Derivation of a true (t 0

+
) quantum transition-state theory.
II. Recovery of the exact quantum rate in the absence of recrossing
Stuart C. Althorpe

and Timothy J. H. Hele

Department of Chemistry, University of Cambridge, Lenseld Road, Cambridge, CB2 1EW, UK.
(Dated: September 5, 2013)
In Part I [J. Chem. Phys. 138, 084108 (2013)] we derived a quantum transition-state theory by
taking the t 0+ limit of a new form of quantum ux-side time-correlation function containing a
ring-polymer dividing surface. This t 0+ limit appears to be unique in giving positive-denite
Boltzmann statistics, and is identical to ring-polymer molecular dynamics (RPMD) TST. Here, we
show that quantum TST (i.e. RPMD-TST) is exact if there is no recrossing (by the real-time quan-
tum dynamics) of the ring-polymer dividing surface, nor of any surface orthogonal to it in the space
describing uctuations in the polymer-bead positions along the reaction coordinate. In practice, this
means that RPMD-TST gives a good approximation to the exact quantum rate for direct reactions,
provided the temperature is not too far below the cross-over to deep tunnelling. We derive these
results by comparing the t limit of the ring-polymer ux-side time-correlation function with
that of a hybrid ux-side time-correlation function (containing a ring-polymer ux operator and
a Miller-Schwarz-Tromp side function), and by representing the resulting ring-polymer momentum
integrals as hypercubes. Together with Part I, the results of this article validate a large number of
RPMD calculations of reaction rates. Copyright (2013) American Institute of Physics. This article
may be downloaded for personal use only. Any other use requires prior permission of the author
and the American Institute of Physics. The following article appeared in The Journal of Chemical
Physics, 139 (2013) 084115, and may be found at http://link.aip.org/link/?JCP/139/084115/1
I. INTRODUCTION
In Part I,
1
we derived a quantum generalization of clas-
sical transition-state theory (TST), which corresponds to
the t 0
+
limit of a new form of quantum ux-side time-
correlation function. This function uses a ring-polymer
2
dividing surface, which is invariant under cyclic permu-
tation of the polymer beads, and thus becomes invariant
to imaginary-time translation in the innite-bead limit.
The resulting quantum TST appears to be unique, in
the sense that the t 0
+
limit of any other known
form of ux-side time-correlation function
1,35
gives ei-
ther incorrect quantum statistics, or zero. Remarkably,
this quantum TST is identical to ring-polymer molecu-
lar dynamics (RPMD) TST,
6
and thus validates a large
number of recent RPMD rate calculations,
719
as well as
the earlier-developed quantum TST method
2025
(which
is RPMD-TST in the special case of a centroid dividing
surface,
6
and which, to avoid confusion, we will refer to
here as centroid-TST
26
).
There are a variety of other methods for estimat-
ing the quantum rate based on short-time
5,2731
or
semiclassical
3242
dynamics. What is dierent about
quantum TST is that it corresponds to the instantaneous
t 0
+
quantum ux through a dividing surface. Clas-
sical TST corresponds to the analogous t 0
+
classical
ux, which is well known to give the exact (classical)
rate if there is no recrossing of the dividing surface;
43,44
in practice, there is always some such recrossing, and
thus classical TST gives a good approximation to the ex-

Corresponding author: sca10@cam.ac.uk

act (classical) rate for systems in which the amount of
recrossing is small, namely direct reactions. The purpose
of this article is to derive the analogous result for quan-
tum TST (i.e. RPMD-TST), to show that it gives the
exact quantum rate if there is no recrossing (by the ex-
act quantum dynamics
45
), and thus that it gives a good
approximation to the exact quantum rate for direct re-
actions.
To clarify the work ahead, we summarize two im-
portant dierences between classical and quantum TST.
First, classical TST gives a strict upper bound to the
corresponding exact rate, but quantum TST does not,
since real-time coherences may increase the quantum ux
upon recrossing.
1
Quantum TST breaks down if such co-
herences are large; one then has no choice but to attempt
to model the real-time quantum dynamics. However, in
many systems (especially in the condensed phase), real-
time quantum coherence has a negligible eect on the
rate. In such systems, quantum TST gives a good ap-
proximation to an upper bound to the exact quantum
rate. This becomes a strict upper bound only in the
high-temperature limit, where classical TST is recovered
as a special limiting case.
Second, when discussing recrossing in classical TST,
one has only to consider whether trajectories initiated
on the dividing surface recross that surface. In quantum
TST, the time-evolution operator is applied to a series
of N initial positions, corresponding to the positions of
the polymer beads. A consequence of this, as we discuss
below, is that one needs to consider, not just recrossing
(by the exact quantum dynamics) of the ring-polymer di-
viding surface, but also of surfaces orthogonal to it in the
(N1)-dimensional space describing uctuations in the
polymer-bead positions along the reaction coordinate. A
a
r
X
i
v
:
1
3
0
7
.
3
0
2
0
v
2


[
p
h
y
s
i
c
s
.
c
h
e
m
-
p
h
]


4

S
e
p

2
0
1
3
2
major task of this article will be to show that the recross-
ing of these surfaces (by the exact quantum dynamics)
causes the long-time limit of the ring-polymer ux-side
time-correlation function to dier from the exact quan-
tum rate. It then follows that the RPMD-TST rate is
equal to the exact quantum rate if there is neither re-
crossing of the ring-polymer dividing surface, nor of any
of these N1 orthogonal surfaces.
We will use quantum scattering theory to derive these
results, although we emphasise that they apply also in
condensed phases (where RPMD has proved particularly
groundbreaking
9,1317
). The scattering theory is em-
ployed merely as a derivational tool, exploiting the prop-
erty that the ux-side plateau in a scattering system ex-
tends to innite time, which makes derivation of the rate
straightforward. The results thus derived can be applied
in the condensed phase, subject to the usual caveat of
there being a separation in timescales between barrier-
crossing and equilibration.
46,47
We have relegated most
of the scattering theory to Appendices, in the hope that
the outline of the derivation can be followed in the main
body of the text.
The article is structured as follows: After summariz-
ing the main ndings of Part I in Sec. II, we introduce
in Sec. III a hybrid ux-side time-correlation function,
which correlates ux through the ring-polymer dividing
surface with the Miller-Schwarz-Tromp
3
side function,
and which gives the exact quantum rate in the limit
t . We describe the N-dimensional integral over
momenta obtained in this limit by an N-dimensional hy-
percube, and note that the t limits of the ring-
polymer and hybrid ux-side time-correlation functions
cut out dierent volumes from the hypercube, thus ex-
plaining why the former does not in general give the exact
quantum rate. In Sec. IV we show that the only parts
of the integrand that cause this dierence are a series of
Dirac -function spikes running through the hypercube.
In Sec. V we show that these spikes disappear if there
is no recrossing (by the exact quantum dynamics
45
) in
the (N1)-dimensional space orthogonal to the divid-
ing surface (mentioned above). It then follows that the
RPMD-TST rate is equal to the exact rate if there is also
no recrossing of the dividing surface itself. In Sec. VI
we explain how these results (which were derived in one
dimension) generalize to multi-dimensions. Section VII
concludes the article.
II. SUMMARY OF PART I
Here we summarize the main results of Part I. To sim-
plify the algebra, we focus on a one-dimensional scatter-
ing system with hamiltonian

H, potential V (x) and mass
m. However, the results generalize immediately to multi-
dimensional systems (see Sec. VI) and to the condensed
phase (see comments in the Introduction).
The ring-polymer ux-side time-correlation function,
introduced in Part I, is
C
[N]
fs
(t) =
_
dq
_
dz
_
d

F[f(q)]h[f(z)]

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|e
i

Ht/
|z
i
_

z
i
|e
i

Ht/
|q
i

i
/2
_
(1)
where N is the number of polymer beads,
N
= /N,
with = 1/k
B
T, and q {q
1
, . . . , q
N
}, with z and
similarly dened. The function f(q) is the ring-polymer
dividing surface, which is invariant under cyclic permuta-
tions of the polymer beads (i.e. of the individual q
i
), and
thus becomes invariant to imaginary-time translation in
the limit N . The operator

F[f(q)] gives the ux
perpendicular to f(q), and is given by

F[f(q)] =
1
2m
N

i=1
_
p
i
f(q)
q
i
[f(q)] +[f(q)]
f(q)
q
i
p
i
_
(2)
Note that we employ here a convention introduced in
Part I, that the rst term inside the curly brackets is in-
serted between e

N

H

q
i
+
i
/2
_
and

q
i
+
i
/2

e
i

Ht/
in Eq. (1), and the second term between e
i

Ht/

q
i
+

i
/2
_
and

q
i
+
i
/2

N

H
. This is done to empha-
sise the form of C
[N]
fs
(t); [Eq. (1) is written out in full in
Part I].
We can regard C
[N]
fs
(t) as a generalized Kubo-
transformed time-correlation function, since it correlates
an operator (in this case

F[f(q)]) on the (imaginary-
time) Feynman paths at t = 0 with another operator (in
this case h[f(z)]) at some later time t, and would re-
duce to a standard Kubo-transformed function if these
operators were replaced by linear functions of position or
momentum operators. The advantage of C
[N]
fs
(t) is that
it allows both the ux and the side dividing surface to
be made the same function of ring-polymer space (i.e.
f), which is what makes C
[N]
fs
(t) non-zero in the limit
t . One can show
1
that the invariance of f(q) to
imaginary time-translation in the limit N ensures
that C
[N]
fs
(t) is positive-denite in the limits t 0
+
and
N . This allows us to dene the quantum TST rate
k

Q
()Q
r
() = lim
t0+
lim
N
C
[N]
fs
(t) (3)
where
k

Q
()Q
r
() = lim
N
1
(2)
N
_
dq
_
dP
0
[f(q)]

_
B
N
(q)
P
0
m
h(P
0
)
_
2
N

2
m
e
P
2
0

N
/2m
N

i=1

q
i1
|e

N

H
|q
i
_
(4)
3
Comparison with refs. 68 shows that k

Q
() is identical
to the RPMD-TST rate. The terms quantum TST and
RPMD-TST are therefore equivalent (and will be used
interchangeably throughout the article).
For quantum TST to be applicable, one must be able
to assume that real-time coherences have only a small
eect on the rate. It then follows that (a good approxi-
mation to) the optimal dividing surface f(q) is the one
that maximises the free energy of the ring-polymer en-
semble. If the reaction barrier is reasonably symmetric,
48
or if it is asymmetric but the temperature is too hot for
deep tunnelling, then a good choice of dividing surface is
f(q) = q
0
q

(5)
where
q
0
=
1
N
N

i=1
q
i
(6)
is the centroid. (This special case of RPMD-TST was in-
troduced earlier
2024
and referred to as quantum TST;
to avoid confusion we refer to it here as centroid-TST
26
)
If the barrier is asymmetric, and the temperature is be-
low the cross-over to deep tunnelling, then a more com-
plicated dividing surface should be used which allows the
polymer to stretch.
6
As mentioned above, f(q) must be
invariant under cyclic permutation of the beads so that
it becomes invariant to imaginary time-translation in the
limit N , and thus gives positive-denite quantum
statistics.
It is assumed above, and was stated without proof in
Part I, that the RPMD-TST rate gives the exact quan-
tum rate in the absence of recrossing, and is thus a good
approximation to the exact rate if the amount of recross-
ing is small. The remainder of this article is devoted to
deriving this result.
III. LONG-TIME LIMITS
A. Hybrid ux-side time-correlation function
To analyze the t limit of C
[N]
fs
(t), we will nd
it convenient to consider the t limit of the closely
related hybrid ux-side time-correlation function:
C
[N]
fs
(t) =
_
dq
_
dz
_
d

F[f(q)]h(z
1
q

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|e
i

Ht/
|z
i
_

z
i
|e
i

Ht/
|q
i

i
/2
_
(7)
Note that we could equivalently have inserted any one
of the other z
i
into the side-function, and also that we
could simplify this expression by collapsing the identities
_
dz
i
e
i

Ht/

z
i
_
z
i

e
i

Ht/
, i = 1 [but we have not done
so in order to emphasise the relation with C
[N]
fs
(t)].
The function C
[N]
fs
(t) does not give a quantum TST,
except in the special case that N = 1 and f(q) = q
1
. In
this case, C
[N]
fs
(t) is identical to C
[1]
fs
(t), whose t 0
+
limit was shown in Part I to be identical to the quan-
tum TST introduced on heuristic grounds by Wigner in
1932.
49
For N > 1, the ux and side dividing surfaces
in C
[N]
fs
(t) are dierent functions of ring-polymer space,
with the result that C
[N]
fs
(t) tends smoothly to zero in the
limit t 0
+
.
1
By taking the t limit of the equivalent side-ux
time-correlation function C
[N]
sf
(t), we show in Appendix
A that
k
Q
()Q
r
() = lim
t
C
[N]
fs
(t) (8)
where k
Q
() is the exact quantum rate, and this expres-
sion holds for all N 1. For N = 1, we have thus proved
that the ux-side time-correlation function that gives the
Wigner form of quantum TST (see above) also gives the
exact rate in the limit t .
50
For N > 1, which
is our main concern here, C
[N]
fs
(t) has the same limits
as the Miller-Schwarz-Tromp
3
ux-side time-correlation
function, tending smoothly to zero as t 0
+
, and giving
the exact quantum rate as t . We can also evaluate
the t limit of C
[N]
fs
(t) directly [i.e. not via C
[N]
sf
(t)].
We apply rst the relation
lim
t
_

dz

x|e
i

Kt/
|z
_
h(z q

z|e
i

Kt/
|y
_
=
_

dp

x|p
_
h(p)

p|y
_
(9)
where

K is the kinetic energy operator and

x|p
_
=
(2)
1/2
exp (ipx); this converts Eq. (7) into a form that
involves applications of the Mller operator
51

lim
t
e
i

Ht/
e
i

Kt/
(10)
onto momentum states

p
i
_
. We then use the relation

p
_
=

p
_
(11)
where

p
_
is the (reactive) scattering wave function
with outgoing boundary conditions,
52
to obtain
lim
t
C
[N]
fs
(t) =
_
dpA
N
(p)h(p
1
) (12)
with
A
N
(p) =
_
dq
_
d

F[f(q)]

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|

pi
_

pi
|q
i

i
/2
_
(13)
4
FIG. 1: Representation of the momentum integrals in
Eqs. (12) and (17) for N = 3. The axes (a) are positioned such
that the origin is at the centre of each of the cubes, which are
cut by (b) the centroid dividing surface h(p
0
) (blue), and (c)
the dividing surface h(p1) (blue). The red arrow represents
the centroid axis. This picture can be generalized to N > 3,
by replacing the cubes with N-dimensional hypercubes.
B. Representation of the ring-polymer momentum
integral
To analyze the properties of Eq. (12) (and of Eq. (17)
given below), we will nd it helpful to represent the
space occupied by the intregrand as an N-dimensional
hypercube,
53
whose edges are the axes p
max
< p
i
<
p
max
, i = 1 . . . N, in the limit p
max
. We assume
no familiarity with the geometry of hypercubes, and in
fact use this terminology mainly to indicate that once a
property of A
N
(p) has been derived for N = 3 (where
the hypercube is simply a cube and thus easily visualised
as in Fig. 1) it generalizes straightforwardly to higher N.
The only formal properties of hypercubes that we need
are, rst that a hypercube has 2
N
vertices, second that
one can represent the hypercube by constructing a graph
showing the connections between its vertices, and third
that the graph for a hypercube of dimension N can be
made by connecting equivalent vertices on the graphs
of two hypercubes of dimension N 1. Figure 2 illus-
trates this last point, showing how the graph for a cube
(N = 3) can be made by connecting equivalent vertices
on the graphs for two squares (N = 2). Figure 2 also
introduces the (self-evident) notation that we will use to
label vertices; e.g. (1, 1, 1) refers to the vertex on an
N = 3 hypercube (i.e. a cube) located at p
1
= p
max
,
p
2
= p
3
= p
max
.
These properties allow one to build up a hypercube by
adding together its subcubes in a recursive sequence. By
subcube we mean that each p
i
is conned to either the
positive or negative axis; there are therefore 2
N
subcubes,
each corresponding to a dierent vertex of the hypercube
(so we can label the subcubes using the vertex notation
introduced above). Figure 3 shows how one can build up
an N = 3 hypercube (i.e. a cube) by adding its subcubes
together recursively, joining rst two individual subcubes
along a line, then joining two lines of subcubes in the form
of a square, and nally joining two squares of subcubes
to give the entire cube. The analogous sequence can be
used to build up a hypercube of any dimension N from
its subcubes, and will be useful in Sec. IV.B.
We now dene the energies
E
i
E

(p
i
) =
p
2
i
2m
+V
prod
p
i
> 0
=
p
2
i
2m
+V
reac
p
i
< 0 (14)
and introduce the notation p
i
, such that
p
i
=
_
p
2
i
+ 2m(V
prod
V
reac
) p
i
> 0
p
i
= +
_
p
2
i
+ 2m(V
reac
V
prod
) p
i
< 0 (15)
where V
reac
and V
prod
are the asymptotes of the potential
V (x) in the reactant (x ) and product (x )
regions; i.e. the tilde has the eect of converting a prod-
uct momentum to the reactant momentum corresponding
to the same energy E
i
, and vice versa. Note that we will
not need to interconvert between the reactant and prod-
uct momenta if one or other of them is imaginary, and
hence the square roots in Eq. (15) are always real.
For a symmetric barrier, it is clear that p
i
= p
i
, and
from this it is easy to show that
A
N
( p) = A
N
(p) for symmetric barriers (16)
where p ( p
1
, p
2
, . . . , p
N
); i.e. A
N
(p) is antisymmetric
with respect to inversion through the origin. Clearly this
antisymmetry ensures that the integration of A
N
( p) over
the entire hypercube (i.e. with the side function omitted)
gives zero. This integral is also zero for an asymmetric
barrier, but there is then no simple cancellation of A
N
(p)
with A
N
( p).
Finally, we note that A
N
(p) is symmetric with respect
to cyclic permutations of the p
i
, and thus has an N-fold
axis of rotational symmetry around the diagonal of the
hypercube on which all p
i
are equal. We will refer to this
diagonal as the centroid axis, since displacement along
this axis measures the displacement of the momentum
centroid p
0
=

N
i=1
p
i
/N.
FIG. 2: Diagram showing how a cube can be built up by
connecting the equivalent vertices on two squares. One can
similarly build up an N-dimensional hypercube by connecting
the equivalent vertices on two (N1)-dimensional hypercubes.
This gure also illustrates the notation used in the text to
label the vertices of a hypercube.
5
FIG. 3: Diagram showing how a cube can be built up recur-
sively in three steps from its eight subcubes. One can similarly
build up an N-dimensional hypercube in N steps from its 2
N
subcubes.
C. Ring-polymer ux-side time-correlation
function
It is straightforward to modify the above derivation
to obtain the t limit of the ring-polymer ux-side
time-correlation function C
[N]
fs
(t). The only change nec-
essary is to replace the side function h(z
1
) by h[f(z)],
which gives
lim
t
C
[N]
fs
(t) =
_
dpA
N
(p)h[f(p)] (17)
where A
N
(p) is dened in Eq. (13), and f(p) is dened
by
lim
t
h[f(pt/m)] = h[f(p)] (18)
i.e. f(p) is the limit of f(p) at very large distances. In the
special case that f(q) = q
0
, we obtain f(p) = p
0
= f(p);
but in general f(p) = f(p). A time-independent limit
of Eq. (18) is guaranteed to exist, since otherwise f(q)
would not satisfy the requirements of a dividing surface.
Whatever the choice of f(q), it is clear that the (per-
mutationally invariant) h[f(p)] encloses a dierent part
of the hypercube than does h(p
1
). For example, if
f(q) = q
0
and N = 3, then h[f(p)] = h(p
0
) cuts out
the half of the cube on the positive side of the hexagonal
cross-section shown in Fig. 1b, whereas h(p
1
) cuts o the
top half of the cube on the p
1
axis (Fig. 1c). Thus we
cannot in general expect the t limits of C
[N]
fs
(t) and
C
[N]
fs
(t) to be the same, unless A
N
(p) satises some spe-
cial properties in addition to those just mentioned. We
will show in the next two Sections that A
N
(p) does sat-
isfy such properties if there is no recrossing of any surface
orthogonal to f(q) in ring-polymer space.
IV. RING-POLYMER MOMENTUM
INTEGRALS
A. Structure of AN(p)
One can show using scattering theory (see Appendix
B) that A
N
(p) consists of the terms
A
N
(p) = a
N
(p)
_
N1

i=1
(E
i+1
E
i
)
_
+r
N
(p) (19)
where a
N
(p) is some function of p, and r
N
(p) satises
r
N
(p
1
, . . . , p
j
, . . . , p
N
) =

p
j
p
j

r
N
(p
1
, . . . , p
j
, . . . , p
N
)
(20)
(where the dots indicate that all the p
i
except p
j
take
the same values on both sides of the equation). Equa-
tion (20) is equivalent to stating that r
N
(p) alternates in
sign between adjacent subcubes (i.e. subcubes that dier
in respect of just one axis), or that r
N
(p) takes oppo-
site signs in even and odd subcubes (where a subcube is
dened to be even/odd if it has an even/odd number of
axes for which p
i
< 0). Note that r
N
(p) = 0 if any p
i
,
i = 1 . . . N, is imaginary (see Appendix B).
The rst term in Eq. (19) describes a set of 2
N
-
function spikes running along all the lines in the hyper-
cube for which the energies E
i
, i = 1 . . . N, are equal.
There is one such line in every subcube. Two of these
lines point in positive and negative directions along the
centroid axis (i.e. the diagonal of the hypercube). The
other 2
N
2 o-diagonal spikes radiate out from this axis.
If the barrier is symmetric, then each o-diagonal spike
is a straight line joining the centre of the hypercube to
one of its vertices. If the barrier is asymmetric, the o-
diagonal spikes are hyperbolae [on account of Eq. (15)].
The o-diagonal spikes are distributed with N-fold rota-
tional symmetry about the centroid axis because of the
invariance of A
N
(p) under cyclic permutations; e.g. for
N = 3, the spikes (1, 1, 1), (1, 1, 1), (1, 1, 1) (where
this notation identies each spike by the subcube that it
runs through) rotate into one another under cyclic per-
mutation of the beads; see Fig. 4.
B. Cancellation of the term rN(p)
We now show that r
N
(p) in Eq. (19) contributes zero
to C
[N]
fs
(t) and C
[N]
fs
(t) in the limits t, N , and may
6
FIG. 4: Plot of the o-diagonal spikes in AN(p) for N = 3,
obtained by looking down the centroid axis (the red arrow in
Fig. 1b).
therefore be ignored when discussing whether C
[N]
fs
(t)
gives the exact quantum rate in these limits. This prop-
erty is easy to show for a symmetric barrier, for which
Eqs. (16) and (20) imply that r
N
(p) is zero for all even
N, and thus that the contribution to the integral from
r
N
(p) tends to zero in the limit N . For an asym-
metric barrier, r
N
(p) is in general non-zero. However, we
now show that the alternation in sign between adjacent
subcubes [Eq. (20)] causes r
N
(p) to cancel out in both
C
[N]
fs
(t) and C
[N]
fs
(t) in the limits t, N .
This cancellation is easy to demonstrate for C
[N]
fs
(t):
One simply notes that the side-function h(p
1
) encloses an
even number of subcubes, which can be added together
in adjacent pairs. For example, if we add together the
adjacent subcubes (1, . . . , 1, 1) and (1, . . . , 1, 1) (where
the dots indicate that the intevening values of 1 and 1
are the same for the two subcubes), we obtain
_

0
dp
1
. . .
_

0
dp
N1
_

0
dp
N
r
N
(p)h(p
1
)
+
_

0
dp
1
. . .
_

0
dp
N1
_
0

dp
N
r
N
(p)h(p
1
) (21)
(where the dots indicate that the integration ranges for
p
i
, i = 2 . . . N 2 are the same in both terms). We
can change the limits on the last integrand to 0
by transforming the integration variable from p
N
to p
N
,
and using the relation p
i
dp
i
= p
i
d p
i
[see Eq. (15)]. Equa-
tion (20) then ensures that the two terms in Eq. (21)
cancel out. Hence the contribution from r
N
(p) cancels
out in the t limit of C
[N]
fs
(t) (for any N > 0).
Using similar reasoning, we can show that the contri-
bution from r
N
(p) to C
[N]
fs
(t) cancels out in the limits
t, N . For nite N, this cancellation is in general
54
only partial, because the function h[f(p)] encloses dif-
ferent volumes in any two adjacent subcubes. However,
one can show that the total mismatch in the volumes en-
closed in the even subcubes and the odd subcubes tends
rapidly to zero as N . The trick is to build up the
hypercube recursively, by extending to higher N the se-
quence shown in Fig. 3 for N = 3. The jth step in this
sequence can be written
S(N) =
_

dp
1
. . .
_

dp
j
_

0
dp
j+1
. . .
_

0
dp
N
r
N
(p)h[f(p)]
=
_

dp
1
. . .
_

dp
j1
_

0
dp
j
. . .
_

0
dp
N
r
N
(p)h[f(p)]
+
_

dp
1
. . .
_

dp
j1
_
0

dp
j

_

0
dp
j+1
. . .
_

0
dp
N
r
N
(p)h[f(p)] (22)
(where the rst set of dots in each term indicates that
the intervening integration ranges are < p
i
< ,
and the second set that they are 0 < p
i
< ). Because
each subcube in the second term is adjacent to its coun-
terpart in the third term, there is an almost complete
cancellation in the r
N
(p) terms. All that is left is the
residue,
S(N) =
_

dp
1
. . .
_

dp
j1
_

0
dp
j
. . .
_

0
dp
N
r
N
(p)

_
h[f(p
1
, . . . , p
j
, . . . , p
N
)]
h[f(p
1
, . . . , p
j
, . . . , p
N
)]
_
(23)
which occupies the volume sandwiched between the two
heaviside functions. Appendix C shows that this volume
is a thin strip on the order of N times smaller than the
volume occupied by r
N
(p) in each of the two terms that
were added together in Eq. (22). Now, each of these
terms was itself the result of a similar addition in the
j 1 th step, which also reduced the volume occupied
by r
N
(p) by a factor on the order of N, and so on. As
a result, the volume occupied by r
N
(p) after the Nth
(i.e. nal) step is on the order of N
N
times smaller than
the volume of a single subcube. The mismatch in volume
between the even and odd subcubes thus tends rapidly
to zero in the limit N , with the result that r
N
(p)
cancels out completely
55
in C
[N]
fs
(t) in the limits t, N
.
C. Comparison of -function spikes
We have just shown that only the rst term in Eq. (19)
contributes to C
[N]
fs
(t) and C
[N]
fs
(t) in the limits t, N
. Any dierence between these quantities can thus be
accounted for by comparing which spikes are enclosed
by the side functions h(p
1
) and h[f(p)]. It is clear that
both h(p
1
) and h(p
0
) enclose the spike that runs along the
centroid axis in a positive direction, and exclude the spike
7
that runs in a negative direction. A little thought shows
that this property must hold for any choice of h[f(p)]
(since the positive spike corresponds to all momenta p
i
travelling in the product direction as t , and vice
versa for the negative spike).
Any dierence between the t, N limits of C
[N]
fs
(t)
and C
[N]
fs
(t) can therefore be explained in terms of which
o-diagonal spikes are enclosed by h(p
1
) and h[f(p)].
These functions will enclose dierent sets of spikes. For
example, for a symmetric barrier, with N = 3, the
function h(p
0
) encloses the o-diagonal spikes (1, 1, 1),
(1, 1, 1) and (1, 1, 1), whereas h(p
1
) encloses (1, 1, 1),
(1, 1, 1) and (1, 1, 1).
We have therefore obtained the result that the t, N
limit of C
[N]
fs
(t) is identical to that of C
[N]
fs
(t) (and thus
gives the exact quantum rate) if the contribution from
each o-diagonal spike to A
N
(p) is individually zero. We
make use of this important result in the next Section.
V. EFFECTS OF RECROSSING
The results just obtained show that quantum TST will
give the exact quantum rate if two conditions are satis-
ed. First, there must be no recrossing of the cyclically
invariant dividing surface f(q) (by which we mean sim-
ply that C
[N]
fs
(t) is time-independent). Second, each of
the o-diagonal spikes [in the rst term of Eq. (19)] must
contribute zero to C
[N]
fs
(t) in the long-time limit. We
now show that this last condition is satised if there is
no recrossing of any dividing surface orthogonal to f(q)
in ring-polymer space.
A. Orthogonal dividing surfaces
A dividing surface g(q) orthogonal to f(q) satises
N

i=1
g(q)
q
i
f(q)
q
i
= 0 (24)
When f(q) = q
0
, the surface g(q) can be any function
of any linear combination of polymer beads orthogonal
to q
0
. For a more general (cyclically permutable) f(q),
g(q) will also take this form close to the centroid axis
(where, by denition, all degrees of freedom orthogonal
to the centroid vanish), and will assume a more general
curvilinear form away from this axis.
By no recrossing of g(q), we mean that the time-
correlation function
M
[N]
fs
(t) =
_
dq
_
dz
_
d

F[f(q)]h[g(z)]

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|e
i

Ht/
|z
i
_

z
i
|e
i

Ht/
|q
i

i
/2
_
(25)
is time-independent. We know from Part I that the
t 0
+
limit of M
[N]
fs
(t) is zero, since the ux and side di-
viding surfaces are dierent. Hence no recrossing of g(q)
implies that M
[N]
fs
(t) is zero for all time t, indicating that
there is no net passage of ux from the initial distribu-
tion on f(q) through the surface g(q). Taking the t
limit (using the same approach as in Sec. III), we obtain
lim
t
M
[N]
fs
(t) =
_
dpA
N
(p)h[g(p)]
= 0 if no recrossing of g(q) (26)
where A
N
(p) is dened in Eq. (13), and g(p) is dened
analogously to f(p), i.e.
lim
t
h[g(pt/m)] = h[g(p)] (27)
In the N limit, the contribution of r
N
(p) to
M
[N]
fs
(t) cancels out (for the same reason that it can-
cels out in C
[N]
fs
(t)see Sec. IV.B). Equation (26) is thus
equivalent to stating that the total contribution to A
N
(p)
from the spikes enclosed by h[g(p)] is zero if there is no
recrossing of g(q).
B. Eect of no recrossing orthogonal to f(q)
If there is no recrossing of any g(q) orthogonal to f(q),
we can use Eq. (26) to generate a set of equations giving
constraints on the spikes. Let us see what eect these
constraints have in the simple case that N = 3 and
f(q) = q
0
.
56
We can construct dividing surfaces g(q)
orthogonal to f(q) by taking any function of the normal
mode coordinates
Q
x
=
1

6
(2q
1
q
2
q
3
)
Q
y
=
1

2
(q
2
q
3
) (28)
Let us take
g
r
(q) =
_
Q
2
x
+Q
2
y
r

g
F
(q) = F[(Q
x
, Q
y
)] (29)
where r

> 0 species the position of surface g

r
(q), and
F can be chosen to be any smooth function
57
of the angle
(Q
x
, Q
y
) = arctan(Q
y
/Q
x
) (30)
8
Clearly g
r
(q) and are polar coordinates in the plane
orthogonal to the centroid axis. If there is no recrossing
of g
r
(q) or g
F
(q), then Eq. (26) will hold with
g
r
(p) = lim
0
_
P
2
x
+P
2
y

g
F
(p) = F[(P
x
, P
y
)] (31)
in place of g(p) [where (P
x
, P
y
) are the combinations of
p
i
analogous to (Q
x
, Q
y
)]. Now, g
r
(p) is a thin cylinder
enclosing the centroid axis, and hence this function gives
the constraint that the contributions to A
N
(p) from the
two spikes lying along this axis (in positive and negative
directions) cancel out.
58
We are then free to choose F
so that h[g
F
(p)] encloses each o-diagonal spike in turn,
since no two o-diagonal spikes pass through the same
angle (see Fig. 4). We do not need to worry about
the spikes along the centroid axis (which appear as a
point at the originsee Fig. 4), since we have just shown
that they cancel out. Equation (26) then gives a set of
constraints, each of which species that the contribution
to A
N
(p) from one of the spikes is individually zero [if
there is no recrossing orthogonal to f(q)].
In Appendix D, we show that this result generalizes
to any N and to any choice of the cyclically invariant
dividing surface f(q). The t, N limit of C
[N]
fs
(t) is
therefore equal to the t limit of C
[N]
fs
(t) if there is no
recrossing orthogonal to f(q). Since the t 0
+
limit of
C
[N]
fs
(t) is by denition equal to its t limit if there is
also no recrossing of f(q), we have therefore derived the
main result of this article: quantum TST (i.e. RPMD-
TST) gives the exact quantum rate for a one-dimensional
system if there is no recrossing of f(q), nor of any surface
orthogonal to it in ring-polymer space. We will show in
Sec. VI that this result generalises straightforwardly to
multi-dimensions.
C. Interpretation
Quantum TST therefore diers from classical TST in
requiring an extra condition to be satised if it is to
give the exact rate: in addition to no recrossing of the
dividing-surface f(q), there should also be no recross-
ing (by the exact quantum dynamics) of surfaces in the
(N 1)-dimensional space orthogonal to f(q). In the
limit t 0
+
, this space describes uctuations in the
positions of the ring-polymer beads. The extra condi-
tion is therefore satised automatically in the classical
(i.e. high temperature) limit, where it is impossible to
recross any surface orthogonal to f(q), since the initial
distribution of polymer beads is localised at a point and
only the projection of the momentum along the centroid
axis is non-zero. For similar reasons, it is also impossi-
ble to recross any surface orthogonal to f(q) = q
0
q

for a parabolic barrier at any temperature (at which the

parabolic-barrier rate is dened). As a result, quantum
TST gives the exact rate in the classical limit and for a
parabolic barrier, provided there is no recrossing of f(q)
(which condition is satised for a parabolic barrier when
q

is located at the top of the barrier).

In a real system, there will always be some recrossing
of surfaces orthogonal to f(q) on account of the anhar-
monicity. However, the amount of such recrossing is zero
in the high temperature limit (see above), and will only
become signicant at temperatures suciently low that
the t 0
+
distribution of polymer beads is delocalised
beyond the parabolic tip of the potential barrier. In prac-
tice, this means that quantum TST (i.e. RPMD-TST)
will give a good approximation to the exact quantum rate
at temperatures above the cross-over to deep-tunnelling
(provided the reaction is not dominated by dynamical
recrossing or real-time coherence eects). On reducing
the temperature below cross-over, the amount of recross-
ing orthogonal to f(q) will increase, with the result that
quantum TST will become progressively less accurate.
Previous work on RPMD
68,10,11,13
and related instan-
ton methods
6,3542
has shown that this deterioration in
accuracy is gradual, with the RPMD-TST rate typically
giving a good approximation to the exact quantum rate
at temperatures down to half the cross-over temperature
and below.
D. Correction terms
An alternative way of formulating the above is to re-
gard the M
[N]
fs
(t) as a set of correction terms, which can
be added to C
[N]
fs
(t) in order to recover the exact quantum
rate in the limits t . The orthogonal surfaces g(q)
should be chosen such that the resulting sum of terms
contains the same set of spikes in the t limit as does
C
[N]
fs
(t). For example, if N = 3 and f(q) = q
0
, we can
dene two time-correlation functions M
1
(t) and M
2
(t)
which use dividing surfaces of the form of g
F
(q), with
F chosen to enclose, respectively, the spikes (1, 1, 1)
and (1, 1, 1). The corrected ux-side time-correlation
function
C
[N=3]
corr
(t) = C
[N=3]
fs
(t) M
1
(t) +M
2
(t) (32)
then contains the same spikes in the t limit as
C
[N]
fs
(t). Since M
1
(t) and M
2
(t) are zero in the limit
t 0
+
, it follows that C
[N=3]
corr
(t) interpolates between
the RPMD-TST rate in the limit t 0
+
, and the exact
quantum rate in the limit t .
56
Clearly M
1
(t) and
M
2
(t) will be zero for all values of t if there is no recross-
ing of surfaces orthogonal to f(q) in ring-polymer space.
This treatment generalizes in an obvious way to N > 3.
An alternative way of stating the result of Sec. V.B is
thus that C
[N]
fs
(t) gives the exact rate in the t limit
when added to correction terms which are zero in the
absence of recrossing.
9
VI. APPLICATION TO MULTI-DIMENSIONAL
SYSTEMS
Here we outline the modications needed to extend
Secs. III-V to multi-dimensional systems. As in Secs. III-
V, we make use of quantum scattering theory, but we
emphasise that the results obtained here apply also
in the condensed phase, provided there is the usual
separation in timescales between barrier-crossing and
equilibration.
46
Following Part I, we represent the space of an F-
dimensional reactive scattering system using cartesian
coordinates q
j
, j = 1 . . . F, and dene ring-polymer coor-
dinates q {q
1
, . . . , q
N
}, where q
i
{q
i,1
, . . . , q
i,F
} is
the geometry of the ith replica of the system. Analogous
generalizations can be made of z, p, , and so on. We
then construct a multi-dimensional version of C
[N]
fs
(t) by
making the replacements

q
i
+
i
/2
_

q
i,1
+
i,1
/2, . . . , q
i,F
+
i,F
/2
_
(33)
in Eq. (1), and integrating over the multi-dimensional co-
ordinates (q, z, ). The dividing surface f(q) is now in-
variant under collective cyclic permutations of the coordi-
nates q, and is thus a permutationally invariant function
of the replicas {
1
(q
1
), . . . ,
N
(q
N
)} of a (classical)
reaction coordinate (q
1
, . . . , q
F
).
It is straightforward to analyze the t behaviour
of C
[N]
fs
(t) by combining the analysis of Secs. III-V with
centre-of-mass-frame scattering theory. All we need to
note is that the relative motion of the reactant or prod-
uct molecules can be described by a one-dimensional scat-
tering coordinate, with all other degrees of freedom be-
ing described by channel functions
51
(which include the
rovibrational states of the scattered molecules, and also
specify whether the system is in the reactant or product
arrangement). We will denote the momentum of the ith
replica along the scattering coordinate as
i
, using the
convention that
i
is negative in the reactant arrange-
ment and positive in the product arrangement. Since all
other internal degrees of freedom are bound, it follows
that
lim
t
h[
i
(p
i
t/m)] = h(
i
) (34)
This last result allows us to construct a multi-
dimensional generalisation of the hybrid function C
[N]
fs
(t)
by replacing h[f(q)] in C
[N]
fs
(t) by h[
i
(q
i
)]. One can
show (by generalizing Appendix A) that the multi-
dimensional C
[N]
fs
(t) gives the exact quantum rate in the
limit t . We then take the t limits of C
[N]
fs
(t)
and C
[N]
fs
(t) by using the scattering relation

i
_

v
i
_
=

i,vi
_
(35)
where

is the (multi-dimensional) Mller operator,

51

i
_
is a momentum eigenstate,

v
i
_
is a reactant or
product channel function, and

i,vi
_
is a scatter-
ing eigenstate satisfying outgoing boundary conditions.
As in one-dimension, we obtain integrals over an N-
dimensional hypercube:
lim
t
C
[N]
fs
(t) =
_
dA
N
()h(
i
) (36)
lim
t
C
[N]
fs
(t) =
_
dA
N
()h[f()] (37)
where {
1
, . . . ,
N
}, and A
N
() is a generalisa-
tion of A
N
(p), obtained by making the replacements
of Eq. (33) in Eq. (13), replacing

pi
_
by

i,vi
_
,
and summing over v
i
. The function f() is a multi-
dimensional generalisation of f(p), and satises
lim
t
h[f(pt/m)] = h[f()] (38)
(which is equivalent to stating that f(q) separates cleanly
the reactants from the products in the limit t ).
The derivation of Appendix B generalizes straightfor-
wardly to multi-dimensions, with the result that A
N
()
has the analogous structure to A
N
(p) in Eq. (19). Fol-
lowing Sec. IV and Appendix C, one can show that only
the -function spikes [corresponding to the rst term in
Eq. (19)] contribute to C
[N]
fs
(t) and C
[N]
fs
(t) in the lim-
its t, N . There are many more of these spikes in
multi-dimensions than in one-dimension, since there is a
spike for every possible pair of (open) reactant or prod-
uct channels. However, it is possible to isolate each o-
diagonal spike by constructing angular functions F (see
Sec. V and Appendix D) in the space orthogonal to f().
It then follows that each o-diagonal spike in A
N
() con-
tributes zero to C
[N]
fs
(t) in the limits t, N , if there is
no recrossing of surfaces orthogonal to f(q) in the space
.
Hence we have obtained the same result in multi-
dimensions as in one-dimension: that the RPMD-TST
rate is equal to the exact quantum rate if there is no
recrossing of the dividing surface, nor of any surface or-
thogonal to it in an (N1)-dimensional space orthogonal
to f(q), which describes (in the t 0
+
limit) the uctu-
ations in the polymer-bead positions along the reaction
coordinate (q
1
, . . . , q
F
). It is impossible to recross these
orthogonal surfaces in the classical (i.e. high-temperature
limit), where RPMD-TST thus reduces to classical TST.
VII. CONCLUSIONS
We have shown that quantum TST (i.e. RPMD-TST)
is related to the exact quantum rate in the same way
that classical TST is related to the exact classical rate;
i.e. quantum TST is exact in the absence of recrossing.
Recrossing in quantum TST is more complex than in
classical TST, since, in addition to recrossing of the ring-
polymer dividing surface, one must also consider recross-
ing through surfaces that describe uctuations in the po-
10
sitions of the polymer beads along the reaction coordi-
nate. Such additional recrossing disappears in the classi-
cal and parabolic barrier limits, and thus becomes impor-
tant only at temperatures below the cross-over to deep
tunnelling. Previous RPMD-TST calculations
6
indicate
that the resulting loss in accuracy increases slowly as the
temperature is reduced below cross-over, such that quan-
tum TST remains within a factor of two of the exact rate
at temperatures down to below half the cross-over tem-
perature. However, it is clear that further work will be
needed in order to predict quantitatively how far one can
decrease the temperature below cross-over before quan-
tum TST breaks down (which will always happen at a
suciently low temperature).
Just as with classical TST, quantum TST will not
work for indirect reactions, such as those involving long-
lived intermediates, or diusive dynamics (e.g. the high-
friction regimes of the quantum Kramers problem
61
).
However, this leaves a vast range of chemical reactions
for which quantum TST is applicable, and for which it
will give an excellent approximation to the exact quan-
tum rate. The ndings in Part I and in this article
thus validate the already extensive (and growing) body
of results from RPMD rate-simulations
719
(which give
a lower bound to the RPMD-TST rate), as well as re-
sults obtained using the older centroid-TST method
2025
(which is a special case of RPMD-TST
6,26
).
Acknowledgments
TJHH is supported by a Project Studentship from the
UK Engineering and Physical Sciences Research Council.
APPENDIX A: Long-time limit of the hybrid
ux-side time-correlation function
Here we derive Eq. (8), which states that C
[N]
fs
(t) gives
the exact quantum rate in the t limit. We use the
property that
C
[N]
fs
(t) = C
[N]
sf
(t) =
d
dt
C
[N]
ss
(t) (A1)
where C
[N]
sf
(t) and C
[N]
ss
(t) are the side-ux and side-side
time-correlation functions corresponding to C
[N]
fs
(t). We
then write C
[N]
sf
(t) as
C
[N]
sf
(t) =
_
dq
_

d
1
h[f(q)]

q
1

1
/2|e

N

H
|q
2
_

i=3

q
i1
|e

N

H
|q
i
_

q
N
|e

N

H
|q
1
+
1
/2
_

q
1
+
1
/2|e
i

Ht/

F(q

)e
i

Ht/
|q
1

1
/2
_
(A2)
where

F(q

) is the ux operator
3

F(q

) =
1
2m
_
p(q q

) +(q q

) p

(A3)
and insert identities of the form e
i

Ht/
e
i

Ht/
to obtain
C
[N]
sf
(t) =
_
dq
_

d
1
h[f(q)]

q
1

1
/2|e
i

Ht/
e

N

H
e
i

Ht/
|q
2
_

i=3

q
i1
|e
i

Ht/
e

N

H
e
i

Ht/
|q
i
_

q
N
|e
i

Ht/
e

N

H
e
i

Ht/
|q
1
+
1
/2
_

q
1
+
1
/2|e
i

Ht/

F(q

)e
i

Ht/
|q
1

1
/2
_
(A4)
This allows us to take the t limit of C
[N]
sf
(t) by
using Eq. (9) together with the relation

p
i
_
=

+
pi
_
(A5)
where

+
lim
t
e
i

Ht/
e
i

Kt/
(A6)
and

+
pi
_
is a (reactive) scattering wave function with
incoming boundary conditions.
51
We then obtain
C
[N]
sf
(t) =
_
dp
_

dp

1
h
_
f
_
p
1
+p

1
2
, p
2
, . . . , p
N
__

+
p

1
|e

N

H
|
+
p2
_

i=3

+
pi1
|e

N

H
|
+
pi
_

+
p
N
|e

N

H
|
+
p1
_

+
p1
|

F(q

)|
+
p

1
_
(A7)
From the orthogonality of the scattering eigenstates,
51
we obtain

+
p
|e

N

H
|
+
p

_
= e
p
2

N
/2m
(p p

) (A8)
11
We also know that
h
_
f(p, p, . . . , p)

= h(p) (A9)
(since otherwise f(q) would not correctly distinguish be-
tween reactants and products in the limit t ). We
thus obtain
lim
t
C
[N]
sf
(t) =
_

dp e
p
2
/2m
h(p)

+
p
|

F(q

)|
+
p
_
(A10)
which is the t limit of the Miller-Schwarz-Tromp
ux-side time-correlation function,
3
from which we ob-
tain Eq. (8).
APPENDIX B: Derivation of the structure of AN(p)
Here we derive Eq. (19) of Sec. IV. We rst dene a
special type of side-side time-correlation function,
P
[N]
l
(E, t) =
_
dq
_
dz
_
d
h[f(q)]
_
_
N

i=1,i=l
h(z
i
q

)
_
_

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|e
i

Ht/
(

H E
i
)|z
i
_

z
i
|e
i

Ht/
|q
i

i
/2
_
(B1)
where E {E
1
, E
2
, . . . , E
N
}, and then consider its t
time-derivative,
Q
[N]
l
(E) = lim
t
d
dt
P
[N]
l
(E, t) (B2)
We can obtain two equivalent expressions for Q
[N]
l
(E),
by evaluating it as either a ux-side or a side-ux time-
correlation function. The ux-side version is
Q
[N]
l
(E) =
_
dq
_
dp
_
d

F[f(q)]
_
_
N

i=1,i=l
h(p
i
)
_
_

i=1

q
i1

i1
/2|e

N

H
|q
i
+
i
/2
_

q
i
+
i
/2|(

H E
i
)|

pi
_

pi
|q
i

i
/2
_
(B3)
which gives
|p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
)
+| p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
)
=
Q
[N]
l
(E)
m
N
N

i=1,i=l
|p
i
| if p
l
real (B4)
and
|p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
) =
Q
[N]
l
(E)
m
N
N

i=1,i=l
|p
i
| if p
l
imaginary (B5)
The side-ux version is
Q
[N]
l
(E) =
_
ds
_
ds

h[f(s +s

)]

_
N

i=1

+
s

i1
|e

N

H
|
+
si
_
_

+
s
l
|(

H E
l
)|
+
s

l
_

j=1,j=l

+
sj
|(

H E
j
)

F(q

)|
+
s

j
_

i=1,i=l
i=j

+
si
|(

H E
i
)

h(q

)|
+
s

i
_
(B6)
The second to fourth lines in this expression contain the
-functions,
(s
l
s

l
)
N

i=1
(s

i1
s
i
)[E
+
(s
i
) E
i
] (B7)
where E
+
(s
i
) is dened the other way round to E

(p
i
)
of Eq. (14), and where the -functions in s
i
and s

i
result
from the orthogonality of the scattering states

+
s
_
.
51
Integrating over s
i
and s

i
, we obtain
Q
[N]
l
(E) = b
N
(p)(E
l+1
E
l
) (B8)
where b
N
(p) is some function of p (which we do not
need to know explicitly). Substituting this expression
into Eqs. (B4) and (B5), we obtain
|p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
)
+| p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
)
= c
N
(p)(E
l+1
E
l
) if p
l
real (B9)
and
|p
l
|
1
A
N
(p
1
, . . . , p
l
, . . . , p
N
)
= c
N
(p)(E
l+1
E
l
) if p
l
imaginary (B10)
where c
N
(p) is some function of p. This derivation was
obtained for the case that p
i
> 0, i = l, but can clearly
be repeated for all combinations of positive and negative
p
i
[by replacing various h(z
i
q

) by h(z
i
+q

)]. Hence
Eqs. (B9) and (B10) holds for all p.
Now, we can obtain alternative expressions for the
righthand side of Eqs. (B9) and (B10) by adding and
subtracting sequences of terms that correspond to fol-
lowing dierent paths through the hypercube. Consider,
12
for example (for the case that p
i
, p
j
are both real), the
sequence
|p
j
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
+| p
j
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
= X
N
(p)(E
j+1
E
j
)
|p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
+| p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
= Y
N
(p)(E
i+1
E
i
)
| p
j
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
+|p
j
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
j
, . . . , p
N
)
= Z
N
(p)(E
j+1
E
j
) (B11)
where each of X
N
(p), Y
N
(p), Z
N
(p) is some (dierent)
function of p. Combining these expressions, we obtain
|p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
N
)
+| p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
N
)
= P
N
(p)(E
i+1
E
i
) +Q
N
(p)(E
j+1
E
j
)
(B12)
where P
N
(p) = |p
j
|| p
j
|
1
Y
N
(p), and Q
N
(p) =
|p
j
|
_
|p
i
|
1
X
N
(p) +| p
i
|
1
Z
N
(p)

.
59
We can repeat this
procedure for each of the N 1 dierent values of j = i.
Because the resulting set of coecients P
N
(p) and Q
N
(p)
are linearly independent
62
, it follows that
|p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
N
)
+| p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
N
)
= d
N
(p)
N1

i=1
(E
i+1
E
i
) if p
i
real
(B13)
and
|p
i
|
1
A
N
(p
1
, . . . , p
i
, . . . , p
N
)
= d
N
(p)
N1

i=1
(E
i+1
E
i
) if p
i
imaginary
(B14)
where d
N
(p) is some function of p. From this, we obtain
Eq. (19) of Sec. IV.
APPENDIX C: Cancellation of the term rN(p) in
the limit N
Because the function f(p) must vary smoothly with
imaginary time and converge in the limit N , it can
be rewritten as a function of a nite number K of the
linear combinations
P
i
=
N

j=1
T
ij
p
j
i = 1, . . . , K (C1)
in which T
ij
N
1
(i.e. P
i
is normalised such that it
converges in the limit N ; e.g. T
0j
= N
1
corre-
sponds to the centroid). It then follows that f(p)/p
j

N
1
, and hence that
lim
N
f(p
1
, . . . , p
j
, . . . , p
N
) = f(p) + ( p
j
p
j
)
f(p)
p
j
(C2)
provided the range of p
j
p
j
is nite [which it is because
r
N
(p) contains Boltzmann factors]. We can therefore
write the N limit of Eq. (23) as
lim
N
S(N) =
_

dp
1
. . .
_

dp
j1
_

0
dp
j
. . .
_

0
dp
N
r
N
(p)
( p
j
p
j
)
f(p)
p
j
[f(p
1
, . . . , p
j
, . . . , p
N
)]
(C3)
which shows that the volume sandwiched between the
two heaviside functions becomes a strip of width ( p
j

p
j
)f(p)/p
j
N
1
in the limit N .
APPENDIX D: Isolating the o-diagonal spikes for
N > 3
It is straightforward to generalize the result obtained
for N = 3 and f(q) = q
0
in Sec. V.B to general N and
to any (cyclically invariant) choice of f(q).
We consider rst the special case of a centroid dividing
surface [f(q) = q
0
]. The space orthogonal to q
0
can be
represented by orthogonal linear combinations Q
i
, i =
1, . . . N 1 of q
i
, analogous to Q
x
and Q
y
in Sec. V.B.
We can then dene a generalized radial dividing surface
g
r
(q) =

_
N1

i=1
Q
2
i
r

(D1)
(which is invariant under cyclic permutation of the q
i
)
and generalized angular dividing surfaces
g
F
(q) = F[(Q
X
, Q
Y
)] (D2)
with
(Q
X
, Q
Y
) = arctan(Q
Y
/Q
X
) (D3)
where (Q
X
, Q
Y
) can be chosen to be any mutually or-
thogonal pair of linear combinations of the Q
i
. From
Eq. (27), the t limits of g
r
(q) and g
F
(q) are
g
r
(p) = lim
0

_
N1

i=1
P
2
i
(D4)
and
g
F
(p) = F[(P
X
, P
Y
)] (D5)
13
where P
i
and (P
X
, P
Y
) are the linear combinations of
p
i
analogous to Q
i
and (Q
X
, Q
Y
). We can then pro-
ceed as for the N = 3 example in Sec. V.B. Substitut-
ing g
r
(p) into Eq. (26), we obtain the constraint that
the spikes along the centroid axis contribute zero (since
g
r
(p) encloses these spikes only). This leaves us free
to construct angular dividing surfaces g
F
(q) in various
planes (Q
X
, Q
Y
) (which need not be mutually orthogo-
nal) in order to enclose individual o-diagonal spikes.
60
Equation (26) then gives a set of constraints, each stating
that the contribution to A
N
(p) from one of these spikes
is zero if there is no recrossing of any surface orthogonal
to f(q).
This reasoning can be applied with slight modication
to a general (cyclically invariant) dividing surface f(q).
By construction, such a surface reduces to a function of
just the centroid near the centroid axis, and hence there
exists a radial coordinate in the (N1)-dimensional curvi-
linear space orthogonal to f(q) which reduces to g
r
(q)
close to the centroid axis. We therefore obtain the con-
straint that the spikes along the centroid axis contribute
zero, and are then free to isolate each of the o-diagonal
spikes by using curvilinear generalisations of the angles
, which sweep over curvilinear surfaces that are orthog-
onal to f(q), and which reduce to the form of Eq. (D3)
close to the centroid axis.
1
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084108 (2013).
2
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3
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Phys. 79, 4889 (1983).
4
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14
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15
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16
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17
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18
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20
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25
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26
To avoid confusion, note that we use the term centroid-
TST to refer to the quantum TST method introduced
in refs. 2023, and not to any application of the cen-
troid molecular dynamics (CMD) method of Voth and co-
workers [such as discussed in, e.g., S. Jang and G.A. Voth,
J. Chem. Phys. 112, 8747 (2000)].
27
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29
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30
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31
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Chem. Phys. 123, 054108 (2005).
32
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33
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35
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40
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41
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Theor. Comput. 7, 690 (2011).
42
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43
D. Chandler, Introduction to Modern Statistical Mechanics
(Oxford University Press, New York, 1987).
44
D. Frenkel and B. Smit, Understanding Molecular Simula-
tion (Academic Press, London, 2002).
45
To clarify, note that we do not mean here the extended
(and ctitious) classical dynamics in ring-polymer space;
we mean applications of the exact quantum time-evolution
operator exp(i

Ht/).
46
D. Chandler, J. Chem. Phys. 68, 2959 (1978).
14
47
An alternative route to deriving these results could be to
use linear response theory [as was done a long time ago
for the Kubo-transformed version of the ux-ux time-
correlation function; see T. Yamamoto, J. Chem. Phys.
33, 281 (1960)].
48
A centroid dividing surface will break down also for a sym-
metric barrier at very low temperatures, when it becomes
necessary to include other (even) normal modes of the ring-
polymer in the dividing surface; see ref. 6.
49
E. Wigner, Z. Phys. Chem. B 19, 203 (1932).
50
This supplies the necessary proof of Eq. (23) of Part I,
which was deferred to the present article.
51
J.R. Taylor, Scattering Theory (Dover, New York, 2006).
52
Meaning that

x|

has the form limx

x|

x|p

+ R(p)

x| p

for p > 0, where R(p) is the re-

ection coecient; see ref. 51.
53
H.S.M. Coxeter, Regular Polytopes (Dover, New York,
1973).
54
One can show that rN(p) cancels exactly for nite, even,
N, in the special case of a centroid dividing surface. This
follows from the property that p
0
(p) = p
0
(p).
55
This statement would be invalid if the integral of rN(p)
over a subcube were to grow at least as rapidly as N
N
.
One can show that this is not the case, by writing out
AN(p) in its side-ux form.
56
We are ignoring that rN(p) does not cancel out perfectly
for N = 3, since our aim is to illustrate the use of the
constraints on the spikes, which are then applied in the
N limit (where rN(p) does cancel out).
57
The function F must be smooth to ensure that the contri-
bution from rN(p) cancels out in the limit N .
58
For the case of a symmetric barrier, the spikes along the
centroid axis will sum to zero even if there is recrossing,
on account of Eq. (16).
59
Note that QN(p) = 0, since otherwise the integral
over AN(p) would sum to zero over every two pairs of
adjacent sub-cubes, giving the (erroneous) result that
limtC
[N]
fs
(t) = 0 for all odd N 3.
60
Surfaces F(p) capable of enclosing each o-diagonal spike
must exist, since otherwise one could map a point on one
o-diagonal spike onto a point on another spike by adding
some function of p
0
and f
r
(p). This is impossible, since
such a function would need to be non-cyclically invariant,
yet both p
0
and f
r
(p) are cyclically invariant.
61
E. Pollak, H. Grabert and P. H anggi, J. Chem. Phys. 91,
4073 (1989).
62
It is conceivable that these coecients might become lin-
early dependent at some value of p but these would be
isolated points and thereby contribute nothing to the inte-
gral.