Orthogonalization vectors geometrical view

Orthogonalization of Vectors and its Relation to Cognitive Phenomena 3. A GEOMETRICAL VIEW... 

In the theory of optics this phenomenon is accounted for in terms of geometrical construction, but the physical picture is less convincing. Double refraction is a well-documented property of most crystals, at its most spectacular in Iceland spar. The double image of an object viewed through the crystal indicates the existence of two independent rays and not the components of a single ray. In mathematical terms the two rays are linearly independent and therefore orthogonal. Any intermediate situation represents a linear combination of the two orthogonal basis vectors and can be resolved into two components. What happens to an individual photon is however, not clear. 

The mutual orthogonality of the character vectors is reminiscent of the axes of a Cartesian coordinate system, and suggests the valuable idea that the character vectors of a group form a basis for the symmetry. Any vector can be resolved into components of different symmetry types. The projection of any vector onto any symmetry species is calculable. So we have returned to the geometrical point of view ... 

Let us now briefly summarize the essentials of MC-SCF theory. It is helpful to take an entirely different approach to the usual formulation of closed shell SCF theory. In fact, it proves to be useful to think of the MC-SCF process in a similar way to geometry optimization. Thus we shall view the orbital and Cl coefficient variables that occur in MC-SCF in the same way as the internal geometrical variables in a geometry optimization. Technically in order to do this we must assume that we have an orthogonal set of starting orbitals o and an orthogonal set of Cl vectors K>. The MC-SCF Cl expansion (for state K) is written as... 

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