The trust-region self-consistent field method: towards a black-box optimization in Hartree-Fock and Kohn-Sham theories

The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E(SCF) of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E(SCF) may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E(SCF) but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.     

Linear-scaling implementation of molecular electronic self-consistent field theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.     

An analysis for the DIIS acceleration method used in quantum chemistry calculations

This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and Hartree–Fock calculations and in the Coupled Cluster method. Taking up results from Harrison (J Comput Chem 25:328, 2003), we show that for the general nonlinear case, DIIS corresponds to a projected quasi-Newton/secant method. For linear systems, we establish connections to the well-known GMRES solver and transfer according (positive as well as negative) convergence results to DIIS. In particular, we discuss the circumstances under which DIIS exhibits superlinear convergence behaviour. For the general nonlinear case, we then use these results to show that a DIIS step can be interpreted as step of a quasi-Newton method in which the Jacobian used in the Newton step is approximated by finite differences and in which the according linear system is solved by a GMRES procedure, and give according convergence estimates.     

Linear-scaling implementation of molecular response theory in self-consistent field electronic-structure theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field theories for the calculation of frequency-dependent molecular response properties and excitation energies is presented, based on a nonredundant exponential parametrization of the one-electron density matrix in the atomic-orbital basis, avoiding the use of canonical orbitals. The response equations are solved iteratively, by an atomic-orbital subspace method equivalent to that of molecular-orbital theory. Important features of the subspace method are the use of paired trial vectors (to preserve the algebraic structure of the response equations), a nondiagonal preconditioner (for rapid convergence), and the generation of good initial guesses (for robust solution). As a result, the performance of the iterative method is the same as in canonical molecular-orbital theory, with five to ten iterations needed for convergence. As in traditional direct Hartree-Fock and Kohn-Sham theories, the calculations are dominated by the construction of the effective Fock/Kohn-Sham matrix, once in each iteration. Linear complexity is achieved by using sparse-matrix algebra, as illustrated in calculations of excitation energies and frequency-dependent polarizabilities of polyalanine peptides containing up to 1400 atoms.     

Open-shell localized Hartree-Fock method based on the generalized adiabatic connection Kohn-Sham formalism for a self-consistent treatment of excited states

An effective exact-exchange Kohn-Sham approach for the treatment of excited electronic states, the generalized adiabatic connection open-shell localized Hartree-Fock (GAC-OSLHF) method is presented. The GAC-OSLHF method is based on the generalized adiabatic connection Kohn-Sham formalism and therefore capable of treating excited electronic states, which are not the energetically lowest of their symmetry. The method is self-interaction free and allows for a fully self-consistent computation of excited valence as well as Rydberg states. Results for atoms and small- and medium-size molecules are presented and compared to restricted open-shell Hartree-Fock (ROHF) and time-dependent density-functional results as well as to experimental data. While GAC-OSLHF and ROHF results are quite close to each other, the GAC-OSLHF method shows a much better convergence behavior. Moreover, the GAC-OSLHF method as a Kohn-Sham method, in contrast to the ROHF approach, represents a framework which allows also for a treatment of correlation besides an exchange by appropriate functionals. In contrast to the common time-dependent density-functional methods, the GAC-OSLHF approach is capable of treating doubly or multiply excited states and can be easily applied to molecules with an open-shell ground state. On the nodal planes of the energetically highest occupied orbital, the local multiplicative GAC-OSLHF exchange potential asymptotically approaches a different, i.e., nonzero, value than in other regions, an asymptotic behavior which is known from exact Kohn-Sham exchange potentials of ground states of molecules.     

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.     

Density-based Globally Convergent Trust-region Methods for Self-consistent Field Electronic Structure Calculations

A theory of globally convergent trust-region methods for self-consistent field electronic structure calculations that use the density matrices as variables is developed. The optimization is performed by means of sequential global minimizations of a quadratic model of the true energy. The global minimization of this quadratic model, subject to the idempotency of the density matrix and the rank constraint, coincides with the fixed-point iteration. We prove that the global minimization of this quadratic model subject to the restrictions and smaller trust regions corresponds to the solution of level-shifted equations. The precise implementation of algorithms leading to global convergence is stated and a proof of global convergence is provided. Numerical experiments confirm theoretical predictions and practical convergence is obtained for difficult cases, even if their geometries are highly distorted. The reduction of the trust region is performed by a strategy that uses the structure of the energy function providing the algorithm with a nice practical behavior. This framework may be applied to any problem with idempotency constraints and for which the derivative of the objective function is a symmetric matrix. Therefore, application to calculations based both on Hartree–Fock or Kohn–Sham density functional theory are straightforward.     

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.     

Berry phase approach to longitudinal dipole moments of infinite chains in electronic-structure methods with local basis sets

The authors provide a reformulation of the modern theory of polarization for one-dimensional stereoregular polymers, at the level of the single determinant Hartree-Fock and Kohn-Sham methods within a basis set of local orbitals. By starting with localization of one-electron orbitals, their approach naturally arrives to the Berry phases of Bloch orbitals. Then they describe a novel numerical algorithm for evaluation of longitudinal dipole moments, computationally more convenient than those presently implemented within the local basis periodic codes. This method is based on the straightforward evaluation of the usual direct space dipole matrix elements between local orbitals, as well as overlap matrices between wave functions at two neighboring k points of the reciprocal space mesh. The practical behavior of the algorithm and its convergence properties with respect to the k-point mesh density are illustrated in benchmark calculations for water chains and fluorinated trans-polyacetylene.     

Truncation of small matrix elements based on the Euclidean norm for blocked data structures

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.     

Solution of three-dimensional reference interaction site model and hypernetted chain equations for simple point charge water by modified method of direct inversion in iterative subspace

We proposed a modified procedure of the direct inversion in the iterative subspace (DIIS) method to accelerate convergence in the integral equation theory of liquids. We update the DIIS basis vectors at each iterative step by using the approximate residual obtained in the DIIS extrapolation. The procedure is tested by solving the 3-dimensional (3-D) generalization of the reference interaction site model equation together with the hypernetted chain closure, as well as their 1-D version. We calculated the 3-D site distribution of water, represented by the simple point charge model, around one water molecule considered as a central particle. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 928–936, 1999     

A recipe for geometry optimization of diradicalar singlet States from broken-symmetry calculations

The equilibrium geometries of the singlet and triplet states of diradicals may be somewhat different, which may have an influence on their magnetic properties. The single-determinantal methods, such as Hartree-Fock or Kohn-Sham density functional theory, in general rely on broken-symmetry solutions to approach the singlet-state energy and geometry. An approximate spin decontamination is rather easy for the energy of this state but is rarely performed for its geometry optimization. We suggest simple procedures to estimate the optimized geometry and energy of a spin-decontaminated singlet, the accuracies of which are tested on a few organic diradicals. This technique can be generalized to interactions between higher-spin units or to multispin systems.     

A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations

A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in Hartree-Fock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.     

Efficiency and accuracy of the elongation method as applied to the electronic structures of large systems

Current state of development of the elongation method originally proposed by Imamura is presented. Recent progress in methodology, including geometry optimization and employment of the fast multiple method, is highlighted. The accuracy and efficiency of the elongation method as compared to exact canonical Hartree-Fock and Kohn-Sham approaches are discussed. Potential applications are illustrated by wide range of calculations for model systems. The elongation calculations are demonstrated to be much more efficient compared to the conventional ones with high accuracy maintained. The elongation CPU time is shown by the model calculations as linear or sub-linear scaling for quasi-one-dimensional systems. Future work of development into post-Hartree-Fock methodologies are pointed out.     

Implementation of divide-and-conquer method including Hartree-Fock exchange interaction

The divide-and-conquer (DC) method, which is one of the linear-scaling methods avoiding explicit diagonalization of the Fock matrix, has been applied mainly to pure density functional theory (DFT) or semiempirical molecular orbital calculations so far. The present study applies the DC method to such calculations including the Hartree-Fock (HF) exchange terms as the HF and hybrid HF/DFT. Reliability of the DC-HF and DC-hybrid HF/DFT is found to be strongly dependent on the cut-off radius, which defines the localization region in the DC formalism. This dependence on the cut-off radius is assessed from various points of view: that is, total energy, energy components, local energies, and density of states. Additionally, to accelerate the self-consistent field convergence in DC calculations, a new convergence technique is proposed.     

Density-cumulant functional theory

Starting point is the energy expectation value as a functional of the one-particle density matrix gamma and the two-particle density cumulant lambda(2). We decompose gamma into a best idempotent approximation kappa and a correction tau, that is entirely expressible in terms of lambda(2). So we get the energy E as a functional of kappa and lambda(2), which can be varied independently. Approximate n-representability conditions, derived by perturbation theory are imposed on the variation of lambda(2). A nonlinear system of equations satisfied by lambda(2) is derived, the linearized version of which turns out to be equivalent to the coupled electron-pair approximation, variant zero. The start for kappa is Hartree-Fock, but kappa is then updated to become the best idempotent approximation of gamma. Relations to density matrix functional theory and Kohn-Sham type density functional theory are discussed.     

An improved iterative solution to solve the electrostatic problem in the polarizable continuum model

 We present a discrete iterative interpolation scheme (DIIS) to improve the convergence rate of electrostatic calculations in the polarizable continuum model (PCM) to describe solvent effects on molecular solutes. The electrostatic calculations may easily become the bottleneck of the calculation when the solute size is large. For large molecules iterative procedures turn out to be computationally more convenient than matrix inversion or closure methods. The DIIS scheme is compared here to another iterative procedure (DAMP) and to the biconjugate gradient (BCG) method. The comparisons show that DIIS leads to a sizeable saving of computational time for the C-PCM and IEF-PCM methods (average 40%) compared to DAMP, and more than 50% with respect to the BCG method. Received: 5 October 2000 / Accepted: 13 November 2000 / Published online: 19 January 2001     

Analytic energy gradients of the optimized effective potential method

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals.     

Kohn-Sham potentials corresponding to Slater and Gaussian basis set densities

A new method based on linear response theory is proposed for the determination of the Kohn-Sham potential corresponding to a given electron density. The method is very precise and affords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function δ _i (r)=(−ɛ _i )ϕ _i (r) when SCF orbitals (either Kohn-Sham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth “average” potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles. Received: 24 February 1997 / Accepted: 18 June 1997     

Efficient evaluation of analytic vibrational frequencies in Hartree-Fock and density functional theory for periodic nonconducting systems

We report a method for the efficient evaluation of analytic energy second derivatives with respect to in-phase nuclear coordinate displacements within Hartree-Fock and Kohn-Sham density functional theories using Gaussian orbitals and periodic boundary conditions. The use of an atomic orbital formulation for all computationally challenging steps allows us to adapt the direct space fast multipole method for the Coulomb-type infinite summations. Our implementation also exploits the local character of the exact Hartree-Fock exchange in nonconducting systems. Exchange-correlation contributions are computed using extensive screening and fast numerical quadratures. We benchmark our scheme for in-phase vibrational frequencies of a trans-polyacetylene chain, a two-dimensional boron nitride sheet, and bulk diamond with the 6-31G** basis set and various density functionals. A study of computational scaling with the size of the unit cell for trans-polyacetylene reveals subquadratic scaling for our scheme.