Basic self-consistent procedure for solving the (discretized) non-linear eigenvector problem H[n]x = E Sx with H real symmetric and S symmetric positive definite ( using non-orthogonal basis functions), and obtaining the ground state electron density n = 2∑ m |x m |2 (with a factor 2 for spin). For the DFT/Kohn-Sham problem, the Hamiltonian , is composed of the Hartree potential (solution of Poisson equation), the exchange-correlation potential , and other external potential including the ionic potential.
Basic FEAST procedure for solving the generalized eigenvalue problem Hx = E Sx of size N with H real symmetric and S symmetric positive definite (spd), and obtaining all the M eigenpairs within a given interval [E min, E max]. The density matrix appears implicitly in Step 2, using the complex contour integration of the Green's function G(Z) = (Z S − H)−1. In practice, the vectors Q in Step 2 can be computed using a high-order numerical integration such as Gauss-Legendre quadrature, where only a small number of linear systems, (Z e S − H)Q e = Y, need to be solved, one for each of a number of specific Gauss nodes Z e (associated with the weights ω e ) along a complex contour , i.e.,
FEAST procedure for solving the non-linear eigenvector problem H[n]x = E Sx of size N with H real symmetric and S symmetric positive definite (spd), and obtaining all the M lowest occupied states within a given interval [E min, E max] and hence the ground state electron density. Here, Step 3-c of the algorithm is making use of a traditional SCF procedure for solving the non-linear reduced system. We note that in Step 3-a, the column vectors of the newly generated subspace are appended to the matrix of the old subspace, increasing the total subspace size by M 0. This can be done such that only a finite number of the most recently generated subspaces are retained, or the subspace size can be increased until convergence. Additionally, a singular value decomposition (SVD) is performed on (i.e., Step 3-b), such that only the left singular vectors U are used in the Rayleigh-Ritz procedure.
Results of numerical experiments comparing the performance of our algorithm to that of DIIS for the silane , benzene , and caffeine molecules. The relative error on the total energy is used here as the measure of convergence. The convergence criteria for NLFEAST has been set to 10−10 while DIIS-SCF was stopped after a maximum of 10 iterations for silane and 15 iterations for both benzene and caffeine. The meaning of outer-iterations is different for both algorithms.
Numerical experiments demonstrating the role of the number of most recent retained Q (k) subspaces for constructing the search subspace used by NLFEAST. The convergence rate depends on this number that here varies between one and “infinity” (i.e., where the search subspace keeps increasing until convergence is reached).
Experiments demonstrating the effect on convergence of the accuracy of the contour integration in NLFEAST. Different curves represent the convergence for different numbers of Gauss points used in the numerical contour integration (Step 2 in Figure 3 ). For all these results, the search subspace keeps increasing indefinitely until convergence.
Results of numerical experiments where the initial guess for the electron density was set to zero, n = 0. The maximum number of subspace to form has been set equal to 10 for all molecules but and where it was set equal to 4. The convergence is reached for an error on the trace smaller than 10−9.
Plots of the density of states for buckminsterfullerene ( ) and caffeine ( ). Our all-electron code is able to capture both the valence states and the core states of each type of atom. From those results, we can clearly identify the low energy core regions of oxygen, nitrogen, and carbon.
Comparison of the number of contour integrations and Poisson system solves required to reach convergence (set at ∼10−8) using NLFEAST and SCF-DIIS and the DFT/Kohn-Sham/LDA model for the three molecules , , and . The number of additional iterations needed to construct the initial subspace (not reported in Figure 4 ) has also been taken into consideration (i.e., one additional contour for NLFEAST and five DIIS iterations for SCF-DIIS). With enough parallelism for FEAST, the cost of each contour integral can be straightforwardly reduced to that of solving a single complex linear system, whereas the solution of Poisson's equation consists of solving a single real-valued linear system.
Convergence rate results for NLFEAST for various molecules (the molecular geometries are obtained from the experimental data 23 ). The convergence criteria is satisfied for NLFEAST when the relative error on trace is below 10−10, which also provides very low non-linear relative residual max m (||H(x m )x m − E m Sx m ||/||H(x m )x m ||), typically here below 10−8. The total energy results using our cubic real-space FEM code are in good agreement with those obtained by NWChem 22 using the basis from to , and the basis for and .
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