Anticommutation rules

The essential feature of the orthogonality of the underlying basis orbitals with respect to second quantization is the adjoint relation  

Both of these equations have their own significance. The adjoint relation is essential when putting down the second quantized correspondances of bra-vectors, cf. Eq. (2.53). The anticommutation rule is important in dealing with matrix elements where one often has to transpose creation and annihilation operators. 

As seen in Sect. 2.6, Eqs. (13.1) and (13.2) hold simultaneously only if the underlying orbital set is orthogonal  

Accordingly, the second quantized formalism can be generalized to the nonorthogonal case in two alternative manners one may keep either the adjoint relation of Eq. (13.1) or the simple anticommutation rule of Eq. (13.2). In the former case the commutation rules become more complicated, while in the latter case the annihilation operators will not be the adjoints of the corresponding creation operators. 

Let us first look at the generalization of the anticommutation rules if we accept Eq. (13.1). For the sake of forthcoming distinctions, the creation operators creating electrons on a nonorthogonal set of spinorbitals % will be denoted by  

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It is required, in accordance with the Fermi character of particles and antiparticles, to be separately antisymmetric in the particle and antiparticle variables, which in turn requires that the operator b and d satisfy the following anticommutation rules ... 

The only operation used for obtaining this partitioning is the anticommutation rule of the fermion operators. Note, that by adding the F and G terms one falls into the unitarily invariant Absar and Coleman partitioning [32,33] which was obtained by using a Group theoretical approach. 

Recalling that the anticommutation rules for the creation and annihilation operators are (App. C)... 

Also, the fermion anticommutation rules interrelate the ROMs with the HRDMs they render these matrices antisymmetric with respect to odd permutations of the row or column indices and, finally, they interrelate them with two other families of matrices the G-matrices and the comelation matrices. 

This implies that the bra expression <0 av - aa is the adjoint of the corresponding ket expression a --- aj 0>. The equivalence of the operator and determi-nantal formalism ensures that all operators act in accordance with the Pauli exclusion principle for fermions. The following anticommutation rules thus follow at once ... 

All the described features of how the creation and annihilation operators act on the Slater determinants constructed from the fixed basis of spin-orbitals are condensed in the set of the anticommutation rules ... 

The cross product obeys anticommutation rules. Similar rules apply to the gradient vector we obtain... 

In an orthogonal basis, the adjoints of creation operators are the annihilation operators an. The electron creation and annihilation operators obey simple anticommutation rules... 

As examples for the simplifications in the Hamilton matrix elements we consider the contributions of the doubly external integrals to the operators pjp.uq (197)) Depending on the number of excitations from the closed shells one can distinguish six distinct cases. Using the relations 7f 0> = 7f 0> = 0 and the anticommutation rules in Eqs (6)-(8) one obtains the following results (the all-internal integral contributions are omitted). 

It follows from the above relations that the operator can be called the creation operator of the states nf, and bnf the annihilation operator for the same states. It also follows from (3.7) that the operators obey the anticommutation rules... 

Now we move the operators a aa through the expression to the right taking account of the anticommutation rules. From (51), (52) and (53) we see that... 

The four matrices 7 are 4x4 matrices, which obey the anticommutation rules... 

In summary, there are three different lanes available for evaluating the operator products on the rhs of the generalized Bloch Eq. (23) and for all effective operators, such as the Hamiltonian (15) or any other interaction operator. These alternatives are displayed in Figure 10.2 beside of the (i) purely algebraic transformation, i.e. the use of the fermion anticommutation rules for the creation and annihilation operators, variety of graphical methods and rules... 

The treatment of orbital overlap in conjunction with the use of nonorthogonal basis sets deserves particular attention in treatments in terms of electron field operators. The definition of creation and annihilation operators and their anticommutation rules are basic for this development. Let s( ) be a set of atomic spin orbitals used to define the creation operators... 

Both types of operators do not commute but fulfill the following anticommutation rules... 

Show that operators bj, b obey the same anticommutation rules as operators... 

According to Eq. (7.20), the second-order density matrix corresponding to a one-determinantal wave function consists of two parts. When F is expressed in terms of P, these two parts differ only in the order of the indices. The first term can be called the Coulomb part, while the second one is the exchange part. This nomenclature is justified because the latter term appears as a consequence of the antisymmetry of the wave function which is reflected by the anticommutation rules in this formahsm. 

Substituting the definition of F and applying the anticommutation rules one may write ... 

At the end of this section, we put down the expressions for the basic anticommutator rules in terms of spatial orbitals. Based on Eq. (9.2), they obviously read as ... 

The two-electron part of the PPP Hamiltonian can be expressed in terms of the particle number operators similarly to the Hubbard case. Applying the relevant anticommutation rules one may write as ... 

Application of the anticommutation rule for the two particle operators gives a minus sign (the system being alternant, m n ) ... 

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