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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  = <   | H |   > S  = <   |   >. Kohn-Sham hamiltonian.

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calculation of matrix elements

Calculation of matrix elements

José M. Soler

Universidad Autónoma de Madrid

schr dinger equation
Schrödinger equation

Hi(r) = Eii(r)

i(r) =  ci (r)

(H-EiS)ci = 0

H = <  | H |  >

S = <  |  >

kohn sham hamiltonian
Kohn-Sham hamiltonian

H = T + VPS+ VH(r) + Vxc(r)

T = -(1/2)2

VPS = Vion(r) + Vnl

Vion(r)  - Zval / r Local pseudopotentiual

Vnl =  |>  <| Kleinman-Bylander

VH(r) =  dr’ (r’) / |r-r’| Hartree potential

Vxc(r) = vxc((r)) Exchange & correlation

long range potentials

H = T + Vion(r) + Vnl+ VH(r) + Vxc(r)

Long range

Two-center

integrals

Grid integrals

H = T + Vnl + Vna(r) + dVH (r)+ Vxc(r)

Long-range potentials

Vna(r) = Vion(r) + VH[ratoms(r)] Neutral-atom potential

dVH(r) = VH[rSCF(r)] - VH[ratoms(r)]

sparsity

5

4

1

2

3

Non-overlap interactions

Basis orbitals

KB pseudopotential projector

Sparsity

1 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andHare sparse

is not strictly sparse but only a sparse subset is needed

two center integrals
Two-center integrals

Convolution theorem

overlap and kinetic matrix elements
Overlap and kinetic matrix elements
  • K2max= 2500 Ry
  • Integrals by special radial-FFT
  • Slm(R) and Tlm (R)calculated and tabulated once and for all
poisson equation

2VH(r) = - 4 (r)

(r) = G G eiGr  VH(r) = G VG eiGr

VG = - 4 G / G2

(r)  G  VG  VH(r)

FFT

FFT

Poisson equation
egg box effect

Grid points

Orbital/atom

E

x

Egg-box effect
  • Affects more to forces than to energy
  • Grid-cell sampling
points and subpoints

Points (index and value stored)

Subpoints (value only)

x  Ecut

Points and subpoints

Sparse storage: (i,i )

extended mesh

3

1

2

3

1

16

17

18

19

20

6

4

5

6

4

11

12

13

14

15

3

1

2

3

1

6

7

8

9

10

6

4

5

6

4

1

2

3

4

5

+1,+5,+6

1  7  8,12,13  2,4,5

6  14  15,19,20  4,3,1

+1,+5,+6

Extended mesh
forces and stress tensor
Forces and stress tensor

Analytical: e.g.

Calculated only in the last SCF iteration