Improved Fermi operator expansion methods for fast electronic structure calculations

Abstract

Articles you may be interested inAtomic spectral methods for molecular electronic structure calculations Explicitly correlated divide-and-conquer-type electronic structure calculations based on two-electron reduced density matrices Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations Linear scaling algorithms based on Fermi operator expansions ͑FOE͒ have been considered significantly slower than other alternative approaches in evaluating the density matrix in Kohn-Sham density functional theory, despite their attractive simplicity. In this work, two new improvements to the FOE method are introduced. First, novel fast summation methods are employed to evaluate a matrix polynomial or Chebyshev matrix polynomial with matrix multiplications totalling roughly twice the square root of the degree of the polynomial. Second, six different representations of the Fermi operators are compared to assess the smallest possible degree of polynomial expansion for a given target precision. The optimal choice appears to be the complementary error function. Together, these advances make the FOE method competitive with the best existing alternatives.

Graph-based linear scaling electronic structure theory

“…The graph-based electronic structure theory combines the natural parallelism of a divide and conquer approach [12][13][14][15][16][17] with the automatically adaptive and tunable accuracy of a thresholded sparse matrix algebra, [18][19][20][21][22][23][24][25][26][27][28][29][30][31] which can be combined with fast, low pre-factor, recursive Fermi operator expansion methods [32][33][34][35][36][37][38][39][40][41] and can be applied to modern formulations of Born-Oppenheimer molecular dynamics. [42][43][44][45][46][47][48][49][50] The article is outlined as follows: first we introduce the graph-based formalism for general sparse matrix polynomials expanded over separate subgraphs, thereafter we apply the methodology to the Fermi-operator expansion in electronic structure theory with demonstrations for a protein-like structure of polyalanine solvated in water, before analyzing applications in molecular dynamics simulations.…”

Linear scaling calculation of band edge states and doped semiconductors

“…To check the performance of our method, we calculate the HOMO and LUMO of BN (5,5) nanotubes with different number of atoms in the supercells. The CPU time used is shown in Fig.…”

“…In the field of computational mathematics, the shift-and-invert Lanczos algorithm is a wellknown method for calculating a pair of eigenvalue and eigenvector near a reference energy. This method was used by Liang et al 5 to obtain the Fermi level in the context of linear scaling Fermi operator expansion method. In this method, the Lanczos method is applied to the socalled shift-and-invert matrix, (H − ǫ ref I) −1 , where H and I are the Hamiltonian and identity matrices, respectively, and ǫ ref is the reference energy.…”

Extended Lagrangian free energy molecular dynamics

“…[37][38][39][40][41][42][43][44][45][46][47] However, in contrast to schemes that are based on serial expansions, the recursive expansion in Eq. (42) leads to a very highorder approximation in only a few number of iteration steps.…”