Post on 11-Apr-2018
Cornell University
Towards a diffusion Monte Carlo study of Mott transition
in MnO under pressure
Wissam A. Al-Saidi
Cornell University
Outline
1. Brief introduction to diffusion Monte Carlo
2. Improving trial wave functions:
Long range periodic Jastrow
silicon interstitials
backflow transformations:
First row atoms
Quantum dots
3. Towards MnO . . .
Pseudopotential validation
1
Cornell University
CollaborationsCyrus Umrigar
MnO
Richard Hennig
Kevin Driver
John Wilkins
Quantum dots
Devrim Guclu
Harold Baranger
Atomic systems
Frank Petruzielo
Julien Toulouse
2
Introduction to DMC
Ground state is obtained by projection from trial wavefunction ΨT (R):
ΨGS(R) = limt−→∞
e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)
R is a 3N dimensional vector and H is Hamiltonian of system.
3
Introduction to DMC
Ground state is obtained by projection from trial wavefunction ΨT (R):
ΨGS(R) = limt−→∞
e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)
R is a 3N dimensional vector and H is Hamiltonian of system.
Direct approach is extremely inefficient and not stable . . . sign problem!
Fixed node approximation: Use the nodes of a trial wave function as a boundary
condition
3-a
Introduction to DMC
Ground state is obtained by projection from trial wavefunction ΨT (R):
ΨGS(R) = limt−→∞
e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)
R is a 3N dimensional vector and H is Hamiltonian of system.
Direct approach is extremely inefficient and not stable . . . sign problem!
Fixed node approximation: Use the nodes of a trial wave function as a boundary
condition
T
Node
GS
Ψ
Ψ
Stable solution:
exact if nodes are exact
variational if nodes are ap-
proximate
Find the best trial WF! VMC optimization
3-b
Trial wavefunctions in DMC
Slater type:
ΨT (R) =ndet∑
i
ci D↑i D↓
i
ci are CI coefficients, and D↑i and D↓
i are spin-up and spin-down determinants
R = (~r1,~r2, . . . ,~rN) 3N-dimensional vector.
4
Trial wavefunctions in DMC
Slater type:
ΨT (R) =ndet∑
i
ci D↑i D↓
i
ci are CI coefficients, and D↑i and D↓
i are spin-up and spin-down determinants
R = (~r1,~r2, . . . ,~rN) 3N-dimensional vector.
Slater-Jastrow type:
ΨT (R) = eJ(R)∑
ci D↑i D↓
i
J(R) = Jee + Jen + Jeen + . . .
J(R) introduces 1-body, 2-body, 3-body, . . . correlations. Forces ΨT (R) to satisfy
cusp conditions.
J(R, α) is parametrized. α = αee, αen, αeen and CI coefficients are
optimized with VMC
Slater-Jastrow type works well for molecular systems and solids.
4-a
Trial wavefunctions in DMC (periodic systems)
Jastrow factors suitable for isolated systems are short ranged!
long-range Jastrow correlations: Prendergast, Bevan, Fahy PRB ’02
JLR(R) = J(1)LR + J
(2)LR
J(1)LR =
∑
stars
nelec∑
i
ps cos(
~k · ~ri
)
+ qs sin(
~k · ~ri
)
J(2)LR =
∑
stars
nelec∑
i,j
rs cos[
~k · (~ri − ~rj)]
rs, ps, and qs are variational parameters optimized with VMC.
5
Trial wavefunctions in DMC (periodic systems)
Jastrow factors suitable for isolated systems are short ranged!
long-range Jastrow correlations: Prendergast, Bevan, Fahy PRB ’02
JLR(R) = J(1)LR + J
(2)LR
J(1)LR =
∑
stars
nelec∑
i
ps cos(
~k · ~ri
)
+ qs sin(
~k · ~ri
)
J(2)LR =
∑
stars
nelec∑
i,j
rs cos[
~k · (~ri − ~rj)]
rs, ps, and qs are variational parameters optimized with VMC.
For one-body J(1)LR all stars consistent with the point group of the crystal are included:
non-inversion symmetry case
symmorphic and non-symmorphic cases
5-a
Silicon interstitials (preliminary)
-66.8
-66.79
-66.78
-66.77
-66.76
-66.75
-66.74
VM
C e
nerg
y (H
artr
ee)
H T X
With LRPJ (cosine)
With LRPJ (cosine+sine)
Short range Jastrow
DMC study using CHAMP by Batista and coworkers PRB ’06
(16 +1) atom supercell- (64 + 4) electrons. Three kinds of interstitials H, T and X
variance reduction: From σ2 = 0.8 to σ2 = 0.6 Hartree2
smaller locality error, efficient simulation, . . .
6
Trial wavefunctions in DMC
Slater type:
ΨT (R) =ndet∑
i
ci D↑i D↓
i
Slater-Jastrow type:
ΨT (R) = eJ(R)∑
ci D↑i D↓
i
J(R) = Jee + Jen + Jeen + . . .
J(R) introduces 1-body, 2-body, 3-body, . . . correlations.
For solids, Slater-Jastrow with periodic long-range correlations:
JLR(R) =∑
s
nelec∑
i
[
ps cos(
~k · ~ri
)
+ qs sin(
~k · ~ri
)]
+∑
s
nelec∑
i,j
rs cos[
~k · (~ri − ~rj)]
With a single determinant, all these WFs have same nodes!
all DMC energies should be the same
7
backflow transformations
Feynman-Cohen backflow (liquid He II): Phs. Rev. ’56
excited-state WF with an excitation of momentum ~q:
∏
i
ei~q·~riφ0 −→∏
i
ei~q·~xiφ0
where ~xi = ~ri +∑
j 6=i η(rij)~rij
e
8
backflow transformations
Feynman-Cohen backflow (liquid He II): Phs. Rev. ’56
excited-state WF with an excitation of momentum ~q:
∏
i
ei~q·~riφ0 −→∏
i
ei~q·~xiφ0
where ~xi = ~ri +∑
j 6=i η(rij)~rij
In Fermionic systems, backflow transformations modify the nodes
Applied to the Fermionic homogeneous electron gas [Schmidt PRL ’81]
Generalized to inhomogeneous systems [Rios PRE ’06]
~xi = ~ri +∑
j 6=i
η(ri, rj , rij)~rij +∑
j 6=i
ξ(ri, rj , rij)~ri
ri
rije
e−−
backflow is highly desirable in solids
implemented in champ and applied to molecular systems and quantum dots
8-a
VMC energies of first row atoms (preliminary)
Li Be B C N O F Ne
-1
-0.8
-0.6
-0.4
-0.2
0
Dev
iatio
ns f
rom
exa
ct v
alue
(eV
)
VMC-1detVMC-BF-1det only VMC with 1-det
Li Be B C N O F Ne40
50
60
70
80
90
100
vari
ance
% c
hang
e Large reduction in variance
9
Quantum dots
Quantum dots are very similar to atoms (artificial atoms)
H =∑
i
(
−1
2∇i
2 +1
2ω2r2
i
)
+∑
i 6=j
1
|ri − rj |
Highly tunable systems (vary ω)— strongly correlated physics (Wigner crystallization)
10
Quantum dots
Quantum dots are very similar to atoms (artificial atoms)
H =∑
i
(
−1
2∇i
2 +1
2ω2r2
i
)
+∑
i 6=j
1
|ri − rj |
Highly tunable systems (vary ω)— strongly correlated physics (Wigner crystallization)
Electron density:
Ω=0.0015625 Ω= 0.0005 Ω=0.0001
10-a
qdots with backflow
N = 7 and w = 0.0015625
L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2
1.255
1.256
1.257
1.258
1.259
1.26
1.261
1.262
1.263
1.264
1.265
1.266
1.267
1.268
Ene
rgy
(Ryd
berg
* )
VMC (SJ)DMC (SJ)
11
qdots with backflow
N = 7 and w = 0.0015625
L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2
1.255
1.256
1.257
1.258
1.259
1.26
1.261
1.262
1.263
1.264
1.265
1.266
1.267
1.268
Ene
rgy
(Ryd
berg
* )
VMC (SJ)DMC (SJ)VMC (BF+SJ)DMC (BF+SJ)
12
qdots with backflow
N = 7 and w = 0.0015625
L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2
1.255
1.256
1.257
1.258
1.259
1.26
1.261
1.262
1.263
1.264
1.265
1.266
1.267
1.268
Ene
rgy
(Ryd
berg
* )
VMC (SJ)DMC (SJ)VMC (BF+SJ)DMC (BF+SJ)
pair density g↓,↓ (N=7, w = 0.0005)
VMC DMC VMC-BF
12-a
Where are we?
1. Brief review of diffusion Monte Carlo
2. Trial wave functions?
3. Long range periodic Jastrow
silicon intersteials
4. backflow transformations:
First row atoms
Quantum dots
5. Towards MnO . . .
Pseudopotentials validation
HF or GGA pseudopotentials
13
Study of anti-ferromagnetic MnO (preliminary)
AF FCC structure: Mn moments anti-parallel along adjacent sheets in the [111] plane
Experimentally:
First-order insulator-metal transition near 100 GPa
volume collapse by ≈ 5% equilbrium volume
moment collapse: 5 −→ 1 bohr magneton
Rich physics Cohen Science ’97, Kasinathan PRB ’06, Kunes Nature materials ’08
14
All-electron equation of state (preliminary)
Calculations done with GGA/PBE using WIEN2k
175 200 225 250 275 300
Volume (Bohr3)
-1233.9
-1233.9
-1233.8
Ene
rgy
(Har
tree
)
Non-magnetic
Ferromagnetic anti-ferromagnetic
Three different phases of MnO (FCC structure)
AFM phase is the ground state under ambient pressures
volume collapse in AFM phase for high pressures
15
Magnetic and volume collapse (all-electron) (preliminary )
160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320
Volume (bohr3)
1
2
3
4
5
Magnetic moment at Mn site
Energy
high-spin to low-spin transition (5/2 −→ 1/2) Kasinathan PRB ’06,
16
Towards a DMC of MnO
Which pseudopotentials to use?
Wide spread belief that HF pseudopotentials are better than DFT ones in QMC
HF pseudopotentials are cleaner in descreening step (DFT fixed by NLCC)
Limited experience shows that HF and GGA PsP are okay but not LDA one
OPIUM can generate HF and DFT pseudopotentials Al-Saidi, Walter and Rappe
PRB’08
small core pseudopotentials
17
Towards a DMC of MnO
Which pseudopotentials to use?
Wide spread belief that HF pseudopotentials are better than DFT ones in QMC
HF pseudopotentials are cleaner in descreening step (DFT fixed by NLCC)
Limited experience shows that HF and GGA PsP are okay but not LDA one
OPIUM can generate HF and DFT pseudopotentials Al-Saidi, Walter and Rappe
PRB’08
small core pseudopotentials
What kind of trial WF?
We want to optimize the orbitals
For high pressures, GGA trial wavefunctions will be okay
Near the equilibrium geometry, LDA+U (or HF) might be best
17-a
Non magnetic phase (preliminary)
90 100 110 120 130 140 150
Volume (Bohr3)
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Ene
rgy
(eV
)AE PsP (GGA)PsP (HF)
GGA/PBE calculations
GGA and HF PsPs
Excellent agreement between all-electron and pseudopotential results
18
AFM magnetic phase (preliminary)
180 200 220 240 260 280 300
Volume (Bohr3)
0
1
2
3
4
5
Ene
rgy
(eV
)
AEGGA (PsP)HF (PsP)
GGA/PBE calculations
GGA and HF PsPs
pseudopotential equilibrium volume is ≈ 1 % smaller than all-electron one
Birch-Murnaghan B0 and B0′ are also in excellent agreement
pseudopotentials do not seem to give a volume collapse????
19
AFM magnetic phase (aligning equilb. positions)
200 250 300
Volume (Bohr3)
0
2
4
6E
nerg
y (e
V)
AEGGA (PsP)HF (PsP)
GGA/PBE calculations
GGA and HF PsPs
High pressures, larger deviations between pseudopotential and all-electron
change in magnetic properties of system
similar behavior has been seen by Kolorenc and Mitas PRB ’07
20
Cornell University
Summary
Most general long-range periodic Jastrow factor for solids
Backflow transformations:
atoms and molecules
quantum dots
solids . . . easily implemented
Investigating different pseudopotentials for our MnO study
Future work
orbital optimization for solids
finite-size correction scheme
al-saidi@cornell.edu
21