Linear Scaling Electronic Structure Calculations Research Articles

Local electronic structure analysis by ab initio elongation method: A benchmark using DNA block polymers.

The ab initio elongation (ELG) method based on a polymerization concept is a feasible way to perform linear-scaling electronic structure calculations for huge aperiodic molecules while maintaining computational accuracy. In the method, the electronic structures are sequentially elongated by repeating (1) the conversion of canonical molecular orbitals (CMOs) to region-localized MOs (RLMOs), that is, active RLMOs localized onto a region close to an attacking monomer or frozen RLMOs localized onto the remaining region, and the subsequent (2) partial self-consistent-field calculations for an interaction space composed of the active RLMOs and the attacking monomer. For each ELG process, one can obtain local CMOs for the interaction space and the corresponding local orbital energies. Local site information, such as the local highest-occupied/lowest-unoccupied MOs, can be acquired with linear-scaling efficiency by correctly including electronic effects from the frozen region. In this study, we performed a local electronic structure analysis using the ELG method for various DNA block polymers with different sequential patterns. This benchmark aimed to confirm the effectiveness of the method toward the efficient detection of a singular local electronic structure in unknown systems as a future practical application. We discussed the high-throughput efficiency of our method and proposed a strategy to detect singular electronic structures by combining with a machine learning technique.

GPU algorithms for density matrix methods on MOPAC: linear scaling electronic structure calculations for large molecular systems.

Purification of the density matrix methods should be employed when dealing with complex chemical systems containing many atoms. The running times for these methods scale linearly with the number of atoms if we consider the sparsity from the density matrix. Since the efficiency expected from those methods is closely tied to the underlying parallel implementations of the linear algebra operations (e.g., P2 = P × P), we proposed a central processing unit (CPU) and graphics processing unit (GPU) parallel matrix-matrix multiplication in SVBR (symmetrical variable block row) format for energy calculations through the SP2 algorithm. This algorithm was inserted in MOPAC's MOZYME method, using the original LMO Fock matrix assembly, and the atomic integral calculation implemented on it. Correctness and performance tests show that the implemented SP2 is accurate and fast, as the GPU is able to achieve speedups up to 40 times for a water cluster system with 42,312 orbitals running in one NVIDIA K40 GPU card compared to the single-threaded version. The GPU-accelerated SP2 algorithm using the MOZYME LMO framework enables the calculations of semiempirical wavefunction with stricter SCF criteria for localized charged molecular systems, as well as the single-point energies of molecules with more than 100.000 LMO orbitals in less than 1h. Graphical abstract Parallel CPU and GPU purification algorithms for electronic structure calculations were implemented in MOPAC's MOZYME method. Some matrices in these calculations, e.g., electron density P, are compressed, and the developed linear algebra operations deal with non-zero entries only. We employed the NVIDIA/CUDA platform to develop GPU algorithms, and accelerations up to 40 times for larger systems were achieved.

Linear-scaling self-consistent field theory based molecular dynamics: application to C60buckyballs colliding with graphite

ABSTRACTIn this work, we investigate the collision of a C fullerene with graphite using large-scale molecular dynamics simulations, where the interatomic forces are computed ‘on-the-fly’ by means of self-consistent tight-binding calculations. This method is based on an exact decomposition of the grand-canonical potential for independent fermions suitable for linear-scaling electronic structure calculations. We observe that at lower collision velocities, the buckyball is rebound from the graphite surface, but that starting from 50 km/s chemisorption processes are occurring that causes the buckyball to sticks to the topmost graphene layer. These outcomes are in stark contrast to previous simulations using empirical interaction potentials, but in very good agreement with experimental measurements.

Corrigendum: A perspective on the density matrix purification for linear scaling electronic structure calculations

Article First Published online: 7 December 2015 DOI: 10.1002/qua.25305 [Article in Int. J. Quantum Chem. 2016, 116, 563-568, DOI: 10.1002/qua.25048] When the article cited above was originally published online, the funding source was not acknowledged. The correct funding statement can be seen below. We acknowledge the support through the National Research Foundation of Korea from the Korean Government (NRF-2015R1A2A1A15055539). The authors apologize for the mistake. This error does not change the scientific conclusions of the article in any way.

A perspective on the density matrix purification for linear scaling electronic structure calculations

Density matrix purification is a linear scaling algorithm to solve the self‐consistent field (SCF) electronic structure problem. A brief review is made of the two usages of the purification method in SCF calculations, namely in direct energy minimization and for iterative calculation of density matrix from Hamiltonian. Density matrix purification can be viewed as a minimization algorithm of the particularly defined objective function such as , and hence, this objective function plays a key role in the developments of purification polynomials. Indeed, in this perspective, we show that a carefully chosen objective function can lead to significantly improved algorithms and numerical results. © 2015 Wiley Periodicals, Inc.

How does it become possible to treat delocalized and/or open-shell systems in fragmentation-based linear-scaling electronic structure calculations? The case of the divide-and-conquer method

The authors have developed a fragmentation-based linear-scaling electronic structure calculation strategy named the divide-and-conquer (DC) method, which has been implemented into the Gamess program package. Although there are many sorts of fragmentation-based linear-scaling schemes, most of them require the charge and spin multiplicity of each fragment a priori. Therefore, their applications to delocalized and/or open-shell systems have been limited. However, the DC method is a notable exception because the distribution of electrons in the entire system is automatically determined by the universal Fermi level. In this perspective, the authors have summarized the performance of the linear-scaling self-consistent field (SCF) and post-SCF calculations of delocalized and/or open-shell systems based on the DC method. Furthermore, some future prospects of the method have been discussed.

Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix.

As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this letter, there is room for improvements. The key is to allow for nonmonotonicity in the recursive polynomial expansion. On the basis of this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.

Assessment of density matrix methods for linear scaling electronic structure calculations

Purification and minimization methods for linear scaling computation of the one-particledensity matrix for a fixed Hamiltonian matrix are compared. This is done by consideringthe work needed by each method to achieve a given accuracy in terms of the difference fromthe exact solution. Numerical tests employing orthogonal as well as non-orthogonal versionsof the methods are performed using both element magnitude and cutoff radius basedtruncation approaches. It is investigated how the convergence speed for the differentmethods depends on the eigenvalue distribution in the Hamiltonian matrix. An expressionfor the number of iterations required for the minimization methods studied is derived,taking into account the dependence on both the band gap and the chemical potential. Thisexpression is confirmed by numerical tests. The minimization methods are found toperform at their best when the chemical potential is located near the center ofthe eigenspectrum. The results indicate that purification is considerably moreefficient than the minimization methods studied even when a good starting guessfor the minimization is available. In test calculations without a starting guess,purification is more than an order of magnitude more efficient than minimization.

Bringing about matrix sparsity in linear‐scaling electronic structure calculations

The performance of linear-scaling electronic structure calculations depends critically on matrix sparsity. This article gives an overview of different strategies for removal of small matrix elements, with emphasis on schemes that allow for rigorous control of errors. In particular, a novel scheme is proposed that has significantly smaller computational overhead compared with the Euclidean norm-based truncation scheme of Rubensson et al. (J Comput Chem 2009, 30, 974) while still achieving the desired asymptotic behavior required for linear scaling. Small matrix elements are removed while ensuring that the Euclidean norm of the error matrix stays below a desired value, so that the resulting error in the occupied subspace can be controlled. The efficiency of the new scheme is investigated in benchmark calculations for water clusters including up to 6523 water molecules. Furthermore, the foundation of matrix sparsity is investigated. This includes a study of the decay of matrix element magnitude with distance between basis function centers for different molecular systems and different methods. The studied methods include Hartree–Fock and density functional theory using both pure and hybrid functionals. The relation between band gap and decay properties of the density matrix is also discussed.

Linear scaling electronic structure calculations with numerical atomic basis set

We discuss in this review recent progress, especially by our group, on linear scaling algorithms for electronic structure calculations with numerical atomic basis sets. The principles of the construction of numerical basis sets and the Hamiltonian are introduced first. Then we discuss how to solve the single-electron equation self-consistently, and how to obtain electronic properties via post-self-consistent-field processes in a linear scaling way. The linear response calculation with linear scaling is also introduced. Numerical implementation is emphasized, with some applications presented for demonstration purposes.

Optimization of selected molecular orbitals in group basis sets

We derive a local basis equation which may be used to determine the orbitals of a group of electrons in a system when the orbitals of that group are represented by a group basis set, i.e., not the basis set one would normally use but a subset suited to a specific electronic group. The group orbitals determined by the local basis equation minimize the energy of a system when a group basis set is used and the orbitals of other groups are frozen. In contrast, under the constraint of a group basis set, the group orbitals satisfying the Huzinaga equation do not minimize the energy. In a test of the local basis equation on HCl, the group basis set included only 12 of the 21 functions in a basis set one might ordinarily use, but the calculated active orbital energies were within 0.001 hartree of the values obtained by solving the Hartree-Fock-Roothaan (HFR) equation using all 21 basis functions. The total energy found was just 0.003 hartree higher than the HFR value. The errors with the group basis set approximation to the Huzinaga equation were larger by over two orders of magnitude. Similar results were obtained for PCl(3) with the group basis approximation. Retaining more basis functions allows an even higher accuracy as shown by the perfect reproduction of the HFR energy of HCl with 16 out of 21 basis functions in the valence basis set. When the core basis set was also truncated then no additional error was introduced in the calculations performed for HCl with various basis sets. The same calculations with fixed core orbitals taken from isolated heavy atoms added a small error of about 10(-4) hartree. This offers a practical way to calculate wave functions with predetermined fixed core and reduced base valence orbitals at reduced computational costs. The local basis equation can also be used to combine the above approximations with the assignment of local basis sets to groups of localized valence molecular orbitals and to derive a priori localized orbitals. An appropriately chosen localization and basis set assignment allowed a reproduction of the energy of n-hexane with an error of 10(-5) hartree, while the energy difference between its two conformers was reproduced with a similar accuracy for several combinations of localizations and basis set assignments. These calculations include localized orbitals extending to 4-5 heavy atoms and thus they require to solve reduced dimension secular equations. The dimensions are not expected to increase with increasing system size and thus the local basis equation may find use in linear scaling electronic structure calculations.

Augmented Orbital Minimization Method for Linear Scaling Electronic Structure Calculations

We present a novel algorithm which can overcome the drawbacks of the conventional linear scaling method with minimal computational overhead. This is achieved by introducing additional constraints, thus eliminating the redundancy of the orbitals. The performance of our algorithm is evaluated in ab initio molecular-dynamics simulations as well as in a model system.

Linear scaling electronic structure calculations and accurate statistical mechanics sampling with noisy forces

Numerical simulations based on electronic structure calculations are finding ever growing applications in many areas of physics. A major limiting factor is however the cubic scaling of the algorithms used. Building on previous work [F. R. Krajewski and M. Parrinello, Phys.Rev. B71, 233105 (2005)] we introduce a novel statistical method for evaluating the inter-atomic forces which scales linearly with system size and is applicable also to metals. The method is based on exact decomposition of the fermionic determinant and on a mapping onto a field theoretical expression. We solve exactly the problem of sampling the Boltzmann distribution with noisy forces. This novel approach can be used in such diverse fields as quantum chromodynamics, quantum Monte Carlo or colloidal physics.

Density Functional Study of the In-Line Mechanism of Methanolysis of Cyclic Phosphate and Thiophosphate Esters in Solution: Insight into Thio Effects in RNA Transesterification

Density functional calculations of thio effects on the in-line mechanism of methanolysis of ethylene phosphate (a reverse reaction model for RNA phosphate transesterification) are presented. A total of 12 reaction mechanisms are examined using the B3LYP functional with large basis sets, and the effects of solvation were treated using the PCM, CPCM, and SM5 solvation models. Single thio substitutions at all of the distinct phosphoryl oxygen positions (2', 3', 5', pro-R) and a double thio substitution at the nonbridging (pro-R/pro-S) positions were considered. Profiles for each reaction were calculated in the dianionic and monoanionic/monoprotic states, corresponding to reaction models under alkaline and nonalkaline conditions, respectively. These models provide insight into the mechanisms of RNA transesterification thio effects and serve as a set of high-level quantum data that can be used in the design of new semiempirical quantum models for hybrid quantum mechanical/molecular mechanical simulations and linear-scaling electronic structure calculations.

Preconditioned iterative minimization for linear-scaling electronic structure calculations

Linear-scaling electronic structure methods are essential for calculations on large systems. Some of these approaches use a systematic basis set, the completeness of which may be tuned with an adjustable parameter similar to the energy cut-off of plane-wave techniques. The search for the electronic ground state in such methods suffers from an ill-conditioning which is related to the kinetic contribution to the total energy and which results in unacceptably slow convergence. We present a general preconditioning scheme to overcome this ill-conditioning and implement it within our own first-principles linear-scaling density functional theory method. The scheme may be applied in either real space or reciprocal space with equal success. The rate of convergence is improved by an order of magnitude and is found to be almost independent of the size of the basis.

Sparse matrix multiplications for linear scaling electronic structure calculations in an atom-centered basis set using multiatom blocks.

A sparse matrix multiplication scheme with multiatom blocks is reported, a tool that can be very useful for developing linear-scaling methods with atom-centered basis functions. Compared to conventional element-by-element sparse matrix multiplication schemes, efficiency is gained by the use of the highly optimized basic linear algebra subroutines (BLAS). However, some sparsity is lost in the multiatom blocking scheme because these matrix blocks will in general contain negligible elements. As a result, an optimal block size that minimizes the CPU time by balancing these two effects is recovered. In calculations on linear alkanes, polyglycines, estane polymers, and water clusters the optimal block size is found to be between 40 and 100 basis functions, where about 55-75% of the machine peak performance was achieved on an IBM RS6000 workstation. In these calculations, the blocked sparse matrix multiplications can be 10 times faster than a standard element-by-element sparse matrix package.

Multiresolution density-matrix approach to electronic structure calculations

A multiresolution density-matrix wavelet approach to electronic-structure calculations is proposed. A separable multidimensional biorthogonal interpolating multiwavelet and scaling representation of the Hamiltonian operator is introduced in which individual operator elements can be calculated locally at any scale. Issues regarding this representation are discussed, such as its low complexity in higher dimensions and its direct relation to finite-difference schemes. The density matrix is calculated via polynomial expansions in terms of the Hamiltonian. The expansions are improvements of the McWeeny purification within a grand canonical ensemble and are shown to be up to 20% more efficient in the number of matrix-matrix multiplications. The efficiency of the multiresolution representation of the density matrix compared to a real-space representation is analyzed. Within the multiresolution wavelet representation it is shown how the sparsity of the density matrix is preserved for localized insulating systems as well as for itinerant metallic systems. This is not possible within a real-space representation of the density matrix which has a very slow decay of its elements in the metallic phase. This makes the multiresolution approach very attractable for linear- or close to linear-scaling electronic-structure calculations.

Comment on: “Transforms for idempotency purification of density matrices in linear-scaling electronic-structure calculations” [Chem. Phys. Lett. 340 (2001) 552–558

In a recent Letter, Holas introduced nth-order generalizations of the McWeeny canonical purification scheme, hoping that these might be useful in linear-scaling electronic structure theory methods that rely on purification of density matrices. In this comment we show that for theories that use purification to guide unconstrained optimizations, the generalizations offer no advantage.

Parallel sparse matrix multiplication for linear scaling electronic structure calculations

Linear-scaling electronic-structure techniques, also called O(N) techniques, rely heavily on the multiplication of sparse matrices, where the sparsity arises from spatial cut-offs. In order to treat very large systems, the calculations must be run on parallel computers. We analyze the problem of parallelizing the multiplication of sparse matrices with the sparsity pattern required by linear-scaling techniques. We show that the management of inter-node communications and the effective use of on-node cache are helped by organizing the atoms into compact groups. We also discuss how to identify a key part of the code called the ‘multiplication kernel’, which is repeatedly invoked to build up the matrix product, and explicit code is presented for this kernel. Numerical tests of the resulting multiplication code are reported for cases of practical interest, and it is shown that their scaling properties for systems containing up to 20,000 atoms on machines having up to 512 processors are excellent. The tests also show that the CPU efficiency of our code is satisfactory.

Transforms for idempotency purification of density matrices in linear-scaling electronic-structure calculations

A family of purification transforms (of the nth-order convergence, n>2) is proposed, which is a generalization of the McWeeny transform (of the second-order convergence). These transforms can be applied in calculational schemes where the number of basis functions exceeds the number of occupied orbitals in the density matrix. Another family of purification transforms, proposed recently by Kryachko [Chem. Phys. Lett. 318 (2000) 210], is shown to be applicable in such schemes only where these numbers are equal. Various properties of these two families are discussed.