(p.329) D Classical mechanics, coordinate transformations
(p.329) D Classical mechanics, coordinate transformations
When the dynamics of a molecular system is studied, we have to choose a set of coordinates to describe the system. There are many possibilities and in the following we will focus on coordinates that are convenient in the analysis of molecular collisions and chemical reactions.
D.1 Diagonalization of the internal kinetic energy
It is often useful to transform from simple Cartesian coordinates to other sets of coordinates when we study collision processes including chemical reactions. n a collision process, it is obvious that the relative positions of the reactants are relevant and not the absolute positions as given by the simple Cartesian coordinates. It is therefore customary to change from simple Cartesian coordinates to a set describing the relative motions of the atoms and the overall motion of the atoms. For the latter motion the center-of-mass motion is usually chosen. In the following we will describe a general method of transformation from Cartesian coordinates to internal coordinates and determine its effect on the expression for the kinetic energy.
A system of N particles is described by the N position coordinates r 1,…,r N and N momenta p 1,…, p N . Here p i = m i r i, since we use Cartesian coordinates.
By a linear transformation, let us introduce a new set of coordinates R i,… RN :
In matrix form this equation may be written
The N × N matrix Am –1 A T will in general not be diagonal, so there will be cross terms of the kind P i P j in the expression for the kinetic energy. This may sometimes be inconvenient, and we shall see in the following how one may choose the matrix A in such a way that the kinetic energy is still diagonal in the new momenta. This leads to the so-called Jacobi coordinates that are often used in reaction dynamics calculations.
First we want to single out the overall motion of the system, where all atoms move by the same amount, so all distances will be preserved. This is done by introducing the following condition on the matrix elements in A:
If all particles are displaced by the amount u to the position ri + u, then coordinates R 1,…, r n–1 are seen from Eq. (D.1) not to change because of the condition in Eq. (D.7), while RN is displaced by u. R 1,…, r n–1 are thus internal coordinates, whereas RN describes the overall position of the system. For the momenta, we get
The moment P N, conjugate to coordinate r n, is therefore the total momentum of the system.
Usually, RN is chosen as the center-of-mass coordinate:
If we develop the matrix product in Eq. (D.6) then we get
The kinetic energy has been divided into two contributions: an internal kinetic energy (the first term) and an external center-of-mass kinetic energy T c.m., where (p.331)
The condition for the internal kinetic energy to be ‘diagonal’, so that there will be no cross term, is seen from Eq. (D.11) to be
If we introduce the notation
Such internal coordinates for which the internal kinetic energy is diagonal are called generalized Jacobi coordinates when more than two particles are considered.
D.1.1 An example
In this example we shall describe a systematic way to choose the matrix elements such that the kinetic energy will be ‘diagonal’, that is, with no cross terms. The procedure is to couple the particles with the aid of the center-of-mass coordinates for larger and larger clusters of particles. Let us illustrate the method for a five-particle system. We start with particles one and two. For the first row in A we find, using Eq. (D.7),
Particles one and two are now coupled in the center-of-mass coordinates for those two particles. This center is now coupled to particle three. Again we use Eq. (D.13) for rows two and three, and Eq. (D.7) for row three to get
The particles one, two, and three are now coupled and we continue with particle four in the same way:
The fifth row is finally given by (p.332)
Here we recognize the mass associated with coordinate R 1 as the reduced mass M 1 for a two-particle system (as in Eq. (2.25)). With these generalized Jacobi coordinates, the internal kinetic energy has the simple form without cross terms according to Eq. (D.15):
Often one chooses to set α1 = α2 = α3 = α4 = 1 in the expressions for the effective masses. Another possibility would be to choose values of αi so that all the effective masses will be equal to one, that is,
When the potential energy, V = V (R 1, R 2, R 3, R 4), only depends on the internal coordinates, Hamilton’s equations of motion are (use Eq. (4.63))
D.1.2 Mass-weighted skewed angle coordinate systems
Sometimes a mass-weighted skewed angle coordinate system rather than a rectangular system is used to plot the potential energy surface and the trajectories for a simple triatomic reaction like
For simplicity, we assume that the collision is collinear, say along the x-axis. The atoms A,B, and C are numbered 1, 2, and 3, respectively, to harmonize the notation with the previous section. Then, according to the analysis above, the following internal Jacobi coordinates are consistent with a ‘diagonal’ kinetic energy:
We note that the coordinate X 1 directly reflects the distance between atoms one and two, whereas the coordinate X 2 reflects a combination of both distances. Therefore, a knowledge of the two coordinates does not directly tell us what the distances are between the involved atoms. Also, for the potential energy function in a collinear collision, the natural variables will be the distances between atoms A and B and atoms B and C. These variables appear as the components along a new set of coordinate axes, if instead of a rectangular coordinate system we use a mass-weighted skewed angle coordinate system.
Let X 1 and X 2 be the coordinates in a rectangular Cartesian coordinate system with X 1 along the ordinate axis and X 2 along the abscissa axis. One of the axes in the new coordinate system is now chosen to be collinear with the abscissa axis in the rectangular coordinate system, and we let the coordinate along this axis be the first term in the expression for X 2 in Eq. (D.22), namely α2 (x 3 – x 2). The situation is sketched in Fig. D.1.1. The other axis with a coordinate proportional to the other distance x 2 – x 1forms an angle φ with the first. This angle is determined from the requirements that the projections of this coordinate on the X 1 coordinate axis is α1(x 2 – x 1) and on the X 2 coordinate axis is α2 m 1(x 2 – x 1)/s2. If we let the proportionality constant of x 2 – x 1 be β, then we have (p.334)
For the case where α1 = α2 = 1, we find
If we consider the potential energy as a function of the Jacobi coordinates X 1 and X 2 and draw the energy contours in the X 1–X 2 plane, then the entrance and exit valleys will asymptotically be at an angle φ to one another and in the mass-weighted skewed angle coordinate system parallel to its axes. So the idea with this coordinate system is that it allows us to directly determine the atomic distances as they develop in time and that it shows us the asymptotic directions of the entrance and exit channels.
It is important to note that the dynamics is done in the Jacobi coordinates, because they make the equations of motion (see Eqs (D.19) and (D.20)) particularly simple. The idea of choosing the αs as in Eq. (D.17) is that all masses M k equal one, so the equations of motion are analogous to those for a particle of mass one with coordinates X 1 and X 2.
In fact, we can construct a mechanical analogue to the trajectories generated by the equations of motion by rolling a particle of mass one on a hard surface under the influence of gravity. The topography of this analogue surface is closely related to that of the potential energy surface. It is determined by realizing that we have to transform from a potential energy surface to a position surface or ‘mountain landscape’, so the energy axis is converted into a position axis with a coordinate giving the ‘height’ of the particle above the X 1–X 2 plane. In Fig. D.1.2(a) a cut through the potential energy surface and the force at some point X 0 are shown. In Fig. D.1.2(b) the same cut through the ‘mountain landscape’ of the analogue surface is shown. The tangential component F g of the gravity force mg (here m = 1) causes the particle to move on the (p.336) analogue surface. We can construct the analogue surface, h(X), by requiring that this force shall equal the force derived from the potential energy function at any point; that is,
Thus, the analogue surface should be constructed in such a way that θ at any point X 1,X 2 satisfies Eq. (D.29).