Next: 7.3 Wave Packet Dynamics
Up: 7. Quantum Mechanical Simulation
Previous: 7.1 Diffusion Monte Carlo
7.2 Path Integral Monte Carlo (PIMC)
So far: only ground state, i.e. .
How to treat quantum systems at finite temperature?
Let a quantum system be in contact with a heat bath of temperature
; it is then in a mixed state made up of the
energy eigenstates:
|
(7.17) |
In classical statistical mechanics the probability density of states
is given by the Boltzmann factor. In quantum mechanics the
density matrix serves the same purpose:
The average of some observable
is given by
|
(7.19) |
where
.
Problem:
The explicit form of
is usually complex or
unknown.
Solution: Reduce the above expression to one that contains only known
and tractable density matrices.
Back to school:
What are the explicit forms of the density matrix for very simple models?
Strategy of PIMC:
Express the density matrix of any given system in terms of the
free particle density 7.24.
This can be done because:
The density matrix may therefore be written as
|
(7.29) |
The integrand in the path integral on the r.h.s.
consists of two density matrices at , i.e. double the original
temperature.
Now we iterate, writing
instead of
:
|
(7.30) |
The number of intermediate steps is called the Trotter number.
If is large () the high temperature will act to damp
the effect of the potential :
with diagonal element
|
(7.32) |
where
|
(7.33) |
and
|
(7.34) |
Three dimensions:
|
(7.35) |
with the same as above, and
|
(7.36) |
Visualization / Classical isomorphism:
The expression 7.35 looks like a classical Boltzmann factor
for a ring polymer of links [CHANDLER 81]:
Each link is under the influence of an external potential
|
(7.37) |
while successive links are coupled by a harmonic bond potential
|
(7.38) |
with .
Figure 7.2:
Classical isomorphism for one particle
|
Therefore: To solve the quantum problem, apply Monte Carlo method to the
classical ring polymer.
Figure 7.3:
PIMC for one particle
|
Note:
- The oscillator strength
increases with larger
Trotter numbers , requiring smaller MC steps
- The external potential
is damped by larger , allowing for larger steps
Move the entire ring polymer by a large random step, then
displace the individual elements by a small amount.
Or:
Displace the center of mass of the chain by a wide random step,
then construct the entire ring polymer anew,
sampling the single element positions from the narrow multivariate Gauss
distribution (see Chapter ``Stochastics''):
|
(7.39) |
interacting quantum objects:
- Assume a pair potential
.
- Each quantum particle is represented by a classical
-element ring chain.
- Let
be the position of element in chain
(=particle)
- The diagonal element of the total density matrix is then
with
.
The pair potential acts only between respective links
() of different chains.
EXERCISE:
Write a PIMC program treating the case of one particle of mass in a
two-dimensional oscillator potential
.
Let the Trotter number vary between and .
Determine the positional probability
of the particle from the
relative frequency of residence at , averaged over all chain links. Noting
that
|
(7.41) |
we would expect for the two-dimensional harmonic oscillator (with
)
|
(7.42) |
(For convenience, put .) Draw several configurations of the ring
polymer that occur in the course of the simulation.
Sample applications:
- Solvated electron in molten KCl [PARRINELLO 84]: localized or not?
Answer: clear localization.
- Solvation of electrons in simple fluids [COKER 87]
- electron in liquid helium: strongly localized
- electron in liquid xenon smeared out (see Figure)
Reason: the shell of He is rigid and difficult to polarize,
repulsing an extra electron; the shell of Xe is easily polarizable,
producing a long-ranged, locally flat dipole potential.
Figure 7.4:
From Coker et al.: solvated electron
a) in liquid helium, b) in liquid xenon
|
- Solid parahydrogen [ZOPPI 91]:
Kinetic energy in the lattice from PIMC agrees well with results from
neutron scattering.
- Recent survey of PIMC applications: [CEPERLEY 95]; see also
the web site [CEPERLEY WWW].
Next: 7.3 Wave Packet Dynamics
Up: 7. Quantum Mechanical Simulation
Previous: 7.1 Diffusion Monte Carlo
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001