Vacuum state annihilation operator applied

We have thus shown the relations (3 28) for the transformation of the annihilation and creation operators to a new spin-orbital basis. We can use these relations to express an arbitrary Slater determinant in the new basis in terms of the determinants in the original basis. In order to do so, we generate the Slater determinant by applying a sequence of creation operators on the vacuum state ... 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

With each creation operator ap, there is an associated conjugate operator ap - known as the annihilation operator -which removes the electron from spin orbital P, undoing the effect of the creation operator. If the annihilation operator ap is applied to any state that does not contain the spin orbital P, the state vanishes. In particular, applying an annihilation operator to the vacuum state, we obtain... 

Electron pair annihilation, n LMi,SMs), and creation, n ( LMiSMs), operators can be defined such that tt creates a normalized two-electron state with the specified orbital and spin angular momenta when applied to the vacuum... 

---