Operator strings

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-... 

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. 

Fiow does Wick s theorem help us in evaluating matrix elements of second-quantized operators Recall that any matrix element of an operator may be written as a vacuum expectation value by simply writing its left- and right-hand determinants as operator strings acting on the vacuum state, I ). The... 

A somewhat more general version of Wick s theorem may be developed which involves products of operator strings, some or all of which may be normal-ordered. The original form of Wick s theorem is only slightly modified in that the contractions need be evaluated only between normal-ordered strings and not within them. For example, for a product of two normal-ordered strings, the generalized Wick s theorem says that... 

The generalized form of Wick s theorem (see Eq. [91]) says that this product of normal-ordered operator strings may be written using only contractions between the two strings. That is,... 

For the second term of the expanded commutator, Ti n> where the operator strings from and T are simply reversed in order, Wick s theorem gives only one term, namely. 

Applying Wick s theorem to the operator strings in the first term on the right-hand side of this equation gives... 

The construction of the coupled cluster amplitude equations is somewhat more complicated than the energy equation in that the latter requires only reference expectation values of the second-quantized operators. For the amplitude equations, we now require matrix elements between the reference, o, on the right and specific excited determinants on the left. We must therefore convert these into reference expectation value expressions by writing the excited determinants as excitation operator strings acting on Oq. For example, a doubly excited bra determinant may be written as... 

The final matrix element therefore requires that we obtain all fully contracted Wick s theorem terms from the product of the operator string above and the elements of H. 

When applied to the operator strings in this expression, Wick s theorem gives only two nonzero contractions, in spite of the relatively large number of construction operators ... 

Extended normal-order sequence of operator strings 201... 

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 form, the wave operator can operate upon the entire space, i.e. the model space M and its complement HQM. Since, namely, the operator strings represent genuine excitations, the wave operator gives nonvanishing results only when it acts onto states which belong to M. 

In the Goldstone program below [cf. Section 4], Wick s theorem is fulfilled implicitly by performing a pairwise permutation of the operators until all operators are in normal order. This ensures that all contractions are taken into account easily. Of course, this step-wise procedure gives the same results as a more sophisticated implementation of Wick s theorem which, in particular, is helpful for large operator strings. 

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. 

Evaluation of the vacuum amplitudes (46) as they arise by combining the operator strings from step (1) and the Fock space representation a)a and a )p of the model space functions from step (2). As described above, there is no summation associated with the indices n, and Uj in the definition of the model-space functions (43). 

Again, the operator string a a moved an electron initially on orbital i to orbital k. Accordingly, such operator strings are often called shift operators. 

Analyzing the left-hand side, it is seen that i and j should be elements of the set 1,2,..N, otherwise or (j)]" could not annihilate. Therefore, the summation restrictions are the same on both sides of this equation. On the left-hand side, two electrons are annihilated from orbitals ( )i and ( )j by operators ( )j" and but two electrons are created by and It is easy to see that an odd number of transpositions is required to bring the operator string on the left-hand side to the same form as that of the string at the right, thus a minus sign is obtained ... 

Here, the Greek labels refer to an aribtrary orthonormalized basis set. Note that the labels a and X appear in a reverse order in the operator string as compared to that in the integral list. This is not a misprint, but a consequence of the eliminated negative sign in Eq. (4.37). 

Consider first the matrix element of the operator string a between orbitals

... 

To begin with, consider the matrix element of the operator string aj aj between a Hartree-Fock wave function, i.e. determinant, built up from orbitals i 2 Vn- Assume i, k g 1,2,... N. We have then ... 

The appearance of the occupation numbers is the only difference one has to keep in mind when evaluating the expectation value of an operator string with respect to the Fermi vacuum. Checking the occupancies may be too time-consuming however, by introducing a simple trick the whole process can be... 

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... 

Although real orbitals are dealt with, in this section asterisks will be put at the relevant places to indicate complex conjugation, making the formulae a bit more transparent. The above formulae permit us to express our operator string in Eq. 

It is usual to write Eq. (10.4) in a simpler form by realizing that the Hermitian conjugate of the operator string ... 

The operator strings can be reordered applying the fermion anticommutation rules. The result is ... 

We may bring this expression into a more transparent form by appropriate interchanges of summation indices in order to have the operator string x TXv all terms. In particular, this requires a v X interchange in the first term, a v a... 

The expectation value of the operator string sandwiched by the Fermi vacuum can be evaluated in several manner. The simplest way is to realize that l k must annihilate the virtuals s r, and i j must re-create q p". Collecting all possibile pairings we get ... 

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