Generalized path integral based quantum transition state theory

A theory for calculating rates of transitions in quantum systems is presented and applied to desorption of H from a 2 Ž . Cu 110 surface. The quantum transition state is defined as a conical dividing surface in the space of closed Feynman paths and a ‘reaction coordinate’ in this extended space is used to parametrize a reversible work evaluation of the free energy barrier. In a low temperature, harmonic limit the theory reduces to instanton theory. Above the cross-over temperature for tunneling, the theory reduces to the centroid density approximation and in the classical limit, variational classical transition state theory is recovered. q 1997 Elsevier Science B.V. Ž . Transition state theory TST is well established and widely used for calculating rates of slow transiw x tions in classical systems 1,2 . It gives an approximation which frequently is quite accurate and it, furthermore, provides a viable computational procedure for obtaining the exact rate constant via dynamical corrections which require only short time trajectories starting at the transtition state. For a system with N degrees of freedom, the transition state is a Ny1 dimensional dividing surface separating reactants and products. The accuracy of the TST approximation depends strongly on the choice of the dividing surface. For classical systems, it can be shown that TST always gives an overstimate of the rate and this provides a variational principle for optimizing w x the location of the dividing surface 2 . Basically, TST transforms the dynamical problem into a statistical one by approximating the transition rate as being proportional to the probability of finding the system in a transition state. A good approximation to the rate constant can be obtained from TST if the transition state is chosen to be a dividing surface representing a tight bottleneck for advancement of the system from reactants to products. The challenge is to generalize TST to transitions in quantum systems. Several experimentally measured transition rates show temperature dependence where below a cross-over temperature the effective w x activation energy is significantly reduced 3 . This is characteristic of the onset of quantum behaviour where thermally assisted tunneling becomes the dominant mechanism. Several versions of quantum w x TST have been proposed 2 . The most widely used Ž . formulation is based on statistical or imaginary time Feynman path integrals where the partition function w x of a quantum system is given by 4 yS w qŽt .xr " E Qs e D q t . 1 Ž . Ž . H Ž . Throughout, q represents an N-dimensional vector. Here S is the Euclidean action, S sH Hdt with E E 0 H being the Hamiltonian. For discretized paths de0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. Ž . PII S0009-2614 97 00886-5 ( ) G. Mills et al.rChemical Physics Letters 278 1997 91–96 92 scribed by P configurations of the system, the action can be approximated as 2 P m q yq jq1 j S q sDt qV q , 2 Ž . Ž . Ž . Ý E j 2 Dt js1 Ž where Dtsb "rP boldface type is used here for . NP-dimensional vectors . This leads to a mathematical analogy between the partition function of a quantum particle and the classical partition function of a Ž . string of P ‘images’ or replicas of the system connected by springs with a temperature dependent spring constant. The path integral formulation provides a practical method for evaluating the quantum statistical mechanics, but the key question is how to define the transition state. In previous theories, the Ž transition state has been defined in terms of a Ny . 1 -dimensional dividing surface in the classical coorw x dinate space 2 . In particular, in the centroid density w x method, proposed by Gillan 5 and later generalized w x by others 6–10 , the transition state constraint is applied to the average, or centroid, of the Feynman Ž . Ž . paths q s 1rb " Hq t dt . This was tested and ̃0 found to work well for transitions involving symmetw x ric barriers 5–7,11 . The centroid constraint, however, does not work well for asymmetric transitions w x at low temperature 12,13 . We present an example of that below. We also present a generalized path integral based quantum TST where the transition Ž state is defined in a more general way, as a NPy . 1 -dimensional cone in the space of all closed Feynman paths with P images. This theory represents a natural, anharmonic generalization of the so-called Ž . ‘instanton theory’ see below . A methodology is described for evaluating the free energy barrier in this higher dimensional space, which we will refer to as ‘action-space’. The technique involves evaluating the reversible work required to shift the system confined to a dividing surface cone from the reactants towards products. We refer to the method as reversible action-space work quantum transition state Ž . theory RAW-QTST . The statistical weight of Feynman paths is given by the Euclidean action and topology of the action w x surface is of central importance 12 . Fig. 1 shows the action surface for a one-dimensional asymmetric w x Eckart barrier 13 and paths described by two Fourier Ž . Ž . components, x t sx qx sin 2ptrb " . At high 0 1 Fig. 1. Contour plots of the Euclidean action, S , for an asymmetE Ž ric Eckart barrier exoergic transition by 0.19 eV and barrier . height 0.25 eV in the space of closed paths represented by two Fourier components, x and x . The dotted line at x s0 repre0 1 1 Ž . sents the collapsed, classical paths. a T s300 K, above the cross-over temperature. As the images spread apart, the action increases. The MAP therefore only includes collapsed paths and Ž . reduces to the MEP. b T s50 K, below the cross-over temperaŽ . ture. A local maximum in S q develops on the x s0 line at E 1 the potential barrier. The projection of the MAP onto the two-dimensional space is shown with a dashed line. The MAP now includes delocalized paths, a signature of tunneling. The dividing surface with maximal action is indicated by thick, solid lines. temperature, the action increases as the Feynman Ž . paths are opened x /0 . But, below a certain 1 Ž . cross-over temperature, T 275 K in this case , the c topology changes as a local maximum of action appears at x s0 in the barrier region. Then, the 1 minimal action in the barrier region is obtained with open, delocalized paths and saddle points appear off Ž . the ‘classical’, collapsed path x s0 axis. A sym1 metry in the action surface corresponding to relabeling of images is evident in Fig. 1b where two symmetrically equivalent saddle points appear. In general, P saddle points form and, in the limit of ( ) G. Mills et al.rChemical Physics Letters 278 1997 91–96 93 Ž continuous representation of the Feynman paths P . TM` , there is a continuum of saddle points. An important concept in classical TST is the Ž . minimum energy path MEP connecting reactants and products. For a quantum system, we generalize Ž . Ž this to the minimum action path MAP see Fig. . 1b . Above the cross-over temperature, the MAP reduces to the MEP. We denote the MAP with Gs where s is a scalar variable parametrizing progression along the MAP. We will choose s as ‘reaction coordinate’, a parameter that shifts the system from reactants to products. This choice is particularly good for numerical sampling since the probability density of Feynman paths is largest at the MAP. Also, we will choose a dividing surface that has X 5 X 5 normal vector tangent to the MAP, n 'G r G ˆ s s s Ž . prime denotes drd s . This ensures that displacement along the unstable mode at saddle points is not included in the dividing surface. A hyperplanar reaction coordinate constraint on the dividing surface can Ž Ž .. be written as d 'd n P qyG . ˆ r p s s Ž . Ž . Since the Feynman paths, q t , are closed, q t Ž . sq b "qt , and the Euclidean action is invariant w Ž .x w Ž under imaginary time translation, S q t sS q t E E .x q t . The origin of the imaginary time is arbitrary. The set of all Feynman paths equivalent by this symmetry form a ‘circle’ in action-space. We, therefore, construct a dividing surface from a sequence of hyperplanes defined by the imaginary time translation of n and G . This family of hyperplanes enˆ s s velops a cone with an axis consisting of all collapsed paths. The time translational symmetry of the system is retained in the conical dividing surface and this ‘zero mode’ can be integrated out separately. This facilitates numerical sampling of the Feynman paths. The contribution of the zero mode to the partition locŽ . function is Q q ' b " z Pq where z ' ˆ ̇ ˆ Ž . 0 s s ̇ ̇ 5 5 G r G and qsdqrdt . The cone dividing surface ̇ s s partition function, Q , can then be evaluated from s yS EŽq .r " loc w x Q s e Q q d d Dq t , 3 Ž . Ž . H s 0 r p 0 Ž Ž .. where d ' d z P q y G . The constraint, ˆ 0 s s loc Ž . Q d d , specifies a NPy2 -dimensional wedge 0 r p 0 of the dividing surface cone. The transition state in our theory is chosen to be the cone corresponding to the tightest statistical bottleneck, i.e. the cone with maximum free energy . The calculation is carried out in terms of the reversible work of shifting the dividing surface from the reactant region towards products, using the reaction coordinate, s, to parametrize the progression. We choose the transition state to be the cone, s, which gives a maximum in the free energy function F . To simplify the notation, we define an effective s potential for the Feynman paths as V q 'yk T ln e EŽq .r Q loc q 4 Ž . Ž . Ž . Ž . eff B 0 Ž . and effective force F q sy= V . The change eff q eff in the free energy, F , as the dividing surface is s shifted is z Pq ˆ ̇ s X X F P n y n n qz z P qyG y . Ž . ˆ ˆ ˆ ˆ ˆ Ž . 1⁄2 5 eff s s s s s s ¦ ; b z Pq ˆ ̇ s