The path sampling procedure results in a collection of true dynamic tran- sition paths. However, these paths do not provide a direct link to experiments.
It would be convenient if one could use the information of the path ensemble for computing experimental kinetic observables, in particular, the rate constants.
Unfortunately, such a computation requires an additional path sampling simu- lation.57,58In this section, we briefly describe the correlation function approach that was introduced in the original TPS paper,53and in “Transition Interface Sampling” we will discuss the more efficient TIS approach.
We start with the correlation function defined earlier C(t)≡
(hA(x0)hB(xt))
(hA(x0)) [102]
whereAandBare defined in the same way as in the TPS procedure. As discussed in the section on “Reactive Flux Methods”, if there is a separation of time scales, then this population correlation function grows linearly in time,C(t)∼kABt, for timesmol< trxn. In that case, the time-dependent reaction rate
kTPSAB(t)= d
dtC(t) [103]
reaches a plateau formol< trxn(c.f. Figure 5).C(t) can be calculated in a fixed-length path-sampling simulation employing the shooting and shifting Monte Carlo moves in combination with an umbrella sampling algorithm in which the final regionB is shrunk slowly from the entire phase space to the
162 Trajectory-Based Rare Event Simulations
final stable stateB.58The disadvantage of such a procedure is that it can take a relatively long timemol beforeC(t) reaches a plateau. This time is, in general, longer than in a transmission coefficient calculation because of trajectories that are not released from the top of the barrier but may start, in principle, anywhere in stable stateA.58All paths in the path sampling should have a minimum length T > mol, causing the system to spend long periods inside the stable-state basins of attraction. Moreover, cancellation of positive and negative terms can slow the convergence of the sampling of the derivative ofC(t). For adjacent regions AandB, the TPS rate approach becomes equivalent to the TST approximation in the limitt→0.58
To compute the rate constant with TPS, we need to express C(t) as a path ensemble average. A path ensemble averageO(t)is an average across all possible trajectories
O(t)AB=
DxP[x]O(t)
DxP[x] [104]
where the integral Dx is across all possible paths of length t. Using this defi- nition, the correlation functionC(t) then can be rewritten as the ratio of two path ensemble averages:
C(t)= hA(x0)hB(xt)
hA [105a]
=
DxP [x]hA(x0)hB(xt)
DxP[x]hA(x0) [105b]
where the denominator is, in fact,tindependent and equal to the equilibrium averagehA. Having defined an order parameter(x) (see Section on “Order Parameters”), we can choose regionBas
B= {x:Bmin< (x)< Bmax} [106]
Substitution of the indicator functionhB(x) in Eq. [105a] leads to C(t)= 1
hA
DxP[x]hA(x0) Bmax
Bmin
dı[−(xt)] [107a]
= Bmax
Bmin
d
DxP[x)]hA(x0)ı[−(xt)]
DxP[x]hA(x0) [107b]
= Bmax
Bmin dı[−(xt)]A [107c]
≡ Bmax
Bmin
dPA(, t) [107d]
where ã ã ãA denotes an average on trajectories starting in A. The function PA(, t) is the probability a path reaches at time t provided it started in A. This probability is, of course, low for the values of spanning region B, as we are investigating a rare event. This problem can be solved by invoking techniques from free energy computation—in this case, the umbrella sampling method20 (see “Free Energy Calculation”). The path sampling equivalent of Eq. [46] is
PAWi(, t)=
DxP[x]hA(x0)hWi(xt)ı[−(xt)]
DxP[x]hA(x0)hWi(xt) [108a]
= ı[−(xt)]AWi [108b]
which is a probability histogram as a function ofcomputed in the ensemble of paths of lengthtstarting inAand ending inWi. Computing the histogram with Eq. [108] in all windowsWiand matching (gluing) the results using (e.g., the WHAM method)87eventually yieldsPA(, t) and, hence, through Eq. [107], the correlation functionC(t) (see Figure 15).
0 0.2 0.4
PAWi(λ,t)
3 4
λ 10-4
10-2 100
PA(λ,t)
Figure 15 Top: An example of histograms obtained via Eq. [108]. Bottom: the matched histograms in a logarithmic plot. The stable-state regions are shaded.
164 Trajectory-Based Rare Event Simulations
In principle, the umbrella sampling procedure should be repeated for ev- erytto get the full correlation functionC(t). This is, however, computationally expensive and turns out not to be necessary because of a convenient factoriza- tion.53,56 For a timet< t, we factorizeC(t) as
C(t)≡ hA(x0)hB(xt)
hA [109a]
= hA(x0)hB(xt) hA(x0)hB(xt)
hA(x0)hB(xt)
hA [109b]
= hA(x0)hB(xt)
hA(x0)hB(xt)C(t) [109c]
This expression shows that knowledge ofC(t) at timetleads toC(t) at all other times t through multiplication by the factor hA(x0)hB(xt)/hA(x0)hB(xt). This latter factor equals
hA(x0)hB(xt)
hA(x0)hB(xt) = hB(xt)AB
hB(xt)AB
[110]
and can be computed in a path sampling simulation with long fixed length T. The factorC(t) in Eq. [109] can be obtained using the previous umbrella sampling scheme for shorter times. The rate constant is then
k(t)≡C(t)˙ = h˙B(xt)AB
hB(xt)AB
C(t) [111]
The first factor can be improved by using a special indicator function that is unity for paths that only visit B but do not have to end there. We refer to reference 58 for more details on this algorithm.
Transition Interface Sampling
The calculation of rate constants as described in the previous section is computer time consuming. A more efficient alternative is to use the TIS method.59TIS is a path sampling scheme using a variable (flexible) path length, thereby limiting the required simulation time steps to the strict necessary mini- mum. The TIS rate equation is based on an effective positive flux formalism and is less sensitive to recrossings. In addition, multidimensional or even discrete order parameters can be implemented in TIS.
Although TIS is specifically a path sampling method, it is based on the measurement of fluxes through multiple dividing surfaces. As such it has a lot in common with the reactive flux methods discussed previously. The reac- tive flux approach employs a single dividing surface defined by(x)=∗. TIS
generalizes this concept and defines a set ofn+1 nonintersecting multidimen- sional interfaces{0,1. . . n}. A straightforward choice isi,i=0 . . . nsuch that i−1< iand that the boundaries of stateAandBare given by0andn, re- spectively. Considering, for the moment, deterministic dynamics, a phase space pointx0determines entirely a single trajectoryxin the forward and backward time direction (TIS is also valid for stochastic dynamics, but the derivation is conceptually simpler for deterministic dynamics). Given this trajectory, we can define a backward timetib(x0) and a forward timetif(x0) as follows:
tbi(x0)≡the time it takes the backward path to reachi fromx0 [112a]
tif(x0)≡the time it takes the forward path to reachifromx0 [112b]
which mark the points of first crossing with interfaceion a backward (forward) trajectory starting inx0at timet=0. Note thattibandtif defined in this way always have positive values. These crossing times can be very long if a large barrier exists betweenx0and the interfacei, but for an ergodic system, these times still will be finite. Now consider the indicator theta functions that depend on two interfacesi=/ j,
hbi,j(x)=(tbj(x)−tbi(x)) [113a]
hfi,j(x)=(tjf(x)−tfi(x)) [113b]
which measure whether the, respectively, backward and forward time evolution ofxwill reach interfaceibeforejor vice versa. Assumingi < j, both these func- tions are always unity for(x)< iand are zero for(x)> j. Only for values i < (x)< j do the indicator functions have to be evaluated. As both inter- facesiandjwill be crossed in finite time,hbi,j(x)+hbj,i(x)=hfi,j(x)+hfj,i(x)=1.
The next step is to divide the phase space into two adjacent regions by defining theoverall regionsAand B with the following two backward characteristic functions, respectively
hA(x)≡hb0,n(x) [114a]
hB(x)≡hbn,0(x) [114b]
Because the dynamics are time-reversible, in principle, one could use also the forward time direction to define the overall regions, but this seems less intuitive.
Thus, a phase point belongs toAif it came directly fromAin the past without having visitedB, and a phase point belongs toBif it came directly fromBin the past without having visitedA(see Figure 16). Taken together, theseoverall regions span the entire phase space (i.e., there is no “no-mans-land”) Although
166 Trajectory-Based Rare Event Simulations
1 i-1 i
A
B
n-1 i+1
Figure 16 The two main concepts of TIS illustrated: (1) TIS divides phase space in several nonintersecting hyper-surfaces, the interfaces, denoted byi. (2) The definition of the overall states in TIS requires the knowledge of the entire path. In the figure, the state points indicated by a filled circle on the top trajectory belong to the overall stateAbecause, when the trajectory is traced back, it reachesAbeforeB. The open circles denote state points that belong toBbecause, when going backward in time, the trajectory reachesBbeforeA. Note that leavingBtemporarily does not change this. For the lowerBAtrajectory, the situation is opposite, with most phase points on the barrier region belonging toB.
these definitions resemble those of Eq. [22], there are some important differ- ences. First, the boundary between Aand B is now very irregular and most likely of fractal nature. Second, the definitions of Eq. [114a] do not depend sensitively on the precise boundaries of the stable statesAandB, meaning that altering the definition of the stable interface0will change the value of the in- dicator functionshb0,n(x) of only a small percentage of the phase points, namely those for which thet0b(x)≈tnb(x).
Using these new characteristic functions, we can write the following cor- relation function similar to Eq. [36]:
C(t)=
(hA(x0)hB(xt))
(hA(x0)) [115]
which exhibits a linear regime ∼kABt for 0< t < trxn.59 This linear regime starts att=0 because, in contrast with Eq. [36], phase pointsx0 located at the boundary ofBcan contribute toC(t). BecausehB(t) changes only when the trajectory entersBfor the first time, taking the derivative of Eq. [115] att=0 yields
kAB=
hb0,n(x0) ˙(x0)ı((x0)−n)
(hA(x0)) [116]
Here, only positive terms contribute to the rate, and hence, TIS is related to the effective positive flux formalism of the section “The Effective Positive Flux”.
i j k
A
0
B
Figure 17 A graphical illustration of Eq. [117]. The flux through the interfacekfor trajectories coming directly fromibefore recrossingkagain (the right phase point) is the same as the flux throughjfor trajectories that go on tokbefore going back toi provided the trajectory comes directly fromi(the left phase point). This translates into Eq. [121], which states that the flux throughkis equal to the flux throughjtimes the probability that the trajectory goes on tokbefore going back toifor trajectories coming directly fromi.
This is shown as follows. Introducing the following flux relation fori< j<
k(see Figure 17)59
hbi,kı((x)˙ −k) =
hbi,jı((x)˙ −j)hfk,i
[117]
takingi=0 andk=n, we can make the connection to the transmission coef- ficient discussed in “The Effective Positive Flux”. The transmission coefficient in terms of TIS definitions becomes
TIS=
hb0,j(x0) ˙(x0)(x˙ 0)
hfn,0(x0)
j
((x˙ 0)(x˙ 0))
j
[118]
for a dividing surface j =∗. Although in principle, (x˙ 0)
is redundant in the numerator of Eq. [118] because hb0,i(x0)=0 if ˙(x0)<0, it is there to highlight that only positive crossings are counted. Comparing Eq. [118]
with Eq. [52], we see that the definition of the transmission coefficient in Eq. [118] is indeed that of the effective positive flux withABepf(x0)=(tjb(x0)− t0b(x0))(t0f(x0)−tfn(x0)). This function is evaluated by following trajectories starting fromx0 on interface j (i.e., at∗) backward in time until they reach the stable regionAor recross the interfaceiand by following them forward in time until they reach one of the stable states. However, unlike the reactive flux procedure, TIS is not restricted to∗.
168 Trajectory-Based Rare Event Simulations
To develop a working algorithm for the rate computation based on Eq. [116] we first introduce the following flux function:
ij(x)≡hbj,i(x)|(x)|ı((x)˙ −i)=hbj,i(x) lim
t→0
1 t
t−tif(x)
[119]
The first equality has the same flux notation as Eq. [116], but the second equality is more useful in practice. A dynamic trajectory crossing interface i never will be exactly on this interface because of its discreteness (as opposed to a transmission coefficient calculation, that is constrained to the dividing surface).
Still,ij(x) has a meaning for discrete paths, where tis the (fixed) time-interval between a slice just before crossingi and the next slice just after crossingi. Thus,ij(x) equals 1/ t if the forward trajectory crossesi in one single t time step, and the backward trajectory crossesj beforei. Otherwise,ij(x) vanishes. With this flux definition, Eq. [116] is immediately rewritten as
kAB= n,0/hA [120]
The flux relation Eq. [117] reads as follows fori< j < k: (ki(x))
=
ji(x)hfk,i(x)
[121]
and thus, the rate constant Eq. [116] becomes kAB=(
n,0
)/( hA)
=
j,0(x)hfn,0(x) /(
hA)
[122]
for eachj with 0≤j≤n. Note the analogy with Eq. [118]. The second step is to define a conditional crossing probabilityPA(j|i) that depends on the location of the two interfacesi, jand the stable state definition ofA:
PA(j|i)≡ i0(x)hfj0(x)/i0(x) [123]
In words, this is the probability for the system to reach interfacejbefore inter- face 0 under the condition that it crosses the interfaceiatt=0, while coming directly from interface 0 (stateA) in the past. Settingi=1 andj=n—the first and last interface, respectively—we can write
n,0 = 1,0PA(n|1) [124]
which relates the flux through∂B(the boundary ofB) to the flux through an interface1 much closer toA. By applying Eq. [121] twice, one can show for i < j< kas follows:
PA(k|i)=PA(k|j)PA(j|i) [125]
Repeated application of this latter equality leads to
PA(n|1)=
n−1
i=1
PA(i+1|i) [126]
Using these definitions turns the TIS rate constant into a form well suited for a computer simulation,59
kAB= (1,0)
(hA) PA(n|1) [127a]
PA(n|1)=
n−1 i=1
PA(i+1|i) [127b]
The flux factor 1,0
hA follows directly from a straightforward MD run when 1 will be close toA. The second factor, the crossing probabilityPA(n|1), is naturally much more difficult to compute because it is very small. However, the factorization suggests that a sequence of path sampling simulations can compute each factor separately, with much better statistics.
In an entirely similar manner, one can derive the expression for the reverse rate constantkBA, which reads
kBA=
(n−1,n
)
(hB) PB(0|n−1) [128]
PB(0|n−1)=
n−1 i=1
PB(i−1|i) [129]
PB(j|i)= in(x)hfjn(x)/in(x) [130]
The Flux Algorithm The flux factor 1,0
hA is the effective flux through1of the trajectories coming from0 (fromA) and is most conveniently computed by setting the first two interfaces to be identical. Although for1=0, the flux1,0
hA seems ill defined, it is correct in the limit1→0. Hence, the effective positive flux equals the