Hydrogen electron energy levels

The theoretical results for the hydrogen-like atom may be related to experimentally measured spectra. Observed spectral lines arise from transitions of the atom from one electronic energy level to another. The frequency v of any given spectral line is given by the Planck relation... 

Fig. 1. Possible positions of electronic energy levels associated with hydrogen in a semiconductor. (a) normal order (acceptor above donor), allowing possible predominance of any of the charge states H+, H°, H. (b) negative-U order (donor above acceptor), H+ or H always predominant.

Erwin Schrodinger developed an equation to describe the electron in the hydrogen atom as having both wavelike and particle-like behaviour. Solution of the Schrodinger wave equation by application of the so-called quantum mechanics or wave mechanics shows that electronic energy levels within atoms are quantised that is, only certain specific electronic energy levels are allowed. 

Fig. 2-44. Energy balance and electron energy levels in the normal hydrogen electrode reaction Ch- = standard gaseous proton level Estd = standard gaseous electron level.

Fig. 4-22. Electron energy levels of the hydrogen electrode in electron-and-ion transfer equilibrium Hjiju) = gaseous hydrogen molecule on electrode eaj>/H2, u) = gaseous redox electron in equilibrium with the hydrogen reaction, + 2e(H-/H p,) Hp, =...

Fig. 10-26. Energy diagram for a cell of photoelectrolytic decomposition of water consisting of a platinum cathode and an n-type semiconductor anode of strontium titanate of which the Fermi level at the flat band potential is higher than the Fermi level of hydrogen redox reaction (snao > epM+zHj) ) he = electron energy level referred to the normal hydrogen electrode ri = anodic overvoltage (positive) of hole transfer across an n-type anode interface t = cathodic overvoltage (negative) of electron transfer across a metallic cathode interface.

We have used the electronic energy levels for atomic hydrogen to serve as a model for other atoms. In a similar way, we can use the interaction of two hydrogen atoms giving the hydrogen molecule as a model for bonding between other atoms. In its simplest form, we can consider the bond between... 

The agreement between this relation and the observed line spectrum of hydrogen was far too good to be a mere coincidence. While the details of the election orbits employed by Bohr in his calculation may not be a part of the cnrrent paradigm, the concept of electronic energy levels is here to stay. 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

Fig. 2.3. Electronic energy level diagram for the hydrogen atom.

The discrete line spectrum of hydrogen shows that only certain energies are possible that is, the electron energy levels are quantized. In contrast, if any energy level were allowed, the emission spectrum would be continuous. 

Calculate the energy required to excite the hydrogen electron from level n = 1 to level n = 2. Also, calculate the wavelength of light that must be absorbed by a hydrogen atom in its ground state to reach this excited state. 

The rel-HFS and rel-HF computer programs allow calculations of electronic energy levels, ionization potentials, and radii of atoms and ions from hydrogen into the superheavy region. In order to arrive at the oxidation states most hkely to be exhibited by each superheavy element and also the relative stabilities of these various oxidation states, we need to be able to relate these properties to calculable electronic properties. The relationship between reduction potentials and the Born-Haber cycle has offered an effective approach to this problem (69, 70). 

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