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Lecture 20: Variational Method Approximation and Linear Varational Method

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  • Recap of Lecture 19

    Last lecture address three aspects: (1) We continued to discussed the complications of electron-electron repulsions in solving Schrodinger equations of atomic systems and showed the "Ignorance is Bliss" approach is really pretty poor. (2) we discussed a qualitative way to address these repulsions by introducing an effective charge interpreted within a shielding and penetration perspective (this was discussed primarily within the context of radial distribution functions). (3) We motivated variational method by arguing the energy of a trial wavefunction will be lowest when it most likely resembles the true wavefunction (the same for the corresponding energies).

    Back to Variational method

    Because of the electron-electron interaction Schrödinger's equation cannot be solved exactly for the helium atom or more complicated atomic or ionic species. However, the ground state energy of the helium atom can be calculated using approximate methods. One of these is the variation method which requires the minimizing of the following variational integral.

    \[E = \dfrac{\int_0^{\infty} \psi_{trial}^* \hat{H} \psi_{trial} d\tau}{ \int_0^{\infty} \psi_{trial}^2 d\tau} \label{7.3.1a}\]

    or via Dirac's Bra-Ket notion

    \[E_{trial} = \dfrac{\langle \psi_{trial}| \hat{H} | \psi_{trial} \rangle }{\langle \psi_{trial}| \psi_{trial} \rangle} \ge E_{true}\label{7.3.1b}\]

    Equation \(\ref{7.3.1a}\) is call the variational theorem and states that for a time-independent Hamiltonian operator, any trial wave function will have an variational energy expectation value that is greater than or equal to the true ground state wave function corresponding to the given Hamiltonian. Because of this, the variational energy is an upper bound to the true ground state energy of a given molecule. The general approach of this method consists in choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible.

    Simplified algorithmic flowchart of the Variational Method approximation

    As is clear from Equation \(\ref{7.3.1b}\), the variational method approximation requires that a trial wavefunction with one or more adjustable parameters be chosen.

    Applying Variational Method to a Helium Atom

    Conceptually, the Hamiltonian for Helium atom consists of the kinetic energy for electron 1 and electron 2, the electron-nucleus attraction, and the electron-electron repulsion.

    \[\hat{H} = -\dfrac {\hbar^2}{2m_e} \left(\bigtriangledown_1^2 + \bigtriangledown_2^2\right) - \dfrac {Ze^2}{4\pi \epsilon_0} \left(\dfrac {1}{r_1} + \dfrac {1}{r_2} \right) + \dfrac {e^2}{4\pi\epsilon_0 r_{12}}\label{3}\]

    Rearranging this equation gives the Hamiltonian in two one-electron equations and one repulsion term.

    \[\hat{H} = \underbrace{\dfrac{-\hbar^2}{2m_e}\bigtriangledown_1^2 - \dfrac {Ze^2}{4\pi\epsilon_0 r_1}}_{\text{electron 1 in Hydrogen Orbital with Z=2}} + \underbrace { \dfrac {-\hbar^2}{2m_e}\bigtriangledown_2^2 - \dfrac {Ze^2}{4\pi\epsilon_0 r_2}}_{\text{electron 2 in Hydrogen Orbital with Z=2}} + \underbrace{\dfrac {e^2}{4\pi\epsilon_0 r_12}}_{\text{annoying electron-electron repulsion}} \label{4}\]

    We use \(\psi_{H_{1s}}\) as our wave function \(\psi_0\). Then we define our trial wavefunction

    \[\phi(r_1,r_2) = \psi_{H_{1s}}(r_1)\psi_{H_{1s}}(r_2)\]

    and set \(\zeta\) as the adjustable parameter.

    This is sort of like the "Ignorance is Bliss" approximation, but with \(\zeta\) that is varied to handle sheilding and better describe the true wavefunction.

    The general idea is that the electron-electron repulsion has the effect of reducing shielding, the effective positive charge of the nucleus. The energy of this trial wavefunction as function of \(Z_{eff} \approx \zeta\) is evaluated to be

    \[E_{trial}(\zeta) = \dfrac {m_e e^4}{16\pi^2 \epsilon_0^2 \hbar^2} \left(\zeta^2 - \dfrac {27}{8}\zeta \right)\label{5}\]

    Upon taking the derivative with respect to \(\zeta\) setting the derivative equal to \(0\)

    \[ \left. \dfrac{d E_{trial}(\zeta) }{ d \zeta}\right |_{\zeta_{min}} = 0\]

    gives a minima at

    \[\zeta_{min} = \dfrac {27}{16} \approx 1.69.\]

    For an unshielded helium system, the expected \(\zeta\) should have a value of two

    \[ \zeta_{min} = Z = 2\]

    This is the origin of the shielding concept.

    According to the variation principle, the minimum value of the energy on this graph is the best approximation of the true energy of the two-electron helium system, and the associated value of \(\zeta\) is the best value for the adjustable parameter.

    Graph of variationl energies for helium atom as a function of the adjustable parameter \(\zeta\), which represents the effective nuclear charge felt by the electrons.

    Using the mathematical function for the energy of a system, the minimum energy with respect to the adjustable parameter can be found by taking the derivative of the energy with respect to that parameter, setting the resulting expression equal to zero, and solving for the parameter, in this case \(\zeta\).

    How do you know when you are correct (i.e., the trial function looks like the real wavefunction when minimized for energy)?

    You do not from the theory alone, but you can get that information from comparing to experiment.

    When this procedure is carried out for the He atom with \(Z=2\), we find \(\zeta_{min} = 1.6875\) and the approximate energy (Equation \ref{5}) we calculate using this optimized paramter is

    \[E \approx = -77.483\; eV.\]

    Table below show that a substantial improvement in the accuracy of the computed binding energy is obtained by using shielding to account for the electron-electron interaction. Including the effect of electron shielding in the wavefunction reduces the error in the binding energy to about 2%. This idea is very simple, elegant, and significant.

    Comparison of the results of three approximation methods to experiment.
    He binding energy (eV)
    He binding energy (hartrees)
    Neglect repulsion between electrons ("ignorance is bliss" approximation)
    Variational Method with variable \(\zeta\)

    We can choose any trail function we like for Variational method

    The closer the trial function looks like the real wavefunction (Solution to the unsolvable, by us at least, Schrödinger equation), the lower its energy will be.

    • First Trial Wavefunction

    A logical first choice for such a function would be to assume that the electrons in the helium atom occupy two identical, but scaled, hydrogen 1s orbitals.

    \[ \psi (1,2) = \Phi (1) \Phi (2) = \exp\left[- \alpha (r_1 +r_2)\right] \label{7.3.2}\]

    \[E = -2.84766 \;E_h\]

    \[ \left| \dfrac{E(\alpha)-E_{\exp}}{E_{\exp}} \right| = \left| \dfrac{-2.84766 \;E_h + 2.90372 \;E_h}{-2.90372 \;E_h} \right| = 1.93 \%\]

    • Second Trial Wavefunction

    Some electron correlation can be built into the wavefunction by assuming that each electron is in an orbital which is a linear combination of two different and scaled hydrogen 1s orbitals.

    \[\Phi = \exp(- \alpha r_1) + \exp(- \beta r_2) \label{7.3.3}\]

    \[\psi (1,2)= \Phi (1) \Phi (2) \label{7.3.4a}\]

    \[= {\exp(- \alpha r_1 )\exp(- \alpha r_2)}+\exp(- \alpha r_1 )\exp(- \beta r_2)+\exp(- \beta r_1 )\exp(- \alpha r_2 )+\exp(- \beta r_1 )\exp(- \beta r_2 ) \label{7.3.4b}\]

    \[E =-2.86035 \;E_h\]

    Deviation from experimental value:

    \[ \left| \dfrac{E(\alpha)-E_{\exp}}{E_{\exp}} \right| = \left| \dfrac{-2.86035 \;E_h + 2.90372 \;E_h}{-2.90372 \;E_h} \right| = 1.49 \%\]

    • Third Trial Wavefunction

    The extent of electron correlation can be increased further by eliminating the first and last term in the second wavefunction (Equation \(\ref{7.3.4b}\)). This yields a wavefunction of the form,

    \[ \psi (1,2) = \exp(- \alpha r_1 )\exp(- \beta r_2 ) + \exp(- \beta r_1 )\exp(- \alpha r_2 ) \label{7.3.5}\]

    This trial wavefunction places the electrons in different scaled hydrogen 1s orbitals 100% of the time this adds further improvement in the agreement with the literature value of the ground state energy is obtained. This result is within 1% of the actual ground state energy of the helium atom.

    \[E = -2.87566 \;E_h\]

    Deviation from experimental value:

    \[ \left| \dfrac{E(\alpha)-E_{\exp}}{E_{\exp}} \right| = \left| \dfrac{-2.87566 \;E_h + 2.90372 \;E_h}{-2.90372 \;E_h} \right| = 0.97 \%\]

    • Fourth Trial Wavefunction

    The third trial wavefunction, however, still rests on the orbital approximation and, therefore, does not treat electron correlation adequately. Hylleraas took the calculation a step further by introducing electron correlation directly into the first trial wavefunction by adding a term, \(r_{12}\), involving the inter-electron separation.

    \[\psi(1,2) = \left(\exp[- \alpha ( r_1 + r_2 )]\right) \left(1 + \beta r_{12} \right) \label{7.3.6} \]

    Now, if the electrons are far apart, then \(r_{12}\) is large and the magnitude of the wave function increases favors that configuration. This modification of the trial wavefunction has further improved the agreement between theory and experiment to within 0.5%.

    \[E = - 2.89112\; E_h\]

    Deviation from experimental value:

    \[ \left| \dfrac{E(\alpha)-E_{\exp}}{E_{\exp}} \right| = \left| \dfrac{- 2.89112 \;E_h + 2.90372 \;E_h}{-2.90372 \;E_h} \right| = 0.43 \%\]

    • Fifth Trial Wavefunction

    Chandrasakar brought about further improvement by adding Hylleraas's \(r_{12}\) term to the third trial wave function (Equation \(\ref{7.3.5}\)) as shown here.

    \[\psi (1,2) = \left[\exp(- \alpha r_1 )\exp(- \beta r_2 ) + \exp(- \beta r_1 )\exp(- \alpha r_2 ) \right][1 + \gamma r _{12} ] \label{7.3.7}\]

    \[E = -2.90143 \;E_h\]

    Deviation from experimental value is

    \[ \left| \dfrac{E(\alpha)-E_{\exp}}{E_{\exp}} \right| = \left| \dfrac{E(\alpha)--2.90372 \;E_h}{-2.90372 \;E_h} \right| = 0.0789 \%\]