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\def\bo{Born-Oppenheimer}
\def\sos{surface of section}
\def\pes{potential energy surface}
\def\pess{potential energy surfaces}
\def\jacs{{\sl J. Am. Chem. Soc. }}
\def\jcp{{\sl J. Chem. Phys. }}
\def\jcph{{\sl J. Comp. Phys. }}
\def\jpc{{\sl J. Phys. Chem. }}
\def\ang{{\sl Angew. Chem. }}
\def\angint{{\sl Angew. Chem. Int. Ed. Engl. }}
\def\acc{{\sl Acc. Chem. Res.}}
\def\jcc{{\sl J. Comp. Chem. }}
\def\jog{{\sl J. Org. Chem. }}
\def\theo{{\sl J. Mol. Struct. (THEOCHEM) }}
\def\cpl{{\sl Chem. Phys. Letters }}
\def\crev{{\sl Chem. Rev. }}
\def\ijqc{{\sl Intern. J. Quantum Chem. }}
\def\mol{{\sl Mol. Phys. }}
\def\ijqcs{{\sl Intern. J. Quantum Chem. Symp. }}
\def\tca{{\sl Theoret. Chim. Acta }}
\def\cp{{\sl Chem. Phys.}}
\def\prl{{\sl Phys. Rev. Lett. }}
\def\cmp{{\it Comm. Math. Phys. }}
\def\ann{{\it Ann. Phys. (N.Y.) }}
\def\prs{{\it Proc. Roy. Soc. London }}
\def\acp{{\it Adv. Chem. Phys. }}
\def\pr{{\it Phys. Rev. }}
%****************************************************************
%********macros**********
\def\bo{Born-Oppenheimer}
\def\sos{surface of section}
\def\pes{potential energy surface}
\def\rt{radiationless transition}
\def\S{SchrUdinger }
\def\as{absorption spectrum}
\def\wp{wave packet}
\def\rs{Raman spectrum}
\def\phs{phase space}
\def\p-s{phase-space}
\def\tdse{time dependent \S equation }
\def\ni{\noindent}
\def\ul{\underline}
\def\cl{\centerline}
\def\ft{Fourier transform}
\def\fc{Franck-Condon}
\def\wf{wavefunction}
\def\ef{eigenfunction}
\def\ap{approximation}
\def\g{Gaussian}
\def\td{time dependent}
\def\mxe{matrix element}
\def\sp{stationary phase}
\def\gf{Green's function}
\def\sc{semiclassical}
\def\se{Schr\"odinger equation}
\def\c{classical}
\def\t{trajectory}
\def\ts{trajectories}
\def\wf{wavefunction}
\def\wp{wavepacket}
\def\li{linearizable}
\def\ni{\noindent}
\def\ho{harmonic oscillator}
\def\nv{non-vertical}
\def\tun{tunneling}
\def\fcf{Franck-Condon factor}
\def\cf{classically forbidden}
\def\corf{correlation function}
\def\acf{autocorrelation function}
\def\et{electronic transition}
\def\nc{nonclassical}
\def\rl{\rangle\langle}
\def\mj{momentum jump}
\def\pj{position jump}
\def\az{accepting zones}
\def\wg{Wigner}
%****************************************************************
% Some new macros for the Dirac bra-ket notation.
%****************************************************************
\def\bra#1{\langle{#1}\vert}
\def\ket#1{\vert{#1}\rangle}
\def\braket#1#2{\langle{#1}\vert{#2}\rangle}
\def\me#1#2#3{\langle{#1}\vert{#2}\vert{#3}\rangle}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}
% constants and functions
\newcommand{\e}{e}% e = 2.718...
\renewcommand{\i}{i}% e^(Pi i/2)
\renewcommand{\Re}{\mathrm{Re}\,}% Re (\hspace{1pt})
\renewcommand{\Im}{\mathrm{Im}\,}% Im
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\newcommand{\eref}[1]{equation~(\ref{#1})}% equations (M)
\newcommand{\eeref}[2]{equations~(\ref{#1}) and (\ref{#2})}% equations (M) and (N)
\newcommand{\tfrac}[2]{\frac{#1}{#2}}% for some reason, definition of tfrac disappeared (?)
% marks
%\newcommand{\init}[1]{#1^{(\mathrm{I})}}% something Initial
%\newcommand{\fin}[1]{#1^{(\mathrm{F})}}% something Final
\newcommand{\init}[1]{{#1}^{(\mathrm{D})}}% something Initial
\newcommand{\fin}[1]{{#1}^{(\mathrm{A})}}% something Final
\newcommand{\initfin}[1]{#1^{(\mathrm{I}\rightarrow\mathrm{F})}}% something Initial->Final
\newcommand{\Left}[1]{{#1}_\mathrm{L}}% something Left
\newcommand{\Right}[1]{{#1}_\mathrm{R}}% something Right
\newcommand{\Land}[1]{{#1}_\mathrm{c}}% something Landau
% buzz words
\newcommand{\ps}{phase space}
\newcommand{\ha}{harmonic approximation}
%\newcommand{\ho}{harmonic oscillator}
\newcommand{\wif}{Wigner function}
\newcommand{\jp}{jumping point}
\newcommand{\bos}{Born - Oppenheimer surface}
\newcommand{\mo}{Morse oscillator}
\newcommand{\pto}{Poeschl - Teller oscillator}
%\newcommand{\se}{Schr\"odinger equation}
%\newcommand{\fcf}{Franck - Condon factor}
\title{Hopping and jumping between potential energy surfaces}
\author{E.J. Heller${}^{1}$\footnote{To
whom correspondence should be addressed
(heller@physics.harvard.edu {\it or} bsegev@bgumail.bgu.ac.il).},
Bilha Segev${}^{2*}$, and A.V. Sergeev${}^{2}$
\\
$1$ \small{\it Departments of Chemistry and Physics, Harvard University,
Cambridge, MA 02138 USA} \\
$2$
\small{\it Department of Chemistry, Ben-Gurion University of the
Negev, POB 653, Beer-Sheva 84105, ISRAEL}}
\begin{document}
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\maketitle
\begin{abstract}
%bs[
Novel approaches to surface hopping (in the case where surfaces cross
in classically allowed regions) and surface ``jumping''
(in cases where they
never cross, or 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 phase-space approach
is applied to multidimensional systems with large energy gaps,
where the traditional semiclassical approaches are more difficult to
apply. The concept of jumps extends the spirit of surface hopping into
new regimes.
%bs]
\end{abstract}
\section{Introduction}
The crossing of potential energy surfaces (and the flow of nuclear
wavefunction amplitudes between them) becomes important in many different
contexts in chemical physics. Photochemical processes often involve
dynamics on surfaces which cross (or can be made to cross in an appropriate
diabatic representation). Photophysical processes involving radiationless
transitions between different Born-Oppenheimer 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 close lying potential energy surfaces.
Some years ago, Tully and Preston \cite{tp} introduced an approach for
approximate treatment of dynamics
at regions of close lying 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 Landau-Zener
theory\cite{lz}, 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 \cite{gelbart} and the exponential momentum gap law of Ewing
\cite{ewing} for vibrational predissociation (where a high
frequency vibrational mode plays the role of the electronic state,
relative to a low frequency Van der Waals mode). In these studies,
considerable effort has been devoted to developing new intuition
for nonclassical Franck-Condon 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 \pes s which control the Franck-Condon factors, which in
turn
control the rates.
The fundamental idea which permits a classical treatment of surface hopping
is contained in the Landau-Zener-Stuckelberg model \cite{lz}, 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 $|\Delta {\bf F}_\perp|$ normal to the intersection of
the two potentials at the crossing, and the speed $p_\perp /m$ with which
the trajectory goes through the crossing region:
\begin{eqnarray}
\label{finalhopeqn}
P=k\tau&=\frac{2\pi W^2} {\hbar^2 \delta E} \frac{1}{ p_\perp} \left[
\frac{m\hbar\delta E} {|\Delta {\bf F}_\perp|}\right] \nonumber\\
&=\frac{2\pi W^2 }{\hbar} \left[\frac{m} {p_\perp |\Delta{\bf
F}_\perp|}\right].
\end{eqnarray}
%bs[
In this paper we study hopping and jumping between potential energy
surfaces. In section 2 Equation (\ref{finalhopeqn}) for the
Landau-Zener-Stuckelberg rate is rederived following Bergsma et.\ al.\
\cite{berg}. In the weak-coupling regime transitions between potential
energy surfaces are controlled by Franck-Condon matrix elements.
The relation between these matrix elements (overlap integrals) and the
hopping or jumping mechanism is the subject of the paper.
In section 3 within the traditional semiclassical perspective,
the transition is shown to occur through a small hop or a large jump to
the point in coordinate space where the two surfaces cross, $x_c$.
In general, $x_c$ defined in this way is complex.
A likewise complex momentum is then associated with this jump. Finding
it is easy in one dimension, but much more complicated in the multidimensional
case. In section 4 the same Franck-Condon integrals are considered in the
Wigner representation. This time, the point that dominates the
Franck-Condon factor is real. The phase-space formulation is
applicable for any dimension. A relation between the jumping point
$(x_m,p_m)$ in real phase space and the complex surface-crossing
coordinate $x_c$ and momentum $p_c$ in the one dimensional case is
suggested and demonstrated for model systems, in limiting cases,
and through numerical examples, in sections 4 and 5 respectively.
Section 6 presents conclusions.
%bs
\section{Trajectory binning method}
Some years ago, Bergsma et. al. \cite{berg} introduced a simple way to
evaluate the 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 \pess, at the
location of the trajectory. If this energy difference becomes less than
some specified amount,
$\delta E$, the trajectory is subjected to hopping at a fixed probability
$\phi$ per unit time. This is easy to arrange
by taking $\phi$ to be, say, 0.05 per time step. The overall rate of
hopping is governed by $\delta E$ and $\phi$,
but the local hopping rate is correctly governed by the above Ansatz. We
now proceed to demonstrate
this.
Consider Fig.~\ref{binproof}, which shows a
contour plot
of $V_A({\bf r}^N) -V_B({\bf r}^N)$ for a two dimensional 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
\begin{equation}
\label{erange}
-\frac{\delta E}{2} \leq E\leq \frac{\delta E}{2},
\end{equation} The amount of time spent
between the two shaded contour lines is proportional to the contribution
this particular trajectory segment will make to the hopping rate.
The time $\tau$ spent between contours is the distance $D$
traversed divided by the velocity, or
\begin{equation}
\tau=\frac{D}{(p/m)} =\frac{md} {p\sin\theta} =\frac{md} {p_{\perp}},
\end{equation}
in which $p$ is the magnitude of the momentum, $p_\perp=p \sin\theta$ is
the magnitude of the momentum perpendicular to the contour lines, and $m$
the mass. The perpendicular separation $d$ between the contours is
\begin{equation}
d=\frac{\delta E}{|\Delta {\bf F}_\perp|}
\end{equation}
where
\begin{equation}
|\Delta{\bf F}_\perp|=|\nabla V_A({\bf r}^N) - \nabla V_B({\bf r}^N)|
\end{equation}
evaluated in the vicinity of the crossing region. $\Delta{\bf F}_\perp$
is the force change between the final and initial potential surfaces
perpendicular to the contour lines of $V_A-V_B$ at the intersection $V_B+
\hbar \omega_m=V_A$.
Taking the rate of crossing per unit time from the initial to the final
surface to be
\begin{equation}
k=\frac{2\pi W^2}{\hbar \delta E},
\end{equation}
in which $W$ is a coupling parameter, the total probability $P$ of crossing
from the initial to the final surface while on the trajectory in the region
of the bin is given by Eq.~(\ref{finalhopeqn}).
Nothing essential depends on our assumption of two degrees of freedom.
Eq.~(\ref{finalhopeqn}) is applicable to many degrees of freedom.
Equation (\ref{finalhopeqn}) is the Landau-Zener-Stuckelberg rate for
radiationless
transition surface crossing\cite{lz} in the case of diabatic surfaces
which 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 Franck-Condon matrix element.
\section{\label{trad} The traditional semiclassical perspective}
\subsection{Hopping at a classically allowed crossing point}
The semiclassical eigenstates may be represented as a sum of
terms of the form (ignoring Maslov phases)
\begin{eqnarray}
\psi(x) \approx {1\over\vert p(x)\vert ^{1/2}}
\exp\left[-{\frac{i}{\hbar} \int^x p(x') \, dx'}\right] \label{8}
\end{eqnarray}
A Franck-Condon overlap between two such states, each on its own
Born-Oppenheimer \pes,
is given by
\begin{eqnarray}
\langle\fin{\psi}\vert\init{\psi} \rangle \approx \int
{1\over \vert p_A(x)\vert ^{1/2}\vert p_D(x)\vert ^{1/2}}
\exp\left[{\frac{i}{\hbar} \int^x p_A(x') \, dx'
-\frac{i}{\hbar} \int^x p_D(x') dx'\,} \, dx \right]
\end{eqnarray}
The \sp\ evaluation of this integral requires
\begin{eqnarray}
\label{stationary}
{d\over dx}\Biggl ( \int^x p_A(x') \, dx'
- \int^x p_D(x') \, dx'\Biggr ) = 0
\end{eqnarray}
and the stationary-phase point is $x_c$ satisfying
\begin{eqnarray}
p_A(x_c) = p_D(x_c)
\end{eqnarray}
which implies, since the total energy is the same on both \pes s,
\begin{eqnarray}
\label{cmplx}
V_A(x_c) = V_D(x_c). \label{12}
\end{eqnarray}
This is the well known result that the semiclassical contribution
arises where \pes s cross. There may of course be more than one
\sp\ point; in what follows, we shall assume we are dealing with
the dominant contribution.
Apart from phases and an overall factor, stationary phase evaluation of the
integral gives,
\begin{eqnarray}
f_{A,D}=\vert \langle\psi_A\vert\psi_D \rangle \vert^2 \sim
\sum \left | {\partial (p_A-p_D)\over \partial x}
{1\over p_A p_D}
\right |_{x=x_c} \sim {1\over p(x_c) \Delta F(x_c)}
\end{eqnarray}
which is the one dimensional version of \eref{finalhopeqn}.
\subsection{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 pure imaginary. (Real momentum {\it and} real position correspond to
the classical regime where only ``hopping'' is required). We assign the name ``position jump'' to label the real quantity, i.e. the
position, which changes in the transition. From the coordinate space perspective the contribution to the integral is coming
from overlapping tails of the wavefunctions. Both
$p_A(x_c)$ and
$p_A(x_c)^*$ are stationary phase points, but one of them
corresponds to exponentially increasing wavefunction and is
discarded.
One might retain the idea of a
hop which 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 stationary phase 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.
%bs[
It often happens that the two real potential surfaces do not cross,
not even at classically forbidden regions.
Analytically continuing the potentials into the complex coordinate space
the crossing (i.e. the stationary phase 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 {\it mostly} imaginary, and the corresponding momentum
mostly real, giving a jump that is largely in momentum.
%bs]
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 pure imaginary momentum this is the same as
$p_c^{*}$,
but generally there are four \sp\ momenta. Two give
rise to increasing rather than decreasing wavefunctions in the \cf\
region and are discarded, leaving
two remaining. There may be constructive or destructive
interference between these distinct but equal magnitude \sp\
amplitudes. This fact was noted by Medvedev
\cite{M82} and separately by Child \cite{child}.
Given two \pes s, it is possible to infer the propensity (as a
function of the energy gap) for position {\it versus} momentum
tunneling or jumping using Eq.~(\ref{cmplx}). This is a very
direct and convenient tool.
Objections might be raised to the representation of the vibrational
state of the upper Born-Oppenheimer 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 \cite{miller},
semiclassical approximations should work well in the deep tunneling regime.
In the classically forbidden region equations
(\ref{8}) through (\ref{12}) hold,
yet with complex momenta, and real exponents.
Here a bound wavefunction
of the Hamiltonian $H(x,p)$ with energy $E$ in the limit of
small $\hbar$ is
\begin{equation}
\psi(x)\sim \exp\left(-\frac{1}{\hbar}\sigma(x)\right),\label{gpsi}
\label{psi}
\end{equation}
where $\sigma(x)= S(x)/\i$ and $S(x)$ is the solution of the Hamilton-Jacobi equation
\begin{equation}
H\left(x,\partial S(x)/\partial x\right)= E, \label{hj}
\end{equation}
for which $S(x)/\i$ tends to infinity as $|x|\rightarrow\infty$.
As long as one considers only
the
classically forbidden region, the wavefunction (\ref{psi}) is just
one of the primitive WKB functions with a real decaying exponent, not
a linear combination with complex-valued exponents
(since we have omitted an unphysical exponentially increasing solution).
We define $w$ proportional to the logarithm of the Franck-Condon
factor squared, so that
\begin{eqnarray}
f_{A,D}=\vert \langle\psi_A\vert\psi_D \rangle \vert^2 \sim
\exp(-{2 w}/\hbar) . \label{wdef}
\end{eqnarray}
We will now proceed to find $w$, by expanding around the
complex crossing point of the initial and final potentials.
\subsection{\label{landau}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 of the initial \pes.
We perform the integration with the
quasiclassical wavefunctions; the dominant contribution comes from
the crossing point.
Let
$\init{V}$ and $\fin{V}$ be the potentials of the donor and the
acceptor \bos s respectively. In the following,
we use mass-weighted coordinates.
We make additional assumptions regarding the
functions $\init{V}(x)$ and $\fin{V}(x)$.
For the initial potential we assume that
\begin{equation}
\init{V}(x)= v(x-x_0), \label{vinit}
\end{equation} which increases monotonically away from its minimum at $\xi=0$.
We assume the final potential $\fin{V}$
has two turning points,
$\Left{x}<\Right{x}$ so that
\begin{equation}
\fin{V}(\Left{x})= \fin{V}(\Right{x})= E, \label{vfin}
\end{equation}
the function increasing monotonically beyond the turning
points. Consider first the case when $x_0<\Left{x}$. It may be
shown that bulk of the overlap integral is concentrated in the
interval $(x_0, \Left{x})$ where the wavefunctions behave for
small $\hbar$ as \cite{Landau:1977:QMN}
\begin{eqnarray}
\init{\psi}(x)&\sim& \exp\left[-\frac1\hbar\int_{x_0}^x
\init{\mu}(x')
d x' \right], \label{psilaninit}\\
\fin{\psi}(x)&\sim&
\exp\left[-\frac1\hbar\int_{\Left{x}}^x \fin{\mu}(x') d x'\right],
\label{psilanfin}
\end{eqnarray}
both functions are real,
and
\begin{eqnarray}
\init{\mu}=i\init{p}(x)&=&
\left[2\left(v(x-x_0)-v_0\right)\right]^{1/2}, \label{pinit}\\
\fin{\mu}=i\fin{p}(x)&=& -\left[2\left(\fin{V}(x)-E\right)\right]^{1/2},
\label{pfin}
\end{eqnarray}
are the classical momenta in the \textit{inverted}\cite{miller} potentials.
The overlap function
$\init{\psi}(x)\fin{\psi}(x)$ has a sharp maximum at the point
$x=x_c$, which is the point of crossing of the potentials,
\begin{equation}
\init{V}(x_c)-v_0= \fin{V}(x_c)-E, \quad
x_0\Right{x}$ is considered in a similar way, giving
\begin{equation}
\Right{w}= \int_{x_0}^{x_c} \init{\mu}(x) d x +
\int_{\Right{x}}^{x_c} \fin{\mu}(x) d x, \label{sland1}
\end{equation}
where $\init{\mu}(x)$ and $\fin{\mu}(x)$ are defined as in
\eeref{pinit}{pfin}, but with an opposite sign of the square root.
In this way, we determine the overlap in cases when $x_0<\Left{x}$,
$x_0>\Right{x}$, and also in the interval $(\Left{x},\Right{x})$
in the vicinities of the points $\Left{x}$ and $\Right{x}$ by analytic continuation
of $\Left{w}$ and $\Right{w}$ as functions of the position of
minimum of the donor potential $x_0$. These functions are zero at
points $x_0=\Left{x}$ and $x_0=\Right{x}$ when the transition
becomes classically allowed (without tunnelling), and are analytic
in the vicinity of these points. For example, $\Left{w}=
\frac{\omega}{2}(\Left{x}-x_0)^2-
\frac{\omega^3}{6V_1}(\Left{x}-x_0)^3+ O(\Left{x}-x_0)^4$ where
$\omega= [v^{\prime\prime}(0)]^{1/2}$ and $V_1=
\fin{V}(\Left{x})$.
Finally,
when $x_0\in (\Left{x},\Right{x})$, we
have found through many examples (but without deriving a general proof)
that the overlap
in this case may be determined by considering
two analytic continuations with respect to the variable $x_0$,
yielding
\begin{equation}
w=\min(\Re \Left{w}, \Re \Right{w}) . \label{winside}
\end{equation}
In section \ref{6} below we will need another form for $\Left{w}$ and
$\Right{w}$ obtained by
considering the crossing point $x_{c}$ as a function
of the position of the minimum $x_0$ of $\init{V}(x)$.
By differentiating
\eref{sland} and taking into account that $\init{p}(x_{c})+
\fin{p}(x_{c})= 0$ we find $d\Left{w}/dx_0=
-\init{p}(q_\mathrm{c})$, so $\Left{w}$ can be rewritten in a more
compact form as
\begin{equation}
\Left{w}= -\int_{\Left{x}}^{x_0} \mu_{c}(x_0^\prime) d
x_0^\prime, \label{salt}
\end{equation}
where
\begin{equation}
\mu_{c}(x_0^\prime)=
\left[2\left(v(x_\mathrm{c}^\prime-x_0^\prime)-v_0\right)\right]^{1/2},
\label{pc}
\end{equation}
and $x_{c}^\prime$ is defined by the equation
\begin{equation}
v(x_\mathrm{c}^\prime-x_0^\prime)-v_0=
\fin{V}(x_\mathrm{c}^\prime)-E, \quad
x_0^\prime \Right{w}$.
Now, let us consider this case within the \p-s\ formalism of
\eref{overlapcl} above.
First, we prove that if $\init{V} \approx v_0$, and
the action $\init{\sigma}(x)$ is small, then only a coordinate jump
is possible. For small momenta, \eref{stph} is solved by
$\eta_s=-\i p/\sigma^{\prime\prime}(x)$. Substituting it in
\eref{wigw} we find that
\begin{equation}
W(x,p)\approx
\sigma(x)+\tfrac12\frac{p^2}{\sigma^{\prime\prime}(x)}.
\label{smallp}
\end{equation}
Under the assumption regarding the shape of the function $v(x)$ in
\eref{vinit} (see the beginning of Subsection \ref{landau}) we
have ${\init{\sigma}}^{\prime\prime}(x)> 0$. Now, it follows from
\eref{smallp} that the function $W(x,p)$ is small at $p=0$ and
rapidly increases when the momentum becomes nonzero. For such a
function, it is obvious that its minimum occurs at $p_m=0$ (no
momentum jump).
We suppose here that the potential of the acceptor is steeper on
the right side, so that
$\init{W}(\Right{x},0)<\init{W}(\Left{x},0)$ and we conclude that
\begin{eqnarray}
x_m&=& \Right{x}\approx x_c , \\
p_m&=&0 =\Re p_c \\
W_m&=&w=\init{\sigma}\left(\Right{x}\right)
\end{eqnarray}
It means that the result of the \p-s\ approach is the same as the
quasiclassical result when the overlap is approximated by
an exponent of the action $\sigma(\Right{x})$.
\subsubsection*{\label{stagnant}Steep potential $\init{V}$}
Consider the opposite case of a steep donor potential $\init{V}$.
Here too we would like to compare the results of the
more traditional analysis in coordinate space with the \p-s\ analysis.
First, following section \ref{trad} we look for $x_c$,
the point of crossing
of the two \pes s, and the \fcf\ as given by \eref{wdef}.
We approximate the donor potential $\init{V}$ near its minimum as
\begin{equation}
\init{V}(x)\sim v_0+ \frac{{\init{\omega}}^2} {2} (x-x_0)^2+
\ldots, \label{steepi}
\end{equation}
where the frequency $\init{\omega}$ is supposed to be large. We
find the crossing point of the potentials by an expansion of
$\fin{V}$ in a Tailor series around the point $x_0$,
\begin{equation}
\fin{V}(x)\sim \fin{V}(x_0)+ {\fin{V}}^\prime(x_0)(x-x_0)+ \ldots,
\label{steepf}
\end{equation}
and by solving \eref{qc}. The result is
\begin{equation}
x_c\approx x_0 \pm\frac{\i}{\init{\omega}}
\left\{2\left[E-\fin{V}(x_0)\right]\right\}^{1/2}+
\frac{{\fin{V}}^\prime(x_0)}{{\init{\omega}}^2} \label{steepqc}
\end{equation}
(we consider here only ``nested" potentials when the square root in
(\ref{steepqc}) is real). The momentum at the crossing point is
\begin{equation}
p_c= i\init{\mu}_c
\approx \pm\ \left\{2\left[E-\fin{V}(x_0)\right]\right\}^{1/2}+i
\frac{{\fin{V}}^\prime(x_0)}{\init{\omega}}. \label{steeppc}
\end{equation}
Using (\ref{steeppc}) and (\ref{salt}) we estimate
\begin{eqnarray}
\Re \Left{w}&=& \int_{x_0}^{\Left{x}} \Re
\mu_c(x_0^\prime) d x_0^\prime \nonumber \\
&\approx& \frac{1}{\init{\omega}} \int_{x_0}^{\Left{x}}
{\fin{V}}^\prime(x_0^\prime) d x_0^\prime=
\frac{E-v_0}{\init{\omega}}. \label{steeps}
\end{eqnarray}
Estimation of $\Re \Right{w}$ is the same. Using (\ref{winside})
we find
\begin{equation}
w\approx \frac{E-v_0}{\init{\omega}}. \label{steepw}
\end{equation}
The \p-s\ approach reproduces (\ref{steeps}) straightforwardly. In
the harmonic approximation the \wif\ is given by
\eref{harmw}. When the frequency is large, the coordinates of the \jp\
satisfying $\fin{H}=E$ and reducing $W_0$ to a minimum
are $x_m=x_0$, $p_m=\left[2(E-v_0)\right]^{1/2}$, so
$W_m =(E-v_0)/\init{\omega}$.
Finally, notice that from \eref{steepqc} follows that
$x_m\approx \Re{x_c}$, and from \eref{steeppc} it follows
that $p_m\approx \pm \Re{p_c}$. The
plus-minus sign appears because of the existence of
two equivalent jumping points.
\section{\label{num}Numerical examples}
Let us consider an example of the harmonic potential of the donor
\begin{equation}
\init{V}(x)= \tfrac12 \omega^2 x^2, \label{vharm}
\end{equation}
and either Morse potential
\begin{equation}
\fin{V}(x)= \tfrac12 \left(J+\tfrac12\right) \left(1-\e^{-\beta
x}\right)^2, \quad \beta= (J+\tfrac12)^{-1/2}, \quad
J=300\label{vmors}
\end{equation}
or Poeschl - Teller potential
\begin{equation}
\fin{V}(x)= \tfrac12 \alpha^{-2} \left(\mathrm{cosh}^{-2}\alpha x-
1\right), \quad \alpha= \left[J(J+1)\right]^{-1/4}, \quad
J=400\label{vposc}
\end{equation}
of the acceptor. There are in total $J$ bound states in potentials
(\ref{vmors}) and (\ref{vposc}) labelled by quantum numbers $n=
0,1,2, \ldots, J-1$. In case of Morse potential we consider the
excited state with the quantum number $n=150$ and energy
$E=112.812$, and in case of Poeschl - Teller potential the quantum
number $n=200$ and energy $E=150.312$.
If the parameter $\omega$ is small, the donor potential is
relatively shallow, and when it is large, $\init{V}(x)$ is steep.
Numerical results are shown in the Table. A similar table was generated for an earlier paper, but the parameters and the comparisons being made were somewhat different\cite{beck}.
\begin{table}[tbp] \centering
\begin{tabular}{cccccccccc}
$\omega$& $\fin{V}$&$\Land{x}$ &$\Land{p}$ &$x_m$ &$p_m$ &$\Land{w}$ &$W_m$ &$w$ &Case \\
0.2& M& -10.90& $2.18\i$& -10.82& 0& 11.8& 11.7& 14.1& shallow
$\init{V}(x)$ \\
1.0& M& -13.34& $13.34\i$& -10.82& 0& 67.6& 58.5& 69.5& \\
1.3& M& -15.56& $20.22\i$& -10.82& 0& 98.6& 76.1& 100.6& \\
5.0& M& $0.01-3.06\i$& $-15.32-0.06\i$& 0& -15.02& 22.9& 22.6& 24.4& steep
$\init{V}(x)$ \\
0.2& PT& 32.52& $6.50\i$& 26.38& 0& 77.9& 69.6& 78.1& shallow
$\init{V}(x)$ \\
0.5& PT& $21.95+15.62\i$& $-7.81+10.97\i$& 22.44& -6.25& 176.2& 165.0& 177.6& \\
1.0& PT& $12.10+16.49\i$& $-16.49+12.10\i$& 0& -17.34& 179.6& 150.3& 180.2& \\
5.0& PT& $3.54\i$& -17.70& 0& -17.34& 30.5& 30.1& 31.9& steep
$\init{V}(x)$ \\
% & & & & & & & & & \\
\end{tabular}
\caption{Numerical results for various frequencies $\omega$ of the
donor potential (\ref{vharm}) and for two options for the acceptor
potential, either Morse (M) or Poeschl - Teller (PT),
\eeref{vmors}{vposc}.
%as Short explanations of $\Land{x}$, $\Land{p}$, $x_m$, $p_m$, $\Land{w}$, $W_m$, $w$
$\Land{x}$ defined by equation $\init{V}(\Land{x})-v_0=
\fin{V}(\Land{x})-E$ is the coordinate where potentials cross, and
$\Land{p}= \pm \left[2(v_0- \init{V}(\Land{x}))\right]^{1/2}= \pm
\i\omega \Land{x}$ is the momentum at the crossing point.
$(x_m,p_m)$ is the point in phase space on the surface of constant
energy $\fin{H}(x,p)=E$ where the Wigner function
$\init{\rho}(x,p)$ reaches its maximum. $w= -\frac{\hbar}{2} \ln
f_{\mathrm{D}\rightarrow\mathrm{A}}$ is proportional to logarithm
of \fcf, $\Land{w}$ and $W_m$ are respectively quasiclassical and
\ps\ approximations to $w$, see Sections \ref{landau} and \ref{6}.
In cases of several equivalent jumping points we give only one of
them.} \label{table1}
\end{table}
For cases of steep or shallow potential $\init{V}(x)$, relations
(\ref{relatq}) and (\ref{relatp}) are satisfied with good
accuracy.
\section{Summary and Conclusions}
The trajectory surface hopping methods initiated by the famous Tully and
Preston paper\cite{tp} have been widely applied in chemistry\cite{tp2}.
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 we discuss are simply instructive alternatives
to other approaches. Our intent in these cases has been to compare the
various techniques with the Wigner phase space approach, since only the
latter generalizes easily to many dimensions. We have been able to give
a rather more solid grounding to the Wigner phase space surface jumping
method\cite{beck,us1,Sergeev:2002:MPP} than had been the case previously,
especially through the semiclassical analysis with complex stationary
phase. It is our belief that for cases of slow radiationless transition,
or in the wings of absorption bands (where again FC factors are very
unfavorable and correspond to tunneling events since there is no nearby
crossing of potential energy surfaces), that the phase space approach may
be the only viable way to do calculations of relative rates and of
promoting and accepting modes.
At the same time, the phase space ``surface jumping'' approach is a very
intuitive one, providing clear pictures of the mechanism for radiationless
or nonvertical radiative transitions, in terms of bond length changes
and/or bond momentum jumps in one or more coordinates.
It may be interesting to combine the generalization of hops into jumps
with some new developments in mixed quantum-classical Lioville
propagation recently introduced by Ciccotti et al. and Martens et al.
\cite{n1}-\cite{n5}. These two approaches and our phase-space surface
jumping approach seem to complement each other in several ways.
In the formalism of Refs.~\cite{n2}-\cite{n4}, for example, for the
specific case of nonradiative nonadiabatic processes, the excess energy
flows in the direction of the nonadiabatic coupling vector and is dumped
into the velocities.
The phase space approach provides a systematic way for predicting the
energy flow during a transition between non-crossing surfaces.
We have recently found, within the phase-space analysis, that for
small energy gaps between the surfaces (as in hopping) the excess
energy flows in the direction of the nonadiabatic coupling vector,
but we have also found that this rule does not apply in general to
large energy gaps \cite{kallush}. In the general case the jump
is sensitive to the shape of the accepting potential and to the initial
distribution on the donor surface. This sensitivity can sometimes reduce
in principle the number of trajectories one needs to consider,
yet it requires some sampling of phase space in order to determine the
relevant jumps. So far, the phase space approach does not take into
account interference from different jumps, although it is clear that
such interferences could be important, in particular between two momentum
jumps of opposite signs. Future work could consider combining the
treatment of coherences as in Ref.~\cite{n1} with the phase-space
jumping mechanism, to properly account for interference.
\vfill
\eject
\begin{thebibliography}{99}
\bibitem{tp}
Tully, J. C.; Preston, R. K. {\em J. Chem. Phys.} 1971, {\bf 55}, 562
\bibitem{lz}
Nikitin, E. F. in {\em Theory Elementary Atomic and Molecular
Processes in Gases}; Clarendon: Oxford, 1974.
\bibitem{gelbart}
Avouris, P.; Gelbart, W. M.; El-Sayed, M. A.
{\it Chemical Reviews} 1977, {\bf 77}, 793
\bibitem{ewing}
Ewing, G. \jpc\ 1987, {\bf 91}, 4662
\bibitem{berg}
Bergsma, J. P.; Behrens, P. H.; Wilson, K. R.; Fredkin, D. R.;
Heller, E. J. {\em J. Phys. Chem.} 1984, {\bf 88}, 612
\bibitem{M82}
Medvedev E. S. {\em Chemical Physics} 1982, {\bf 73}, 243-251
\bibitem{child}
Hunt, P. M.; Child, M. S. \cpl\ 1978, {\bf 58} 202
\bibitem{miller}
Miller, W. H. {\em Adv. Chem. Phys.} 1974, {\bf XXV}, 69
\bibitem{Berry:1977:RTP}
Berry, M. V. {\em Proceedings of the Royal Society of London Series A -
Mathematical Physical and Engineering Sciences} 1977, {\bf 287}, 237
\bibitem{Landau:1977:QMN}
Landau, L. D.; Lifshitz, E. M.
{\em Quantum Mechanics: Non-Relativistic Theory};
{Pergamon}: {Oxford}, {1977}.
\bibitem{beck}
Heller, E. J.; Beck; D. {\em Chem. Phys. Lett.} 1993, {\bf 202} 350
\bibitem{us1}
Segev, B.; Heller, E. J. {\em J. Chem. Phys.} 2000, {\bf 112}, 4004
\bibitem{Sergeev:2002:MPP}
{Sergeev}, {A. V.}; {Segev}, {B.} {\em J. Phys. A: Math. Gen.} 2002,
{\bf 35}, 1769
\bibitem{Heller:1978:QCC}
{Heller}, {E. J.}; {\em J. Chem. Phys.} 1978, {\bf 68}, {2066}
\bibitem{Hupper:1998:USE}
{H{\"u}pper}, B.; {Eckardt}, B. {\em Phys. Rev. A} 1998, {\bf 57},
{1536}
\bibitem{Berry:1970:DCP}
{Berry}, R. S.; {Nielsen}, S. E.
{\em Phys. Rev. A} 1970, {\bf 1},
{383}
\bibitem{tp2}
Tully, J. C. in {\em Modern Methods for Multidimensional Dynamics
Computations in Chemistry}, edited by Thompson, D. L.;
World Scientific: Singapore 1998; p. 34.
%bs[
\bibitem{n1}
Donoso, A.; Martens, C. C. {\em J. Phys. Chem. A} 1998, {\bf 102}, 4291
\bibitem{n2}
Kapral, R.; Ciccotti, G. {\em J. Chem. Phys.} 1999, {\bf 110}, 8919
\bibitem{n3}
Neilsen, S.; Kapral, R.; Ciccotti, G. {\em J. Chem. Phys.} 2000, {\bf 112}, 6543
\bibitem{n4}
Neilsen, S.; Kapral, R.; Ciccotti, G. {\em J. Chem. Phys.} 2001, {\bf 115} 5805
\bibitem{n5}
Donoso, A.; Martens, C. C. {\em Phys. Rev. Lett} 2001, {\bf 87} 223202-1
\bibitem{kallush}
Kallush, S.; Segev, B.; Sergeev, A. V.; Heller E. J.;
{\em J. Phys. Chem. A} 2002, (in press)
%bs]
\end{thebibliography}
\begin{figure}
\centerline{\includegraphics*[width=5.2in]{binningc.pdf}}
\caption{Contours of the difference potential and a trajectory at
equal time steps} \label{binproof}
\end{figure}
\begin{figure}
\centerline{\includegraphics*[width=5.2in]{curves.pdf}}
\caption{Three cases showing relative positions of minima and relative shapes of potential
energy surfaces resulting in very different regimes of FC overlap. In (a), we show the overlap
of two ground states leading to a position jump,; in (b), a case of a shallow donor potential leading to a position jump, and in (c), nested potentials with a steep donor potential leading to a momentum jump.}
\label{cases}
\end{figure}
\end{document}