Electronic States and Potential Energy Functions Quiz
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Questions and Answers

What role do the electronic energies, Ee(R), play in the nuclear motion?

  • Represent the potential energy functions experienced by the nuclei (correct)
  • Act as a kinetic energy term in the Schrödinger equation
  • Contribute to the Coulomb repulsion between nuclei
  • Determine the vibrational and rotational motion of the nuclei
  • What does the approximation for the nuclear problem involve?

  • Obtaining rotation-vibration states for each electronic state
  • Solving the Schrödinger equation for nuclear motion with electronic energies (correct)
  • Solving the Schrödinger equation for electronic states
  • Calculating the Coulomb repulsion between fixed nuclei
  • What are En in the context of the electronic states?

  • Energy levels associated with nuclear vibrational and rotational motion (correct)
  • Antibonding orbitals in molecular structures
  • Set of energy levels associated with electron motion
  • Potential energy functions at different nuclear separations
  • Why is the described scenario an approximation according to the text?

    <p>Neglecting electron-nuclei interactions</p> Signup and view all the answers

    What does Term 1 in the full Schrödinger equation represent?

    <p>Potential energy due to electron-nuclei interactions</p> Signup and view all the answers

    Which part of the equation is responsible for accounting for kinetic energy contributions?

    <p>Ĥnχn(R) = T̂n + Ee(R) χn(R)</p> Signup and view all the answers

    What does the 1 ± SAB term in Ee± ensure in the context of bonding orbitals?

    <p>Antibonding orbitals are more antibonding than bonding orbitals are bonding.</p> Signup and view all the answers

    How does increasing the number of basis functions in Ψe affect the LCAO approximation results?

    <p>It improves the accuracy towards the exact result.</p> Signup and view all the answers

    What is a suggested way to improve basis functions in quantum mechanics?

    <p>Optimizing ξ to minimize the energy.</p> Signup and view all the answers

    In the context of electronic states, what does the overlap density V(R) provide a qualitative guide to?

    <p>The nature of the binding and order of electronic states.</p> Signup and view all the answers

    What is a significant advantage of using the variation principle in quantum mechanics?

    <p>Improving accuracy by optimizing coefficients to minimize energy.</p> Signup and view all the answers

    How do atomic orbitals get labeled according to quantum numbers?

    <p>Using principal and orbital angular momentum quantum numbers.</p> Signup and view all the answers

    What is the purpose of using the Born-Oppenheimer approximation in quantum mechanics?

    <p>To consider the motion of electrons and nuclei separately.</p> Signup and view all the answers

    Which wavefunction describes the motion of the nuclei in the context of the given text?

    <p>χn(R)</p> Signup and view all the answers

    In the context of quantum mechanics, what does the electronic Hamiltonian Ĥe represent?

    <p>A Hamiltonian for stationary nuclei.</p> Signup and view all the answers

    What are adiabatic electronic states a result of in quantum mechanics?

    <p>Born-Oppenheimer approximation.</p> Signup and view all the answers

    Which equation focuses on the electronic motion at fixed positions of the nuclei?

    <p>$Ĥe Ψe (r|R) = Ee (R)Ψe(r|R)$</p> Signup and view all the answers

    In quantum mechanics, what factor allows for a separable product form of the total wavefunction?

    <p>Adiabatic separation</p> Signup and view all the answers

    Study Notes

    Schrödinger Equation for H2

    • The Schrödinger equation for H2 in the centre-of-mass (CM) frame is: Ĥ = T̂e + T̂n + V̂en + V̂nn + V̂ee
    • T̂e is the electronic kinetic energy, T̂n is the nuclear kinetic energy, V̂en is the electron-nucleus potential, V̂nn is the nucleus-nucleus potential, and V̂ee is the electron-electron potential

    Born-Oppenheimer Approximation

    • The approximation assumes that electrons move much faster than nuclei and adapt quickly to changes in nuclear motion
    • The total wavefunction is written as a separable product: Ψ(r, R) = χn(R)Ψe(r|R)
    • χn(R) is the wavefunction describing the motion of the nuclei, and Ψe(r|R) is the electronic wavefunction (dependent on nuclear coordinates)

    Electronic Hamiltonian and Adiabatic Separation

    • The electronic Hamiltonian is: Ĥe = Ĥ - T̂n
    • Solution of the simplified Schrödinger equation for electronic motion at fixed R yields a set of electronic states with energies Ee(R) that depend on nuclear positions
    • Ee(R) represents the potential energy functions (curves) experienced by the nuclei at a given separation, R

    Nuclear Problem

    • For each electronic state, the Schrödinger equation for nuclear motion is: Ĥnχn(R) = T̂n + Ee(R) χn(R)
    • The electronic energies, Ee(R), play the role of the potential energy, V(R), experienced by the nuclei
    • Solution of the equation yields a set of energy levels, En, associated with vibrational and rotational motion of nuclei in a specific electronic state

    Potential Energy Curves

    • Each electronic state is described by a different potential energy curve, with its own set of rotation-vibration states
    • The potential energy curves, V(R), include the Coulomb repulsion between the (fixed) nuclei

    Limitations of the Born-Oppenheimer Approximation

    • The approximation is not exact, as the full Schrödinger equation is not separable
    • The approximation ignores the nuclear kinetic energy operator acting on the electronic wavefunction

    Linear Combination of Atomic Orbitals (LCAO) Approximation

    • The LCAO approximation is used to simplify the electronic problem
    • The approximation has an error of ~8% in Re and ~46% in De for the ground electronic state of H2+
    • The accuracy can be improved by increasing the number of basis functions and optimizing the coefficients using the variation principle

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    Description

    Test your understanding of electronic states, potential energy functions, and the Schrödinger equation for nuclear motion. Explore how nuclei experience potential energy curves and the Coulomb repulsion between fixed nuclei.

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