Development of parallel density functional program using distributed matrix to calculate all-electron canonical wavefunction of large molecules

We developed a new parallel density-functional canonical molecular-orbital program for large molecules based on the resolution of the identity method. In this study, all huge matrices were decomposed and saved to the distributed local memory. The routines of the analytical molecular integrals and numerical integrals of the exchange-correlation terms were parallelized using the single program multiple data method. A conventional linear algebra matrix library, ScaLAPACK, was used for matrix operations, such as diagonalization, multiplication, and inversion. Anderson's mixing method was adopted to accelerate the self-consistent field (SCF) convergence. Using this program, we calculated the canonical wavefunctions of a 306-residue protein, insulin hexamer (26,790 orbitals), and a 133-residue protein, interleukin (11,909 orbitals) by the direct-SCF method. In regard to insulin hexamer, the total parallelization efficiency of the first SCF iteration was estimated to be 82% using 64 Itanium 2 processors connected at 3.2 GB/s (SGI Altix3700), and the calculation successfully converged at the 17-th SCF iteration. By adopting the update method, the computational time of the first and the final SCF loops was 229 min and 156 min, respectively. The whole computational time including the calculation before the SCF loop was 2 days and 17 h. This study put the calculations of the canonical wavefunction of 30,000 orbitals to practical use.     

Large scale Hartree-Fock calculations with conventional SCF algorithm. Influence of integral and index compression on Fock matrix construction

It is shown that a compression of two-electron integrals and their indices significantly improves efficiency of the conventional self-consistent field (SCF) algorithm for a solution of the Hartree-Fock equation by decrease the Fock matrix calculation time. The improvement is reached not only due to a reduction of the integral file size, but mainly because data compression reduces or even can eliminate a cache conflict in data transfer from the hard drive to the main computer memory. Thus, the conventional SCF algorithm with the data compression becomes very efficient and permits to carry out large-scale Hartree-Fock calculations. The largest Hartree-Fock calculations have been performed for RNA 433D structure from the PDB data bank with 6080 basis functions formed from 6-31G basis on a workstation with 1 GHz Alpha processor.     

Connected moments expansion calculations of the correlation energy in small molecules

The second-order connected moments expansion (CMX(2)) approach to calculation of the correlation energy is tested numerically on several closed-shell di- and tri-atomic molecules. Benchmark computations performed within 6–31G⁷ basis set reveal that CMX(2) usually recovers more than 50% of the MP3 correlation energy and improves the SCF molecular geometries at a cost comparable to the MP3 calculations.     

Calculations using a set of evenly spaced S-type gaussian functions

A basis set of evenly spaced S-type Gaussian functions with common exponents is examined. Formulas for common one- and two-electron integrals are derived. Because of thesymmetry of this basis set, a very compact two-electron integral list is produced. The number of two-electron integrals that must be stored is approximately eight times the number of basis functions. Use of this basis set in an SCF calculation is examined. Numerical results show that this approach works well for molecules containing only small atoms such as hydrogen, helium, or lithium, but that the method has problems with the core orbitals of heavier atoms. Procedures for augementing this basis set in calculations involving heavier atoms are examined.     

Ab initio calculations on large molecules: The multiplicative integral approximation

In the multiplicative integral approximation (MIA), two-electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self-consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA-assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock-matrix allows important time savings. Results are presented for a number of test cases.     

Ab initioLCAO-MO-SCF Calculation of chlorpromazine and promazine

Ab initio LCAO -MO -SCF Gaussian basis function calculations have been performed for chlorpromazine and promazine. By prescreening for the size of the integrals before calculation, it was only necessary to calculate 12 million out of the possible 38.5 million integrals for chlorpromazine. By a novel procedure of processing the integral tapes for the SCF it was possible to cut down significantly on the amount of time for the SCF . The SCF calculations converged smoothly for both promazine and chlorpromazine. There is a sizeable energy gap between the energy of the highest occupied molecular orbitals in these molecules (which is of the order of ?0.3 a.u.) and the lowest unoccupied molecular orbital (which is of the order of + 0.15 a.u.). The gross atomic populations of chlorpromazine and promazine resemble each other and differ only somewhat on the carbon atom to which the substituent is attached and the carbons and their hydrogens adjacent to it.     

Improved intermolecular SCF theory and the BSSE problem

Analytical and numerical studies are performed concerning the exclusion of the basis set superposition error (BSSE ) from the SCF calculations of intermolecular interactions. Based on these studies a new procedure is proposed, which consists of the following steps: (1) determine the orbitals by the SCF scheme based on the recent “chemical Hamiltonian approach” (CHA-SCF method), i.e., excluding the delocalization effects caused by BSSE , and then (2) calculate the usual energy expectation value. (This gives results superior to those obtained by the previous nonsymmetric CHA energy formula.) The actual numerical calculations performed for different simple systems (He₂, water dimer) by using various basis sets indicate that the CHA/CE (CHA with “conventional energy” formula) potential curves are well-balanced and are close to those obtained by the Boys–Bernardi (BB ) method and usually (but not necessarily) go slightly beyond the latter. So our method gives results better than (or close to) those given by the BB method by performing only a single ～N⁴ calculation at each geometrical arrangement of the system.     

A new theory for symmetry orbital and tensor (Ⅱ)——Symmetric reduction of molecular integrals and self-consistent field calculations

The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation scheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries.     

Computational chemistry with transputers: a direct SCF program

By using transputers it is possible to build up networks of parallel processors with varying topology. Due to the architecture of the processors it is appropriate to use the MIMD (multiple instruction multiple data) concept of parallel computing. The most suitable programming language is OCCAM. We investigate the use of transputer networks in computational chemistry, starting with the direct SCF method. The most time consuming step, the calculation of the two electron integrals is executed parallelly. Each node in the network calculates whole batches of integrals. The main program is written in OCCAM. For some large-scale arithmetic processes running on a single node, however, we used FORTRAN subroutines out of standard ab-initio programs to reduce the programming effort. Test calculations show, that the integral calculation step can be parallelized very efficiently. We observe a speed-up of almost 8 using eight network processors. Even in consideration of the scalar part of the SCF iteration, the speed-up is not less than 7.1.     

A new theory for symmetry orbital and tensor (II)

The symmetry orbital-symmetry orbital tensor methtd is applied to the evaluation of molecular integrals (one-electron) and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation sheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries. Projcct supported by the National Natural Science Foundation of China (Grant No. 29473119).     

Molecular orbital studies on small molecules using H2+-type elliptical basis orbitals. Application to H2+, H2, He2++ and H3+

H₂⁺-type elliptical orbitals are defined in Section 1. These orbitals, which in elliptical coordinates involve a factor (1 + ξ)^σ, are employed in variational calculations on the ground states of H₂⁺ and H₂ (Sections 2 and 3). Various choices of σ are explored for H₂⁺, while two choices are used for H₂ : the “boundary condition” (Equation 6) and the “cusp condition” (Equation 9) values. Variational energies are calculated and compared to the results of similar calculations. Section 3 concludes by employing the H₂⁺-type orbitals in LCETO-MO-SCF calculations on the ground states of H₂ and He₂⁺⁺. For both molecules a four-function basis set with two (nonlinear) variational parameters yields more than 99% of the Hartree-Fock limit. Section 4 deals with LCETO-MO-SCF calculations on triangular H₃⁺. Three four-function basis sets are used, and the best energy is -1.2306 a.u., which is in reasonable agreement with the Hartree-Fock limit, -1.2999 a.u. Our best basis set is a four-term two-center expansion of the wave function with only one nonlinear variational parameter. Section 5 concludes the paper with a summary of the methods used to evaluate the integrals which arise in SCF calculations in the H₂⁺-type elliptical orbital basis.     

Semiempirical NDDO calculations with STO-3G and 4-31G basis sets

Possible refinements of semiempirical methods include the use of larger basis sets and of correlated wave functions. These possibilities are investigated in semiempirical NDDO SCF calculations with the STO-3G and 4-31G basis sets, and in correlated calculations at the STO-3G level. The present approach is characterized by the analytical evaluation of all one-center terms and two-electron integrals, and the semiempirical adjustment of the remaining one-electron integrals and the nuclear repulsions. The NDDO SCF results tend to reproduce the correspondingab initio results more closely than experimental data, even if they are parametrized with respect to experiment. The explicit inclusion of electron correlation at the STO-3G level improves the calculated results only slightly.     

An integral direct, distributed-data, parallel MP2 algorithm

Summary A scalable integral direct, distributed-data parallel algorithm for four-index transformation is presented. The algorithm was implemented in the context of the second-order M?ller-Plesset (MP2) energy evaluation, yet it is easily adopted for other electron correlation methods, where only MO integrals with two indices in the virtual orbitals space are required. The major computational steps of the MP2 energy are the two-electron integral evaluationO(N ⁴) and transformation into the MO basisO(ON ⁴), whereN is the number of basis functions, andO the number of occupied orbitals, respectively. The associated maximal communication costs scale asO(n _Σ O ² V N), whereV andn _Σ denote the number of virtual orbitals, and the number of symmetry-unique shells. The largest local and global memory requirements areO(N ²) for the MO coefficients andO(OV N) for the three-quarter transformed integrals, respectively. Several aspects of the implementation such as symmetry-treatment, integral prescreening, and the distribution of data and computational tasks are discussed. The parallel efficiency of the algorithm is demonstrated by calculations on the phenanthrene molecule, with 762 primitive Gaussians, contracted to 412 basis functions. The calculations were performed on an IBM SP2 with 48 nodes. The measured wall clock time on 48 nodes is less than 15 min for this calculation, and the speedup relative to single-node execution is estimated to 527. This superlinear speedup is a result of exploiting both the compute power and the aggregate memory of the parallel computer. The latter reduces the number of passes through the AO integral list, and hence the operation count of the calculation. The test calculations also show that the evaluation of the two-electron integrals dominates the calculation, despite the higher scaling of the transformation step.     

A Theoretical Study of Complex Formation between Formaldehyde and Lithium

Ab initio SCF and CI calculations on the cationic and neutral complexes of formaldehyde and lithium are reported. For the cationic complex CH₂O/Li⁺, the stabilization energy of 41.7 kcal/mol obtained from the SCF calculation increases to 51.6 kcal/mol if a configuration interaction is introduced. For the neutral complex CH₂O^?/Li⁺, the C_2v-conformer of the ²A₁-state with the equilibrium bond distances of d(C? O) = 1.23 Å and d (O? Li) = 1.90 Å is calculated to be more stable than the ₂B₁-state with d (C? O) = 1.34 Å, and d (O? Li) = 1.65 Å. Charge transfer and polarization effects upon complex formation are discussed.     

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.     

Direct relativistic MP2: properties of ground state CuF, AgF and AuF

A computer program for the calculation of the MP2 energy correction for a Kramers-restricted Dirac-Hartree-Fock four component wave-function is presented. In the spirit of the integral-driven direct SCF scheme the algorithm has been developed as direct MP2, calculating integrals as they are needed and avoiding the integral storage bottle-neck of conventional MP2. Relativistic MP2 is applied to ground state (¹Σ⁺) CuF, AgF and AuF. Received: 15 December 1996 / Accepted: 2 April 1997     

Vector processing algorithm for electron repulsion integrals in Ab Initio HF calculation based upon the PK supermatrix

An efficient vector processing algorithm generating PK supermatrices has been developed, in particular aiming at calculations on large molecules. The algorithm utilizes the recurrence relations for electron repulsion integrals. The PK supermatrices are listed in a nearly canonical order so that the Fock matrix generation is efficiently vectorized, no temporary ERI and PK files being used. This is effected by partition of the basis set (atomic orbitals) into subsets of certain appropriate sizes, and the partition approach is named as the three-dimensional partial space method. A high-speed Hartree–Fock calculation including integrals and SCF procedures is achieved. © 1992 by John Wiley & Sons, Inc.     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.