Hydrogen atom kinetic energy

The sum of two operators is an operator. Thus the Hamiltonian operator for the hydrogen atom has — j as the kinetic energy part owing to its single election plus — 1/r as the electiostatic potential energy part, because the charge on the nucleus is Z = 1, the force is atrtactive, and there is one election at a distance r from the nucleus... 

In order to determine the operator, we first write down the classical energy expression in terms of the coordinates and momenta. For the electron in a hydrogen atom, the classical energy is the sum of the kinetic energy and the mutual potential energy of the eleetron and the nucleus (a proton)... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

This is the energy required to remove the electron from the ground state of a hydrogen atom to a state of zero kinetic energy at infinity and is also known as the ionization potential of the hydrogen atom. 

This, at first glance innocuous-looking functional FHK[p] is the holy grail of density functional theory. If it were known exactly we would have solved the Schrodinger equation, not approximately, but exactly. And, since it is a universal functional completely independent of the system at hand, it applies equally well to the hydrogen atom as to gigantic molecules such as, say, DNA FHK[p] contains the functional for the kinetic energy T[p] and that for the electron-electron interaction, Eee[p], The explicit form of both these functionals lies unfortunately completely in the dark. However, from the latter we can extract at least the classical Coulomb part J[p], since that is already well known (recall Section 2.3),... 

There is one more problem which is typical for approximate exchange-correlation functionals. Consider the simple case of a one electron system, such as the hydrogen atom. Clearly, the energy will only depend on the kinetic energy and the external potential due to the nucleus. With only one single electron there is absolutely no electron-electron interaction in such a system. This sounds so trivial that the reader might ask what the point is. But... 

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