Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

1QUANTUM STATISTICAL PHYSICS APPROACH TO EST CALCULATIONS

WITH PATH-INTEGRAL MONTE CARLO

Tapio T. Rantala,Department of Physics, Tampere University of Technology

http://www.tut.fi/semiphys

CONTENTS1. MOTIVATION2. PI APPROACH TO QUANTUM MECHANICS3. PI APPROACH TO STATISTICAL MECHANICS4. MONTE CARLO SAMPLING OF PATHS5. THE METHOD: PIMC6. CASE STUDIES

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

1. MOTIVATION

Electronic structure is the key quantity to materials properties and related phenomena:Mechanical, thermal, electrical, optical,... .

Conventional ab initio / first-principles type methods• suffer from laborious description of electron–

electron correlations (CI, MCHF, DFT-funtionals)• typically ignore nuclear (quantum) dynamics and

coupling of electron–nuclei dynamics (Born–Oppenheimer approximation)

• give the zero-Kelvin description, only. So, how about the effects from finite temperature?

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

2. PATH-INTEGRAL APPROACH TO QUANTUM MECHANICS

2.1. CLASSICAL PATH

Let us consider particle dynamics from a to b.Lagrangian formulation of classical mechanics for finding the path/trajectory leads to equations of motion from minimization (extremum) of action

where the Lagrangian L = T–V.

=> =>

For example, the classical action of the free-particle is

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S = L(x,x, t)dtta

tb

∫ ,

S = 12m Δx

Δt"

#$

%

&'2

Δt = 12m (xb −xa )

2

(tb − ta ).

δS = 0ddt∂L∂x

−∂L∂x

= 0

a = (xa , ta )

b = (xb, tb )

Δx, Δt

−∂V∂x

=mx

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

2.2. QUANTUM PATH

Usually, the most probable quantum path is the classical one, but other paths contribute, too, with a certain probability. Quantum probability of the particle propagation is P(b,a) = |K(b,a)|2 ,the absolute square of the probability amplitude K.The probability amplitude is the sum over all oscillating phase factors of the paths xab as

where the phase is proportional to the action

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K(b,a) = φ[xab ]all xab

∑

φ[x(t)]= A×exp iS[x(t)]

#

$%

&

'(

φ

PARTICLE PHOTON

Class. Geom. mech. optics

Quantum Physicalmechanics optics

λ

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

2.3. PATH-INTEGRAL

Now, let us define the sum over all paths as a path-integral

We call this ”kernel” or ”propagator” or ”Green’s function”.For example, the free-particle propagator takes the form

This kernel satisfies the free-particle Schrödinger equation in space {xb,tb}.

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K(b,a) = e(i/ )S[b,a ]a

b

∫ Dx(t).

K0(b,a) =m

2πi(tb − ta )#

$%

&

'(

1/2

exp im(xb −xa )2

2(tb − ta )#

$%

&

'(.

Feynman–Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965)Feynman R.P., Rev. Mod. Phys. 20, 367–387 (1948)Feynman R.P., Statistical Mechanics (Westview, Advance Book Classics, 1972)

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

3. PATH-INTEGRAL APPROACH TO STATISTICAL MECHANICS

3.1. THERMAL EQUILIBRIUM, NVT ENSEMBLE

Let us consider thermal equilibrium in NVT ensemble, where we have the Boltzmann distribution exp(–E/kT) of the system total energy E. In the occupied density-of-states, note possible degeneracy.Normalization of Boltzmann probability

is done with the partition function

Alternatively, normalization can be done with the Helmholtz free energy

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pn =1Ze−βEn p(E) = 1

Ze−βE

Z = e−βEn

n∑ .

pn = e−β(En−F ) Z = e−βF .

β =1kT

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

3.2. SOME THERMODYANMICS

Expected total energy or internal energy becomes as

All standard thermodynamic quantities and relations can be derived from the partition function Z or free energy F.

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E =U =1Z

Ene−βEn

n∑ = kT2 ∂(lnZ)

∂T=∂(βF)∂β

.

∂F∂T

= −S

∂F∂V

= −P F =U−TS

dU = −PdV+dQ

dQ = TdS

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

3.3. DENSITY MATRIX

Considering all states of the particle, probability p(x) of finding the particle/system in configuration space at x, we have

Now, define the mixed state density matrix (in position presentation)

(Or more generally for operators ) Thus, we find

and normalization implies

(Trace, Spur)

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φn (x)

P(x) = pn (x)n∑ =

1Z

φ*n (x)φn (x)n∑ e−βEn .

ρ(x ',x) = φ*n (x ')φn (x)n∑ e−βEn .

Z = ρ(x,x)∫ dx = Tr(ρ).

P(x) = 1Zρ(x,x)

ρ(β) = e−βH.

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

3.4. PATH-INTEGRAL EVALUATION

Now, compare

and

in equilibrium (constant hamiltonian) and tb > ta.Replacing (tb – ta) by –iℏβ or β = i(tb – ta)/ℏ (imaginary time period) we

obtain ρ(b,a) for which Cf.

for a time independent hamiltonian. Thus, we can evaluate the density matrix from path-integral similarly

where the imaginary time action is

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K(b,a) = φ*n (a)φn (b)n∑ e−(1/ )En ( tb−ta )

ρ(x ',x) = φ*n (x ')φn (x)n∑ e−βEn

∂ρ(b,a)∂β

= −Hb ρ(b,a).∂K(b,a)∂tb

= −iHb K(b,a)

S[x(u);β,0]= m2x2(u)+V(x(u))

"

#$%

&'0

β

∫ du.

ρ(xb,xa;β) = e(− i/ )S[β,0 ]all x(u )∫ Dx(u),

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

4. MONTE CARLO SAMPLING OFIMAGINARY TIME PATHS

For the density operator we can write

if the kinetic and potential energies in the hamiltonian H = T + Vcommute. This becomes exact at the limit of imaginary time period goes to zero, the high temperature limit, because the potential energy approaches constant in position representation.Thus, we can write

where and M is called the Trotter number.This allows numerical sampling of the imaginary time paths with a Monte Carlo method.

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ρ(β) = e−βH = e−β/2He−β/2H ,

ρ(r0 , rM;β) = ρ(r0 , r1; τ)ρ(r1, r2; τ) ...ρ(rM−1, rM; τ)∫∫∫ dr1dr2 ...drM−1,

τ =β /M

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

DISCRETIZE THE PATH ...

using Feynman path-integral

where M is the Trotter number andWith short enough τ = β/Mthe action

approaches that of free particleas .

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

... FOR EVALUATION WITH MONTE CARLO

MC allows straightforward numerical procedure for evaluation of multidimensional integrals.

Metropolis Monte Carlo• NVT (equilibrium) ensemble• yields mixed state density matrix with almost classical transparency

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

5. QUANTUM STATISTICAL PHYSICS PIMC APPROACH

An ab initio electronic structure approach with• FULL ACCOUNT OF CORRELATION• TEMPERATURE DEPENDENCE• BEYOND BORN–OPPENHEIMER

APPROXIMATION• DESCRIPTION OF A CHEMICAL

REACTION

• So far, without the exchange interaction

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

5.1. SIZE AND TEMPERATURE SCALES COMPARISON

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T / K

100 000

10 000

1 000RT

100

10

10 100 1000 10 000 100 000 size/atomsDFT

QCDFT

MOLDY

PIMC PATH INTEGRAL MONTE CARLO APPROACH

Ab initio-MOLDY

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

5.2. QUANTUM / CLASSICAL APPROACHESTO DYNAMICS COMPARISON

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time dependent

T > 0equilibrium

T = 0

MOLECULAR

Wave packet approaches ?

TDDFT

Car–Parrinello

and

DYNAMICS

Metropolis Monte Carlo Rovibrational

!ab initio MOLDY

Molecular mechanics

approaches ab initio Quantum

Chemistry / DFT /

semiemp.

electronic dyn.:

nuclear dyn.:

Class

Q

Q

Q

Q Q

Class

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

6.1. CASEH3

+ MOLECULE

Quantum statistics of two electrons and three nuclei (five-particle system) as a function of temperature:

• Structure and energetics:• quantum nature of nuclei• pair correlation functions,

contact densities, ...• dissociation temperature

• Comparison to the data from conventional quantum chemistry.

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e–

p+ p+

p+

e–

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

ENERGETICS AT ZERO KELVIN LIMIT: FINITE NUCLEAR MASS AND ZERO-POINT MOTION

Total energy of the H3+ ion up to the dissociation temperature.Born–Oppenheimer approximation, quantum nuclei andclassical nuclei.

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

MOLECULAR GEOMETRY AT ZERO KELVIN LIMIT: ZERO-POINT MOTION

Internuclear distance. Quantum nuclei, classical nuclei, with FWHM of distrib.

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Snapshot from simulation, projection to xy-plane. Trotter number 216.

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PARTICLE–PARTICLE CORRELATIONS:ELECTRON–NUCLEI COUPLING

Pair correlation functions. Quantum p (solid), classical p (dashed) and Born–Oppenheimer (dash-dotted).

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p–p

e–ep–p

p–e

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0 5000 10000 15000

ï1.30

ï1.20

ï1.10

ï1.00

ï0.90

ï0.80

H3+

H2+H+

H2++H

2H+H+

Barrier To Linearity

T (K)

Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

ENERGETICS AT HIGH TEMPERATURES:ELECTRON–NUCLEI COUPLING

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Total energy of the H3+ beyond dissociation temperature.Lowest density,mid density,highest density

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

DISSOCIATION–RECOMBINATION EQUILIBRIUM REACTION

The molecule and its fragments:

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The equilibrium composition of fragments at about 5000 K.

ï1.3 ï1.2 ï1.1 ï1.00

20

40

60

80

100

120

140

160 H3+ H2+H+ H2

++H 2H+H+

BTL

E (units of Hartree)

Num

ber o

f Mon

te C

arlo

Blo

cks

We also evaluate molecular free energy, entropy and heat capacity!

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

6.2. CASEDIPOSITRONIUM Ps2

• Pair approximation and matrix squaring

• Bisection moves• Virial estimator for the kinetic energy

• same average ”time step” for all temperatures ‹ τ › = β/M = 0.015, M ≈ 105

! M = 217 ... 214

e–

e–

e+

e+

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

Ps2 @ 0K – 900K

We find:• EPs = – 0.250• EPs2

= – 0.5154• ED = 0.0154 ≈ 0.435 eV :≈ 5000 K

Ps2 @ T ≤ 900 K

Ps

Ps2 @ T = 0 K

T ≤ 900 K

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

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”APPARENT” DISSOCIATION ENERGY

Dissociation energy from the equilibrium total energies as

DT = 2 EPs – EPs2

Low density limit !

ED = 0.0154 ≈ 0.44 eV ≈ 5000 K

@ 950 KkT ≈ 0.082 eV

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Department of Physics, TCOMP-EST NEST 15–17 Aug 2012Tapio Rantala: QUANTUM STATISTICAL PHYSICS APPROACH WITH PIMC

CASE REFERENCES25

0 5000 10000 15000

ï1.30

ï1.20

ï1.10

ï1.00

ï0.90

ï0.80

H3+

H2+H+

H2++H

2H+H+

Barrier To Linearity

T (K)

For more details see http://www.tut.fi/semiphys1. Ilkka Kylänpää, PhD Thesis (Tampere University of Technology, 2011).2. Ilkka Kylänpää and Tapio T. Rantala, Phys. Rev. A 80, 024504, (2009); I. Kylänpää, M. Leino and T.T. Rantala, Phys. Rev. A 76, 052508, (2007).3. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 133, 044312 (2010).4. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 135, 104310 (2011).

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