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22.2.2: ii. Exercises

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    1. In quantum chemistry it is quite common to use combinations of more familiar and easy- to-handle "basis functions" to approximate atomic orbitals. Two common types of basis functions are the Slater type orbitals (STO's) and gaussian type orbitals (GTO's). STO's have the normalized form:
      \[ \left( \dfrac{2\xi}{a_0} \right)^{n+\dfrac{1}{2}} \left( \dfrac{1}{(2n)!} \right)^{\dfrac{1}{2}} r^{n-1} e^{\left( \dfrac{-\xi r}{a_0} \right)} Y_{l,m}\left( \theta , \phi \right), \]
      whereas GTO's have the form:
      \[ N r^1 e^{\left( -\xi r^2 \right)} Y_{l,m} ( \theta ,\phi ). \]
      Orthogonalize (using Löwdin (symmetric) orthogonalization) the following 1s (core), 2s (valence), and 3s (Rydberg) STO's for the Li atom given:
      \[ Li_{1s} \xi = 2.6906 \]
      \[ Li_{2s} \xi = 0.6396 \]
      \[ Li_{3s} \xi = 0.1503. \]

      Express the three resultant orthonormal orbitals as linear combinations of these three normalized STO's.

    2. Calculate the expectation value of r for each of the orthogonalized 1s, 2s, and 3s Li orbitals found in Exercise 1.
    3. Draw a plot of the radial probability density (e.g., \(r^2 [R_{nl}(r)]^2\) with R referring to the radial portion of the STO) versus r for each of the orthonormal Li s orbitals found in Exercise 1.
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