Hamiltonian second quantization

Hamiltonian in the second-quantization fomi, only one //appears in this fmal so-called equation of motion (EOM) f//, <7/]+ = AJr 7 p(i e. in the second-quantized fomi, // and //are one and the same). 

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 the second quantization representation, the Hamiltonian Hl describing the motion of the reactive -oscillator in the left potential well has the form... 

The Russian school of ETR (Levich, 1966 Dogonadze, 1971 Vorotyntsev et al, 1970) treats the medium polarization by a second-quantized Hamiltonian, written as... 

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. 

Recent application of the TB method to transition metal clusters often made use of a convenient formulation in the language of second quantization.14 In this formalism, the TB Hamiltonian in the unrestricted Hartree-Fock approximation can be written as a sum of diagonal and nondiagonal terms15... 

The very simplest theoretical approach, with linear electron-phonon coupling, is in terms of a two-center (a,b) one-electron Hamiltonian (27), with just one harmonic mode, u>, associated with each center. This is (in second quantized notation, with H = 1) ... 

The Dirac-Coulomb-Breit Hamiltonian rewritten in second-quantized... 

The next step is that we find inverse transformations to (25-28) and substitute these inverse transformations into eq. (22) and then applying Wick theorem, we requantize the whole Hamiltonian (16) in a new fermions and bosons [14]. This leads to new V-E Hamiltonian (we omit sign on the second quantized operators)... 

In previous part we developed canonical transformation (through normal coordinates) by which we were able to pass from crude adiabatic to adiabatic Hamiltonian. We started with crude adiabatic molecular Hamiltonian on which we applied canonical transformation on second quantized operators... 

Within second quantization [41] the Hamiltonian operator may be expressed as... 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

The reduced matrices and V represent a partitioning of the Hamiltonian into one- and two-electron parts. Rearranging the second-quantized operators and using the definition of the 2- and 3-RDMs,... 

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

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, ... 

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

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... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Consequently, with second quantization, the approach using Hamiltonian (2.1)-(2.7) and relativistic wave functions (2.15) differs from the approach using Hamiltonian (1.16)—(1.22) and the non-relativistic wave... 

Second Quantization Photodissociation Hamiltonian, if we consider a system containing many molecules and fragments, it is convenient to use second quantization formalism. We have introduced above the matrix element for photodissociation (see eqs. 50 and 53-57). Based on it, one can write the total... 

One can introduce the tunneling Hamiltonian in a second quantization formalism (39) which can be written in the form... 

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. 

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,... 

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

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