Single excitation operators

All occupation number vectors in F(m,N) can be obtained from an occupation number vector I n> with N electrons by applying one or several elementary excitation operators on I n>. If a single excitation operator is applied we obtain a single excitation, if two excitation operators are involved, we obtain a double excitation, etc. 

These operator relations allow manipulating the operators independently of the function they are operating on. In general we will work with products of the operators. These can then often be simplified by the use of (3 5) or relations derived from them. Important operator products are those that preserve the number of particles. They always contain equally many annihilation and creation operators. A basic operator of this kind is the single excitation operator, which excites an electron from orbital i to orbital j ... 

Application of the time-independent Wick theorem to the single-excitation operator X% Xiy present in both the description of singly excited determinants with respect to the Fermi vacuum and in the cluster operator T, gives... 

Contractions between the creation operators or the annihilation operators vanish identically because of the Fermi-Dirac statistics obeyed by electrons [cf. Eq. (43)] and as in the single-excitation operator case, contractions between creation and annihilation operators are zero, because the indices belong to disjoint sets [cf. Eq. (44)]. Hence, Eq. (62) becomes... 

The next step to recover the fifth-order energy diagrams is to operate with on the T(3) amplitude in order to obtain T. Since at the third-order level we have a single excitation operator and at the fourth-order level we need a double excitation operator, we have to use the form of the Vj operator that increases the level of excitation, i.e., -—V- When com-... 

Eq. (107) as the form of a single excitation operator that preserves the and eigenvalues. The operator e ,u may be written as... 

The differences between the contracted and uncontracted procedure are best illustrated by giving two examples with a very simple reference wave function = X hhadb -I- fi hhabb. In the first place, we will apply the single excitation operator involving the occupied orbital h and the unoccupied orbitals p and p. The uncontracted wave function reads... 

Ti being a single excitation operator, T2 a double excitation operator and so on up to Tn for an n-electron excitation. 

In Box 13.1, we have listed all nonzero conunutators between the Hamiltonian operator and the single-excitation operators. Usually, we are not so much interested in the commutators themselves as in their application to the Hartree-Fock state see Box 13.2. In these boxes [22], the operators l carry out permutations of the pair indices in the following manner ... 

In Boxes 13.1 and 13.2, we have listed only those commutators that contain the single-excitation operators. Commutators involving double and higher exeitations are easily expressed in terms of the single excitations, for instance ... 

Box 13.1 The nonzero conunutators between the Hamiltonian and the single-excitation operators... 

Box 13.2 The Hamiltonian and its nonzero commutators with the single-excitation operators applied to the Hartree-Fock state. The commutators are listed in Box 13.1... 

Thus, in addition to the density-matrix elements for lla and lcr ) determined in Exercise 5.2.1, we must consider the transition-density elements. The one-electron transition-density elements in (5S.2.7) vanish since a single-excitation operator cannot connect two determinants that differ by more than a single occupation. By contrast, because of the double-excitation operator present in the two-electron transition-density elements... 

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