Inatomic physics, atwo-electron atom orhelium-like ion is aquantum mechanical system consisting of onenucleus with acharge ofZe and just twoelectrons. This is the first case of many-electron systems where thePauli exclusion principle plays a central role.

It is an example of athree-body problem.

The first few two-electron atoms are:

The Schrödinger equation for any two-electron system, such as the neutralHelium atom (He,Z = 2), the negativeHydrogenion (H⁻,Z = 1), or the positiveLithium ion (Li⁺,Z = 3) is:^[1] For a more rigorous mathematical derivation of the Schrödinger equation, see also.^[1]$${\displaystyle E\psi =-{\hslash }^2\left[\frac{1}{2\mu }\left({\mathrm{\nabla }}_1^2+{\mathrm{\nabla }}_2^2\right)+\frac{1}{M}{\mathrm{\nabla }}_1\cdot {\mathrm{\nabla }}_2\right]\psi +\frac{e^2}{4\pi {\epsilon }_0}\left[\frac{1}{r_{12}}-Z\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\right]\psi }$$wherer₁ 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 massM$${\displaystyle \mu =\frac{m_{\text{e}}M}{m_{\text{e}}+M}}$$andZ is theatomic number for the element (not aquantum number).

The cross-term of two Laplacians$${\displaystyle \frac{1}{M}{\mathrm{\nabla }}_1\cdot {\mathrm{\nabla }}_2}$$is known as themass polarization term, which arises due to the motion ofatomic 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 by¹S₀¹P₁¹D₂¹F₃ etc., and for the triplets, some energy levels are split:³S₁³P₂³P₁³P₀³D₃³D₂³D₁³F₄³F₃³F₂.^[2]Alkaline earths andmercury also have spectra with similar features, due to the two outer valence electrons.^[2]

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

• Atoms
• Quantum models

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