Electrons creation and annihilation

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

Second-quantization. Electron creation and annihilation operators... 

For commutation relations of this operator with electron creation and annihilation operators, instead of (13.29) and (13.30), we obtain... 

For some one-determinant state ai,...,ajy) we can completely change over from particle description to hole description if, instead of electron creation and annihilation operators, we introduce, respectively, annihilation and creation operators for holes... 

Utilization of the tensorial properties of the electron creation and annihilation operators allows us to obtain expansions in terms of irreducible tensors of any operators in the second-quantization representation. So, using the Wigner-Eckart theorem (5.15) in (14.11) and (14.12), then coupling ranks of second-quantization operators by (5.12) and utilizing (14.10), we can represent one-shell operators of angular momentum in the irreducible tensor form... 

The anticommutation relations between the electron creation and annihilation operators, accounting for (14.14), become... 

Using the anticommutation properties of electron creation and annihilation operators we can establish any necessary commutation relations for their tensorial products. For example, [103]... 

The sums of products of CFP obey some additional relations. In fact, the operators of particle number N, of orbital L and spin S momenta are expressed in terms of tensorial products of electron creation and annihilation operators - relationships (14.17), (14.15) and (14.16), respectively. We can expand the submatrix elements of such tensorial products using (5.16) and then go over, using (15.21) and (15.15), to the CFP. On the other hand, these submatrix elements are given by the quantum numbers of the states of the lN configuration. Then we obtain... 

By computing the commutators of the components of the quasispin operator with electron creation and annihilation operators, we can directly see that the latter behave as the components of a tensor of rank q = 1/2 in quasispin space and obey the relationship of the type (14.2)... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

Considering the tensorial properties of the electron creation and annihilation operators in quasispin space, we shall introduce the double tensor... 

We will consider only zero temperature. It is convenient to switch [19] to the interaction representation H —> H — (.irNr — iirNl- This transformation induces time dependence in the electron creation and annihilation operators. As a result, 2U2nkF cos(n g

[Pg.152]

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

Often configuration interaction calculations are performed after an initial Hartree-Fock calculation. In such cases the one-electron creation and annihilation operators would refer to eigenfunctions of the Fock operator, and the Fock operator would have a simple form similar to that shown in equation (53). 

In the vast majority of the quantum chemistry literature, Slater determinants have been used to express antisymmetric N-electron wavefunctions, and explicit dilTerential and multiplicative operators have been used to write the electronic Hamiltonian. More recently, it has become quite common to express the operators and state vectors that arise in considering stationary electronic states of atoms and molecules (within the Born-Oppenheimer approximation) in the so-called second quantization notation (Linderberg and Ohrn, 1973). The electron creation ) and annihilation... 

Although Eqs. (1.2)-(1.5) contain all of the fundamental properties of the Fermion (electron) creation and annihilation operators, it may be useful to make a few additional remarks about how these operators are used in subsequent applications. In treating perturbative expansions of N-electron wavefunctions or when attempting to optimize the spin-orbitals appearing in such wavefunctions, it is often convenient to refer to Slater determinants that have been obtained from some reference determinant by replacing certain spin-orbitals by other spin orbitals. In terms of second-quantized operators, these spin-orbital replacements will be achieved by using the replacement operator as in Eq. (1.9). 

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