the radial Schrödinger equations. The calculated spectroscopic constants of the neutral and positive ion system accord well with the experimental values; however for the negative ion system, the calculation of equilibrium internuclear distance needs further improving, and other spectroscopic constants are consistent well with the experimental values. The present computational results indicate that the ground state wave functions of neutral and positive ion systems have obvious single reference configuration characteristics, while the ground state of negative ion molecule system should be described with multireference configuration wave functions. The vibration-rotational energy levels of ground state with different theoretical methods are in good agreement with the experimental values. The vibrational-rotational energy levels and spectroscopic constants of neutral and positive ion systems are well reproduced, and some experimental information about spectrum is still lacking. Although the difference among the equilibrium internuclear distances for the ground state of the negative ion, obtained from different theoretical methods are still existent, the results of the vibrational level interval accord well with each other. This study provides useful information about spectrum for accurately investigating the electronic structures and spectra of the ground state of Li<sub>2</sub> molecular system and its two isotopic molecules, especially for the negative ion system with little information about spectrum.

First-order properties from internally contracted multireference coupled-cluster theory with particular focus on hyperfine coupling tensors.

Internally contracted multireference coupled-cluster (icMRCC) theory is extended to the computation of first-order properties (expectation values). We use the previously defined Lagrange formulation of the energy functional to derive the required equations for the Lagrange multipliers and arrive at an expression for first-order properties according to the generalized Hellmann-Feynman theorem, analogous to single-reference coupled-cluster theory. The present formulation does not include orbital relaxation, but in line with previous experience in coupled-cluster theory, the single-excitation cluster operator can recover a significant portion of orbital relaxation. Further aspects of the theory that arise from the internal contraction approach are discussed. Using automated derivation techniques, we have implemented a pilot code for icMRCCSD and icMRCCSDT for testing the method numerically. We find good agreement with full configuration interaction for several properties of boron monohydride and dipole moment curves of hydrogen fluoride and chromium hydride. A particular focus is given to spin-dependent properties: The hyperfine coupling tensors of Σ and Π radicals have been computed and compared to experiment and previous computations. We discuss the problem of describing spin polarization with properly spin-adapted wavefunctions, which requires either including pseudo-triple excitations or employing sufficiently flexible reference functions.

Perspective: Multireference coupled cluster theories of dynamical electron correlation.

Predicting the electronic structure and properties of molecular systems that display strong electron correlation effects continues to remain a fundamental theoretical challenge. This perspective discusses the recent progress and current challenges in multireference wave function methods for dynamical electron correlation, focusing on systematically improvable methods that go beyond the limitations of configuration interaction and perturbation theory.

Spectroscopy of Na I and K I atoms embedded in weakly coupled plasma environment

The effect of the plasma environment on the structure and properties of Na I and K I atoms is studied using the Fock-space multireference coupled-cluster theory in the relativistic framework. The Debye-Hückle model is used to account the effect of the plasma environment on the atoms. A significant change in the ionization energies, transition energies, transition probabilities, and oscillator strengths of Na I and K I atoms is seen when they are immersed in plasma. The number of bound states and the binding energies of plasma embedded Na I and K I reduce notably in comparison to the isolated plasma free atoms, and these changes tend the system towards gradual instability. The transition spectra of Na I and K I show red shifts in the presence of plasma screening, whereas the oscillator strengths show the red and blue shifts for the same and different principal quantum number transitions, respectively. Furthermore, the presence of plasma screening causes the suppression of spontaneous transition probabilities of plasma embedded Na I and K I atoms. These results are useful to interpret the spectral lines of astrophysical plasma and to get insights into the physical processes that are operative in the celestial objects.

How To Arrive at Accurate Benchmark Values for Transition Metal Compounds: Computation or Experiment?

With the objective of analyzing which kind of reference data is appropriate for benchmarking quantum chemical approaches for transition metal compounds, we present the following, (a) a collection of 60 transition metal diatomic molecules for which experimentally derived dissociation energies, equilibrium distances, and harmonic vibrational frequencies are known and (b) a composite computational approach based on coupled-cluster theory with basis set extrapolation, inclusion of core-valence correlation, and corrections for relativistic and multireference effects. The latter correction was obtained from internally contracted multireference coupled-cluster (icMRCC) theory. This composite approach has been used to obtain the dissociation energies and spectroscopic constants for the 60 molecules in our data set. In accordance with previous studies on a subset of molecules, we find that multireference corrections are rather small in many cases and CCSD(T) can provide accurate reference values, if the complete basis set limit is explored. In addition, the multireference correction improves the results in cases where CCSD(T) is not a good approximation. For a few cases, however, strong deviations from experiment persist, which cannot be explained by the remaining error in the computational approach. We suggest that these experimentally derived values require careful revision. This also shows that reliable reference values for benchmarking approximate computational methods are not always easily accessible via experiment and accurate computations may provide an alternative way to access them. In order to assess how the choice of reference data affects benchmark studies, we tested 10 DFT functionals for the molecules in the present data set against experimental and calculated reference values. Despite the differences between these two sets of reference values, we found that the ranking of the relative performance of the DFT functionals is nearly independent of the chosen reference.

Internally Contracted Multireference Coupled Cluster Calculations with a Spin-Free Dirac-Coulomb Hamiltonian: Application to the Monoxides of Titanium, Zirconium, and Hafnium.

We combine internally contracted multireference coupled cluster theory with a four-component treatment of scalar-relativistic effects based on the spin-free Dirac-Coulomb Hamiltonian. This strategy allows for a rigorous treatment of static and dynamic correlation as well as scalar-relativistic effects, which makes it viable to describe molecules containing heavy transition elements. The use of a spin-free formalism limits the impact of the four-component treatment on the computational cost to the non-rate-determining steps of the calculations. We apply the newly developed method to the lowest singlet and triplet states of the monoxides of titanium, zirconium, and hafnium and show how the interplay between electronic correlation and relativistic effects explains the electronic structure of such molecules.

Multireference Perturbation Theory and Multireference Coupled Cluster Theory Based on the "Block-Correlation" Framework

Multireference Perturbation Theory and Multireference Coupled Cluster Theory Based on the "Block-Correlation" Framework

Perturbative universal state-selective correction for state-specific multi-reference coupled cluster methods.

In this work, we report an extension of our previous development of the universal state-selective (USS) multireference coupled-cluster (MRCC) formalism. It was shown [Brabec et al., J. Chem. Phys. 136, 124102 (2012)] and [Banik et al., J. Chem. Phys. 142, 114106 (2015)] that the USS(2) approach significantly improves the accuracy of Brillouin-Wigner and Mukherjee MRCC formulations, however, the numerical and storage costs associated with calculating highly excited intermediates pose a significant challenge, which can restrict the applicability of the USS(2) method. Therefore, we introduce a perturbative variant of the USS(2) approach (USS(pt)), which substantially reduces numerical overhead of the full USS(2) correction while preserving its accuracy. Since the new USS(pt) implementation calculates the triple and quadruple projections in on-the-fly manner, the memory bottleneck associated with the need of storing expensive recursive intermediates is entirely eliminated. On the example of several benchmark systems, we demonstrate accuracies of USS(pt) and USS(2) approaches and their efficiency in describing quasidegenerate electronic states. It is also shown that the USS(pt) method significantly alleviates problems associated with the lack of invariance of MRCC theories upon the rotation of active orbitals.