Exponential map

Obviously an exponential mapping looks also good in the sense of the first criterion. One sees easily that J r) is independent of the choice of j, such that we may as well take 7 = 1. Of course, this is only a plausiblity argument and we need a rigorous criterion for the optimum mapping. We come back to this problem in the conclusions. 

The exponential map gives a T-equivariant isomorphism between a neighborhood of 0 G T X and a neighborhood oi x G X. This shows that is a submanifold of X and that the tangent space at x is given by = F(0). Moreover, / is approximated around X by the map... 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

Our use of exponential mappings starts by noting that for an arbitrary hermitian matrix k, the matrix... 

These operators are used to define the isomorphism p(g) which characterizes the unitary irreducible and infinite-dimensional representation of H", through the exponential map... 

The central idea of the LEM [4], [5] is to apply a parametric exponential mapping, which in this case is of the form... 

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