We present an approach to compute accurate correlation energies for atoms and molecules using an adaptive discontinuous spectral-element multiresolution representation for the two-electron wave function. Because of the exponential storage complexity of the spectral-element representation with the number of dimensions, a brute-force computation of two-electron (six-dimensional) wave functions with high precision was not practical. To overcome the key storage bottlenecks we utilized (1) a low-rank tensor approximation (specifically, the singular value decomposition) to compress the wave function, and (2) explicitly correlated R12-type terms in the wave function to regularize the Coulomb electron-electron singularities of the Hamiltonian. All operations necessary to solve the Schrödinger equation were expressed so that the reconstruction of the full-rank form of the wave function is never necessary. Numerical performance of the method was highlighted by computing the first-order Møller-Plesset wave function of a helium atom. The computed second-order Møller-Plesset energy is precise to ~2 microhartrees, which is at the precision limit of the existing general atomic-orbital-based approaches. Our approach does not assume special geometric symmetries, hence application to molecules is straightforward.
We present an approach to compute accurate correlation energies for atoms and molecules using an adaptive discontinuous spectral-element multiresolution representation for the two-electron wave function. Because of the exponential storage complexity of the spectral-element representation with the number of dimensions, a brute-force computation of two-electron (six-dimensional) wave functions with high precision was not practical. To overcome the key storage bottlenecks we utilized (1) a low-rank tensor approximation (specifically, the singular value decomposition) to compress the wave function, and (2) explicitly correlated R12-type terms in the wave function to regularize the Coulomb electron-electron singularities of the Hamiltonian. All operations necessary to solve the Schrödinger equation were expressed so that the reconstruction of the full-rank form of the wave function is never necessary. Numerical performance of the method was highlighted by computing the first-order Møller-Plesset wave function of a helium atom. The computed second-order Møller-Plesset energy is precise to ~2 microhartrees, which is at the precision limit of the existing general atomic-orbital-based approaches. Our approach does not assume special geometric symmetries, hence application to molecules is straightforward.
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.
Using the finite simulation-cell homogeneous electron gas (HEG) as a model, we investigate the convergence of the correlation energy to the complete-basis-set (CBS) limit in methods utilizing plane-wave wave-function expansions. Simple analytic and numerical results from second-order M\o{}ller-Plesset theory (MP2) suggest a 1/M decay of the basis-set incompleteness error where M is the number of plane waves used in the calculation, allowing for straightforward extrapolation to the CBS limit. As we shall show, the choice of basis-set truncation when constructing many-electron wave functions is far from obvious, and here we propose several alternatives based on the momentum transfer vector, which greatly improve the rate of convergence. This is demonstrated for a variety of wave-function methods, from MP2 to coupled-cluster doubles theory and the random-phase approximation plus second-order screened exchange. Finite basis-set energies are presented for these methods and compared with exact benchmarks. A transformation can map the orbitals of a general solid state system onto the HEG plane-wave basis and thereby allow application of these methods to more realistic physical problems. We demonstrate this explicitly for solid and molecular lithium hydride.
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.
Post-Hartree-Fock methods are wavefunction-based quantum chemistry approaches that explicitly account for the electron correlation beyond the mean-field approximation, allowing accurate determination of electron orbitals and energies in a given system. However, their computational cost scales as the fourth power or higher with respect to the number of atoms, severely limiting their practical applications. This paper proposes an efficient GPU implementation of the Møller-Plesset method, a post-Hartree-Fock approach based on perturbation theory. We focus on the computation of second- and thirdorder perturbation energies (MP2 and MP3) for closed-shell systems, which incorporate electron correlation on top of the Hartree-Fock energy. To accelerate the extensive contraction operations arising in the Møller-Plesset method, our implementation employs a formulation that maximizes the use of highly optimized matrix multiplication routines. Furthermore, we effectively utilize computational capabilities of GPU architecture through a hierarchical reduction using warp-level primitives and shared memory, as well as contiguous coalesced access to matrix elements. Performance evaluation on an NVIDIA A100 GPU shows up to 255.5 and 71.1 times speedups for the MP2 correlation energy computation, compared with the widely used quantum chemistry software package PySCF and its GPU plugin GPU4PySCF, respectively. In the MP3 calculations, our DGEMM-based implementation achieves a maximum speedup of 96.2 times over PySCF.
In this paper we derive a numerical tensor method to solve the many-body Schrödinger equation for three-particle systems. The Schrödinger equation is first represented in a canonical tensor format using a triple orthogonal Laguerre-based wave function in scaled perimetric coordinates. We demonstrate that the antisymmetry of the wave function can be explicitly included in the tensor method formulation by coupling the indices associated with the coordinates representing the identical fermions, e.g., electrons or muons. It is shown that energies accurate to the picohartree can be achieved in full tensor format in less than 3300 s for atomic and muonic molecular systems. Furthermore, this CPU time is reduced to less than 300 s, without compromising the accuracy of the energy or quality of the wave function, using a low-rank tensor-train decomposition.
The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17x10(9) determinants in the active space, could not be performed.