Matrix elements creation operator

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

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... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

The relation to the density matrix elements in the spin-orbital occupation numbers representation recovers from noticing that the rows of indices of spin-orbitals ki,k2,..., / y = K (defining a row of creation operators. ..a a , forming a basis Slater determinant) can be in the same manner considered as a set of electronic coordinates in the spin-orbital representation as is the list xi, X2,. .., xjv, ... 

Where s (R, annihilation operators for an electron spin transfer integral is a matrix element connecting one-electron functions ... 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

Since the left- and right-hand states may be written simply as single annihilation and creation operators acting on the vacuum, the desired matrix element of A may be rewritten as the vacuum expectation value of a new operator, = a Aal. Therefore, we need only rewrite B in normal order and select only the terms that contain no annihilation or creation operators, as we did in Eq. [77]. After much algebraic manipulation, which we shall omit here, it can be shown that... 

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. 

It may be readily verified that the matrix exp( - iA) is in fact unitary, provided the matrix A is Hermitian. The fact that the matrix exp( — iA) is unitary also means that the operator exp( — iA) is unitary and its matrix representation in the full ket expansion space, with matrix elements , is a unitary matrix. An analogous relation holds for transformations of the electron annihilation operators a, but it is the creation operator expansion that is most important for the MCSCF method. Substitution of the operator transformation into the expression of an arbitrary determinant gives the relation... 

Here k is the momentum of the quasi-free electrons, whose single-particle energies include the effect of electron-electron repulsion renormalization and is the occupation number operator for state k). In Eq. (6.94) the electronic coupling between electrode and redox center is included which is governed by the matrix element between states k> and

cj and are the creation operators for the states in the redox system and the metal, respectively, whereas c and are the corresponding annihilation operators. Creation and annihilation means that an orbital k in the metal becomes occupied by an electron or emptied, respectively here in the presence of the electronic coupling. 

The creation/annihilation operators aj /a, denote the one-particle operators which diagonalize the Hamiltonian Hen. The summation indices i, j, k, l denote the usual set of one-electron quantum numbers and run over positive-energy states only. The quantities Vjju are two-electron Coulomb matrix elements and the quantities biju denote two-electron Breit matrix elements, respectively. We specify their static limit (neglecting any frequency dependence) ... 

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. 

While evaluating the matrix elements of H exp(T ) between cind (j> , it becomes convenient to rewrite H in normal order with respect to d>v as vacuum. This simplification of computation of matrix-elements was noted earlier by Hose and Kaldor [17] and has since been exploited by many workers [14,15,25,26]. Since T" has only excitations out of it has only hole-particle creation operators defined with respect to d>v and consequently exp(T ) is in normal order with respect to (jx. Using Wick s theorem, we then find... 

All other operators can now be expressed as sums of products of matrix elements (complex numbers) with these creation or annihilation operators. The Dirac Hamiltonian becomes... 

It is now useful to analyze the density matrix elements [15] that enter into these equations. Each of the T+( j) operators contains an odd number of creation or annihilation operators, and the Hamiltonian H contains two (i.e. f+y) or four (i.e. such... 

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