modynamic quantities that agree with the quantum corrections known from

statistical mechanics.

3.5.1 Feynman-Hibbs potential

In Sections 3.3.7 (page 55) and 3.3.8 (page 57) the intrinsic quantum widths

of free and of harmonically-bound particles were derived. Both are Gaussian

3.5 Quantum corrections to classical behavior 71

distributions, with variances:

2

2

σ= (free particle) (3.106)

12 mkB T

1 ω kT

’

σ2 = coth (harmonic particle) (3.107)

mω 2 2kB T ω

These can be used to modify pair potentials. We shall avoid the complica-

tions caused by the use of a reference potential,8 needed when the harmonic

width is used, and only use the free particle distribution. Feynman and

Hibbs (1965) argued that each pair interaction Vij (rij ) = U (r) between

two particles i and j with masses mi and mj should be modi¬ed by a 3D

convolution with the free-particle intrinsic quantum distribution:

s2

2 ’3/2

ds U (|r + s|) exp ’ 2 ,

FH

Vij (r) = (2πσ ) (3.108)

2σ

where

r = r ij = r i ’ r j , (3.109)

σ2 = , (3.110)

12μkB T

m1 m2

μ= . (3.111)

m1 + m2

This is the Feynman“Hibbs potential. It can be evaluated for any well-

behaved interaction function U (r) from the integral (we write z = cosine of

the angle between r and s):

√ ∞ 1

s2

’3

+ ’ 2rsz) exp ’ 2 .

FH 2

r2 s2

Vij (r)= (σ 2π) ds dz 2πs U (

2σ

’1

0

(3.112)

Some insight is obtained by expanding U to second order in s/r. Using

1 s2 s3

s

’ 2rsz = r 1 ’ z + (1 ’ z ) + O( 3 ) ,

2

r2 s2

+ (3.113)

2 r2

r r

√

U ( r2 + s2 ’ 2rsz) expands as

1 U (r)

U = U (r) ’ szU (r) + s2 (1 ’ z 2 ) + z 2 U (r) . (3.114)

2 r

Evaluating the integral (3.112), we ¬nd

2 2U (r)

FH

Vij (r) = U (r) + + U (r) . (3.115)

24μkB T r

8 See Mak and Andersen (1990) and Cao and Berne (1990) for a discussion of reference potentials.

72 From quantum to classical mechanics: when and how

It is left to Exercises 3.2 and 3.3 to evaluate the practical importance of

the potential correction. For applications to Lennard-Jones liquids see Ses´

e

(1992, 1993, 1994, 1995, 1996). Guillot and Guissani (1998) applied the

Feynman“Hibbs approach to liquid water.

3.5.2 The Wigner correction to the free energy

The Wigner 2 corrections to the classical canonical partition function Q

and Helmholtz free energy A are treated in Section 17.6 with the ¬nal result

in terms of Q given in (17.102) on page 476. Summarizing it is found that

Q = Qcl (1 + fcor ), (3.116)

A = Acl ’ kB T fcor , (3.117)

2 1 1

fcor = ’ ∇2 V ’ (∇i V )2 . (3.118)

i

2T2 mi 2kB T

12 kB i

The two terms containing potential derivatives can be expressed in each

other when averaged over the canonical ensemble:

(∇i V )2 = kB T ∇2 V , (3.119)

i

as we shall prove below. Realizing that the force F i on the i-th particle is

equal to ’∇i V , (3.118) can be rewritten as

2 1

fcor = ’ F2 . (3.120)

mi i

3

24 kB T 3 i

This is a convenient form for practical use. For molecules it is possible

to split the sum of squared forces into translational and rotational degrees

of freedom (see Powles and Rickayzen, 1979). These are potential energy

corrections; one should also be aware of the often non-negligible quantum

corrections to the classical rotational partition function, which are of a ki-

netic nature. For formulas the reader is referred to Singh and Sinha (1987),

Millot et al. (1998) and Schenter (2002). The latter two references also give

corrections to the second virial coe¬cient of molecules, with application to

water.

Proof We prove (3.119). Consider one particular term, say the second

derivative to x1 in ∇V :

‚ 2 V ’βV

e dx1 dr ,

‚x2 1

3.5 Quantum corrections to classical behavior 73

where the prime means integration over all space coordinates except x1 .

Now, by partial integration, we obtain

x1 =∞ 2

‚V ‚V

’βV

e’βV dr.

e dr +

‚x1 ‚x1

x1 =’∞

The ¬rst term is a boundary term, which vanishes for a ¬nite system where

the integrand goes to zero at the boundary if the latter is taken beyond

all particles. It also vanishes for a periodic system because of the equality

of the integrand at periodic boundaries. Since every term in ∇2 V can be

equally transformed, (3.119) follows.

3.5.3 Equivalence between Feynman“Hibbs and Wigner

corrections

We now show that application of the Feynman-Hibbs potential (3.115) yields

the same partition function and free energy as application of the Wigner

correction (3.118). We start by rewriting (3.118), using (3.119):

2 12