Web Release Date: June 27,
Departments of Chemistry and Physics, Harvard University, Cambridge, Massachusetts 02138, and Department of Chemistry, BenGurion University of the Negev, POB 653, BeerSheva 84105, Israel
Received: May 2, 2002
In Final Form: June 4, 2002
Abstract:
Novel approaches to surface hopping (in the case in which surfaces cross in classically allowed regions) and surface "jumping" (in cases in which they never cross or they cross in classically forbidden regions) are discussed. Classically forbidden transitions necessarily involve discontinuous "jumps" in position or momentum or both (but so as to preserve energy). In general, the jumps are discontinuous changes in nuclear positions or momenta on the time scale of the electronic transition. After reviewing various approaches in one dimension, a phasespace approach is applied to multidimensional systems with large energy gaps, in which the traditional semiclassical approaches are more difficult to apply. The concept of jumps extends the spirit of surface hopping into new regimes.
The crossing of potential energy surfaces (and the flow of nuclear wave function amplitudes between them) becomes important in many different contexts in chemical physics. Photochemical processes often involve dynamics on surfaces that cross (or can be made to cross in an appropriate diabatic representation). Photophysical processes involving radiationless transitions between different BornOppenheimer potential energy surfaces are extremely common. Finally, once we "dress" an initial potential energy surface with the energy of a photon, radiative processes involving absorption or emission of a photon are also seen to involve transitions between crossing or closelying potential energy surfaces.
Some years ago, Tully and Preston^{1} introduced an approach for approximate treatment of dynamics at regions of closelying potential energy surfaces. In many subsequent trials and refinements, it has proved a worthy computational tool, simple to implement and also very intuitive. The "surface hopping" method has its roots in the LandauZener theory^{2} but goes beyond it in dealing with multiple crossings and with many degrees of freedom.
Related theories include the exponential energy gap law of radiationless transitions^{3} and the exponential momentum gap law of Ewing^{4} for vibrational predissociation (in which a highfrequency vibrational mode plays the role of the electronic state, relative to a lowfrequency van der Waals mode). In these studies, considerable effort has been devoted to developing new intuition for nonclassical FranckCondon factors. The emphasis has been on the mode competition problem and the overall dependence of rates on the energy or quantum number gaps. Our emphasis is instead directly on the features of the potential energy surfaces, which control the FranckCondon factors, which in turn control the rates.
The fundamental idea that permits a classical treatment of
surface hopping is contained in the LandauZenerStuckelberg
model,^{2} which shows that the hops from one surface to another
are localized to the region where the surfaces cross or almost
cross. For weak coupling, the probability for hopping is given
in terms of the overall coupling, W, between the surfaces, the
difference in slopes, F_{}, normal to the intersection of the
two potentials at the crossing, and the speed, p_{}/m, with which
the trajectory goes through the crossing region:
Some years ago, Bergsma et al.^{5} introduced a simple way to evaluate the hopping probability by a somewhat different stochastic approach. We call it proximity hopping.
The idea is as follows. For a given trajectory, we monitor the energy difference between the two potential energy surfaces at the location of the trajectory. If this energy difference becomes less than some specified amount, E, the trajectory is subjected to hopping at a fixed probability per unit time. This is easy to arrange by taking to be, say, 0.05 per time step. The overall rate of hopping is governed by E and , but the local hopping rate is correctly governed by the above Ansatz. We now proceed to demonstrate this.
Consider Figure 1, which shows a contour plot of V_{A}(r^{N}) 
V_{B}(r^{N}) for a twodimensional case, together with a particular
trajectory on the V_{A} potential surface. This trajectory is plotted
as dots spaced at equal times, which could be the sample times
used to inquire whether the trajectory is in the hopping energy
range. The space between a pair of contour lines corresponds
to an energy range
Figure 1 Contours of the difference potential and a trajectory at equal time steps. 
Equation 1 is the LandauZenerStuckelberg rate for radiationless transition surface crossing^{2} in the case of diabatic surfaces that intersect with small coupling W. (The adiabatic crossing probability would be 1  P.) The "proximity hopping" method may be restricted to the weak coupling limit, when P is small per encounter with a crossing. It has the advantage of making clear the origin of the role of the velocity, the angle of approach to the surface intersection, and the difference in the magnitudes of the slopes of the two potential energy surfaces. These are all purely geometric factors, which simply determine how much time the trajectories spend "exposed" to the possibility of hopping between the surfaces. In the limit where P is small, the hopping probability is related to perturbation theory, in particular, a FranckCondon matrix element.
3.1. Hopping at a Classically Allowed Crossing Point. The
semiclassical eigenstates may be represented as a sum of terms
of the form (ignoring Maslov phases)
Apart from phases and an overall factor, stationaryphase
evaluation of the integral gives
3.2. Generalized Crossings. What if there is no crossing at classically allowed values of x? There are two possibilities for a crossing in the nonclassical regime. If it occurs at a real (but classically inaccessible) value of x_{c}, then the momentum at the crossing, p_{A}(x_{c}) = p_{D}(x_{c}), is purely imaginary. (Real momentum and real position correspond to the classical regime in which only "hopping" is required). We assign the name "position jump" to label the real quantity, that is, the position, that changes in the transition. From the coordinate space perspective, the contribution to the integral is coming from overlapping tails of the wave functions. Both p_{A}(x_{c}) and p_{A}(x_{c})* are stationaryphase points, but one of them corresponds to exponentially increasing wave function and is discarded.
One might retain the idea of a hop that simply occurs at the forbidden crossing, but from the perspective of the classical regime, this is a finite jump to a new location. More importantly, in many dimensions, the jumping idea turns out to be quite tractable, while the stationaryphase evaluation become extremely difficult. Thus, the "complex intersection" idea will give way to another type of evaluation of the integral, wherein it is noticed that the accepting states are much higher in vibrational energy than the donor, making a great simplification possible.
It often happens that the two real potential surfaces do not cross, not even at classically forbidden regions. If one analytically continues the potentials into the complex coordinate space, the crossing (i.e., the stationaryphase point) is then at complex values of x. The momentum is generally also complex (and not purely imaginary). One finds in these cases that the position of the crossing is sometimes mostly imaginary and the corresponding momentum mostly real, giving a jump that is largely in momentum. If the crossing happens at p = p_{c}, from the form of the Hamiltonian it also must occur for opposite sign p = p_{c}. For purely imaginary momentum, this is the same as , but generally there are four stationaryphase momenta. Two give rise to increasing rather than decreasing wave functions in the classically forbidden region and are discarded, leaving two remaining. There may be constructive or destructive interference between these distinct but equal magnitude stationaryphase amplitudes. This fact was noted by Medvedev^{6} and separately by Hunt and Child.^{7}
Given two potential energy surfaces, it is possible to infer the propensity (as a function of the energy gap) for position versus momentum tunneling or jumping using eq 11. This is a very direct and convenient tool.
Objections might be raised to the representation of the vibrational state of the upper BornOppenheimer surface by a semiclassical form, because we often take it to be the ground state, which is seemingly a dubious candidate for semiclassical approximation. At a very simple level, we may note that we are using the semiclassical form only in the classically forbidden region and normally deeply within it. As pointed out long ago by Miller,^{8} semiclassical approximations should work well in the deep tunneling regime.
In the classically forbidden region, eqs 711 hold yet with
complex momenta and real exponents. Here, a bound wave
function of the Hamiltonian, H(x,p), with energy E in the limit
of small is
We define w proportional to the logarithm of the FranckCondon factor squared so that
3.3. Transitions between Bound States. Below, we derive some general formulas for transitions between two bound states that will be needed in what follows. We suppose that the donor state is the ground vibrational state of the initial potential energy surface. We perform the integration with the quasiclassical wave functions; the dominant contribution comes from the crossing point.
Let V^{(D)} and V^{(A)} be the potentials of the donor and the
acceptor BornOppenheimer surfaces, respectively. In the
following, we use massweighted coordinates. We make additional assumptions regarding the functions V^{(D)}(x) and V^{(A)}(x).
For the initial potential, we assume that
New insight can be gained by studying the FranckCondon
factors in the Wigner representation:
Here, we define (x_{m}, p_{m}) to be the phasespace point dominating the phasespace overlap integral. The Wigner function is real so that we find (x_{m}, p_{m}) by finding the maximum of the phasespace integrand. As in coordinate space, there may be more than one maximum point, yet we shall assume that we are dealing with the dominant contribution. By definition, (x_{m}, p_{m}) are real and thus easily lend themselves to interpretation as the nuclear coordinates and momenta at the hopping or jumping point.
In previous publications, we have introduced this concept and demonstrated that it gives a useful interpretation.^{11,12} First, we showed for a model that the jumping point found in this way gives the correct initial conditions for dynamics on the accepting surface.^{12,13} Second, we showed that in a multidimensional system, the specific nature of the jump singles out the coordinate system most appropriate for analyzing the jump. For example, for radiationless transitions, one can find in this way the dominant accepting mode(s).^{12,13} Work is in progress to calculate overall transition rates (as contrasted with the abovementioned relative phasespace propensities) in multidimensional systems. This may be accomplished by expanding the integrand around the jumping point.
The phasespace approach has several advantages over the traditional semiclassical methods. The phasespace picture of the jump trivially extends to any number of dimensions. One just has to find the multidimensional point in phase space that has the maximal value of the integrand of eq 29 with respect to each coordinate and momentum. Extension of the traditional semiclassical approach to many dimensions is much more difficult. The phasespace approach allows for both small hops between close surfaces and large jumps, say, between nested potential energy surfaces when it is impossible to hop.
While the power of the phasespace picture comes forth in multidimensional systems and for large jumps, we focus below on onedimensional transitions between close surfaces. The examples that we consider are chosen so as to allow for a comparison between the two approaches in cases accessible to both. We hope to achieve two goals by doing so: first, to verify that the new approach reproduces correctly the results of the traditional methods and, second, to gain some insight as to the physical meaning of complex crossing points.
We have defined the point dominating the phasespace
integral, (x_{m}, p_{m}), as the "jumping point"the point in phase
space where the transition between the two potential energy
surfaces occurs. Below, we focus on the relations between the
more traditional hopping point x_{c} and our new definition for
the jumping point at (x_{m}, p_{m}). Instead of formally studying all
possible cases, we focus on specific examples in which both x_{c}
and (x_{m}, p_{m}) are well defined and show that for these examples
There are strong analogies between looking for a point in complex coordinate space dominating the FC factor integral and looking for a point in real phase space dominating the same integral in a different representation. Both methods sample the same phase space. We note in passing that in both cases the number of real variables on which the search for a dominant point is carried out is twice the number of degrees of freedom. In the first, more traditional approach, the momentum p_{c} is calculated after the complex coordinate x_{c} is found. In the phasespace approach, x_{m} and p_{m} are found simultaneously yet are both real. Equations 31 and 32 make the connection between these two seemingly different pictures. Positions and momenta are related by Fourier transforms, and real and imaginary parts of physical quantities are related by analyticity. We would like to assign physical significance to the jumping point, at least in the semiclassical limit and for cases in which the jumping point is well defined. It is difficult to assign physical meaning to a complex coordinate. Equations 31 and 32, when they apply, assign such physical meaning to x_{c} through the following prescription: one must calculate p_{c} from x_{c} and then take the real parts of both. This gives the physical coordinates and momenta of the transition between the potential surfaces.
We now proceed to justify eqs 31 and 32. We would like to find the jumping point (x_{m}, p_{m}) that is the extremum point of the integrand in eq 29. In section 5, we do so numerically for some examples. Here, we take an analytic approach that requires that we first find some explicit form for the Wigner functions, ^{(D)}(x,p) and ^{(A)}(x,p). We do so for the examples depicted in Figure 2.
4.1. Quasiclassical Approximation for the Donor Wigner Function, ^{(D)}(x,p). Let the initial state be the ground vibrational state of the donor potential energy surface so that the wave function is given by eq 13. The classically forbidden region is the entire real axis except for the region between the two classical turning points. Most of the cases that we consider are dominated by the region where eq 13 is valid, in the deeply forbidden region. Note that under these assumptions, (x) is real for real x.
Using the definition of the Wigner function (eq 30) and the
quasiclassical formula (eq 13), we express the Wigner function
of the donor state as
Wigner Function for Harmonic and Anharmonic Oscillators.
As an example of the utility of the analysis above, let us consider
a harmonic oscillator perturbed by cubic anharmonicity,
The function '(x) is the same as the classical momentum in
the inverted potential. By integration, we find (x) = x^{2}/2 +
x^{3}/(18) (neglecting terms of order ^{2}). By solving eq 38, we
get _{s} = ip/ + ixp/(3^{3}) and
4.2. Quasiclassical Approximations for the Acceptor
Wigner Function ^{(A)}(x,p). We consider two different approximations for the acceptor Wigner function, ^{(A)}(x,p). If the
final state is the ground state of the acceptor potential energy
surface, it is best approximated by eq 40. If, on the other hand,
it is some excited vibrational state or, as is often the case, a
manifold of degenerate excited states, a useful approximation
gives
4.3. Quasiclassical Estimation of the FranckCondon
Factor for Transitions between Two Ground States. Let both
donor and acceptor states be the ground states, each on its
respective potential energy surface. In this case
Eliminating p from two eqs 38, one for the initial and another
for the final state, we have
4.4. Transitions between the Ground Vibrational State of
the Donor Potential Energy Surface and a Manifold of
Vibrationally Excited States of the Acceptor Potential
Energy Surface. Let the donor state be the ground state on its
potential energy surface and the acceptor state be an excited
vibrational state or a manifold of degenerate excited states. In
this case,
The FranckCondon factor (eq 29) is then roughly (without
a prefactor) expressed as
Note that we define the "jumping point" (x_{m}, p_{m}) as the point where the integrand of the phasespace overlap integral is maximal. This notation is the same as that in section 4.3, although, in general, the two points need not be the same because they are the extremum points of different approximations for the integrand.
Let us establish the relation between the jumping point (x_{m}, p_{m}) of eq 61 and the crossing point of the potential energy surfaces (x_{c}, p_{c}) and between the logarithm of the FranckCondon factor w as calculated in section 3 within the traditional approach and the logarithm of the FranckCondon factor W_{m} as calculated here within the phasespace approach. We restrict ourselves to two cases, the case when the potential V^{(D)} is shallow or that when it is a steep function in comparison with the function V^{(A)}. We show below that in these limiting cases eqs 31 and 32 hold and in addition W_{m} = w. The phasespace approach agrees then with the traditional semiclassical method of section 3 above.
Shallow Minimum Potential V^{(D)}. For a shallow minimum
potential when V^{(D)} v_{0}, the turning point and the crossing
point, x_{R} and x_{c} defined by eqs 17 and 22, almost coincide, x_{R}
x_{c}_{}, so that the method of section 3 gives for the logarithm of
the FranckCondon factor
Now, let us consider this case within the phasespace
formalism of eq 61. First, we prove that, if V^{(D)} v_{0} and the
action ^{(D)}(x) is small, then only a coordinate jump is possible.
For small momenta, eq 38 is solved by _{s} = ip/' '(x).
Substituting it in eq 40, we find that
We suppose here that the potential of the acceptor is steeper
on the right side, so that W^{(D)}(x_{R},0) < W^{(D)}(x_{L},0), and we
conclude that
Steep Potential V^{(D)}. Consider the opposite case of a steep donor potential V^{(D)}. Here too, we would like to compare the results of the more traditional analysis in coordinate space with the phasespace analysis.
First, following section 3, we look for x_{c}, the point of crossing
of the two potential energy surfaces, and the FranckCondon
factor as given by eq 15. We approximate the donor potential
V^{(D)} near its minimum as
Let us consider an example of the harmonic potential of the
donor
If the parameter is small, the donor potential is relatively shallow, and when it is large, V^{(D)}(x) is steep. Numerical results are shown in Table 1. A similar table was generated for an earlier paper, but the parameters and the comparisons being made were somewhat different.^{11} For cases of steep or shallow potential V^{(D)}(x), relations 58 and 59 are satisfied with good accuracy.
The trajectory surface hopping methods initiated by the
famous Tully and Preston paper^{1} have been widely applied in
chemistry.^{17} In this paper, we have provided perspective, new
analysis, and testing of nonstandard approaches to surface
hopping and jumping. For the cases of surface hopping in one
or many dimensions or surface jumping in one dimension, the
techniques that we discuss are simply instructive alternatives
to other approaches. Our intent in these cases has been to
compare the various techniques with the Wigner phasespace
approach, because only the latter generalizes easily to many
dimensions. We have been able to give a rather more solid
grounding to the Wigner phasespace surface jumping method^{1113}
It may be interesting to combine the generalization of hops
into jumps with some new developments in mixed quantumclassical Lioville propagation recently introduced by Ciccotti
et al. and Martens et al.^{1822}
This research was supported by a grant (2000118) from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel.
Part of the special issue "John C. Tully Festschrift".
* To whom correspondence should be addressed. Email addresses: heller@physics.harvard.edu and bsegev@bgumail.bgu.ac.il.
Harvard University.
BenGurion University of the Negev.
1. Tully, J. C.; Preston, R. K. J. Chem. Phys. 1971, 55, 562.
2. Nikitin, E. F. Theory Elementary Atomic and Molecular Processes in Gases; Clarendon: Oxford, U.K., 1974.
3. Avouris, P.; Gelbart, W. M.; ElSayed, M. A. Chem. Rev. 1977,
77, 793.
4. Ewing, G. J. Phys. Chem. 1987, 91, 4662.
5. Bergsma, J. P.; Behrens, P. H.; Wilson, K. R.; Fredkin, D. R.;
Heller, E. J. J. Phys. Chem. 1984, 88, 612.
6. Medvedev, E. S. Chem. Phys. 1982, 73, 243251.
7. Hunt, P. M.; Child, M. S. Chem. Phys. Lett. 1978, 58, 202.
8. Miller, W. H. Adv. Chem. Phys. 1974, XXV, 69.
9. Berry, M. V. Proc. R. Soc. London, Ser. A 1977, 287, 237.
10. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics: NonRelativistic Theory; Pergamon: Oxford, U.K., 1977.
11. Heller, E. J.; Beck, D. Chem. Phys. Lett. 1993, 202 350.
12. Segev, B.; Heller, E. J. J. Chem. Phys. 2000, 112, 4004.
13. Sergeev, A. V.; Segev, B. J. Phys. A: Math. Gen. 2002, 35, 1769.
14. Heller, E. J. J. Chem. Phys. 1978, 68, 2066.
15. Hüpper, B.; Eckardt, B. Phys. Rev. A 1998, 57, 1536.
16. Berry, R. S.; Nielsen, S. E. Phys. Rev. A 1970, 1, 383.
17. Tully, J. C. In Modern Methods for Multidimensional Dynamics Computations in Chemistry; Thompson, D. L., Ed.; World Scientific: Singapore, 1998; p 34.
18. Donoso, A.; Martens, C. C. J. Phys. Chem. A 1998, 102, 4291.
19. Kapral, R.; Ciccotti, G. J. Chem. Phys. 1999, 110, 8919.
20. Neilsen, S.; Kapral, R.; Ciccotti, G. J. Chem. Phys. 2000, 112, 6543.
21. Neilsen, S.; Kapral, R.; Ciccotti, G. J. Chem. Phys. 2001, 115 5805.
22. Donoso, A.; Martens, C. C. Phys. Rev. Lett. 2001, 87, 2232021.
23. Kallush, S.; Segev, B.; Sergeev, A. V.; Heller, E. J. J. Phys. Chem.
A, in press.
V^{(A)} 
x_{c} 
p_{c} 
x_{m} 
p_{m} 
w_{c} 
W_{m} 
w 
case 

0.2 
M 
10.90 
2.18i 
10.82 
0 
11.8 
11.7 
14.1 
shallow V^{(D)}(x) 
1.0 
M 
13.34 
13.34i 
10.82 
0 
67.6 
58.5 
69.5 

1.3 
M 
15.56 
20.22i 
10.82 
0 
98.6 
76.1 
100.6 

5.0 
M 
0.01  3.06i 
15.32  0.06i 
0 
15.02 
22.9 
22.6 
24.4 
steep V^{(D)}(x) 
0.2 
PT 
32.52 
6.50i 
26.38 
0 
77.9 
69.6 
78.1 
shallow V^{(D)}(x) 
0.5 
PT 
21.95 + 15.62i 
7.81 + 10.97i 
22.44 
6.25 
176.2 
165.0 
177.6 

1.0 
PT 
12.10 + 16.49i 
16.49 + 12.10i 
0 
17.34 
179.6 
150.3 
180.2 

5.0 
PT 
3.54i 
17.70 
0 
17.34 
30.5 
30.1 
31.9 
steep V^{(D)}(x) 
^{a} x_{c}, defined by the equation V^{(D)}(x_{c})  v_{0} = V^{(A)}(x_{c})  E, is the coordinate where potentials cross, and p_{c} = ±[2(v_{0}  V^{(D)}(x_{c}))]^{1/2} = ±ix_{c} is the momentum at the crossing point. (x_{m}, p_{m}) is the point in phase space on the surface of constant energy H^{(A)}(x,p) = E where the Wigner function, ^{(D)}(x,p), reaches its maximum. w = (/2) ln f_{D}_{}_{A} is proportional to the logarithm of the FranckCondon factor; w_{c} and W_{m} are, respectively, the quasiclassical and phase space approximations to w, see sections 3.3 and 4.4. In the cases of several equivalent jumping points, we give only one of them.