title: Two-electron atom
text: # 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: For a more rigorous mathematical derivation of Schrödinger's equation, see also.
  
  $E\psi =-\hbar ^{2}\left[{\frac {1}{2\mu }}\left(\nabla _{1}^{2}+\nabla _{2}^{2}\right)+{\frac {1}{M}}\nabla _{1}\cdot \nabla _{2}\right]\psi +{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left[{\frac {1}{r_{12}}}-Z\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}\right)\right]\psi$
  
  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
  
  $|\mathbf {r} _{12}|=|\mathbf {r} _{2}-\mathbf {r} _{1}|$
  
  μ is the two-body reduced mass of an electron with respect to the nucleus of mass M
  
  $\mu ={\frac {m_{e}M}{m_{e}+M}}$
  
  and Z is the atomic number for the element (not a quantum number).
  
  The cross-term of two Laplacians
  
  ${\frac {1}{M}}\nabla _{1}\cdot \nabla _{2}$
  
  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:
  
  $\psi =\psi (\mathbf {r} _{1},\mathbf {r} _{2})$
  
  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 $^1S_0$ $^1P_1$ $^1D_2$ $^1F_3$ etc., and for the triplets, some energy levels are split: $^3S_1$ $^3P_2$ $^3P_1$ $^3P_0$ $^3D_3$ $^3D_2$ $^3D_1$ $^3F_4$ $^3F_3$ $^3F_2.$ Alkaline earths and mercury also have spectra with similar features, due to the two outer valence electrons.
  
  