Electron Repulsion Integrals Research Articles

A novel Rys quadrature algorithm for use in the calculation of electron repulsion integrals

The half-range Rys polynomials, Rn(x), are orthonormal with respect to weight function w(x)=e-cx2 on the interval x∈[0,1] and defined with the set of coefficients, αn and βn, in the three term recurrence relation for the polynomials. Full range Rys polynomials, Jn(x), are orthonormal with respect to w(x)=e-cx2 on the interval x∈[-1,1]. They are defined with the set of β̂n recurrence coefficients as α̂n=0. The Gauss-Rys quadrature defined with the Rys polynomials are used to evaluate electron repulsion integrals in quantum chemistry computer codes. The present paper proposes a new algorithm for the efficient computation of the Rys quadrature weights and points versus the parameter c in the weight function. The method is based on the full range Rys polynomials and a novel method for the calculation of the positive quadrature points and related weights.

A new mixed self-consistent field procedure

A new approach for the calculation of three-centre electronic repulsion integrals (ERIs) is developed, implemented and benchmarked in the framework of auxiliary density functional theory (ADFT). The so-called mixed self-consistent field (mixed SCF) divides the computationally costly ERIs in two sets: far-field and near-field. Far-field ERIs are calculated using the newly developed double asymptotic expansion as in the direct SCF scheme. Near-field ERIs are calculated only once prior to the SCF procedure and stored in memory, as in the conventional SCF scheme. Hence the name, mixed SCF. The implementation is particularly powerful when used in parallel architectures, since all RAM available are used for near-field ERI storage. In addition, the efficient distribution algorithm performs minimal intercommunication operations between processors, avoiding a potential bottleneck. One-, two- and three-dimensional systems are used for benchmarking, showing substantial time reduction in the ERI calculation for all of them. A Born–Oppenheimer molecular dynamics calculation for the Na+55 cluster is also shown in order to demonstrate the speed-up for small systems achievable with the mixed SCF.

Libcint: An efficient general integral library for Gaussian basis functions.

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian-type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one-electron overlap and nuclear attraction, two-electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4-component relativistic Hartree-Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth-order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi-CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20-core node, we are able to achieve an average output 8.3 GB/s for C60 molecule with cc-pVTZ basis.

Accompanying coordinate expansion and recurrence relation method using a transfer relation scheme for electron repulsion integrals with high angular momenta and long contractions.

An efficient algorithm for the rapid evaluation of electron repulsion integrals is proposed. The present method, denoted by accompanying coordinate expansion and transferred recurrence relation (ACE-TRR), is constructed using a transfer relation scheme based on the accompanying coordinate expansion and recurrence relation method. Furthermore, the ACE-TRR algorithm is extended for the general-contraction basis sets. Numerical assessments clarify the efficiency of the ACE-TRR method for the systems including heavy elements, whose orbitals have long contractions and high angular momenta, such as f- and g-orbitals.

Periodic local MP2 method employing orbital specific virtuals.

We introduce orbital specific virtuals (OSVs) to represent the truncated pair-specific virtual space in periodic local Møller-Plesset perturbation theory of second order (LMP2). The OSVs are constructed by diagonalization of the LMP2 amplitude matrices which correspond to diagonal Wannier-function (WF) pairs. Only a subset of these OSVs is adopted for the subsequent OSV-LMP2 calculation, namely, those with largest contribution to the diagonal pair correlation energy and with the accumulated value of these contributions reaching a certain accuracy. The virtual space for a general (non diagonal) pair is spanned by the union of the two OSV sets related to the individual WFs of the pair. In the periodic LMP2 method, the diagonal LMP2 amplitude matrices needed for the construction of the OSVs are calculated in the basis of projected atomic orbitals (PAOs), employing very large PAO domains. It turns out that the OSVs are excellent to describe short range correlation, yet less appropriate for long range van der Waals correlation. In order to compensate for this bias towards short range correlation, we augment the virtual space spanned by the OSVs by the most diffuse PAOs of the corresponding minimal PAO domain. The Fock and overlap matrices in OSV basis are constructed in the reciprocal space. The 4-index electron repulsion integrals are calculated by local density fitting and, for distant pairs, via multipole approximation. New procedures for determining the fit-domains and the distant-pair lists, leading to higher efficiency in the 4-index integral evaluation, have been implemented. Generally, and in contrast to our previous PAO based periodic LMP2 method, the OSV-LMP2 method does not require anymore great care in the specification of the individual domains (to get a balanced description when calculating energy differences) and is in that sense a black box procedure. Discontinuities in potential energy surfaces, which may occur for PAO-based calculations if one is not careful, virtually disappear for OSV-LMP2. Moreover, due to much increased compactness of the pair-specific virtual spaces, the OSV-LMP2 calculations are faster and require much less memory than PAO-LMP2 calculations, despite the noticeable overhead of the initial OSV construction procedure.

Calculation of the molecular integrals with the range-separated correlation factor.

Explicitly correlated quantum chemical calculations require calculations of five types of two-electron integrals beyond the standard electron repulsion integrals. We present a novel scheme, which utilises general ideas of the McMurchie-Davidson technique, to compute these integrals when the so-called "range-separated" correlation factor is used. This correlation factor combines the well-known short range behaviour resulting from the electronic cusp condition, with the exact long-range asymptotics derived for the helium atom [Lesiuk, Jeziorski, and Moszynski, J. Chem. Phys. 139, 134102 (2013)]. Almost all steps of the presented procedure are formulated recursively, so that an efficient implementation and control of the precision are possible. Additionally, the present formulation is very flexible and general, and it allows for use of an arbitrary correlation factor in the electronic structure calculations with minor or no changes.

Acceleration of High Angular Momentum Electron Repulsion Integrals and Integral Derivatives on Graphics Processing Units.

We present an efficient implementation of ab initio self-consistent field (SCF) energy and gradient calculations that run on Compute Unified Device Architecture (CUDA) enabled graphical processing units (GPUs) using recurrence relations. We first discuss the machine-generated code that calculates the electron-repulsion integrals (ERIs) for different ERI types. Next we describe the porting of the SCF gradient calculation to GPUs, which results in an acceleration of the computation of the first-order derivative of the ERIs. However, only s, p, and d ERIs and s and p derivatives could be executed simultaneously on GPUs using the current version of CUDA and generation of NVidia GPUs using a previously described algorithm [Miao and Merz J. Chem. Theory Comput. 2013, 9, 965-976.]. Hence, we developed an algorithm to compute f type ERIs and d type ERI derivatives on GPUs. Our benchmarks shows the performance GPU enable ERI and ERI derivative computation yielded speedups of 10-18 times relative to traditional CPU execution. An accuracy analysis using double-precision calculations demonstrates that the overall accuracy is satisfactory for most applications.

Two-photon absorption cross sections within equation-of-motion coupled-cluster formalism using resolution-of-the-identity and Cholesky decomposition representations: Theory, implementation, and benchmarks.

The equation-of-motion coupled-cluster (EOM-CC) methods provide a robust description of electronically excited states and their properties. Here, we present a formalism for two-photon absorption (2PA) cross sections for the equation-of-motion for excitation energies CC with single and double substitutions (EOM-CC for electronically excited states with single and double substitutions) wave functions. Rather than the response theory formulation, we employ the expectation-value approach which is commonly used within EOM-CC, configuration interaction, and algebraic diagrammatic construction frameworks. In addition to canonical implementation, we also exploit resolution-of-the-identity (RI) and Cholesky decomposition (CD) for the electron-repulsion integrals to reduce memory requirements and to increase parallel efficiency. The new methods are benchmarked against the CCSD and CC3 response theories for several small molecules. We found that the expectation-value 2PA cross sections are within 5% from the quadratic response CCSD values. The RI and CD approximations lead to small errors relative to the canonical implementation (less than 4%) while affording computational savings. RI/CD successfully address the well-known issue of large basis set requirements for 2PA cross sections calculations. The capabilities of the new code are illustrated by calculations of the 2PA cross sections for model chromophores of the photoactive yellow and green fluorescent proteins.

Approximating Matrices with Multiple Symmetries

If a tensor with various symmetries is properly unfolded, then the resulting matrix inherits those symmetries. As tensor computations become increasingly important it is imperative that we develop efficient structure preserving methods for matrices with multiple symmetries. In this paper we consider how to exploit and preserve structure in the pivoted Cholesky factorization when approximating a matrix $A$ that is both symmetric ($A=A^T$) and what we call perfect shuffle symmetric, or perf-symmetric. The latter property means that $A = \Pi A\Pi$, where $\Pi$ is a permutation with the property that $\Pi v = v$ if $v$ is the vec of a symmetric matrix and $\Pi v = -v$ if $v$ is the vec of a skew-symmetric matrix. Matrices with this structure can arise when an order-4 tensor $\cal A$ is unfolded and its elements satisfy ${\cal A}(i_{1},i_{2},i_{3},i_{4}) = {\cal A}(i_{2},i_{1},i_{3},i_{4}) ={\cal A}(i_{1},i_{2},i_{4},i_{3}) ={\cal A}(i_{3},i_{4},i_{1},i_{2}).$ This is the case in certain quantum chemistry applications where the tensor entries are electronic repulsion integrals. Our technique involves a closed-form block diagonalization followed by one or two half-sized pivoted Cholesky factorizations. This framework allows for a lazy evaluation feature that is important if the entries in $\cal A$ are expensive to compute. In addition to being a structure preserving rank reduction technique, we find that this approach for obtaining the Cholesky factorization reduces the work by up to a factor of 4 (see [G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013]).

A systematic way for the cost reduction of density fitting methods.

We present a simple approach for the reduction of the size of auxiliary basis sets used in methods exploiting the density fitting (resolution of identity) approximation for electron repulsion integrals. Starting out of the singular value decomposition of three-center two-electron integrals, new auxiliary functions are constructed as linear combinations of the original fitting functions. The new functions, which we term natural auxiliary functions (NAFs), are analogous to the natural orbitals widely used for the cost reduction of correlation methods. The use of the NAF basis enables the systematic truncation of the fitting basis, and thereby potentially the reduction of the computational expenses of the methods, though the scaling with the system size is not altered. The performance of the new approach has been tested for several quantum chemical methods. It is demonstrated that the most pronounced gain in computational efficiency can be expected for iterative models which scale quadratically with the size of the fitting basis set, such as the direct random phase approximation. The approach also has the promise of accelerating local correlation methods, for which the processing of three-center Coulomb integrals is a bottleneck.

Speeding up local correlation methods.

We present two techniques that can substantially speed up the local correlation methods. The first one allows one to avoid the expensive transformation of the electron-repulsion integrals from atomic orbitals to virtual space. The second one introduces an algorithm for the residual equations in the local perturbative treatment that, in contrast to the standard scheme, does not require holding the amplitudes or residuals in memory. It is shown that even an interpreter-based implementation of the proposed algorithm in the context of local MP2 method is faster and requires less memory than the highly optimized variants of conventional algorithms.

HONPAS: A linear scaling open‐source solution for large system simulations

HONPAS is an ab initio electronic structure program for linear scaling or O(N) first‐principles calculations of large and complex systems using standard norm‐conserving pseudopotentials, numerical atomic orbitals (NAOs) basis sets, and periodic boundary conditions. HONPAS is developed in the framework of the SIESTA methodology and focuses on the development and implementation of efficient O(N) algorithms for ab initio electronic structure calculations. The Heyd–Scuseria–Ernzerhof (HSE) screened hybrid density functional has been implemented using a NAO2GTO scheme to evaluate the electron repulsion integrals (ERIs) with NAOs. ERI screening techniques allow the HSE functional calculations to be very efficient and scale linearly. The density matrix purification algorithms have been implemented, and the PSUTC2 and SUTC2 methods have been developed to deal with spin unrestricted systems with or without predetermined spin multiplicity, respectively. After the self‐consistent field (SCF) process, additional O(N) post‐SCF calculations for frontier molecular orbitals and maximally localized Wannier functions are also developed and implemented. Finally, an O(N) method based on the density matrix perturbation theory has been proposed and implemented to treat electric field in solids. This article provides an overall introduction to capabilities of HONPAS and implementation details of different O(N) algorithms. © 2014 Wiley Periodicals, Inc.

Robust and efficient variational fitting of Fock exchange.

We propose a new variational fitting approach for Fock exchange that requires only the calculation of analytical three-center electron repulsion integrals. It relies on localized molecular orbitals and Hermite Gaussian auxiliary functions. The working equations along with a detailed description of the implementation are presented. The computational performance of the new algorithm is analyzed by benchmark calculations on systems with different dimensionality. Comparison with standard four-center and three-center electron repulsion integral Hartree-Fock calculations shows an excellent accuracy-performance relation.

Derivation of general analytic gradient expressions for density-fitted post-Hartree-Fock methods: an efficient implementation for the density-fitted second-order Møller-Plesset perturbation theory.

General analytic gradient expressions (with the frozen-core approximation) are presented for density-fitted post-HF methods. An efficient implementation of frozen-core analytic gradients for the second-order Møller-Plesset perturbation theory (MP2) with the density-fitting (DF) approximation (applying to both reference and correlation energies), which is denoted as DF-MP2, is reported. The DF-MP2 method is applied to a set of alkanes, conjugated dienes, and noncovalent interaction complexes to compare the computational cost of single point analytic gradients with MP2 with the resolution of the identity approach (RI-MP2) [F. Weigend and M. Häser, Theor. Chem. Acc. 97, 331 (1997); R. A. Distasio, R. P. Steele, Y. M. Rhee, Y. Shao, and M. Head-Gordon, J. Comput. Chem. 28, 839 (2007)]. In the RI-MP2 method, the DF approach is used only for the correlation energy. Our results demonstrate that the DF-MP2 method substantially accelerate the RI-MP2 method for analytic gradient computations due to the reduced input/output (I/O) time. Because in the DF-MP2 method the DF approach is used for both reference and correlation energies, the storage of 4-index electron repulsion integrals (ERIs) are avoided, 3-index ERI tensors are employed instead. Further, as in case of integrals, our gradient equation is completely avoid construction or storage of the 4-index two-particle density matrix (TPDM), instead we use 2- and 3-index TPDMs. Hence, the I/O bottleneck of a gradient computation is significantly overcome. Therefore, the cost of the generalized-Fock matrix (GFM), TPDM, solution of Z-vector equations, the back transformation of TPDM, and integral derivatives are substantially reduced when the DF approach is used for the entire energy expression. Further application results show that the DF approach introduce negligible errors for closed-shell reaction energies and equilibrium bond lengths.

Ab initio nonadiabatic dynamics of multichromophore complexes: a scalable graphical-processing-unit-accelerated exciton framework.

Conspectus Although advances in computer hardware and algorithms tuned for novel computer architectures are leading to significant increases in the size and time scale for molecular simulations, it remains true that new methods and algorithms will be needed to address some of the problems in complex chemical systems, such as electrochemistry, excitation energy transport, proton transport, and condensed phase reactivity. Ideally, these new methods would exploit the strengths of emerging architectures. Fragment based approaches for electronic structure theory decompose the problem of solving the electronic Schrodinger equation into a series of much smaller problems. Because each of these smaller problems is largely independent, this strategy is particularly well-suited to parallel architectures. It appears that the most significant advances in computer architectures will be toward increased parallelism, and therefore fragment-based approaches are an ideal match to these trends. When the computational effort involved scales with the third (or higher) power of the molecular size, there is a large benefit to fragment-based approaches even on serial architectures. This is the case for many of the well-known methods for solving the electronic structure theory problem, especially when wave function-based approaches including electron correlation are considered. A major issue in fragment-based approaches is determining or improving their accuracy. Since the Achilles' heel of any such method lies in the approximations used to stitch the smaller problems back together (i.e., in the treatment of the cross-fragment interactions), it can often be important to ensure that the size of the smaller problems is "large enough." Thus, there are two frontiers that need to be extended in order to enable molecular simulations for large systems and long times: the strongly coupled problem of medium sized molecules (100-500 atoms) and the more weakly coupled problem of decomposing ("fragmenting") a molecular system and then stitching it back together. In this Account, we address both of these problems, the first by using graphical processing units (GPUs) and electronic structure algorithms tuned for these architectures and the second by using an exciton model as a framework in which to stitch together the solutions of the smaller problems. The multitiered parallel framework outlined here is aimed at nonadiabatic dynamics simulations on large supramolecular multichromophoric complexes in full atomistic detail. In this framework, the lowest tier of parallelism involves GPU-accelerated electronic structure theory calculations, for which we summarize recent progress in parallelizing the computation and use of electron repulsion integrals (ERIs), which are the major computational bottleneck in both density functional theory (DFT) and time-dependent density functional theory (TDDFT). The topmost tier of parallelism relies on a distributed memory framework, in which we build an exciton model that couples chromophoric units. Combining these multiple levels of parallelism allows access to ground and excited state dynamics for large multichromophoric assemblies. The parallel excitonic framework is in good agreement with much more computationally demanding TDDFT calculations of the full assembly.

Magnetizability and rotational g tensors for density fitted local second-order Møller-Plesset perturbation theory using gauge-including atomic orbitals.

In this paper, we present theory and implementation of an efficient program for calculating magnetizabilities and rotational g tensors of closed-shell molecules at the level of local second-order Møller-Plesset perturbation theory (MP2) using London orbitals. Density fitting is employed to factorize the electron repulsion integrals with ordinary Gaussians as fitting functions. The presented program for the calculation of magnetizabilities and rotational g tensors is based on a previous implementation of NMR shielding tensors reported by S. Loibl and M. Schütz [J. Chem. Phys. 137, 084107 (2012)]. Extensive test calculations show (i) that the errors introduced by density fitting are negligible, and (ii) that the errors of the local approximation are still rather small, although larger than for nuclear magnetic resonance (NMR) shielding tensors. Electron correlation effects for magnetizabilities are tiny for most of the molecules considered here. MP2 appears to overestimate the correlation contribution of magnetizabilities such that it does not constitute an improvement over Hartree-Fock (when comparing to higher-order methods like CCSD(T)). For rotational g tensors the situation is different and MP2 provides a significant improvement in accuracy over Hartree-Fock. The computational performance of the new program was tested for two extended systems, the larger comprising about 2200 basis functions. It turns out that a magnetizability (or rotational g tensor) calculation takes about 1.5 times longer than a corresponding NMR shielding tensor calculation.

Extension of accompanying coordinate expansion and recurrence relation method for general-contraction basis sets.

An algorithm of the accompanying coordinate expansion and recurrence relation (ACE-RR), which is used for the rapid evaluation of the electron repulsion integral (ERI), has been extended to the general-contraction (GC) scheme. The present algorithm, denoted by GC-ACE-RR, is designed for molecular calculations including heavy elements, whose orbitals consist of many primitive functions with and without higher angular momentum such as d- and f-orbitals. The performance of GC-ACE-RR was assessed for (ss|ss)-, (pp|pp)-, (dd|dd)-, and (ff|ff)-type ERIs in terms of contraction length and the number of GC orbitals. The present algorithm was found to reduce the central processing unit time compared with the ACE-RR algorithm, especially for higher angular momentum and highly contracted orbitals. Compared with HONDOPLUS and GAMESS program packages, GC-ACE-RR computations for ERIs of three-dimensional gold clusters Aun (n = 1, 2, …, 10, 15, 20, and 25) are more than 10 times faster.

Communication: Acceleration of coupled cluster singles and doubles via orbital-weighted least-squares tensor hypercontraction.

We apply orbital-weighted least-squares tensor hypercontraction decomposition of the electron repulsion integrals to accelerate the coupled cluster singles and doubles (CCSD) method. Using accurate and flexible low-rank factorizations of the electron repulsion integral tensor, we are able to reduce the scaling of the most vexing particle-particle ladder term in CCSD from O(N6) to O(N5), with remarkably low error. Combined with a T1-transformed Hamiltonian, this leads to substantial practical accelerations against an optimized density-fitted CCSD implementation.

Density-fitted singles and doubles coupled cluster on graphics processing units

We adapt an algorithm for singles and doubles coupled cluster (CCSD) that uses density fitting or Cholesky decomposition (CD) in the construction and contraction of all electron repulsion integrals (ERIs) for use on heterogeneous compute nodes consisting of a multicore central processing unit (CPU) and at least one graphics processing unit (GPU). The use of approximate three-index ERIs ameliorates two of the major difficulties in designing scientific algorithms for GPUs: (1) the extremely limited global memory on the devices and (2) the overhead associated with data motion across the bus. For the benzene trimer described by an aug-cc-pVDZ basis set, the use of a single NVIDIA Tesla C2070 (Fermi) GPU accelerates a CD-CCSD computation by a factor of 2.1, relative to the multicore CPU-only algorithm that uses six highly efficient Intel Core i7-3930K CPU cores. The use of two Fermi GPUs provides an acceleration of 2.89, which is comparable to that observed when using a single NVIDIA Kepler K20c GPU (2.73).

Efficient calculation of two‐electron integrals for high angular basis functions

The computation of electron repulsion integrals (ERIs) is the most time‐consuming process in the density functional calculation using Gaussian basis set. Many temporal ERIs are calculated, and most are stored on slower storage, such as cache or memory, because of the shortage of registers, which are the fastest storage in a central processing unit (CPU). Moreover, the heavy register usage makes it difficult to launch many concurrent threads on a graphics processing unit (GPU) to hide latency. Hence, we propose to optimize the calculation order of one‐center ERIs to minimize the number of registers used, and to calculate each ERI with three or six co‐operating threads. The performance of this method is measured on a recent CPU and a GPU. The proposed approach is found to be efficient for high angular basis functions with a GPU. When combined with a recent GPU, it accelerates the computation almost 4‐fold. © 2014 Wiley Periodicals, Inc.