December 16, 2007

When chemical reaction in a linear collision of three molecules A+BC→AB+C is considered, two degrees of freedom, or two coordinates, are needed to describe the internal mechanics of the reaction, apart from one degree of freedom for the center-of-mass motion. If we imagine a 2-dimensional space spanned by the two coordinates, then the chemical reaction is understood by an imaginary particle motion in the space under some potential due to the interaction between the molecules. In this report, we introduce an attempt to analyze such motion with vibrationally adiabatic approximation by classical mechanics [1].

The outline of this report is as follows: we first describe the qualitative feature of a particular type of potential energy we want to discuss; next, the Hamilton-Jacobi equation is constructed after the coordinate system suitable for the potential energy surface of the problem is briefly explained; we then solve the equation by separating variables and using adiabatic approximation for vibrational motion. Even though the adiabatic approximation is not valid for many actual cases, we focus on the adiabatic limit for simplicity.

The potential energy surface in the 2-dimensional space obtained in the procedure described above typically has following features. First of all, there should be two local minima, each corresponding to reactants (A+BC) and products (AB+C). Secondly, there is a relatively low energy smooth path (we call it C) which connects the reactants state with the products state. The path C is usually called "reaction path" in the literature, and the maximum energy state in C is identified as the transition state of the chemical reaction. Finally, states along C are stable against perturbation in the direction perpendicular to C and a particle's motion in this direction is thus oscillatory.

Figure 1 is an example of a contour plot of potential surface geometry. Note that the Fig. 1 merely serves to explain the features listed above and does not represent potential energy surface of any real chemical reaction [2].

In a realistic calculation, the potential energy surface is often plotted by using skewed coordinate axes. This is in order to simplify the kinetic energy by removing a cross term which exists in center-of-mass Cartesian coordinates in addition to the normal terms proportional to the square of each momentum. Moreover, the mass along each axis can become the same by the choice of the skewed axis [1]. We therefore should emphasize that the calculation in this report does not assume Cartesian coordinate system in the potential energy surface plot. Note that the Fig. 1 was drawn with Cartesian coordinates just for ease.

Since we expect that the motion of the imaginary
particle in the 2-D space is confined only to the vicinity of Curve C,
the coordinate system along Curve C seems a natural choice.
Specifically, the new coordinate system is constructed by the following
procedure: we first find the point P_{0} which is on C and the
closest to the point P of the particle; the distance x between P and
P_{0} is the one of the new coordinates; the length s along C
from a fixed point O_{C} to P_{0} is the other one. This
coordinate system (x, s) is in fact very convenient in order to separate
variables and apply adiabatic approximations, which will be explained
later. In addition, by transforming the initail coordinate system into
the coordinate system along C, we no longer need to worry about the
initial (possibly skewed) coordinate system.

Before proceeding further, let us explain the
notations used in the following discussion. The vector from the point of
origin O to P is **r**(s), and the vector from O to P_{0}is
**r**_{0}(s). The unit vector pointing from P_{0}
toward P is **e**_{x}(s) and the unit vector from
P_{0} along C is **e**_{s}(s) (see also Fig. 1). The
relations between these vectors are listed below:

(1) | ||

(2) | ||

(3) |

where κ is the curvature of C at P_{0}.
The equation (2) follows directly from the definition of curvature or
from the Serret-Frenet formula in elementary differential geometry
[3].

What we need next in order to write down Hamiltonian
is the canonical momentum of the new coordinate system. We use the
contact transformation method by the generating function ψ′ =
**p**•**r** to obtain the momentum [4]. By taking the
derivative of ψ′ with respect to coordinates, it is shown
that

(4) |

and

(5) |

On the other hand, it is convenient to separate the
potential energy V into two parts V_{1}(s) and V_{2}(x,
s). V_{1}(s) is defined as the potential along C, while
V_{2}(x, s) ≡ V - V_{1}. V_{2}(0, s) =0
due to the definition of V_{1}(s).

Now we are ready to write down the Hamiltonian as

(6) |

This Hamiltonian immediately leads to the Hamilton-Jacobi equation [5];

(7) |

where W(x, s, α) is the Hamilton's
characteristic function and α_{1} is a constant of motion
representing the total energy. The momenta are related to W by the
following equations:

(8) |

and

(9) |

The equation (7) is of course not easy to solve because it involves two coordinates (x and s). We therefore try to separate the variables in this section. Specifically we seek for a solution in the following form:

(10) |

The reason why we used "≅" instead of "=" is simply because, as we will see later, without adiabatic approximation we cannot obtain the solution in the form (10).

After substitution of (10) into (7) and some algebra, the equation (7) becomes

(11) |

In the equation (11), we immediately see that the
left-hand side does not contain x, which allows us to set the both sides
equal to a function α_{2}(s) of only s. Thus (11) becomes
the following two equations:

(12) |

and

(13) |

where

(14) |

The two equations (12) and (13) appear to be two separate Hamilton-Jacobi equations, one (12) for s and the other (13) for x, except for the fact that (13) includes s. (12) corresponds to the motion of s along C, while (13) represents the vibrational motion of x in a direction perpendicular to C. From this point of view, U(x, s) serves as an effective potential for the vibrational motion.

The physical meaning of the adiabatic approximation is to assume that the time scale of x, that is, the period of the oscillatory motion is much faster than the time scale of s, that is, the motion along C. With this approximation, the slow motion of s is influenced not directly by the fast oscillatory motion of x but only by the time average of x. This is the reason why (12) does not explicitly include x.

The consideration in the previous paragraph implies
what we should do next. We need to calculate the x_{0} which
minimizes U(x, s). Since x_{0} can be looked on as the average,
or the center, of x through vibrational motion, we can expand functions
of x including U(x, s) around x = x_{0}. After the expansion, we
can effectively average out the vibrational motion and will obtain an
essential equation to determine α(s).

Let us follow the procedure above. The derivative of
(14) with respect to x should be zero at x = x_{0}, so

(15) |

By solving (15) with respect to x_{0}, we
obtain x_{0} using α_{2}. We next expand functions
of x in (11) around x_{0} and, by using (8) and (14), we
have

(16) |

The equation (16) is very convenient because x is not contained except in the first bracket, which represents vibrational energy. We can then solve the equation

(17) |

and take the average of E over the time scale longer
than the period of vibration in order to obtain E_{0}. This
E_{0} is substituted into (16), giving

(18) |

Note that introducing an action variable for the
vibrational motion makes it easier to obtain E_{0}, though it is
not essential. This procedure is explained in [1].

Since we have all the essential equations, we here
summarize the solution procedure of the standard problem in which s is
to be determined as a function of t given V_{1}, V_{2}
and initial conditions. We have already noted that (15) gives the
functional form of x_{0}(s) by using α_{2}. This
x_{0} can be substituted into (18) and we can determine
α_{2}(s). (12) then becomes very easy integration problem
and W_{1}(s, α) is thus obtained. Finally, W_{1}
is substituted into (8) to give the relation between momentum
p_{s} and coordinate s. Since Hamilton's equation suggests that
p_{s} = μ(1 + κx_{0})ds/dt, s(t) is thus
determined.

© 2007 Ko Munakata. The author grants permission to copy, distribute and display this work in unaltered form, with attributation to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] R. A. Marcus, J. Chem. Phys. **45**, 4500
(1966).

[2] Potential energy surface plots obtained by
simulating realistic systems are found in many articles about kinetic
theory of chemical reaction. See, for example, Fig. 3 in R. E. Weston,
J. Chem. Phys. **31**, 892 (1959).

[3] C. G. Gibson, *Elementary Geometry of
Differentiable Curves: An Undergraduate Introduction* (Cambridge,
2001).

[4] H. C. Corben and P. Stehle, *Classical
Mechanics* (Wiley, 1960). Section 58 and 59 of this book treat
various contact transformation methods by generating function, while
sec. 95 explains a topic similar to the discussion leading to the
Hamiltonian (6).

[5] H. Goldstein, C. Pole and J. Safko, *Classical
Mechanics* (Addison-Wesley, 2002).