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Chemistry LibreTexts

423: Variation Method for the Rydberg Potential

  • Page ID
    135913
  • Approximate Methods: The Rydberg Potential

    For unit mass the Rydberg potential function has the following energy operator in atomic units.

    • Kinetic energy operator: \[- \dfrac{1}{2} \frac{d^{2}}{dx^{2}} \blacksquare\]
    • Potential energy operator: \[ V(x) = 2 -2 (1 + x) exp(-x)\]

    Limits of integration: xmin := -3 xmax := 5 \( \int_{x_{min}}^{x_{max}} \blacksquare ~dx\)

    Display potential energy:

    Screen Shot 2019-02-08 at 12.01.05 PM.png

    Suggested trial wave function:

    \[ \psi (x, \beta ) = ( \frac{2 \beta}{ \pi})^ \frac{1}{4} exp (- \beta x^{2})\]

    \[ \int_{- \infty}^{ \infty} \psi (x, \beta )^{2} dx~assume,~ \beta > 0 \rightarrow 1\]

    Evaluate the variational energy integral.

    \[ E( \beta ) := \int_{x_{min}}^{x_{max}} \psi (x, \beta ) - \frac{1}{2} \frac{d^{2}}{dx^{2}} \psi (x, \beta ) dx + \int_{x_{min}}^{x_{max}} \psi (x, \beta ) V(x) \psi (x, \beta ) dx\]

    Minimize the energy with respect to the variational parameter \( \beta\) and report its optimum value and the ground-state energy.

    β := 1 β := Minimize(E, β) β := 0.86327 E(β) = 0.789456

    Plot the optimum wave function and the potential energy on the same graph.

    Screen Shot 2019-02-08 at 11.57.42 AM.png

    Numerical Solution for the Rydberg Potential

    Compare the variational result to energy obtained by numerically integrating Schrödinger's equation for the Rydberg potential. For all practical purposes the numerical solution can be considered to be exact.

    Numerical integration of Schrödinger's equation:

    Given:

    \[ \frac{-1}{2} \frac{d^{2}}{dx^{2}} \Phi (x) + V(x) \Phi (x) = Energy \Phi (x)\]

    with \( \Phi (x_{min} = 0)\) and \( \Phi '(x_{min} = 0.1\)

    \( \Phi\) = Odesolve(x, x_{max}) \]

    Normalize wave function:

    \[ \Phi (x) := \frac{ \Phi (x)}{ \sqrt{ \int_{x_{min}}^{x_{max}} \Phi (x)^{2} dx}}\]

    Enter energy guess: Energy = 0.64752

    Screen Shot 2019-02-08 at 11.59.07 AM.png

    Compare the variational and numerical solutions for the Morse oscillator by putting them on the same graph.

    Screen Shot 2019-02-08 at 12.00.15 PM.png