Pair creation operators, annihilation

The creation operators in Eqs. [22] and [23] are restricted to act only on the virtual orbitals, and the annihilation operators may act only on the occupied orbitals. Therefore, by Eq. [21], the creation-annihilation operator pairs exactly anticommute ... 

For spin-orbitals, the creation and annihilation operator pair is a generator of the unitary group, and so Eq. (52) can be written... 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

As usual in degenerate time-independent diagrammatic methods,512 27 37 38 excited determinants are represented by external space operators rather than by pair creation-annihilation operators. Diagrammati-cally, an external line is a ray, whereas an internal line is a line segment. The four possible types of rays and line segments that arise in a CCSDT method are shown in Fig. 5. The indexing convention we will use for the rest of the chapter is that u, v, and w will denote external holes i and j internal holes /3, y, and e external particles (i.e., electrons) and a and b internal particles. 

In eq.(28), we have first rewritten qi s in terms of Q,-, and then reordered the product QiQjQkQi as a sum of normal products, using as the vacuum. The traditional Wick s theorem applied to the products of Q, s will lead to pair contractions in the traditional sense between groups of creation-annihilation operators in one Q, with one or more Qy s. If these contractions completely exhaust all the operators present in the composites Q,-, Qj- etc. involved in the contraction, we denote them by bars and centred or filled circles. Joining by some operators will lead to terms with carets and open circles. Thus, the second and the third term in the braces involve incomplete contractions. The second term has connections between operators of Q and Qj and between Qk and Qi. The third term involves connections between Q,- and Q and between Qj and Q(. The fourth term involves complete connections between all the operators of Qi and Qj. The fifth term involves contraction of all the operators of Q,- with those of Qj and of some between Qk and Q . The sixth term involves complete contractions between operators of Q,- and Qj and of Qk and Qi. The seventh term QiQjQkQi indicates that all the operators of Qii Qji Qk and Qi are contracted among themselves which cannot be factored out to pairs such as QiQj QkQi etc. 

To extract the linearly independent excitations, we shall have to use the so-called singular value decomposition of the valence density matrices generated by the creation/annihilation operators with valence-labels which axe present in the particular excitation operator in T. To illustrate this aspect, let us take an example. For any excitation operator containing the destruction of a pair of active orbitals from V o the overlap matrix of all such excited functions factorize, due to our new Wick s theorem, into antisymmetric products of one-body densities with non-valence labels and a two-particle density matrix ... 

The bending vibration of a linear ABC molecule may be treated as a two-dimensional isotropic harmonic oscillator. The v,l) basis set is particularly convenient, where v and l are respectively the vibrational and vibrational angular momentum quantum numbers. The allowed values of l are —v, —v + 2,. .. v. Since there are two quantum numbers, two pairs of creation, annihilation operators are needed to generate all basis states from the v = 0, 1=0) zero-point state. These are, following the notation of Cohen-Tannoudji, et al., (1977), a, ay and at,a9, where... 

Acetylene, H-C=C-H, has two identical HCC two-dimensional isotropic local-bender harmonic oscillators. In the normal mode basis set, the bending basis states are specified by four quantum numbers, v4l4V5l5), where mode 4 is the 7r9-symmetry trans-bend and mode 5 is the ttu-symmetry cis-bend. (The normal mode basis states do not have identical harmonic frequencies, u>5 — w4 = 121 cm-1.) There are four pairs of normal mode creation, annihilation operators, two for mode 4, a. d, a.dd, and a. g, a.ig, and two for mode 5, a, 5d,... 

Then the creation/annihilation operators are transformed into pair of operators qi = (a/ - - a )fy/2 and pi = i a — aj)f /2, which satisfy The total Hamiltonian (in diabatic basis) becomes. 

We now introduce a notation involving a normal product with one or more contracted pairs of factors. If U, V, denote a set of free-field creation and annihilation operators, we define the mixed product by... 

The analytical description of high-frequency line shapes becomes possible in the low-temperature limit, i.e., at n((uK) exp -p/jcoK l, which represents an experimentally important case. In this situation, the Wick coupling for the operators of low-frequency modes in expression (A3.19) involves only the terms in which the annihilation operator is to the left of the creation operator in all but one operator pair. Then Eq. (A3.19) can be written as ... 

Here AB is the difference in ionization potentials of AT and GC base pairs, b is the transfer integral, c, and c, are the creation and annihilation operators for a hole at the f-th site, respectively, index i labels DNA base pairs in the sequence, and the sum E is taken over GC sites only. 

All these formulas for a single pair of creation and annihilation operators obviously apply to a more general situation of dim R pairs. The matrix elements... 

All four creation and annihilation operators for electrons in the pairing state (a,/ ) can be expressed via the tensor in (15.38) at various values of the projections v and m. The anticommutation relations (18.8) and... 

Any products of creation and annihilation operators for electrons in a pairing state can be expanded in terms of irreducible tensors in the space of quasispin and isospin. So, for the operators (18.10) and (15.35) we have, respectively,... 

There is a second, alternative approach. One could assume that the unpaired neutron and the unpaired proton form a quasibound state. The total number of components of the angular momenta of this quasi-bound state is given by n n v. Then we introduce a pair of new bosonic creation and annihilation operators associated with each level of this subsystem, cj, Cj, I,J =... 

We now introduce creation and annihilation operators ajj and an which create/annihilate e-h pairs at a given combination of sites n = (n, n1), i.e., 41°) = 14 = nen h), where 0) is the ground state. Using these operators, a generic monoexcitation configuration interaction Hamiltonian can be formulated as follows in second quantization notation,... 

The narrow stripon band splits in the SC state, through the Bogoliubov transformation, into the EL(ft) and +(k) bands, given in Eq. (20). The states in these bands are created, respectively, by pl(k) and p+(k), which are expressed in terms of creation and annihilation operators of stripons of the two pairing subsets [see Eq. (16)] through equations of the form ... 

We give here a brief account of the quantum dynamics in L-space explicitly for the system described above. The relation between H- and L-space is summarized in Table 1. The creation operator in the //-space becomes a pair of creation superoperators, and similarly for the annihilation operator, in L-space. They are denoted by... 

Next, insert the resolution of the identity between the pairs of creation and annihilation operators in the two-electron term. Clearly, the sum must run over (N — 2)-electron states. The expression for a becomes... 

In the present paper, we shall discuss a method for generating many-electron states of a given symmetry using Kramers pair creation operators and other symmetry-preserving pair creation and annihilation operators. We will first develop the formalism for the case where orthonormality between the orbitals of different configurations can be assumed. Afterwards we will extend the method to cases where this orthonormality is lost, so that the method also can be used in generalized Sturmian calculations [11-13] and in valence bond calculations. 

Thus, the Kramers pair creation and annihilation operators defined by equations (36) and (37) preserve the symmetry of the states on which they act. 

Using the anticommutation relations (32), we can obtain the following commutation relations for the Kramers pair creation and annihilation operators [8] ... 

A contraction (or pairing) of two creation or annihilation operators is defined by... 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

In second quantization, the numerical vector-coupling coefficients (the Aff and ) appear as matrix elements of creation and annihilation operators X] and jc> The operator X creates an electron in an orthonormal spin orbital io), where /(j) = /) (j), and (T = a or p. Similarly, operator destroys an electron in the orthonormal spin orbital ia). In quantum chemistry problems in which the number of particles is conserved, the Xj and will always occur in pairs. The role of these operators is easily illustrated by showing their operation on a specific type of CSF, namely a Slater determinant. Thus, as an example, for the determinant... 

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