M= , (17.188)

DT C

the kinetic energy now equals

1

K = q T Fq ,

™ ™ (17.189)

2

leading to conjugate momenta

™

p = Fq , (17.190)

which di¬er from the momenta in full space. The canonical average of a

variable A(q ) obtained in the constrained system is

A(q )|F|1/2 exp[’βV (q )] dq

A = . (17.191)

c

|F|1/2 exp[’βV (q )] dq

The same average is obtained from constrained dynamics in cartesian coor-

dinates:

A(r)Πs δ(σs (r)) exp[’βV (r)] dr

Ac= , (17.192)

Πs δ(σs (r)) exp[’βV (r)] dr

where σs (r) = 0, s = 1, . . . , nc , are the constraint equations that remain

satis¬ed by the algorithm.

Compare this with a classical physical system where q are near con-

straints that only negligibly deviate from constants, for example restrained

by sti¬ oscillators. The di¬erence with mathematical constraints is that the

near constraints do contribute to the kinetic energy and have an additional

potential energy Vc (q ) as well. The latter is a harmonic-like potential with

a sharp minimum. In order to obtain averages we need the full con¬guration

space, but within the integrand we can integrate over q :

Q= exp[’βVc (q )] dq . (17.193)

The average over the near-constraint canonical ensemble is

A(q )|M (q , c)|1/2 Q exp[’βV (q ) dq ]

A = . (17.194)

nc

|M (q , c)|1/2 Q exp[’βV (q ) dq ]

Q may depend on q : for example, for harmonic oscillators Q depends on

the force constants which may depend on q . But this dependence is weak

and often negligible. In that case Q drops out of the equation and the

weight factor is simply equal to |M|1/2 . Since |F | = |M |, the weight factor

17.9 Canonical distribution functions 501

in the constraint ensemble is not equal to the weight factor in the classical

physical near-constraint case. Therefore the constraint ensemble should be

corrected by an extra weight factor (|M|/|F|)1/2 . This extra weight factor

can also be expressed as exp[’βvc (q )] with an extra potential

|M|

1

vc (q ) = ’ kB T ln . (17.195)

|F|

2

The ratio |M|/|F| seems di¬cult to evaluate since both M and F are

complicated large-dimensional matrices, even when there are only a few

constraints. But a famous theorem by Fixman (1979) saves the day:

|M| |Z| = |F|, (17.196)

where Z is the (q q ) part of the matrix M’1 :

1 ‚qs ‚qt

XY

M’1 = ·

, Zst (q ) = . (17.197)

YT Z mi ‚r i ‚r i

i

Z is a low-dimensional matrix which is generally easy to compute. We ¬nd

for the extra weight factor |Z|’1/2 , or

1

vc (q ) = kB T ln |Z|.. (17.198)

2

The corrected constrained ensemble average of an observable A can be ex-

pressed in cartesian coordinates as

|Z|’1/2 A(r)Πs δ(σs (r)) exp[’βV (r)] dr

A= . (17.199)

|Z|’1/2 Πs δ(σs (r)) exp[’βV (r)] dr

For completeness we give the ingenious proof of Fixman™s theorem.

Proof From

XY FD

M’1 M = = 1,

YT Z DT C

we see that

XF + YDT = 1,

YT F + ZDT = 0.

Consider

F0 F0 XY F0

= MM’1 =M

DT 1 DT 1 YT Z DT 1

XF + YDT Y 1Y

=M =M . (17.200)

YT F + ZDT Z 0Z

502 Review of statistical mechanics

Hence |F| = |M| |Z|.

This e¬ect is often referred to as the metric tensor e¬ect, which is not a

correct name, since it is not the metric tensor proper, but the mass(-metric)

tensor that is involved.

The e¬ect of the mass tensor is often zero or negligible. Let us consider a

few examples:

• A single distance constraint between two particles: q = r12 = |r 1 ’ r 2 |.

The matrix Z has only one component Z11 :

1 ‚r12 ‚r12 1 ‚r12 ‚r12 1 1

· ·

Z11 = + = + . (17.201)

m1 ‚r 1 ‚r 1 m2 ‚r 2 ‚r 2 m1 m2

This is a constant, so there is no e¬ect on the distribution function.

• A single generalized distance constraint that can be written as q = R =

| i ±i r i |, with ±i being constants. For this case Z11 = i ±i /mi is also

2

a constant with no e¬ect.

• Two distance constraints r12 and r32 for a triatomic molecule with an

angle φ between r 12 and r 32 . The matrix Z now is a 2 — 2 matrix for

which the determinant appears to be (see Exercise 17.9):

1 1 1 1 1

|Z| = ’ 2

+ + 2 cos φ. (17.202)

m1 m1 m2 m3 m2

This is a nonzero case, but the weight factor is almost constant when the

bond angle is nearly constant.

More serious e¬ects, but still with potentials not much larger than kB T , can

be expected for bond length and bond angle constraints in molecular chains

with low-barrier dihedral angle functions. It seems not serious to neglect the

mass tensor e¬ects in practice (as is usually done). It is, moreover, likely

that the correction is not valid for the common case that constrained degrees

of freedom correspond to high-frequency quantum oscillators in their ground

state. That case is more complicated as one should really use ¬‚exible con-

straints to separate the quantum degrees of freedom rather than holonomic

constraints (see page (v).)

17.10 The generalized equipartition theorem

For several applications it is useful to know the correlations between the

¬‚uctuations of coordinates and momenta in equilibrium systems. Statements

like “the average kinetic energy equals kB T per degree of freedom” (the

equipartition theorem) or “velocities of di¬erent particles are not correlated”

17.10 The generalized equipartition theorem 503

or the virial theorem itself, are special cases of the powerful generalized

equipartition theorem, which states that

‚H

zi = kB T δij , (17.203)

‚zj

where zi , i = 1, . . . , 2n, stands, as usual, for any of the canonical variables

{q, p}. We prove this theorem for bounded systems in the canonical ensem-

ble, but it is also valid for other ensembles. Huang (1987) gives a proof for

the microcanonical ensemble.

Proof We wish to prove