Calculation of matrix elements

José M. SolerUniversidad Autónoma de Madrid

Schrödinger equation

Hi(r) = Eii(r)

i(r) = ci (r)

(H- EiS)ci = 0

H = < | H | >

S = < | >

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

VH(r) = VH[SCF(r)] - VH[atoms(r)]

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

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

Two-center

integralsGrid integrals

Sparsity

KB pseudopotential projector

Basis orbitalsNon-overlap interactions

1 23

4

5

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 and H are sparse

is not strictly sparse

but only a sparse subset is needed

Two-center integrals

kkkR

rrk

rRrrR

kR

kr

deS

de

dS

i

i

)()()(

)()2(

1)(

)()()(

21

3/2

2121

Convolution theorem

Atomic orbitals

)ˆ(Y)()(

)ˆ(Y)ˆ(Y)(4

)()(/2)()(

)ˆ(Y)()()ˆ(Y)()(

)()()(

21

0

*

0

0

2

21

RR

Rk

kkrr

kkkR

kR

kR

lm

ll

l

l

lmlm

lmlmll

l

lm

li

lll

l

lmllml

i

RSS

kRjie

rrjdrrik

kr

deS

Overlap and kinetic matrix elements

)()()(2

1)(

)()()()(

)ˆ(Y)()(

21

0

4

21

0

2

0

max

2211

max

2211

21

kkkRjdkkGRT

kkkRjdkkGRS

RSS

l

k

lmmlmllm

l

k

lmmlmllm

lm

ll

l

l

lmlm

RR

•K2max = 2500 Ry

•Integrals by special radial-FFT

•Slm(R) and Tlm (R) calculated and tabulated once and for all

Real spherical harmonics

xzy

mllmlm

pppml

mm

mmPCY

,,1,0,1,1

0if)cos(

0if)sin()(cos),(

Grid work

)(r(r)φφρ(r)ψρ(r)

ccρ

(r)φc(r)ψ

νμμν

μνi

2i

iνi

iμμν

μμ

iμi

(r)δVδρ(r)

(r)ρ(r)ρδρ(r)

(r)Vρ(r)

H

atomsSCF

xc

FFT

)(r

)(r

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

GGA

)()(),(,)(),(

)(

])'(),'([)(

2 rrrrrV

r

rrErv

GGA

GGAxc

i

xcixc

GGAxcii

ii

Erv

EExxx

)(

,..., 2111

11

Egg-box effect

•Affects more to forces than to energy

•Grid-cell sampling

Ex

Grid points

Orbital/atom

Grid fineness: energy cutoff

x kc=/x Ecut=h2kc2/2me

Grid fineness convergence

2)/( xEcut

Points and subpointsSparse storage: (i,i )

Points (index and value stored)

Subpoints (value only)

x Ecut

Extended mesh

46546

13213

46546

13213

54321

109876

1514131211

2019181716

1 7 8,12,13 2,4,5

6 14 15,19,20 4,3,1

+1,+5,+6

+1,+5,+6

Forces and stress tensor

Analytical: e.g.

rrr

rrrrrr

rr

rrrr

r

with

)()()(

)()()(

3

3

yx

TT

dV

drVV

xy

i

Calculated only in the last SCF iteration