Superlinear scaling in master-slave quantum chemical calculations using in-core storage of two-electron integrals

We describe the implementation of a parallel, in-core, integral-direct Hartree-Fock and density functional theory code for the efficient calculation of Hartree-Fock wave functions and density functional theory. The algorithm is based on a parallel master-slave algorithm, and the two-electron integrals calculated by a slave are stored in available local memory. To ensure the greatest computational savings, the master node keeps track of all integral batches stored on the different slaves. The code can reuse undifferentiated two-electron integrals both in the wave function optimization and in the evaluation of second-, third-, and fourth-order molecular properties. Superlinear scaling is achieved in a series of test examples, with speedups of up to 55 achieved for calculations run on medium-sized molecules on 16 processors with respect to the time used on a single processor.     

Polarizability and optical rotation calculated from the approximate coupled cluster singles and doubles CC2 linear response theory using Cholesky decompositions

A new implementation of the approximate coupled cluster singles and doubles CC2 linear response model using Cholesky decomposition of the two-electron integrals is presented. Significantly reducing storage demands and computational effort without sacrificing accuracy compared to the conventional model, the algorithm is well suited for large-scale applications. Extensive basis set convergence studies are presented for the static and frequency-dependent electric dipole polarizability of benzene and C60, and for the optical rotation of CNOFH2 and (-)-trans-cyclooctene (TCO). The origin-dependence of the optical rotation is calculated and shown to persist for CC2 even at basis set convergence.     

Integral algorithm and density matrix integration scheme for ab initio band structure calculations on polymeric systems

A new program for band structure calculations of periodic one-dimensional systems has been constructed. It is distinguishable from other codes by the efficient two-electron integral evaluation and the integration schemes of the density matrix in the first Brillouin zone. The computation of polymeric two-electron integrals is based on the McMurchie Davidson algorithm and builds batches of the different cell indices included in the polymeric system. Consequently it presents efficient scaling with respect to the number of unit cells taken into account. Our algorithm takes into account fully the polymeric symmetry rather than the molecular symmetry. A semidirect procedure where only exchange integrals are computed at each SCF cycle is proposed in order to maintain balance between computation time and disk space. In addition, the integration of the density matrix over a large number of cell indices can be performed by different methods, such as Gauss-Legendre, Clenshaw-Curtis, Filon, and Alaylioglu-Evans-Hyslop. This last scheme is able to obtain an accuracy of 10(-13) a.u. on each individual density matrix element for all cell indices with only 48 k-points.     

Evaluation of Hylleraas-CI atomic integrals. III. Two-electron kinetic energy integrals

This paper is the part III of a series about the evaluation of Hylleraas-Configuration Interaction (Hy-CI) integrals by the method of direct integration over the interelectronic coordinates. The two-electron kinetic-energy integrals have been derived using the Hamiltonian in Hylleraas coordinates. We have improved the algorithm used in part II of this series and obtained general expressions. The method used for the two-electron integrals can be used in the same fashion for the evaluation of the three-electron ones. The formulas shown here have been tested in actual Hy-CI calculations of two-electron systems. The two-electron kinetic energy integrals values agree with the ones obtained using the Kolos and Roothaan transformation. The effectiveness of the different methods is discussed.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

Large scale polarizability calculations using the approximate coupled cluster model CC2 and MP2 combined with the resolution-of-the-identity approximation

We present an implementation of static and frequency-dependent polarizabilities for the approximate coupled cluster singles and doubles model CC2 and static polarizabilities for second-order Mo?ller-Plesset perturbation theory. Both are combined with the resolution-of-the-identity approximation for electron repulsion integrals to achieve unprecedented low operation counts, input-output, and disc space demands. To avoid the storage of double excitation amplitudes during the calculation of derivatives of density matrices, we employ in addition a numerical Laplace transformation for orbital energy denominators. It is shown that the error introduced by this approximation is negligible already with a small number of sampling points. Thereby an implementation of second-order one-particle properties is realized, which avoids completely the storage of quantities scaling with the fourth power of the system size. The implementation is tested on a set of organic molecules including large fused aromatic ring systems and the C(60) fullerene. It is demonstrated that exploiting symmetry and shared memory parallelization, second-order properties for such systems can be evaluated at the CC2 and MP2 level within a few hours of calculation time. As large scale applications, we present results for the 7-, 9-, and 11-ring helicenes.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. II. Four-electron integrals

An alternative procedure to the classical method for evaluating the four-electron Hylleraas-CI integrals is given. The method consists of direct integration over the r ₁₂ coordinate and integration over the coordinates of one of the electrons, reducing the integrals to lower order. The method based on the earlier work of Calais and L?wdin and of Perkins is extended to the general angular case. In this way it is possible to solve all of the four-electron integrals appearing in the Hylleraas-CI method. The four-electron integrals are expanded in three-electron ones which are in turn expanded in two-electron integrals. Finally the two-electron integrals are expanded into two-electron auxiliary integrals which usually have one negative power. The use of three- and four-electron electron auxiliary integrals is avoided. Some special cases lead to one- and two-electron auxiliary integrals with negative powers which do not converge individually but do converge in combination with others. These relations and their solutions are presented, together with results of various kinds of integrals.     

A new analytical method for computing solvent-accessible surface area of macromolecules and its gradients

In the calculation of thermodynamic properties and three-dimensional structures of macromolecules, such as proteins, it is important to have an efficient algorithm for computing the solvent-accessible surface area of macromolecules. Here, we propose a new analytical method for this purpose. In the proposed algorithm we consider the transformation that maps the spherical circles formed by intersection of the atomic surfaces in three-dimensional space onto the circles on a two-dimensional plane, and the problem of computing the solvent-accessible surface area is reduced to the problem of computing the corresponding curve integrals on the plane. This allows to consider only the integrals along the circular trajectories on the plane. The algorithm is suitable for parallelization. Testings on many proteins as well as the comparison to the other analogous algorithms have shown that our method is accurate and efficient.     

An economic storage and processing method for two-electron integrals in LCAO-MO calculations

An economic algorithm for storing and processing two-electron integrals arising in LCAO-MO calculations is presented. The integrals are sorted prior to the SCF iterative scheme, classified according to equivalences in the orbital indices and finally stored on separate files that contain only integrals of one type. The novel approach of physically separating the integrals according to category is shown to be more efficient than random storage. Actual computing times for the new technique are tabulated for a representative number of molecular systems and compared with times obtained using previously reported methods.     

Parallelization of the integral equation formulation of the polarizable continuum model for higher-order response functions

We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.     

Resolutions of the Coulomb operator. VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb operator [T. Limpanuparb, A. T. B. Gilbert, and P. M. W. Gill, J. Chem. Theory Comput. 7, 830 (2011)] and the long-range Ewald operator [T. Limpanuparb and P. M. W. Gill, J. Chem. Theory Comput. 7, 2353 (2011)]) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use.     

Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules

The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater-type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one- and two-center integrals (both one- and two-electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three- and four-center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds.     

On bound states in two-body systems

The application of the Σ-separation method to the calculation of multicenter two-electron molecular integrals with Slater-type basis functions is reported. The approach is based on the approximation of a scalar component of the two-center atomic density by a two-center expansion over Slater-type functions. A least-squares fit was used to determine the coefficients of the expansion. The angular multipliers of the atomic density were treated exactly. It is shown that this approach can serve as a sufficiently accurate and fast algorithm for the calculation of multicenter two-electron molecular integrals with Slater-type basis functions. © 1995 John Wiley & Sons, Inc.     

The SHARK integral generation and digestion system

In this paper, the SHARK integral generation and digestion engine is described. In essence, SHARK is based on a reformulation of the popular McMurchie/Davidson approach to molecular integrals. This reformulation leads to an efficient algorithm that is driven by BLAS level 3 operations. The algorithm is particularly efficient for high angular momentum basis functions (up to L = 7 is available by default, but the algorithm is programmed for arbitrary angular momenta). SHARK features a significant number of specific programming constructs that are designed to greatly simplify the workflow in quantum chemical program development and avoid undesirable code duplication to the largest possible extent. SHARK can handle segmented, generally and partially generally contracted basis sets. It can be used to generate a host of one- and two-electron integrals over various kernels including, two-, three-, and four-index repulsion integrals, integrals over Gauge Including Atomic Orbitals (GIAOs), relativistic integrals and integrals featuring a finite nucleus model. SHARK provides routines to evaluate Fock like matrices, generate integral transformations and related tasks. SHARK is the essential engine inside the ORCA package that drives essentially all tasks that are related to integrals over basis functions in version ORCA 5.0 and higher. Since the core of SHARK is based on low-level basic linear algebra (BLAS) operations, it is expected to not only perform well on present day but also on future hardware provided that the hardware manufacturer provides a properly optimized BLAS library for matrix and vector operations. Representative timings and comparisons to the Libint library used by ORCA are reported for Intel i9 and Apple M1 max processors.     

Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R(2)·n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the M?ller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N(1.8)) for the AO integrals, O(N(1.4)) for the MO integrals, and O(N(1.2)) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as ε = 10(-2). This leads to an error in the MP2 energy in the order of mHartree.     

Some aspects of parallelization of two-electron integrals in molecular orbital programs

Evaluation of two-electron integrals forms a substantial part of the CPU time for any ab initio molecular orbital program. This part of the package, “MICROMOL”, is parallelized. However, this parallelization leads to only sublinear speedups (typically 3 on a 4-node machine). In view of these results, the task of development of an efficient program for two-electron integrals suitable for the parallel environment has been taken up. The program is written in FORTRAN considering specific symmetry features and application of rigorous bounds. This program is further parallelized with a good load balancing strategy. The molecules used as the test cases are: trans-butadiene, benzene, nitrobenzene, naphtalene and cytosine, with 3G and 4–31G basis sets. The results indicate that the parallel version of this program gives a typical speedup of 3.6 for a 3G basis set and approximately 3.4 for a 4–31G basis set for all the molecules tested. The sequential version of this program is ～1.2 times faster than the sequential version of MICROMOL, whereas the parallel version is ～1.4 times faster than the parallelized MICROMOL.     

Ab initio calculation of magnetic properties by the “direct” IGLO method

A direct method for the ab initio calculation of the magnetic susceptibility and chemical shielding tensors based on the individual gauge for localized molecular orbitals (IGLO) formalism is introduced. “Direct” in this context means we avoid storing the two-electron repulsion integrals in favor of recalculating them whenever necessary. In conjunction with the Direct-SCF package TURBOMOLE Direct IGLO (DIGLO) permits calculation of magnetic second-order properties for large molecules by minimizing peripheral disc storage requirements. The size of the molecules to be treated is limited only by the amount of CPU time available. The performance of DIGLO is demonstrated for some selected examples.     

Avoiding the use of integral lists in the ab initio LCAO SCF MO method

The suggestion is made that the storage of long lists of two-electron repulsion integrals in the LCAO SCF MO method could be avoided if such integrals were evaluated at each SCF cycle and an initial set of eigenvectors were available to lead to rapid convergence. Results are presented showing that the simulated ab initio molecular orbital (SAMO) technique provides an excellent set of initial eigenvectors for the SCF procedure.     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.