Operator electron creation

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

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... 

The electron creation operator is introduced in such a way that the one-electron wave function produced by it would appear in the first line of the determinant with the same sign. Thereby the sign of the one-determinant wave function is uniquely specified also in the case where the resultant one-particle wave function is in any line of the determinant... 

It should be stressed that in the literature one can come across a wide variety of notations for creation and annihilation operators. In this book we follow the authors [14, 95] who attach the sign of Hermitian conjugation to the electron annihilation operator, but not to the electron creation operator. Although the opposite notation is currently in wide use, it is inconvenient in the theory of the atom, since it is at variance with the common definitions of irreducible tensorial quantities. 

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... 

Rearranging the electron creation operators in the right side of this expression and using the symmetry properties of Clebsch-Gordan coefficients... 

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

We shall also provide the commutation relations for the electron creation operator with tensor (14.40) in the irreducible tensor form... 

In the general case, the second-quantized operator linear combination of irreducible tensorial products of electron creation operators. The combination must be selected so that a classification of states according to additional quantum numbers be provided for. Without loss of generality, all the numerical coefficients in the linear combinations can be considered real. Then, from (14.14), we can introduce the operators... 

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... 

Specifically, this relationship holds for the electron creation operator a(jfj which, by (15.49), is a component v = 1/2 of the triple tensor 0. If we take into consideration that the submatrix element of the creation operator is expressed in terms of the CFP from (15.21), we have... 

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

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

We shall now introduce the electron creation operators a and b and the electron annihilation operators a = (—b = (—1 y mb for the electrons in the subshells nihjNl and n2hjNl, respectively. They are irreducible tensors of rank t = 1 /2... 

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]

Due to the fact that the SLG wave function belongs to the GF approximation (Section 1.7), it is subject to numerous selection rules characteristic of GF. Their explicit form can be easily obtained using the second quantization formalism. Since the operators of electron creation on the right and left HOs satisfy usual anticommutation relations for orthogonal basis and the number of particle operators have the usual form ... 

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

This gives a sign factor of 1 if there are an even number of electrons occurring before spin orbital (f), and a factor of — 1 for an odd number of electrons. It is also convenient to define an electron creation operator, written al, which acts to create an electron in spin orbital (f), or to give zero if such an electron already exists. The general expression for an arbitrary ket is... 

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