Centre-of-mass

We collect syimnetry operations into various syimnetry groups , and this chapter is about the definition and use of such syimnetry operations and symmetry groups. Symmetry groups are used to label molecular states and this labelling makes the states, and their possible interactions, much easier to understand. One important syimnetry group that we describe is called the molecular symmetry group and the syimnetry operations it contains are pemuitations of identical nuclei with and without the inversion of the molecule at its centre of mass. One fascinating outcome is that indeed for... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

We obtain a coordinate set more suitable for describing translational synnnetry by mtroducing the centre of mass coordinates... 

Figure Al.4.4. The definition of the Euler angles (0, ( ), x) that relate the orientation of the molecule fixed (x, y, z) axes to the (X, Y, Z) axes. The origin of both axis systems is at the nuclear centre of mass O, and the node line ON is directed so that a right handed screw is driven along ON in its positive direction by twisting it from Z to z through 9 where 0 < 9 < n. ( ) and x have the ranges 0 to In. x is measured from the node line.

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

Consider now the aquo-complexes above, and let v be the distance of the centre of mass of the water molecules constituting the iimer solvation shell from the central ion. The binding mteraction of these molecules leads to vibrations... 

Flere 5. = 1 (0) is the Kronecker delta for/= (f ) and /r. is the wavenumber for the collision, related to the initial relative centre of mass translational energy. before tlie collision... 

A bimoleciilar reaction can be regarded as a reactive collision with a reaction cross section a that depends on the relative translational energy of the reactant molecules A and B (masses and m ). The specific rate constant k(E ) can thus fonnally be written in tenns of an effective reaction cross section o, multiplied by the relative centre of mass velocity... 

Here one has the thennal average centre of mass velocity... 

Figure A3.5.2. The Ar photofragment energy spectmm for the dissociation of fiions at 752.5 mn. The upper scale gives the kinetic energy release in the centre-of-mass reference frame, both parallel and antiparallel to the ion beam velocity vector in the laboratory.

Several reactivity trends are worth noting. Reactions that are rapid frequently stay rapid as the temperature or centre-of-mass kinetic energy of the reactants is varied. Slow exothenuic reactions almost always show behaviour such tliat... 

Consider the collision of an atom (denoted A) with a diatomic molecule (denoted BC), with motion of the atoms constrained to occur along a line. In this case there are two important degrees of freedom, the distance R between the atom and the centre of mass of the diatomic, and the diatomic intemuclear distance r. The Flamiltonian in tenns of these coordinates is given by ... 

Another reason why mass-scaled coordinates are useful is that they simplify the transfomiation to the Jacobi coordinates that are associated with the products AB + C. If we define. S as the distance from C to the centre of mass of AB, and s as the AB distance, mass scaling is accomplished via... 

Consider collisions between two molecules A and B. For the moment, ignore the structure of the molecules, so that each is represented as a particle. After separating out the centre of mass motion, the classical Hamiltonian that describes tliis problem is... 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

B2.2.2.3 ENERGY AND ANGULAR MOMENTUM CENTRE OF MASS AND RELATIVE VELOCITY... 

The velocity of the centre of mass (CM) of the projectile and target particles of respective masses M. and Mg... 

B2.2.2.7 CENTRE-OF-MASS TO LABORATORY CROSS SECTION CONVERSION... 

Theorists calculate cross sections in the CM frame while experimentalists usually measure cross sections in the laboratory frame of reference. The laboratory (Lab) system is the coordinate frame in which the target particle B is at rest before the collision i.e. Vg = 0. The centre of mass (CM) system (or barycentric system) is the coordinate frame in which the CM is at rest, i.e. v = 0. Since each scattering of projectile A into (v[i, (ji) is accompanied by a recoil of target B into (it - i[/, ([) + n) in the CM frame, the cross sections for scattering of A and B are related by... 

In a crossed-beam experiment the angular and velocity distributions are measured in the laboratory coordinate system, while scattering events are most conveniently described in a reference frame moving with the velocity of the centre-of-mass of the system. It is thus necessary to transfonn the measured velocity flux contour maps into the center-of-mass coordmate (CM) system [13]. Figure B2.3.2 illustrates the reagent and product velocities in the laboratory and CM coordinate systems. The CM coordinate system is travelling at the velocity c of the centre of mass... 

Figure B2.3.2. Velocity vector diagram for a crossed-beam experiment, with a beam intersection angle of 90°. The laboratory velocities of the two reagent beams are and while the corresponding velocities in the centre-of-mass coordinate system are and U2, respectively. The laboratory and CM velocities for one of the products (assumed here to be in the plane of the reagent velocities) are denoted if and u, respectively.

In the bimolecular collision of the photolytically generated reagent, assumed to have a mass m. and laboratory speed Vp the centre-of-mass speed will be... 

Figure B2.3.16. Velocity diagram for die reaction of a photolytically generated reagent with an assumed stationary co-reagent. In this case, the relative velocity of the reagents is parallel to the velocity c of the centre of mass.

There are several practical limitations to the use of equation (B2.3.16) for the detennination of CM angidar distributions. The optimum kinematics for the use of this equation is the case where the speed c of the centre of mass is approximately equal to the product CM speed u hi the limiting case where the latter is small, the... 

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