Computing Reaction Path: A New Perspective of Old Problem Bijoy Dey “Basic research is what I am doing when I don’t know what I am doing.” Wernher von Braun

Reaction Path Reaction path is usually calculated by following several steepest-descents on the PES from the TSs and stringing them together. Path (R P)=(IRP)1+(IRP)2+….+(IRP)6 Comments on this slide This is a static path, that is, kinetic energy does not play a role in the definition.

Normal Laboratory scenario (expected) Input: Reactant state only. Goal: Predict product, transition state (s), reactive intermediates (if any). (mechanism) Comments on the slide In TST, the highest saddle point determines the rate. Hence, it is important to compute the path tracing all of the transition state (s).

What is different about the present method? Our method computes the entire path in a single run, no matter how many TSs or intermediates are there. No steepest descent on the PES. No prior knowledge of the TSs or Product state is required. Only an approximate reactant state needs to be supplied. Only a single point potential energy calculation is needed. No need for the gradient or Hessian of the Potential energy (many of the existing methods needed them). Comments on the slide Can we apply this to protein whose PES has many TSs and stable conformers?

The Method The method constitutes two parts: A. Wave-front expansion: Solve a Hamilton-Jacobi type equation, | ∇ S (n) (Q)|= (2 (E − V (Q))) n/2 Initial Condition: Γ int (a 0 )={Q int ∈ R N ; S (n) = a 0 }. Usually, a 0 =0. Definition: Γ (a) = {Q ∈ R N ; S (n) = a} is the wave-front, a set of points, {Q}, for which S (n) =a.

B. Steepest-descent on S (n) surface Mathematically, dQ MERP (s)/ds = − ∇ S (n) /| ∇ S (n) |; Q MERP (0) = Q Pictorially,

Front evolution in space by HJ equation HJ equation has many solution for S (n) (Q) since this depends on how the front evolves from Q int to get to the point, Q. Local speed function (distant travelled per unit spent of S (n) ) is v(Q)=1/ (2 (E − V (Q))) n/2.

Suppose the front, Γ int, arrives at a point, Q, following two different ways: (a) : Q int Q 1 Q; (b) : Q int Q 2 Q For n - large number, v(Q 1 ) » v(Q 2 ) if V(Q 1 ) < V(Q 2 ) Integral form of HJ equation is Clearly, S (n) (Q) [(a)] < S (n) (Q) [(b)] So, the low value of S (n) at the grid points means that the front travels faster and that the direction of the front expansion is such that the potential energy is low along the direction.

Numerical Method Discretize the HJ equation as (Rouy & Tourin SIAM J. Num. Ana, 1992) Rouy and Tourin showed that the solution of this equation will give least- S (n). Comments on this slide This requires iterative procedure. Fast marching algorithm of Sethian (SIAM Rev 1999) efficiently solves this without the need for iteration.

Fast Marching Algorithm Define Γ int.This is assumed a fixed point, Q int, typically a reactant state. Construct 1 st narrow band, around Q int where S (n) are known (crudely). Tag different points as follows: Select a trial point from 1 st narrow band with smallest S (n), tag it alive. Construct 2 nd narrow band by including the nearest neighboring points of the trial point. Values of S (n) at these neighboring points are updated based on the above quadratic equation. Continue this until the narrow bands are empty, that is, until all points become alive.

Pictorial description of the algorithm

Some Notes: n = -large number gives MERP n=- 1 gives minimum time path n=1 gives minimum action path (the correspoding equation is the classical HJ equation. (minimum action path is true dynamical path (cf. Newton’s equation of motion).) n=0 gives minimum distant path. Note on this slide: A series of different paths are obtained in the present method. Of these minimum action path tend to lie in the high potential energy region.

Some Results on 2D PES A. Potential Well Potential :

B. Muller-Brown Potential This shows that initial states can be known only approximately. Still the path traces through the TSs.

C. Isomerization of Malonaldehyde:

D. Rotation of BH_2 This is an example for multiple paths involving multiple TSs.

E. F+H2  FH + H reaction: An example of interfacing with Gaussian-03.

F. SN2 reaction (CH3Cl+F): An example of interfacing with Gaussian-03.

Conclusion The method is good for finding multiple pathways and transition states. The method requires solving a quadratic equation only. The most expensive part of the method is the computation of the potential energy. Question? Can we apply the method to biomolecules? QM/MM Sub-space method Compute only the alive points Coarse-graing Parallelization ls

Acknowledgement Paul W. Ayers Annie Liu (working on further development) Herschel Rabitz Paul Brumer Eberhard Gross Attila Askar B. M. Deb