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3: Nuclear Motion

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    The Application of the Schrödinger Equation to the Motions of Electrons and Nuclei in a Molecule Lead to the Chemists' Picture of Electronic Energy Surfaces on Which Vibration and Rotation Occurs and Among Which Transitions Take Place.

    • 3.1: The Born-Oppenheimer Separation of Electronic and Nuclear Motions
      Many elements of chemists' pictures of molecular structure hinge on the point of view that separates the electronic motions from the vibrational/rotational motions and treats couplings between these (approximately) separated motions as 'perturbations'. It is essential to understand the origins and limitations of this separated-motions picture
    • 3.2: Time Scale Separation
      The physical parameters that determine under what circumstances the BO approximation is accurate relate to the motional time scales of the electronic and vibrational/rotational coordinates.
    • 3.3: Vibration/Rotation States for Each Electronic Surface
      The BO picture is what gives rise to the concept of a manifold of potential energy surfaces on which vibrational/rotational motions occur.
    • 3.4: Rotation and Vibration of Diatomic Molecules
      For a diatomic species, the vibration-rotation kinetic energy operator can be expressed in terms of the bond length and the angles that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system. Solutions require the vibrational wavefunction and energy to respond to the presence of the 'centrifugal potential'  to obey the full coupled V/R equations.
    • 3.5: Separation of Vibration and Rotation
      It is common, in developing the working equations of diatomic-molecule rotational/vibrational spectroscopy, to treat the coupling between the two degrees of freedom using perturbation theory.
    • 3.6: The Rigid Rotor and Harmonic Oscillator
      Treatment of the rotational motion at the zeroth-order level described above introduces the so-called 'rigid rotor' energy levels and wavefunctions that arise when the diatomic molecule is treated as a rigid rotor. These harmonic-oscillator solutions predict evenly spaced energy levels. Of course,  molecular vibrations display anharmonicity (i.e., the energy levels move closer together at higher energies) and that quantized vibrational motion ceases once the bond dissociation energy is reached.
    • 3.7: The Morse Oscillator
      The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. The solutions to the Morse solutions display anharmonicity characteristic of true bonds.
    • 3.8: Rotation of Polyatomic Molecules
      For a non-linear polyatomic molecule, again with the centrifugal couplings to the vibrations evaluated at the equilibrium geometry, the following terms form the rotational part of the nuclear-motion kinetic energy. Molecules with all three principal moments of inertia  are equal are called 'spherical tops' and with  two of the three principal moments of inertia are equal are called symmetric top molecules.
    • 3.9: Rotation of Linear Molecules
      To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles. For any non-linear molecule, three angles are needed. Hence the rotational Schrödinger equation for a nonlinear molecule is a differential equation in three-dimensions. There are 3M-6 vibrations of a non-linear molecule containing M atoms; a linear molecule has 3M-5 vibrations. The linear molecule requires two angular coordinates to describe its orientation with respect to a laboratory axis system.
    • 3.E: Exercises
    • 3.10: Rotation of Non-Linear Molecules
      The proper rotational eigenfunctions for the polyatomic molecules are known as 'rotation matrices' and depend on three angles (the three Euler angles needed to describe the orientation of the molecule in space) and three quantum numbers: J, M, and K.
    • 3.11: Chapter Summary
      The solution of the Schrödinger equation governing the motions and interparticle potential energies of the nuclei and electrons of an atom or molecule (or ion) can be decomposed into two distinct problems:  solution of the electronic Schrödinger equation for the electronic wavefunctions and energies, both of which depend on the nuclear geometry and solution of the vibration/rotation Schrödinger equation for the motion of the nuclei on any one of the electronic energy surfaces.

    This page titled 3: Nuclear Motion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jack Simons via source content that was edited to the style and standards of the LibreTexts platform.