Skip to main content
Chemistry LibreTexts

Solutions 5

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)


    Basis sets:

    Orbital Type a) Generic Formula c) Limitations
    Atomic orbitals \(R_n(r)Y^m_l(\theta,\phi)\) Accurate for only one electron system
    Slater Type Orbitals \(\frac{ (2 \zeta)^{n+\frac{1}{2}}}{[(2n)!]^{\frac{1}{2}}}r^{n-1} exp(-\zeta r)Y^m_l(\theta , \phi)\) Integrals are computationally expensive
    Gaussian orbitals \(N_nr^{n-1}exp(-\alpha r^2)Y^m_l(\theta,\phi)\) Less accurate for shorter distances

    b). Several examples exist, specific to combinations for quantum numbers n, l and m.


    The similarities of the spatial parts to each wavefunction is that they decrease for longer distances. Both the AOs and STOs decrease exponentially whereas the Gaussian decreases even faster. One major difference between the AO and STO is that only AOs have nodes.


    The \(\zeta\) term exists to correctly model the decrease in the STO with increasing distance to match those tof he AOs.


    The resulting solutions matter more than the specific basis set used. The end results are what we compare directly to experiment. The only other metric considered other than accuracy is computation time.

    Solutions 5 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?