Normal-ordered second-quantized operators

Three of the five terms in the final rearrangement contain operator strings of reduced length, and the first term contains only Kronecker delta functions. Note also that all the operator strings on the right-hand side of the final equality are normal-ordered by Merzbacher s definition. If we now evaluate the quan- 

on the other hand, we wish to evaluate a matrix element of A involving determinants other than I ) on the left and right, normal ordering simplifies this analysis as well. For example, consider the matrix element of A between the single-particle states (tj)J and l( ) )  

Since the left- and right-hand states may be written simply as single annihilation and creation operators acting on the vacuum, the desired matrix element of A may be rewritten as the vacuum expectation value of a new operator, = a Aal. Therefore, we need only rewrite B in normal order and select only the terms that contain no annihilation or creation operators, as we did in Eq. [77]. After much algebraic manipulation, which we shall omit here, it can be shown that 

By rearranging a given string of annihilation and creation operators into a normal-ordered form, matrix elements of such operators between determinan-tal wavefunctions may be evaluated in a relatively algorithmic manner. However, such an approach based on the direct application of the anticommutation relations can be quite tedious even for relatively short operator strings, and many opportunities for error may arise. 

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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... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

Wave functions (13.1) form an orthonormal set, but their normalization factors are defined only up to a sign. The fact is that the wave function (13.1) is antisymmetric not only under coordinate permutations, but also under permutations of one-electron quantum numbers. Thus, to fix the sign of a wave function requires a way of ordering the set of quantum numbers (a) = ai, 0C2,..., ajy. There exists, however, a convenient formalism that allows us to include the constraints imposed by the requirement that the wave functions be antisymmetric in a simple operator form. This formalism became known as the second-quantization method. This chapter gives a detailed description of the fundamentals of the second-quantization method. 

In general, a given sequence of creation and annihilation operators is said to be normal ordered, if all the creation operators appear left of all annihilation operators. Such an ordering of the operator strings simplifies the manipulation of operator products as well as the evaluation of their matrix elements, as the action of these operators can be read off immediately. In the particle-hole formalism, its hereby obvious that we can annihilate only those particles or holes which exist initially in fact, an existing hole is nothing else than that there is no electron in this hole state. In this formalism, therefore, an operator in second quantization is normal ordered with regard to the reference state [Pg.190]

In particular, having the model operator (34) and the wave operator (40) in second quantization, we can evaluate the commutator on the Ihs of the Bloch Eq. (16) and bring it into its normal-order form by analyzing term by term,... 

Evaluates the normal-order form [A-B— of a product of two or more operators in second quantization. 

All the properties of Slater determinants are contained in the anticommutation relations between two creation operators (Eq. (2.194)), between two annihilation operators (Eq. (2.208)), and between a creation and an annihilation operator (Eq. (2.217)). In order to define a Slater determinant in the formalism of second quantization, we introduce a vacuum state denoted by >. The vacuum state represents a state of the system that contains no electrons. It is normalized. 

Finally, in expectation values sequences of annihilation and creation operators stemming from the second-quantized Hamiltonian and from the states in bra and ket of the full bra-ket must be evaluated for which rules such as Wick s theorem, which implements the anticommutation relations of operator pairs to obtain a relation to normal ordered operator products, can be beneficial [65,353]. 

Along this line, in a recent paper [37] we introduced the so-called quasiparticle-based MR CC method (QMRCC). The mathmatical structure of QMRCC is more or less the same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques can be used, the method is extensive, etc. The point where the MR description appears is the application of quasiparticle slates instead of the ordinary molecular orbitals. These quasiparticles are second-quantized many-particle objects introduced by a unitary transformation which allows us to represent the reference CAS function in a determinant-like form. As it is shown in the cited paper, on one hand the QMRCC method has some advantages with respect to the closely related SR-based MR CC theory [22, 31, 34] (more... 

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). 

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

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