Quantum Monte Carlo simulations of solid helium:Keeping cool under pressure

Robert Hinde, Dept. of Chemistry, University of Tennessee

1. Low-temperature solids2. Monte Carlo methods3. Quantum Monte Carlo4. Three-body and four-body interactions in solid helium

Quantum Monte Carlo simulations of solid helium:Keeping cool under pressure

Robert Hinde, Dept. of Chemistry, University of Tennessee

Quantum Monte Carlo simulations of solid helium:Keeping cool under pressure

Robert Hinde, Dept. of Chemistry, University of Tennessee

Quantum Monte Carlo simulations of solid helium:Keeping cool under pressure

Robert Hinde, Dept. of Chemistry, University of Tennessee

The physical systems that we study are low-temperature crystals of He(and also of H2 and D2 molecules). Weakly attractive pair interactionsand small masses mean that quantum effects are important.

In these systems, quantum effects manifest themselves as large-amplitude zero-point motions of the particles around their nominallattice sites, accompanied by strong nearest-neighbor correlations.

H2 : Vmol = 23.16 cm3

D2 : Vmol = 19.95 cm3

H2 : κ = 0.54 kbar –1

D2 : κ = 0.31 kbar –1

Silvera, Rev. Mod.Phys. (1980)

0 10

1

What’s thearea?

0 10

16 blocks wide and 6 blocks tall

0 10

1

The basic idea is at least a couple of centuries old:

Buffon’s Needle (1777) is a method for estimating the value of π by

repeatedly dropping a needle on a hardwood floor:

The basic idea is at least a couple of centuries old:

Buffon’s Needle (1777) is a method for estimating the value of π by

repeatedly dropping a needle on a hardwood floor:

H hits in N attempts → π ≈ 2 N / H

Let’s watch a short movie of a Buffon’s Needle experiment . . .

“state”

relativeprobability

a b c d e f

“state”

relativeprobability

a b c d e f

. . . a c a b e c b f e b f f e b a e f . . .

“state”

relativeprobability

a b c d e f

. . . a c a b e c b f e b f f e b a e f . . .

“state”a b c d e f

What is the probability thatwe will observe the system instate b? In state d?

abcdef

12π

3–14

State Happiness

How happy is thesystem on average?

abcdef

12π

3–14

State Happiness

How happy is thesystem on average?

1 × P(a) + 2 × P(b) + π × P(c) + 3 × P(d) – 1 × P(e) + 4 × P(f)

Metropolis and coworkers developed a simulation method that allowsus to generate long sequences of states:

. . . a b f c b f a e f c f a e c b b a c f e f f a b c a f . . .

in which, for a long enough sequence, each state appears with thecorrect relative probability. The method is often called MetropolisMonte Carlo simulation, or sometimes simply Monte Carlo simulation.

Metropolis and coworkers developed a simulation method that allowsus to generate long sequences of states:

. . . a b f c b f a e f c f a e c b b a c f e f f a b c a f . . .

in which, for a long enough sequence, each state appears with thecorrect relative probability. The method is often called MetropolisMonte Carlo simulation, or sometimes simply Monte Carlo simulation.

The method made possible the first realistic simulations of atomicand molecular condensed phases (such as liquid water). In thesesimulations, the physical system of interest has an infinite number ofpossible states, whose relative probabilities vary over many orders ofmagnitude. Even sketching the probability distribution function isimpossible!

Metropolis and coworkers developed a simulation method that allowsus to generate long sequences of states:

. . . a b f c b f a e f c f a e c b b a c f e f f a b c a f . . .

in which, for a long enough sequence, each state appears with thecorrect relative probability. The method is often called MetropolisMonte Carlo simulation, or sometimes simply Monte Carlo simulation.

0. Pick an initial state.1. Choose a prospective new state.2. If the new state has equal or higher relative probability, go there.3. If the new state has lower relative probability, calculate the ratio R = P (new) / P (old), which will be less than 1. Go to the new state with probability R, and stay where you are otherwise.4. Return to step 1 and repeat.

“state”

relativeprobability

a b c d e f

Let’s try it out . . .

Atomic positions cannot be measured (or known) with infiniteprecision (remember Heisenberg’s uncertainty principle) . . .

So to build in quantum mechanicaleffects, we create many replicas . . .

force constant k ∝ m

Thank you,Dr. Feynman!

By following the motions of many,many replicas, we get a statisticalpicture of the quantum mechanical

probability density function for acomplicated many-atom system.

We don’t know the probability densityas a function that we can write down,

but maybe we don’t need to!

Observable properties are integralsover the probability density . . . So we

can just use Monte Carlo ideas!

The fundamental simulation objectis a sequence of replicas of thephysical system . . .

with particles in adjacent replicasconnected by harmonic springs.

In the simple case shown here, eachreplica has 8 Cartesian coordinates(4 atoms in 2 dimensions):

x1

y1

x2

y2

x3

y3

x4

y4

. . . .. . . .

–pΔτ –Δτ 0 Δτ pΔτ

imaginary time

R

We can loosely interpret the replica chain as a polymer whose “atoms”move in a very high-dimension space. (Here the number of atoms N =864, and each atom has 3 Cartesian coordinates. 3 × 864 = 2592!)

Ψ22592

5,000 to 10,000 replicas

. . . .. . . .

–pΔτ –Δτ 0 Δτ pΔτ

imaginary time

R

k

Ψ2

That’s an awful lot of data to specify the state of the polymer! Thesimulation is going to be slow! Is there any way to speed things up?

We notice that every replica only “talks” to two others:

So we have one big advantage:

All “even” replicas can be moved simultaneously (as can all “odd”replicas), so it’s easy to exploit parallel architectures.

Alternate the simulation between group A and group B replicas.

A

B

What can we do now? Calculate the quantum mechanical energy ofa helium crystal under many different conditions — in this case, atmany different values of the density.

Why helium? The He–He interaction potential is known to very highaccuracy. This makes it solid helium a good arena to test quantumchemical calculations of three-body interactions.

+ +

≠

effect of 3-bodyinteractions

2+3-body

2-body

What can we do now? Calculate the quantum mechanical energy ofa helium crystal under many different conditions — in this case, atmany different values of the density.

effect of 3-bodyinteractions

2+3-body

2-body

Thermodynamics then lets us calculate the external pressureassociated with a given density from our knowledge of how the energydepends on density.

Remember the First Law of Thermodynamics: dE = – p dV + T dS

effect of 3-bodyinteractions

2+3-body

2-body

Thermodynamics then lets us calculate the external pressureassociated with a given density from our knowledge of how the energydepends on density.

Remember the First Law of Thermodynamics: dE = – p dV + T dS

At zero Kelvin,dE = – p dV orp = –(dE / dV)T=0

Compute E(ρ) and fit to a polynomial in ρ, then differentiate to obtainpressure as a function of density:

Experimental results from Driessen et al. (Phys. Rev. B, 1986)

Three-body interactions clearly become important at moderatelyhigh densities.

Three-body interactions clearly become important at moderatelyhigh densities.

Four- and many-body interactions manifest themselves at evenhigher compressions. . . A new challenge for quantum chemistry!

Quantum simulations of solid helium:Keeping cool under pressure

Robert Hinde, Dept. of Chemistry, University of Tennessee

1. Low-temperature solids2. Monte Carlo methods3. Quantum Monte Carlo4. Three-body and four-body interactions in solid helium

Ellen Brown

William Werner

Matt Wilson