Coordinate systems diatomic

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

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). 

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

Figure 6.1 Binding and antibinding regions for a heteronuclear diatomic molecule consisting of two nuclei A and B with ZA = ZB. The coordinate system is superimposed. The distance from a point with coordinates (x,y,z) to nucleus A is rA and to nucleus B is rB. the distance between the nuclei is RAb To obtain the 3D binding and antibinding regions rotate the figure about the intemuclear axis.

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

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 ... 

First suppose that a spherical surface of unit radius is drawn and the center of this sphere is taken as the origin of a spherical polar coordinate system. Suppose further that p(0), the initial orientation of a diatomic molecule, is represented by the unit vector, k, along the positive Z axis of... 

Simultaneously with the oscillations with respect to the internuclear axis, a diatomic molecule rotates as a whole with a frequency ft (not to be confused with the quantum number of the projection of the total momentum, as in Section 1.2) around an axis which is perpendicular to the direction of the internuclear axis, exhibiting a total angular momentum J see Fig. 1.3. Hence in the laboratory coordinate system, with respect to which the light wave possesses a certain polarization, electrons participate simultaneously in two types of motion dipole oscillations with respect to the internuclear axis, and rotation of the molecule as a whole. 

We note that the atom-diatom dispersion coefficients are usually expressed in the center-of-mass coordinate system by an expansion in terms of Legendre polynomials Pn(cos0). For example, for the A---(BC) interaction, one gets... 

In the case of an interaction like O + OH, the center of geometry of the diatomic molecule may still be adopted as the reference provided one introduces the modifications connected with the change of origin. For example, if we define (r, 9) and (r 0 ) as the coordinates of the center of mass and center of geometry, respectively, and if z represents the distance between those centers, then the change from one coordinate system into the other will involve the following changes ... 

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

In conclusion we summarise the total Hamiltonian (excluding nuclear spin effects), written in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, for a diatomic molecule with electron spin quantised in the molecular axis system. We number the terms sequentially, and then describe their physical significance. The Hamiltonian is as follows ... 

Let us consider a nucleus with I > 1 in atom 1 of a diatomic molecule. We shall use a local coordinate system (x, y, z) with its origin at the nucleus and z lying along the molecular bond. An unscreened electric charge q at a distance r from the nucleus gives rise to an electrostatic potential... 

Fig. 5. Coordinate system used for one-electron diatomic atoms. The two nuclei are on the z axis with charges Z, and Z2, separated by a distance R, and the electron (e ) is a distance r from the origin. See text for details.

Figure 1. Center-of-mass coordinate systems for atom-diatom van der Waals molecules. The primed axes are used in the space-fixed (SF) formulation, the unprimed axes in the body-fixed (BF) formulation.

In general the Hartree-Fock equations for any molecular system form a set of 3-dimensional partial differential equations for orbitals, Coulomb and exchange potentials. In the case of diatomic molecules the prolate spheroidal coordinate system can be used to describe the positions of electrons and one of the coordinates (the azimuthal angle) can be treated analytically. As a result one is left with a problem of solving second order partial differential equations in the other two variables, (rj and ). 

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

To describe tunnelling in reactive systems,34,35 A -I- BC - AB + C, the above procedure must be modified somewhat. If r is the vibrational coordinate of diatom BC, then the above procedure for choosing the time path is followed with regard to the variable ra until A and BC reach their distance of closest approach. At this point the complex time path is chosen to cause the reaction to occur i.e. one wants rc(t), the vibrational coordinate of AB, to head toward its equilibrium value. Thus the same procedure is used to choose the time path but with regard to rc, the vibrational coordinate of the new diatom. 

---