We present in this contribution the first application of local correlation in the context of multicomponent methods. Multicomponent approaches allow for the targeted simulation of electrons together with other Fermions (most commonly protons) as quantum particles. These methods have become increasingly popular over the last years, particularly for the description of nuclear quantum effects (in strong hydrogen bonds, proton tunneling, and many more). However, most implementations are still based on canonical formulations of wave function theory, which we know for decades to be computationally inefficient for capturing dynamical correlation effects. Local correlation approaches, particularly with the use of pair natural orbitals (PNOs), enable asymptotically linear scaling of computational costs with very little impact on the overall accuracy. In this context, the efficient use of density fitting approximations in the integral calculation proves essential. We start by discussing our implementation of density-fitted NEO-MP2 and NEO-PNO-LMP2, upgrading the electronic correlation treatment up to PNO local coupled cluster level of theory. Several challenging examples are provided to benchmark the method in terms of accuracy as well as computational cost scaling. Following appropriate protocols, anharmonic corrections to localized X-H stretches can be applied routinely with little computational overhead.
We present in this contribution the first application of local correlation in the context of multicomponent methods. Multicomponent approaches allow for the targeted simulation of electrons together with other Fermions (most commonly protons) as quantum particles. These methods have become increasingly popular over the last years, particularly for the description of nuclear quantum effects (in strong hydrogen bonds, proton tunneling, and many more). However, most implementations are still based on canonical formulations of wave function theory, which we know for decades to be computationally inefficient for capturing dynamical correlation effects. Local correlation approaches, particularly with the use of pair natural orbitals (PNOs), enable asymptotically linear scaling of computational costs with very little impact on the overall accuracy. In this context, the efficient use of density fitting approximations in the integral calculation proves essential. We start by discussing our implementation of density-fitted NEO-MP2 and NEO-PNO-LMP2, upgrading the electronic correlation treatment up to PNO local coupled cluster level of theory. Several challenging examples are provided to benchmark the method in terms of accuracy as well as computational cost scaling. Following appropriate protocols, anharmonic corrections to localized X-H stretches can be applied routinely with little computational overhead.
This work targets the development and application of low-order scaling multicomponent methods for efficient simulation of nuclear quantum effects (NQE) within complex systems. The awareness of NQEs is increasingly growing across various fields, including chemistry, biology, and materials science, where light nuclei, like protons, play an essential role in molecular interactions and properties. This thesis presents a systematic development and application of multicomponent methods based on the Nuclear-Electronic Orbital (NEO) framework developed by Hammes-Schiffer and coworkers. In general, the focus is placed on the implementation and optimization of multicomponent Hartree-Fock, density functional and Møller-Plesset perturbation theory, as well as composite correlation approaches. Overall, by addressing computational bottlenecks and notably expanding the scope of multicomponent methods, this work contributes to the development of an extended standard quantum chemistry toolbox and paves the way for routine simulations of NQEs.
A well-parallelized local singles and doubles coupled-cluster (LCCSD) method using pair natural virtual orbitals (PNOs) is presented. The PNOs are constructed using large domains of projected atomic orbitals (PAOs) and orbital specific virtual orbitals (OSVs). We introduce a hierarchy of close, weak, and distant pairs, based on pair energies evaluated by local Møller–Plesset perturbation theory (LMP2). In contrast to most previous implementations of LCCSD methods, the close and weak pairs are not approximated by LMP2 but treated by higher-order methods. This leads to greatly improved accuracy for large systems, in particular when long-range dispersion interactions are important. Close pair amplitudes are optimized using approximate LCCSD equations, in which slowly decaying terms that mutually cancel at long-range are neglected. For weak pairs, the same approximations are used, but in addition, the nonlinear terms are neglected (coupled electron pair approximation). Distant pairs are treated by spin-component scaled (SCS)-LMP2 using multipole approximations. For efficiency reasons, various projection approximations are also introduced. The impact of these approximations on absolute and relative energies depends on the PNO domain sizes. The errors are found to be negligible, provided that sufficiently large PNO domains are used for close and weak pairs. For the selection of these domains the usual natural orbital occupation number criterion is found to be insufficient, and an additional energy criterion is used. For extended one-dimensional systems, the computational effort of the method scales nearly linearly with the number of correlated electrons, but the linear scaling regime is usually not reached in real-life applications for three-dimensional systems. Nevertheless, due to the parallelization that is efficient up to about 100–200 CPU cores (dependent on the molecular size), accurate calculations for three-dimensional molecules with about 100 atoms and augmented triple-ζ basis sets (e.g., cc-pVTZ-F12) can be carried out in 1–3 h of elapsed time (depending on the molecular structure and the number of CPU cores, excluding the time for Hartree–Fock). The convergence of the results with respect to the thresholds and options that control the domain, pair, and projection approximations is extensively tested. Benchmark examples for several difficult and large cases are presented, which demonstrate that the errors of relative energies (e.g., reaction energies, barrier heights) caused by the pair and projection approximations can be reduced to below 1 kJ mol –1 . The remaining errors are mainly caused by the finite PAO domains. The larger these are made, the more intramolecular or intermolecular basis set superposition errors are introduced, and the canonical results are approached only very slowly. This problem is eliminated in the explicitly correlated variant of our method (PNO-LCCSD-F12), which will be described in a separate paper, along with a larger set of benchmark calculations.
We present a formulation of the explicitly correlated second-order Møller-Plesset (MP2-F12) energy in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the domain based local pair-natural orbital-MP2 (DLPNO-MP2) method by us recently [Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes), the ON DLPNO-MP2-F12 method becomes less expensive than the conventional ON(5) MP2-F12 for n between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C200H402, in def2-TZVP basis, the observed computational complexity is N(∼1.6), largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation.
Multicomponent systems are defined as chemical systems that require a quantum mechanical description of two or more different types of particles. Non-Born-Oppenheimer electron-nuclear interactions in molecules, electron-hole interactions in electronically excited nanoparticles, and electron-positron interactions are examples of physical systems that require a multicomponent quantum mechanical formalism. The central challenge in the theoretical treatment of multicomponent systems is capturing the many-body correlation effects that exist not only between particles of identical types (electron-electron) but also between particles of different types (electron-nuclear and electron-hole). In this work, the development and implementation of multicomponent coupled-cluster (mcCC) theory for treating particle-particle correlation in multicomponent systems is presented. This method provides a balanced treatment of many-particle correlation in a general multicomponent system while maintaining a size-consistent and size-extensive formalism. The coupled-cluster ansatz presented here is the extension of the electronic structure CCSD formulation for multicomponent systems and is defined as ∣ΨmcCC⟩=eT1I+T2I+T1II+T2II+T11I,II+T12I,II+T21I,II+T22I,II∣0I0II⟩. The applicability of the mcCC method was demonstrated by computing biexciton binding energies for multiexcitonic systems and benchmarking the results against full configuration interaction calculations. The results demonstrated that connected cluster operators that generate simultaneous excitation in type I and type II space are critical for capturing electron-hole correlation in multiexcitonic systems.
One of the major obstacles in modern quantum chemistry is the need to employ exceedingly large one-electron basis sets in electron correlation methods to obtain reliably accurate results. The family of explicitly correlated wave function methods circumvents this issue by introducing terms that explicitly depend on the interelectronic distance, thereby enforcing the correct electron–electron cusp behavior, albeit at the cost of additional complicated many-electron integrals when evaluated without further approximation. The main focus of the present work is the efficient, low-scaling evaluation of explicitly correlated approaches, examined here through the explicitly correlated correction to second-order Møller–Plesset perturbation theory (MP2-F12) as a representative example. While these methods are highly effective in reducing the basis set incompleteness error, the cost associated with the corresponding correction often far exceeds that of the underlying correlation method. To overcome these limitations, several techniques and strategies are introduced to significantly reduce computational demand and resolve the main bottlenecks that arise. A particularly straightforward approach is provided by the scaled MP2-F12 ansatz, which introduces a single empirically determined factor that scales the direct-type geminal–geminal terms to closely reproduce the MP2-F12 correction, thereby omitting the most expensive exchange-type contributions. Furthermore, the computation of Fock matrix elements spanning multiple orbital spaces is drastically accelerated by extending techniques originally developed for Hartree–Fock and density functional theory. The direct contributions are evaluated using a modified resolution-of-the-identity Coulomb (RI-J) approach employing the J-engine ansatz for integral construction, while the exchange contributions are treated via three-dimensional numerical quadrature in the form of the seminumerical sn-LinK method. These approaches reduce the computational cost by more than three orders of magnitude compared to the approximation-free evaluation while maintaining high accuracy, with additional speedups achieved by exploiting the massive parallelism of graphics processing units. The central focus of this work is the efficient decomposition of complicated exchange-type multi-electron integrals, where numerical quadrature plays a key role in improving performance. The complementary auxiliary basis set resolution of the identity (CABS-RI) reduces the most expensive exchange-type integral to a three-electron integral, which is then efficiently decomposed via numerical quadrature. In this context, a highly efficient batch-wise, distance-dependent integral screening exploiting the short-range behavior of F12 operators is introduced, significantly improving performance and enabling near-linear-scaling evaluations while retaining high accuracy. These strategies are extended to all exchange-type contributions in F12 theory, leading to an alternative formulation of integrals better suited for efficient factorization. A novel combination of numerical quadrature with density fitting is presented for the efficient computation of products of four-center two-electron integrals, alongside algorithms covering all exchange-type contributions, including advantageous batching strategies. In general, the formal computational cost with system size M is reduced from O(M⁵) to O(M⁴), with further improvements appearing feasible through the use of localized molecular orbitals combined with advanced integral screening techniques.
The central problem in the modern electronic structure theory is the calculation of cor- relation energy, possibly by an approach that would account for both static and dynamic correlation in an efficient, balanced and accurate way. In this thesis, I present a collection of methods that combine the effective treatment of dynamic correlation by the coupled cluster theory with density matrix renormalization group, a well-established technique for calculations of strongly correlated systems. The connection between them is achieved via the tailored coupled clusters (TCC) ansatz, which conveniently does not impose any ad- ditional computational costs. After the successful initial assessment, we developed more efficient implementations of these methods by employing the local approaches based on pair natural orbitals. This way, we extended the range of possible applications to larger systems with thousands of basis functions. To assess the accuracy of TCC as well as its local counterparts, we performed a variety of benchmark calculations ranging from small, yet challenging systems such as the nitrogen molecule or tetramethyleneethane diradical, to larger molecules like oxo-Mn(Salen) or Fe(II)-porphyrin model. 1