Second-quantized operators

One can, for example, express T in temis of a superposition of configrirations = Y.jCj whose amplitudes Cj have been detemiined from an MCSCF, Cl or MPn calculation and express Q in temis of second-quantization operators Offt that cause single-, double-, etc, level excitations (for the IP (EA)... 

The next step is that we find inverse transformations to (25-28) and substitute these inverse transformations into eq. (22) and then applying Wick theorem, we requantize the whole Hamiltonian (16) in a new fermions and bosons [14]. This leads to new V-E Hamiltonian (we omit sign on the second quantized operators)... 

In previous part we developed canonical transformation (through normal coordinates) by which we were able to pass from crude adiabatic to adiabatic Hamiltonian. We started with crude adiabatic molecular Hamiltonian on which we applied canonical transformation on second quantized operators... 

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... 

As in Eqs.(14) and (15) for the 2-positive metric matrices, the 3-positive metric matrices are connected by linear mappings, which can be derived by rearranging the second-quantized operators. A 2-RDM is defined to be 3-positive if it arises from the contraction of a 3-positive 3-RDM ... 

In contrast to the and metric matrices in 3-positivity, the strength of the T2 matrix as a 2-RDM A -representability condition is not completely invariant upon altering the order of the second-quantized operators in C, j - For example, a slightly different metric matrix T2 can be defined by exchanging the operators a, and a in Eq. (40) to obtain... 

The generalized f 2 matrix is contained in Erdahl s original theoretical treatment of these conditions [38], although the recent applications to atoms and molecules [27, 34] employ either T2 or f 2- The condition 72 > 0 implies both T2 >0 and f 2 > 0 as well as any other conditions from different orderings of the second-quantized operators. 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

The connected stmcture of the CSE has also been explored by Yasuda [23] using Grassmann algebra, by Kutzelnigg and Mukheijee [27] using a cumulant version of second-quantized operators, and by Herbert and Harriman [30] using a diagrammatic technique. 

The reduced matrices and V represent a partitioning of the Hamiltonian into one- and two-electron parts. Rearranging the second-quantized operators and using the definition of the 2- and 3-RDMs,... 

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... 

We assume at this stage that we have all necessary second-quantized operator matrices (associated with Ileft, RIGHT, and the partition orbital) to enable the construction of H and other physical operators of interest. 

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,... 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

The recipe for constructing a two-electron second quantization operator is thus given by Eqs. (2.6) and (2.9). 

Second quantization operators have been constructed for one- and two-... 

The above development indicates that commutation relationships that hold for first quantization operators do not necessarily hold for second quantization operators in a finite one-electron basis. Consider the canonical commutators... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

Let us now turn to the commutation relations between second-quantization operators. Acting in succession with two different creation operators on a one-determinant wave function, from (13.2), we get... 

The above anticommutation relations for second-quantization operators... 

The expressions that define the action of second-quantization operators <4 and aa on wave functions a) and 0) can be presented in terms of the conventional relation... 

The pertinent transformation of the second-quantization operators now yields creation and annihilation operators for holes (13.34) and (13.35). 

Second-quantization operators as irreducible tensors. Tensorial properties of electron creation and annihilation operators... 

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. 

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... 

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. 

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. 

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... 

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

Let us now put (14.59) into the general formula (13.23) and rearrange the creation and annihilation operators so that when the second-quantization operators are placed side by side the rank projections enter into the same Clebsch-Gordan coefficient. A summation over the projections gives... 

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. 

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