Density-functional expansion methods: generalization of the auxiliary basis

The formulation of density-functional expansion methods is extended to treat the second and higher-order terms involving the response density and spin densities with an arbitrary single-center auxiliary basis. The two-center atomic orbital products are represented by the auxiliary functions centered about those two atoms, and the mapping coefficients are determined from a local constrained variational procedure. This two-center variational procedure allows the mapping coefficients to be pretabulated and splined as a function of internuclear separation for efficient look up. The splines of mapping coefficients have a range no longer than that of the overlap integrals, and the auxiliary density appears as a single point-multipole expansion to all nonoverlapping atoms, thus allowing for the trivial implementation of a linear-scaling algorithm. The method is tested using Gaussian multipole expansions, and the effect of angular and radial completeness is explored. Several auxiliary basis sets are parametrized and compared to an auxiliary basis analogous to that used in the self-consistent-charge density-functional tight-binding model, and the method is demonstrated to greatly improve the representation of the density response with respect to a reference expansion model that does not use an auxiliary basis.     

Methods for constructing Stewart atoms

We compare four methods for generating Stewart atoms, the spherically-symmetric nuclear-centred functions whose sum best fits a given electron density. We find that projecting a molecular density onto an atom-centred basis is a more subtle and difficult problem than is generally recognized and we conclude that new approaches, based on integral equations, may be more satisfactory than traditional projection methods.     

Magnetizability tensors from auxiliary density functional theory

The working equations for the calculation of the magnetizability tensor in the framework of auxiliary density functional theory with gauge including atomic orbitals (ADFT-GIAO) are derived. Unlike in the corresponding conventional density functional theory implementations the numerical integration of the GIAOs is avoided in ADFT-GIAO. Our validation shows that this simplification has no effect on the accuracy of the methodology. As a result, a reliable and efficient implementation for the calculation of magnetizabilities of systems with more than 1000 atoms and 14 000 basis functions is presented.     

NMR shielding tensors from auxiliary density functional theory

The working equations for the calculation of NMR shielding tensors in the framework of auxiliary density functional theory are derived. It is shown that in this approach the numerical integration over gauge-including atomic orbitals can be avoided without the loss of accuracy. New integral recurrence relations for the required analytic electric-field-type integrals are derived. The computational performance of the resulting formalism permits shielding tensor calculations of systems with more than 1000 atoms and 15,000 basis functions.     

Performance of parallel TURBOMOLE for density functional calculations

The parallelization of density functional treatments of molecular electronic energy and first-order gradients is described, and the performance is documented. The quadrature required for exchange correlation terms and the treatment of exact Coulomb interaction scales virtually linearly up to 100 nodes. The RI-J technique to approximate Coulomb interactions (by means of an auxiliary basis set approximation for the electron density) even shows superlinear speedup on distributed memory architectures. The bottleneck is then linear algebra. Demonstrative application examples include molecules with up to 300 atoms and 3000 basis functions that can now be treated in a few hours per geometry optimization cycle in C₁ symmetry. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1746–1757, 1998     

Low‐memory iterative density fitting

A new low‐memory modification of the density fitting approximation based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the number of iterations needed for convergence by up to one order of magnitude. The matrix‐vector products needed within the iterative algorithm are calculated using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the standard density fitting implementation, up to 15‐fold reduction of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing density functional theory calculations for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12‐core CPU workstation. © 2015 Wiley Periodicals, Inc.     

Electron density fitting for the Coulomb problem in relativistic density-functional theory

A density fitting approach for the Coulomb matrix representation within the four-component formulation of relativistic density-functional theory is presented. Our implementation, which uses G-spinor basis sets, shares all the advantages of those found in nonrelativistic quantum chemistry. We show that very accurate Coulomb energies may be obtained using a modest number of scalar auxiliary basis functions for molecules containing heavy atoms. The efficiency of this new implementation is demonstrated in a detailed study of the spectroscopic properties of the gold dimer, and its scaling behavior has been tested by calculations of some closed-shell gold clusters (Au2, Au3+, Au4, Au5+). The algorithm is found to scale as O(N3), just as it does in the nonrelativistic case, and represents a dramatic improvement in efficiency over the conventional approach in the calculation of the Coulomb matrix, with computation times that are reduced to less than 3% for Au2 and up to 1% in the case of Au5+.     

Automatically generated Coulomb fitting basis sets: design and accuracy for systems containing H to Kr

For intermediate sized chemical systems the use of an auxiliary basis set (ABS) to fit the charge density provides a useful means of accelerating the performance of various quantum chemical methods. As a consequence much effort has been devoted to the design of various ABSs. This paper explores a fundamentally new approach where the ABS is created dynamically based on the specific orbital basis set (OBS) being used. The new approach includes a parameter that is used to coalesce candidate fitting functions together but which can also be used to provide some coarse grain control over the number of functions in the ABS. The accuracy of the new automatically generated ABS (auto-ABS) is systemically studied for a variety of small systems containing the elements H-Kr. Errors in the Coulomb energy computed using auto-ABS and with a variety of OBSs are shown to be small compared to errors in the Hartree-Fock energy due to incompleteness in the OBS. In contrast to fixed size ABSs, the use of auto-ABS is shown to lead to smaller errors as the size (quality) of the OBS is expanded. The performance of auto-ABS is also compared with the use of the recently proposed universal fitting sets [Weigend, Phys. Chem. Chem. Phys. 8, 1057 (2006)] for 180 compounds containing atoms from H to Kr.     

Possible criterion for the balance of basis sets in quantum-chemical calculations

We propose to characterize the width of a basis set (BS) by the number of basis functions falling within one electron of the considered atomic or molecular systems. It is established that for atoms, this value (the electron function endowment, or EFE) undergoes drastic changes as the atomic number of a periodic system element increases. It is shown that widening the BS through the addition of valence and polarizational functions increases the imbalance of the basis sets of various atoms in terms of EFE. A scheme of construction is proposed and an example of a BS balanced according to the EFE value is given. The properties of LiH and HF molecules are calculated by the density functional UB3LYP method with the use of standard and molecular-optimized (relaxed) BSes with segmented and general contraction of the Gaussian functions. It is established that there are uniform dependences for the error of calculating the properties of both molecules from the EFE. We conclude that the accuracy of calculating the equilibrium distance, ionization energy, electron affinity, atomization energy, dipole moment, and frequency of normal vibration increases steadily as the EFE value of a molecule rises.     

Density of non-Bloch electron states in perfect cubic crystals

This paper gives an abbreviated method for the calculation of the density of states of a crystal on the basis of that band theory in which the crystal electron states are represented by the standinglike wave functions classified according to the point-group symmetry species. The crystal is a large but finite sphere filled regularly with atoms, and the wave functions are quantized at the boundary of the sphere. The Bloch theorem is not satisfied in this theory since the wave functions are not basis functions of the irreducible representations of the translation subgroup. On the other hand, a theorem is established that the density of states can be made up of contributions given by all irreducible representations of the crystal point group, any contribution being proportional to the square of the dimension of the irreducible representation. In distinction to a former approach, the band structure is calculated solely from the energy eigenvalues obtained with the aid of the diagonalization process of the Wannier–Slater differential operator. A simple cubic lattice with an s atomic orbital on each lattice site is taken as an example, and the results are compared with Bloch's theory.     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

From linear combinations to integrals: A new approach to the basis function problem

A new formalism is presented, based upon the finite element method, that permits a dual representation of orbitals in terms of exponential or Gaussian functions as both an integral over the space of exponential parameters and as a linear combination of basis functions. The method has been implemented for the atomic Hartree–Fock problem using exponential functions and test calculations made for atoms ranging from B to Cl. Accurate and consistent results can be obtained for a variety of atoms in a simple way using computational schemes that are systematic and hierarchic in nature. The new formalism is promising for any method where the calculation of integrals is not a major problem, such as some approaches of the density functional method and the pseudospectral formulation of ab initio methods. © 1992 by John Wiley & Sons, Inc.     

Hartree–Fock High-Precision Analytical Functions of Open-Shell Atoms

The algorithm of high-precision optimization of basis functions suggested previously for calculating the analytical Hartree–Fock orbitals of closed-shell atoms is generalized to open-shell systems described by the Roothaan method (1960). Expressions for the first (free gradient) and second (Hesse matrix) derivatives of the system's energy with respect to the nonlinear parameters (orbital exponents) of the basis functions are derived in terms of density matrices for the filled and open shells. An algorithm is proposed for high-precision optimization of the nonlinear parameters using these equations based on Murtagh–Sargent and Newton minimization procedures. To illustrate the application of this algorithm, we give optimization of the basis sets of Slater type functions for atoms from the second row, as well as for Al, Si, P, K, Sc, and Fe atoms. The analytical Hartree–Fock orbitals giving nearly Hartree–Fock energies are calculated with a high degree of accuracy.     

Numerical fitting of molecular properties to Hermite Gaussians

A procedure is presented to fit gridded molecular properties to auxiliary basis sets (ABSs) of Hermite Gaussians, analogous to the density fitting (DF) method (Dunlap; et al. J. Chem. Phys. 1979, 71, 4993). In this procedure, the ab initio calculated properties (density, electrostatic potential, and/or electric field) are fitted via a linear- or nonlinear-least-squares procedure to auxiliary basis sets (ABS). The calculated fitting coefficients from the numerical grids are shown to be more robust than analytic density fitting due to the neglect of the core contributions. The fitting coefficients are tested by calculating intermolecular Coulomb and exchange interactions for a set of dimers. It is shown that the numerical instabilities observed in DF are caused by the attempt of the ABS to fit the core contributions. In addition, this new approach allows us to reduce the number of functions required to obtain an accurate fit. This results in decreased computational cost, which is shown by calculating the Coulomb energy of a 4096 water box in periodic boundary conditions. Using atom centered Hermite Gaussians, this calculation is only 1 order of magnitude slower than conventional atom-centered point charges.     

Cholesky decomposition of the two-electron integral matrix in electronic structure calculations

A standard Cholesky decomposition of the two-electron integral matrix leads to integral tables which have a huge number of very small elements. By neglecting these small elements, it is demonstrated that the recursive part of the Cholesky algorithm is no longer a bottleneck in the procedure. It is shown that a very efficient algorithm can be constructed when family type basis sets are adopted. For subsequent calculations, it is argued that two-electron integrals represented by Cholesky integral tables have the same potential for simplifications as density fitting. Compared to density fitting, a Cholesky decomposition of the two-electron matrix is not subjected to the problem of defining an auxiliary basis for obtaining a fixed accuracy in a calculation since the accuracy simply derives from the choice of a threshold for the decomposition procedure. A particularly robust algorithm for solving the restricted Hartree-Fock (RHF) equations can be speeded up if one has access to an ordered set of integral tables. In a test calculation on a linear chain of beryllium atoms, the advocated RHF algorithm nicely converged, but where the standard direct inversion in iterative space method converged very slowly to an excited state.     

Poisson-transformed density fitting in relativistic four-component Dirac-Kohn-Sham theory

We present recent developments in the implementation of the density fitting approach for the Coulomb interaction within the four-component formulation of relativistic density functional theory [Belpassi et al., J. Chem. Phys. 124, 124104 (2006)]. In particular, we make use of the Poisson equation to generate suitable auxiliary basis sets and simplify the electron repulsion integrals [Manby and Knowles, Phys. Rev. Lett. 87, 163001 (2001)]. We propose a particularly simple and efficient method for the generation of accurate Poisson auxiliary basis sets, based on already available standard Coulomb fitting sets. Just as is found in the nonrelativistic case, we show that the number of standard auxiliary fitting functions that need to be added to the Poisson-generated functions in order to achieve a fitting accuracy equal or, in some cases, better than that of the standard procedure is remarkably small. The efficiency of the present implementation is demonstrated in a detailed study of the spectroscopic properties and energetics of several gold containing systems, including the Au dimer and the CsAu molecule. The extraction reaction of a H(2)O molecule from a Au(H(2)O)(9) (+) cluster is also calculated as an example of mixed heavy-light-atom molecular systems. The scaling behavior of the algorithm implemented is illustrated for some closed shell gold clusters up to Au(5) (+). The increased sparsity of the Coulomb matrices involved in the Poisson fitting is identified, as are potential computational applications and the use of the Poisson fitting for the relativistic exchange-correlation problem.     

Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials

We present auxilliary basis sets for the atoms H to At – excluding the Lanthanides – optimized for an efficient treatment of molecular electronic Coulomb interactions. For atoms beyond Kr our approach is based on effective core potentials to describe core electrons. The approximate representation of the electron density in terms of the auxilliary basis has virtually no effect on computed structures and affects the energy by less than 10⁻⁴ a.u. per atom. Efficiency is demonstrated in applications for molecules with up to 300 atoms and 2500 basis functions. Received: 17 December 1996 / Accepted: 8 May 1997