Two-electron atom
In atomic physics, a two-electron atom or helium-like ion is a quantum mechanical system consisting of one nucleus with a charge of Ze and just two electrons. This is the first case of many-electron systems where the Pauli exclusion principle plays a central role.
It is an example of a three-body problem.
The first few two-electron atoms are:
Z=1: | H^{–} | hydrogen anion |
Z=2: | He | helium atom |
Z=3: | Li^{+} | lithium ion |
Z=4: | Be^{2+} | beryllium ion |
Z=5: | B^{3+} | boron ion |
Schrödinger equation
The Schrödinger equation for any two-electron system, such as the neutral Helium atom (He, Z = 2), the negative Hydrogen ion (H^{–}, Z = 1), or the positive Lithium ion (Li^{+}, Z = 3) is:^{[1]} For a more rigorous mathematical derivation of Schrödinger's equation, see also.^{[1]}
where r_{1} is the position of one electron (r_{1} = |r_{1}| is its magnitude), r_{2} is the position of the other electron (r_{2} = |r_{2}| is the magnitude), r_{12} = |r_{12}| is the magnitude of the separation between them given by
μ is again the two-body reduced mass of an electron with respect to the nucleus of mass M, so this time
and Z is the atomic number for the element (not a quantum number).
The cross-term of two laplacians
is known as the mass polarization term, which arises due to the motion of atomic nuclei. The wavefunction is a function of the two electron's positions:
There is no closed form solution for this equation.
Spectrum
The optical spectrum of the two electron atom has two systems of lines. A para system of single lines, and an ortho system of triplets (closely spaced group of three lines). The energy levels in the atom for the single lines are indicated by ^{1}S_{0} ^{1}P_{1} ^{1}D_{2} ^{1}F_{3} etc, and for the triplets, some energy levels are split: ^{3}S_{1} ^{3}P_{2} ^{3}P_{1} ^{3}P_{0} ^{3}D_{3} ^{3}D_{2} ^{3}D_{1} ^{3}F_{4} ^{3}F_{3} ^{3}F_{2}.^{[2]} Alkaline earths and Mercury also have spectra with similar features, due to the two outer valence electrons.^{[2]}
References
- ↑ ^{1.0} ^{1.1} Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
- ↑ ^{2.0} ^{2.1} {{#invoke:citation/CS1|citation |CitationClass=book }}