Submitted to The Journal of Physical Chemistry
Semiclassical Nonadiabatic Surface Hopping Wave Function Expansion at
Low Energies: Hops in the Forbidden Region
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The Journal of Physical Chemistry
jp-2008-04937q
Special Issue Article
04-Jun-2008
Herman, Michael; Tulane University, Chemistry
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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 70118
Abstract
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 S
and all orders in the nonadiabatic coupling. It 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. In the numerical study reported here, transition probabilities are
evaluated for energies well below the crossing point energy. It is found that the expansion
provides accurate results for transition probabilities as small as 10-12.
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I. Introduction
Semiclassical methods often offer attractive alternatives to fully quantum approaches for
the calculation of collision processes in atomic and molecular systems.1-14 A variety of
semiclassical techniques have been proposed for problems in which more than one adiabatic
electronic state play a significant role.15-47 It has recently been shown33 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 (TISE) 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 Q is formally exact in the sense that all terms have been shown
to cancel when it is inserted into (H - E)Q, 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 TISE
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.23,28,33
Previous calculations have only considered transitions where the avoided crossings of the
adiabatic electronic energy surfaces are in the classically allowed region.23,28,33 In this work, the
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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. It 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 II. A. A one dimensional model curve crossing problem is
presented in section III, 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 IV.
II. Theory
This work numerically studies the behavior of the semiclassical surface hopping
expansion of the time independent wave function23,33 for one dimensional problems with turning
points. Let the incoming flux be on adiabatic surface W1(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.33 The energy determines the magnitude of the momentum after the
hop, but not its sign. If the momentum has the same sign before and after the hop, this hop is
referred to as a transmission or T-type hop. If the sign of the momentum changes, it is referred to
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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.23,33,48 These single surface reflection terms are neglected in this work, as they have been in
previous numerical calculations,23,27,33 resulting in a semiclassical approximation for the multistate wave function. Each trajectory from initial point x0 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/S), and an amplitude for each hop. For instance, the
contribution to the wave function on W2 from a trajectory that starts at x0 on surface W1 with
energy E, has a T-type hop to surface W2 at x1, and then continues to x, has the form
(1)
where pj is the momentum on Wj. One dimensional expressions are used here. The many
dimensional generalization for problems with any number of adiabatic states is provided
elsewhere.33 The amplitude for the T-type hop is given by23,33
(2)
where p1 and p2 are evaluated at x, 021 = < n2 | dn1/dx > is the nonadiabatic coupling, nj is the jth
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 R(x0, x1,
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x) over all possible hopping points x1; i.e., for x0 < x1 < x. The amplitude for an R-type hop is23,33
(3)
In numerical problems, the trajectory is divided into small steps of length )x. It is
convenient to express the wave function on surface Wj as Aj1Pj1, where Aj1 = [p1(x0)/pj(x)]½ is the
usual semiclassical prefactor and the initial incoming wave function is on surface W1. After the
first )x step, P11 can be approximated by the no hop term, P11(x0+)x) . exp[(i/S)Ip1dxN]P11(x0),
where the integral is evaluated from x0 to x0+)x, and P12 can be approximated using Eq. (1) as
(4)
where (12 is the integral of J12 from x0 to x0+)x. As long as )x is small, Eq.(4) provides a good
numerical approximation to the integral over all single T-type hop trajectories that move from x0
to x0+)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 x0 and x0+)x.23,28 All these
trajectories end on W2 and make contributions to P21(x0+)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. (4). Summing all of these
terms results in the replacement of (12 in Eq. (4) with sin((12).23 A similar summation of all
terms with an even number of T-type hops (that is, all terms ending on W1) gives P11(x0+)x) =
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cos((12) exp[(i/S)Ip1dxN] P11(x0), where the p1 integral is taken over the )x interval.23
Generalizing this analysis to any )x interval and initial state, it is found that
(5)
where xn = x0 + n)x and j can be 1 or 2. The elements of the step matrix An are given by
(6)
where
(7)
and 012 = - 021 has been used to obtain Eq. (6). For one dimensional problems of the type
considered here, the wave function amplitudes at x0+n)x can be expressed in terms of their
values at x0 as
(8)
Eq. (8) sums all trajectories from x0 and xn with energy E and any number of T-type hops. This
matrix multiplication method allows for all contributions to the Pkj(x) to be summed. In higher
dimensional problems, this summation over all hopping trajectories would generally be
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performed using a Monte Carlo procedure.28-30 Previous calculations indicate that only T-type
hops in the classically allowed region need to be considered if E > Ec to get very good transition
probabilities, where Ec is the energy at the crossing point in the diabatic energy surfaces.49,50
Defining B(n)(x0,)x) / An An-1 ... A1, then Bj1(n)(x0,)x) is the amplitude for the component of the
wave function for adiabatic state j at x0 + n)x if the initial wave function has an amplitude of
unity for state one and an amplitude of zero for state two at x0. Note that Bij(n)(xn,-)x) =
Bji(n)(x0,)x), given that (12 changes sign when the sign of )x is reversed, that (21 = -(12, and that
Sij for the trajectory from xm to xm+)x equals Sji for the reversed trajectory from xm+)x to xm.
Now consider the case in which both adiabatic surfaces, W1(x) and W2(x), approach
constant values at large x and increase rapidly for small x. At a given energy E, there is a turning
point, xtj, for classical motion on each surface Wj. If W1 is the upper adiabatic surface, then xt1 >
xt2. Let )x = (x0 - xt1)/n and choose the incoming wave function to be on surface one, R1(in)(x) =
exp[-ip10(x-x0)/S], where x0 is in the large x asymptotic region and p10 = |p1(x0)|. In this case,
Bj1(n)(x0,-)x) / bj1(C-) is the incoming wave function amplitude on surface Wj at xt1. Throughout
this work, the region x < xt2 is referred to as region A, xt2 < x < xt1 is referred to as region B, and
x > xt1 is referred to as region C. The superscript C- on bj1(C-) denotes that this amplitude
accounts for propagation in the negative direction across region C to xt1. If j =1, this contribution
to the wave function picks up the usual e-iB/2 turning point phase factor51,52 at xt1 and a factor of
bk1(C+) /Bk1(n)(xn,)x) for the propagation back from xt1 to x0 ending in state k. If j = 2, the
trajectory on W2 at xt1 continues in the negative direction for x < xt1. If hopping in the region
between xt2 and xt1 is ignored, as in previous calculations, then the component of the wave
function on W2 picks up a semiclassical phase factor of ei"/S as the trajectory travels from xt1 to
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xt2, where " = Ip2dxN with p2 < 0 and the integration is from xt1 to xt2. This yields an incoming
amplitude at xt2 of ei"/S b21(C-). This contribution to the wave function picks up a factor of e-iB/2 at
the turning point, xt2, another factor of ei"/S for the propagation from xt2 back to xt1, and a factor of
bk2(C+) for the propagation from xt1 to x0. Summing these contributions yields bk1(C+)e-iB/2 b11(C-) +
bk2(C+)e2i"/S e-iB/2 b21(C-) as the outgoing wave function amplitude on surface k at x0. This expression
contains contributions from all classically allowed hopping and non-hopping trajectories,
including only T-type hops. It is the expression used in previous computations.23,28,33
It is typical for the nonadiabatic coupling to be peaked around a crossing point, xc, for the
diabatic surfaces.49,50 At high energies, xc is sufficiently larger than xt1 for the type of problem
considered here, and the coupling is largely in the classically allowed region. If 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, xt1 increases. At some energy, xt1 = xc. Below
this energy, xt1 > xc and the coupling is peaked in the forbidden region. If this is the case, it is
necessary to include hops for x < xt1 in order to obtain accurate results. If xt2 < x < xt1, then p1 =
(2m[E-W1(x)])1/2 is imaginary, and p2 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 q1 = -i|p1| when traveling in the negative direction
and q1 = i|p1| when traveling in the positive direction. The expression for the transition amplitude
J21, Eq. (2), becomes
(9)
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and J12(x) = iJ21(x). The factor of i in J12 and it absence in J21 can be understood as follows. An
analysis of the surface hopping wave function in the Schrodinger equation23,33 shows that J21(x) =
[-(p1+p2)/p2][A1/A2]021 for a W1 to W2 hop, where Aj = [|pi(x0)/pj(x)|]½ is the semiclassical
prefactor at the hopping point x for a trajectory starting on initial surface i. The ratio A1/A2 gives
a factor of [|p2/p1|]½ yielding Eq. (2) if p1 is real and Eq. (9) if p1 is imaginary. If the hop is from
surface 2 to surface 1, then J12(x) = [-(p1+p2)/p1][A2/A1]012. Using (p1+p2)/p1 = (|p2| + i|p1|)/i|p1|
for imaginary p1, together with 012 = - 021, it is readily shown that J12 = iJ21.
In the region between the two turning points, the amplitude for R-type hops is given by
D21 = - J21* and D12 = - iD21, as can be seen by substituting for the imaginary p1 and comparing
Eqs.(2), (3), and (9). Thus, the magnitude of the R-type transition amplitude, D21, is equal to the
magnitude for the T-type transition amplitude, J21, in this region, in contrast to the allowed region
where |D21| is always less than |J21|. Consequently, the R-type hops cannot be ignored if hops in
region B are included in the calculations. In this region, the T-type transitions give rise to a step
matrix An, which is calculated using Eqs.(6) and (7) with p1 = -i|p1| and p2 = -|p2|. Since J12 is
complex and J21 = -iJ12 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((12) factors in Eq.(6) are
replaced with (12 and the cos((12) factors are dropped in the calculations for this region. These
lower order expressions should still result in a good approximation for small )x. If only T-type
hops were included, B(n)(xt1,-)x) / An An-1 ... A1 / b(B-), where )x = (xt1-xt2)/n, accounts for all
hopping trajectories traveling in the negative direction between xt1 and xt2. Similarly, b(B+) =
B(n)(xt2,)x) accounts for all (T-type only) hopping trajectories traveling in the positive direction
between xt2 and xt1. Since J21 = -iJ12, and accounting for the change in the sign of )x, b21(B+) =
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ib12(B-), b12(B+) = -ib21(B-), b11(B+) = b11(B-), and b22(B+) = b22(B-).
To simplify the calculation with R-type hops, trajectories with at most one R-type hop
between xt1 and xt2 are included in the calculations. The total contribution from all trajectories
with a single R-type hop in region B is a sum over all )x intervals between xt1 and xt2 of the
contribution with an R-type hop in that interval
(10)
where tij(k-1,+) = Bij(k-1)(xk-1,)x), xk-1 = xt1 - (k-1))x, tij(k-1,!) = Bij(k-1)(xt1,-)x),
,
(11)
r21(k) = ir12(k), and (D is integral of D12 over the )x interval. Eq. (10) includes trajectories with one
R-type hop and up to 2(n-1) T-type hops in region B.
If x < xt2, a trajectory is classically forbidden on both adiabatic surfaces. Both p1 and p2
are imaginary, and the amplitudes for T-type and R-type transitions are given by Eqs.(2) and (3).
The contribution corresponding to a trajectory decays as it moves in the negative direction. A Rtype hop reflects the trajectory back, and the contribution again decays as the trajectory moves in
the forward direction. The resulting amplitude, Rij(A-), has the form of Eq.(10) with
(12)
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and r21(k) = r12(k). Rij(A-) includes contributions from all trajectories that travel in the negative
direction from xt2 on surface j to the point xk-1 = xt2 - (k-1))x with k > 0 and with up to k-1 Ttype hops, have a R-type hop in the kth )x inteval, and then travel in the positive direction from
xk-1 to xt2 with up to k-1 T-type hops. In this region, tij(k-1,!) = tji(k-1,+), as is the case in the allowed
region.
The standard analysis for matching semiclassical expressions across turning points give
the following results.51,52 An incoming trajectory in the allowed region results in outgoing
trajectories in both the allowed and forbidden regions when it encounters the turning point, xt,
where incoming (outgoing) means traveling toward (away from) xt when discussing turning
points. The amplitude associated with the outgoing trajectory in the forbidden region is equal to
the amplitude associated with the incoming trajectory multiplied by a factor of e-iB/4 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-iB/2. 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 eiB/4. This
incoming trajectory from the forbidden region also gives rise to an outgoing trajectory in
forbidden region, but this trajectory, which is reflected back into the forbidden region, is ignored
in this work to simplify the calculations.
These turning point phase factors (i.e., e-iB/4, e-iB/2, and eiB/4) are used in the multi-state
calculations presented below. Let Pkj(+)(x) and Pkj(-)(x) be the state j to state k wave function
amplitudes corresponding to propagation in the forward and backward directions at x,
respectively. At the turning point, xt1, the matching conditions give P1j(+)(xt1+) = P1j(-)(xt1+)e-iB/2 +
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P1j(+)(xt1-)eiB/4 and P1j(-)(xt1-) = P1j(-)(xt1+)e-iB/4, where xt1- (xt1+) is a point x infinitesimally smaller
(larger) than xt1. The quantities P1j(-)(xt1+) and P1j(+)(xt1-) represent incoming wave functions
amplitudes at xt1, while P1j(-)(xt1-) and P1j(+)(xt1+) are the outgoing wave function amplitudes. The
contribution to the outgoing wave in the forbidden region, P1j(-)(xt1-), due to the incoming wave
from the forbidden region, P1j(+)(xt1-), is ignored in these expressions. The wave function
amplitudes on surface two, P2j(+) and P2j(-) are continuous at xt1. The same analysis is applied at
the surface two turning point, xt2, with the subscripts 1 and 2 interchanged.
The amplitudes for propagation in the negative direction at xt1- (i.e., traveling into region
B) are given by P1j(-)(xt1-) = e-iB/4[b11(C-) P1j(-)(x0) + b12(C-) P2j(-)(x0)] and P2j(-)(xt1-) = b21(C-) P1j(-)(x0) +
b22(C-) P2j(-)(x0). These give rise to amplitudes for propagation in the negative direction at xt2- (i.e.,
traveling into region A) of P1j(-)(xt2-) = b11(B-) P1j(-)(xt1-) + b12(B-) P2j(-)(xt1-) and P2j(-)(xt2-) = e-iB/4[b21(B-)
P1j(-)(xt1-) + b22(B-) P2j(-)(xt1-)], where the e-iB/4 is due to the turning point xt2. These lead to forward
propagating amplitudes at xt2+ of P1j(+)(xt2+) = R11(A-) P1j(-)(xt2-) + R12(A-) P2j(-)(xt2-) and P2j(+)(xt2+) =
e-iB/2 P2j(-)(xt2+) + eiB/4[R21(A-) P1j(-)(xt2-) + R22(A-) P2j(-)(xt2-)] at xt2+. The forward propagating
amplitudes at xt1+ are then given by P1j(+)(xt1+) = e-iB/2P1j(-)(xt1+) + eiB/4[b11(B+) P1j(+)(xt2+) + b12(B+)
P2j(+)(xt2+) + R11(B-)P1j(-)(xt1-) + R12(B-) P2j(-)(xt1-)] and P2j(+)(xt1+) = b21(B+) P1j(+)(xt2+) + b22(B+) P2j(+)(xt2+) +
R21(B-) P1j(-)(xt1-) + R22(B-) P2j(-)(xt1-). Finally, the outgoing wave function amplitudes at x0 can be
expressed as P1j(+)(x0) = b11(C+) P1j(+)(xt1+) + b12(C+) P2j(+)(xt1+) and P2j(+)(x0) = b21(C+) P1j(+)(xt1+) +
b22(C+) P2j(+)(xt1+). These expressions can be combined to obtain the outgoing amplitudes at x0,
P1j(+)(x0) and P2j(+)(x0), in terms of the incoming amplitudes at x0, P1j(-)(x0) and P2j(-)(x0). 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,
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PS = p2(x0)|A21(x0)P21(+)(x0)|2/[p1(x0)|A11(x0)P11(-)(x0)|2] = |P21(+)(x0)|2/|P11(-)(x0)|2.
(13)
It 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.
III. Results
In this section, semiclassical transition probabilities are compared to quantum results for
the one dimensional, two state model defined by the diabatic49,50 energy surfaces V11(x) = A11
exp(-a11x) + V0, V22(x) = A22 exp(-a22x), and V12(x) = A12{1- tanh[a12(x-x12)]} with A11 = 1.0, A22
= 5.0, A12 = 0.01, a11 = 1.0, a22 = 2.0, a12 = 2.0, x12 = 2.0, and V0 = 0.1. These are plotted in figure
1. The diabatic surfaces V11 and V22 cross at xc = 1.2975 and their value at the crossing point is
Ec = 0.3732. The adiabatic state energies and wave functions are obtained by diagonalizing the
diabatic energy matrix, giving W1(x) = Vav(x) + R(x)/2, and W2(x) = Vav(x) - R(x)/2, where
Vav(x) = (V11+V22)/2 and R(x) = [(V11-V22)2+4V122]12. The nonadiabatic coupling is given by
021(x) = <n2|dn1/dx> = [(V11-V22) dV12/dx - V12 (dV11/dx - dV22/dx)]/R2. Since W1 is the upper
adiabatic state, the classical turning point for this surface, xt1, is larger than xt2, the classical
turning point for W2, 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 shown23,28,33 to provide accurate
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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. If 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 Ec, 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 xmin 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 xf = 10, which is well into the asymptotic regime.
One of the two calculations begins with the wave function solely on V11 at xmin, while the other
calculation starts with the wave function on V22 at xmin. 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 # xf 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 V22 and an
incoming component of the wave function on V11 of R1(in)(x) = exp[-i|p1|(x-xf)]. The transition
probability is evaluated as the ratio of the outgoing flux on V22 and the incoming flux, PQ =
p2|R2|2/p1, where pj is the magnitude of the momentum on surface Vjj corresponding to the energy
E, and all quantities are evaluated at xf.
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Transition probabilities are presented in table 1 for energies above the energy of the
crossing point for the diabatic surfaces. Three levels of approximation are shown for the
semiclassical results. PS1 is the semiclassical transition probability when hops in regions A and B
are neglected. This is the level of approximation employed in previous calculations. PS2 includes
hops in region B, but not in region A (i.e., x < xt2), while PS3 includes hops in all regions. Hops
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 xt1 approaches the crossing
point. Transition probabilities for energies below the crossing point energy are shown in table 2.
The PS2 transition probabilities, which include hops in regions B and C, but not in region A, are
somewhat more accurate than the PS3 results. These results are consistently within 20% of the
quantum results for transition probabilities as small as 10-12 and within a factor of a little more
than 2 for transition probabilities down to less than 10-15. The results in the last three rows of
table 2 are numerically accurate to only about ±2 in the last place shown, due to the large
cancellation between the contributions from regions B and C.
The PS2(FO) results in table 2 are similar to the PS2, except that they include at most one
hop in region B; i.e., all contributions to b21(B±) and R21(B-) have one hop and contributions to b11(B±)
and b22(B±) have no hops. Any number of hops are still included in the calculation of the b(C+) and
b(C+) 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. In these calculations, all the terms included in
the outgoing wave function amplitude P21(+)(x0) contain one and only one of the following: b21(B-),
b21(B+), b21(C-), b21(C+), or R21(B-), while all other factors in each term are diagonals element (b11 or
b22) from the matrices b(B-), b(B+), b(C-), or b(C+). If the b(C-), or b(C+) were restricted to no hop and
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single hop terms, then this would be a first order (FO) PS2 calculation. Restricting the calculation
to single hop trajectories in the forbidden region would simplify the calculation in higher
dimensions. The accuracy of the PS2(FO) results is very similar to the full PS2 calculation when E
< Ec. At the lowest two energies considered, it provides results that are somewhat more accurate
than the full PS2 calculation, although this improvement may be fortuitous given that it is only
seen at the two lowest energies.
IV. 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. (2) and (3), are singular at turning
points in the classical motion, although these singularities are integrable. It 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
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considerable cancellation between terms with hops in the allowed regions and those that have
hops for xt2 < x < xt1. This is most easily seen in the PS2(FO) calculations in table 2. There are
four terms in these calculations, each of which have one b21 or R21(B-) amplitude. These four
terms have the form b21(C+)e-iB/2b11(C-), b22(C+)b22(B+)e-iB/2b22(B-) b21(C-), b22(C+)b22(B+)e-iB/2b21(B-)e-iB/4b11(C-),
and b22(C+)R21(B-)e-iB/4b11(C-) . 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 PS2(FO)
calculation at E = 0.2 are given in table 3 along with their sum. The value of the contribution
arising from R-type hops for x < xt2 , which is included in the PS3 calculation, is also given in the
table. The summation of the four terms in PS2(FO) results in almost three orders of magnitude in
cancellation. Because the interaction is relatively weak in the allowed region at this energy, there
is only a small difference between the elements of b(C-), or b(C+), and the corresponding elements
that would be obtained in a true first order (i.e., no multiple hop trajectories) approximation. If
the expressions for the second and third terms in table 3 are summed in the first order limit, this
can be reexpressed as an integral with the integrand of the form of Eq.(1) and limits of
integration of xt2 and x0. Likewise, terms one and four can also be combined into a single
integral from xt2 to x0 in the first order limit. If 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 W1 and W2 are
equal. Such a deformation of the integration path is shown in figure 2. The deformed path
continues from xt2 a small distance along the real axis to a minimum value xm < xt2 and then
follows a contour below the real axis back to x0 on the real axis. This deformed path avoids the
singularities at xt1 and xt2 and the large cancellation that occurs when the integration is taken
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along the real axis. If 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 0.95 x 10-10. The
value of the integral is numerically found to be independent of the choice of xm, as it should be,
as long as the contour does not go around the branch point for the adiabatic surfaces, xb. The
squared magnitude of the sum of terms two and three in table 3 is 0.98 x 10-10. 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 3 include multiple hops in region C.
R-type hops are ignored for x > xt1 in these calculations, as has been the case in previous
work. The contribution from R-type hops in region C, R21(C-), is presented for E = 0.2 and E = 1.0
in the last line of table 3. For E = 1.0, the two terms involving T-type hops in region C dominate
the transition probability, and the R21(C-) 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 = 0.2.
If it were included in the calculation at this energy, the transition probability would be on the
order of 10-5, while the correct answer is on the order of 10-10. Unlike the terms with T-type hops
in region C, there is no compensating term from region B to largely cancel R21(C-), which is
dominated by the region near xt1 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 < Ec.
The surface hopping expansion33 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
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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.
In 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. It 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 10-12. 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. In
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-0715333.
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Table 1. Comparison of quantum (PQ) and semiclassical (PS1, PS2, and PS3) transition
probabilities for E > Ec. PS1 only includes T-type hops in region C. PS2 includes T-type hops in
region C, and T and R-type hops in region B. PS3 includes T-type hops in region C, and T and Rtype hops in regions A and B.
E
PQ
PS1
PS2
PS3
0.38
0.618
0.440
0.576
0.587
0.40
0.951
0.819
0.918
0.918
0.45
0.142
0.179
0.143
0.143
0.50
0.835
0.761
0.838
0.838
0.55
1.02x10-2
7.76x10-3
1.00x10-2
1.00x10-2
0.60
0.543
0.508
0.544
0.544
0.65
0.599
0.589
0.599
0.599
0.70
1.42x10-2
1.77x10-2
1.41x10-2
1.41x10-2
0.75
0.356
0.348
0.356
0.356
0.80
0.637
0.650
0.638
0.638
0.85
0.132
0.137
0.132
0.132
0.90
0.118
0.120
0.118
0.118
1.00
0.327
0.327
0.327
0.327
1.20
1.87x10-2
1.77x10-2
1.86x10-2
1.86x10-2
1.40
0.184
0.182
0.184
0.184
1.60
0.485
0.487
0.485
0.485
1.80
3.44x10-2
3.41x10-2
3.43x10-2
3.43x10-2
2.00
0.359
0.359
0.359
0.359
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Table 2. Comparison of quantum (PQ) and semiclassical (PS2, and PS3) transition probabilities for
E < Ec. PS2 and PS3 are defined as in table 1. The numbers in the last three rows are converged
only to within ±2 in the last place given. PS2(FO) includes the same hops as PS2, but only
includes at most one hop for x #xt1.
E
PQ
PS2
PS3
0.36
0.275
0.261
0.290
0.288
0.34
8.65x10-2
8.54x10-2
0.105
8.89x10-2
0.32
1.93x10-2
1.94x10-2
2.51x10-2
1.97x10-2
0.30
3.00x10-3
3.03x10-3
4.00x10-3
3.05x10-3
0.28
3.16x10-4
3.19x10-4
4.26x10-4
3.20x10-4
0.26
2.14x10-5
2.16x10-5
2.92x10-5
2.15x10-5
0.24
8.54x10-7
8.55x10-7
1.19x10-6
8.56x10-7
0.22
1.77x10-8
1.76x10-8
2.60x10-8
1.81x10-8
0.20
1.49x10-10
1.74x10-10
2.81x10-10
1.42x10-10
0.19
8.50x10-12
7.07x10-12
1.39x10-11
7.10x10-12
0.18
3.06x10-13
1.3x10-13
4.1x10-13
3.4x10-13
0.17
5.33x10-15
2.4x10-15
4.8x10-14
8.5x10-15
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PS2(FO)
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Table 3. Contributions to semiclassical transition probabilities. (See text for details.)
Term
E = 0.2
E = 1.0
b21(C+)e-iB/2b11(C-)
6.50398x10-3 + 3.05368x10-3 i
-3.45899x10-2 + 3.85245x10-1 i
b22(C+)b22(B+)e-iB/2b22(B-)b21(C-)
6.18492x10-3 - 3.65697x10-3 i
3.80373x10-1 + 7.01941x10-2 i
b22(C+)b22(B+)e-iB/2b21(B-)e-iB/4b11(C-) -6.17858x10-3 + 3.66455x10-3 i
6.94549x10-4 - 5.31652x10-4 i
b22(C+)R21(B-)e-iB/4b11(C-)
-6.49839x10-3 - 3.06182x10-3 i
-6.98721x10-4 + 5.26158x10-4 i
1.19235x10-5 - 5.66904x10-7 i
3.45779x10-1 + 4.55434x10-1 i
Sum of terms 1 - 4
b22(C+)b22(B+)R21(A-)b11(B-)e-iB/4b11(C-) 2.82697x10-6 - 1.24408x10-7 i
R21(C-)
-3.34076x10-3 + 3.73151x10-3 i
1.87570x10-23 + 2.47054x10-23 i
-1.26083x10-5 + 5.40694x10-4 i
Figure Captions
Figure 1. The diabatic potential surfaces V11, V22, and V12 are plotted. V12 is multiplied by a
factor of 10.
Figure 2. Integration contours in complex plane. The dark line along the real axis runs from xm
to x0. (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 pj = (2m[E Wj(x)]) for x > xtj with the correct branch of pj for x < xtj. 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 xm to x0 and does not
enclose the branch point for the adiabatic surfaces, xb.
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