Massively parallel algorithm and implementation of RI‐MP2 energy calculation for peta‐scale many‐core supercomputers

A new parallel algorithm and its implementation for the RI‐MP2 energy calculation utilizing peta‐flop‐class many‐core supercomputers are presented. Some improvements from the previous algorithm (J. Chem. Theory Comput. 2013, 9, 5373) have been performed: (1) a dual‐level hierarchical parallelization scheme that enables the use of more than 10,000 Message Passing Interface (MPI) processes and (2) a new data communication scheme that reduces network communication overhead. A multi‐node and multi‐GPU implementation of the present algorithm is presented for calculations on a central processing unit (CPU)/graphics processing unit (GPU) hybrid supercomputer. Benchmark results of the new algorithm and its implementation using the K computer (CPU clustering system) and TSUBAME 2.5 (CPU/GPU hybrid system) demonstrate high efficiency. The peak performance of 3.1 PFLOPS is attained using 80,199 nodes of the K computer. The peak performance of the multi‐node and multi‐GPU implementation is 514 TFLOPS using 1349 nodes and 4047 GPUs of TSUBAME 2.5. © 2016 Wiley Periodicals, Inc.     

General orbital invariant MP2-F12 theory

A general form of orbital invariant explicitly correlated second-order closed-shell Moller-Plesset perturbation theory (MP2-F12) is derived, and compact working equations are presented. Many-electron integrals are avoided by resolution of the identity (RI) approximations using the complementary auxiliary basis set approach. A hierarchy of well defined levels of approximation is introduced, differing from the exact theory by the neglect of terms involving matrix elements over the Fock operator. The most accurate method is denoted as MP2-F12/3B. This assumes only that Fock matrix elements between occupied orbitals and orbitals outside the auxiliary basis set are negligible. For the chosen ansatz for the first-order wave function this is exact if the auxiliary basis is complete. In the next lower approximation it is assumed that the occupied orbital space is closed under action of the Fock operator [generalized Brillouin condition (GBC)]; this is equivalent to approximation 2B of Klopper and Samson [J. Chem. Phys. 116, 6397 (2002)]. Further approximations can be introduced by assuming the extended Brillouin condition (EBC) or by neglecting certain terms involving the exchange operator. A new approximation MP2-F12/3C, which is closely related to the MP2-R12/C method recently proposed by Kedzuch et al. [Int. J. Quantum Chem. 105, 929 (2005)] is described. In the limit of a complete RI basis this method is equivalent to MP2-F12/3B. The effect of the various approximations (GBC, EBC, and exchange) is tested by studying the convergence of the correlation energies with respect to the atomic orbital and auxiliary basis sets for 21 molecules. The accuracy of relative energies is demonstrated for 16 chemical reactions. Approximation 3C is found to perform equally well as the computationally more demanding approximation 3B. The reaction energies obtained with smaller basis sets are found to be most accurate if the orbital-variant diagonal Ansatz combined with localized orbitals is used for the first-order wave function. This unexpected result is attributed to geminal basis set superposition errors present in the formally more rigorous orbital invariant methods.     

Implementation of empirical dispersion corrections to density functional theory for periodic systems

A recently developed empirical dispersion correction (Grimme et al., J. Chem. Phys. 2010, 132, 154104) to standard density functional theory (DFT‐D3) is implemented in the plane‐wave program package VASP. The DFT‐D3 implementation is compared with an implementation of the earlier DFT‐D2 version (Grimme, J. Comput. Chem. 2004, 25, 1463; Grimme, J. Comput. Chem. 2006, 27, 1787). Summation of empirical pair potential terms is performed over all atom pairs in the reference cell and over atoms in shells of neighboring cells until convergence of the dispersion energy is obtained. For DFT‐D3, the definition of coordination numbers has to be modified with respect to the molecular version to ensure convergence. The effect of three‐center terms as implemented in the original molecular DFT‐D3 version is investigated. The empirical parameters are taken from the original DFT‐D3 version where they had been optimized for a reference set of small molecules. As the coordination numbers of atoms in bulk and surfaces are much larger than in the reference compounds, this effect has to be discussed. The results of test calculations for bulk properties of metals, metal oxides, benzene, and graphite indicate that the original parameters are also suitable for solid‐state systems. In particular, the interlayer distance in bulk graphite and lattice constants of molecular crystals is considerably improved over standard functionals. With the molecular standard parameters (Grimme et al., J. Chem. Phys. 2010, 132, 154104; Grimme, J. Comput. Chem. 2006, 27, 1787) a slight overbinding is observed for ionic oxides where dispersion should not contribute to the bond. For simple adsorbate systems, such as Xe atoms and benzene on Ag(111), the DFT‐D implementations reproduce experimental results with a similar accuracy as more sophisticated approaches based on perturbation theory (Rohlfing and Bredow, Phys. Rev. Lett. 2008, 101, 266106). © 2012 Wiley Periodicals, Inc.     

Array files for computational chemistry: MP2 energies

A simple message‐passing implementation for distributed disk storage, called array files (AF), is described. It is designed primarily for parallelizing computational chemistry applications but it should be useful for any application that handles large amounts of data stored on disk. AF allows transparent distributed storage and access of large data files. An AF consists of a set of logically related records, i.e., blocks of data. It is assumed that the records have the typical dimension of matrices in quantum chemical calculations, i.e., they range from 0.1 to ～32 MB in size. The individual records are not striped over nodes; each record is stored on a single node. As a simple application, second‐order Møller‐Plesset (MP2) energies have been implemented using AF. The AF implementation approaches the efficiency of the hand‐coded program. MP2 is relatively simple to parallelize but for more complex applications, such as Coupled Cluster energies, the AF system greatly simplifies the programming effort. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007.     

Improved third-order Møller-Plesset perturbation theory

Based on a partitioning of the total correlation energy into contributions from parallel‐ and antiparallel‐spin pairs of electrons, a modified third‐order Møller–Plesset (MP) perturbation theory is developed. The method, termed SCS–MP3 (SCS for spin‐component‐scaled) continues previous work on an improved version of MP2 (S. Grimme, J Chem Phys 2003, 118, 9095). A benchmark set of 32 isogyric reaction energies, 11 atomization energies, and 11 stretched geometries is used to assess to performance of the model in comparison to the standard quantum chemical approaches MP2, MP3, and QCISD(T). It is found, that the new method performs significantly better than usual MP2/MP3 and even outperforms the more costly QCISD method. Opposite to the usual MP series, the SCS third‐order correction uniformly improves the results. Dramatic enhancements are especially observed for the more difficult atomization energies, some of the stretched geometries, and reaction and ionization energies involving transition metal compounds where the method seems to be competitive or even superior to the widely used density functional approaches. Further tests performed for other complex systems (biradicals, C₂₀ isomers, transition states) demonstrate that the SCS–MP3 model yields often results of QCISD(T) accuracy. The uniformity with which the new approach improves for very different correlation problems indicates significant robustness, and suggests it as a valuable quantum chemical method of general use. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1529–1537, 2003     

An efficient parallel algorithm for the calculation of unrestricted canonical MP2 energies

We present details of our efficient implementation of full accuracy unrestricted open‐shell second‐order canonical Møller–Plesset (MP2) energies, both serial and parallel. The algorithm is based on our previous restricted closed‐shell MP2 code using the Saebo–Almlöf direct integral transformation. Depending on system details, UMP2 energies take from less than 1.5 to about 3.0 times as long as a closed‐shell RMP2 energy on a similar system using the same algorithm. Several examples are given including timings for some large stable radicals with 90+ atoms and over 3600 basis functions. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Accurate calculation of the heats of formation for large main group compounds with spin-component scaled MP2 methods

Three MP2-type electron correlation treatments and standard density functional theory (DFT) approaches are used to predict the heats of formation for a wide variety of different molecules. The SCF and MP2 calculations are performed efficiently using the resolution-of-the-identity (RI) approximation such that large basis set (i.e., polarized valence quadruple-zeta quality) treatments become routinely possible for systems with 50-100 atoms. An atom equivalent scheme that corrects the calculated atomic energies is applied to extract the "real" accuracy of the methods for chemically relevant problems. It is found that the spin-component-scaled MP2 method (SCS-MP2, J. Chem. Phys, 2003, 118, 9095) performs best and provides chemical accuracy (MAD of 1.18 kcal/mol) for a G2/97 test set of molecules. The computationally more economical SOS-MP2 variant, which retains only the opposite-spin part of the correlation energy, is slightly less accurate (MAD of 1.36 kcal/mol) than SCS-MP2. Both spin-component-scaled MP2 treatments perform significantly better than standard MP2 (MAD of 1.77 kcal/mol) and DFT-B3LYP (MAD of 2.12 kcal/mol). These conclusions are supported by results obtained for a second test set of complex systems containing 70 molecules, including charged, strained, polyhalogenated, hypervalent, and large unsaturated species (e.g. C60). For this set, DFT-B3LYP performs badly (MAD of 8.6 kcal/mol) with many errors >10-20 kcal/mol while the spin-component-scaled MP2 methods are still very accurate (MAD of 2.8 and 3.7 kcal/mol, respectively). DFT-B3LYP shows an obvious tendency to underestimate molecular stability as the system size increases. Out of six density functionals tested, the hybrid functional PBE0 performs best. All in all, the SCS-MP2 method, together with large AO basis sets, clearly outperforms current DFT approaches and seems to be the most accurate quantum chemical model that routinely can predict the thermodynamic properties of large main group compounds.     

An improved algorithm for analytical gradient evaluation in resolution-of-the-identity second-order Møller-Plesset perturbation theory: application to alanine tetrapeptide conformational analysis

We present a new algorithm for analytical gradient evaluation in resolution‐of‐the‐identity second‐order Møller‐Plesset perturbation theory (RI‐MP2) and thoroughly assess its computational performance and chemical accuracy. This algorithm addresses the potential I/O bottlenecks associated with disk‐based storage and access of the RI‐MP2 t‐amplitudes by utilizing a semi‐direct batching approach and yields computational speed‐ups of approximately 2–3 over the best conventional MP2 analytical gradient algorithms. In addition, we attempt to provide a straightforward guide to performing reliable and cost‐efficient geometry optimizations at the RI‐MP2 level of theory. By computing relative atomization energies for the G3/99 set and optimizing a test set of 136 equilibrium molecular structures, we demonstrate that satisfactory relative accuracy and significant computational savings can be obtained using Pople‐style atomic orbital basis sets with the existing auxiliary basis expansions for RI‐MP2 computations. We also show that RI‐MP2 geometry optimizations reproduce molecular equilibrium structures with no significant deviations (>0.1 pm) from the predictions of conventional MP2 theory. As a chemical application, we computed the extended‐globular conformational energy gap in alanine tetrapeptide at the extrapolated RI‐MP2/cc‐pV(TQ)Z level as 2.884, 4.414, and 4.994 kcal/mol for structures optimized using the HF, DFT (B3LYP), and RI‐MP2 methodologies and the cc‐pVTZ basis set, respectively. These marked energetic discrepancies originate from differential intramolecular hydrogen bonding present in the globular conformation optimized at these levels of theory and clearly demonstrate the importance of long‐range correlation effects in polypeptide conformational analysis. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007     

Acceleration of the GAMESS‐UK electronic structure package on graphical processing units

The approach used to calculate the two‐electron integral by many electronic structure packages including generalized atomic and molecular electronic structure system‐UK has been designed for CPU‐based compute units. We redesigned the two‐electron compute algorithm for acceleration on a graphical processing unit (GPU). We report the acceleration strategy and illustrate it on the (ss|ss) type integrals. This strategy is general for Fortran‐based codes and uses the Accelerator compiler from Portland Group International and GPU‐based accelerators from Nvidia. The evaluation of (ss|ss) type integrals within calculations using Hartree Fock ab initio methods and density functional theory are accelerated by single and quad GPU hardware systems by factors of 43 and 153, respectively. The overall speedup for a single self consistent field cycle is at least a factor of eight times faster on a single GPU compared with that of a single CPU. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Analysis of B3LYP and MP2 conformational population distributions in trans‐nicotine,acetylcholine, and ABT‐594

This work presents an analysis of the equivalence of MP2 and DFT (B3LYP functional) conformational populations. As a test case, we select three cholinergic agents (trans‐nicotine, acetylcholine, and the nicotinic analgesic ABT‐594), where the minima on the conformational energy hypersurfaces expand a large range of energies (～0–30 kJ mol^?1). From energetic and structural data obtained in vacuo at the MP2 and B3LYP/cc‐pVDZ levels, we build conformational partition functions, including the effect of the conformational kinetic energy and the rotovibrational coupling. Our results at a physiological temperature (37°C) show qualitative agreement in all cases. Quantitative agreement, however, is only found for trans‐nicotine and ABT‐594. In the first case, energy minima differ by <0.2 kJ mol^?1. Therefore, the equivalence of structural results translates in the equivalence of the conformational distribution. For ABT‐594, the minima are separated by as much as 8.0 kJ mol^?1, and the conformational energy determines the conformational distribution. In this case, the slight relative variation of conformational energy, between B3LYP and MP2, does not affect the population, since the secondary minima are high in energy and very low in population. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Intermolecular interactions of formic acid with benzene: Energy decomposition analyses with ab initio MP2 and double‐hybrid density functional computations

The intermolecular interactions of formic acid (HCOOH) with benzene (C₆H₆) have been investigated using localized molecular orbital energy decomposition analyses (LMO‐EDA) with ab initio MP2 and several double‐hybrid density functionals. The molecular geometries of five HCOOH…C₆H₆ complexes and corresponding benchmark total interaction energies at the CCSD(T)/CBS level are taken from literature (Zhao et al., J. Chem. Theory Comput. 2009, 5, 2726). According to the results of LMO‐EDA with the MP2 method, the dispersion energies are found to be as important as the electrostatic energies for the total interaction energies of the five HCOOH…C₆H₆ complexes. Based on LMO‐EDA with the double‐hybrid density functionals of B2PLYP, B2K‐PLYP, B2T‐PLYP, and B2GP‐PLYP computations, two new parameters for the framework of B2PLYP are extrapolated. These two new parameters are tested with other 10 complexes involving C₆H₆ (Crittenden, J. Phys. Chem. A 2009, 113, 1663), and they perform well on predicting the corresponding total interaction energies. Interestingly, these two new parameters for the framework of B2PLYP also perform well on the noncovalent complexation energies database (NCCE31/05) developed by Truhlar's group (Zhao and Truhlar, J. Phys. Chem. A 2005, 109, 5656). Therefore, these two new parameters appear to be suitable for investigating the noncovalent interactions, and they are denoted as B2N‐PLYP, where N stands for the noncovalent interaction. This study is expected to provide new insight into the derivation of double‐hybrid density functionals for studying the noncovalent interactions. © 2013 Wiley Periodicals, Inc.     

The MC‐DFT approach including the SCS‐MP2 energies to the new minnesota‐type functionals

We have applied the multicoefficient density functional theory (MC‐DFT) to four recent Minnesota functionals, including M06‐2X, M08‐HX, M11, and MN12‐SX on the performance of thermochemical kinetics. The results indicated that the accuracy can be improved significantly using more than one basis set. We further included the SCS‐MP2 energies into MC‐DFT, and the resulting mean unsigned errors (MUEs) decreased by approximately 0.3 kcal/mol for the most accurate basis set combinations. The M06‐2X functional with the simple [6–311+G(d,p)/6–311+G(2d,2p)] combination gave the best performance/cost ratios for the MC‐DFT and MC‐SCS‐MP2|MC‐DFT methods with MUE of 1.58 and 1.22 kcal/mol, respectively. © 2014 Wiley Periodicals, Inc.     

Analytic derivatives for perturbatively corrected "double hybrid" density functionals: theory, implementation, and applications

A recently proposed new family of density functionals [S. Grimme, J. Chem. Phys. 124, 34108 (2006)] adds a fraction of nonlocal correlation as a new ingredient to density functional theory (DFT). This fractional correlation energy is calculated at the level of second-order many-body perturbation theory (PT2) and replaces some of the semilocal DFT correlation of standard hybrid DFT methods. The new "double hybrid" functionals (termed, e.g., B2-PLYP) contain only two empirical parameters that have been adjusted in thermochemical calculations on parts of the G2/3 benchmark set. The methods have provided the lowest errors ever obtained by any DFT method for the full G3 set of molecules. In this work, the applicability of the new functionals is extended to the exploration of potential energy surfaces with analytic gradients. The theory of the analytic gradient largely follows the standard theory of PT2 gradients with some additional subtleties due to the presence of the exchange-correlation terms in the self-consistent field operator. An implementation is reported for closed-shell as well as spin-unrestricted reference determinants. Furthermore, the implementation includes external point charge fields and also accommodates continuum solvation models at the level of the conductor like screening model. The density fitting resolution of the identity (RI) approximation can be applied to the evaluation of the PT2 part with large gains in computational efficiency. For systems with approximately 500-600 basis functions the evaluation of the double hybrid gradient is approximately four times more expensive than the calculation of the standard hybrid DFT gradient. Extensive test calculations are provided for main group elements and transition metal containing species. The results reveal that the B2-PLYP functional provides excellent molecular geometries that are superior compared to those from standard DFT and MP2.     

Linear‐scaling self‐consistent field calculations based on divide‐and‐conquer method using resolution‐of‐identity approximation on graphical processing units

Graphical processing units (GPUs) are emerging in computational chemistry to include Hartree?Fock (HF) methods and electron‐correlation theories. However, ab initio calculations of large molecules face technical difficulties such as slow memory access between central processing unit and GPU and other shortfalls of GPU memory. The divide‐and‐conquer (DC) method, which is a linear‐scaling scheme that divides a total system into several fragments, could avoid these bottlenecks by separately solving local equations in individual fragments. In addition, the resolution‐of‐the‐identity (RI) approximation enables an effective reduction in computational cost with respect to the GPU memory. The present study implemented the DC‐RI‐HF code on GPUs using math libraries, which guarantee compatibility with future development of the GPU architecture. Numerical applications confirmed that the present code using GPUs significantly accelerated the HF calculations while maintaining accuracy. © 2014 Wiley Periodicals, Inc.     

New parallel algorithm for MP2 energy gradient calculations

A new parallel algorithm has been developed for calculating the analytic energy derivatives of full accuracy second order Møller‐Plesset perturbation theory (MP2). Its main projected application is the optimization of geometries of large molecules, in which noncovalent interactions play a significant role. The algorithm is based on the two‐step MP2 energy calculation algorithm developed recently and implemented into the quantum chemistry program, GAMESS. Timings are presented for test calculations on taxol (C₄₇H₅₁NO₁₄) with the 6‐31G and 6‐31G(d) basis sets (660 and 1032 basis functions, 328 correlated electrons) and luciferin (C₁₁H₈N₂O₃S₂) with aug‐cc‐pVDZ and aug‐cc‐pVTZ (530 and 1198 basis functions, 92 correlated electrons). The taxol 6‐31G(d) calculations are also performed with up to 80 CPU cores. The results demonstrate the high parallel efficiency of the program. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007