Hamiltonian second-quantized 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] ... 

In (4.28) and (4.30), we have achieved our aim of expressing the Hamiltonian in the appropriate second quantized form for acting on the state vectors in Fock space. 

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, ... 

An approach to constructing CSFs and matrix elements of the Hamiltonian that initially appears quite different from the symmetric group approach can be developed by considering the second-quantized form of the Hamiltonian. If we have an orthonormal... 

This hamiltonian has cylindrical symmetry and may be used to introduce trigonal or tetragonal anisotropy, depending on whether the principal z axis is oriented along a C3 or C4 symmetry axis. The second-quantized form of the intra-r29 part of this operator is given in Eq. 39. 

Finally, in the second quantized form the tight-binding Hamiltonian is... 

For applications the tunneling Hamiltonian (54) should be formulated in the second quantized form. We introduce creation and annihilation Schrodinger operators c fc, cRk, (hq, cRq. Using the usual rules we obtain... 

These moves allow us to write the electronic Hamiltonian in the second quantized form with respect to the basis of (spin-)orbitals ok (x) introduced above ... 

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

The one- and two-electron integrals appearing in the second-quantized form of the Hamiltonian carry all information about the specific features of the quantum system. The one-electron integrals are defined 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... 

The formulation of the relativistic CASPT2 method is almost the same as the nonrelativistic CASPT2 in the second quantized form. In this section, firstly we express the relativistic Hamiltonian in the second quantized form, and then, we give a summary of the CASPT2 method [11, 12],... 

The total electronic Hamiltonian (6.1) is rewritten in the second quantized form... 

We can thus conveniently express the standard PPP Hamiltonian in a modified (particle number preserving) second quantized form... 

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

In the above equations, hpv are the usual one-electron integrals while [juv Ao] and [juA vo] are the standard bare and antisymmetrized two-electron integrals, respectively. To derive these formulae, one has merely to substitute the second quantized form of the total Hamiltonian and apply the above rules for the density matrix elements. The analogy of Eq. (27) to the corresponding HF formula is obvious. 

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,... 

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 )... 

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] ... 

Hamiltonian in the second-quantization form, only one //appears in this final so-called equation of motion (EOM) [ //, = AEQ P(i e. in the second-quantized form, // and //are one and the same). 

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... 

For reasons that will become clear in Section V, the operators and D in (2.95) have basically the same physical meaning as the coordinate and momentum operators. The final result of this replacement is that one can write the second-quantized form of the spherical invariant Hamiltonian operator in terms of U(4) coupled boson operators (up to two-body interactions and with the compact notation h j = b =pI-3> k = 2, 3,... 

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. 

The Second Quantized Form of the Born-Oppenheimer Hamiltonian... 

In Sect. 4.1 we have learned how to convert the one-electron part into the second quantized form [cf. Eq. (4.27)]. Similiary, the representation of the two-electron part is given by Eq. (4.38) of Sect. 4.2. Thus, the Hamiltonian in the second quantized form can be written down immediately as ... 

It is important to observe that the spin labels are not eliminated from the second quantized form of the Hamiltonian. They do not appear in the list of the integrals, however, which corresponds to the fact that the first quantized Hamiltonian is spin-independent and permits one to use the spin-free formalism. But it is essential to realize that creation and annihilation operators cannot be specified merely for spatial orbitals. 

Some Model Hamiltonians in Second Quantized Form... 

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