Numerical Aspects of Many-Body Theory • Choice of basis for crystalline solids • Local orbital versus Plane wave • Plane waves e i(q+G).r • Complete (in practice for valence space) • No all electron treatment (PAW?) • Large number of functions x.10 4 • Slow for HF exchange • Straightforward to code (abundance of Dirac delta’s) • Local orbital (x - A x ) i (y - A y ) j e –(x - A) 2 • Incomplete (needs care in choice of basis) • All electron possible and relatively inexpensive • Relatively small number of functions permits large unit cells to be treated • Relatively fast for HF exchange in gapped materials • Difficult to code (lattice sum convergence, exploitation of symmetry, ..) G q q+G IBZ
q. G. q + G. IBZ. Numerical Aspects of Many-Body Theory. Choice of basis for crystalline solids Local orbital versus Plane wave Plane waves e i ( q + G ). r Complete (in practice for valence space) No all electron treatment (PAW?) Large number of functions x.10 4 - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Numerical Aspects of Many-Body Theory
• Choice of basis for crystalline solids• Local orbital versus Plane wave
• Plane waves ei(q+G).r
• Complete (in practice for valence space)• No all electron treatment (PAW?)• Large number of functions x.104
• Slow for HF exchange• Straightforward to code (abundance of Dirac delta’s)
• Local orbital (x - Ax)i(y - Ay)j e–(x - A)2
• Incomplete (needs care in choice of basis)• All electron possible and relatively inexpensive• Relatively small number of functions permits large unit cells to be treated• Relatively fast for HF exchange in gapped materials• Difficult to code (lattice sum convergence, exploitation of symmetry, ..)
G q
q+G
IBZ
Numerical Aspects of Many-Body Theory
• Coulomb Energy in real and reciprocal spaces
• Coulomb interaction
• Ewald form of Coulomb interaction
'
)'()('ddECoul rr
rrrr
G q Gq
rrGqq
q
rrqq
rr BZ23
space reciprocal all
23
)').(i(e
2
d)'.(ie
2
d
'
1
dxdk
2
(k)I dx I dx Idx
x'e
2
'
1
R3
R
000
22
rr
rr
r r’
r r’
Numerical Aspects of Many-Body Theory
• Density Matrix Representation of Charge Density
kkk
GkGkGkGG'
GG'GG'
gigikgik
GGkkGkk
kkkk
k
rrgkkrrr
k
kkr
rrr
rr
rrrr
rG GrGkrG k
gk
r GrGk
rk
2*2112
g2
0112
g2
01
0g12
,n,*
',n,,n,
,,n,,,n
,n,n,n,n
nnnn
c c c)(P
)()( .ie )(P)()(P)ρ(
c cc)(P
e )(Pe)e(P)ρ(
)(e )( c)(ψ
ec )(uec)(ψ
)(u )(ψe )(u)(ψ
).'().'().(
.
.
).(
.
tscoefficien expansion orbital local
tscoefficien expansion orbital waveplane
tionrepresenta waveplane
cell gth in site ith on orbital local
tionrepresenta waveplane
function cellperiodic function Bloch
iii
i
ii
i
r
Numerical Aspects of Many-Body Theory
• Coulomb Energy with real space representation of charge density
34g
12403
g2
01
34g
12h4
h3
g2
01Coul
PP)'-()'(- '-
1)()(
PP)- - '() -'('-
1)()( E
rrhrr
g-rr
hrhrrr
g-rr
hhrr
lattice space real on functionperiodic - '-
1
h
r r’
e.convergenc absolute rapid, Overall,
R. limit upperby off cut space real in Part
rapidly. converges FT - smooth space reciprocal in Part integral. Split
dxdk F(k) dx I dx Idx x - '
e2
- '
1
R
R
000
22
hrr
hrr hh
0Gk,kkkk
kGkG
kGkG
kGkGkk
kkkGkk
kkkkGk, q
G q
rrGrrrGr
rrrGqr
rrr
rrGqrrrGqr q
rGq
rrGqr
q
GGGq
-GGGq
G--kGkGq
rGkrGqrGk
rGrGkrk
n,'n''n'nn2Coul
3''n
*'n
3''n
*'n
''n*
'nnn
nnnnn
'n'nnnn, BZ
3
2BZ
3Coul
)'('.ie)'()(.ie)(G
1 E
)(2 cc
)(2 cc
eeed cc)().i(e)(
ec)(uec)e(u)(
)'(').i(-e)'()().i(e)(2
d
)'()').(i(e
)(2
d E
''- ' 0
'''
'''
).''().().'(
. ).(.
iii
iii
Numerical Aspects of Many-Body Theory
• Coulomb Energy with reciprocal space representation of interaction
r r’
Numerical Aspects of Many-Body Theory
• Exchange Energy with real space representation of interaction
• No Ewald transformation possible since h sum is split• 3 lattice sums instead of 2• Absolute convergence neither guaranteed nor rapid
34g
12h4
g2
h3
01Exch PP) - - '() - '(
'-
1)()( E hrgr
rrh -r r r r’
34g
12h4
h3
g2
01Coul PP)- - '() - '(
'-
1)()( E hrhr
rrg-rr r r’
• Exchange Energy with reciprocal space representation of interaction
• q + G lattice sum instead of just G• Absolute convergence not guaranteed nor rapid