Tutorial: Quantum Dynamics of Four- Atom Reactions within the Real Wave Packet Framework Stephen Gray Chemistry Division Argonne National Laboratory Argonne, Illinois 60439

Outline Overview Representation and Evaluation of H q Initial Conditions, Propagation, Analysis Computational Experiments

Overview We will learn the “nuts and bolts” of six- dimensional AB + CD  ABC + D reactive scattering Within the RWP framework but many issues apply to other propagation schemes System: H 2 (v 1, j 1 ) + OH(v 2, j 2 )  H 2 O + H using the old, but much studied “WDSE” potential

Representation

R, r 1 : H 2 -- OH distance and H 2 internuclear distance : Large, evenly spaced grids r 2 : OH distance -- remains bound so a PODVR is convenient since it is a (small) set of grid points consistent with a set of vibrational states Rotational basis: j 1 = 0, 2, 4,.. ; j 2 = 0, 1,.. ; k 1 = determined o be consistent. E.g., for K = 0, even parity, k 1 = 0, 1,.., min(j 1, j 2 )

H q = (T d +T rot ) q + V q Use Dispersion Fitted Finite Differences (DFFD’s) to evaluate R and r 1 terms in T d If C(i R, i 1, i 2, j 1, j 2, k 1 ) denotes the wave packet, the R kinetic energy would involve do over k 1, j 2, j 1, i 2, i 1 do i R = 1, N R do s = -n, n C’’(i R,..) = C’’(i R,..) + d(s) C(i R +s,..)

PODVR for r 2 : A small kinetic energy matrix in the PODVR points acts on the r 2 part of C T rot : Not strictly diagonal with our rotational basis -- tridiagonal in k 1 -- however this is irrelevant in terms of actual numerical effort which is dominated by T d and V Numerical effort for T d q ? N rot {N 1 N 2 N R (2n+1) + N 2 N R N 1 (2n+1) + N R N 1 N 2 2 } = N tot { 4n+ 2 + N 2 }, near linear scaling

V q Diagonal in the three radial distances, so loop over them and (i) transform from rotational basis to angular grid :

Multiply by a diagonal V and convert back to basis : Effort : about 2 N R N 1 N 2 N j1 N j2 N k1 (N j1 +N j2 +N k1 ) -- again near linear

Propagation and Analysis

Analysis Use H 2 distance to separate reactants from products : S(r 1 ) = r 1 – r 1 * = 0 Write out q(R, r 1 *, r 2, j 1,j 2,k 1 ) and  q(R, r 1, r 2, j 1,j 2,k 1 )/  r 1 | r 1 =r 1 * every L time (or iteration) steps

Computational Experiments Incoming Gaussian wave packet centered at R = 10 a o, with  = 0.25 eV and a width,  = 0.3 a o Reactants in ground vibration-rotation states r 1 * = 3 a o for analysis line Absorption in last 3 a o of R and r 1 grids

Run 1 : Radial representation R/ a o : 1 – 14, N R = 130 points r 1 /a o :0.5 – 6, N 1 = 35points. 2 PODVR points in r 2, based, on diagonalization of a primitive grid Hamiltonian with r 2 /a o : 1 – 4, 32 points (r 2e = 1.85 a o ; the PODVR pts are 1.79 and 2.07 a o ) N j1 = 5 H 2 rotational states, j 1 = 0, 2,.., 4 N j2 = 10 OH rotational states, j 2 = 0, 1,.., 9 A total of 180 rotational states. (The number of angular grid points should be about 10 for each angle.) Rotational Representation

Reaction Probability

Run 1 requires 2.6 hrs on a 1 GHz Linux workstation (and 65 MB) The reaction probabilitiy P(E) is essentially converged over the E = 0.5 to 1.1 eV total energy range (energy relative to separated H 2 + OH) General aim of the experiments: to experiment with the representation details to see how they affect P(E) and if smaller grids or bases can be used for some purposes

Run 2 : Try just one PODVR point, N 2 = 1 all else the same. Result: good, but still rather long for our purposes. (PODVR pt 1.90 a o, CPU, memory both halved.)

Run 3: N R = 80 and N 1 = 25, N 2 = 1 CPU time has been further reduced by 0.5 to 0.6 hrs or 36 minutes. Result -- still reasonably good

Run 4 : Like Run 3 but N j1 = 4 (j 2 = 0, 2, 4 and 6) and N j1 = 9 (j 1 = 0,1,..,8) the total number of allowed rotational states decreases from 180 to 94. Result: CPU time of just 19 minutes. Noticeable high error in P(E) :

Run 5 : Like Run 4 but now we have reduced the number of R points from N R = 80 to N R = 60. This leads to about a 33% speedup and the calculation requires about 14 min.

Consider a setup like either Run 4 or Run 5. Experiment further with reductions in the number of grid points in R and r 1. Investigate, with Run 4 (or 5), how the quality of the calculated reaction probability varies with DFFD approximation. Experiment, with Run 4 (or 5), the role of rotational excitation in the reactants Questions and Further Runs

Appendix I: Making and Running the Codes - - See also “README” files To make the propagation program, abcd.x : make -f makefile.abcd To run it : abcd.x abcd.run5.out& Making, running the (flux-based) probability program : g77 -O3 prodflux.f -o prodflux.x prodflux.x prodflux.run5.out

Appendix II : DFFD files Subdirectory dffd has various (2n+1) DFFD’s of various overall accuracies  E. g., fd11.e-3 is a (2n+1) = 11 DFFD with accuracy See Gray-Goldfield paper for more details [JCP 115, 8331 (2001).]

Appendix III: PODVR’s (Echave and Clary; Wei and Carrington) Define a finite grid representation of some 1D potential problem in x: H o = T o + V o Diagonalize, obtain numerical eigenstates {, E v }, x i = grid, v = 0, 1,.. Now represent “x” in a finite vibrational basis, x v,v’ =, v,v’ = 0,.., Npo-1 The eigenvalues of the x v,v’ matrix are the PODVR grid pts

Can use, e.g., T 0 = H 0 - V 0 where H 0 = N po x N po rep. of H 0 in PODVR eigenstates and V 0 = diagonal potential in PODVR to approximate KE operator for the x degree of freedom within the PODVR.