Tensor Structure of the Many-Electron Hamiltonian and Wave Function

Tensor Structure of the Many-Electron Hamiltonian and Wave Function 

The solution of the electronic Schrodinger equation, Eq. (8.76), requires the integration of a complicated partial differential equation depending on 3N electronic coordinates as variables. The complexity of this problem increases with the number of electrons in the molecule. In order to establish a general solution strategy it is mandatory to first study the underlying formal framework of many-particle quantum mechanics, namely that of a linear vector space — the Hilbert space (cf. section 4.1). The N-electron Hilbert space, which hosts the total quantum mechanical state vector, is then constructed by direct multiplication of the one-electron Hilbert spaces. 

According to the standard representation of the Dirac theory, the one-electron operators considered in chapters 5 and 6 are (4 x 4)-matrices acting on the one-particle Hilbert space 

For many-electron Hamiltonians defined as in Eq. (8.73) the situation is more involved [283]. The Hilbert space of an N-electron system is the tensor product (or direct product) of the N one-particle Hilbert spaces i.e., 

Any two-electron entries for not necessarily neighboring i and j thus enter the proper places in the 2 super-matrix Hamiltonian. Because of Eq. (8.86) a total quantum mechanical state Y possesses the structure 

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