Supplementary Information
Below we discuss various quantitative aspects of energy transfer between two DOFs mediated by
kinetic energy via the direct mechanism.
Energy exchange between PEC-i and KECs
To understand the mechanism of energy transfer mediated by kinetic energy, we need to examine
the energy exchanges between a PEC and the KECs in the system, which are determined by
Lagrange equations. From the Lagrange equation for qi , we obtain
(S1) .
By comparing Eq. (S1) with changes in energy of different KECs, we can identify different
terms in Eq. (S1) as the energy exchanges between PEC-i and different KECs. KECs can be
categorized into self-terms (e.g.
) and cross-terms (e.g.
) and will be discussed
separately.
The change in the energy content of a self-term KEC (i, i) is
.
. They are “one-body” terms in that
The relevant terms in Eq. (S1) are
2
¶x
they only contain velocity and acceleration of qi , although aii = å ma ( a ¶qi ) , where ma is the
a
mass of atom a and the summation is over all Cartesian coordinates xa of the system, is a
1
function of the configuration of the overall system. One-body terms are analogous to autocorrelation functions commonly used for studying many-body systems.
2
Note that aii qi =
¶aii 3
¶a
qi + å ii qi2 q j , we have:
¶qi
j¹i ¶q j
(S2).
Equation (S2) shows that PEC-i is equipped with all the possible means, both structure-based (
) and velocity-based (
), for changing the energy content of KEC (i, i). Thus it can be
considered the major provider for changes in KEC (i, i), whereas the other PECs modify KEC (i,
i) only through the structure-based “correction” terms
, which are “two-body”
terms. Two-body terms are analogous to cross-correlation functions and are of high magnitude
only if there is strong correlation between qi and q j . By comparing with Eq. (S1), we can see
that
belongs to d W j . Thus KEC (i, i) provides a potential site for direct
energy exchange between PEC-i and PEC-j if there is strong correlation between qi and q j .
The change in a cross-term KEC (i, j) is
. By comparing
with Eq. (S1), it is easy to see that the contribution to
from d Wi is
, whereas the contribution from d W j is
2
. Thus PEC-i and PEC-j contribute to
symmetric manner––we can recover
in a
by switching indices i and j in the expression for
. Adding these two contributions together, we obtain:
(S3).
Recall that aij qi q j =
¶aij
¶qi
qi2 q j +
¶aij
¶q j
qi q 2j + å
k¹i, j
¶aij
¶qk
qi q j qk , we have
. Following similar arguments as above,
PEC-i and PEC-j can be considered as the major providers for changes in KEC (i, j) with equal
importance, whereas other PECs can modify KEC (i, j) only through the structure-based
“correction” terms
. A term like
¶aij
qi q j qk is a three-body terms, which is
¶qk
usually of small magnitude and in general a daunting challenge to analyze. Thus we will limit
ourselves to considering only one-body and two-body terms in the subsequent analyses. Under
this approximation,
is essentially determined by PEC-i and PEC-j, whereas
contributions from all other d Wk (k ¹ i, j) are neglected, as they are three-body terms. By the
same token, the impact of d W j on any
is neglected as well.
From the discussions above, we can summarize the general rules for energy exchanges between
PECs and KECs as follows. 1) PEC-i is the major provider for energy change in KEC (i, i).
PEC-j can make significant modifications to KEC (i, i) if there is strong correlation between qi
and q j . In such a case, KEC (i, i) provides a potential site for direct energy exchange between
PEC-i and PEC-j. The energy exchange between PEC-i and KEC (i, i) is
3
; the energy exchange between PEC-j and KEC (i, i) is
. 2) PEC-i and PEC-j are the two major providers for energy change in KEC (i,
j), thus KEC (i, j) provides a potential site for direct energy exchange between them. The energy
exchange between PEC-i and KEC (i, j) is
KEC (i, j) is
; the energy exchange between PEC-j and
.
Bookkeeping energy transfer of PEC-i using kinetic virial
Based on the discussions above, two PECs can exchange energy at multiple KECs, and the net
energy transfer between them via the direct mechanism is the sum over the exchanges at all these
KECs. To keep track of the total energy transfer of PEC-i, it is convenient to gather in a single
place all the KECs at which PEC-i can conduct significant energy exchange. Kinetic virial of qi
satisfies this purpose quite well. From the previous discussions, PEC-i serves as a major
provider for KEC (i, i) and KECs (i, j) ( j ¹ i ), which are the components of Ki =
1
2
åa q q .
j
ij i
j
In addition, it is reasonable to partition KEC (i, j) equally between PEC-i and PEC-j since they
contribute to KEC (i, j) in a symmetric manner. Below we show that d Wi - d Ki provides a
convenient means for bookkeeping the overall energy transfer into or out of PEC-i.
For the purpose of simplicity and clarity, we discuss the situation where significant amount of
energy is transferred from PEC-i ( d Wi > 0 ) into PEC-j ( d Wj < 0 ) via the direct mechanism. We
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show that the energy transferred at all the three KECs, (i, i), (j, j) and (i, j), can be captured by
d Wi - d Ki .
To have energy transfer at KEC (j, j), PEC-i needs to add energy of the amount
- 1 2d t
¶a jj 2
qi q j > 0 into it. This amount of energy will be fully consumed by PEC-j, which
¶qi
æ
1 ¶a jj 3 ö
extracts energy of the amount d t ç a jj q 2j + a jj q j q j q j out of KEC (j, j), and
2 ¶q j ÷ø
è
æ
d t ç a jj q 2j + a jj q j q j è
¶a
1 ¶a jj 3 ö 1
q j ÷ > 2 d t jj qi q 2j . Thus the amount of energy transferred from
2 ¶q j ø
¶qi
¶a
PEC-i to PEC-j at KEC (j, j) is - 1 d t jj qi q 2j . This term is in d Wi but not d K i , thus it will be
2 ¶q
i
¶a
correctly recorded as - 1 d t jj qi q 2j in d Wi - d Ki .
2 ¶q
i
To have energy transfer at KEC (i, i), PEC-j needs to extract energy of the amount
1 ¶a
- d t ii qi2 q j out of it. This amount of extraction is fully supplied by PEC-i, which adds
2 ¶q j
æ
1 ¶aii 3 ö
energy of the amount d t ç aii qi2 + aii qi qi qi into KEC (i, i), and
2 ¶qi ÷ø
è
æ
d t ç aii qi2 + aii qi qi è
1 ¶aii 3 ö 1 ¶aii 2
qi > d t
qi q j . Thus the amount of energy transferred from
2 ¶qi ÷ø 2 ¶q j
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PEC-i to PEC-j at KEC (i, i) is
recorded as
1 ¶aii 2
dt
qi q j . This term is in d K i but not d Wi , thus it will be
2 ¶q j
1 ¶aii 2
dt
qi q j in d Wi - d Ki , which is again the correct value.
2 ¶q j
Energy transfer via KEC (i, j) has more variations. For the case of the most efficient transfer of
energy from PEC-i to PEC-j, we have d (aij qi q j )i = -d (aij qi q j ) j , meaning all the energy added
into KEC (i, j) by PEC-i is fully absorbed by PEC-j. In this case, d Wi registers an amount of
d (aij qi q j )i , whereas d K i registers an amount of
1
2
[d (aij qi q j )i + d (aij qi q j ) j ] = 0 . Therefore
d Wi - d Ki correctly records an energy transfer of the amount d (aij qi q j )i . On the opposite limit
where there is absolutely no energy transfer, we have both PEC-i and PEC-j contribute the same
amount energy to KEC (i, j): d (aij qi q j )i = d (aij qi q j ) j ; we have both d Wi and d K i register the
same amount of d (aij qi q j )i . Therefore d Wi - d Ki records zero energy transfer, which is again
correct. For intermediate situations, d Wi - d Ki registers an amount of
1
2
[d (aij qi q j )i - d (aij qi q j ) j ] being transferred from PEC-i to PEC-j, which is an upper bound of
the energy transfer because only the amount min( d (aij qi q j )i , d (aij qi q j ) j ) can be fully
transferred––PEC-j cannot obtain more energy from PEC-i than what PEC-i adds into KEC (i, j)
through the direct mechanism.
To summarize, the amount of energy transferred between PEC-i and PEC-j recorded by
d Wi - d Ki is d Wi - d Ki = - 1 2 d t
¶a jj
1 ¶a
qi q 2j + d t ii qi2 q j + 1 2 [d (aij qi q j )i - d (aij qi q j ) j ], which
¶qi
2 ¶q j
will be positive and significant in magnitude if PEC-i indeed transferred significant amount of
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energy to PEC-j via the direct mechanism. By the same token, energy transfer in the amount of
1
2
d Wj - d K j = - d t
¶a
¶aii 2
qi q j + 1 2 d t jj qi q 2j + 1 2 [d (aij qi q j ) j - d (aij qi q j )i ] = - (d Wi - d Ki ) will be
¶q j
¶qi
recorded by d W j - d K j .
Signature of a simplest scenario of energy transfer via the direct mechanism
The same analysis applies for energy transfer between PEC-i and any PEC-k ( k ¹ i ), thus
d Wi - d Ki keeps track of the net transfer of energy between PEC-i and all the other PECs in the
system. As a result, if the energy transfer between PEC-i and two different PECs j and k differ
in sign, their contributions to the record in d Wi - d Ki cancel with one another. Therefore, if
d Wi d Ki (e.g.
in Fig. S6), it suggests either that PEC-i does not
conduct any significant energy transfer or that PEC-i is an intermediate step for energy transfer
via an indirect mechanism. The former case is not interesting, whereas the latter case is too
complicated. On the other hand, if d Wi - d Ki is of large magnitude, then it is likely that PEC-i
has significant energy transfer via the direct mechanism, though it is not necessarily easy to
identify the other partners of the energy transfer.
Out of all the scenarios of energy transfer via direct mechanism, one particularly simple situation
is easy to identify and shows a strong and clear signature. This is when PEC-i ( d Wi > 0 )
transfers significant amount of energy to only one PEC-j ( d Wj < 0 ), and PEC-j receives energy
only from PEC-i as well. For this case, the following set of conditions are satisfied
simultaneously:
7
(S4).
8
Figure S1: The averaged work ( DWx ( pB ) ) as functions of the committor for all the bond angles
and dihedrals in the system. The coordinates discussed in the main text are marked by thick lines
with color coding the same as those defined in Fig. 3 (black: f , red: q1, green: y , magenta: b ).
Work along bond lengths are not shown as they are of much smaller magnitude.
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Figure S2: The values of a (black thin line) and b (red thin line) along three randomly chosen
reactive trajectories. The blue dashed lines are the committor scaled and shifted vertically to
indicate the barrier crossing region. The thick lines are a and b smoothed by performing
running average over 50 fs to show the underlying trend of the changes. Note the increase in a
and b when f cross the barrier ( f always closely correlates with the committor in the barriercrossing region as shown in the upper panel in Fig. S3), conveying a picture that they are pushed
to open up by f to reduce the barrier that f needs to cross. Also, the increase in b always
slightly precedes the similar change in a .
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Figure S3: A demonstration of the effect of q1 on f along a rare reactive trajectory that contains
a failed attempt of barrier crossing. (Upper) The values of f (black) and q1 (red) as functions
of time. The green dashed line is the committor to indicate the actual progression of the
transition process. The horizontal dashed line marks the value of f when it was bounced back
by q1 in its failed attempt to cross the barrier. The two vertical dashed line mark the critical
barrier crossing period. The first vertical dashed line intersects with the horizontal dashed line
where f reached the same position it reached when it was bounced backward by q1 during its
first failed attempt to cross the barrier. Note difference in the value of q1at the same value of f
between the failed and the final successful attempt to cross the barrier. (Middle) The total force
on f (black) and the contribution to it from q1 (red). Notice the large negative contribution
from q1 when it bounced f backward away from the transition barrier. (Lower) The total work
on f (black) and the contribution from q1 (red).
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Figure S4: The momenta of f (black) and q1 (red) along the same reactive trajectories shown in
Fig. S2. The dashed green lines are the committors along the reactive trajectories to indicate the
critical transition region. Notice the close correlation between pf and pq1 around the transition
region. Both pf and pq1 decrease right before the system starts to cross the barrier. The blue
asterisks mark the points where this barrier-crossing related change in the momentum start. Note
that the change in pf always precedes the similar change in pq1 .
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Figure S5: The velocities of f (black) and q1 (red) along the same reactive trajectories shown in
Fig. S2 around the transition region. The green dashed lines are the committor scaled and shifted
to mark the progression of the barrier crossing process. Note the anti-correlation behavior of vf (
f ) and vq ( q1). Also note that vf shows more oscillations than vq does, as vf has more
1
1
maxima in the same region.
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Figure S6: The averaged kinetic virial ( DK x ( pB ) ) as functions of the committor for all the
bond angles and dihedrals in the system. The coordinates discussed in the main text are marked
by thick lines with color coding the same as those defined in Fig. 3 (black: f , red: q1, green: y ,
blue: a , magenta: b ). The green dashed line denotes DWy ( pB ) . Work along bond lengths
are not shown as they are of much smaller magnitude.
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