SCF Algorithm Research Articles

An inexact Matrix-Newton method for solving NEPv

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a Newton-based method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.

Design of bitstream-based adaptive-connected array architecture for mmWave Multiuser MIMO systems

We investigate a bitstream-based adaptive-connected massive multiple-input multiple-output (MIMO) architecture that trades off between high-power full-connected and low-performance sub-connected hybrid precoding architectures. The proposed adaptive-connected architecture which enables each data stream to be computed independently and in parallel, consists of fewer phase shifters (PS) and switches than the other adaptive-connected architectures. With smaller array groups, the proposed architecture uses fewer PS and switches, so that its power consumption gradually decreases in millimeter wave (mmWave) Multiuser MIMO (MU-MIMO) system. To fully demonstrate the performance of the proposed architecture in mmWave MU-MIMO system with practical constraints, we combine the connection-state matrix with the hybrid precoders and combiners to maximize energy efficiency (EE) of the system equipped with the proposed architecture. We then propose the hybrid precoding and combining (HPC) scheme suitable for multi-user and multi-data streams which utilizes the SCF algorithm to obtain the constant modulus of the analog precoder at convergence. In the digital precoding and combining stage, the digital precoder and combiner are designed to reduce the amount of computation by utilizing the singular value decomposition (SVD) of corresponding equivalent channel. In the mmWave MU-MIMO-OFDM system equipped with the proposed architecture, with the increase of the total number of data streams, simulation results demonstrate that we can exploit the proposed HPC scheme to achieve better EE than the traditional hybrid full-connected architecture exploiting some existing schemes.

Holomorphic Hartree-Fock Theory: An Inherently Multireference Approach.

We investigate the existence of holomorphic Hartree–Fock solutions using a revised SCF algorithm. We use this algorithm to study the Hartree–Fock solutions for H2 and H42+ and report the emergence of holomorphic solutions at points of symmetry breaking. Finally, we find these holomorphic solutions for H4 and use them as a basis for Non-Orthogonal Configuration Interaction at a range of rectangular geometries and show them to produce energies in good agreement with Full Configuration Interaction.

Conformational analysis and parallel QM/MM X-ray refinement of protein bound anti-Alzheimer drug donepezil.

The recognition and association of donepezil with acetylcholinesterase (AChE) has been extensively studied in the past several decades because of the former's use as a palliative treatment for mild Alzheimer disease. Herein we examine the conformational properties of donepezil and we re-examine the donepezil-AChE crystal structure using combined quantum mechanical/molecular mechanical (QM/MM) X-ray refinement tools. Donepezil's conformational energy surface was explored using the M06 suite of density functionals and with the MP2/complete basis set (CBS) method using the aug-cc-pVXZ (X = D and T) basis sets. The donepezil-AChE complex (PDB 1EVE) was also re-refined through a parallel QM/MM X-ray refinement approach based on an in-house ab initio code QUICK, which uses the message passing interface (MPI) in a distributed SCF algorithm to accelerate the calculation via parallelization. In the QM/MM re-refined donepezil structure, coordinate errors that previously existed in the PDB deposited geometry were improved leading to an improvement of the modeling of the interaction between donepezil and the aromatic side chains located in the AChE active site gorge. As a result of the re-refinement there was a 93% reduction in the donepezil conformational strain energy versus the original PDB structure. The results of the present effort offer further detailed structural and biochemical inhibitor-AChE information for the continued development of more effective and palliative treatments of Alzheimer disease.

Restrained optimization of broken‐symmetry determinants

AbstractThe goal of this work is to provide an overview on how Lagrange multipliers can be used in self‐consistent‐field algorithms for convergence control. In particular, a restrained optimization scheme is proposed with the purpose to guide any set of initial guess molecular orbitals to those of a Slater determinant which (1) has certain expectation values (i.e., molecular properties) within a predefined range and (2) corresponds to a true (local) energy minimum, if such a minimum exists within the property range chosen. The most important practical application of such an algorithm is the calculation of minimum‐energy broken‐symmetry determinants, which are often used in Kohn‐Sham density functional calculations to model antiferromagnetically coupled states. This is achieved by (a) combining a constrained optimization with a subsequent optimization where the constraints are released, i.e., the Lagrange multipliers are set to zero. Alternatively, (b) a modified scheme may be employed, where the constraints are released as soon as the energy starts to rise in the SCF algorithm. Both alternatives are combined with both intermediate and final automatic control of whether the properties of interest are within the desired range. A parameter study using scheme (a) is carried out for a set of small test systems. Applications to a synthetically available trinuclear copper complex and to Crabtree and Brudvig's water‐oxidizing complex demonstrate the value and limitations of restrained optimization. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Large scale Hartree–Fock calculations. Compression of two-electron integrals and their indices

The new efficient algorithm for a compression of the two-electron repulsion integrals and their indices is proposed and employed in the conventional SCF algorithm of the Hartree–Fock method. Results of numerical investigations of this algorithm obtained in the molecular Hartree–Fock calculations including the largest known one's are presented and discussed.

The distributed data SCF

This paper describes a distributed data parallel SCF algorithm. The distinguishing features of this algorithm are: (a) columns of density and Fock matrices are distributed evenly among processors, (b) pair-wise dynamic load balancing is developed to achieve excellent load balance, (c) network communication time is minimized via careful analysis of data flow in the SCF algorithm. The developed performance models and benchmarking results illustrate good performance of the distributed data SCF algorithm.

A BSSE-free SCF algorithm for intermolecular interactions. IV. Generalization for open-shell systems

The simple and efficient a priori BSSE-free SCF method (CHA/F) proposed in the previous parts of this series is extended to the case of open-shell systems treated at the single-determinant UHF level of theory. The appropriate equations were derived and sample calculations are presented on three different (H3+(DOTTED BOND)H, H2O(DOTTED BOND)H, and CH4(DOTTED BOND)H) systems. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 151–158, 1998

Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations

The Roothaan equations have been modified to compute molecular interactions between weakly bonded systems at the SCF level of theory without the basis set superposition error (BSSE). The increase in complication with respect to the usual SCF algorithm is negligible. Calculation of the SCF energy on large systems, such as nucleic acid pairs, does not pose any computational problem. At the same time, it is shown that a modest change in basis-set quality from 3-21G to 6-31G changes the binding energy by about 50% when computed according to standard SCF “supermolecule” techniques, while remaining practically constant when computed without introducing BSSE. Bader analysis shows that the amount of charge transferred between the interacting units is of the same order of magnitude when performed on standard SCF wave functions and those computed using the new method. The large difference between the corresponding computed energies is thus ascribed to the BSSE. © 1996 John Wiley & Sons, Inc.

A BSSE-free SCF algorithm for intermolecular interactions. III. Generalization for three-body systems and for using bond functions

The simple BSSE-free SCF method (CHA/F) introduced in the previous parts of this series is extended to the case of three subsystems, which may be either three weakly interacting molecules or a bimolecular system described by using bond functions. The CHA/F formalism is formulated in a more transparent manner, indicating also a straightforward way for generalization to the case of an arbitrary number of subsystems. The illustrative calculations show the viability of using the CHA/F scheme for three-component systems. © 1996 John Wiley & Sons, Inc.

BSSE-Free SCF Algorithm for Treating Several Weakly Interacting Systems

The BSSE-free SCF procedure “CHA/F” for calculating intermolecular interactions has been generalized to the case when more than two subsystems are involved. This permits the calculations to be performed for systems consisting of several weakly interacting molecules, and it offers the possibility of introducing “bond functions” in the CHA framework. Sample calculations are presented for hydrogen fluoride chains of different lengths, which show the “collective effect” of the hydrogen bonding and its saturation as the chain length increases.

SCF calculations on MIMD type parallel computers

One of the key methods in quantum chemistry, the Hartree-Fock SCF method, is performing poorly on typical vector supercomputers. A significant acceleration of calculations of this type requires the development and implementation of a parallel SCF algorithm. In this paper various parallelization strategies are discussed comparing local and global communication management as well as sequential and distributed Fock-matrix updates. Programs based on these algorithms are bench marked on transputer networks and two IBM MIMD prototypes. The portability of the code is demonstrated with the portation of the initial Helios version to other operating systems like Parallel VM/SP and PARIX. Based on the PVM libraries, a platform-independent version has been developed for heterogeneous workstation clusters as well as for massively parallel computers.

Network supercomputing: A distributed-concurrent direct SCF scheme

The direct SCF algorithm has been parallelized such that the computation can be split over a number of computers. The server (“slave”) machines use all their processors in parallel to execute the tasks distributed by the client (“master”), a mode of operation best described asdistributed-concurrent parallel computing. Relatively high speedups resulting in performance of several GigaFLOPS have been demonstrated in realistic applications with clusters of Cray computers in dedicated mode. Since the communication and synchronization requirements are modest, machines that are accessible via communication services like Internet can be networked. GigaFLOPS performances during regular production hours have been obtained when combining computers located at different centers, even if these were on both sides of the Atlantic.

A BSSE‐free SCF algorithm for intermolecular interactions. II. Sample calculations on hydrogen‐bonded complexes

AbstractThe usefulness and reliability of the recent BSSE‐free SCF algorithm based on the “chemical Hamiltonian approach” (CHA/F) is demonstrated by calculating potential curves for several hydrogen‐bonded complexes with 4‐31G, 6‐31G, and 6‐31G** basis sets. It is concluded that the CHA/F scheme gives results that are numerically close to those of the Boys–Bernardi a posteriori correction scheme but are free from the “overcompensation” characteristic of the latter at smaller distances and given basis sets. © 1992 John Wiley & Sons, Inc.

Principles of AB initio SCF calculations with minimal storage requirements

A new SCF algorithm which allows substantial savings of the external storage and the IO time, while increasing the CPU time by only a small amount, is described. The technique is based on splitting the two-electron integrals into approximate parts and the corrections. Only the corrections that are large enough are stored. The approximate integrals are rapidly recomputed in every SCF cycle. Partitioning of the integrals is made possible by the multipole annihilation technique. The approximate integrals are calculated from a limited number of terms of the multipole expansion. The remainder is shown to decay extremely rapidly with the increasing distance between orbitals. A numerical example is given which shows that a tenfold reduction of the external storage can be easily achieved.

Ab initio calculations on the hydration of dimethylpyrazole and indazole. Solvent effects on tautomeric energies.

The effect of hydration upon the energies of the tautomeric pairs of indazoles and dimethylpyrazoles was investigated (i) by attaching one water molecule to each nitrogen, and (ii) by the continuum model. In the calculation (i) the STO-3G basis set was used. The geometry optimization resulted in strongly hydrogen bonded cyclic structures with notable co-operative effect. This was explored in more detail with 3,4-dimethylpyrazole hydrate using the 3-21G basis set and correction for the basis set superposition error. In the calculation (ii) on the indazole pair the 6-31G basis set was used together with the continuum based SCF algorithm of Tomasi and co-workers [1]. The results of both approaches are in good agreement. The tautomeric energy of the indazole tautomers calculated by the extended basis set (10 kcal mol −1) is 2.9 kcal smaller than that obtained from STO-3G calculations. The continuum solvent effects reduces it to 5 kcal mol −1.

ChemInform Abstract: The Influence of the Basis Set on the Evaluation of Conformational Energies for Small Organic Solutes in Aqueous Solutions

Abstract Conformational energies of a small set of double rotor molecules (CH 3 CH 2 OH, CH 3 CHFOH(S), CH 2 FCH 2 OH, NH 2 CH 2 OH, CH 3 CH 2 OF) computed in vacuo and in aqueous solution with three basis sets (STO-3G, 3-21G, 4-31G) are compared in order to examine the basis set dependence of the results. Solvent effects are introduced with an SCF algorithm which relies on a continuum description of the solvent. The interaction operator depends on the description of the solute charge distribution and, consequently, on the basis set employed in the calculations.

The influence of the basis set on the evaluation of conformational energies for small organic solutes in aqueous solutions

Conformational energies of a small set of double rotor molecules (CH 3CH 2OH, CH 3CHFOH(S), CH 2FCH 2OH, NH 2CH 2OH, CH 3CH 2OF) computed in vacuo and in aqueous solution with three basis sets (STO-3G, 3-21G, 4-31G) are compared in order to examine the basis set dependence of the results. Solvent effects are introduced with an SCF algorithm which relies on a continuum description of the solvent. The interaction operator depends on the description of the solute charge distribution and, consequently, on the basis set employed in the calculations.

Intrinsic convergence in closed-shell SCF calculations. A general criterion

It is shown that the intrinsic convergence of the classical SCF algorithm for closed shells is governed by the eigenvalues of a supermatrix Q whose elements involve orbital excitation energies and electron repulsion integrals over occupied and virtual orbitals. In general, the SCF process is intrinsically convergent if all eigenvalues of Q have absolute values less than unity, and intrinsically divergent if one ore more is greater than unity. It is also shown how the symmetry preserving features of the SCF algorithm enable one to avoid certain divergences provided one starts a calculation with symmetry adapted orbitals. In cases of intrinsic convergence it is proved that the ratio of successive energy increments produced by the SCF algorithm is equal to the square of the largest eigenvalue of Q. The implications of the theory are illustrated by calculations on linear H8, on the cyclic planar ion S4N3+, and on model Q matrices filled with random numbers over the interval (−0.5,0.5).