We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N4), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.
We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N4), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.
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.
We present a low-prefactor, cubically scaling scaled-opposite-spin second-order Møller-Plesset perturbation theory (SOS-MP2) method which is highly suitable for massively parallel architectures like graphics processing units (GPU). The scaling is reduced from O(N⁵) to O(N³) by a reformulation of the MP2-expression in the atomic orbital basis via Laplace transformation and the resolution-of-the-identity (RI) approximation of the integrals in combination with efficient sparse algebra for the 3-center integral transformation. In contrast to previous works that employ GPUs for post Hartree-Fock calculations, we do not simply employ GPU-based linear algebra libraries to accelerate the conventional algorithm. Instead, our reformulation allows to replace the rate-determining contraction step with a modified J-engine algorithm, that has been proven to be highly efficient on GPUs. Thus, our SOS-MP2 scheme enables us to treat large molecular systems in an accurate and efficient manner on a single GPU-server.
First-principles methods are a valuable tool to study the electronic structure of systems and explore the quantum-mechanical mechanisms driving them on the microscopic scale. Semi-local Density-Functional Approximations (DFAs) are still very popular due to the good balance between accuracy and computational efficiency. However, the instantaneous electronic interactions are described with an averaged potential in these methods, which is often insufficient to describe charge transfers or subtle many-body correlations. A promising way to improve the description of such interactions is the usage of advanced correlation methods which explicitly treat the electronic interactions in a many-body picture. Well known methods form this class are e.g. the Random- Phase Approximation (RPA), 2nd Order Møller-Plesset Perturbation Theory (MP2) and Coupled-Cluster Theory (CC). Unfortunately, the time and memory requirements for all these methods suffer from a steep scaling with system size in their canonical formulations. Several lower-scaling reformulations exist for these methods, but the at least cubic memory requirements and the increased algorithmic complexity prevented any efficient parallelization of these methods so far. In my thesis I developed a new lower-scaling technique for advanced correlation methods to address this challenge. The key components of this technique are the Laplace-Transformation (LT) and a new local Resolution of Identity (RI) strategy, named RI-LVL. Together, they significantly reduce the memory requirement without sacrificing accuracy and allow for an efficient parallelization by means of à priori known sparsity patterns. In combination with established integral screening techniques, the computational scaling can also be reduced considerably. The potential of this new approach is demonstrated at the example of an in-memory LT-MP2 with a MPI/OpenMP hybrid parallelization. It features an at worst cubic computational scaling, a quadratic memory scaling and a high parallel efficiency. Based on the Numeric Atom-Centered Orbital (NAO) framework used in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) code, this implementation can handle both molecular and periodic systems with the same numerical footing. I will also discuss in detail how the combination of the LT and RI- LVL can be applied to RPA calculations. The accuracy and performance of my LT- MP2 is demonstrated in detail for both molecular and periodic systems. In addition, the convergence of the MP2 level adsorption energy of water on a rutile titanium dioxide (TiO2) surface in the low coverage limit is explored. This new LT-MP2 recovers more than 99.9% of the canonical MP2 correlation energy and can handle TiO2 surfaces with up to 270 atoms in the unit cell.
We propose progressive downsampling of wave vectors in the Brillouin zone integrations occurring in the second-order many-body or Møller-Plesset perturbation (MP2) method for extended systems of one-dimensional periodicity. Higher-lying unoccupied and lower-lying occupied Bloch orbitals are subject to downsampling by an exponentially increasing factor (with base n), making the total number of Bloch orbitals included in the MP2 lattice sums grow only logarithmically with respect to the number of basis functions per unit cell. Unlike the mod n downsampling scheme proposed earlier, this log n scheme reduces the scaling of the computational cost and thus achieves a greater speedup as the unit cell size increases. Correct band indexing is essential for accuracy. Two-electron integrals entering the MP2 energy and quasiparticle energy expressions must be multiplied by quadrature weights that are a function of the energy bands involved, and an algorithm to compute the weights is proposed. A combined use of the log n and mod n schemes can speedup the MP2/6-31G** calculation of polyacetylene typically by a factor of 20 with an error in the correlation energy within a few percent relative to the conventional calculation. Similar combinations can reproduce the MP2 quasiparticle energy bands accurately at a fraction of the usual computational cost.
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.
Second-order Møller-Plesset perturbation theory using the Resolution-of-the-Identity approximation (RI-MP2) is a state-of-the-art approach to accurately estimate many-body electronic correlation effects. This is critical for predicting the physicochemical properties of complex molecular systems; however, the scale of these calculations is limited by their extremely high computational cost. In this paper, a novel many-GPU algorithm and implementation of a molecular-fragmentation-based RI-MP2 method are presented that enable correlated calculations on over 180,000 electrons and 45,000 atoms using up to the entire Summit supercomputer in 12 minutes. The implementation demonstrates remarkable speedups with respect to other current GPU and CPU codes, excellent strong scalability on Summit achieving 89.1% parallel efficiency on 4600 nodes, and shows nearly-ideal weak scaling up to 612 nodes. This work makes feasible ab initio correlated quantum chemistry calculations on significantly larger molecular scales than before on both large supercomputing systems and on commodity clusters, with a potential for major impact on progress in chemical, physical, biological and engineering sciences.