General formulas for matrix elements

Part of the Lecture Notes in Chemistry book series (LNC, volume 42)


In this section I am rederiving in a slightly generalized form a general formula given already by Corson(1951) and Kotani et al (1955) — nothing new under the sun. There are two pitfalls one should avoid: one is not being general enough to cover all the cases and the other being so general that the results become impractical (cf Seligman 1981). We are really not interested in any formulas, we would rather like to have graphical rules to calculate matrix elements, but one has to start from something. In the previous section the physical part of matrix element calculation has been separated from the structural one, therefore it is enough to calculate the elements of the shift operators, i.e. structure constants. However, the projection operator approach presented in the last section is clearly not the most general, for example it does not work for the two—slope, Ŝ2—adapted graph of Fig 20c. Here we should use the \(\hat{E}_{{ij}}^{{ \uparrow \uparrow }},\hat{E}_{{ij}}^{{ \downarrow \downarrow }}\) shift operators (for typographical convenience Ê i j is written as \(\hat{E}_{{ij}}^{{ \downarrow \downarrow }}\)) but we can not define the one—particle states \(\left| {i \uparrow \rangle ,\left| i \right.} \right. \downarrow \rangle\). The arrows refer to the spin couplings and have meaning only in context of many—particle states. Although we may formally write:
$$\left| {\Phi \rangle = \left| {2{{p}_{0}}} \right.} \right. \uparrow \rangle \left| {2{{p}_{{ - 1}}}} \right. \uparrow \rangle \left| {2{{p}_{{ - 1}}}} \right. \downarrow \rangle \left| {2s \downarrow \rangle \left| {2s \uparrow \rangle } \right.} \right.$$


Matrix Element Symmetric Group Shift Operator Double Coset Occupied Orbital 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • W. Duch
    • 1
    • 2
  1. 1.Max-Planck-Institut für Physik und AstrophysikGarching bei MünchenDeutschland
  2. 2.Instytut FizykiUMKToruńPoland

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