In second-quantization form

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

Many-electron wave functions in second-quantization form can conveniently be represented in an operator form. To this end, we shall introduce the vacuum state 0), i.e. the state in which there are no particles. We shall define it by... 

The PPP Hamiltonian can then be written in second quantized form as... 

This general notation is deceptively simple. The bra is an excited determinant. There is an equation for each excited determinant, and each level of excitation leads to a different type of equation. Furthermore, the equations are all coupled, and they are non-linear in the amplitudes. However, they may be formulated in a quasilinear manner [27], and they have been solved for a wide range of CC schemes. The operator HN is the Hamiltonian written in second-quantized form minus the energy of the reference determinant, i.e. HN = H— < 0 /7 0 >. The subscript C restricts the operator product of HN and eT to connected terms. Once the CC equations have been solved, the CC correlation energy can be calculated from... 

It is easiest to see this relationship by writing the Hamiltonian in second quantized form ... 

It is more convenient to express the different contribntions in second quantized form. Thns, we have for the electrode and its interaction with the reactant ... 

The first two terms are the molecular Hamiltonian and the radiation field Hamiltonian. The molecular Schrodinger equation for the first term in (5.2) is assumed solved, with known eigenvalues and eigenfunctions. Solutions for the second term in (3.4) in vacuo are taken in second-quantized form. Hint can be taken in minimal-coupling form (5.3) allowing for the variation of the radiation field over the extent of the molecule,... 

We begin by describing Olsen s expressions for the a vector. In second quantized-form (cf. section 2.3.2), H becomes... 

A unique feature of the occupation number representation is that the number of electrons does not appear in the definition of the Hamiltonian operator in this form as it does in the wavefunction form. This is because all of the occupation information resides in the bras and kets. This is true for any operator in second quantized form. This feature is used to advantage in theories that allow the number of particles to change, and to a more limited extent in the calculation of electron affinities and ionization potentials. It is less important to the MCSCF method but it is useful to remember that the bras and kets contain all of the occupation information. Other details of the wavefunction, such as the AO and MO basis set information, are included in the integrals that are used as expansion coefficients in the second quantized representation of the operator. 

The Hamiltonian is assumed to be spin-independent. It can then be written, in second quantized form, in terms of the spin-averaged excitation operators (the generators of the unitary group )... 

These relationships can also be expressed in second-quantized form [4-8] by introducing Fermion creation and annihilation operators, which obey the anticommutation relations... 

We next turn to the question of whether the total spin operator commutes with Wj v In Ref. [8] it is shown that the components of the total spin operator can be expressed in second-quantized form by means of the relationships ... 

Because lO, A ) and lAT, + 1) contain different numbers of electrons, it is convenient and most common in developing EOM theories of EAs to express the electronic Hamiltonian H in second-quantized form [13] ... 

To make further progress, the zero-order Hamiltonian and the perturbation must be written in second quantized form. Recall that the annihilation operator, a and the creation operator, a], satisfy the following anticommutation relations... 

In second quantized form the zero-order Hamiltonian may be written in terms of creation and annihilation field operators in the form... 

The Breit interaction in second-quantized form, written in normal order relative to a closed-shell core is... 

The general Hamiltonian of a molecule interacting with an external field in second quantization form reads "... 

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

Some Model Hamiltonians in Second Quantized Form... 

In second-quantization form, the corresponding operator will be... 

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