Automatic Selection of Integral Thresholds by Extrapolation in Coulomb and Exchange Matrix Constructions.

We present a method to compute Coulomb and exchange matrices with predetermined accuracy as measured by a matrix norm. The computation of these matrices is fundamental in Hartree-Fock and Kohn-Sham electronic structure calculations. We show numerically that, when modern algorithms for Coulomb and exchange matrix evaluation are applied, the Euclidean norm of the error matrix ε is related to the threshold value τ as ε = cτ(α). The presented extrapolation method automatically selects the integral thresholds so that the Euclidean norm of the error matrix is at the requested accuracy. This approach is demonstrated for a variety of systems, including protein-like systems, water clusters, and graphene sheets. The proposed method represents an important step toward complete error control throughout the self-consistent field calculation as described in [J. Math. Phys. 2008, 49, 032103].

[1]  Eric Schwegler,et al.  Fast assembly of the Coulomb matrix: A quantum chemical tree code , 1996 .

[2]  Paweł Sałek,et al.  The trust-region self-consistent field method: towards a black-box optimization in Hartree-Fock and Kohn-Sham theories. , 2004, The Journal of chemical physics.

[3]  J. Ángyán,et al.  Spherical harmonic expansion of short-range screened Coulomb interactions , 2006 .

[4]  Christian Ochsenfeld,et al.  Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals. , 2005, The Journal of chemical physics.

[5]  Emanuel H. Rubensson,et al.  Density matrix purification with rigorous error control. , 2008, The Journal of chemical physics.

[6]  Matt Challacombe,et al.  Linear scaling computation of the Fock matrix. V. Hierarchical Cubature for numerical integration of the exchange-correlation matrix , 2000 .

[7]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[8]  Sara Zahedi,et al.  Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification. , 2008, The Journal of chemical physics.

[9]  Marco Häser,et al.  Improvements on the direct SCF method , 1989 .

[10]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build , 1997 .

[11]  Michael J Frisch,et al.  Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems. , 2006, The Journal of chemical physics.

[12]  Emanuel H. Rubensson,et al.  Hartree-Fock calculations with linearly scaling memory usage. , 2008, The Journal of chemical physics.

[13]  Emanuel H. Rubensson,et al.  Rotations of occupied invariant subspaces in self-consistent field calculations , 2008 .

[14]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix , 1997 .

[15]  G. Scuseria,et al.  A black-box self-consistent field convergence algorithm: One step closer , 2002 .

[16]  Christian Ochsenfeld,et al.  Linear and sublinear scaling formation of Hartree-Fock-type exchange matrices , 1998 .

[17]  Paweł Sałek,et al.  The trust-region self-consistent field method in Kohn-Sham density-functional theory. , 2005, The Journal of chemical physics.

[18]  Paweł Sałek,et al.  Efficient implementation of the fast multipole method. , 2006, The Journal of chemical physics.

[19]  M. Head‐Gordon,et al.  A multipole acceptability criterion for electronic structure theory , 1998 .

[20]  Christian Ochsenfeld,et al.  Locality and Sparsity of Ab Initio One-Particle Density Matrices and Localized Orbitals , 1998 .