Calculation of matrix elements José M. Soler Universidad Autónoma de Madrid.

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Calculation of matrix elements José M. Soler Universidad Autónoma de Madrid

Schrödinger equation H  i (r) = E i  i (r)  i (r) =   c i    (r)  (H  - E i S  )c i = 0 H  = S  =

Kohn-Sham hamiltonian H = T + V PS + V H (r) + V xc (r) T = -(1/2)  2 V PS = V ion (r) + V nl V ion (r)  - Z val / r Local pseudopotentiual V nl =   |   >   <   | Kleinman-Bylander V H (r) =  dr’  (r’) / |r-r’| Hartree potential V xc (r) = v xc (  (r)) Exchange & correlation

Long-range potentials H = T + V ion (r) + V nl + V H (r) + V xc (r) Long range  V H (r) = V H [  SCF (r)] - V H [  atoms (r)] V na (r) = V ion (r) + V H [  atoms (r)] Neutral-atom potential H = T + V nl + V na (r) +  V H (r) + V xc (r) Two-center integrals Grid integrals

Sparsity KB pseudopotential projector Basis orbitals Non-overlap interactions with 1 and 2 2 with 1,2,3, and 5 3 with 2,3,4, and 5 4 with 3,4 and 5 5 with 2,3,4, and 5 S  and H  are sparse   is not strictly sparse but only a sparse subset is needed

Two-center integrals Convolution theorem

Atomic orbitals

Overlap and kinetic matrix elements K 2 max = 2500 Ry Integrals by special radial-FFT S lm (R) and T lm (R) calculated and tabulated once and for all

Real spherical harmonics

Grid work FFT

Poisson equation  2 V H (r) = - 4   (r)  (r) =  G  G e iGr  V H (r) =  G V G e iGr V G = - 4   G / G 2  (r)   G  V G  V H (r) FFT

GGA

Egg-box effect Affects more to forces than to energy Grid-cell sampling E x Grid points Orbital/atom

Grid fineness: energy cutoff  x  k c =  /  x  E cut =h 2 k c 2 /2m e

Grid fineness convergence

Points and subpoints Sparse storage: (i,  i ) Points (index and value stored) Subpoints (value only)  x  E cut

Extended mesh  7  8,12,13  2,4,5 6  14  15,19,20  4,3,1 +1,+5,+6

Forces and stress tensor Analytical: e.g. Calculated only in the last SCF iteration