Order(N) methods in QMC

Dario Alfè [d.alfe@ucl.ac.uk]

2007 Summer School on Computational Materials Science Quantum Monte Carlo: From Minerals and Materials to MoleculesJuly 9 –19, 2007 • University of Illinois at Urbana–Champaignhttp://www.mcc.uiuc.edu/summerschool/2007/qmc/

• Evaluation of multidimensional integrals:• Quadrature methods, error ~ M-4/d

• Monte Carlo methods, error ~ M-0.5

I = d ( ) g∫R R

I = lim

M→ ∞

1M

f (Rm)m=1

M

∑⎧⎨⎩

⎫⎬⎭≈

1M

f (Rm)m=1

M

∑

( ) 0; d ( )=1 ℘ ≥ ℘∫R R R

I = dRf (R)℘(R); f (R)=g(R) ℘(R) ∫

Monte Carlo methods

*

0*

ˆ( ) ( )

( ) ( )

T T

V

T T

H dE E

d

Ψ Ψ= ≥

Ψ Ψ∫∫

R R R

R R R

Variational Monte Carlo

2 1

2

ˆ( ) ( ) ( )

( )

T T T

V

T

H dE

d

−⎡ ⎤Ψ Ψ Ψ⎣ ⎦=Ψ

∫∫

R R R R

R R

℘(R) =ΨT (R)

2ΨT (R)

2dR∫

E

V≈

1M

EL(Rm)m=1

M

∑ ; EL(Rm) =ΨT (Rm)−1 HΨT (Rm)

( )( , ) ˆ ( , )T

tH E t

i t

∂Ψ− = − Ψ

∂

RR

Extracting the ground state: substitute = it

0, ( , ) ( )τ τ→ ∞ Ψ → ΦR R

Fixed nodes:

0, ( , ) ( )FNτ τ→ ∞ Ψ → ΦR R

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Diffusion Monte Carlo

• N electrons• N single particle orbitals i • ~ N basis functions for each i if

φi (r j ) = cG exp{−iG⋅r j}

G∑

Cost proportional to N3

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Cost of evaluating ΨT

Cost of total energy proportional to N4

Variance of E is proportional to N

HOWEVERVariance of E/atom is proportional to 1/N

Cost of energy/atom proportional to N2

(relevant to free energies in phase transitions, surface energies, ….)

Cost of evaluating Energy E

• N electrons• N single particle orbitals i • ~ N basis functions for each i if

φi (r j ) = cG exp{−iG⋅r j}

G∑

Cost proportional to N3

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Cost of evaluating ΨT

• N electrons• N single particle orbitals i

• ~ o(1) basis functions for each i if

Cost proportional to N2

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Cost of evaluating ΨT

φi (r j ) = cl fl (r j )

l∑ if fl(r) is

localised

Cost of total energy proportional to N3

Variance of E is proportional to N

HOWEVERVariance of E/atom is proportional to 1/N

Cost of energy/atom proportional to N

(relevant to free energies in phase transitions, surface energies, ….)

Cost of evaluating Energy E

B-splines: Localised functions sitting at the points of a uniform grid

( )

32

3

3 3( ) 1 0 1

2 41

2 1 24

f x x x x

x x

= − + ≤ ≤

= − ≤ ≤

E. Hernàndez, M. J. Gillan and C. M. Goringe, Phys. Rev. B, 20 (1997)

D. Alfè  and M. J. Gillan, Phys. Rev. B, Rapid Comm., 70, 161101, (2004)

• N electrons• N single particle orbitals i

• ~ o(1) basis functions for each i if

Cost proportional to N2

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Cost of evaluating ΨT

φi (r j ) = cl fl (r j )

l∑

• N electrons• ~ o(1) single particle localised orbitals i,if i is localised• ~ o(1) basis functions for each i if

Cost proportional to N (Linear scaling)

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

Cost of evaluating ΨT

φi (r j ) = cl fl (r j )

l∑

Cost of total energy proportional to N2

Variance of E is proportional to N

HOWEVERVariance of E/atom is proportional to 1/N

Cost of energy/atom independent on N !(relevant to free energies in phase transitions, surface energies, ….)

Cost of evaluating Energy E

( )

32

3

3 3( ) 1 0 1

2 41

2 1 24

f x x x x

x x

= − + ≤ ≤

= − ≤ ≤

Xi=ia

Θ(x−Xi ) = f ((x−Xi ) / a)Grid spacing a:

B-splines (blips)

Grid spacing 1:

• Defined on a uniform grid• Localised: f(x)=0 for |x|>2• Continuous with first and second derivative continuous

ψ n(r) = ani

i∑ Θ(r −R i )

Blips in three dimensions:

Θ(r −R i ) =Θ(x−Xi )Θ(y−Yi )Θ(z−Zi )

For each position r there are only 64 blip functions that are non zero, for any size and any grid spacing a. By contrast, in a plane wave representation the number of plane waves is proportional to the size of the system, and also depends on the plane wave cutoff:

Single-particle orbital representation:

ψ n(r) = cnG

G∑ eiG⋅r

For example, the number of plane waves in MgO is ~ 3000 per atom!(HF pseudopotentials, 100 Ha PW cutoff)

One dimension:L

/a L N

↑=

( ) ( )j x x jaΘ ≡Θ −^

χn(x) = Θ j (x)e

2π ijn/N

j=0

N−1

∑ − 12 N + 1 ≤ n ≤

12 N

max

2 /2 / ; /( ) n

n

ik xinx Lk n L k an x e eπ

π πφ = == =

Approximate equivalence between blips and plane waves

fµ(q) = f(x)e−iqxdx

−∞

+∞

∫ = f(x)e−iqxdx−2

+2

∫ =3q4 3−4cos(q) + cos(2q)[ ]

φn χ m = dx θ j (x)e2π ijm /Ne−ikn x =j=0

N −1

∑0

L

∫ kn =2π n

aN

= e2π i(m−n) j /N

j=0

N −1

∑ θ j (x)0

L

∫ e2π in( j−x /a)/Ndx = (use PBC) x − ja → x,Θ j → Θ0

= e2π i(m−n) j /N

j=0

N −1

∑Nδnm

1 244 34 4

θ0 (x)0

L

∫ e−2π inx /aNdx = Nδ nm f (x / a)0

L

∫ e−2π inx /aNdx =

=Nδ nma f (y)

0

L /a

∫ e−2π iny/Ndy = Lδ nm fµ(2π n / N )

Now consider:

αn =φn χ n

φn φn

2χ n χ n

2

If χn and n were proportional then αn = 1, and blip waves would be identical to plane waves. Let’s see what value of αn has for a few values on n.

n kn αn

0 0 1.00

1 π/6a 0.9986

3 π/2a 0.9858

6 π/a 0.6532

Take N=12, for example

kn= 2πn / Na

kmax

=π / a

Therefore if we replace n with χn we make a small error at low k but maybe a significant error for large values of k. Let’s do it anyway, then we will come back to the size of the error and how to reduce it.

ψ n(r) = cnG

G∑ eiG⋅r

Representing single particle orbitals in 3 d

χG (r) = Θ j (r)

j∑ eiG⋅R j

eiG⋅r ; γGχG (r)

ψ n(r) ; γG

G∑ cnGχG (r) = cnG

G∑ γG Θ j (r)e

iG⋅R j

j∑ = anj

j∑ Θ j (r)

a

nj= γGcnG

G∑ eiG⋅R j (note that in DA & MJG, PRB 70, 16101 (2004) γ

Gshould be 1/γ

G)

ψ n(r) = anj

j∑ Θ j (r)

max k

/aπ

max k

/aπmax

/a kπ=

max /a kπ<

Improving the quality of the B-spline representation

May not be good enoughReduce a and achieve better quality

Blips are systematically improvable

B splines PWn n

nB splines B splines PW PWn n n n

ψ ψα

ψ ψ ψ ψ

−

− −=

x y z .999977 .892597 .991164 .991164 .996006

.99999998 .995315 .999988 .999988 .999993

1 max( / )a kα π=

α1(a=π / 2kmax)

Example: Silicon in the ß-Sn structure, 16 atoms, 1st orbital

PW Blips (a=/kmax) Blips (a=/2kmax)

Ek (DFT) 43.863

Eloc(DFT) 15.057

Enl(DFT) 1.543

Ek(VMC) 43.864(3) 43.924(3) 43.862(3)

Eloc(VMC) 15.057(3) 15.063(3) 15.058(3)

Enl(VMC) 1.533(3) 1.525(3) 1.535(3)

Etot(VMC) -101.335(3) -101.277(3) -101.341(3)

(VMC) 4.50 4.74 4.55

T(s/step) 1.83 0.32 0.34

Etot(DMC) -105.714(4) -105.713(5) -105.716(5)

(VMC) 2.29 2.95 2.38

T(s/step) 2.28 0.21 0.25

Energies in QMC (eV/atom) (Silicon in ß-Sn structure, 16 atoms, 15 Ry PW cutoff)

PW Blips (a=/kmax) Blips (a=/2kmax)

Ek(VMC) 178.349(49) 178.360(22) 178.369(22)

Eloc(VMC) -225.191(50) -225.128(24) -225.177(23)

Enl(VMC) -17.955(25) -17.974(11) -17.976(11)

Etot(VMC) -227.677(8) -227.648(4) -227.669(4)

(VMC) 14 15 14.5

T(s/step) 7.8 5.6x10-2 7.1x10-2

Energies in QMC (eV/atom) (MgO in NaCl structure, 8 atoms, 200 Ry PW cutoff)

Blips and plane waves give identical results, blips are 2 orders of magnitude faster on 8 atoms, 3 and 4 order of magnitudes faster on 80 ad 800 atoms respectively (on 800 atoms: Blips 1 day, PW 38 years)

Ωω

Maximally localised Wannier functions (Marzari Vanderbilt) [Williamson et al., PRL, 87, 246406 (2001)]

φi (r) = cmiψm(

m=1

N

∑ r) Linear scaling obtained by truncating orbitals to zero outside their localisation region.

Unitary transformation:

And:

D{φi (r)} =D{ψm(r)}

Achieving linear scaling

ΨT (R) =exp[J (R)] cn

n∑ Dn

↑{ψ i (r j )}Dn↓{ψ i (r j )}

=exp[J (R)] cn

n∑ Dn

↑{φi (r j )}Dn↓{φi (r j )}

The VMC and DMC total energies are exactly invariant with respect to arbitrary (non singular) linear combinations of single-electron orbitals! Transformation does not have to be unitary.

φi (r) = cmiψm(

m=1

N

∑ r)

Arbitrary linear combination:

And:

D{φm(r)} =D{cij}D{ψ i (r)}

EL(R) =ΨT (R)−1 HΨT (R)

Provided D{c

ij}≠0

We have set of orbitals extending over region .( )nψ r Ω ΩωWant to make orbital

maximally localised in sub-region .

Maximise “localisation weight” P:

( ) ( )n nn

cφ ψ= ∑r r

ω∈Ω

( ) ( )22

P d dω

φ φΩ

= ∫ ∫r r r r

Express P as:, ,

m mn n m mn nm n m n

P c A c c A cω∗ ∗ Ω=∑ ∑where: , mn m n mn m nA d A dω

ω

ψ ψ ψ ψ∗ Ω ∗

Ω

= =∫ ∫r r

Weight P is maximal when cn eigenvector of mn n mn nn n

A c A cωαλ

Ω=∑ ∑associated with maximum eigenvalue , and maxλ maxP λ=

Linear scaling obtained by truncating localised orbitals to zero outside their localisation region . Then for given r number of non-zero orbitals is O(1).

ω

( )mφ r

Convergence oflocalisation weight Q=1-P in MgO. Squares: present method (D. Alfè  and M. J. Gillan, J. Phys. Cond. Matter 16, L305-L311 (2004))

Diamonds: MLWF(Williamson et al. PRL, 87, 246406 (2001)

Localisation weight in MgO

Convergence oflocalisation weight Q=1-P in MgO. Squares: present method (D. Alfè  and M. J. Gillan, J. Phys. Cond. Matter 16, L305-L311 (2004))

Diamonds: MLWF(Williamson et al. PRL, 87, 246406 (2001)

Localisation weight in MgO

Convergence of VMC energy.Squares: Non-orthogonal orbitals. Diamons: MLWF

Convergence of DMC energy, non-orthogonal orbitals

VMC and DMC energies in MgO

Convergence of VMC energy.Squares: Non-orthogonal orbitals. Diamons: MLWF

Convergence of DMC energy, non-orthogonal orbitals

VMC and DMC energies in MgO

Spherical cutoff at 6 a.u on a 7.8 a.u. cubic box (64 atoms): - WF evaluation 4 times faster- Memory occupancy 4 times smaller