Space-fixed coordinate system SFCS

Lei us consider first a diatomic molecule with the nuclei labeUed by a, fc, and n electrons. Let us choose a Cartesian coordinate system in our laboratory (called the space-fixed coordinate system, SFCS) with the origin located at an arbitrarily chosen point and with arbitrary orientation of the axes. The nuclei have the following positions Ra = (Xa, Ya,Za) and Rt = (Xi, Yb,Zi,), while electron i has the coordinates x -, y -, z[. 

When we are not interested in collisions of our molecule with a wall or similar obstruction, we may consider a separation of the motion of the centre-of-mass, then forget about the motion and focus on the rest, i.e. on the relative motion of the particles. 

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Xi represents the veetor identifying nucleus / in a space-fixed coordinate system, SFCS, cf.. Appendix I available at booksite.elsevier.com/978-0-444-59436-5). First, let us define Tk. ... 

The exact kinetic energy expression for a polyatomic molecule in a space-fixed coordinate system (SFCS cf. Appendix I available at booksite.elsevier.com/978-0-444-59436-5) bas been given in Chapter 6 (Eq. 6.39). After separation of the center-of-mass motion, the Hamiltonian is equal to H = T + V. where V represents the electronic energy playing the role of the potential energy for the motion of the nuclei (an analog of (/ ) from Eq. (6.8), and we assume the Bom-Oppenheimer approximation). In the hyperspherical democratic coordinates, we obtain ... 

A planetoid (or molecule) moves through empty space, and we are observing it from our (inertial ) spaceship. In order to carry out observations of the planetoid (molecule), we have to install some gear on our spaceship and install a Cartesian coordinate system that will enable us to describe the planetoid whatever happens to it. This is the space-fixed coordinate system (SFCS), whose orientation with respect to distant stars does not change with time. 

The kinetic energy in a space-fixed coordinate system (SFCS) is composed of ... 

Each of the coordinate systems (let us label them k = 1,2,3) highlights two atoms close to each other (i, j) and a third distant (k). Now, let us choose a pair of vectors r, for each of the choices of the Jacobi coordinates by the following procedure (X/ represents the vector identifying nucleus i in a space-fixed coordinate stem, SFCS, cf. Appendix I). First, let us define... 

Let us consider a space fixed Cartesian coordinate i stem (SFCS, see Appendix 1 on p. 971), and vector Rcm indicating the centre of mass of a molecule composed of M atoms, Fig. 6.5. Let us construct a Cartesian coordinate i stem (Body-Fixed Coordinate System, BFCS) with the origin in the centre of mass and the axes parallel to those of the SFCS (the third possibility in Appendix I). 

Fig. 6.5. Space- and Body-Fixed Cooidinate Systems (SFCS and BFCS). (a) SFCS is a Cartesian coordinate system arbitrarily chosen in space (left). The origin of the BFCS is located in the centre of mass of the molecule (right). The centre of mass is shown by the vector Rcm front the SFCS. The nuclei of the atoms are indicated by vectors ri,r2,rj... from the BFCS. Fig. (b) shows what happens to the velocity of atom a, when the system is rotating with the angular velocity given as vector ta. In such a case the atom acquires additional velocity ta x r . Fig. (c) shows that if the molecule vibrates, then atomic positions ta differ from the equilibrium positions a by the displacements...

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