How does one do this in practice, particularly if one starts with the results of some high level quantum chemistry calculation? Last week Seth Olsen gave a nice cake meeting talk about a method due to Cederbaum, Schirmer and Meyer.
Basically, one wants to block diagonalise the Hamiltonian and then focus on just one of the blocks which defines the effective Hamiltonian. But there an infinite number of ways of doing this. What criteria does one use to decide that a specific block diagonalisation is the "best one"?
The key mathematical results are in this paper. I give a brief summary of a few of the key things I learnt from Seth's talk.
The central result of the paper is
The key to getting this result is using a theorem which says that if one has a non-singular matrix M then the closest unitary matrix to M is M(M^dagger M)^-{1/2}. This related to matrix polar decomposition and is just the matrix analogue of the fact that for a vector v the closest unit vector to v is a normalised vector parallel to v.
Another way of looking at T is that S diagonalises H and one follows this with the inverse of a block diagonalisation to give the transformation U=S(S_BD)^-1. The matrix T is the closest unitary transformation to T.
A key consequence of practical significance is:
This is particularly useful with State-Averaged Complete Active Space -Self Consistent Field (SA-CAS-SCF) calculations. Then one has very accurate eigenvectors for the few quantum states that one performed the state averaging over.
For a concrete implementation of all this for a specific molecule see A diabatic three-state representation of photoisomerization in the green fluorescent protein chromophore
One curious historical aside: a similar approach was considered more than 50 years ago by Jacques des Cloizeaux (of Bethe ansatz fame) in a paper in the journal Nuclear Physics.
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