1. Path integral Monte Carlo
Charusita Chakravarty
Department of Chemistry
Indian Institute of Technology Delhi

2. 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

3. PARTITION FUNCTIONS AND THERMAL DENSITY MATRICES
Canonical Partition Function
Density Operator
Expectation values

4. Coordinate Representation of the Density Matrix

5. Free Particle Density Matrix

6. Semiclassical Approximation for the High-Temperature Density Matrix

7. Path integral Representation of the Density Matrix

8. 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

9. 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.
Potential energy at
temperature e=b/M
Integral over Maxwell-Boltzmann
distribution of NM particles 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

10. 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

13. 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

20. 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

21. 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.

22. Marx-Parrinello Approach:
Primitive approximation + normal mode transformation + molecular dynamics + density functional theory (

23. 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 A
Weht et al, J. Chem. Phys., 1998

24. Identical Particle Exchange
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

25. 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.