Finite Size Effects
Ceperley Finite Size effects
Periodic boundary conditions
• Minimum Image Convention:take the closest distance
|r|M = min ( r+nL)
Potential is cutoff so that V(r)=0 for r>L/2 since force
needs to be continuous. Remember perturbation theory.
• Image potential
VI =  v(ri-rj+nL)
For long range potential this leads to the Ewald image
potential. You need a back ground and convergence
method.
Ceperley Finite Size effects
The electron gas
D. M. Ceperley, Phys. Rev. B 18, 3126 (1978)
• Standard model for
electrons in metals
• Basis of DFT.
• Characterized by 2
dimensionless
parameters:
– Density
– Temperature
2
H 


2m
i
2
i


i j
1
rij
lo g ( rs )
rs  a / a 0
  e / Ta
2
• What is energy?
• When does it freeze?
• What is spin
polarization?
• What are properties?
  rs
cla ssica l O C P
Ceperley Finite Size effects
  1 7 5 cla ssica l m e ltin g
lo g (  )
Charged systems
How can we handle charged systems?
• Just treat like short-ranged potential: cutoff potential at r>L/2.
Problems:
– Effect of discontinuity never disappears: (1/r) (r2) gets bigger.
– Will violate Stillinger-Lovett conditions because Poisson equation
is not satisfied
• Image potential solves this:
VI = Σv(ri-rj+nL)
– But summation diverges. We need to resum. This gives the ewald
image potential.
– For one component system we have to add a background to make
it neutral.
– Even the trial function is long ranged and needs to be resummed.
Ceperley Finite Size effects
Ewald summation method
• Key idea is to split potential into k-space part and realspace part. We can do since FT is linear.
V=
å f (r -r +nL)
i
i<j,L
V=
åf
k
k
(
j
2
rk - N
)
where rk = å e
ik iri
i
1
and j k = ò dre ik irf (r)
W
4p e 2
2
For f (r)=e /r Þ j k = 2
k
• Bare potential converges slowly at large r (in r-space) and
at large k (in k-space)
Ceperley Finite Size effects
Classic Ewald
• Split up using Gaussian charge
distribution
 (r ) 
erfc ( r )
decays fast at large r
r
k 
4 e
 ( k / 2 )
k
2
2
decays fast at large k
 = convergence param eter
• If we make it large enough we can
use the minimum image potential in
2
r-space.
V dipole 
(2   1) 
• Extra term for insulators:
Ceperley Finite Size effects
2

i
i
How to do it
•
•
•
•
r-space part same as short-ranged potential
k-space part:
1. Compute exp(ik0xi)=(cos (ik0xi), sin (ik0xi)), k0=2/L  i
2. Compute powers exp(i2k0xi)= exp(ik0xi )*exp(ik0xi) etc.
This way we get all values of exp(ik . ri) with just
multiplications.
3. Sum over particles to get k all k.
4. Sum over k to get the potentials.
5. Forces can also be done by taking gradients.
Constant terms to be added.
Checks: perfect lattice: V=-1.4186487/a (cubic lattice).
Ceperley Finite Size effects
O(N3/2)
O(N)
O(N3/2)
O(N3/2)
O(N1/2)
O(N3/2)
O(1)
Optimized Ewald
J. Comput. Physics 117, 171 (1995).
• Division into Long-range and short-ranged function is
convenient but is it optimal? No
• Trial functions are also long-ranged but not simply 1/r. We
need a procedure for general functions.
• Natoli-Ceperley procedure. What division leads to the
highest accuracy for a given radius in r and k?
• Leads to a least squares problem.
• FITPN code does this division.
– Input is fourier transform of vk
on grid appropriate to the supercell
– Output is a spline of vsr(r)
and table of long ranged function.
Ceperley Finite Size effects
Problems with Image potential
• Introduces a lattice structure which may not be appropriate.
• Example: a charge layer.
– We assume charge structure continues at large r.
– Actually nearby fluid will be anticorrelated.
– This means such structures will be penalized.
• One should always consider the effects of boundary conditions,
particularly when electrostatic forces are around!
• You need to have a continuum model to understand the results of a
microscopic simulation.
Ceperley Finite Size effects
•
•
•
Jastrow factor for the e-gas
Look at local energy either in r space or k-space:
r-space as 2 electrons get close gives cusp condition: du/dr|0=-1
K-space, charge-sloshing or plasmon modes.
2  uk 
•
•
k2

1
k
2
Can combine 2 exact properties in the Gaskell form. Write EV in terms structure
factor making “random phase approximation.” (RPA).
2  uk  
•
Vk
1
Sk

1
2
Sk

Vk
k2
S k  id eal stru ctu re facto r
Optimization can hardly improve this form for the e-gas in either 2 or 3 dimensions.
RPA works better for trial function than for the energy.
NEED EWALD SUMS because potential trial function is long range, it also decays
as 1/r, but it is not a simple power.
lim r  
 r 1
 1 / 2
u (r )   r
 log( r )

3D
2D
1D
Long range properties important
•Give rise to dielectric properties
•Energy is insensitive to uk at
small k
•Those modes converge t~1/k2
Ceperley Finite Size effects
Derivation of the e-gas Jastrow
For simplicity, consider boson trial function
 R  e
EL  R  

 u  rij 
i j
Find local energy.
  v  r   2   u  r    
2
ij
ij
i j
i


    i u  rij  
 j

W rite in term s of phonon variables:  k 

e
2
i kri
i
EL  R  
1

2
k
2
 vk  2  k 2u k  


k

k
 p  k  p  kp u k u p
k,p
discard term s k   p : R P A approxi m ation
EL  R  

k
2
 v k  2  k 2 u k  N  k 2 u k2 


k
S olve for u k so that
 0
2  uk  1 
Ceperley Finite Size effects
1
vk
k2
Generalized Feynman-Kacs formula
• Let’s calculate the average population resulting from
DMC starting from a single point R0 after a time `t’.
t
P  R0 ; t  
 R
 dR   R 
R e
 t  H  ET


 dtE L ( t )
R0 
e
0
0

expand the density m atrix in term s of ex act eigenstates
P  R0 ; t  
 dR
 R
  R0 
lim t   P  R 0 ; t  
    R     R 0 e
*

0  R0 
  R0 
 0
t
0  R0 
  R0 

 dt
~e
EL (t )

0
Ceperley Finite Size effects
 t  E  E T

Wavefunctions beyond Jastrow
smoothing
1
 n H n 
• Use method of residuals construct  n  1 ( R )   n ( R ) e
a sequence of increasingly better
i  k r
trial wave functions. Justify from
0  e
the Importance sampled DMC.
• Zeroth order is Hartree-Fock
E0  V (R )
wavefunction
U  R 



e
1
0
• First order is Slater-Jastrow pair
2
wavefunction (RPA for electrons
E1  U ( R )    W ( R )   i  k j   r j   jY  R  
gives an analytic formula)
j
• Second order is 3-body backflow
wavefunction
• Three-body form is like a squared
force. It is a bosonic term that does
not change the nodes.
j
j
j
e x p {  [   ij ( rij )(ri  r j )] }
2
i
j
Ceperley Finite Size effects
Backflow wave function
• Backflow means change the
coordinates to quasi- coordinates.
D et {e
ik i r j
}  D et {e
x i  ri 

ij
ik i x j
}
( rij )(ri  r j )
j
• Leads to a much improved energy
and to improvement in nodal
surfaces. Couples nodal surfaces
together.
Kwon PRB 58, 6800 (1998).
Ceperley Finite Size effects
3DEG
Dependence of energy on wavefunction
3d Electron fluid at a density rs=10
Kwon, Ceperley, Martin, Phys. Rev. B58,6800, 1998
-0.107
-0.1075
Energy
• Wavefunctions
– Slater-Jastrow (SJ)
– three-body (3)
– backflow (BF)
– fixed-node (FN)
• Energy < |H| > converges to ground
state
• Variance < [H-E]2 > to zero.
• Using 3B-BF gains a factor of 4.
• Using DMC gains a factor of 4.
-0.108
FN -SJ
-0.1085
FN-BF
-0.109
Ceperley Finite Size effects
0
0.05
Variance
0.1
Comparison of Trial functions
• What do we choose for the trial function in VMC and DMC?
• Slater-Jastrow (SJ) with plane wave orbitals :
  u ij ( rij )
 2 ( R )  D e t { k  r } e i  j
j
• For higher accuracy we need to go beyond this form.
• Need correlation effects in the nodes.
• Include backflow-three body.
Example of incorrect physics within SJ
Ceperley Finite Size effects
Analytic backflow
Holzmann et al, Phys. Rev. E 68, 046707:1-15(2003).
• Start with analytic Slater-Jastrow using Gaskell trial function
• Apply Bohm-Pines collective coordinate transformation and express
Hamiltonian in new coordinates
• Diagonalize resulting Hamiltonian.
• Long-range part has Harmonic oscillator form.
• Expand about k=0 to get backflow and 3-body forms.
• Significant long-range component to BF
OPTIMIZED BF
ANALYTIC BF
rs=1,5,10,20
• 3-body term is non-symmetric
 2 ( R ) e x p {   iW y  R   iW u  R }
Ceperley Finite Size effects
i
Results of Analytic tf
Analytic form EVMC better for rs<20 but not for rs20.
Optimized variance is smaller than analytic.
Analytic nodes always better! (as measured by EDMC)
Form ideal for use at smaller rs since it will minimize optimization
noise and lead to more systematic results vs N, rs and polarization.
• Saves human & machine optimization time.
• Also valuable for multi-component system of metallic hydrogen.
•
•
•
•
Ceperley Finite Size effects
Twist averaged boundary conditions
• In periodic boundary conditions (
point), the wavefunction is
periodicLarge finite size effects for
metals because of shell effects.
• Fermi liquid theory can be used to
correct the properties.
• In twist averaged BC we use an
arbitrary phase  as r r+L
• If one integrates over all phases the
momentum distribution changes from
a lattice of k-vectors to a fermi sea.
• Smaller finite size effects
PBC
TABC
Ceperley Finite Size effects
  e
ik r
k L  2 n  
kx
i
 ( x  L)  e  ( x )
A 

1
 2 
3


3
d    A 
Twist averaged MC
• Make twist vector dynamical by changing during the
random walk.
   i  
i= 1 ,2 ,3 
• Within GCE, change the number of electrons
• Within TA-VMC
– Initialize twist vector.
– Run usual VMC (with warmup)
– Resample twist angle within cube
– (iterate)
• Or do in parallel.
Ceperley Finite Size effects
Grand Canonical Ensemble QMC
• GCE at T=0K: choose N such that E(N)-N is minimized.
• According to Fermi liquid theory, interacting states are related to noninteracting states and described by k.
• Instead of N, we input the fermi wavevector(s) kF. Choose all states
with k < kF (assuming spherical symmetry)
• N will depend on the twist angle . = number of points inside a
randomly placed sphere.
kn 
2
L
n

L
kn  kF
• After we average over  (TA) we get a sphere of filled states.
• Is there a problem with Ewald sums as the number of electrons varies?
No! average density is exactly that of the background. We only work
with averaged quantities.
Ceperley Finite Size effects
Single particle size effects
• Exact single particle properties with TA within HF
• Implies momentum distribution is a continuous curve with a sharp
feature at kF.
• With PBC only 5 points
• Holzmann et al. PRL 107,110402 (2011)
2

3
2
• No size effect within single particle theory!
T  d k
k n(k )
2m
• Kinetic energy will have much smaller size effects.
Ceperley Finite Size effects
Potential energy
•
Write potential as integral over structure function:
4
i(r r
3
V   d k 2 S (k )
S ( k )    k  k  1  ( N  1) e
k
1
•
•
•
2
)k
Error comes from 2 effects.
2
– Approximating integral by sum
S HF ( k )  1 
 q  q ' k

– Finite size effects in S(k) at a given k.
N q ,q '
Within HF we get exact S(k) with TABC.
Discretization errors come only from non-analytic points of S(k).
– the absence of the k=0 term in sum. We can put it in by hand since we know
the limit S(k) at small k (plasmon regime)
– Remaining size effects are smaller, coming from the non-analytic behavior of
S(k) at 2kF.
Ceperley Finite Size effects
3DEG at rs=10
TABC
TABC+1/N
PBC
GC-TABC+1/N
We can do simulations with N=42!
Size effects now go like:
We can cancel this term at special values of N!
N= 15, 42, 92, 168, 279,…
Ceperley Finite Size effects
Brief History of Ferromagnetism
in electron gas
What is polarization state of fermi liquid at low density?
 
N  N
N  N
• Bloch 1929 got polarization from exchange interaction:
– rs > 5.4 3D
– rs > 2.0 2D
• Stoner 1939: include electron screening: contact interaction
• Herring 1960
• Ceperley-Alder 1980 rs >20 is partially polarized
• Young-Fisk experiment on doped CaB6 1999 rs~25.
• Ortiz-Balone 1999 : ferromagnetism of e gas at rs>20.
• Zong et al Redo QMC with backflow nodes and TABC.
Ceperley Finite Size effects
Ceperley, Alder ‘80
T=0 calculations
with FN-DMC
3d electron gas
• rs<20 unpolarized
• 20<rs<100 partial
• 100<rs Wigner crystal
Energies are very close together at low density!
More recent calculations of Ortiz, Harris
and Balone PRL 82, 5317 (99) confirm this
result but get transition to crystal at
rs=65.
Ceperley Finite Size effects
Polarization of 3DEG
• We see second order partially
polarized transition at rs=52
• Is the Stoner model (replace
interaction with a contact potential)
appropriate? Screening kills long
range interaction.
• Wigner Crystal at rs=105
•Twist averaging makes calculation
possible--much smaller size effects.
•Jastrow wavefunctions favor the
ferromagnetic phase.
•Backflow 3-body wavefunctions
more paramagnetic
Polarization
transition
Ceperley Finite Size effects
Phase Diagram
• Partially polarized
phase at low density.
• But at lower energy
and density than
before.
• As accuracy gets
higher, polarized
phase shrinks
• Real systems have
different units.
Ceperley Finite Size effects
Recent calculations in 2D
Tanatar,Ceperley ‘89
Rapisarda, Senatore ‘95
Kwon et al ‘97
T=0 fixed-node calculation:
Also used high quality backflow
wavefunctions to compute energy
vs spin polarization.
Energies of various phases are
nearly identical
Attaccalite et al:
PRL 88, 256601 (2002)
2d electron gas
•rs <25 unpolarized
•25< rs <35 polarized
•rs >35 Wigner crystal
para
Ceperley Finite Size effects
ferro
solid
Polarization of 2D electron gas
• Same general trend in 2D
• Partial polarization before freezing
Results using phase averaging and BF-3B wavefunctions
rs=10
rs=20
Ceperley Finite Size effects
rs=3
0
Linear response for the egas
• Add a small periodic potential.
• Change trial function by replacing plane waves with solutions to the
Schrodinger Eq. in an effective potential.
• Since we don’t care about the strength of potential use trial function to
find the potential for which the trial function is optimal.
• Observe change in energy since density has mixed estimator problems.
Ceperley Finite Size effects
Fermi Liquid parameters
• Do by correlated sampling: Do one long MC random walk with a
guiding function (something overlapping with all states in question).
• Generate energies of each individual excited state by using a weight
function
 (R)
w ( R ) 
G 
2

 G (R)
 |  ( R ) |
2

• “Optimal Guiding function” is
• Determine particle hole excitation energies by replacing
columns:fewer finite size effects this way. Replace columns in slater
matrix
• Case where states are orthogonal by symmetry is easier, but nonorthogonal case can also be treated.
• Back flow needed for some excited state since Slater Jastrow has no
coupling between unlike spins.
Ceperley Finite Size effects