Commutation relations excitation operators

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

The commutator relation (3 8) for the excitation operators together with the adjoint relation (3 9) leads to the following symmetry properties for the coupling coefficients (for real-valued wave functions) ... 

We start with the equation for the orbital gradient (4 9). We then have to compute the commutator between the Hamiltonian (3 24) and an excitation operator E. To do that we use the commutator relation for the excitation operators as given in equation (3 8). The result of the computation is ... 

Consider the excitation operator k (k corresponding to the elementary transition relating k (root state) to a target state k) F,f acting from the right to commutator [H, k)(k ] yields ... 

Linear terms are absent because of the Brillouin theorem. The coefficients Ap p. and Bap p, can be calculated by equating the nonzero matrix elements of the RPA Hamiltonian [Eq. (122)], in the basis of Eq. (121), to the corresponding matrix elements of the exact Hamiltonian [Eq. (23)] in the same basis. From the translational symmetry of the mean field states it follows that the A and B coefficients do not depend on the complete labels P = n, i, K and P = n, /, K1, but only on the sublattice labels /, AT and /, K. The second ingredient of the RPA formalism is that we assume boson commutation relations for the excitation and de-excitation operators (Raich and Etters, 1968 Dunmore, 1972). 

We note that the excitation operator Dt defined by (2.41) is expressed in terms of the linearly independent operators in the sets BT and TBr T, which form a new basis we note that the set BT as before contains exactly p linearly independent elements. It is evident that the coefficients in this basis are no longer necessarily optimal, and we may hence again start out with the double-commutator relations (1.49) and (1.50), etc.,... 

Here n = (n, a) labels the molecules in the crystal, where n is the lattice vector and a enumerates the molecules in a unit cell and / are the creation and annihilation operators of an exciton on molecule n, obeying Pauli commutation relations. Eq is the renormalized excitation energy in the monomer (see Section 3.2), and Mnm is the matrix element of the excitation energy transfer from the molecule m to the molecule n. 

Equations (15.1)—(15.3) are combinations of commutation rules of Fermi type (when s = s ) and of Bose type (when s s ). The appearance of commutation relations for Fermi operators when s = s means that the number of molecular excitations, i.e. the eigenvalue of the operator Pj-fp/, takes either the value 0 (the molecule in the ground state) or 1 (the molecule is excited). In turn, the Bose commutation rules for s A s are related to the fact that the operators with different indices s are acting on different arguments of the crystal wavefunction. 

Each excitation operator in (13.2.5) excites at least one electron from an occupied to a virtual spin orbital. Since there are N electrons in the system, the expansion (13.2.5) terminates after T/v. We further note that the excitation operators given by (13.2.6) and (13.2.7) satisfy the commutation relation (13.1.9) and that the following relationship holds... 

First, the cluster operator may be chosen to include only those excitation operators that conserve the spin projection of the RHF state so as to satisfy the commutation relation... 

The commutator relations (1.8.11) and (1.8.12) may now be used to simplify more complicated commutators. Invoking (1.8.4), we obtain for the commutator between two excitation operators... 

The singlet excitation operators play an important role in the second-quantization treatment of molecular electronic structure. More generally, they are known as the generators of the unitary group, satisfying the same commutation relations... 

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