Two-electron atom

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]

${\displaystyle E\psi =-\hbar ^{2}\left[{\frac {1}{2\mu }}\left(\nabla _{1}^{2}+\nabla _{2}^{2}\right)+{\frac {1}{M}}\nabla _{1}\cdot \nabla _{2}\right]\psi +{\frac {e^{2}}{4\pi \epsilon _{0}}}\left[{\frac {1}{r_{12}}}-Z\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}\right)\right]\psi }$

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

${\displaystyle |\mathbf {r} _{12}|=|\mathbf {r} _{2}-\mathbf {r} _{1}|\,\!}$

μ is the two-body reduced mass of an electron with respect to the nucleus of mass M

${\displaystyle \mu ={\frac {m_{e}M}{m_{e}+M}}\,\!}$

and Z is the atomic number for the element (not a quantum number).

The cross-term of two laplacians

${\displaystyle {\frac {1}{M}}\nabla _{1}\cdot \nabla _{2}\,\!}$

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:

${\displaystyle \psi =\psi (\mathbf {r} _{1},\mathbf {r} _{2})}$

There is no closed form solution for this equation.

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 ¹S₀ ¹P₁ ¹D₂ ¹F₃ etc., and for the triplets, some energy levels are split: ³S₁ ³P₂ ³P₁ ³P₀ ³D₃ ³D₂ ³D₁ ³F₄ ³F₃ ³F₂.[2] Alkaline earths and mercury also have spectra with similar features, due to the two outer valence electrons.[2]

• Hydrogen-like atom
• Hydrogen molecular ion
• Helium atom
• Lithium atom