Published: May 20, 2011 r2011 American Chemical Society 6603 dx.doi.org/10.1021/jp110799m | J. Phys. Chem. A 2011, 115, 6603–6609 ARTICLE pubs.acs.org/JPCA Algorithms for Sampling a Quantum Microcanonical Ensemble of Harmonic Oscillators at Potential Minima and Conical Intersections Kyoyeon Park, † Joshua Engelkemier, ‡ Maurizio Persico, § Paranjothy Manikandan, † and William L. Hase* ,† † Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States ‡ Department of Chemistry, Iowa State University, Ames, Iowa 50011-3111, United States § Dipartimento di Chimica e Chimica Industriale, Universit  a di Pisa, Pisa, Italy I. INTRODUCTION Classical trajectory chemical dynamics simulations are widely used 1,2 to study a wide variety of chemical processes including unimolecular 3 and bimolecular 4 reactions, intramolecular vibra- tional energy redistribution (IVR), 5,6 gas-phase collisional en- ergy transfer, 7 and collisions of projectiles with surfaces. 8,9 To compare with experiment or a theoretical model for chemical dynamics, it is important to choose proper initial conditions (i.e., coordinates and momenta) 10 for the ensemble of trajectories propagated for the simulation. The appropriate distribution functions must be sampled to model the chemical system and process under investigation. In previous work an algorithm was described for Monte Carlo sampling of a microcanonical ensemble of classical harmonic oscillators. 11 Because of the energy continuity of classical mechanics, it was possible to derive an analytic expres- sion for sampling this ensemble. This algorithm is important for selecting trajectories to test for intrinsic non-RRKM (RiceRamspergerKasselMarcus) behavior in classical un- imolecular dissociation 3,12 and for modeling experiments for which the dynamics is well-represented by classical mechanics. 13 However, for many situations, the classical description of the initial states is not correct, and the quantum representation is necessary. 10 In this work an algorithm is presented for sampling a quantum microcanonical ensemble of harmonic oscillators about a potential energy minimum. Also presented is an algorithm for sampling a microcanonical ensemble for a conical intersection (CoI). 14,15 This latter work was in part motivated by the manner in which Burghardt and co-workers 16,17 partitioned modes in their studies of nonadiabatic dynamics. Methods for sampling a microcanonical ensemble for a CoI, di fferent than the algorithm presented here, have been used in previous chemical dynamics simulations. 1821 Lischka and co- workers 18 have used two models in which the kinetic energy of the atoms is added randomly at the CoI, and the chemical dynamics is then propagated on the ground state potential energy surface (PES). For one model, the kinetic energy was only added to the two degrees of freedom for the g, h branching space defining the CoI, while for the other the kinetic energy was added to all the degrees of freedom. Bowman and co-workers 1921 performed an appropriate microca- nonical sampling of momenta at the CoI for two different models of quenching OH by H 2 , followed by ground state trajectories. The CoI microcanonical sampling algorithm presented here is different than those described above. The sampling is not performed at the CoI and, instead, is performed about the CoI for the excited electronic state. It is meant to generate initial conditions for the nonadiabatic dynamics, which is not necessa- rily very fast when the starting point is substantially higher in energy than the CoI (the two limiting cases of slow and fast decay through a CoI correspond to the “adiabatic” and “diabatic” models of Bowman and co-workers). 1921 The remainder of this article is organized as follows: Section II describes sampling a microcanonical ensemble for a collection of normal mode harmonic oscillators. Section III describes the more complex algorithm that samples a microcanonical ensemble for a CoI. Section IV is a summary. Received: November 11, 2010 Revised: May 6, 2011 ABSTRACT: Algorithms are presented for sampling quantum microcanonical ensembles for a potential energy minimum and for the conical intersection at the minimum energy crossing point of two coupled electronic states. These ensembles may be used to initialize trajectories for chemical dynamics simulations. The unim- olecular dynamics of a microcanonical ensemble about a potential energy minimum may be compared with the dynamics predicted by quantum RiceRamspergerKasselMarcus (RRKM) theory. If the dynamics is non-RRKM, it will be of particular interest to determine which states have particularly long lifetimes. Initializing a microcanonical ensemble for the electronically excited state at a conical intersection is a model for electronic nonadiabatic dynamics. The trajectory surface-hopping approach may be used to study the ensuing chemical dynamics. A strength of the model is that zero-point energy conditions are included for the initial nonadiabatic dynamics at the conical intersection.