Centre of mass coordinate systems

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.

More immediate kinematic evidence for the formation of a complex which shows an isotropic decay in the centre-of-mass coordinate system was obtained by Gislason et al. [138] in the endothermic reaction... 

As discussed in the previous section, the problem is solving a differential equation with respect to either the position (classical) or wave function (quantum) for the particles in the system. The standard method of solving differential equations is to find a set of coordinates where the differential equation can be separated into less complicated equations. The first step is to introduce a centre of mass coordinate system, defined as the mass-weighted sum of the coordinates of all particles, which allows the translation... 

The use of the spectral resolution of the identity in this form is not fully justified. A sudden cut in the electrie field may leave the molecule with a non-zero translational energy. However, in the above Spectral resolution one has the stationary states computed in the centre-of-mass coordinate system, and therefore translation is not taken into account. 

The function koAf is the total wave function written in the centre-of-mass coordinate system (a special body-fixed coordinate system, see Appendix I), in which the total angular momentum operators and Jz are now defined. The three operators H,J and Jz commute in any space-fixed or body-fixed coordinate system (including the centre-of-mass coordinate system), and therefore the corresponding physical quantities (energy and angular momentum) have exact values. In this particular coordinate system p = pcM = 0. We may say, therefore, that... 

The kinetic energy T transferred to the lattice atom during an elastic collision can be expressed by the following equation in the classical hard sphere approximation, which can be applied to metals in the centre-of-mass coordinate system [43] ... 

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... 

For a three-body system such as A + BC AB + C, there are 3 x 3 = 9 degrees of freedom in the problem. However, three degrees of freedom can be eliminated by considering the centre of mass coordinates. If we consider... 

The state of the system consisting of n electrons and two nuclei is now specified in terms of 3n + 6 spatial coordinates referred to the centre of mass plus three centre-of-mass coordinates. Three of the spatial coordinates are redundant and can therefore be eliminated. Various choices of redundant coordinates can be made we choose to... 

The coordinates (XpTpZj) are redundant since they ean be determined from the eondition that the X, Y, Z) axis system has origin at the molecular centre of mass. Obviously, the translational symmetry operation diseussed above has the effect of changing the centre of mass coordinates... 

This describes the interparticle correlations and gives access to the interaction between the entities. Ri is the vector to the centre of mass coordinate of particle i. The structure factor is close to unity at all Q values for dilute systems and, hence, Eq. 63 can be written as ... 

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. 

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... 

At this point, it is wise to see an example, and I will take H2O with a bond length of 95.6 pm and bond angle of 104.5°. If I take the coordinate system such that the molecule lies in the yz-plane with coordinate origin given by the centre of mass, and the z-axis as the symmetry axis, then the Cartesian coordinates are given in Table 8.4. 

The coordinate system may be oriented so that the atom A and centre of mass BC lie in the Y-Z plane. This gives the simple forms... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

Each selected configuration is translated and rotated in such a way that all of the solvent coordinates can be referred to a reference system centred on the centre of mass of the solute with the coordinate axes parallel to the principal axes of inertia of the solute. 

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