Submitted to The Journal of Physical Chemistry

Transcription

1 Submitted to The Journal of Physical Chemistry Semiclassical Nonadiabatic Surface Hopping Wave Function Expansion at Low Energies: Hops in the Forbidden Region Journal: Manuscript D: Manuscript Type: Date Submitted by the Author: Complete List of Authors: The Journal of Physical Chemistry jp-00-0q Special ssue Article 0-Jun-00 Herman, Michael; Tulane University, Chemistry

2 Page of Submitted to The Journal of Physical Chemistry Abstract Semiclassical Nonadiabatic Surface Hopping Wave Function Expansion at Low Energies: Hops in the Forbidden Region Michael F. Herman Department of Chemistry Tulane University New Orleans, LA 0 The accuracy of a semiclassical surface hopping expansion of the time independent wave function for problems in which the nonadiabatic coupling is peaked in the classically forbidden regions is studied numerically for a one dimensional curve crossing problem. This surface hopping expansion has recently been shown to satisfy the Schrodinger equation to all orders in and all orders in the nonadiabatic coupling. t has also been found to provide very accurate transition probabilities for problems in which the avoided crossing points of the adiabatic energy surfaces are classically allowed. n the numerical study reported here, transition probabilities are evaluated for energies well below the crossing point energy. t is found that the expansion - provides accurate results for transition probabilities as small as.

3 Submitted to The Journal of Physical Chemistry Page of ntroduction Semiclassical methods often offer attractive alternatives to fully quantum approaches for - the calculation of collision processes in atomic and molecular systems. A variety of semiclassical techniques have been proposed for problems in which more than one adiabatic - electronic state play a significant role. t has recently been shown that a semiclassical surface hopping expansion for multi-state, multi-dimensional nonadiabatic problems can be developed into a formally exact solution to the time independent Schrodinger equation (TSE) for the motion of the nuclei by including single surface correction terms that allow for energy conserving changes in the direction of the momentum along classical trajectories. This expansion for the wave function is formally exact in the sense that all terms have been shown to cancel when it is inserted into (H - E), giving zero for the answer. However, the expansion, each term of which has a primitive semiclassical prefactor and phase function, has the usual semiclassical divergence at classical turning points and caustics. As a result it is not convergent near these points. Nonetheless, the fact that this expansion formally satisfies the TSE demonstrates that it includes all appropriate hopping and momentum reversal events and that the phases for all trajectories are correctly calculated. Furthermore, the semiclassical version of the surface hopping expansion, which ignores the single surface correction terms but keeps the hopping terms, has been shown to provide very accurate transition probabilities, even for problems where there is considerable interference between different avoided crossing regions.,, Previous calculations have only considered transitions where the avoided crossings of the,, adiabatic electronic energy surfaces are in the classically allowed region. n this work, the

4 Page of Submitted to The Journal of Physical Chemistry application of the formalism to one-dimensional problems in which the avoided crossing is in the forbidden region is considered. Accurate results are obtained for transition probabilities down to very small values, corresponding to highly forbidden processes. The extension of the trajectories into the forbidden regions and the inclusion of hops in these regions are required to obtain accurate results. t is found that the small transition probability results from significant cancellation between contributions from hops in the allowed region and hops in the forbidden region. This paper is organized as follows. The surface hopping expansion is presented for the one dimensional case in section. A. A one dimensional model curve crossing problem is presented in section, and the semiclassical surface hopping transition probabilities are compared with the results from exact quantum calculations. These results are discussed, and an analysis in terms of contour integrations in the complex plane is presented in section V.. Theory This work numerically studies the behavior of the semiclassical surface hopping, expansion of the time independent wave function for one dimensional problems with turning points. Let the incoming flux be on adiabatic surface W (x). At each point along a trajectory contributing to the surface hopping wave function, this trajectory can continue to evolve classically or it can undergo a momentum reversal, or it can undergo an energy conserving hop to the other adiabatic surface. The energy determines the magnitude of the momentum after the hop, but not its sign. f the momentum has the same sign before and after the hop, this hop is referred to as a transmission or T-type hop. f the sign of the momentum changes, it is referred to

5 Submitted to The Journal of Physical Chemistry Page of as a reflection or R-type hop. After the hop, the trajectory continues to evolve classically until it undergoes another hop and/or momentum reversal. Momentum reversals without a hop correct for the use of semiclassical expressions for the contribution to the wave functions between,, hops. These single surface reflection terms are neglected in this work, as they have been in,, previous numerical calculations, resulting in a semiclassical approximation for the multi- state wave function. Each trajectory from initial point x 0 to final point x with any number of hops gives rise to a contribution to the wave function at x. This contribution contains a prefactor A, a phase factor of the form exp(is/ ), and an amplitude for each hop. For instance, the contribution to the wave function on W from a trajectory that starts at x 0 on surface W with energy E, has a T-type hop to surface W at x, and then continues to x, has the form where p is the momentum on W. One dimensional expressions are used here. The many j j dimensional generalization for problems with any number of adiabatic states is provided, elsewhere. The amplitude for the T-type hop is given by where p and p are evaluated at x, = < d /dx > is the nonadiabatic coupling, j is the j th adiabatic electronic wave function, and <... > denotes integration over the electronic coordinates. The contribution from all trajectories with a single T-type hop is obtained by integrating (x 0, x, () ()

6 Page of Submitted to The Journal of Physical Chemistry x) over all possible hopping points x ; i.e., for x 0 < x < x. The amplitude for an R-type hop is, n numerical problems, the trajectory is divided into small steps of length x. t is ½ convenient to express the wave function on surface W j as Aj j, where A j = [p (x 0)/p j(x)] is the usual semiclassical prefactor and the initial incoming wave function is on surface W. After the first x step, can be approximated by the no hop term, (x 0+ x) exp[(i/ ) pdx ] (x 0), where the integral is evaluated from x 0 to x 0+ x, and can be approximated using Eq. () as where is the integral of from x 0 to x 0+ x. As long as x is small, Eq.() provides a good numerical approximation to the integral over all single T-type hop trajectories that move from x 0 to x 0+ x. The nonadiabatic coupling can become very large near the avoided crossing points of the adiabatic surfaces. Near these points it is useful to approximately sum the contributions from, all trajectories with an odd number of T-type hops between x 0 and x 0+ x. All these trajectories end on W and make contributions to (x 0+ x). This summation can be performed by treating all hops as occurring at the midpoint of the interval, so that the phase function for all trajectories is the same as that appearing in the exponential in Eq. (). Summing all of these terms results in the replacement of in Eq. () with sin( ). A similar summation of all terms with an even number of T-type hops (that is, all terms ending on W ) gives (x 0+ x) = () ()

7 Submitted to The Journal of Physical Chemistry Page of cos( ) exp[(i/ ) pdx ] (x 0), where the p integral is taken over the x interval. Generalizing this analysis to any x interval and initial state, it is found that where x n = x 0 + n x and j can be or. The elements of the step matrix A n are given by where and = - has been used to obtain Eq. (). For one dimensional problems of the type considered here, the wave function amplitudes at x 0+n x can be expressed in terms of their values at x 0 as Eq. () sums all trajectories from x 0 and x n with energy E and any number of T-type hops. This matrix multiplication method allows for all contributions to the kj(x) to be summed. n higher dimensional problems, this summation over all hopping trajectories would generally be () () () ()

8 Page of Submitted to The Journal of Physical Chemistry performed using a Monte Carlo procedure. Previous calculations indicate that only T-type hops in the classically allowed region need to be considered if E > E c to get very good transition,0 probabilities, where E c is the energy at the crossing point in the diabatic energy surfaces. (n) Defining B (x, x) A A... A, then B (x, x) is the amplitude for the component of the (n) 0 n n- j 0 wave function for adiabatic state j at x 0 + n x if the initial wave function has an amplitude of (n) unity for state one and an amplitude of zero for state two at x 0. Note that B ij (x n,- x) = B (x, x), given that changes sign when the sign of x is reversed, that = -, and that (n) ji 0 S ij for the trajectory from x m to x m+ x equals S ji for the reversed trajectory from x m+ x to x m. Now consider the case in which both adiabatic surfaces, W (x) and W (x), approach constant values at large x and increase rapidly for small x. At a given energy E, there is a turning point, x tj, for classical motion on each surface W j. f W is the upper adiabatic surface, then x t > (in) x t. Let x = (x 0 - x t)/n and choose the incoming wave function to be on surface one, (x) = exp[-ip (x-x 0)/ ], where x 0 is in the large x asymptotic region and p = p (x 0). n this case, B (x,- x) b is the incoming wave function amplitude on surface W at x. Throughout (n) (C-) j 0 j j t this work, the region x < x t is referred to as region A, x t < x < x t is referred to as region B, and x > x is referred to as region C. The superscript C- on b denotes that this amplitude t accounts for propagation in the negative direction across region C to x t. f j =, this contribution -i /, to the wave function picks up the usual e turning point phase factor at x t and a factor of (C+) (n) k k n t 0 (C-) j b B (x, x) for the propagation back from x to x ending in state k. f j =, the trajectory on W at x t continues in the negative direction for x < x t. f hopping in the region between x and x is ignored, as in previous calculations, then the component of the wave t t i / function on W picks up a semiclassical phase factor of e as the trajectory travels from x t to

9 Submitted to The Journal of Physical Chemistry Page of x t, where = pdx with p < 0 and the integration is from x t to x t. This yields an incoming i / (C-) -i / amplitude at x t of e b. This contribution to the wave function picks up a factor of e at i / the turning point, x t, another factor of e for the propagation from x t back to x t, and a factor of (C+) (C+) -i / (C-) b k for the propagation from x t to x 0. Summing these contributions yields bk e b + (C+) i / -i / (C-) bk e e b as the outgoing wave function amplitude on surface k at x 0. This expression contains contributions from all classically allowed hopping and non-hopping trajectories,,, including only T-type hops. t is the expression used in previous computations. t is typical for the nonadiabatic coupling to be peaked around a crossing point, x c, for the,0 diabatic surfaces. At high energies, x is sufficiently larger than x for the type of problem c considered here, and the coupling is largely in the classically allowed region. f this is the case, including only hops that occur in the classically allowed region, as described above, is a very good approximation. As the energy is lowered, x t increases. At some energy, x t = x c. Below this energy, x > x and the coupling is peaked in the forbidden region. f this is the case, it is t c necessary to include hops for x < x t in order to obtain accurate results. f x t < x < x t, then p = / (m[e-w (x)]) is imaginary, and p is still real. The surface hopping wave function expansion is still valid in this region, and it is useful to think of the trajectory as continuing into this region with x real, time imaginary, and momentum q = -i p when traveling in the negative direction and q = i p when traveling in the positive direction. The expression for the transition amplitude, Eq. (), becomes t ()

10 Page of Submitted to The Journal of Physical Chemistry and (x) = i (x). The factor of i in and it absence in can be understood as follows. An, analysis of the surface hopping wave function in the Schrodinger equation shows that (x) = ½ [-(p +p )/p ][A /A ] for a W to W hop, where A j = [ p i(x 0)/p j(x) ] is the semiclassical prefactor at the hopping point x for a trajectory starting on initial surface i. The ratio A /A gives ½ a factor of [ p /p ] yielding Eq. () if p is real and Eq. () if p is imaginary. f the hop is from surface to surface, then (x) = [-(p +p )/p ][A /A ]. Using (p +p )/p = ( p + i p )/i p for imaginary p, together with = -, it is readily shown that = i. n the region between the two turning points, the amplitude for R-type hops is given by * = - and = - i, as can be seen by substituting for the imaginary p and comparing Eqs.(), (), and (). Thus, the magnitude of the R-type transition amplitude,, is equal to the magnitude for the T-type transition amplitude,, in this region, in contrast to the allowed region where is always less than. Consequently, the R-type hops cannot be ignored if hops in region B are included in the calculations. n this region, the T-type transitions give rise to a step matrix A n, which is calculated using Eqs.() and () with p = -i p and p = - p. Since is complex and = -i in this region, the summation of all even (odd) hop terms do not approximate a real valued cosine (sine) series. For this reason, the sin( ) factors in Eq.() are replaced with and the cos( ) factors are dropped in the calculations for this region. These lower order expressions should still result in a good approximation for small x. f only T-type (n) hops were included, B (x,- x) A A... A b, where x = (x -x )/n, accounts for all (B-) t n n- t t (B+) hopping trajectories traveling in the negative direction between x and x. Similarly, b = (n) B (x t, x) accounts for all (T-type only) hopping trajectories traveling in the positive direction (B+) between x t and x t. Since = -i, and accounting for the change in the sign of x, b = t t

11 Submitted to The Journal of Physical Chemistry Page of (B-) (B+) (B-) (B+) (B-) (B+) (B-) ib, b = -ib, b = b, and b = b. To simplify the calculation with R-type hops, trajectories with at most one R-type hop between x and x are included in the calculations. The total contribution from all trajectories t t with a single R-type hop in region B is a sum over all x intervals between x and x of the contribution with an R-type hop in that interval (k-,+) (k-) (k-, ) (k-) where t ij = B ij (x k-, x), x k- = x t - (k-) x, t ij = B ij (x t,- x), (k) (k) t t (), () r = ir, and is integral of over the x interval. Eq. () includes trajectories with one R-type hop and up to (n-) T-type hops in region B. f x < x t, a trajectory is classically forbidden on both adiabatic surfaces. Both p and p are imaginary, and the amplitudes for T-type and R-type transitions are given by Eqs.() and (). The contribution corresponding to a trajectory decays as it moves in the negative direction. A R- type hop reflects the trajectory back, and the contribution again decays as the trajectory moves in the forward direction. The resulting amplitude, R (A-) ij, has the form of Eq.() with ()

12 Page of Submitted to The Journal of Physical Chemistry (k) (k) (A-) and r = r. R ij includes contributions from all trajectories that travel in the negative direction from x t on surface j to the point x k- = x t - (k-) x with k > 0 and with up to k- T- th type hops, have a R-type hop in the k x inteval, and then travel in the positive direction from x to x with up to k- T-type hops. n this region, t = t, as is the case in the allowed (k-, ) (k-,+) k- t ij ji region. The standard analysis for matching semiclassical expressions across turning points give, the following results. An incoming trajectory in the allowed region results in outgoing trajectories in both the allowed and forbidden regions when it encounters the turning point, x t, where incoming (outgoing) means traveling toward (away from) x t when discussing turning points. The amplitude associated with the outgoing trajectory in the forbidden region is equal to -i / the amplitude associated with the incoming trajectory multiplied by a factor of e at the turning point, and the amplitude associated with the outgoing trajectory in the allowed region is equal to the incoming trajectory s amplitude multiplied by a factor of e -i /. An incoming trajectory in the forbidden region results in an outgoing trajectory in the allowed region, the amplitude of which is equal to the amplitude for this incoming trajectory at the turning point multiplied by e incoming trajectory from the forbidden region also gives rise to an outgoing trajectory in i /. This forbidden region, but this trajectory, which is reflected back into the forbidden region, is ignored in this work to simplify the calculations. -i / -i / i / These turning point phase factors (i.e., e, e, and e ) are used in the multi-state calculations presented below. Let (x) and (x) be the state j to state k wave function (+) (-) kj kj amplitudes corresponding to propagation in the forward and backward directions at x, (+) (-) -i / respectively. At the turning point, x t, the matching conditions give j (x t+ ) = j (x t+ )e +

13 Submitted to The Journal of Physical Chemistry Page of (+) i / (-) (-) -i / j (x t-)e and j (x t-) = j (x t+ )e, where x t- (x t+ ) is a point x infinitesimally smaller (-) (+) (larger) than x t. The quantities j (x t+ ) and j (x t-) represent incoming wave functions (-) (+) amplitudes at x t, while j (x t-) and j (x t+ ) are the outgoing wave function amplitudes. The contribution to the outgoing wave in the forbidden region, from the forbidden region, (x ), due to the incoming wave (-) j t- (x ), is ignored in these expressions. The wave function (+) (-) amplitudes on surface two, j and j are continuous at x t. The same analysis is applied at the surface two turning point, x t, with the subscripts and interchanged. (+) j t- The amplitudes for propagation in the negative direction at x t- (i.e., traveling into region (-) -i / (C-) (-) (C-) (-) (-) (C-) (-) B) are given by j (x t-) = e [b j (x 0) + b j (x 0)] and j (x t-) = b j (x 0) + (C-) (-) b j (x 0). These give rise to amplitudes for propagation in the negative direction at x t- (i.e., (-) (B-) (-) (B-) (-) (-) -i / (B-) traveling into region A) of j (x t-) = b j (x t-) + b j (x t-) and j (x t-) = e [b (-) (B-) (-) -i / j (x t-) + b j (x t-)], where the e is due to the turning point x t. These lead to forward (+) (A-) (-) (A-) (-) (+) propagating amplitudes at x t+ of j (x t+ ) = R j (x t-) + R j (x t-) and j (x t+ ) = -i / (-) i / (A-) (-) (A-) (-) e j (x t+ ) + e [R j (x t-) + R j (x t-)] at x t+. The forward propagating (+) -i / (-) i / (B+) (+) (B+) amplitudes at x t+ are then given by j (x t+ ) = e j (x t+ ) + e [b j (x t+ ) + b (+) (B-) (-) (B-) (-) (+) (B+) (+) (B+) (+) j (x t+ ) + R j (x t-) + R j (x t-)] and j (x t+ ) = b j (x t+ ) + b j (x t+ ) + (B-) (-) (B-) (-) R j (x t-) + R j (x t-). Finally, the outgoing wave function amplitudes at x 0 can be (+) (C+) (+) (C+) (+) (+) (C+) (+) expressed as j (x 0) = b j (x t+ ) + b j (x t+ ) and j (x 0) = b j (x t+ ) + (C+) (+) b j (x t+ ). These expressions can be combined to obtain the outgoing amplitudes at x 0, (+) (+) (-) (-) j (x 0) and j (x 0), in terms of the incoming amplitudes at x 0, j (x 0) and j (x 0). The semiclassical transition probability for a transition from state one to state two is given by the ratio of the outgoing flux to the incoming flux,

14 Page of Submitted to The Journal of Physical Chemistry (+) (-) (+) (-) P S = p (x 0) A (x 0) (x 0) /[p (x 0) A (x 0) (x 0) ] = (x 0) / (x 0). () t should be noted that the approximation presented here allows for, at most, one R-type hop in each of the regions A and B, although it allows for multiple T-type hops. This significantly simplifies the calculations, compared with calculations that include any number of momentum reversals accompanying hops. Calculations that allow any number of R-type hops have been performed, but this did not qualitatively change the accuracy of the method.. Results n this section, semiclassical transition probabilities are compared to quantum results for,0 the one dimensional, two state model defined by the diabatic energy surfaces V (x) = A exp(-ax) + V 0, V (x) = A exp(-ax), and V (x) = A {- tanh[a (x-x )]} with A =.0, A =.0, A = 0.0, a =.0, a =.0, a =.0, x =.0, and V 0 = 0.. These are plotted in figure. The diabatic surfaces V and V cross at x c =. and their value at the crossing point is E c = 0.. The adiabatic state energies and wave functions are obtained by diagonalizing the diabatic energy matrix, giving W (x) = V av(x) + R(x)/, and W (x) = V av(x) - R(x)/, where V (x) = (V +V )/ and R(x) = [(V -V ) +V av ]. The nonadiabatic coupling is given by (x) = < d /dx> = [(V-V ) dv /dx - V (dv /dx - dv /dx)]/r. Since W is the upper adiabatic state, the classical turning point for this surface, x, is larger than x, the classical turning point for W, at all energies. The semiclassical surface hopping calculations are performed using the matrix multiplication method as outlined in the previous section. The R-type terms are neglected in the,, classically allowed region, region C. This has consistently been shown to provide accurate t t

15 Submitted to The Journal of Physical Chemistry Page of results in previous calculations for energies at which the crossing point of the diabatic surfaces is in the allowed region. Both T-type and R-type hops are included in the calculations in region B and in region A. f R-type hops were neglected in region A, the trajectories would continue to travel away from the allowed region with a decaying amplitude. The inclusion of the R-type hop reverses the direction of the trajectory, bringing it back toward the allowed region. Without these R-type terms the effect of the nonadiabatic coupling in region A would be completely neglected in the calculations. For energies below E c, the nonadiabatic coupling is largest in region A. Quantum calculations are performed for comparison with the semiclassical calculations. At each energy, two independent quantum calculations are started at a point x min in region A, and the two surface Schrodinger equation is integrated in the diabatic representation using a fourth order Runge-Kutta routine until x is equal to x f =, which is well into the asymptotic regime. One of the two calculations begins with the wave function solely on V at x min, while the other calculation starts with the wave function on V at x min. Care is taken to avoid numerical problems due to overflows and the potential near linear dependence of the two solutions of the two state problem due to the exponential growth of the wave functions in the classically forbidden region. After both solutions have been evaluated for all x x f at a given energy, the appropriate linear combination is taken of the two independent solutions. This linear combination corresponds to no incoming component of the wave function on V and an incoming component of the wave function on V of (x) = exp[-i p (x-x )]. The transition (in) f probability is evaluated as the ratio of the outgoing flux on V and the incoming flux, P Q = p /p, where p j is the magnitude of the momentum on surface V jj corresponding to the energy E, and all quantities are evaluated at x f.

16 Page of Submitted to The Journal of Physical Chemistry Transition probabilities are presented in table for energies above the energy of the crossing point for the diabatic surfaces. Three levels of approximation are shown for the semiclassical results. P S is the semiclassical transition probability when hops in regions A and B are neglected. This is the level of approximation employed in previous calculations. P S includes hops in region B, but not in region A (i.e., x < x ), while P includes hops in all regions. Hops t in the forbidden regions contribute very little at higher energies, as expected, but they do improve the accuracy of the calculations at lower energies as the turning point x t approaches the crossing point. Transition probabilities for energies below the crossing point energy are shown in table. The P S transition probabilities, which include hops in regions B and C, but not in region A, are somewhat more accurate than the P S results. These results are consistently within 0% of the - quantum results for transition probabilities as small as and within a factor of a little more than for transition probabilities down to less than - S. The results in the last three rows of table are numerically accurate to only about ± in the last place shown, due to the large cancellation between the contributions from regions B and C. The P (FO) results in table are similar to the P, except that they include at most one S (B±) (B-) (B±) hop in region B; i.e., all contributions to b and R have one hop and contributions to b (C+) and b have no hops. Any number of hops are still included in the calculation of the b and (B±) (C+) b matrices, since it is known that multiple hops are needed in region C to get accurate result when E is near or above the crossing point energy. n these calculations, all the terms included in (+) (B-) the outgoing wave function amplitude (x 0) contain one and only one of the following: b, (B+) (C-) (C+) (B-) b, b, b, or R, while all other factors in each term are diagonals element (b or (B-) (B+) (C-) (C+) (C-) (C+) b ) from the matrices b, b, b, or b. f the b, or b were restricted to no hop and S

17 Submitted to The Journal of Physical Chemistry Page of single hop terms, then this would be a first order (FO) P S calculation. Restricting the calculation to single hop trajectories in the forbidden region would simplify the calculation in higher dimensions. The accuracy of the P (FO) results is very similar to the full P calculation when E S < E c. At the lowest two energies considered, it provides results that are somewhat more accurate than the full P S calculation, although this improvement may be fortuitous given that it is only seen at the two lowest energies. V. Discussion The results presented in the previous section demonstrate that the semiclassical surface hopping expansion employed is able to provide accurate transition probabilities for the case where the crossing point and, therefore, the maximum in the coupling are well within the forbidden region. These calculations include both T-type and R-type hops in the forbidden region. The results indicate that the inclusion of hops in the region where the lower adiabatic surface is classically allowed and the upper surface is forbidden is crucial in obtaining good transition probabilities. These contributions also lead to noticeably improved results at energies slightly above the crossing point energy. On the other hand, ignoring hops in the region where both surfaces are classically forbidden generally yields slightly better results at energies below the crossing point energy than when these hops are included. The amplitudes for the T-type and R-type hops, Eqs. () and (), are singular at turning points in the classical motion, although these singularities are integrable. t seems surprising that a primitive semiclassical method with turning point singularities should so accurately reproduce the quantum results. The small transition probability at low energies results from the S

18 Page of Submitted to The Journal of Physical Chemistry considerable cancellation between terms with hops in the allowed regions and those that have hops for x t < x < x t. This is most easily seen in the P S(FO) calculations in table. There are (B-) four terms in these calculations, each of which have one b or R amplitude. These four (C+) -i / (C-) (C+) (B+) -i / (B-) (C-) (C+) (B+) -i / (B-) -i / (C-) terms have the form b e b, b b e b b, b b e b e b, (C+) (B-) -i / (C-) and b R e b. The first and second of these have their hop in region C, while the third and fourth have their hop in region B. The values for these four terms in the P S(FO) calculation at E = 0. are given in table along with their sum. The value of the contribution arising from R-type hops for x < x, which is included in the P calculation, is also given in the t table. The summation of the four terms in P S(FO) results in almost three orders of magnitude in cancellation. Because the interaction is relatively weak in the allowed region at this energy, there (C-) (C+) is only a small difference between the elements of b, or b, and the corresponding elements that would be obtained in a true first order (i.e., no multiple hop trajectories) approximation. f the expressions for the second and third terms in table are summed in the first order limit, this can be reexpressed as an integral with the integrand of the form of Eq.() and limits of integration of x t and x 0. Likewise, terms one and four can also be combined into a single integral from x t to x 0 in the first order limit. f the integration path is deformed into the lower half of the complex plane, the value of the integral is not altered, if the deformation avoids the square root branch points in the complex plane where the adiabatic surfaces W and W are equal. Such a deformation of the integration path is shown in figure. The deformed path continues from x t a small distance along the real axis to a minimum value x m < x t and then follows a contour below the real axis back to x 0 on the real axis. This deformed path avoids the singularities at x and x and the large cancellation that occurs when the integration is taken t t S

19 Submitted to The Journal of Physical Chemistry Page of along the real axis. f the integral corresponding to the first order limit of terms two and three is calculated along this deformed contour, the squared magnitude of the result is 0. x -. The value of the integral is numerically found to be independent of the choice of x m, as it should be, as long as the contour does not go around the branch point for the adiabatic surfaces, x b. The squared magnitude of the sum of terms two and three in table is 0. x -. The slight difference between this result and the one from the integration along the deformed contour is due to the fact that the values in table include multiple hops in region C. R-type hops are ignored for x > x t in these calculations, as has been the case in previous (C-) work. The contribution from R-type hops in region C, R, is presented for E = 0. and E =.0 in the last line of table. For E =.0, the two terms involving T-type hops in region C dominate (C-) the transition probability, and the R term is several orders of magnitude smaller. However, this term increases somewhat as the energy is lowered, and it is no longer negligible at E = 0.. f it were included in the calculation at this energy, the transition probability would be on the order of, while the correct answer is on the order of - - in region C, there is no compensating term from region B to largely cancel R. Unlike the terms with T-type hops (C-), which is dominated by the region near x t where the semiclassical approximation is inaccurate and diverging. The fact that there is no cancellation between this near-turning-point contribution and a term with hops in region B is the reason why the transition probability would be very poor if this term were included for E < E c. The surface hopping expansion is a multi-state, multidimensional formalism. Since it is necessary to accurately account for the significant cancellation between contributions from allowed and forbidden regions, the accurate application of this formalism to highly forbidden

20 Page of Submitted to The Journal of Physical Chemistry transitions in multi-dimensional problems remains an open question at this time. On the other hand, this formalism should certainly be able to provide useful corrections and better accuracy for multi-dimensional problems where the avoided crossing region is near the turning point region. n summary, the semiclassical surface hopping expansion presented previously has been applied to the calculation of transition probabilities for a one dimensional problem in which the avoided crossing is in the classically forbidden region. t is found that the surface hopping calculations are capable of providing very good values of the transition probabilities down to values on the order of -. This seemingly surprising result, given the singular nature of the semiclassical expression at the turning points in the classical motion, is readily understood when the integrations over hopping points are viewed as contour integrations in the complex plane. n order to obtain accurate results, it is necessary to include both transmission and reflection type hops in the region where the upper adiabatic surface is classically forbidden and to ignore reflection type hops in the classically allowed region. Acknowledgment This work is funded by NSF grant CHE-0. References. W. H. Miller, W. H. Adv. Chem. Phys.,, ; (), 0,.. R. A. Marcus, J. Chem. Phys.,, ;,, ;,,.. E. J. Heller, Acc. Chem. Res.,,.. E. J. Heller, J. Chem. Phys.,,.

21 Submitted to The Journal of Physical Chemistry Page 0 of M. F. Herman and E. Kluk, Chem. Phys.,,.. E. Kluk, M. F. Herman, and H. L. Davis, J. Chem. Phys.,,.. K. G. Kay, Annu. Rev. Phys. Chem. 00,,.. D. Zor and K. G. Kay, Phys. Rev. Lett.,, 0.. M. Madhusoodanan and K. G. Kay, J. Chem. Phys.,,.. X. Sun and W. H. Miller, J. Chem. Phys.,,.. X. Sun, H. B. Wang, and W. H. Miller, J. Chem. Phys.,, 0.. H. Wang, M. Thoss, K. L Sorge, R. Gelabert, X. Gimenez, and W. H. Miller, J. Chem. Phys. 00,,.. J. Liu and W. H. Miller, J. Chem. Phys. 00,,.. J. Shao and N. Makri, J. Phys. Chem. A,, ;,,.. J. B. Delos and W. R. Thorson, Phys. Rev. A,, 0;,,.. W. H. Miller and T. F. George, J. Chem. Phys.,,.. H. D. Meyer and W. H. Miller, J. Chem. Phys., 0, ;,, ; 0,,.. G. Stock and M. Thoss, Phys. Rev. Lett.,,.. G. Stock and M. Thoss, Phys Rev. A,,. 0. S. Bonella and D. F. Coker, J. Chem. Phys. 00,, ; 00,, 0.. R. K. Preston and J. C. Tully, J. Chem. Phys.,,.. J. C. Tully and R. K. Preston, J. Chem. Phys.,,.. M. F. Herman, J. Chem. Phys.,,.. M. F. Herman, J. Chem. Phys.,, ;,,.

22 Page of Submitted to The Journal of Physical Chemistry M. F. Herman, J. Chem. Phys.,, ;,, 0.. M. F. Herman, O. El Akramine, M. P. Moody, J. Chem. Phys. 00,,.. M. P. Moody, F. Ding, and M. F. Herman, J. Chem. Phys. 00,, 0.. M. F. Herman and M. P. Moody, J. Chem. Phys. 00,, 0.. Y. Wu and M. F. Herman, J. Chem. Phys. 00,,. 0. G. Yang, and M. F. Herman, J. Phys. Chem. B 00,,.. Y. Wu and M. F. Herman, J. Chem. Phys. 00,,.. Y. Wu and M. F. Herman, J. Chem. Phys. 00,, 0.. M. F. Herman and Y. Wu, J. Chem. Phys. 00,,.. J. C. Tully, J. Chem. Phys. 0,,.. S. Hammes-Shiffer and J. C. Tully, J. Chem. Phys.,,.. J. C. Burant and J. C. Tully, J. Chem. Phys. 000,, 0.. F.-Y. Fang and S. Hammes-Shiffer, J. Chem. Phys.,, ;, 0,.. F. Webster, E. T. Wang, P. J. Rossky, and R. A. Friesner, J. Chem. Phys., 0,.. E. R. Bittner and P. J. Rossky, J. Chem. Phys.,, 0;,,. 0. M. Ben-Nun and T. J. Martinez, J. Chem. Phys.,,.. Y. L Volovuev, M. D. Hack, M. S. Topaler, and D. G. Truhlar, J. Chem. Phys. 000,,.. M. D. Hack, A. M. Wensmann, D. G. Truhlar, M. Ben-Nun, and T. J. Martinez, J. Chem. Phys. 00,,.. A. W. Jasper, M. D. Hack, and D. G. Truhlar, J. Chem. Phys. 00,, 0.. A. W. Jasper, S. N. Stechmann, and D. G. Truhlar, J. Chem. Phys. 00,,.

23 Submitted to The Journal of Physical Chemistry Page of A. W. Jasper and D. G. Truhlar, Chem. Phys. Lett. 00,, 0.. C. Zhu, H. Kamisaka, and H. Nakamura, J. Chem. Phys. 00,,.. A. Kondorshiy and H. Nakamura, J. Chem. Phys. 00,,.. H. Bremmer, Commun. Pure Appl. Math.,,.. J. B. Delos, Rev. Mod. Phys.,, 0. F. T. Smith, Phys. Rev.,,.. V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel Publishing, Dordrecht, ).. L.. Schiff, Quantum Mechanics (McGraw-Hill, New York, ), pp. -.

24 Page of Submitted to The Journal of Physical Chemistry Table. Comparison of quantum (P Q) and semiclassical (P S, P S, and P S) transition probabilities for E > E c. P S only includes T-type hops in region C. P S includes T-type hops in region C, and T and R-type hops in region B. P S includes T-type hops in region C, and T and R- type hops in regions A and B. E P Q P S P S P S x x.00x.00x x x.x.x x - -.x.x.x x.x.x.x

25 Submitted to The Journal of Physical Chemistry Page of Table. Comparison of quantum (P Q) and semiclassical (P S, and P S) transition probabilities for E < E c. P S and P S are defined as in table. The numbers in the last three rows are converged only to within ± in the last place given. P (FO) includes the same hops as P, but only includes at most one hop for x x t. E P Q P S P S P S(FO) x.x 0..x x.x.x -.x x.0x.00x.0x x.x.x.0x x.x.x.x x - -.x.x.x x -.x.0x.x x.x.x.x x -.0x.x -.x x.x.x.x x.x.x -.x S S

26 Page of Submitted to The Journal of Physical Chemistry Table. Contributions to semiclassical transition probabilities. (See text for details.) Term E = 0. E =.0 (C+) -i / (C-) b e b.0x +.0x i -.x +.x i (C+) (B+) -i / (B-) (C-) b b e b b.x -.x i.0x +.0x i (C+) (B+) -i / (B-) -i / (C-) b b e b e b -.x +.x i.x -.x i (C+) (B-) -i / (C-) b R e b -.x -.0x i -.x +.x i Sum of terms -.x -.0x i.x +.x i (C+) (B+) (A-) (B-) -i / (C-) b b R b e b.x -.0x i.0x +.0x i (C-) R -.0x +.x i -.0x +.0x i Figure Captions Figure. The diabatic potential surfaces V, V, and V are plotted. V is multiplied by a factor of. Figure. ntegration contours in complex plane. The dark line along the real axis runs from x m to x 0. (See text for details.) The semicircular sections along this line near the turning points are shown, since these small deviations from the real axis connect the correct branch of p j = (m[e - W j(x)]) for x > x tj with the correct branch of p j for x < x tj. The actual integrations are performed in the limit in which the radius of the semicircle goes to zero (i.e., along the real axis). The dashed line is a contour in the lower complex half plane which connects x m to x 0 and does not enclose the branch point for the adiabatic surfaces, x b.

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