Wave Function Continuity and the Diagonal Born-Oppenheimer Correction at Conical Intersections

Transcription

1 Wave Function Continuity and the Diagonal Born-Oppenheimer Correction at Conical ntersections Garrett A. Meek and Benjamin G. Levine* Department of Chemistry, Michigan State University, East Lansing, M 4884 * to whom correspondence should be addressed: levine@chemistry.msu.edu Abstract We demonstrate that, though exact in principle, the expansion of the total molecular wave function as a sum over adiabatic Born-Oppenheimer (BO) vibronic states makes inclusion of the second-derivative nonadiabatic energy term near conical intersections practically problematic. n order to construct a well-behaved molecular wave function that has density at a conical intersection, the individual BO vibronic states in the summation must be discontinuous. When the second-derivative nonadiabatic terms are added to the Hamiltonian, singularities in the diagonal BO corrections (DBOCs) of the individual BO states arise from these discontinuities. n contrast to the well-known singularities in the first-derivative couplings at conical intersections, these singularities are non-integrable, resulting in undefined DBOC matrix elements. Though these singularities suggest that the exact molecular wave function may not have density at the conical intersection point, there is no physical basis for this constraint. nstead, the singularities are artifacts of the chosen basis of discontinuous functions. We also demonstrate that continuity of the total molecular wave function does not require continuity of the individual adiabatic nuclear wave functions. We classify nonadiabatic molecular dynamics methods according to the constraints placed on wave function continuity and analyze their formal properties. Based on our analysis, it is recommended that the DBOC be neglected when employing mixed quantum-classical methods and certain approximate quantum dynamical methods in the adiabatic representation. 1

2 Key Words: conical intersection, nonadiabatic molecular dynamics, second-derivative coupling, diagonal Born-Oppenheimer correction, well-behaved molecular wave function 1. ntroduction Nonadiabatic molecular dynamics simulations have become a powerful theoretical tool for understanding chemical processes involving multiple electronic states Such simulations are typically performed in either the adiabatic or diabatic representation of electronic states. When the nuclear-electronic time-dependent Schrodinger equation is solved exactly, the results are independent of the choice of representation, but with some exceptions, approximate methods often yield results that depend strongly on this choice. 0,1 The primary advantages of the adiabatic representation are that it is uniquely defined and the coupling between electronic states is typically local. This locality simplifies the accurate treatment of coherent nuclear processes. (Note that it may be possible to construct diabatic bases with similarly local couplings.) Working in the adiabatic representation is not without its challenges, however. The firstderivative nonadiabatic coupling between electronic states becomes singular at conical intersections, points in nuclear configuration space at which adiabatic electronic states become degenerate. -7 representation, 8-43 This difficulty is often circumvented by working in the diabatic though graceful handling of these singularities within the adiabatic representation can also achieve accurate predictions of population transfer probabilities A second-derivative nonadiabatic term, the diagonal of which is referred to as the diagonal Born- Oppenheimer correction (DBOC), also arises. Though methods to compute the DBOC have been widely applied in high-accuracy quantum chemical calculations, with rare exceptions 64-

3 68 the second derivative coupling is neglected in nonadiabatic molecular dynamics simulations. On its surface this approximation appears to be questionable, however, as the DBOC is singular at a conical intersection and very large in its neighborhoods. 5,69 ntuitively, one might expect inclusion of the DBOC to significantly reduce the rate of nonadiabatic transitions by repelling population away from the conical intersection seam, for example. However, recently it has been argued that neglect of the DBOC in surface hopping simulations is justified by the fact that, in practice, errors introduced by this approximation are cancelled by errors arising from neglect of geometric phase effects. 66,70 Perhaps most troubling is that the energy singularities associated with the DBOC do not arise in the diabatic representation. Wave functions with density at the intersection point appear to be energetically forbidden in the adiabatic representation but not in the diabatic representation. Should the probability of finding density at the intersection geometry not be independent of representation? n this work, we demonstrate that the singularity in the DBOC at a conical intersection is an artifact of the representation of the wave function and therefore is not reflected in the exact quantum dynamics of a real physical system. We also analyze the conditions for continuity of the molecular wave function at conical intersections, and discuss the implications that our analysis has for the construction of nonadiabatic molecular dynamics methods based on the adiabatic representation.. ntegrability of the DBOC in the Adiabatic Approximation First we review the definition of the DBOC in the context of the adiabatic approximation. Consider the total molecular wave function in the adiabatic approximation, r, R, t ( R, t) ( r; R ), (1) ad where r and R are the electronic and nuclear coordinates, respectively, t is time, and ( R, t) and ( rr ; ) are the respective nuclear and electronic wave functions associated with adiabatic 3

4 electronic state,. (Note that throughout this work we use the adjective molecular to refer to a wave function that describes both nuclear and electronic motion, and we use the term adiabatic nuclear wave function to refer to a nuclear wave function associated with a single adiabatic electronic state, such as ( R, t).) The DBOC, defined E 1 DBOC ( ) ( ) M ( ) r R R R, () arises from the operation of the nuclear kinetic energy operator on the R-dependent electronic wave function of a particular adiabatic electronic state. Here η indexes nuclear degrees of freedom, M η is the mass associated with a particular degree of freedom, and the subscript r indicates that integration is performed over electronic degrees of freedom only. Atomic units are used throughout this work. The DBOC is typically a small correction to the Born-Oppenheimer (BO) potential energy surface (PES) that is computed only when highly accurate energetics are required, but as noted above this is not the case at the geometry of a conical intersection, R C. At the intersection point, ( rr ; ) is discontinuous with respect to R, and this discontinuity results in a singularity in EDBOC ( R ). We will now prove that, unlike the singularity in the first-derivative coupling, the singularity in the second-derivative term is not integrable. Consider a two-dimensional system described in polar coordinates, R and θ, with a conical intersection between adiabatic states and at the origin, R=0. For convenience we define a normalized branching angle coordinate,. Thus is in units of distance, the -domain is 0, R R, and the volume element for integration is simply dv ddr. t has been shown that the integral of the derivative coupling projected onto an infinitesimal closed path encircling a conical intersection will have a value of 4

5 . 71 A circle of small radius R centered at the origin is an example of such a path, thus for small R R d, ( R, ) d (3) 0 where we arbitrarily choose positive π, and d, ( R, ) is the -component of the firstderivative coupling vector, d, ( R, ) /. (4) By insertion of a resolution of the identity into the expression for the -component of the DBOC of state, it can be shown that / d ( R, ). (5) K, K (For conciseness we do not include fundamental constants or masses and we assume the electronic wave functions to be real.) The average value of d, ( R, ) on some closed circular path, d, ( R, ), is bounded from below by the square of the average of d, ( R, ) along that same path,,, d ( R, ) d ( R, ). (6) Dividing both sides of Eq. (3) by the path length, πr, yields d, therefore ( R, ) R d ( R, ) d 1, (7) R R, 0 d R d, ( R, ) d 0 1, ( R, ) R 4R (8) 5

6 and R d, ( R, ) d 0. (9) R One can demonstrate that the -component of the first-derivative coupling vector is integrable at the intersection by integrating over some neighborhood of radius R, R R d, ( R, ) ddr R. (10) 0 0 By similarly integrating over the lower bound to the -term of the second-derivative nonadiabatic term one can demonstrate that the DBOC is not integrable on the same domain: d R d dr dr R R R R, (, ) (11) This last integral diverges, therefore the DBOC is not integrable over domains including the intersection. Thus the total DBOC contribution to the energy of the BO molecular wave function, ( t) EDBOC ( t), will diverge whenever (, ) R R t is non-zero at the intersection. (For brevity, throughout this work we refer to population that is located at R C and corresponds to one of the intersecting adiabatic electronic states as at the intersection.) Stated differently, because of the discontinuity in ( rr ; ) at R C there can be no continuous molecular wave function, ad rr,, t, for which ( R, t) is non-zero at R C. Thus, as previously demonstrated by several authors, the probability of the molecule being at a conical intersection is zero in all wellbehaved molecular wave functions in the adiabatic approximation. 7,73 (n fact, past proofs that the DBOC is integrable 65,7 were based on the assumption of a vibronic wave function that 6

7 decays rapidly to zero at the intersection, and thus do not contradict our proof above, which is not based on this assumption.) 3. Wave Function Continuity and the DBOC Beyond the Adiabatic Approximation states: Now consider the exact molecular wave function represented as a sum over BO vibronic BO t t t r, R, ( r, R, ) ( R, ) ( r; R ). (1) BO We emphasize that in this work we use the term BO vibronic states,,, t rr, to refer to the products of a nuclear wave function with an adiabatic electronic wave function. f each BO vibronic state is taken to be a well-behaved wave function, then the total molecular wave ad function would have the same properties as,, t rr above, i.e. the probability of being at a conical intersection is zero. There is clearly no physical basis for this constraint, however, when we consider other representations of the exact molecular wave function. For example, consider a similar sum over vibronic wave functions in a diabatic electronic basis, with the diabatic states chosen to be smoothly varying at the intersection point. Such states do not lead to analogous singularities in second-derivative nonadiabatic terms because their electronic wave functions do not have analogous discontinuities. Thus, in the diabatic representation it is trivial to construct a well-behaved molecular wave function with density at the conical intersection. The same lack of singularities at the intersection point can be seen in the exact decomposition of the time-dependent molecular wave function, r, R, t ( R, t) ( r; R, t). (13) t can be proven that any well-behaved molecular wave function can be factored such that both ( rr ;, t) and ( R, t) are continuous functions of R.77 Thus ( ;, t) rr is certain to be 7

8 continuous regardless of whether a conical intersection is present or not. Therefore, the exact molecular wave function may have density at a conical intersection. We also note that there are no discontinuities in ( ;, t) rr to give rise to singularities in the second-derivative nonadiabatic energy of ( rr ;, t), E R t R t R t, (14) 1 ESDEC (, ) (, ) M (, ) r which we now call the exact second-derivative energy correction (ESDEC) to distinguish it from the DBOC, which corresponds to a particular adiabatic electronic state. (Notice that the ESDEC is time-dependent due to the time-dependence of ( ;, t) rr. An identical term arises in the exact time-dependent PES described by Abedi, Maitra, and Gross. 74,75 ) The ESDEC provides a concise description of second-derivative nonadiabatic effects and thus is a useful point of reference for further discussion below. Consider now the apparent inconsistency that arises from the facts established thus far: 1) any continuous molecular wave function in the adiabatic approximation (i.e. a continuous BO vibronic wave function) has no density at a conical intersection, ) the exact molecular wave function may be expanded as a sum over BO vibronic wave functions, and 3) the exact molecular wave function may have density at a conical intersection. We reconcile these facts by recognizing that there is no requirement that the individual terms in Eq. (1) be well-behaved, only that the total molecular wave function be so. Thus, for the three points above to hold true, the BO vibronic wave functions of the intersecting states in the summation in Eq. (1) must be discontinuous at a conical intersection point if the probability of being at that intersection is nonzero. As such, whenever density is present at a conical intersection, the DBOC matrix elements of the individual BO vibronic states, ( t) EDBOC ( t), diverge as described above. Though R 8

9 the individual DBOC matrix elements diverge, when they are summed with the appropriate offdiagonal second-derivative coupling matrix elements to form the ESDEC, this energy obviously must be finite for any well-behaved wave function. This cancelation is a reflection of the fact that the discontinuities in the individual BO wave functions exactly cancel one another out when they are summed to form the well-behaved total molecular wave function. 4. Continuity Conditions for the Molecular and Adiabatic Nuclear Wave Functions While discontinuity of the intersecting BO vibronic states at conical intersections is a necessary condition for a continuous molecular wave function, it is obviously not sufficient, so it is instructive to consider the constraints on ( R, t) and ( R, t) near conical intersections. Noting that the adiabatic electronic wave functions of the intersecting states, ( rr ; ) and ( rr ; ), are discontinuous at R C, the adiabatic nuclear wave functions, ( R, t) and ( R, t), must be constructed such that lim,,t RR C rr is independent of the direction in which the limit is taken. (The intersection may be approached from any direction in the branching plane.) n some special cases, this condition may be satisfied when ( R, t) and ( R, t) are continuous functions of R, but there is no a priori reason to expect them to be continuous, and in the most general case ( R, t) and ( R, t) will be discontinuous at R C. n this section we examine the relationship between the conditions for continuity of the molecular wave function and those for the continuity of the individual adiabatic nuclear wave functions using an illustrative example. Consider a two-dimensional system described in polar coordinates, R { R, }, with a conical intersection at the origin. The electronic wave function can be represented in a basis of two real-valued diabatic electronic wave functions, A( rr ; ) and B( rr. ; ) These diabatic wave 9

10 functions are continuous and differentiable in R at all R. Diagonalization of the Hamiltonian gives the adiabatic basis i/ i/ ( rr ; ) e cos ( ; ) e sin A rr i/ i/ ( ; ) ie sin ( ; ) ie cos rr B rr. (15) n this formulation the nuclear coordinates are defined such that the mixing angle defining the diabat-to-adiabat transformation is /, and the θ-dependent complex phase in the transformation matrix elements ensures continuity and single-valuedness of the adiabatic electronic wave functions at all points except for the origin. 78 Now consider a well-behaved nuclear wave function, ( R, t), that is nonzero at the origin. The nuclear wave function can be written as a sum of two orthogonal terms ( R, t) ( R, t) ( R, t) (16) where c s c R t e R t (17) and i / (, ) cos (, ) s R t ie R t. (18) i / (, ) sin (, ) We choose this basis for its convenience. For example, a well-behaved molecular wave function confined to diabat A can be represented ( R, t) ( r; R) ( R, t) ( r; R) ( R, t) ( r; R ). (19) A c s Note that the derivatives of ( R, t) and ( R, t) with respect to θ diverge at the origin, but c these functions are continuous and differentiable everywhere else. Now consider the total molecular wave function s r, R, t c ( R, t) ( r; R) c ( R, t) ( r; R) c ( R, t) ( r; R) c ( R, t) ( r; R ). (0) c c c c s s s s 10

11 The wave function is defined by the four-dimensional complex vector, c. The condition for normalization (noting that ( R, t) and ( R, t) are neither orthogonal nor normalized) is X, c cx cx c c cx sx c s sx sx s s s c c ( t) ( t) Re c c ( t) ( t) c c ( t) ( t) 1. (1) As noted above, continuity requires that the limit of the total molecular wave function be the same regardless of the direction from which the origin in approached. For simplicity we consider one specific instance of this condition, R0 R0 lim r, R, 0, t lim r, R,, t. () Applying Eq. () to the definition of the total molecular wave function in Eq. (0) yields c c c (3) s and c c c. (4) s t can be shown that the two conditions Eqs. (3) and (4) are sufficient for continuity of the total molecular wave function, and therefore, along with (1), they define the space of continuous and normalized molecular wave functions. Now consider the representation of rr,,t as a sum over BO vibronic wave functions in the adiabatic representation (Eq. (1)). The adiabatic nuclear wave functions are ( R, t) c ( R, t) c ( R, t) (5) and c c s s ( R, t) c ( R, t) c ( R, t). (6) c c s s Continuity of ( R, t) and ( R, t) does not follow from the conditions for continuity of the total molecular wave function; the conditions for continuity of these functions are, 11

12 c c c (7) s and c c c. (8) s Eq. (8) follows from (3), (4), and (7), thus continuity of the individual adiabatic nuclear wave functions introduces a single additional independent constraint on c. Thus ( R, t) and ( R, t) are continuous in only an infinitesimal fraction of possible continuous molecular wave functions. This result has important consequences for quantum dynamical simulations at conical intersections. Many widely used wave function ansatzes require ( R, t) and ( R, t) to be continuous, but this requirement is highly restrictive. Most often continuity of the total molecular wave function will instead be achieved via compensating discontinuities in ( R, t), ( R, t), ( rr ; ), and ( rr ; ). Thus the simulation of exact quantum dynamics in the adiabatic basis requires a representation of ( R, t) and ( R, t) flexible enough to describe discontinuities at the intersection. These discontinuities in ( R, t) and ( R, t) result in undefined nuclear kinetic energy matrix elements, e.g. 1 M. These undefined matrix elements are distinct from the undefined DBOC matrix elements described in previous sections. 5. mplications for Approximate Simulations of Dynamics at Conical ntersections To summarize, we have demonstrated that the singularities in the DBOC at conical intersections arise due to the fact that the BO vibronic wave functions summed over in Eq. (1) are necessarily discontinuous, though the total molecular wave function is well-behaved. These singularities are not integrable, and therefore result in undefined DBOC matrix elements. n contrast, the ESDEC does not exhibit singularities at conical intersections. Thus, undefined 1

13 DBOC matrix elements at conical intersections are artifacts that reflect discontinuities in the chosen basis functions, and do not indicate that the true probability of a molecule being at a conical intersection is zero. n other words, difficulties arise because one is representing a differentiable function as a sum of two (or more) non-differentiable functions, and then attempting to compute the derivative of the total function as the sum of the derivatives of the non-differentiable ones. Now we discuss the implications of the above arguments for nonadiabatic molecular dynamics simulations. Quantum dynamical ansatzes based on the adiabatic representation can be divided into three classes based on which continuity conditions are enforced at the conical intersection: A) Ansatzes in which the continuity of the total molecular wave function is enforced at conical intersections, but individual adiabatic nuclear wave functions are allowed to be discontinuous, B) Ansatzes in which the continuity of both the total molecular wave function and the individual adiabatic nuclear wave functions is enforced, and C) Ansatzes in which the continuity of the individual adiabatic wave functions is enforced, but continuity of the total molecular wave function is not. As we demonstrated in the previous section, only type A ansatzes may be exact. Obviously an exact treatment of quantum dynamics in the adiabatic representation requires the inclusion of the DBOC term in the Hamiltonian. Though potentially exact, construction of a type A ansatz would not be straight forward, as it would require careful treatment of the various discontinuities and resulting undefined matrix elements. Employing the diabatic representation is a simpler alternative to a type A ansatz for near-exact simulations near conical intersections. 13

14 n practice, the majority of quantum dynamical ansatzes based on the adiabatic representation fall into classes B and C. When a type C ansatz is employed the total molecular wave function will generally be discontinuous, and this discontinuity will manifest itself in the DBOC energy terms. The energy arising from the DBOC of a discontinuous wave function diverges, as described in section. For this reason, the inclusion of the DBOC into fully quantum mechanical nonadiabatic molecular dynamics simulations of this type is effectively impossible; it leads to infinite error. Neglect of the DBOC is necessary to correct the energy for the unphysical discontinuities in the wave function. On the other hand, in a type B ansatz the DBOC matrix elements are finite, but there is a cost to be paid for this convenience. As described in section 4, a type B ansatz may only be constructed by dramatically limiting the flexibility of the wave function, either by reducing the dimensionality of the space of possible states that have population at the intersection or by eliminating the possibility of having population at the intersection all together. One can imagine that this reduction in the space of possible states at the intersection may significantly affect the dynamics. Our arguments say nothing about the expected magnitude of E ( R, t), so complete neglect of second-derivative nonadiabatic terms may still be ill-advised. One can imagine that complete neglect of the ESDEC may be a good approximation in many cases but not in others. Kendrick, et al. demonstrated that the non-removable portion of the second-derivative nonadiabatic coupling is similar in magnitude to that in regions of the PES where the BO ESDEC approximation is valid. 79 This non-removable second-derivative coupling is equivalent to the ESDEC for population confined to a single diabatic PES. Thus, when a system undergoes diabatic population transfer between states, one can imagine that the ESDEC is similar in magnitude to the non-removable component of the second-derivative coupling, and thus 14

15 negligible for most purposes. More generally, however, the exact R-dependent electronic wave function, ( ;, t) rr, will depend on the specific dynamics of the system being studied, therefore E ( R, t) is dependent on those dynamics as well. Thus, the accuracy of neglecting ESDEC the DBOC may not only be system-dependent but also dependent on the particular dynamics of the system (e.g. the nature of the initial excitation). More work is needed to investigate the magnitude of the ESDEC in realistic systems. Mixed quantum-classical methods are among the most widely employed classes of methods for modeling dynamics at conical intersections. These methods do not involve a nuclear wave function, and thus the above discussion of continuity cannot be directly applied. However, our arguments suggesting that the DBOC should be neglected in type C quantum dynamics methods can be extended to apply to mixed quantum-classical methods as well. The singularity in the DBOC arises only when taking the semiclassical limit of a discontinuous and thus unphysical vibronic wave function, similar to those that arise in type C methods. Additionally, inclusion of the DBOC in mixed quantum-classical simulations ensures that no population reaches the conical intersection seam itself, though such a constraint has no physical basis. For these reasons, inclusion of the DBOC in mixed quantum-classical simulations is discouraged. The semiclassical limit of the exact wave function will necessarily have a finite second-derivative energy correction. Because ( ;, t) rr cannot be easily determined from a single classical trajectory, however, there is no immediately apparent recipe to incorporate the ESDEC into widely used methods like surface hopping. However, Gross and coworkers have recently reported a promising mixed quantum-classical approach that is based on the exact factorization of the molecular wave function and naturally includes the ESDEC

16 The above arguments suggest new avenues for research. We hope that they will stimulate further investigations into the magnitude of the ESDEC in real systems and the degree to which it can determine dynamics near conical intersections and elsewhere. Additionally, though the above arguments suggest that the adiabatic representation is highly problematic when second-derivative nonadiabatic matrix elements are incorporated, its uniqueness and the locality of the associated interstate couplings still strongly recommend it. Further work may suggest means to achieve a best-of-both-worlds approach, which maintains the advantages of the adiabatic representation to a great degree while circumventing its difficulties. Acknowledgements This material is based upon work supported by the National Science Foundation under CHE The authors are also grateful to Michigan State University for startup funding. BGL thanks Bob Cukier, Artur zmaylov, Todd Martínez, Piotr Piecuch, Don Truhlar, and David Yarkony for productive discussions. References C. Tully and R. K. Preston,. Chem. Phys. 55, 56 (1971). H. D. Meyer and W. H. Miller,. Chem. Phys. 70, 314 (1979). H. Koppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984).. C. Tully,. Chem. Phys. 93, 1061 (1990). S. Hammes-Schiffer and. C. Tully,. Chem. Phys. 101, 4657 (1994). B.. Schwartz, E. R. Bittner, O. V. Prezhdo, and P.. Rossky,. Chem. Phys. 104, 594 (1996).. Burghardt, H. D. Meyer, and L. S. Cederbaum,. Chem. Phys. 111, 97 (1999). M. Ben-Nun,. Quenneville, and T.. Martinez,. Phys. Chem. A 104, 5161 (000). C. Y. Zhu, S. Nangia, A. W. asper, and D. G. Truhlar,. Chem. Phys. 11, 7658 (004). X. S. Li,. C. Tully, H. B. Schlegel, and M.. Frisch,. Chem. Phys. 13, (005). C. F. Craig, W. R. Duncan, and O. V. Prezhdo, Phys. Rev. Lett. 95, (005). X. Chen and V. S. Batista,. Chem. Phys. 15, (006). G. A. Worth, M. A. Robb, and B. Lasorne, Mol. Phys. 106, 077 (008). 16

17 M. Richter, P. Marquetand,. Gonzalez-Vazquez,. Sola, and L. Gonzalez,. Chem. Theory Comput. 7, 153 (011). A.. White, V. N. Gorshkov, R. X. Wang, S. Tretiak, and D. Mozyrsky,. Chem. Phys. 141, (014). W.. Ouyang, W.. Dou, and. E. Subotnik,. Chem. Phys. 14, (015). M. D. Hack, A. M. Wensmann, D. G. Truhlar, M. Ben-Nun, and T.. Martinez,. Chem. Phys. 115, 117 (001). A. W. asper, S. Nangia, C. Y. Zhu, and D. G. Truhlar, Acc, Chem. Res. 39, 101 (006). L.. Wang, A. E. Sifain, and O. V. Prezhdo,. Chem. Phys. 143, (015). E. Neria and A. Nitzan,. Chem. Phys. 99, 1109 (1993). O. V. Prezhdo and P.. Rossky,. Chem. Phys. 107, 85 (1997). D. R. Yarkony, Rev. Mod. Phys. 68, 985 (1996). F. Bernardi, M. Olivucci, and M. A. Robb, Chem. Soc. Rev. 5, 31 (1996). D. R. Yarkony,. Phys. Chem. A 105, 677 (001). G. A. Worth and L. S. Cederbaum, Annu. Rev. Phys. Chem. 55, 17 (004). B. G. Levine and T.. Martinez, Annu. Rev. Phys. Chem. 58, 613 (007). W. Domcke and D. R. Yarkony, Annu. Rev. Phys. Chem. 63, 35 (01). M. Baer, Chem. Phys. 15, 49 (1976). A. Macias and A. Riera, ournal of Physics B-Atomic Molecular and Optical Physics 11, L489 (1978). C. A. Mead and D. G. Truhlar,. Chem. Phys. 77, 6090 (198). T. Pacher, L. S. Cederbaum, and H. Koppel,. Chem. Phys. 89, 7367 (1988). R.. Cave and M. D. Newton, Chem. Phys. Lett. 49, 15 (1996). G.. Atchity and K. Ruedenberg, Theor. Chem. Acc. 97, 47 (1997). A.. Dobbyn and P.. Knowles, Mol. Phys. 91, 1107 (1997). D. R. Yarkony,. Chem. Phys. 11, 111 (000). H. Koppel,. Gronki, and S. Mahapatra,. Chem. Phys. 115, 377 (001). H. Nakamura and D. G. Truhlar,. Chem. Phys. 115, (001). H. Nakamura and D. G. Truhlar,. Chem. Phys. 117, 5576 (00). V. C. Mota and A.. C. Varandas,. Phys. Chem. A 11, 3768 (008). X. Zhu and D. R. Yarkony,. Chem. Phys. 13, (010). C. Evenhuis and T.. Martinez,. Chem. Phys. 135, 4110 (011). A. Sirjoosingh and S. Hammes-Schiffer,. Phys. Chem. A 115, 367 (011).. E. Subotnik, E. C. Alguire, Q. Ou, B. R. Landry, and S. Fatehi, Acc, Chem. Res. 48, 1340 (015). S. Fernandez-Alberti, A. E. Roitberg, T. Nelson, and S. Tretiak,. Chem. Phys. 137, (01). T. Nelson, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, Chem. Phys. Lett. 590, 08 (013). L.. Wang and O. V. Prezhdo,. Phys. Chem. Lett. 5, 713 (014). G. A. Meek and B. G. Levine,. Phys. Chem. Lett. 5, 351 (014). G. A. Meek and B. G. Levine, Chem. Phys. 460, 117 (015). B. C. Garrett and D. G. Truhlar,. Chem. Phys. 8, 4543 (1985). B. H. Lengsfield and D. R. Yarkony,. Chem. Phys. 84, 348 (1986). N. C. Handy, Y. Yamaguchi, and H. F. Schaefer,. Chem. Phys. 84, 4481 (1986). P. Saxe and D. R. Yarkony,. Chem. Phys. 86, 31 (1987). 17

18 P. Piecuch, X. Z. Li, and. Paldus, Chem. Phys. Lett. 30, 377 (1994). L. Wolniewicz and K. Dressler,. Chem. Phys. 100, 444 (1994). S. Mahapatra, H. Koppel, and L. S. Cederbaum,. Phys. Chem. A 105, 31 (001). D. W. Schwenke,. Phys. Chem. A 105, 35 (001). S. L. Mielke, K. A. Peterson, D. W. Schwenke, B. C. Garrett, D. G. Truhlar,. V. Michael, M. C. Su, and. W. Sutherland, Phys. Rev. Lett. 91, (003). E. F. Valeev and C. D. Sherrill,. Chem. Phys. 118, 391 (003). M. S. Schuurman, S. R. Muir, W. D. Allen, and H. F. Schaefer,. Chem. Phys. 10, (004). A. Tajti, P. G. Szalay, A. G. Csaszar, M. Kallay,. Gauss, E. F. Valeev, B. A. Flowers,. Vazquez, and. F. Stanton,. Chem. Phys. 11, (004).. Gauss, A. Tajti, M. Kallay,. F. Stanton, and P. G. Szalay,. Chem. Phys. 15, (006). G. Czako, B. C. Shepler, B.. Braams, and. M. Bowman,. Chem. Phys. 130, (009). C. P. Hu, O. Sugino, and K. Watanabe,. Chem. Phys. 135, (011).. Morelli and S. Hammes-Schiffer, Chem. Phys. Lett. 69, 161 (1997). D. R. Yarkony,. Chem. Phys. 114, 601 (001). R. Gherib,. G. Ryabinkin, and A. F. zmaylov,. Chem. Theory Comput. 11, 1375 (015). A. Abedi, F. Agostini, and E. K. U. Gross, Epl 106, (014). L. K. McKemmish, R. H. McKenzie, N. S. Hush, and. R. Reimers, Physical Chemistry Chemical Physics 17, 4666 (015). T. C. Thompson, D. G. Truhlar, and C. A. Mead,. Chem. Phys. 8, 39 (1985).. G. Ryabinkin, L. oubert-doriol, and A. F. zmaylov,. Chem. Phys. 140, (014). D. R. Yarkony,. Phys. Chem. A 101, 463 (1997). C. A. Mead,. Chem. Phys. 78, 807 (1983). A.. C. Varandas and Z. R. Xu, Chem. Phys. Lett. 316, 48 (000). A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 1300 (010). A. Abedi, N. T. Maitra, and E. K. U. Gross,. Chem. Phys. 137, a530 (01). L. S. Cederbaum,. Chem. Phys. 138, 4110 (013). See supplemental material at [URL will be inserted by AP] for proof of the continuity of the exact nuclear and electronic wave functions. C. A. Mead and D. G. Truhlar,. Chem. Phys. 70, 84 (1979). B. K. Kendrick, C. A. Mead, and D. G. Truhlar, Chem. Phys. 77, 31 (00). 18