In the present work, we show the feasibility of using single precision in quantum chemistry, especially regarding the computation of electron correlation energy. On the example of the MP2 method, we clearly demonstrate that single precision arithmetic is sufficient for evaluating the molecular two-electron integrals by the use of the Cholesky decomposition method. The evaluation of integrals with single precision arithmetic introduces a negligible error into the computed MP2 correlation energy. In particular, the corresponding error in the MP2 correlation energy amounts to only 10(-7)Eh for the 113-atom taxol molecule in double-valence basis set (1099 basis functions). The practical relevance of our result is that 50% performance gain and 50% reduction in memory demands can be achieved by only minor changes in the existing codes. Our finding opens intriguing perspectives for doing accurate correlated quantum chemistry on specialized floating-point mathematical coprocessors.
In the present work, we show the feasibility of using single precision in quantum chemistry, especially regarding the computation of electron correlation energy. On the example of the MP2 method, we clearly demonstrate that single precision arithmetic is sufficient for evaluating the molecular two-electron integrals by the use of the Cholesky decomposition method. The evaluation of integrals with single precision arithmetic introduces a negligible error into the computed MP2 correlation energy. In particular, the corresponding error in the MP2 correlation energy amounts to only 10(-7)Eh for the 113-atom taxol molecule in double-valence basis set (1099 basis functions). The practical relevance of our result is that 50% performance gain and 50% reduction in memory demands can be achieved by only minor changes in the existing codes. Our finding opens intriguing perspectives for doing accurate correlated quantum chemistry on specialized floating-point mathematical coprocessors.
We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Møller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.
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 Cholesky decomposition technique is commonly used to reduce the memory requirement for storing two-particle repulsion integrals in quantum chemistry calculations that use atomic orbital bases. However, when quantum methods use multicomponent bases, such as nuclear–electronic orbitals, additional challenges are introduced due to asymmetric two-particle integrals. This work proposes several multicomponent Cholesky decomposition methods for calculations using nuclear–electronic orbital density functional theory. To analyze the errors in different Cholesky decomposition components, benchmark calculations using water clusters are carried out. The largest benchmark calculation is a water cluster (H 2 O) 27 where all 54 protons are treated quantum mechanically. This study provides energetic and complexity analyses to demonstrate the accuracy and performance of the proposed multicomponent Cholesky decomposition method.
We review the current capabilities of local electron correlation methods up to the coupled cluster model with single, double, and perturbative triple excitations [CCSD(T)], which is a gold standard in quantum chemistry. The main computational aspects of the local method types are assessed from the perspective of applications, but the focus is kept on how to achieve chemical accuracy (i.e., <1 kcal/mol uncertainty), as well as on the broad scope of chemical problems made accessible. The performance of state-of-the-art methods are also compared, including the most employed DLPNO and, in particular, our local natural orbital (LNO) CCSD(T) approach. The high accuracy and efficiency of the LNO method makes chemically accurate CCSD(T) computations accessible for molecules of hundreds of atoms with resources affordable to a broad computational community (days on a single CPU and 10–100 GB of memory). Recent developments in LNO-CCSD(T) enable systematic convergence and robust error estimates even for systems of complicated electronic structure or larger size (up to 1000 atoms). The predictive power of current local CCSD(T) methods, usually at about 1–2 order of magnitude higher cost than hybrid density functional theory (DFT), has become outstanding on the palette of computational chemistry applicable for molecules of practical interest. We also review more than 50 LNO-based and other advanced local-CCSD(T) applications for realistic, large systems across molecular interactions as well as main group, transition metal, bio-, and surface chemistry. The examples show that properly executed local-CCSD(T) can contribute to binding, reaction equilibrium, rate constants, etc. which are able to match measurements within the error estimates. These applications demonstrate that modern, open-access, and broadly affordable local methods, such as LNO-CCSD(T), already enable predictive computations and atomistic insight for complicated, real-life molecular processes in realistic environments.
Determining the energy and properties of an N-electron molecule through a two-electron variational optimization has been a dream for more than half a century. While optimizations, using two-electron reduced density matrices constrained to represent N electrons, have recently been achieved, the computational costs are prohibitive. In this report an efficient algorithm with an order-of-magnitude reduction in floating-point operations and memory usage is presented. Because the optimization occurs on the space of two electrons, this method automatically treats strong, multireference correlation. Application is made to N2 and H6 where the method yields consistent accuracy at all geometries.
Quantum Monte Carlo (QMC) methods have received considerable attention over the last decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems.Thanks to their low scaling with number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations.In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult.In this Perspective, we present a systematic application of large scale multi-determinant expansions in QMC, and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set.We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions.When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them.In fact, only the explicitly-correlated CCSD(T) method with a large basis set produces more accurate results.Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.