Derivation/equation for solid angle factor correction
Summary: I want to determine a correction for the Solid Angle Factor (SAF) due to partially overlapping 'outer' spheres (of different sizes), as perceived/viewed from the center of a 'central' sphere. The distances of these spheres from the 'central sphere' are not equal, i.e. they are not equidistant. The coordinates, distances and relatives angles of all the spheres can be determined via software, i.e. they are known.
“The Solid Angle Factor (SAF) is defined as the solid angle of the ligand cone comprising the metal at the apex and the primary coordinating atom or group (the first-order SAF) or the whole ligand (the second-order SAF) divided by 4π. Geometrically, it refers to the ratio of the projected area to 4π, i.e. the area of the sphere surface. It actually represents the size of the ligand as viewed from the metal centre towards the ligand. The sum of the values of SAF of all the ligands coordinated to the metal centre represents the total occupancy of the ligands in the coordination sphere. It is apparent that this occupancy should not reach unity because there are gaps and holes among the ligands.” (Polyhedron Vol. 6, No. 5, pp. 104-1048, 1987) (See https://www.dropbox.com/s/g2f97bn97yq56vu/xi-zhang1987.pdf?dl=0)
In other words the solid angle is related to the projection of an 'outer' 'ligand' (sphere of known radius) onto the surface of a central ('metal' atom) sphere, as viewed from the center of the central sphere. Solid angle factor (SAF) equation:
$$SAF=\frac{2\pi (1-\cos \Theta )}{4\pi } = \frac{1}{2}(1-\cos \Theta)$$
$$\theta =\sin^{-1}(\frac{\nu }{\iota })$$
, where ν = radius of the sphere of ligand/coordinating atom (This is known), l = distance between the center of the 'ligand' spheres to the center of the metal atom/ 'sphere'.
A correction (approximation) has been supplied for the case where 2 identical 'ligand' spheres (i.e. the spheres have identical radii) are partially overlapping and are equidistant from the metal center:
$$\Delta SAF = \frac{2(\frac{2\varphi (\pi \nu ^{2})}{360}-d\nu \sin \varphi )}{4\pi (\iota \cos \eta )^{^{2}}}$$
(See link above and the following link: https://www.dropbox.com/s/7w3kyvf1xq9e320/Solid%20angle%20factor_details_1_2%20%281%29.pdf?dl=0 . These diagrams/equations illustrate what I understand of the the given correction equation, as well as the definition of the above symbols)
Derivation/equation for solid angle factor correction: However, since there are multiple 'ligand' spheres around the metal several may partially overlap (or be perceived to overlap as viewed from the central 'metal' sphere) due to their close proximity to the metal, thereby resulting in the ‘observed’ solid angle factor being different from the calculated solid angle factor (without correction) from the metal. The correction must consider:
- The distance from the central 'metal' sphere to the center of each 'ligand' sphere. (They are not necessarily equidistant)
- The radius of each 'ligand' sphere (since they may have different (known) radii).
How would such an equation be derived and what would the equation be? Perhaps by considering 2 'ligand' spheres at a time? I am interested in the corrected (i.e. accurate) sum of all the SAF (known as the Solid Angle Sum (SAS)) of all the surrounding 'ligand' spheres, which should be < or = 1.0.
I look forward to any helpful responses. $$Thank you$$