This suggests the rewriting
Q
=
∑
ν
e
−
βE
ν
≈
∑
R
i
e
−
βE
R,i
≡
∑
R
e
−
β
˜
E
R
=
e
−
β
˜
E
R
where
∑
R
is a sum over nuclear states (i.e.
position and momentum – treat
with classical mechanics) and
∑
i
e
−
βE
R,i
is a sum over electronic states (use
quantum mechanics).
The second line is a
definition
of
˜
E
R
, the effective energy (actually free
energy, since it is
β
dependent) of just the nuclear states with the electronic
states averaged out. This is an example of the notion of “coarse-graining” in
statistical mechanics
small objects
(electrons, nuclei)
“fundamental”
interactions
−→
coarse-
grain
larger objects
(atoms)
“effective”
interactions
We have now made a lot of progress in
getting rid of the quantum mechanics
in our problem, assuming that we can evaluate the effective interactions between
atoms that enter
˜
E
R
. Indeed, this is what “ab initio quantum chemistry” tries
to do.
There are now a large number of user-friendly software packages for
computing the effective interactions between atoms and molecules that are very
useful in deducing classical descriptions of ﬂuids.
B. Classical phase space averages
What now are the “classical states”
R
that we are supposed to sum over? In
classical mechanics, the state
of a particle is determined by specifying its position
r
= (
x, y, z
) and momentum
p
= (
p
x
, p
y
, p
z
). (We can then figure out its future
state by integrating Newton’s equation
F
=
m
a
=
˙p
.)
Thus, for an
N
-atom
gas,
R
↔
(
r
1
, . . . ,
r
N
;
p
1
, . . . ,
p
N
)
and
R
is a 6
N
-dimensional “phase space”.
We expect that
Q
=
R
e
−
β
˜
E
R
→
C
d
r
1
. . .
d
r
N
d
p
1
. . .
d
p
N
e
−
β
˜
E
R
where
C
is a constant prefactor. We adopt the shorthands:
(
r
1
, . . . ,
r
N
;
p
1
, . . . ,
p
N
) = (
r
N
,
p
N
)
2