Products of second-quantization operators

Using second-quantization, it is often necessary to transform complicated tensorial products of creation and annihilation operators. If, to this end, conventional anticommutation relations (14.19) are used, then one proceeds as follows write the irreducible tensorial products in explicit form in terms of the sum over the projection parameters of conventional products of creation and annihilation operators, then place these operators in the required order, and finally sum the resultant expression again over the projection parameters. On the other hand, the use of (14.21) enables the irreducible tensorial products of second-quantization operators to be transformed directly. 

Apart from irreducible tensors (14.30) we can also introduce other operators that are expressed in terms of irreducible tensorial products of second-quantization operators, and establish commutation relations for them. As was shown in [12, 102, 103], using relations of this kind, we can relate standard quantities of the theory, which at first sight seem totally different. Consider the operator... 

In Chapter 15, for the CFP with a detached electrons, we obtained a relationship (15.27) whose right side has the form of a vacuum average of a certain product of second-quantized operators q>. To obtain algebraic formulas for CFP, it is necessary to compute this vacuum average by transposing all the annihilation operators to the right side of the creation operators. So, for N = 3, we take into account (for non-repeating terms) the explicit form of operators (15.2) and (15.5), which produce pertinent wave functions out of vacuum, and find (cf. [107])... 

These formulas include irreducible tensorial products of second-quantization operators belonging both to one and the same shell and to different shells. The expressions for the operators can, if necessary, be transformed so that the ranks of second-quantization operators for one shell are coupled first. For example, the operator that enters into (17.30),... 

In an analogous way, we can consider the submatrix elements of irreducible tensorial products of second-quantization operators. So operator (14.40) will be... 

The reduction of products of second-quantized operators makes use of commutators of the replacement operators. The various commutators of Kramers single-replacement operators are given by application of bar reversal to the basic commutator relation... 

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

In O Eqs. 28.25 and O 28.26 the subscript C indicates that only the so-called connected terms are considered, when the operators are expressed in terms of normal ordered product of second-quantization creation and annihilation operators (Bartlett and Musial 2007). 

The conditions that a 3-RDM be 3-positive follow from writing the operators in Eq. (8) as products of three second-quantized operators [16, 17]. The resulting basis functions lie in four vector spaces according to the number of creation operators in the product the four sets of operators defining the basis functions in Eq. (8) are... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

The number of tensorial operators with different possible values of ranks k,K and appropriate values of the projections is predetermined by the number of possible projections of the second-quantization operators that enter into the tensorial product. This number is equal to (4/ + 2)2. 

If in the irreducible tensorial product of operators (14.40) and (14.42) we interchange the second-quantization operators connected with an arrow,... 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

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... 

Atomic spectral moments can be expressed in terms of the averages of the products of the relevant operators. Let Oi,C>2, --,Ok be the operators of interactions in definite shells or the operators of electronic transitions between definite shells in the second quantization form. The average of... 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

The form [Eq. (3)] of the perturbation operator points out that formally we obtain a double perturbation expansion with the two-electron V2 and one-electron Vi perturbations. However, in the case of a Hartree-Fock potential the one-electron part of the perturbation is exactly canceled by some terms of the two-electron part. This becomes more transparent when we switch to the normal product form of the second-quantized operators2-21 indicated by the symbol. ... We define normal orders for second-quantized operators by moving all a ( particle annihilation) and P ( hole annihilation) operators to the right by virtue of the usual anticommutation relations [a b]+ = 8fl, [i j] = 8y since a 0) = f o) = 0. Then... 

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