1549380323-Statistical Mechanics Theory and Molecular Simulation

Thermodynamics 457

extra factor ofP in the force constant and the fact that eqn. (12.2.29) hasP− 1
integrations in one spatial dimension, we obtain

ρ(x,x′;β) =

(

m

2 πβ ̄h^2

) 1 / 2

exp

[

−

m

2 β ̄h^2

(x−x′)^2

]

. (12.2.30)

Interestingly, thePdependence completely disappears so that the limit can be taken
trivially. Moreover, by substitutingβ=it/ ̄h, the quantum propagator for a free par-
ticle can also be deduced from eqn. (12.2.30):

U(x,x′;t) =

( m

2 πi ̄ht

) 1 / 2

exp

[

im

2 ̄ht

(x−x′)^2

]

. (12.2.31)

Analytical evaluation of the path integral is only possible for general quadratic
potentials. Nevertheless, the path integral formalism renders quantum statistical me-
chanical calculations tenable with modern computers, even for large systems for which
determination of the eigenvalues ofHˆ is intractable. Of course, such computations
can only be performed numerically for finiteP, which leads to discrete path integral
representations of the density matrix and partition function.Pshould be large enough
that the difference between the discrete path integral and the formal limitP→∞is
negligible. Methodology for performing path integral calculations in imaginary time
will be discussed in Sections 12.6.1 and 12.6.2. We will see that such calculations are a
little more complicated than analogous calculations in the classical canonical ensemble
(see Section 4.8) but straightforward, in principle. Moreover, they converge on time
scales similar to those of classical calculations. Unfortunately, thesame is not true for
the quantum propagator in eqn. (12.2.23) due to the complex exponential in the in-
tegrand. The latter causes numerical calculations to oscillate wildly as different paths
are sampled, leading to a severe convergence problem known as thedynamical sign
problem. Thus, while computing quantum equilibrium properties via path integrals
has become routine, the calculation of dynamical properties from path integrals re-
mains one of the most challenging problems in computational physics and chemistry.
As of the writing of this book, no truly satisfactory solution has been achieved.

12.3 Thermodynamics and expectation values from the path integral

Path integral expressions for expectation values of Hermitian operators follow from
the basic relation

〈Aˆ〉=

1

Q(L,T)

Tr

[

Aˆe−βHˆ

]

. (12.3.1)

Performing the trace in the coordinate basis gives

〈Aˆ〉=

1

Q(L,T)

∫

dx〈x|Aˆe−β

Hˆ

|x〉. (12.3.2)

(We will not continue to include the spatial domainD(L) in the expressions, but it
must be remembered that the spatial integrals carry this restriction implicitly.) A