Army Ants Tunneling for Classical Simulations

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1 Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons Jngjng Zheng, Xuefe Xu, Rubén Meana-Pañeda, and Donald G. Truhlar* Department of Chemstry, Chemcal Theory Center, and Supercomputng Insttute, Unversty of Mnnesota, Mnneapols, M Corrected: August 19, 2014 Contents Detals of the tunnelng calculaton S-2 Illustraton of the effect of varyng the parameter η S-6

2 S-2 Detals of the tunnelng calculaton Because the tunnelng path s along an nternal coordnate, the dsplacements of Cartesan coordnates along the path are generated by Wlson s A matrx usng eq. 1 Δx = A(x)ΔR (S1) where A(x) s a generalzed nverse matrx of Wlson B matrx at the current geometry x (where x s a vector of 3 Cartesan coordnates, where s the number of atoms), and ΔR s a column vector of nternal coordnate dsplacements. Then the Cartesan coordnates are converted to sonertal coordnates q by eq 2 of the man text. The end of the tunnelng path s the geometry q! at whch # $ V ( q!) V (q 0 )% & becomes zero agan (t s zero at the begnnng of the tunnelng path, wth coordnates q 0, then postve, then comes back to zero). The locaton of the center of mass s unchanged by the tunnelng process because t s carred out n nternal coordnates, and n the presentaton here we place the center of mass at the orgn. Before calculatng a tunnelng path, ts length s unknown. To calculate the magnary acton ntegral effcently n the general case, we predefne a long enough tunnelng path (longer than all the tunnelng paths n the trajectores) and dvde ths predefned path nto segments; n the current studes, the predefned path s 3.6 bohrs for bond length and 180 degrees for torson angle, and the whole path s dvded nto 18 segments. The relatve potental energy of the end pont of each segment relatve to the startng pont of the tunnelng path s calculated, and f the relatve potental energy s postve, the segment should be fully ncluded n the real tunnelng path; f segment M s the frst segment whose end pont has negatve energy, a small step (10-3 bohr for bond

3 S-3 length and 0.1 degree for torson angle) s used to search the precse endng pont of the tunnelng path. To calculate the dstances ξ of ponts from the start of a curved path n sonertal coordnates, an evenly spaced fne grd s created n nternal coordnates along the tunnelng path and the dstance ξ nt n nternal coordnates s calculated for each grd pont. Then ξ n sonertal coordnates s approxmated as a sum of small chord lengths,.e. ξ = q j q j 1. Gauss-Legendre quadrature wth 6 or more nodes s appled to the j=1 whole tunnelng path. For a gven Gauss-Legendre node ξ k that falls between ξ and ξ 1, we use lnear nterpolaton to calculate the correspondng length n nternal nt nt coordnate ξ k,.e., nt (ξ ξk = k ξ 1 )ξ + nt (ξ ξ k )ξ 1 ξ ξ 1. If more than one nternal coordnate s used n the defnton of the tunnelng drecton, ths expresson s used for each nternal coordnate. Once all ξ k nt are known, the Cartesan coordnates of node k are calculated usng Wlson s A matrx teratvely, and then the potental energy s calculated for that Cartesan geometry. To conserve total angular momentum and total energy at the end of the tunnelng path, the fnal atomc momenta are adjusted to satsfy x! p! = x 0, p 0, (S2) =1 =1 p! = p 0, = 1,, (S3) where x 0, and p 0, denote respectvely the ntal poston vector of atom n the unscaled Cartesan coordnates and the ntal momentum of atom. (ote that x 0 n the

4 S-4 man text s a vector of length 3 obtaned by jonng the three components of all x 0, nto a sngle vector.) The prmed varables n eqs. S2 and S3 denote the same quanttes as x 0, and p 0, but at the end of the tunnelng step. The adjustment s accomplshed as follows. The total angular momentum J, whch must be conserved, s J = x p (S4) =1 where x can be x 0, or x! and where p can be p 0, or p!. The change of Cartesan coordnates for atom along the whole tunnelng path s Δx so that x! = x 0, + Δx = 1,, (S5) Equaton S3 conserves the magntudes of the atomc momenta, but not ther drectons. We denote the ntal and fnal atomc momenta as p 0, = p 0, u 0, = p 0, u 0, = 1,, (S6) p! = p 0, u! = p 0, u! = 1,, (S7) where we have used eq. S3, and where u 0, and u! are unt vectors. We choose to mnmze the changes n drecton subject to the constrants of eqs. S2 and S3. Thus we mnmze the quantty f = u 0, " u 2 =1 ( u γ ) 2 =1 γ=x, y,z = u 0,γ " (S8)

5 S-5 subject to the constrant of eq. S2. ote that u 0,γ and u' γ are drecton cosnes. Addng three Lagrange multplers ( λ, =1, 2,3) to enforce the constrant gves a new objectve functon: g = ( u 0,γ u γ # ) 2 & ) + λ ( 1 J x p 0, ( x y # u z # x z # u y # ) + ( + γ=x, y,z ' * & ) & ) +λ ( 2 J y p 0, ( x z # u x # x x # u z # ) + ( + + λ ( 3 J z p 0, ( x x # u y # x y # u x # ) + ( + ' * ' * (S9) Then we combne all the fnal drecton cosnes nto a sngle algebrac vector: v 1 = u 1x,! v 2 = u 1y,! v 3 = u 1z,! v 4 = u 2x,! etc. (S10) Then the equatons to be solved for the fnal drecton cosnes u γ! are g v j = 0, j = 1,, 3 (S11) g λ k = 0. k = 1, 2, 3 (S12) Equatons S11 and S12 consttute nonlnear equatons, and they can be solved teratvely by the ewton-raphson method for the 3 component of v and the three components of λ. Usng the resultng v along wth eqs. S7 and S10, one obtans the momentum components after the tunnelng event.

6 S-6 Illustraton of the effect of varyng the parameter η Fgure S1. Lnearty of the decay curves after nducton tme for varous η values and the lnear fttng results for obtanng rate constant.

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