Second shifting theorem

The second-shifting theorem, which permits the mathematical treatment of lag or dead time, is defined in the following manner ... 

Despite the strong MO mixings indicated by the Ae splittings, one may question to what extent the MO energy variations are reflected in measurable physical properties. As described in Section 3.2.4, the interactions of filled NBOs lead to symmetric second-order energy shifts with no net effect on total energy, wavefunc-fion, and other properties. However, the assumptions of Koopmans theorem imply that the vertical ionization potential (IP) is related to HOMO orbital energy by... 

Second, since the ID for an arbitrary set of vibrational occupation numbers, mi, m2,..., mjv. Hi, H2,. .., wjv, can be obtained by equating coefficients of the IN expansion variables in the infinite multiple power series of Gn, the IDs can be easily convoluted with themselves through the use of simple rules for multiplication of power series. In this way, some addition theorems concerning the normal mode shifts and the order (dimension) of the IDs can be derived (see Sections 4.1.1 and 4.2.2). So the convolution of two or more IDs of given orders leads to a joined ID, the order ofwhich is equal to the sum of the orders of the factors. This important aspect of the ID is of practical use in treating dense manifolds of vibronic states that occur in the statistical limit. 

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