3: Nuclear Motion
- Page ID
- 60511
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
- This page explains the framework for understanding molecular structure in chemistry, focusing on the separation of electronic, vibrational, and rotational motions. It introduces the full Schrödinger equation and the electronic Hamiltonian \(H_e\). The completeness of electronic wavefunctions is highlighted, enabling the full wavefunction to be expressed in terms of electronic states.
- 3.2: Time Scale Separation
- The Born-Oppenheimer approximation effectively separates electronic and nuclear motions due to their differing time scales, as electrons move faster than nuclei. This enables electronic states to adjust smoothly as nuclei vibrate or rotate. However, in scenarios involving loosely bound electrons, such as molecular Rydberg states or anions, this separation may not hold, potentially compromising the accuracy of the model due to comparable electronic and vibrational frequencies.
- 3.3: Vibration/Rotation States for Each Electronic Surface
- This page explains the Born-Oppenheimer (BO) approximation, focusing on how it affects nuclear motion on potential energy surfaces linked to various electronic states. It notes the variance in vibrational and rotational motions between ground and excited states due to differing chemical bonds.
- 3.4: Rotation and Vibration of Diatomic Molecules
- This page covers the kinetic energy operator for vibration-rotation in diatomic molecules, focusing on the effects of bond length and angles. It separates the wavefunction into angular and radial parts, consolidating quantum numbers to J, M, and v. The page outlines equations for vibrational states, particularly when J=0, where rotational energy is absent, providing a simplified wavefunction.
- 3.5: Separation of Vibration and Rotation
- This page explores the application of perturbation theory to diatomic-molecule rotational and vibrational spectroscopy, emphasizing the expansion of centrifugal coupling around the equilibrium bond length \(R_e\) based on the angular momentum quantum number \(J\).
- 3.6: The Rigid Rotor and Harmonic Oscillator
- This page covers the treatment of rotational and vibrational motion in diatomic molecules through a zeroth-order approximation. It introduces the rigid rotor model, detailing energy levels and wavefunctions tied to rotation, and examines vibrational motion via a harmonic potential approximation, leading to specific energy characteristics.
- 3.7: The Morse Oscillator
- This page discusses the Morse oscillator model, which enhances the harmonic oscillator approximation by factoring in bond dissociation energy and a parameter linked to the second derivative of potential energy. It highlights the anharmonicity of the Morse oscillator's energy levels, noted by a negative term in the energy formula dependent on the quantum number v.
- 3.8: Rotation of Polyatomic Molecules
- This page covers the rotational kinetic energy in non-linear polyatomic molecules, focusing on the moment of inertia tensor. It categorizes molecules into spherical tops and symmetric tops, explaining their rotational Hamiltonians and energy levels. Spherical tops feature a high degeneracy of \((2J+1)^2\), while symmetric tops have lower degeneracies affected by the quantum number \(K\).
- 3.9: Rotation of Linear Molecules
- This page discusses the rotational motion of linear polyatomic molecules, building on concepts from diatomic molecules. It introduces the rotational wavefunctions \(Y_{J,M} (\theta,\phi)\) and defines the energy levels as \(E^0_J = \hbar^2 \dfrac{J(J+1)}{2I}\), where I is the total moment of inertia. The degeneracy of these rotational levels, given by the quantum number J, is (2J+1), indicating the different M-values corresponding to each J level.
- 3.10: Rotation of Non-Linear Molecules
- This page explores the rotational kinetic energy of non-linear polyatomic molecules, emphasizing the moment of inertia tensor. It details rotational energy formulas for spherical top molecules, highlighting degenerate energy levels due to quantum number independence. The discussion extends to symmetric top molecules, clarifying prolate and oblate types based on moment of inertia equality, and specifies energy expressions and degeneracies related to quantum numbers J and K.
- 3.11: Chapter Summary
- This page outlines the decomposition of the Schrödinger equation into two main problems: determining electronic wavefunctions and energies based on nuclear geometry, and analyzing nuclear motion on electronic energy surfaces. This foundational separation is vital for understanding molecular structure and spectroscopy. It emphasizes the need for calculating electronic energy levels for varying nuclear configurations before delving into nuclear motion.