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9.5: Evaluating the Overlap Integral (Optional)

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    Learning Objectives
    • Demonstrate how to use elliptical coordinates to analytically solve the overlap integral for \(\ce{H2^{+}}\) as a function of intermolecular distance.

    The overlap integral is critical to finding the energy in the LCAO-MO approach. For the ground state of the \(\ce{H_2^{+}} \) molecule we stated the end result was:

    \[\begin{align} S(R) &= \left \langle 1s_A | 1s_B \right \rangle \nonumber \\[4pt] &= \int 1s_A 1s_B d\tau \\[4pt] &= e^{-R/a_o} \left( 1 +\dfrac{R}{a_o} + \dfrac{R^2}{3a_o^2} \right) \label{eq1} \end{align} \]

    The difficulty in evaluating this integral is a result of the separate spherical coordinate system we use to describe each atom. Thus when we combine the two atoms to make a molecule we end up with three different distances:

    • the internuclear separation \(R\) and
    • the electron position relative to nucleus \( r_a\)
    • the electron position relative to nucleus \( r_b\)

    and four angles \(\phi_A\), \(\theta_A\), \(\phi_B\), and \(\theta_B\) two from each nucleus.

    Figure \(\PageIndex{1}\): Definitions of distances in the \(\ce{H2+}\) molecular ion. (CC BY-NC; Ümit Kaya via LibreTexts)

    There are two ways to address the coordinate system of the overlap integral and the approach discussed below is to change from two spherical coordinates into a single elliptic coordinate system. The natural coordinates for a two-center system like \(H_2\) are elliptical and provides the easiest path to evaluating the integral, however many students are not aware of this method because elliptical coordinates are rarely taught in other math and physics classes (Figure \(\PageIndex{1}\)).

    Figure \(\PageIndex{1}\): Grid for elliptical coordinates. Elliptical grid lines are in even steps of \(μ\) of 0.1 units. Hyperbolic grid lines are in even steps of \(\pi/12\) for the interval [0,2pi]. (CC BY-SA 4.0; SharkD via Wikipedia).

    Two-Center Elliptical Coordinates

    In the two-center elliptical system the any point \(P\) can be expressed by three coordinates \(\lambda, \mu, \phi\) where:

    • \(1\leq \lambda < \infty\)
    • \(-1 \leq \mu \leq 1 \)
    • \(0 \leq \theta \leq 2\pi\)

    With this system

    \[\lambda = \dfrac{r_a + r_b}{R} \label{eq10} \]

    \[\mu = \dfrac{r_a - r_b}{R} \nonumber \]

    and the Jacobian (i.e., volume element) is:

    \[d\tau = \dfrac{R^3}{8}(\lambda^2 - \mu^2) \, d\lambda\, d\mu\, d\theta \nonumber \]

    Back to the definition of the overlap integral \(S\)

    \[S = \int 1s_A 1s_B d\tau \nonumber \]

    The \(1s\) wavefunction is

    \[\psi_{100} = \dfrac {1}{\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} e^{-Zr/a_0} \nonumber \]

    Putting the \(1s_A\) and \(1s_B\) wave functions into the overlap integral still in two sets spherical coordinates we have:

    \[\begin{align*} S &= \int 1s_A 1s_B d\tau\ \\[4pt] &= \int \left( \dfrac {1}{\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} e^{-Zr_a/a_0}\right) \left( \dfrac {1}{\sqrt {\pi}} \left(\dfrac {Z}{a_0}\right)^{\frac {3}{2}} e^{-Zr_b/a_0}\right)  d\tau \end{align*} \]

    Still in two sets of spherical coordinates:

    \[\begin{align} S &= \int 1s_A 1s_B d\tau \nonumber \\[4pt] &= \dfrac {Z^3}{a_0^3\pi} \int e^{-Zr_a/a_0} e^{-Zr_b/a_0} d \tau \nonumber \\[4pt] &= \dfrac {Z^3}{a_0^3\pi} \int e^{\frac{-Z(r_a + r_b)}{a_0}} \label{eqS}d \tau \end{align} \]

    Notice that we only have the \(r_a\) and \(r_b\) coordinates represented. The \(R\) distance in the integral is not present.

    Change Coordinates

    From our definition in Equation \ref{eq10}

    \[R\lambda = r_a + r_b \nonumber \]

    Now expand Equation \ref{eqS}

    \[ \begin{align*} S &= \int 1s_A 1s_B d\tau \\[4pt] &= \dfrac {Z^3}{a_0^3\pi} \int e^{\frac{-RZ\lambda}{a_0}} d \tau \end{align*} \]

    Look how \(R\) becomes part of the integral naturally when the coordinate system changes. There is no need to convert the spherical angles \(\phi_A\), \(\theta_A\), \(\phi_B\), and \(\theta_B\) into elliptical coordinates because they are not present in the original spherical integral, or in the wavefunctions.

    Put in the Jacobian and:

    \[\begin{align} S &= \int 1s_A 1s_B d\tau\ \\[4pt] &= \dfrac {Z^3}{a_0^3\pi} \dfrac{R^3}{8} \int_1^{\infty} \int_{-1}^1 \int_0^{2\pi} e^{\frac{-RZ\lambda}{a_0}} (\lambda^2 - \mu^2) \, d\lambda\, d\mu\, d\theta \\[4pt] & = \dfrac {Z^3}{a_0^3\pi} \dfrac{R^3}{8} \int_0^{2\pi} d\theta \left[ \int_1^{\infty} \int_{-1}^1 \lambda^2 e^{\frac{-RZ\lambda}{a_0}} \, d\lambda\, d\mu - \int_1^{\infty} \int_{-1}^1 \mu^2 e^{\frac{-RZ\lambda}{a_0}} \, d\lambda\, d\mu \right] \end{align} \nonumber \]

    Lets simplify this a little bit by noting that \(Z=1\) for hydrogen and evaluating the angular part.

    \[S = \int 1s_A 1s_B d\tau\ = \dfrac {1}{a_0^3} \dfrac{R^3}{4} \left[ \int_1^{\infty} \int_{-1}^1 \lambda^2 e^{\frac{-R\lambda}{a_0}} \, d\lambda\, d\mu - \int_1^{\infty} \int_{-1}^1 \mu^2 e^{\frac{-R\lambda}{a_0}} \, d\lambda\, d\mu \right] \nonumber \]

    Next the \(d\mu\) integrals

    \[S = \int 1s_A 1s_B d\tau\ = \dfrac {1}{a_0^3} \dfrac{R^3}{4} \left[ 2 \int_1^{\infty} \lambda^2 e^{\frac{-R\lambda}{a_0}} d\lambda - \dfrac{2}{3} \int_1^{\infty} e^{\frac{-R\lambda}{a_0}} d\lambda \right] \nonumber \]

    To finish we need the excellent integral

    \[\int_t^{\infty} z^n e^{-bz} = \dfrac{n!}{b^{n+1}} \left( 1 + bt + \dfrac{b^2t^2}{2!} + .... \dfrac{b^nt^n}{n!} \right) e^{-R/a_0} \nonumber \]

    where \(n = 0,1,2,3,4,... \) and \(b>0\).

    In our case \(n = 0 \) and \(2\) with \(t = 1\) and \(b = R/a_0\)

    \[S = \int 1s_A 1s_B d\tau\ = \dfrac {1}{a_0^3} \dfrac{R^3}{4} \left[ 2 \dfrac{2a_0^3}{R^3} e^{-R/a_0}\left(1 + \dfrac{R}{a_0} + \dfrac{R^2}{2a^2_0} \right) - \dfrac{2}{3} \dfrac{a_0}{R} e^{-R/a_0} \left( 1 \right) \right] \nonumber \]

    Distribute the constant out front

    \[S = \int 1s_A 1s_B d\tau\ = \left[ e^{-R/a_0} \left(1 + \dfrac{R}{a_0} + \dfrac{R^2}{2a^2_0} \right) - \dfrac {R^2}{6a_0^2} e^{-R/a_0} \right] \nonumber \]

    Pull out the exponent and ....

    \[S = \int 1s_A 1s_B d\tau\ = e^{-R/a_0}\left[ 1 + \dfrac{R}{a_0} + \dfrac{R^2}{2a^2_0} - \dfrac {R^2}{6a_0^2} \right] \nonumber \]

    And the answer is as expected (Equation \ref{eq1})

    \[S(R) = \int 1s_A 1s_B d\tau\ = e^{-R/a_0}\left[ 1 + \dfrac{R}{a_0} + \dfrac{R^2}{3a^2_0} \right] \nonumber \]


    • Recursion formulae for calculation of overlap integrals, January 1987. 10.1002/jcc.540080102
    • Formulas and Numerical Tables for Overlap Integrals R. S. Mulliken, C. A. Rieke, D. Orloff and H. Orloff, J. Chem. Phys. 17, 1248 (1949);

    9.5: Evaluating the Overlap Integral (Optional) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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