Point nucleus Hamiltonian

The non-relativistic Hamiltonian for an idealized system made up of a point nucleus of infinite mass and charge Ze, surrounded by N electrons of mass m and charge - e is ... 

We start from the Dirac one-particle Hamiltonian for an electron bound to a point nucleus... 

The total energy of a system consisting of a point nucleus with an infinite mass, surrounded by N electrons can be represented by the Hamiltonian (19),... 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

When formulating the Hamiltonian in atomic units, we prefer to have the choice between the original Hartree system (with 6 = 1/c) and an SI-based system or a system based on esu (with 6=1). The latter two lead formally to the same Hamiltonian, and for all three choices the potential of a point nucleus is —Z/r. 

The problem is that the turn-over rule is valid only if the integrand vanishes at the boundaries. This is the case, e.g. for a homogeneous magnetic field, but not for the magnetic field created by a (point) nucleus. In the former case we get the same result as from the Pauli Hamiltonian (128)... 

Combining all the terms, the final form of Ihe Breil-Pauli Hamiltonian for a point nucleus can be written as follows ... 

We now have an apparently well-behaved operator, in which no expansion has been made that is invalid at small r. That is not to say that the magnitude of the perturbation is always small, because we still have V jc as the perturbation, which becomes infinite at r = 0 for a point nucleus. The rate of convergence of the perturbation series will still be related to this feature. However, the fact that the perturbation operator is large in some region of space does not mean that the integrals over the operator are also large in value, as discussed above in relation to the Pauli Hamiltonian. 

This partitioning is valid for all values of the operators provided the inverses exist. For the case we have just considered, that is, A = 2mc — V,B = E with V = —Zjr, the partitioning is valid for all > —2mc. For E < —2mc the left-hand side is always singular for some value of r, as pointed out in chapter 4 however, the partitioning remains valid. As a further note, it can be shown (van Lenthe 1996) that the ZORA Hamiltonian is only bounded below for Z < c, due to the fact that the Dirac equation has no bound solutions for Z > c for a point nucleus. 

The first two terms are the point nucleus ZORA Hamiltonian. In the last term, the expression between the (expectation value of this operator is positive. We can conclude that the energy for a general potential is always greater than for the bare Coulomb potential. Even if Vi is negative somewhere, such as in a negative ion or a polar molecule, the last two terms may still have a positive expectation value and the bound will still be valid. 

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

Moreover, because the nuclei are effectively point charges, it should be obvious that their positions correspond to local maxima in tlie election density (and these maxima are also cusps), so the only issue left to completely specify the Hamiltonian is die assignment of nuclear atomic numbers. It can be shown diat diis information too is available from the density, since for each nucleus A located at an electron density maximum Fa... 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

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