Hamiltonian second-quantization formalism

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 ... 

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. 

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... 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

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 second quantization formalism also greatly simplifies the treatment of the Hamiltonian and allows its analysis pertinent to the GF approximation for the wave function.23 Indeed, the total electron Hamiltonian H can be rewritten using the second quantization formalism according to the division of the orbital basis set into carrier subspace basis sets as introduced above ... 

Closely inspecting the operator terms entering the electronic Hamiltonian eq. (1.27) one can easily see that they are sums of equivalent contributions dependent on coordinates of one or two electrons only. Analogously in the second quantization formalism only the products of two and four Fermi operators appear in the Hamiltonian. Inserting the trial. Y-electron wave function of the (ground) state into the expression for the electronic energy yields its expectation value in terms of the expectation values of the one- and two-electron operators ... 

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is comprised of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is ... 

There have been a number of means proposed for circumventing superposition error. Mayer et al. advocated what they term a chemical Hamiltonian approach, which separates the physical part of this operator from that responsible for BSSE using a nonorthogonal second quantization formalism. However, the physical Hamiltonian is no longer variational and the wavefunction is constructed from orthonormalized molecular spin orbitals. Surjan et al. " further developed this approach and performed pilot applications on small complexes. 

Since the spectrum of the DCB Hamiltonian is not bounded from below it is not possible to optimize the wave function by minimization of the energy. The unphysical unboundedness is due to the fact that not all possible normalizible antisymmetric wave functions of N coordinates are states of an N-electron system. The set of possible solutions also contains wavefunctions in which one or more negative energy levels are occupied and it is the mixing with such states that gives rise to unphysical arbitrarily low energies. One needs the second quantization formalism of quantum electrodynamics (QED) for a proper treatment of these states. As this is discussed in more depth elsewhere in this... 

The eigenfunctions of the zeroth order Hamiltonian define the projection of the DCB equation onto the subspace of electronic solutions. This is a first and necessary step to apply QED theory in quantum chemistry. The resulting second quantized formalism is compatible with the non-relativistic spin-orbital formalism if the connection (unbarred spinors <-> alpha-spinorbitals) and (barred spinors beta spinorbitals) is made. This correspondence allows transfer to the relativistic domain of non-relativistic algorithms after the differences between the two formalism are accounted for. 

Another useful feature of the second quantized formalism is that the second quantized representant of the Hamiltonian (and any other physical observable) is independent of the number of electrons, N, in contrast to the first quantized form of the Hamiltonian, Eq. (1.2). Thus, chemical systems containing different numbers of electrons, for example, can be described by one and the same (or very similar) Hamiltonian. 

By means of the first- and second-order density matrices introduced above it is a trivial task to derive the expression of the electronic energy in the Hartree-Fock theory. The goal is simply to evaluate the expectation value of the Hamiltonian H, which in the second quantized formalism is given by Eq. (4.40) ... 

Various levels of NDO methods correspond to second quantized Hamiltonians which differ from each other only in the list of integrals defining the model Hamiltonian. This feature calls the attention to one of the important advantages of the second quantized formalism, namely, that it permits one to put down the model Hamiltonians corresponding to various approximate quantum chemical... 

The expression for the matrix elements F y can be derived in the second quantized formalism in an elegant manner. This derivation relies on the physical picture behind the Hartree-Fock approximation. As known, in this model the electrons interact only in an averaged manner, so correlational effects are excluded. To derive such an averaged operator, we start again with the usual Hamiltonian ... 

It is of some interest to consider whether there is a possibility to find a theoretical formalism permitting one to extend the applicability of the Hellmann-Feynman theorem to such cases. In what follows we shall take advantage of the second quantized formalism to investigate this point. In this framework one has a model Hamiltonian for which quantum-chemical wave functions expanded in a finite orbital basis obey a formal Hellmann-Feynman theorem. 

We shall mention that using the mixed second quantized formalism of Ref. [1], already mentioned, it is possible to present the chemical Hamiltonian (32)-(34) in a form in which each term of the Hamiltonian contains only creation and annihilation operators assigned to the corresponding atom or pair of atoms. To save place, we shall illustrate that only by considering the first term of Eq. (33)— all the other terms can be treated analogously. The first term in question is... 

As it is well known proper many body methods including Feynman diagrammatic techniques, developed in elementary particle physics, were transferred to solid-state physics many years ago. The introduction to quantum chemistry followed later, but only on the electronic level. So the question then appears Is it possible to formulate the full quantum chemical electron-vibrational Hamiltonian in a second quantization formalism The answer is negative. In fact the author did spend many years attempting to construct ideal representations by means of appropriate quasiparticle transformations (cf. equivalent FrOhlich type unitary transformations), but all variants, being either adiabatic- or nonadiabatic representations, did indeed fail. The reason lies actually on a deeper level than one would initially imagine. 

Reconstruction of the Total Hamiltonian in the Second Quantization Formalism... 

The final electron-hypervibrational Hamiltonian in the second quantization formalism has now the form... 

The only way-out insists in the requirement, to take into account the whole electron-vibration-rotation-translational Hamiltonian. It means, that the total Hamiltonian in the crude representation, expressed in the second quantization formalism, has explicitly to contain not only the vibrational energy quanta, but also the rotational and translational ones, which originate from the kinetic secular matrix. 

Wick s theorem (3.109), which gives the prescription for handling products of creation and annihilation operators may, of course, be applied to the Hamiltonian operator, 5f, when it is expressed in the second-quantization formalism, eq. (3.91) ... 

The second-quantization formalism has the advantage that these terms can be calculated easily by making use of diagrammatic techniques. We have already shown that the Hamiltonian... 

A critical step in the development of any perturbation expansion is the division of the Hamiltonian 5f into a zero-order part Jfo and a perturbation JKi. In second quantized formalism, the Hamiltonian for the Mpller-Plesset theory is written as... 

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

In Box 1.1, we summarize the fundamentals of the second-quantization formalism. In Section 1.4, we proceed to discuss the second-quantization representation of standard first-quantization operators such as the electronic Hamiltonian. 

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... 

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