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Two-electron atom

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Helium-like quantum systems that have a nucleus and two electrons

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 beryllium ion
Z=5: B 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: For a more rigorous mathematical derivation of Schrödinger's equation, see also. E ψ = − ℏ 2 [ 1 2 μ ( ∇ 1 2 + ∇ 2 2 ) + 1 M ∇ 1 ⋅ ∇ 2 ] ψ + e 2 4 π ε 0 [ 1 r 12 − Z ( 1 r 1 + 1 r 2 ) ] ψ {\displaystyle E\psi =-\hbar ^{2}\left\psi +{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left\psi } where r1 is the position of one electron (r1 = |r1| is its magnitude), r2 is the position of the other electron (r2 = |r2| is the magnitude), r12 = |r12| is the magnitude of the separation between them given by | r 12 | = | r 2 − r 1 | {\displaystyle |\mathbf {r} _{12}|=|\mathbf {r} _{2}-\mathbf {r} _{1}|} μ is the two-body reduced mass of an electron with respect to the nucleus of mass M μ = m e M m e + M {\displaystyle \mu ={\frac {m_{e}M}{m_{e}+M}}} and Z is the atomic number for the element (not a quantum number).

The cross-term of two Laplacians 1 M ∇ 1 ⋅ ∇ 2 {\displaystyle {\frac {1}{M}}\nabla _{1}\cdot \nabla _{2}} 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: ψ = ψ ( r 1 , r 2 ) {\displaystyle \psi =\psi (\mathbf {r} _{1},\mathbf {r} _{2})}

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 S0 P1 D2 F3 etc., and for the triplets, some energy levels are split: S1 P2 P1 P0 D3 D2 D1 F4 F3 F2. Alkaline earths and mercury also have spectra with similar features, due to the two outer valence electrons.

See also

References

  1. ^ Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
  2. ^ Herzberg, Gerhard; J W T Spinks (1944). Atomic Spectra and Atomic Structure (2nd ed.). New York: Dover Publications. p. 75.
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