Diatomic coordinate frame

Coefficients Rlx form an orthogonal matrix iC transforming the (/-orbitals under rotation of the laboratory coordinate frame (LCF) to the local coordinate frame related to the ligand / and constructed such that its Oz axis is going through the metal atom and the ligand atom (DCF - diatomic coordinate frame). The perturbation caused by the ligand has a matrix representation elxx, in the DCF with A = a,7r(x),n(y), 5 xy), S(x2 - y2). These quantities are considered parameters of the AOM. 

The resonance interaction can be recast in the form of interaction between the hybridization tetrahedra, which in its turn depends on the distance between the centers of the tetrahedra, on their mutual orientation, and on their orientation with respect to the bond axis - that connecting the centers of the tetrahedra involved (the nuclei). The latter can be proven by the following construction consider the m-th two-center bond and the 4 x 4 matrix of the resonance integrals between the AOs in the diatomic coordinate frame (DCF) which is defined by setting its t-axis to be directed along the RmLm two center bond (the bond axis) ... 

Linear response of hybridization to bond elongation. First the relation between hybridization and elongation of the C-H bond is considered. For this we need the mixed second order derivatives coupling the bond stretching with the hybridization ESVs. For every C-H bond in methane we can introduce diatomic coordinate frame with the t-axis directed along the bond and express the resonance integral related to this bond as ... 

Now we restrict our consideration to those complexes where each donor atom bears a single LP. We consider the geometries where the z axis of the LFCF coincides with the z axis of the diatomic coordinate frame and the x- and y-axes in the two systems are parallel. Under these assumptions the components f of the vector tL differ from... 

Figure 2. The space-fixed (XYZ) and body-fixed xyz) frames in a diatomic molecule AB. The nuclei are at A and B, and 1 represents the location of a typical electron. The results of inversions of their SF coordinates are A A, B B, and 1 1, respectively. After one executes only the reinversion of the electronic SF coordinates, one obtains 1 — 1. The net effect is then the exchange of the SF nuclear coordinates alone.

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,... 

Mass-scaled Jacobi coordinates associated to a generic arrangement X — a for A -I- BC, /I for B + CA and ) for C + AB) cU c denoted by r (diatom vector) and R (atom-molecule vector). They are used in the definition of hyperspherical coordinates which parametrize the nuclear motion of the system, namely the principal axis body frame hyperspherical coordinates [3, 4, 5]. These coordinates are ... 

Figure 5.1. Schematic representation of a diatomic molecule, consisting of atoms A and B, in the body-fixed molecular frame where the origin is taken to be the center of mass and the z axis is along the principal axis of the molecule, za and zb denote the z coordinates of atoms A and B, respectively, and Zab the bond length.

The Born-Oppenheimer diagonal correction is given in Eq. (2a). In that equation, the gradients refer to space fixed frame (SFF) coordinates. For diatomic molecules, considerable savings result from a transformation to body fixed frame (BFF) coordinates. This transformation is accomplished in two steps. The SFF coordinates are transformed to center of mass fixed frame (CMFF) coordinates and then the CMFF coordinates are transformed to BFF coordinates. The details of the transformation are beyond the scope of this review. Here we sketch the ideas involved. A detailed treatment, based on the pioneering work of Kronig, can be found in Ref. 7. In particular, first the rigorously removable center of mass of the nuclei and... 

To study the atom-diatom reactive scattering, A -f BC(u, y, ) AB(n/, 7/) + C/AC ( /,7/) + Bata state-to-state level, body-fixed (BF) frame Jacobi coordinates are... 

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