Download Path integral Monte Carlo

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 HighTemperature 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 e=b/M.
Integral over Maxwell-Boltzmann
distribution of NM particles at
temperature e=b/M
Potential energy at
temperature e=b/M
Harmonic potential with
Temperature 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 interpolymer 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 nuclei
and electronic structure methods for electrons
• When do we require such methods?
Light atoms: H, He, Li, B, C or when interatomic
potential cannot be readily parametrised because of
polarisation effects or delocalised electronic orbitals
• Possible systems: Lithium clusters/Hydrogen-bonded
solids e.g. ice
• Coordinate basis/finite temperature for nuclei and ground
Born-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 Clusters
PIMC 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
 I ( x, x' ; b ) = (1 / N!)   D ( x, Px' ; b )
P
P
Density matrix for indistinguishable particles can be written as a sum over
permutations. The path integral will now contain paths which end at x’ as well
as all permutational variants of x’.
Classical Limit: only identity
permutation will survive
Bosons: sign factor will always
be positive. Must sample over
permutations as well as paths.
Not problematic in principle
Fermions: 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.