Many-particle Hamiltonian Coulombic

The Coulombic Many-Particle Hamiltonian.- For a system of atomic nuclei and electrons under the influence of electrostatic forces, the many-particle Hamiltonian takes the form ... 

The simplest example of a complex symmetric operator is the non-relativistic many-particle Hamiltonian H for an atomic, molecular, or solid-state system, which consists essentially of the kinetic energy of the particles and their mutual Coulomb interaction. Since such a Hamiltonian is both self-adjoint and real, one obtains... 

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

The many-electron Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian is neither gauge invariant — it does not even contain vector potentials — nor Lorentz covariant as these symmetries have explicitly been broken in section 8.1. Moreover, the first-quantized relativistic many-particle Hamiltonians suffer from serious conceptual problems [217], which are solely related to the unbounded spectrum of the one-electron Dirac Hamiltonian h. ... 

A chemical molecule, by contrast consists of many particles. In the most general case N independent constituent electrons and nuclei generate a molecular Hamiltonian as the sum over N kinetic energy operators. The common wave function encodes all information pertaining to the system. In order to constitute a molecule in any but a formal sense it is necessary for the set of particles to stay confined to a common region of space-time. The effect is the same as on the single confined particle. Their behaviour becomes more structured and interactions between individual particles occur. Each interaction generates a Coulombic term in the molecular Hamiltonian. The effect of these terms are the same as of potential barriers and wells that modify the boundary conditions. The wave function stays the same, only some specific solutions become disallowed by the boundary conditions imposed by the environment. 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

Due to the Coulomb repulsion between the electrons, this hamiltonian is obviously not a sum of single particle hamiltonians Hj. In principle, this means that the total wave function depends on the positions of aU the electrons, the coordinates of which are all correlated. For a system with many... 

To apply the Dirac theory to the many-particle system the one-particle Dirac operator (8.36) is augmented by the Coulomb or Coulomb-Breit operator as the two-particle term, to produce the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from the quantum electrodynamics [496 98] ... 

An assembly of nuclei and electrons could be described very accurately within QED. There would still just be a cluster of particles, our molecule, and any structure would have to arise out of the dynamics of the system. For reasons pointed out earlier, QED—if viable at all— would be a very expensive path to calculation of the electronic stracture and chemical properties of molecules. For electrons, we circumvented this problem by going to a many-particle treatment based on the Dirac equation, as discussed in chapter 5, and we could presumably do the same here for our cluster of electrons and nuclei. In doing this, we choose a Hamiltonian description of the system, but alternative approaches based on a Lagrangian formalism are also possible. In this process we draw a formal distinction between the molecule and the electromagnetic field, which leaves us with the normal Coulomb interactions between the particles in the molecule and the radiation field as an entity external to the molecule. 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spin-orbit interaction is already included in the one-electron Dirac Hamiltonian but the two-electron interaction should also include interactions classically known as spin-other-orbit, spin-spin etc... Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 emd we just concentrate on some practical considerations. 

We have encountered a variety of techniques (Cl, lEPA, CCA, CEP A) for calculating the correlation energy of a many-electron system, and in Chapter 6 we will discuss still another approach based on perturbation theory. The complexity of these formalisms and of the many-electron problem itself is the result of the two-particle nature of the coulomb repulsion between electrons. If the Hamiltonian contained only single particle interactions, there would be no need for sophisticated many-electron theories since we could solve the problem exactly simply by diagonalizing the Hamiltonian in a basis of one-electron functions (i.e., the orbital picture would be exact). Nevertheless, it is instructive to apply the formalism of many-electron theories to an N-electron problem described by a Hamiltonian that contains... 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

Relativistic Hamiltonian for many-electron systems empoying a sum of one-particle Dirac operators and the Coulomb and Breit operators for the electron interaction. 

Many-electron atoms present a more complicated picture because of the electron-electron interaction terms that enter the Hamiltonians for the Schrodinger equations. These interaction terms are the Coulombic repulsive interaction of like charged particles. This couples the motions of the different electrons, and that precludes separation of variables. Nonetheless, important understanding of the features of the wavefunctions of many-electron atoms can be recognized on the basis of the one-electron atoms we have considered so far. 

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