Coordinate system laboratory fixed

Fig. 2.1.1 Idealized molecular-beam experiment for the reaction A(i, va) + B(y, vb) ( (l, ( ) + D(m. I m, ). The coordinate system is fixed in the laboratory. The reactants move with the relative speed v = a — wb. ...

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by

Where XYZ stands for a specified Cartesian coordinate frame. Thus once a Cartesian coordinate frame is chosen, the nine spherical components can be determined using Eq. (7.4.1) from the nine Cartesian components. Clearly the spherical components and the Cartesian components change if the coordinate axes are rotated. Suppose we know the values of the Cartesian components of the polarizability tensor in a coordinate frame rigidly fixed within the molecule6 (the body-fixed frame OXY Z). Then the problem confronting us is to determine the Cartesian components of the polarizability tensor in a coordinate system rigidly fixed in the laboratory (the laboratory frame OX Y Z ). The relative orientations of the molecular and laboratory-fixed... 

Kinematics in the center-of-mass system Kinematics means the description of the motion. Here we consider the motion of two point particles, 1 and 2, in the laboratory and in the c.m. system. In the laboratory the origin of the coordinate system is fixed and the location of the two particles is described by two vectors Ri and R2. The (vector) velocity is, as usual, the rate of change of the position vector, for example, vi = dRi /dt and similarly for V2. The relative position vector of the two molecules is R = Ri -R2. Therefore, the relative velocity v = dR/df is the difference... 

There are 3M-6 vibrations of a non-linear molecule containing M atoms a linear molecule has 3M-5 vibrations. The linear molecule requires two angular coordinates to describe its orientation with respect to a laboratory-fixed axis system a non-linear molecule requires three angles. 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

These simple relations motivate a more formal approximation in which we first re-expand the interaction potential in a space-fixed ("laboratory-frame") coordinate system as... 

The equations of internal motion defined within the local (molecular) fixed coordinate system have to be transformed to the laboratory fixed coordinate system in which all experiments are performed. Thus, introducing Euler s angles, (b, 0, and X, the coordinates of the electrons (e) and nuclei (n) are transformed in the following... 

For a diatomic species, the vibration-rotation (V/R) kinetic energy operator can be expressed as follows in terms of the bond length R and the angles 0 and < ) that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system ... 

We have considered above the deflection of the trajectory in the center-of-mass coordinate system. The deflection angle x is the angle between the final and initial directions of the relative (velocity) vector between the two particles. Experimental observations normally take place in a coordinate system that is fixed in the laboratory, and the scattering angle 0 measured here is the angle between the final and the incident directions of the scattered particle. These two angles would be the same only if the second particle had an infinite mass. Thus, we need a relation between the angles in the two coordinate systems in order to be able to compare calculations with experiments. 

Fig. 4.1.11 The deflection angle (in radians) of the scattered particle in a laboratory fixed coordinate system. The angle is given as a function of the impact parameter b, for the collision between two hard spheres A and B with the average diameter d. Particle B is initially at rest, and the five curves correspond to the mass ratios (mA/mn) 0, 0.5, 1, 1.5, and 2. The o at b = 0 and toa/tob = 1 indicates that the deflection angle is undefined in this case, since va = 0, that is, A is at rest after the collision.

In a collision process, it is the relative position of the atoms that matters, not the absolute positions, when external fields are excluded, and the potential energy E will depend on the distances between atoms rather than on the absolute positions. It will therefore be natural to change from absolute Cartesian position coordinates to a set that describes the overall motion of the system (e.g., the center-of-mass motion for the entire system) and the relative motions of the atoms in a laboratory fixed coordinate system. This can be done in many ways as described in Appendix D, but often the so-called Jacobi coordinates are chosen in reactive scattering calculations because they are convenient to use. The details about their definition are described in Appendix D. The salient feature of these coordinates is that the kinetic energy remains diagonal in the momenta conjugated to the Jacobi coordinates, as it is when absolute position coordinates are used. 

We consider here the relation between volume elements in phase space in particular, the relation between dqdp and dQdP, where dq = dq dqn refers to Cartesian coordinates in a laboratory fixed coordinate system, dQ = dQ dQn refers to normal-mode coordinates, and p and P are the associated generalized conjugate momenta. 

To describe the motion or the position of the two spins (such as 13C and H in Fig. 1) with respect to the external field (or the laboratory-fixed coordinate system), it is customary and useful in treating DD relaxation to introduce relevant space functions. These functions, F(q)(t), appear in the time-dependent Hamiltonian describing DD interactions, which can be written as a product of two second-rank tensors25 A and F ... 

The alignment effect is seen to emanate from Eq. (12.57) by noting that, in molecules, the induced dipole is not necessarily parallel to the field that induced it. In fact, the induced dipole can have three perpendicular components, d, dY, andi J dz, in the X, Y, Z molecular-fixed coordinate system. Given these components, h 3j can express % de/e, the projection of the transition dipole onto the laboratory-fixed axis, appearing in Eq. (12.55) as [450] Jg... 

These rotations are performed sequentially and a rotation which takes one along an axis in the sense of a right-handed screw is defined as being positive. The nuclei are labelled so that the molecule-fixed z axis points from nucleus 1 to nucleus 2. It must be appreciated that this rotating coordinate system is a completely new one it was not mentioned in section 2.3 where all the various coordinate systems have a fixed orientation in laboratory space. 

In molecular quantum mechanics, we often find ourselves manipulating expressions so that one of a pair of interacting operators is expressed in laboratory-fixed coordinates while the other is expressed in molecule-fixed. A typical example is the Stark effect, where the molecular electric dipole moment is naturally described in the molecular framework, but the direction of an applied electric field is specified in space-fixed coordinates. We have seen already that if the molecule-fixed axes are obtained by rotation of the space-fixed axes through the Euler angles (, 6, /) = >, the spherical tensor operator in the laboratory-fixed system Tkp(A) can be expressed in terms of the molecule-fixed components by the standard transformation... 

Much of the following exposition was already presented in chapter 2, but it is fundamental and can bear repetition. The coordinate system employed to describe the motion of the particles in a molecule, both electrons and nuclei, is illustrated in figure 8.11.0 is an arbitrary laboratory-fixed origin and c.m. is the centre-of-mass of the many-particle system R0 is the vector from O to the centre-of-mass. The position of each particle i (electron or nucleus) is defined by the vectors R, and rt from the origin O and the centre-of-mass respectively. 

Figure 8.11. Cartesian coordinate system for describing the position vectors of the particles (electrons and nuclei) in a molecule. 0(X, Y, Z) is the laboratory-fixed frame of arbitrary origin, and c.m. is the centre-of-mass in the molecule-fixed frame. For the purposes of illustration four particles are indicated, but for most molecular systems there will be many more than four.

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