Path integral Monte Carlo

Charusita Chakravarty

Department of Chemistry

Indian Institute of Technology Delhi

Path integral Monte Carlo Methods

• Finite-temperature ensembles

• Metropolis Monte Carlo

• Path Integral formulation of the density matrix

• Discretised Path integral Methods

• Fourier Path integral methods

• Bosons: The superfluid transition

•Fermions

PARTITION FUNCTIONS AND THERMAL DENSITY MATRICES

Canonical Partition Function

Density Operator

Expectation values

Coordinate Representation of the Density Matrix

Free Particle Density Matrix

Semiclassical Approximation for the High-Temperature Density Matrix

Path integral Representation of the Density Matrix

•Multiple paths connecting initial and final points•Contributions from all possible paths are weighted by the exponential of the Euclidean action• Can be sampled by Monte Carlo methods because of the real exponential•Paths with high action will have high kinetic energy (large slopes) and/or high potential energy.•Classical limit: only the path of least action will survive.•Quantum delocalisation effects are indexed by the thermal de Broglie wavelength

Partition Function:The Quantum-Classical Isomorphism

System of N interacting, distinguishable quantum particles is transformed to a classical, NM-particle system at temperature /M.

Integral over Maxwell-Boltzmanndistribution of NM particles at temperature /M

Potential energy at temperature /M

Harmonic potential withTemperature and Trotter index dependent force constant

Suitable form for simulation by Metropolis Monte Carlo with increased dimensionality because of auxilliary coordinates

Bead-Polymer Picture

• Each quantum particle maps over to a cyclic polymer with M beads

• Bead-bead interactions have different intra- and inter- polymer components

• Adjacent beads on the same polymer are con-nected by harmonic springs

• Beads on different polymers interact with potential V(x) if they correspond to the same position in imaginary time or the same value of Trotter index

• Radius of gyration of quantum polymer approximately equal to thermal de Broglie wavelength of quantum particle

Sampling of Quantum Paths

• Naive Sampling:•Displace beads individually•Large quantum effects imply stiff interpolymer linkages•Ergodicity of random walk difficult to ensure•Very inefficient at ensuring collective motion of polymer chain•Permutation moves will rarely be sampled

• Normal mode transformation•Displace collective modes of quantum polymer•Simple to implement but will not work if quantum•effects are very large

• Bisection•Very effective and will work also for bosons

• Molecular dynamics•Dynamical scheme for sampling configuration space

Ab initio Path Integral Simulations

• Finite-temperature path integral treatment for nucleiand electronic structure methods for electrons• When do we require such methods? Light atoms: H, He, Li, B, C or when interatomic Light atoms: H, He, Li, B, C or when interatomic

potential cannot be readily parametrised because of potential cannot be readily parametrised because of polarisation effects or delocalised electronic orbitalspolarisation effects or delocalised electronic orbitals

• Possible systems: Lithium clusters/Hydrogen-bonded solids e.g. ice

• Coordinate basis/finite temperature for nuclei and groundBorn-Oppenheimer state for electronic degrees of freedom

Molecular Dynamics

• By introducing NM classical particles, each of which is assigned a fictitious mass and momentum, one can write a molecular dynamics scheme will generate the same configurational averages as the MC scheme

• The dynamics will be entirely fictitious and unrelated to the true quantum dynamics.

• Smart computational tricks developed for classical MD can be used to generate more efficient collective motion through configuration space.e.g. multi-ple time-step MD algorithms

• Quantum statistics cannot be incorporated because permutation space is discrete.

• Ergodicity is problematic specially for high Trotter numbers. Require elaborate thermostatting schemes.

• Efficient higher-order propagators cannot be as easily used.

Marx-Parrinello Approach:Primitive approximation + normal mode transformation + molecular

dynamics + density functional theory (www.cpmd.org)

Lithium ClustersPIMC technique:

– Discretised path integral– Takahashi-Imada approximation– Normal-mode sampling– Thermodynamic estimators

Electronic structure calculations– Density functional theory.– Gradient-corrected exchange-correlation functionals.– Localised Gaussian basis set– Basis sets: 3-21G, 6-311G, 6-311G*– Double zeta plus polarization– Large uncontracted basis set.

Results for Li4 and Li5+• Quasiclassical regime- spatial correlation functions are broadened but no tunneling is seen.• HOMO and LUMO eigenvalue distributions also broadened.• Radius of gyration for Li atoms 0.15 A Weht et al, J. Chem. Phys., 1998

Identical Particle Exchange

);',()!/1();',( PxxNxx DP

PI

Density matrix for indistinguishable particles can be written as a sum overpermutations. The path integral will now contain paths which end at x’ as wellas all permutational variants of x’.

Classical Limit: only identitypermutation will surviveBosons: sign factor will always be positive. Must sample over permutations as well as paths.Not problematic in principleFermions: The sign problem

Superfluid Transition in Liquid Helium

Typical ‘‘paths’’ of six helium atoms in 2D. The filled circles are markers for the (arbitrary) beginning of the path. Paths that exit on one side of the square reenter on the other side. The paths show only the lowest 11 Fourier components.(a) shows normal 4He at 2 K (b) superfluid 4He at 0.75 K.