Relativistic Second-Quantized Hamiltonians

There are several points to be noted about this operator. First, the second term creates an electron-positron pair, and the third term annihilates an electron-positron pair. This means that the Hamiltonian connects states with different particle numbers, that is, particle number is not conserved, though charge is. The existence of these terms embodies the idea of an infinitely-many-body problem that arose from the filling of the negative-energy states in Dirac s interpretation. Second, the order of the operators in the fourth term means that the vacuum expectation value of this operator is not zero, but 

Thus the vacuum has an energy that is equal to the sum of the energies of the negative solutions of the Dirac equation, as is expected from Dirac s interpretation. Note that the matrix elements are the same as in the Dirac equation, so the sum is negative and infinite. This Hamiltonian operator therefore represents the first stage of the Dirac reinterpretation with the negative-energy states all filled. 

We write the matrix elements of the one-particle Hamiltonian with a lower case h, regardless of the case of the operator. Upper case H is reversed for many-particle Hamiltonian matrix elements. 

To avoid the negative infinite vacuum energy, the vacuum expectation value is subtracted from the Hamiltonian to define a new, QED Hamiltonian  

The subtraction of the vacuum expectation value is therefore equivalent to permuting all annihilation operators to the right as if all anticommutators had vanished. This is called normal ordering, and operators that are normal-ordered are enclosed in colons  

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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 ... 

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... 

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],... 

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]). 

A rigorous mathematical model for the relativistic electron-positron field in the Hartree-Fock approximation has been recently proposed (Bach et al. 1999). It describes electrons and positrons with the Coulomb interaction in second quantization in an external field using generalized Hartree-Fock states. It is based on the standard QED Hamiltonian neglecting the magnetic interaction A = 0 and is motivated by a physical treatment of this model (Chaix and Iracane 1989 Chaix et al. 1989). 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

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. 

The reader may note that the general theory of molecular properties of section 2 has been developed with very few references to relativity. This is an important point and signals that the methods used in the relativistic domain has essentially the same structure as in the non-relativistic domain. This point can be emphasized further by rewriting the zeroth order Hamiltonian (169) in second quantization [80]... 

An important observation is that the relativistic electronic Hamiltonian has exactly the same generic form (3.1) as the non-relativistic one. This feature becomes perhaps even more manifest when expressing the electronic Hamiltonian on second-quantized form... 

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... 

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