Creation operator strings

The occupation number vectors can be written as a product of alpha creation operators, an alpha string, times a product of beta creation operators, a beta string... 

The form of the wave operators need not be defined, but, in principle, they can describe any type of wavefunction, for example, Hartree-Fock or coupled-cluster wavefunctions. However, at their core, they always consist of strings of creation operators. We define the supermolecular wavefunction as... 

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. 

Using the anticommutation relations of Eqs. [19], [20], and [21], an arbitrary string of annihilation and creation operators can be written as a linear combination of normal-ordered strings (most of which contain reduced num-... 

In addition, a third combination in which A is a creation operator and B is an annihilation operator is also zero, since the string is again already normal-ordered ... 

Now we rewrite the string of annihilation and creation operators from the two-electron part of H as... 

Although Handy was the first to use alpha and beta strings, we will employ the subsequent notation of Olsen et al.46 An alpha string is defined as an ordered product of creation operators for spin orbitals with alpha spin. If Ia contains a list, j,... A of the Na occupied spin orbitals with alpha spin in determinant 7), then the alpha string a(Ia) is ajQaja... aj.a. A beta string is defined similarly. Thus a Slater determinant /) in terms of alpha and beta strings is... 

In the second-quantized operators (31) and (32), the summation over the particle indices i,j,... runs over all the electron states of the (complete one-electron) spectrum. If these operators act to the right upon the reference state, i.e. the many-electron vacuum of the particle-hole formalism, some of these (strings of) creation and annihilation operators create excitations while other gives simply zero, i.e. no contribution. For the pure vacuum, in particular, the behavior of the second-quantized operators can be read off quite easily because the creation operators appear left of the annihilation operators in expressions (31) and (32), respectively. 

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]

Of course, the number of permutations, which are required to bring a given operator string into a certain sequence, determines the sign of the expression Each interchange of two (particle-hole) creation or annihilation operators leads to an additional phase factor (—1) and, possibly, to a contraction of terms. For applications, therefore, it is reasonable to define a (so-called) normal-order sequence which specifies the relative position of the (particle-hole) creation or annihilation operators more precisely we write this sequence symbolically as... 

In this notation, a )a means an operator string of creation and annihilation operators which just creates the model state (pa) out of the (many-electron) vacuum ... 

With this brief discussion of the three basic steps (a)-(c) from Subsecfion 4.1, we have arrived at our original destination to represent all physical quantities of interest as sum (of products) of one- and two-parficle amplifudes. In practice, each of these expansions are often lengthy and the complexity of fhese expansions increases rapidly if the number of particle and holes is increased in the valence shells. The latter can be seen easily from the fact that each (valence-shell) particle of hole introduces an additional creation or annihilation operator into the operator strings a)a and (a ), respectively. In contrast to other, e.g. multiconfigurational, expansions of the wave functions, however, the explicit form of the approximate states IV a) cannot be derived so easily in MBPT or the CCA. For this reason also, a straightforward and simply handling of the perturbation expansions decides how successfully the theory can be applied to open-shell atoms and molecules in the future. 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

In the general case, evaluating the matrix element of any string of creation and annihilation operators, one should consider all possible pairing of annihilation and creation operators, and by successive application of transpositions one should try to bring the expression into the following form (note the order of operators) ... 

It is to be emphasized that HF> behaves as the Fermi vacuum only for those strings of creation and annihilation operators which refer to the same orbitals as those constituting the determinant HF>. More precisely, the orbitals corresponding to the operator string, and those present in HF>, are required to form an orthonormalized set altogether. Specifically speaking, at (uj) in Eq. (5.19) should create (annihilate) an electron on a molecular orbitals, and not an atomic orbital, for example. This is required because otherwise the... 

The same holds for the string of creation operators (X g t ), too. In general, this result is always true if the string consists of an odd number of anticommuting operators. If we have an even number of anticommution operators, a cyclic permutation will change its sign. 

The two products of creation operators are in nonrelativistic theory termed a and p strings. To avoid confusion with these operators, but also to retain the analogy between Kramers pairs and spin-orbital pairs, we will term these A and B strings ... 

The subscripts I and J on the strings are indices of the particular set of Kramers pairs from which the creation operators that make up the string are taken. The spinors that make up the Kramers pairs can, in this notation, be labeled A and B spinors. The many-particle state can now be written... 

This means that when acting to right on a string of creation operators in a ket , operator (p behaves as a conventional annihilation operator does, and analogously, when acting to left on a string of annihilation operators... 

Comparison with Eq. (21) indicates that in overlapping basis the elements of the spin-dependent first- and second-order density matrix can be obtained as expectation values of operator strings constructed from biorthogonal creation and annihilation operators ... 

In second quantization. Slater determinants are expressed as products or strings of creation operators aj, working on the vacuum state... 

There is one creation operator aj, for each spin orbital in the basis and each such operator is viewed as creating an electron in the associated spin orbital. Within a given orbital basis, any wavefunction may then be expressed as a linear combination of strings of creation operators working on the vacuum state. 

Next, we have to define the normal product or n-product, and contraction or pairing. The simplification of matrix elements requires that we move creation operators to the left of annihilation operators. A normal product is a reordered operator string which satisfies this requirement. A contraction or pairing of creation and/or annihilation operators is their vacuum expectation value. 

Second quantization treats operators and wave functions in a unified way - they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4)... 

Before considering the evaluation and simplihcadon of commutators and anticommutators, it is useful to introduce the concepts of operator rank and rank reduction. The (particle) rank of a string of creation and annihilation operators is simply the number of elementary operators divided by 2. For example, Ihe rank of a creation operator is 1/2 and the rank of an ON operator is 1. Rank reduction is said to occur when the rank of a commutator or anticommutator is lower than the combined rank of the operators commuted or anticommuted. Consider the basic anticommutation relation... 

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