Wick s theorem

Since DN(UV-- -XY)Z = N(DUV- - -XY)Z, the theorem is proved for n + 1 factors. This lemma can be generalized by multiplying both sides of Eq. (10-196) by an arbitrary number of contracted factors, and using Eq. (10-195) to bring these factors within the N products. Wick s theorem now states that a T product can be decomposed into a unique sum of normal products as follows ... 

The application of Wick s theorem to the above expansion indicates that it consists of a sum of contributions that can be represented diagrammatically in the form shown in fig. 10-2. 

As a specific application of Wick s theorem and the above contraction... 

The contractions that are encountered in applying Wick s theorem in the computation of the -matrix are given by... 

Without going into details we just remark that Eq. (2.45) can be further simplified by applying Wick s theorem to the electronic sector, utilizing the KS propagator (2.25). Taking into account the explicit form (2.8) for it is then possible to eliminate a further class of diagrammatic contributions (the interested reader is referred to [19] for details). Eq. (2.45), which provides an exact representation of terms of the KS... 

This result represents the most important advantage of the particle-hole formalism. Many-body perturbation theory (MBPT) consists mainly in the evaluation of expectation values (with respect to the physical vacuum) of products of excitation operators. This is easily done by means of Wick s theorem in the particle-hole formalism. 

To proceed further, we have to know how to handle the products of creation and annihilation operators. It is Wick s theorem which tells us how to deal with the products of these operators. Before presenting Wick s theorem we have to introduce some necessary definitions and relations. The creation and annihilation operators satisfy the anticommutation relation... 

Now we are ready to formulate the time independent Wick s theorem ... 

Expressed in words, the product of creation and annihilation operators is equal to a normal product of these operators plus the sum of normal products with one contraction plus the sum of normal products with two contractions etc., up to the normal product where all operators are contracted. This theorem can be generalized (to the so-called generalized Wick s theorem) in the way that also a product of the following form can be handled ... 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

We already know that in order to calculate expression (74) using the generalized Wick s theorem, we have to perform contractions between the -products on the right hand side of Eq. (74). We also know that according to Eq. (37) all the operators have to be contracted. Here the diagrams can be introduced because these contractions can be represented diagrammatically. 

According to the generalized Wick s theorem, we can form new diagrams ( -diagrams in notation of Ref.48)) on the left hand side of Eq. (126) that connect the H-and T-diagrams. We can distinguish two types of R-diagrams ... 

It is apparent that different types of electrons should be averaged separately. According to Wick s theorem [1,4,6-8], the averages of the multiple products of the operators in Eq. (60) can be decoupled into the product of Green s functions (paired operators). To find a Dyson equation, we regroup the infinite sums in the following manner ... 

The element (p apaq asar]v) may be written as a vacuum expectation value of a string of creation and annihilation operators such as vacuum expectation values can be expressed as a sum over all totally contracted terms, each of which depends only on the overlap between the orbitals. Since these are 5pq for orthonormal orbitals, we see that the vacuum... 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

Lindgren/71/ confined himself to the particular N —valence model space, and consequently the cluster amplitudes of various valence ranks are not necessarily linearly independent. This situation is entirely analogous to the one discussed in Sec. 7.1. Using Wick s theorem, eq. (7.3.6) with eq. (7.3.5), leads to... 

Wick s theorem + provides a recipe by which an arbitrary string of annihilation and creation operators, ABC XYZ, may be written as a linear combination of normal-ordered strings. Schematically, Wick s theorem is... 

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... 

In many-electron theories such as configuration interaction or coupled cluster theory, it is more convenient to deal with the -electron reference determinant, IOq), rather than the true vacuum state, I ). In the evaluation of matrix elements using Wick s theorem as described above, even the use of normal-ordered strings would be tremendously tedious if one had to include the complete set of operators required to generate ld>o) from the true vacuum (i.e., lOo) = aUjat I )). 

This new definition of normal ordering changes our analysis of the Wick s theorem contractions only slightly. Whereas before, the only nonzero pairwise contraction required the annihilation operator to be to the left of the creation operator (cf. Eq. [84]), now the only nonzero contractions place the q -particle operator to the left of the -particle creation operator. There are only two ways this can occur, namely. 

The concepts of normal ordering and Wick s theorem provide the mathematical tools needed to derive programmable coupled cluster equations from the more formal expressions given in Eqs. [50] and [51]. If we truncate the cluster operator such that T = Tj + T2 insert it into the similarity-transformed normal-ordered Hamiltonian, H = e lij e, we obtain... 

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