Comparison of self-consistent field convergence acceleration techniques

The recently proposed ADIIS and LIST methods for accelerating self-consistent field (SCF) convergence are compared to the previously proposed energy-DIIS (EDIIS) + DIIS technique. We here show mathematically that the ADIIS functional is identical to EDIIS for Hartree-Fock wavefunctions. Convergence failures of EDIIS + DIIS reported in the literature are not reproduced with our codes. We also show that when correctly implemented, the EDIIS + DIIS method is generally better than the LIST methods, at least for the cases previously examined in the literature. We conclude that, among the family of DIIS methods, EDIIS + DIIS remains the method of choice for SCF convergence acceleration.     

Improved convergence in block copolymer self-consistent field theory by Anderson mixing

A modification to real space polymeric self-consistent field theory algorithms that greatly improves the convergence properties is presented. The method is based on Anderson mixing [D. G. Anderson, J. Assoc. Comput. Mach. 12, 547 (1965)], and each iteration computed takes negligibly longer to perform than with other methods, but the number of iterations required to reach a high accuracy solution is greatly reduced. No a priori knowledge of possible phases is required to apply this method. We apply our approach to a standard diblock copolymer melt, and demonstrate iteration reductions of more than a factor of 5 in some cases.     

Non self-consistent field theory

Roothaan's SCF method [2] is reformulated so that two non-SCP methods are developed to solve the eigenequation. The results from these methods can be used as starting eigenfunctions for Roothaan's SCF method.

---

Zusammenfassung Die self-consistent field Methode von Roothaan [2] wird neu formuliert: dabei werden zwei Nicht-SCF Methoden zur Lösung der Eigenwertgleichung entwickelt. Deren Resultate können als Anfangsfunktionen bei Rechnungen mit der SCF-Methode von Roothaan benutzt werden.

---

Résumé La méthode SCF de Roothaan [2] est reformulée en développant deux méthodes non-SCF pour la solution de l'équation aux valeurs propres. Les résultats de ces deux méthodes peuvent être utilisées comme fonctions de départ pour la méthode SCF de Roothaan.

---

---

The title Theoretical Chemistry has been transferred to the Division of Physical Chemistry.     

Semidirect parallel self-consistent field: the load balancing problem in the input/output intensive self-consistent field iterations

The full capacity of contemporary parallel computers can, in the context of iterative ab initio procedures like, for example, self-consistent field (SCF) and multiconfigurational SCF, only be utilized if the disk and input/output (I/O) capacity are fully exploited before the implementation turns to an integral direct strategy. In a recent report on parallel semidirect SCF http://www.tc.cornell.edu/er/media/1996/collabrate.html, http://www.fp.mcs.anl.gd/grand-challenges/chem/nondirect/index.html it was demonstrated that super-linear speedups are achievable for algorithms that exploit scalable parallel I/O. In the I/O-intensive SCF iterations of this implementation a static load balancing, however, was employed, dictated by the initial iteration in which integral evaluation dominates the central processing unit activity and thus determines the load balancing. In the present paper we present the first implementation in which load balancing is achieved throughout the whole SCF procedure, i.e. also in subsequent iterations. The improved scalability of our new algorithm is demonstrated in some test calculations, for example, for 63-node calculation a speedup of 104 was observed in the computation of the two-electron integral contribution to the Fock matrix.Contribution to the Björn Roos Honorary Issue Acknowledgement.We thank J. Nieplocha for valuable help and making the toolkit (including ChemIO) available to us. R.L. acknowledges the Intelligent Modeling Laboratory and the University of Tokyo for financial support during his stay in Japan.     

Size-extensive vibrational self-consistent field method

The vibrational self-consistent field (VSCF) method is a mean-field approach to solve the vibrational Schro?dinger equation and serves as a basis of vibrational perturbation and coupled-cluster methods. Together they account for anharmonic effects on vibrational transition frequencies and vibrationally averaged properties. This article reports the definition, programmable equations, and corresponding initial implementation of a diagrammatically size-extensive modification of VSCF, from which numerous terms with nonphysical size dependence in the original VSCF equations have been eliminated. When combined with a quartic force field (QFF), this compact and strictly size-extensive VSCF (XVSCF) method requires only quartic force constants of the ?(4)V/?Q(i)(2)?Q(j)(2) type, where V is the electronic energy and Q(i) is the ith normal coordinate. Consequently, the cost of a XVSCF calculation with a QFF increases only quadratically with the number of modes, while that of a VSCF calculation grows quartically. The effective (mean-field) potential of XVSCF felt by each mode is shown to be harmonic, making the XVSCF equations subject to a self-consistent analytical solution without matrix diagonalization or a basis-set expansion, which are necessary in VSCF. Even when the same set of force constants is used, XVSCF is nearly three orders of magnitude faster than VSCF implemented similarly. Yet, the results of XVSCF and VSCF are shown to approach each other as the molecular size is increased, implicating the inclusion of unnecessary, nonphysical terms in VSCF. The diagrams of the XVSCF energy expression and their evaluation rules are also proposed, underscoring their connected structures.     

Basis set selections for relativistic self-consistent field calculations

Practical methods of generating reliable and economic basis sets for relativistic self-consistent fields (RSCF) calculations are developed. Large component basis sets are generated from constrained optimizations of exponents in the nonrelativistic atomic calculations for light atoms. For heavy atoms, large component basis sets for inner core orbitals are generated by fitting numerical atomic spinors of Dirac-Hartree-Fock calculations with appropriate number of Slater-type functions. Small component basis sets are obtained by using the kinetic balance condition and other computational criteria. With judicious selections of the basis sets, virtual orbitals in RSCF calculations become very similar to those in nonrelativistic calculations, implying that relativistic virtual orbitals can be used in electron correlation calculations in the same manner as the conventional nonrelativistic virtual orbitals. It is also evident that the Koopmans' theorem is also valid in RSCF results.     

Complete multi-configuration self-consistent field theory

The two-configuration self-consistent field formalism previously presented in this Journal is extended and the CMC SCF LCAO MO (complete multi-configuration self-consistent field LCAO MO) technique is presented. The single Slater determinant for a 2n electron system is replaced by a combination of determinants built from two sets of MO's, one containingn orbitals; the second, ( —n) orbitals. All the possible double excitations from the (n) set to the ( —n) set are considered. The orbitals as well as the linear combination of determinants are simultaneously optimized making use of the self-consistent field technique.

---

Zusammenfassung Es wird die Methode des self-consistent field für eine Gesamtheit mehrerer Konfigurationen in der LCAO-Näherung entwickelt. Ein 2n-Elektronensystem wird nicht mehr durch eine einzige Slaterdeterminante, sondern durch eine Kombination von Determinanten beschrieben, die aus zwei Sätzen von Molekülfunktionen mitn bzw. ( —n) Orbitalen aufgebaut werden. Alle möglichen zweifachen Anregungen vom (n) zum ( —n) Satz werden berücksichtigt. Mit Hilfe der SCF-Teehnik werden sowohl die Orbitale als auch die Kombination der Determinanten gleichzeitig optimiert.

---

Résumé On expose la méthode du champ self-consistant pour un ensemble complet de plusieurs configurations, dans l'approximation LCAO-MO (CMC SCF LCAO MO). Le déterminant de Slater pour un système de 2n électrons est remplacé par une combinaison de déterminants construits a partir de deux ensembles d'orbitales moléculaires, l'un contenant (n) orbitales et l'autre (-n) orbitales. On considère toutes les doubles excitations possibles, de l'ensemble (n) à l'ensemble (-n). La technique du champ self-consistant permet d'optimiser simultanément les orbitales ainsi que les coefficients dans la combinaison linéaire de determinants. La méthode CMC SCF tient plainement compte de la corrélation associée à chaque paire d'électrons et fait intervenir toutes les interactions paire-paire. L'optimisation simultanée des orbitales des deux ensembles (n) et (-n) garantit une convergence rapide.

     

Symmetry broken self-consistent field approximation for coupled oscillators

A self-consistent field approximation allowing for a change of coordinates is applied to a model of two coupled oscillators considered by Davis and Heller [J. Chem. Phys. 75 , 246 (1981)]. The model accounts well for the unsymmetrical nonstationary states involved in the quantum dynamical tunneling phenomenon.     

New vibrational self-consistent field program for large molecules

A recently developed, general computer program that performs vibrational self-consistent field (VSCF) calculations for large molecules is described. The program, which we refer to as VSCF―95, requires as its only input a force field in mass-scaled normal coordinates. Currently, it is limited to a maximum of 200 normal modes, and the force field is limited to coupling terms involving a maximum of six normal modes, with a maximum order of six in any normal mode. As output the program returns VSCF energies for specified quantum states. We illustrate the code with two new applications. The first is to HCO, for which we use a full sixth-order force field. The second is to a model of the fullerene, C₆₀, for which we have calculated a 75,731-term force field, which includes all anharmonic terms up to fifth order, and all two-mode coupling terms up to fourth order. © 1996 by John Wiley & Sons, Inc.     

Simplified time-dependent self-consistent field approximation for intramolecular dynamics

The time-dependent Hartree approximation is applied to intramolecular dynamics of polyatomics with smooth, locally quadratic potential surfaces. It is shown that the full quantum solution is obtained from a certain single self-consistent trajectory. An extremely simple model results, pertinent to intramolecular energy transfer, vibrational lineshapes and unimolecular decay.     

NDDO fragment self-consistent field approximation for large electronic systems

A semi-empirical NDDO method, generalized from a similar scheme at the CNDO/2 level developed previously, is presented to treat very large molecules. The extended molecular system is divided into a relatively small subsystem where substantial chemical changes take place and an environment remaining more-or-less unperturbed during the process. Expanding the wave function on an atomic hybrid basis an SCF procedure is performed for the subsystem in the field of the iteratively determined electronic distribution of the environment. A computer program has been written for the IBM RISC System/6000 530 computer and several test calculations were done for a variety of large classical molecules, like substituted aliphatic hydrocarbons, water oligomers, and a heptapeptide. Protonation energies, proton transfer potential curves, rotational barriers, atomic net charges, and HOMO and LUMO energies, as computed by the exact version of the NDDO method, are fairly well reproduced by our approximation if the subsystem is appropriately defined. © 1992 by John Wiley & Sons, Inc.     

The modified hartree-fock self-consistent field equation

The electron density at the pyridine nitrogen atom and the π-electron density of the pyridine ring were calculated by the CNDO /2-MO method assuming standard bond lengths and angles. The indices were found to correlate with the pK_a values of pyridine derivatives and with the electronic substituent constants of the Hammett type. The correlations were best for four-substituted pyridines, making the routine CNDO /2-MO calculations useful for studies of quantitative relations between structure and biological activity of the compounds.     

Coarse-grained force field for the nucleosome from self-consistent multiscaling

A coarse-grained simulation model for the nucleosome is developed, using a methodology modified from previous work on the ribosome. Protein residues and DNA nucleotides are represented as beads, interacting through harmonic (for neighboring) or Morse (for nonbonded) potentials. Force-field parameters were estimated by Boltzmann inversion of the corresponding radial distribution functions obtained from a 5-ns all-atom molecular dynamics (MD) simulation, and were refined to produce agreement with the all-atom MD simulation. This self-consistent multiscale approach yields a coarse-grained model that is capable of reproducing equilibrium structural properties calculated from a 50-ns all-atom MD simulation. This coarse-grained model speeds up nucleosome simulations by a factor of 10(3) and is expected to be useful in examining biologically relevant dynamical nucleosome phenomena on the microsecond timescale and beyond.     

Incorrectness of the self-consistent field approximation for the argon crystal

Division of Physicomechanical Problems in Power Engineering, Urals Branch, Academy of Sciences of the USSR. Translated from Zhurnal Strukturnoi Khimii, Vol. 29, No. 2, pp. 178–179, March–April, 1988.     

Semiclassical self-consistent field perturbation theory for the hydrogen atom in a magnetic field

A recently developed perturbation theory for solving self-consistent field equations is applied to the hydrogen atom in a strong magnetic field. This system has been extensively studied using other methods and is therefore a good test case for the new method. The perturbation theory yields summable large-order expansions. The accuracy of the self-consistent field approximation varies according to field strength and quantum state but is often higher than the accuracy from adiabatic approximations. A new derivation is presented for the asymptotic adiabatic approximation, the most useful of the adiabatic approaches. This derivation uses semiclassical perturbation theory without invoking an adiabatic hypothesis. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 183–192, 1998     

Accurate ab initio density fitting for multiconfigurational self-consistent field methods

Using Cholesky decomposition and density fitting to approximate the electron repulsion integrals, an implementation of the complete active space self-consistent field (CASSCF) method suitable for large-scale applications is presented. Sample calculations on benzene, diaquo-tetra-mu-acetato-dicopper(II), and diuraniumendofullerene demonstrate that the Cholesky and density fitting approximations allow larger basis sets and larger systems to be treated at the CASSCF level of theory with controllable accuracy. While strict error control is an inherent property of the Cholesky approximation, errors arising from the density fitting approach are managed by using a recently proposed class of auxiliary basis sets constructed from Cholesky decomposition of the atomic electron repulsion integrals.     

Globally convergent trust-region methods for self-consistent field electronic structure calculations

As far as more complex systems are being accessible for quantum chemical calculations, the reliability of the algorithms used becomes increasingly important. Trust-region strategies comprise a large family of optimization algorithms that incorporates both robustness and applicability for a great variety of problems. The objective of this work is to provide a basic algorithm and an adequate theoretical framework for the application of globally convergent trust-region methods to electronic structure calculations. Closed shell restricted Hartree-Fock calculations are addressed as finite-dimensional nonlinear programming problems with weighted orthogonality constraints. A Levenberg-Marquardt-like modification of a trust-region algorithm for constrained optimization is developed for solving this problem. It is proved that this algorithm is globally convergent. The subproblems that ensure global convergence are easy-to-compute projections and are dependent only on the structure of the constraints, thus being extendable to other problems. Numerical experiments are presented, which confirm the theoretical predictions. The structure of the algorithm is such that accelerations can be easily associated without affecting the convergence properties.     

Continuous polydispersity in a self-consistent field theory for diblock copolymers

An efficient algorithm is presented for numerically evaluating a self-consistent field theoretic (SCFT) model of an AB diblock copolymer that incorporates continuous polydispersity in one of the blocks. An interesting segregation effect is found in which chains of intermediate molecular weight are concentrated at domain interfaces. This model of continuous polydispersity is also implemented in the random phase approximation (RPA) to study the order-disorder transition and predicts that the stability of the disordered, homogeneous phase decreases as the polydispersity in one of the blocks increases. The RPA predictions are confirmed by SCFT calculations. Our approach and results are particularly relevant to block copolymers prepared by quasiliving synthesis techniques, where the polymerization of one block is much more controlled than the other block.