vasp6: total energies beyond dftco adsorption on d-metal surfaces. (all energies in ev) schimkaet...
TRANSCRIPT
VASP6: total energies beyond DFT
Martijn Marsman
University Vienna & VASP Software GmbH
Outline
• Beyond DFT• The need to go beyond DFT• HF/DFT hybrid functionals• The Random-Phase-Approximation
• ! "# implementation of the RPA• Cubic-scaling RPA total energies• Minimax frequency and time integration• Cubic-scaling RPA quasi-particles
• Forces in the RPA
• Total energy differences with”chemical accuracy”(1 kcal/mol ≈ 40 meV/atom):Atomization-, formation energies,reaction barriers, etc
• Van der Waals interactions
0.5 1 2 4 8 16Experiment (eV)
0.25
0.5
1
2
4
8
16
The
ory
(eV
)
PBEHSE03PBE0
PbSePbS
PbTe Si
GaAs
Ne
AlP
CdS
SiC
ZnO
GaNZnS
LiFAr
C BN MgO
Figure 8. Band gaps from PBE, PBE0, and HSE03 calculations,plotted against data from experiment.
Need to go beyond DFT and DFT/HF hybrids?
SiCMgO
NaClLiCl
NaFLiF
PBE
PBE0
HSE03
B3LYP-30
-20
-10
0
Rel
ativ
e er
ror
(%)
-40
10
Figure 7. Relative error in the PBE, PBE0, HSE03, and B3LYPheats of formation with respect to experiment.
BandgapsHeats of formation
• Bandstructure of metals and largish gapsystems, and some problematic casesinbetween
CO@Cu(111): �170 meVCO@Rh(111): �40 meVCO@Pt(111): �100 meV
• DFT incorrectly predicts that CO prefers thehollow site: P. Feibelman et al.,J. Phys. Chem. B 105, 4018 (2001)
• This error is relatively large.Best DFT/PBE calculations:
CO adsorption on d-metal surfaces
• CO HOMO-LUMO gap too small in DFT:
CO @ top fcc hcp �Cu(111) PBE 0.709 0.874 0.862 �0.165
PBE0 0.606 0.579 0.565 0.027HSE03 0.561 0.555 0.535 0.006exp. 0.46-0.52
Rh(111) PBE 1.870 1.906 1.969 �0.099PBE0 2.109 2.024 2.104 0.005HSE03 2.012 1.913 1.996 0.016exp. 1.43-1.65
Pt(111) PBE 1.659 1.816 1.750 �0.157PBE0 1.941 1.997 1.944 �0.056HSE03 1.793 1.862 1.808 �0.069exp. 1.43-1.71
A. Stroppa et al., PRB 76, 195440 (2007); A. Stroppa and G. Kresse, NJP 10, 063020 (2008).
CO adsorption on d-metal surfaces
(All energies in eV)
Schimka et al., Nature Materials 9, 741 (2010)
• DFT does well for the metallicsurface, but not for the CO:2"∗ (LUMO) too close to theFermi level.
• HSE does well for the CO,but not for the surface:d-metal bandwidth too large.
Hybrid functionals reduce the tendency to stabilize adsorption at thehollow site w.r.t. the top site.
reduced CO 2π* ⎯ metal-d interaction
The Random-Phase-Approximation: GW and ACFDT
• Quasiparticles from GW: “electronic structure” in the RPA.
• Total energies from the ACFDT in the RPA.
• Are the above related?
DFT: Kohn-Sham eq.
DFT-HF hybrid functionals: Roothaan eq.
GW: quasi-particle eq.
One-electron/QP picture
⇣�1
2�+ Vext(r) + VH(r) + Vxc [⇢] (r)
⌘ nk(r) = ✏nk nk(r)
<latexit sha1_base64="CXioMq33FCZ1KOj9JGyQhdaRzNg=">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</latexit>
⇣�1
2�+ Vext(r) + VH(r)
⌘ nk(r) +
ZVX [{
o
}] (r, r0) nk(r
0)dr0 = ✏nk nk(r)
<latexit sha1_base64="p52S1soj6JNSChdMchGrYyNEpNQ=">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</latexit>
⇣�1
2�+ Vext(r) + VH(r)
⌘ nk(r) +
Z⌃ [{ , E}] (r, r0, Enk) nk(r
0)dr0 = Enk nk(r)
<latexit sha1_base64="AzEgT9+bH61oaA3QplgUo0z9Epw=">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</latexit>
✓�1
2�+ Vext(r) + VH(r)
◆ nk(r) +
Z⌃(r, r0, Enk) nk(r
0)dr0 = Enk nk(r)
The quasi-particle equation:
⌃ = iGWThe “self-energy is given by:
or more explicitlyGreen’s function: G
screened Coulombinteraction: W
Compare to Fock-exchange:
VX
(r, r0) = �occ.X
n
n(r) ⇤n(r
0)⇥ e2
|r� r0|
⌃(r, r0, E) =i
2⇡
Z 1
�1d!
allX
n
n(r) ⇤n(r
0)
! � E � En + i⌘ sgn(En � Efermi)⇥
⇥ e2Z
dr00✏�1(r, r00,!)
|r00 � r0|
bare Coulombinteraction: v
GW
The Green’s function !(#, #′, & − &′) describes the propagation of a particlefrom (#, &) to #′, &′ : i.e., provided we have particle at position r at time t,!(#, #′, & − &′) is the chance of finding it at position r’ at time t’.
G0(r, r0,!) =
allX
n
⇤n(r) n(r0)
! � ✏n + i⌘ sgn(✏n � µ)
G0(1, 2) =vir.X
n
n(r1)⇤ n(r2)e
�i(✏n�µ)(t2�t1)
G0
(1, 2) =occ.X
n
⇤n(r1) n(r2)e
�i(✏n�µ)(t1�t2)
(r1, t1) = 1 (r2, t2) = 2
(r2, t2) = 2 (r1, t1) = 1
• Particle propagator: !* 1,2 = !*(#., #/, &/−&.) for &/ > &.:
• Hole propagator: !* 1,2 for &. > &/:
The Green’s function: physical interpretation
W = ⌫ + ⌫�0⌫ + ⌫�0⌫�0⌫ + ⌫�0⌫�0⌫�0⌫ + ... = ⌫ (1� �0⌫)�1
| {z }✏�1
In the RPA, the screened Coulomb interaction is computed from !" as:
1. The bare Coulombinteraction betweentwo particles
2. The electronic environmentreacts to the field generatedby a particle: induced changein the density !"#, that givesrise to a change in the Hartreepotential: #!"#.
3. The electrons reactto the induced change inthe potential: additionalchange in the density, !"#!"#,and corresponding change inthe Hartree potential: #!"#!"#.
and so on,and so on …
geometricalseries
W in the Random-Phase-Approximation
Key: the “irreducible polarizability in the independent particle picture” !" (or !%&):
�0(r, r0,!) :=@⇢ind(r,!)
@ve↵(r0,!)
W = ✏�1⌫ ✏�1 = 1 + ⌫� � = �0 + �0⌫�
The “irreducible polarizability in the independent particle picture” !" (or !#$):
Adler and Wiser derived expressions for !"
�0(r1, t1, r2, t2) = �(1, 2) = �G0(1, 2)G0(2, 1)
Or in terms of Green’s functions (propagators):
�0(r, r0,!) :=@⇢ind(r,!)
@ve↵(r0,!)
�0
(r1
, r2
,!) =occ.X
i
virt.X
a
h a|r1| iih i|r2| ai✏i � ✏a � !
+occ.X
i
virt.X
a
h i|r1| aih a|r2| ii✏a � ✏i � !
%&, (& %), ()
a
i
*"(1,2)
The IP-polarizability: !"
=+ + + + …
GW in the RPA
E[n] = TKS [{ i}] + EH [n] + EX [{ i}] + Eion�el
[n] + Ec
The “RPA” total energy is given by:
with the RPA correlation energy
Ec =X
q
Z 1
0
d!
2⇡Tr{ln[1� �0(q, i!)⌫] + �0(q, i!)⌫}
�0G,G0(q,!) =
1
⌦
X
nn0k
2wk(fn0k+q � fn0k)
⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘
and the Independent-Particle polarizability:
ln(1 − &'()+&'( = − Tr[&'(&'(]2 − Tr[&'(&'(&'(]
3 − Tr[&'(&'(&'(&'(]4
�0(r, r0,!) :=@⇢ind(r,!)
@ve↵(r0,!)
RPA total energies (ACFDT)
…
+ + + …=
2nd. order 3rd. order 4th. order
ln(1 − )*+)+)*+ = −Tr[)*+)*+]2 − Tr[)*+)*+)*+]3 − Tr[)*+)*+)*+)*+]4
In terms of diagrams
+ + + …
The derivate w.r.t. Gyields GW!
@Ec
@G= ⌃
<latexit sha1_base64="i+XT2UasZQZ1ud5FEFqQWRmR7qw=">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</latexit>
Deviations w.r.t. experiment(corrected for zero-point vibrations)
MRE MAREPBE 1.2 1.2HF 1.1 1.1MP2 0.2 0.4
RPA 0.5 0.4
(in %)
RPA: lattice constantsJ. Harl et al., PRB 81, 115126 (2010)
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
304050100 25
corr
elat
ion
ener
gy(e
V)
volume (ų)
Neon
Argon
Krypton
20 30 40 50 60 70 80 90Volume [ ų]
-150
-100
-50
0
50
Coh
esiv
e En
ergy
[meV
]
LDAPBEACFDTMP2
Argon
C6 coe�cients
RPA(LDA) RPA(PBE) Exp.
Ne 62 53 47
Ar 512 484 455
Kr 1030 980 895
RPA: noble gas solidsJ. Harl and G. Kresse, PRB 77, 045136 (2008)
QMC(Galli)
RPA EXP
d(Å) 3.426 3.34 3.34
C33 36 36-40
E(meV) 56 48 43-50
1/d4 behavior at short distances
J. Harl, G. Kresse, PRL 103, 056401 (2009).S. Lebeque, et al., PRL 105, 196401 (2010).
Graphite vs. Diamond
PBE Hartree-Fock
RPA EXP
LiF 570 664 609 621NaF 522 607 567 576NaCl 355 433 405 413MgO 516 587 577 603MgH2 52 113 72 78AlN 262 350 291 321SiC 51 69 64 69
Example: Mg(bulk metal) + H2→ MgH2
Heats of formation w.r.t. normal state at ambient conditions (in kJ/mol)
RPA: heats of formationJ. Harl and G. Kresse, PRL 103, 056401 (2009)
RPA:• increases surface energy
and• decreases adsorption energy
RPA: CO @ Pt(111) and Rh(111)Schimka et al., Nature Materials 9, 741 (2010)
Schimka et al., Nature Materials 9, 741 (2010)
• DFT does well for the metallicsurface, but not for the CO:2"∗ (LUMO) too close to theFermi level.
• HSE does well for the CO,but not for the surface:d-metal bandwith too large.
• GW gives a good descriptionof both the metallic surfaceas well as of the CO 2"∗ (LUMO).The CO 5% and 1" are slightlyunderbound.
Schimka et al., Nature Materials 9, 741 (2010)
RPA:
• Right site preference
• Good adsorption energies
• Excellent lattice constants
• Good surface energies
Cubic scaling RPA
�0G,G0(q,!) =
1
⌦
X
nn0k
2wk(fn0k+q � fn0k)
⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘
As shown by Adler and Wiser, the IP polarizability !" can be straightforwardlycalculated from #, % :
Expensive: this scales as N4!
Now the worst scaling steps are the evaluation of the Greens function (N3), and
(which scales as N3 due to the diagonalization involved in evaluating the “ln”).
Evaluate the Green's function in “imaginary” time:
and the polarizability as:
Followed by a cosine-transform:
But storing G and χ is expensive! → we need small sets of cleverly chosen “τ” and “ω”
Cubic-scaling RPAM. Kaltak, J. Klimeš, and G. Kresse, PRB 90, 054115 (2014)
Ec =
Z 1
0
d!
2⇡Tr{ln[1� �0(i!)⌫] + �0(i!)⌫}
<latexit sha1_base64="MpgHT16zMz7sjYFeugcfTlUSzYE=">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</latexit>
G(r, r0, i⌧) =X
n
n(r) ⇤n(r
0)e�✏n⌧
<latexit sha1_base64="vd8vvVlVid+OCG3tu1fO9rm7zpk=">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</latexit>
�0(r, r0, i⌧) = �G(r, r0, i⌧)G(r, r0,�i⌧)
<latexit sha1_base64="hbsu+syT++hRXU5XV0w77kufbPc=">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</latexit>
�0(r, r0, i⌧)CT��! �0(r, r0, i!)
<latexit sha1_base64="MCU36+EEEXPiJMKq4y7NIDDoKrg=">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</latexit>
The evaluation of the polarizability scales quadratically (N2)!
The “minimax” fit
||⌘||1 := max
axb
|⌘(~↵, ~�, x)|
<latexit sha1_base64="ugZktatsYD60barwZ91ZmsnY7yQ=">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</latexit>
⌘(~↵, ~�, x) = f(x)�X
i
�i'(↵i, x)
<latexit sha1_base64="vXlhMmIuuFvajkq/dI5+l4Fd0GQ=">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</latexit>
Express a function ! " in a basis of N functions # $%, " ' = 1, … , + ,such that the error:
is “as small as possible”.
Choose $%, ,% ' = 1, … , + so that the maximum error:
is minimized.
This solution, $%∗, ,%∗ ' = 1, … , + , is the so-called ”minimax” solution
The “minimax” frequency gridM. Kaltak, J. Klimeš, and G. Kresse, JCTC 10, 2498 (2014)
�(i!) =X
µ
�µ�(!, xµ)
<latexit sha1_base64="cWdrHM6G5sImfdsFhRdfRWYh6Bo=">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</latexit>
�µ(=ia) = hi|r|aiha|r0|ii
<latexit sha1_base64="Wu2zQxduoftxDWs3VYfLFH4cljE=">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</latexit>
xµ(=ia) = ✏a � ✏i
<latexit sha1_base64="GMH5OpI2HEEEise4QHIw6YEFGPM=">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</latexit>
�(!, x) =2x
x
2 + !
2
<latexit sha1_base64="N7NVeJ9LRQPl5knIClVcrfKC1oQ=">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</latexit>
E(2)c = � 1
8⇡
Z 1
�1d!{�(i!)⌫}2
<latexit sha1_base64="G/8IpZ3llhz0IPw7sv1I6/WfC78=">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</latexit>
The IP polarizability can be written as (Adler&Wiser):
where
Determine a frequency grid and quadrature weights by expressing the“2nd-order direct Møller-Plesset energy”
in the basis ! "#, % & = 1,… , * in the minimax sense:
⌘(~�, ~!, x) =1
x
� 1
⇡
NX
k=1
�k�2(!k, x)
<latexit sha1_base64="IoNYfVkTuG5bGr+vnCtMPaVTDCw=">AAACeXicbZFLbxMxEICd5VXCK8CRiyGqlKIS7YZKwKFSJS6cUJFIWylOV7PecWKt7V3Z3pJotf+QP8Df4NoecB4IaBnJ0ucZz9j6nFVKOh/HPzrRrdt37t7bud998PDR4ye9p89OXFlbjmNeqtKeZeBQSYNjL73Cs8oi6EzhaVZ8XNVPL9A6WZqvflnhVMPMSCE5+JBKe4KhBzpgF8jZDLSGfbrmUuMs8GKPHlImLPAmaZtFS9/82bFKtszV+vxz2hSHSUs3A9KCsmouz0eDzZC02F/spb1+PIzXQW9CsoU+2cZx2rtieclrjcZzBc5Nkrjy0wasl1xh22W1wwp4ATOcBDSg0U2btY+W7oZMTkVpwzKerrN/dzSgnVvqLJzU4Ofuem2V/F9tUnvxftpIU9UeDd9cJGpFfUlXcmkuLXKvlgGAWxneSvkcgi8fvqC7yywa/MbLYMnkDROgpVrmKKBWPvh04jcHXcl1OTfhZDRMDoYfvhz0j0ZbcTvkBXlFBiQh78gR+USOyZhw8p38JJfkqnMZvYwG0evN0aiz7XlO/ono7S9kfcMQ</latexit>
{(�⇤k ,!
⇤k)|k = 1, .., N}
<latexit sha1_base64="M06MiGP3+4XUDHvd7p7EBwnsUa8=">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</latexit>
Eg x max(✏a � ✏i)
<latexit sha1_base64="VfkIrO967P2/TvbsIf68htmFGqA=">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</latexit>
E(2)c = � 1
8⇡
Z 1
�1d!{�(i!)⌫}2
<latexit sha1_base64="G/8IpZ3llhz0IPw7sv1I6/WfC78=">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</latexit>
E(2)c ⇡ � 1
8⇡
NX
k=1
�⇤k{�(i!⇤
k)⌫}2
<latexit sha1_base64="FbYjP+9kJyLKx6LI+pB8wPsroxw=">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</latexit>
The “minimax” time gridM. Kaltak, J. Klimeš, and G. Kresse, JCTC 10, 2498 (2014)
�µ(=ia) = hi|r|aiha|r0|ii
<latexit sha1_base64="Wu2zQxduoftxDWs3VYfLFH4cljE=">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</latexit>
xµ(=ia) = ✏a � ✏i
<latexit sha1_base64="GMH5OpI2HEEEise4QHIw6YEFGPM=">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</latexit>
Alternatively the IP polarizability can be written as:
where
Determine a frequency grid and quadrature weights by expressing the“2nd-order direct Møller-Plesset energy”
in the basis !" #$, & ' = 1, … , + in the minimax sense:
Eg x max(✏a � ✏i)
<latexit sha1_base64="VfkIrO967P2/TvbsIf68htmFGqA=">AAACPnicbZDLSgMxFIYz3q23qks3wSIoYpkRQd0JIrhUsCp0SjmTOVODSWZMMmoZ+hA+jVt9DF/Anbh14cL0ong7EPLx/+dwkj/KBDfW95+8oeGR0bHxicnS1PTM7Fx5fuHUpLlmWGOpSPV5BAYFV1iz3Ao8zzSCjASeRZf7Xf/sGrXhqTqx7QwbElqKJ5yBdVKzvH7QbNFQ4BW97V+hhNvVEDPDhfNh4wv5WrNc8at+r+hfCAZQIYM6apbfwzhluURlmQBj6oGf2UYB2nImsFMKc4MZsEtoYd2hAommUfQ+1aErTolpkmp3lKU99ftEAdKYtoxcpwR7YX57XfE/r57bZKdRcJXlFhXrL0pyQW1KuwnRmGtkVrQdANPcvZWyC9DArMuxtBJqVHjDUilBxUWYgOSiHWMCubCdIjTJJ7u4gt/h/IXTzWqwVd093qrsbQ6CmyBLZJmskoBskz1ySI5IjTByR+7JA3n0Hrxn78V77bcOeYOZRfKjvLcPvuewMg==</latexit>
�̂(i⌧) =X
µ
�µ�̂(⌧, xµ)
<latexit sha1_base64="zXe5UiJ+MMQS/aZk4veJxK0+Qos=">AAACU3icbZBRaxQxFIWzY9W6Wrvqoy/BpbAFWWZKQX0QFnzxsYLbFjbLcCdzpxOaZIbkRl2G/Tv+Gl8r+Fv0wcx2BW29EPg451ySnKLVylOa/hgkd3bu3ru/+2D48NHe4/3Rk6envglO4lw2unHnBXjUyuKcFGk8bx2CKTSeFZfvev/sEzqvGvuRVi0uDVxYVSkJFKV8NBM1EBeyVnyiBEHgh/wt58IHk3fChPXGyyPxPinaWk14n3vJv/TqYT4ap9N0M/w2ZFsYs+2c5KNfomxkMGhJavB+kaUtLTtwpKTG9VAEjy3IS7jARUQLBv2y2/x0zQ+iUvKqcfFY4hv1740OjPcrU8SkAar9Ta8X/+ctAlWvl52ybSC08vqiKmhODe9r46VyKEmvIoB0Kr6VyxocSIrlDg+EQ4ufZWMM2LITFRilVyVWEDStO+GrPxzrym6WcxtOj6bZ8fTNh+Px7Ghb3C57zl6wCcvYKzZj79kJmzPJvrJv7Ip9H1wNfiZJsnMdTQbbnWfsn0n2fgOdubRb</latexit>
�̂(⌧, x) = e
�x|⌧ |
<latexit sha1_base64="quUVGVAI5FDkLzkUTKjc+IoY1bc=">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</latexit>
⌘̂(~�,~⌧ , x) =1
2x�
NX
k=1
�k�̂2(⌧k, x)
<latexit sha1_base64="9TXhPAUVj1stXrcIBRweJkYRQXo=">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</latexit>
{(�⇤k, ⌧
⇤k )|k = 1, .., N}
<latexit sha1_base64="YgMBv6oX/c06UH3q3yVCokqBZTg=">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</latexit>
E(2)c ⇡ �1
4
NX
k=1
�⇤k{�̂(i⌧⇤k )⌫}2
<latexit sha1_base64="G/Ne2/uKdSbTITysDzcLQsbEAFY=">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</latexit>
E(2)c = �1
4
Z 1
�1d⌧{�̂(i⌧)⌫}2
<latexit sha1_base64="yIpMA1vR/7NUIq8WKvD1fhOFMPE=">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</latexit>
E(2)c = �1
4
Z 1
�1d⌧{�̂(i⌧)⌫}2
<latexit sha1_base64="yIpMA1vR/7NUIq8WKvD1fhOFMPE=">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</latexit>
�(i!k) =NX
l=1
akl�̂(i⌧l)
<latexit sha1_base64="QjsHRhJkWZzAUZcU+q5XLlTYLxM=">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</latexit>
The non-uniform cosine transformM. Kaltak, J. Klimeš, and G. Kresse, JCTC 10, 2498 (2014)
⌘
c(~wk, x) = �(!
⇤k, x)�
NX
l=1
wkl cos(!⇤k⌧
⇤l )
ˆ
�(⌧
⇤l , x)
<latexit sha1_base64="Gd52GqemObQngmuo+4bSKm6uSSs=">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</latexit>
{w⇤kl|k, l = 1, .., N}
<latexit sha1_base64="IyzUD9l+hrTbc/DjlEtFXW8bqPk=">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</latexit>
akl = wkl cos(!⇤k⌧
⇤l )
<latexit sha1_base64="i3wA9e9vS4XGxKVzwYgmfjxWL8Q=">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</latexit>
�̂(i⌧) =1
⇡
Z 1
0d!�(i!) cos(!⌧)
<latexit sha1_base64="QLXKyjNiOYnj1NzZxJvaVYpUntk=">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</latexit>
�(i!) = 2
Z 1
0d⌧ �̂(i⌧) cos(!⌧)
<latexit sha1_base64="8SMKajtP7S62SLUDWqSjdSwu/H8=">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</latexit>
The cosine transform of ! "# ↔! "% :
is reformulated on the minimax frequency and time grids as
where the coefficients '() are the minimax solution to the following fitting problem:
minimax fit
for fixed #(∗ and %(∗ , and for +, ≤ . ≤ max 23 − 25 .
How good are the grids: ZnO and Si
• N4 RPA calculations entirely in frequency § N2 calculation of polarizability in time, then transformation to frequency
(200 atoms on 200 cores in about one hour)
M. Kaltak, J. Klimeš, and G. Kresse, JCTC 10, 2498 (2014); PRB 90, 054115 (2014).
Si defect calculations: 64-216 atoms
New RPA code (VASP6):
• Scales linearly in the numberof k-points (as DFT), instead ofquadratically as for conventionalRPA and hybrid functionals
• Scales cubically in system size (as DFT).
Prefactors are much larger than in DFT, but calculations for 200 atoms take lessthan 1 hour (128 cores)
Scaling
PBE HSE HSE(+vdW) QMC RPA
Dumbbell X 3.56 4.43 4.41 4.4(1) 4.28
Hollow H 3.62 4.49 4.40 4.7(1) 4.44
Tetragonal T 3.79 4.74 4.51 5.1(1) 4.93
Vacancy 3.65 4.19 4.38 4.40
pictures and HSE+vdW:
Gao and Tkatchenko,
PRL 111, 045501 (2013).
QMC:
Parker, Wilkins, and Hennig,
Phys. Status Solidi B 248, 267 (2011).
Defect formation energies in Si
RPA total energy calculation: flow chart
DFTgroundstate
HFenergy
DFTvirtual orbitals
RPAenergy
EDIFF = 1E-8ISMEAR = 0 ; SIGMA = 0.1
NBANDS = ⩽ maximum # of plane wavesALGO = Exact ; NELM = 1ISMEAR = 0 ; SIGMA = 0.1 LOPTICS = .TRUE. [ ; LPEAD = .TRUE. ]
ALGO = Eigenval ; NELM = 1LWAVE = .FALSE.LHFCALC = .TRUE. ; AEXX= 1.0ISMEAR = 0 ; SIGMA = 0.1
NBANDS = ⩽ maximum # of plane wavesALGO = RPAR | ACFDTRNOMEGA = 12-16[ PRECFOCK = Normal ]
… or all in one go!
RPAenergy
EDIFF = 1E-8ISMEAR = 0 ; SIGMA = 0.1LOPTICS = .TRUE. ; LPEAD = .TRUE.
ALGO = RPAR | ACFDTRNOMEGA = 12-16
[ PRECFOCK = Normal ]
[ NBANDSEXACT = # of bands in G ]
When one does not specify NBANDS, VASP wil run all consecutive steps in one go:
In this case the number of bands used to compute the Greens function is specifiedby means of the NBANDSEXACT tag (optional).
An example: SiC
KPOINTS:
Automatically generated mesh0
Gamma3 3 30 0 0
POSCAR:
SiC4.350.5 0.5 0.00.0 0.5 0.50.5 0.0 0.51 1cart0.00 0.00 0.000.25 0.25 0.25
INCAR:
ALGO = ACFDTRNOMEGA = 16NBANDSEXACT = 120PRECFOCK = NORMAL
LOPTICS = .TRUE. ; LPEAD = .TRUE.
ISMEAR = 0 ; SIGMA = 0.05EDIFF = 1E-8
LWAVE = .FALSE.
The GW potentials: *_GW POTCAR files
stdout
running 8 mpi-ranks, with 1 threads/rankdistrk: each k-point on 8 cores, 1 groupsdistr: one band on 1 cores, 8 groupsusing from now: INCARvasp.6.1.2 22Jul20 (build Jul 22 2020 23:58:52) complex
POSCAR found : 2 types and 2 ionsscaLAPACK will be usedLDA part: xc-table for Pade appr. of PerdewPOSCAR, INCAR and KPOINTS ok, starting setupFFT: planning ...WAVECAR not read
N E dE d eps ncg rms rms(c)DAV: 1 -0.807695689957E+00 -0.80770E+00 -0.25358E+03 80 0.517E+02DAV: 2 -0.160589369121E+02 -0.15251E+02 -0.12187E+02 80 0.986E+01DAV: 3 -0.166458202722E+02 -0.58688E+00 -0.58688E+00 96 0.179E+01DAV: 4 -0.166475623089E+02 -0.17420E-02 -0.17420E-02 64 0.947E-01DAV: 5 -0.166475838631E+02 -0.21554E-04 -0.21554E-04 104 0.120E-01 0.992E+00DAV: 6 -0.146217932388E+02 0.20258E+01 -0.80384E+00 96 0.241E+01 0.883E+00DAV: 7 -0.145343752600E+02 0.87418E-01 -0.37382E-01 88 0.483E+00 0.577E+00DAV: 8 -0.145238445286E+02 0.10531E-01 -0.25946E-02 80 0.163E+00 0.429E-01DAV: 9 -0.145238173443E+02 0.27184E-04 -0.60800E-04 96 0.212E-01 0.972E-02DAV: 10 -0.145240248823E+02 -0.20754E-03 -0.27113E-04 88 0.148E-01 0.118E-02DAV: 11 -0.145240278179E+02 -0.29356E-05 -0.22309E-05 96 0.467E-02 0.104E-02DAV: 12 -0.145240279045E+02 -0.86635E-07 -0.18940E-06 80 0.143E-02 0.134E-03DAV: 13 -0.145240279035E+02 0.10211E-08 -0.32364E-08 88 0.170E-03
The Fermi energy was updated, please check that it is located mid-gapvalues below the HOMO (VB) or above the LUMO (CB) will cause erroneous energiesE-fermi : 9.6656optical routinesimaginary and real dielectric functiondirection 1direction 2direction 3
“DFT groundstatecalculation”
“Exact diagonalizationand LOPTICS step”
stdout (cont.): set up “minimax” grids
responsefunction array rank= 120
NTAUPAR set to 8 based on available memory per rank (MAXMEM= 2710 MB)
set NWRITE>2 for more details about minimax calculationnumber of imaginary grid points requested: 16number of distinct grids requested: 4quadrature errors minimized for energies in [ 0.250E+01, 0.186E+03 ]time grid (T=0) determined with error: 0.6315E-12fermionic grid (T=0, re) determined with error: 0.1995E-11bosonic grid (T=0, re) determined with error: 0.1995E-11fermionic grid (T=0, im) determined with error: 0.2003E-11bosonic grid (re): 0.1995E-11
0.3979 1.2370 2.2122 3.4344 5.04817.2506 10.3202 14.6578 20.8519 29.7850
42.8243 62.2141 92.0388 141.1698 235.3471502.7281
fermionic grid (re): 0.1995E-110.3979 1.2370 2.2122 3.4344 5.04817.2506 10.3202 14.6578 20.8519 29.7850
42.8243 62.2141 92.0388 141.1698 235.3471502.7281
fermionic grid (im): 0.2003E-110.8065 1.7011 2.7842 4.1811 6.06008.6543 12.2972 17.4731 24.9000 35.6703
51.5185 75.4212 113.2309 179.3773 325.91881022.1852
time grid: 0.6315E-120.0009 0.0048 0.0121 0.0234 0.03980.0628 0.0950 0.1402 0.2039 0.29340.4195 0.5973 0.8486 1.2074 1.73332.5647
“frequency”
“time”
NTAUPAR is set automaticallybased on the available memory:either set manually by means ofthe MAXMEM-tag, or read from/proc/meminfo (MemAvailable)
stdout (cont.): distribution over ⍵ and "Distributing 16 bosonic points into 8 group(s)Group: 1 has 1 cores and 2 bosonic point(s)Group: 2 has 1 cores and 2 bosonic point(s)Group: 3 has 1 cores and 2 bosonic point(s)Group: 4 has 1 cores and 2 bosonic point(s)Group: 5 has 1 cores and 2 bosonic point(s)Group: 6 has 1 cores and 2 bosonic point(s)Group: 7 has 1 cores and 2 bosonic point(s)Group: 8 has 1 cores and 2 bosonic point(s)
Distributing 16 fermionic points into 8 group(s)Group: 1 has 1 cores and 2 fermionic point(s)Group: 2 has 1 cores and 2 fermionic point(s)Group: 3 has 1 cores and 2 fermionic point(s)Group: 4 has 1 cores and 2 fermionic point(s)Group: 5 has 1 cores and 2 fermionic point(s)Group: 6 has 1 cores and 2 fermionic point(s)Group: 7 has 1 cores and 2 fermionic point(s)Group: 8 has 1 cores and 2 fermionic point(s)
Distributing 16 time points into 8 group(s)Group: 1 has 1 cores and 2 time point(s)Group: 2 has 1 cores and 2 time point(s)Group: 3 has 1 cores and 2 time point(s)Group: 4 has 1 cores and 2 time point(s)Group: 5 has 1 cores and 2 time point(s)Group: 6 has 1 cores and 2 time point(s)Group: 7 has 1 cores and 2 time point(s)Group: 8 has 1 cores and 2 time point(s)
min. memory requirement per mpi rank 448.9 MB, per node 3591.2 MB
NOMEGAPAR = 8
NTAUPAR = 8
(Here running on 8 MPI-ranks)
Performance-wise it is optimal to have NTAUPAR = NOMEGAPAR = NOMEGA.Prerequisite: there has to be enough memory for NTAUPAR = NOMEGA,and the number of MPI-ranks must be a multiple of NOMEGA.
OUTCAR: plane wave basis set limit
cutoff energy smooth cutoff RPA correlation Hartree contr. to MP2---------------------------------------------------------------------------------
182.607 146.086 -11.6224735165 -18.3767216933173.912 139.129 -11.5729431431 -18.3208811769165.630 132.504 -11.5167771988 -18.2570942952157.743 126.194 -11.4562166727 -18.1878044293150.232 120.185 -11.3922444110 -18.1141897888143.078 114.462 -11.3188094759 -18.0290907645136.264 109.012 -11.2261996896 -17.9208940815129.776 103.821 -11.1299318178 -17.8073101088
linear regressionconverged value -12.3629185205 -19.2342034424
ENCUTGW (= 2/3 ENCUT) ENCUTGWSOFT (= 0.8 ENCUTGW)
“Extrapolated to infinite basis set limit”: Ec(G) = Ec(1) +A
G3
<latexit sha1_base64="rIrLwMqjMYnjy2pJgX2nnSpa9Y0=">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</latexit>
Convergence tests:• Investigate convergence w.r.t. ENCUT (not ENCUTGW or ENCUTGWSOFT).• If possible converge energy differences not absolute correlation energies.
OUTCAR: RPA total energy
HF energy using frozen KS orbitalsFree energy of the ion-electron system (eV)---------------------------------------------------alpha Z PSCENC = 10.57645478Ewald energy TEWEN = -285.31033112-Hartree energ DENC = -32.94127495-exchange EXHF = 74.67798493-V(xc)+E(xc) XCENC = 0.00000000PAW double counting = 1473.50548166 -1473.60386908entropy T*S EENTRO = -0.00000000eigenvalues EBANDS = -21.25334570atomic energy EATOM = 229.23716772---------------------------------------------------HF-free energy TOTEN = -25.11173175 eVexchange ACFDT corr. = -0.00000000 see jH, gK, PRB 81, 115126HF+correlation energy = -36.73420527HF+E_corr(extrapolated)= -37.47465027
diagrammatic approximations of correlation energy (eV)-----------------------------------------------------direct MP2 EDMP2 = -19.23420344random phase ERPA = -12.36291852GW Galitskii-Migdal EGWGM = 0.00000000-----------------------------------------------------
“Extrapolated to infinitebasis set limit”
Without extrapolation
RPA: important INCAR tags
• ALGO
• NOMEGA• PRECFOCK• NTAUPAR• NOMEGAPAR• MAXMEM
• NBANDS• NBANDSEXACT
• ENCUTGW
Visit our WIKI!
Especially the parts on:Cubic scaling RPA calculations
Cubic-Scaling GWP. Liu, M. Kaltak, J. Klimeš, and G. Kresse, PRB 94, 165109 (2016)
G0(i⌧)
<latexit sha1_base64="0F/8U/msdwqL5vXL4USdUJ/zIck=">AAACHHicbZDLahtBEEV7HMeR5cRRnKU3TYTA2YgZY4izM2QRL2WwbIFGiJqeGqtRP4buGgcx6DOyVX4mu5BtIP/iRVoPgx+50HC4VUVV36xU0lMc/422Xmy/3HnV2G3uvX6z/7b17uDK28oJ7AurrBtk4FFJg32SpHBQOgSdKbzOpl+W9etbdF5ac0mzEkcabowspAAK1vDrOOZHMiWoPo5b7bgbr8SfQ7KBNtuoN27dpbkVlUZDQoH3wyQuaVSDIykUzptp5bEEMYUbHAY0oNGP6tXJc94JTs4L68IzxFfuw4katPcznYVODTTxT2tL83+1YUXF6aiWpqwIjVgvKirFyfLl/3kuHQpSswAgnAy3cjEBB4JCSs1O6tDgN2G1BpPXaQFaqlmOBVSK5nXqi3sOcSVPw3kOV8fd5KT7+eKkfXa8Ca7BDtkHdsQS9omdsXPWY30mmGXf2YL9iBbRz+hX9HvduhVtZt6zR4r+/APZtaKE</latexit>
�0(i⌧)
<latexit sha1_base64="r1YVGwknZ6xN8WiiLu3/cE7qmto=">AAACH3icbZBLSwMxFIUzvq2vqks3wSLopsyIoO4ENy4VrAqdWu5k7thgkhmSO0oZ+j/c6p9xJ279Ly5MawVfBwIf59xLLicplHQUhm/BxOTU9Mzs3HxtYXFpeaW+unbh8tIKbIlc5fYqAYdKGmyRJIVXhUXQicLL5PZ4mF/eoXUyN+fUL7Cj4cbITAogb13Hoie7Id+WMUG50603wmY4Ev8L0RgabKzTbv09TnNRajQkFDjXjsKCOhVYkkLhoBaXDgsQt3CDbY8GNLpONbp6wLe8k/Ist/4Z4iP3+0YF2rm+TvykBuq539nQ/C9rl5QddCppipLQiM+PslJxyvmwAp5Ki4JU3wMIK/2tXPTAgiBfVG0rtmjwXuRag0mrOAMtVT/FDEpFgyp22Rf7uqLf5fyFi91mtNc8PNtrHO2Oi5tjG2yTbbOI7bMjdsJOWYsJZtkDe2RPwWPwHLwEr5+jE8F4Z539UPD2AYEIo+s=</latexit>
�0(i!)
<latexit sha1_base64="VxZaRVEB/0R1WxmFBBFWDoQfdRY=">AAACI3icbZA7SwNBFIVnfRtfq5Y2g0HQJuxKQO0EG0sFo0I2hLuzd5PBeSwzs0pY8k9s9c/YiY2F/8TCSYzg68DAxzn3MpeTFoJbF0WvwdT0zOzc/MJibWl5ZXUtXN+4tLo0DFtMC22uU7AouMKW407gdWEQZCrwKr05GeVXt2gs1+rCDQrsSOgpnnMGzlvdMExYn3cjussTLbEHe92wHjWisehfiCdQJxOddcP3JNOslKgcE2BtO44K16nAOM4EDmtJabEAdgM9bHtUINF2qvHlQ7rjnYzm2vinHB273zcqkNYOZOonJbi+/Z2NzP+ydunyw07FVVE6VOzzo7wU1Gk6qoFm3CBzYuABmOH+Vsr6YIA5X1ZtJzGo8I5pKUFlVZKD5GKQYQ6lcMMqsfkX+7ri3+X8hcv9RtxsHJ0368f7k+IWyBbZJrskJgfkmJySM9IijNySe/JAHoOH4Cl4Dl4+R6eCyc4m+aHg7QOTNqTv</latexit>
W (i!)
<latexit sha1_base64="J936UEzwywDiNJwYYd5RncUz0Ls=">AAACHHicbZDLSgNBEEV7fBtfUZduGkNAN2EmCOpOcONSwRghE0JNT01s0o+hu0cJQz7Drf6MO3Er+C8u7MQIvi40HG5VUdU3yQW3LgzfgpnZufmFxaXlysrq2vpGdXPryurCMGwxLbS5TsCi4ApbjjuB17lBkInAdjI4Hdfbt2gs1+rSDXPsSugrnnEGzludNt3jsZbYh/1etRY2wonoX4imUCNTnfeq73GqWSFROSbA2k4U5q5bgnGcCRxV4sJiDmwAfex4VCDRdsvJySNa905KM238U45O3O8TJUhrhzLxnRLcjf1dG5v/1TqFy466JVd54VCxz0VZIajTdPx/mnKDzImhB2CG+1spuwEDzPmUKvXYoMI7pqUElZZxBpKLYYoZFMKNythmX+zjin6H8xeumo3ooHF8cVA7aU6DWyI7ZJfskYgckhNyRs5JizCiyT15II/BQ/AUPAcvn60zwXRmm/xQ8PoBRtGixA==</latexit>
W (i⌧)
<latexit sha1_base64="vCCyz+tsW8TJjS7BlpHV4u2iI8I=">AAACGnicbZBNSwMxEIazflu/qh69BIugl7IrgnoTvHhUsFbpFpnNzmowyS7JrFKW/gqv+me8iVcv/hcPprWCXy8EHt6ZYSZvUijpKAzfgrHxicmp6ZnZ2tz8wuJSfXnlzOWlFdgSucrteQIOlTTYIkkKzwuLoBOF7eTmcFBv36J1Mjen1Cuwq+HKyEwKIG9dtPmmjAnKrct6I2yGQ/G/EI2gwUY6vqy/x2kuSo2GhALnOlFYULcCS1Io7Nfi0mEB4gausOPRgEbXrYYH9/mGd1Ke5dY/Q3zofp+oQDvX04nv1EDX7ndtYP5X65SU7XUraYqS0IjPRVmpOOV88HueSouCVM8DCCv9rVxcgwVBPqPaRmzR4J3ItQaTVnEGWqpeihmUivpV7LIv9nFFv8P5C2fbzWinuX+y0zjYHgU3w9bYOttkEdtlB+yIHbMWE0yze/bAHoOH4Cl4Dl4+W8eC0cwq+6Hg9QO1MaHx</latexit>
⌃(i⌧)
<latexit sha1_base64="qmQbQCdBzO4pdJ/yti303jcyWtQ=">AAACH3icbZBLSwMxFIUzvq2vqks3wSLopsyIoO4ENy4VrQqdKncyd2owyQzJHaUM/R9u9c+4E7f+FxemtYKvA4GPc+4ll5MUSjoKw7dgbHxicmp6ZrY2N7+wuFRfXjl3eWkFtkSucnuZgEMlDbZIksLLwiLoROFFcns4yC/u0DqZmzPqFdjR0DUykwLIW1fxqexq4JsyJii3ruuNsBkOxf9CNIIGG+n4uv4ep7koNRoSCpxrR2FBnQosSaGwX4tLhwWIW+hi26MBja5TDa/u8w3vpDzLrX+G+ND9vlGBdq6nEz+pgW7c72xg/pe1S8r2OpU0RUloxOdHWak45XxQAU+lRUGq5wGElf5WLm7AgiBfVG0jtmjwXuRag0mrOAMtVS/FDEpF/Sp22Rf7uqLf5fyF8+1mtNPcP9lpHGyPiptha2ydbbKI7bIDdsSOWYsJZtkDe2RPwWPwHLwEr5+jY8FoZ5X9UPD2Ac+JpBk=</latexit>
⌃(i!)
<latexit sha1_base64="i8pVg44KfMGaaHMHEbBrAn0W0bw=">AAACI3icbZA7SwNBFIVnfRtfq5Y2g0HQJuxKQO0EG0tFo0I2hLuzd+PgPJaZWSUs+Se2+mfsxMbCf2LhJEbwdWDg45x7mctJC8Gti6LXYGJyanpmdm6+trC4tLwSrq5dWF0ahi2mhTZXKVgUXGHLcSfwqjAIMhV4md4cDfPLWzSWa3Xu+gV2JPQUzzkD561uGCZnvCeBbvNES+zBTjesR41oJPoX4jHUyVgn3fA9yTQrJSrHBFjbjqPCdSowjjOBg1pSWiyA3UAP2x4VSLSdanT5gG55J6O5Nv4pR0fu940KpLV9mfpJCe7a/s6G5n9Zu3T5fqfiqigdKvb5UV4K6jQd1kAzbpA50fcAzHB/K2XXYIA5X1ZtKzGo8I5pKUFlVZKD5KKfYQ6lcIMqsfkX+7ri3+X8hYvdRtxsHJw264e74+LmyAbZJNskJnvkkByTE9IijNySe/JAHoOH4Cl4Dl4+RyeC8c46+aHg7QPiE6Ud</latexit>
⌃(!)
<latexit sha1_base64="D5JUFEzjPRNPOLvxU4tWczA2CP8=">AAACInicbZBLSwMxFIUzPmt9VV26CRZBN2VGCupOcONS0arQKeVO5k4N5jEkGaUM/SVu659xJ64Ef4oL01rB14HAxzn3kstJcsGtC8PXYGp6ZnZuvrJQXVxaXlmtra1fWl0Yhi2mhTbXCVgUXGHLcSfwOjcIMhF4ldwej/KrOzSWa3Xh+jl2JPQUzzgD561ubTU+5z0JdCfWEnuw263Vw0Y4Fv0L0QTqZKLTbu09TjUrJCrHBFjbjsLcdUowjjOBg2pcWMyB3UIP2x4VSLSdcnz4gG57J6WZNv4pR8fu940SpLV9mfhJCe7G/s5G5n9Zu3DZQafkKi8cKvb5UVYI6jQdtUBTbpA50fcAzHB/K2U3YIA531V1Ozao8J5pKUGlZZyB5KKfYgaFcIMyttkX+7qi3+X8hcu9RtRsHJ4160d7k+IqZJNskR0SkX1yRE7IKWkRRgryQIbkMRgGT8Fz8PI5OhVMdjbIDwVvHwb4pKo=</latexit>
G(i!)
<latexit sha1_base64="GjblSusOK818FPGPJuX3X5MhbyU=">AAACG3icbZDLSgNBEEV7fMb4irp0MxgCugkzElB3ARe6jGBUzIRQ01MTG/sxdPcoYchfuI0/407cuvBfXNiJEXxdaDjcqqKqb5xxZmwQvHkzs3PzC4ulpfLyyuraemVj88KoXFNsU8WVvorBIGcS25ZZjleZRhAxx8v49nhcv7xDbZiS53aQYVdAX7KUUbDOuj7ZZZES2Ie9XqUa1IOJ/L8QTqFKpmr1Ku9RomguUFrKwZhOGGS2W4C2jHIclqPcYAb0FvrYcShBoOkWk4uHfs05iZ8q7Z60/sT9PlGAMGYgYtcpwN6Y37Wx+V+tk9v0sFswmeUWJf1clObct8off99PmEZq+cABUM3crT69AQ3UupDKtUijxHuqhACZFFEKgvFBgink3A6LyKRf7OIKf4fzFy7262GjfnTWqDb3p8GVyDbZIbskJAekSU5Ji7QJJZI8kBF59Ebek/fsvXy2znjTmS3yQ97rB877ooo=</latexit>
� = �GG
<latexit sha1_base64="3WVnXIguqtGLGgDjbEr/xK15Qf4=">AAACHHicbZDLSgNBEEV7fMb4irp00xgEN4YZCagLQXARlwpGhUyQmp4a09iPobtHCUM+w63+jDtxK/gvLuzECL4uNBxuVVHVN8kFty4M34KJyanpmdnKXHV+YXFpubayem51YRi2mRbaXCZgUXCFbcedwMvcIMhE4EVyczSsX9yisVyrM9fPsSvhWvGMM3De6sSsx+kB3aat1lWtHjbCkehfiMZQJ2OdXNXe41SzQqJyTIC1nSjMXbcE4zgTOKjGhcUc2A1cY8ejAom2W45OHtBN76Q008Y/5ejI/T5RgrS2LxPfKcH17O/a0Pyv1ilcttctucoLh4p9LsoKQZ2mw//TlBtkTvQ9ADPc30pZDwww51OqbsYGFd4xLSWotIwzkFz0U8ygEG5Qxjb7Yh9X9Ducv3C+04iajf3TZv1wZxxchayTDbJFIrJLDskxOSFtwogm9+SBPAYPwVPwHLx8tk4E45k18kPB6wciJaIW</latexit>
W =⌫
1� �⌫
<latexit sha1_base64="5I2dI5qaZhSO+856Rpf6f0FkJ2A=">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</latexit>
⌃ = GW
<latexit sha1_base64="bxFwPDPLQJI0oVkUfIV6IbGIRe8=">AAACGnicbZBNS8NAEIY3ftb6VfXoJVgETyUpgnoQCh70WNHaSlNkspm0i7ubsLtRSuiv8Fr/jDfx6sX/4sHth6DWFxYe3plhZt8w5Uwbz/tw5uYXFpeWCyvF1bX1jc3S1vaNTjJFsUETnqhWCBo5k9gwzHBspQpBhByb4f3ZqN58QKVZIq9NP8WOgK5kMaNgrHUbXLGugNPz5l2p7FW8sdxZ8KdQJlPV70qfQZTQTKA0lIPWbd9LTScHZRjlOCgGmcYU6D10sW1RgkDdyccHD9x960RunCj7pHHH7s+JHITWfRHaTgGmp//WRuZ/tXZm4uNOzmSaGZR0sijOuGsSd/R7N2IKqeF9C0AVs7e6tAcKqLEZFfcDhRIfaSIEyCgPYhCM9yOMIeNmkAc6/mYbl/83nFm4qVb8w8rJ5WG5Vp0GVyC7ZI8cEJ8ckRq5IHXSIJQI8kSG5NkZOi/Oq/M2aZ1zpjM75Jec9y8/7KJC</latexit>
G =1
G�10 � ⌃
<latexit sha1_base64="8rm3w/f2WDkxkiGXr7ytuMIOsD8=">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</latexit>
CT
CT
CT+ST
analyticcontinuation EQP
nk
<latexit sha1_base64="7uie9F+pjMuYOfZt3I7UEfJjxSI=">AAACKHicbZBLSwMxFIUzvq2vqrhyEyyCqzIjgroTRHBZwarQqeVO5o6GJpkhySglzI9xq3/Gnbj1d7gwrRV8HQh8nHMvuZykENzYMHwNJianpmdm5+ZrC4tLyyv11bULk5eaYZvlItdXCRgUXGHbcivwqtAIMhF4mfSPh/nlHWrDc3VuBwV2JdwonnEG1lu9+sbJtYu1pGetqueUi5OM9quqV2+EzXAk+heiMTTIWK1e/T1Oc1ZKVJYJMKYThYXtOtCWM4FVLS4NFsD6cIMdjwokmq4bnV/Rbe+kNMu1f8rSkft9w4E0ZiATPynB3prf2dD8L+uUNjvoOq6K0qJinx9lpaA2p8MuaMo1MisGHoBp7m+l7BY0MOsbq23HGhXes1xKUKmLM5BcDFLMoBS2crHJvtjXFf0u5y9c7Dajvebh2V7jaHdc3BzZJFtkh0RknxyRU9IibcKIIw/kkTwFj8Fz8BK8fo5OBOOddfJDwdsH5Gmnxg==</latexit>
SolveQP-equation
• Minimax grids:M. Kaltak, J. Klimeš, and G. Kresse, JCTC 10, 2498 (2014); PRB 90, 054115 (2014).
• Cubic-scaling RPA total energies (ACFDT):M. Kaltak, J. Klimeš, and G. Kresse, PRB 90, 054115 (2014).
• Cubic-scaling RPA quasi-particles (GW):P. Liu, M. Kaltak, J. Klimeš, and G. Kresse, PRB 94, 15109 (2016).
• Finite temperature grids (!):M. Kaltak and G. Kresse, PRB 101, 205145 (2020)
Dr. Merzuk Kaltak(VASP Software GmbH)
@E
@R= h |@H
@R| i+ h @
@R|H| i+ h |H| @
@Ri
Non-Hellmann-Feynman!
h @ @R
|H| i+ h |H| @ @R
i �!✓Z
G0(i!)@ERPA
@G
����i!
G0(i!)d!
◆@VKS
@R
Hellmann-Feynman theorem: when the orbitals are eigenstates of the Hamiltonian,the second and third terms on the RHS are zero.
In the RPA we need to consider ”non-Hellmann-Feynman” contributions!
These take the following form:
@Ec
@G= ⌃
@EH
@G= VH
@EX
@G= VX
“from linear response”
Forces in the RPA
(Diamond 8 atom super cell)
A test: ”RPA forces” (lines) vs. energy derivatives from finite differences (symbols):
Forces in the RPA
Diamond (128 atom super cell)
Graphite (128 atom super cell)
Summary
• Beyond DFT: the RPA• Well balanced description of all bond types (metallic, covalent, ionic, vdW)• Unfortunately energy differences can still not be predicted with chemical accuracy• But at the moment the optimal balance between accuracy and computational effort
• Cubic-Scaling RPA• Both for total energies as well as for quasi-particles• Minimax frequency and time grids• Near future: finite temperature RPA
• Forces in the RPA
The End
Thank you for your attention!