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nist-gov/www.nist.gov/physical-measurement-laboratory/methane-symmetry-operations
Scott Williams 4063bef310 Initital commit.
2025-03-05 18:59:57 +00:00

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<p><a href="https://www.nist.gov/pml/methane-symmetry-operations-version-history">Version History</a></p><p><strong>J. T. Hougen</strong><br>Optical Technology Division<br><br><strong>Gloria Wiersma, HTML conversion</strong><br>Office of Electronic Commerce in Scientific and Engineering Data<br>National Institute of Standards and Technology<br>Gaithersburg, Maryland USA 20899<br><br><strong>Abstract</strong><br>Catalog of explicit symmetry operations for the vibrational, rotational and nuclear spin functions of methane, and the use of these operations in determining symmetry labels and selection rules.</p><h3>Table of Contents</h3><ol><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-introduction"><strong>Introduction</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-geometric-operations"><strong>Familiar Point Group Ideas</strong></a><br>2.1&nbsp;<a href="https://www.nist.gov/pml/methane-symmetry-operations-geometric-operations">Geometric operations</a><br>2.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-algebraic-operations">Algebraic operations</a><br>2.3 <a href="https://www.nist.gov/pml/methane-symmetry-operations-representation-matrices">Representation matrices</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-permutation-inversion"><strong>Permutation-Inversion Group Ideas</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-relation-point-group-operations-permutation-inversion-group"><strong>Relation of Point-Group Operations to Permutation-Inversion Group Operations</strong></a><br>4.1<a href="https://www.nist.gov/pml/methane-symmetry-operations-relation-point-group-operations-permutation-inversion-group"> Algebraic equation relating laboratory-fixed Cartesian coordinates to rotational angles and vibrational displacement vectors</a><br>4.2 <a href="https://www.nist.gov/pml/proper-rotations">Proper rotations</a><br>4.3 <a href="https://www.nist.gov/pml/methane-symmetry-operations-improper-rotations">Improper rotations</a><br>4.4 <a href="https://www.nist.gov/pml/methane-symmetry-operations-explicit-eulerian-angle-transformations">Explicit Eulerian-Angle transformations</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-subgroup-and-full-group"><strong>The Subgroup <em>D</em><sub>2<em>d</em></sub><em>vs</em>. the Full Group <em>T<sub>d</sub></em></strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-symmetry-properties-vibrational-functions"><strong>Symmetry Properties of Vibrational Functions</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-symmetry-properties-rotational-wave-functions-and-direction-cosines"><strong>Symmetry Properties of Rotational Wave Functions and Direction Cosines</strong></a><br>7.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-symmetry-properties-rotational-wave-functions-and-direction-cosines#7.1">Symmetric-top rotational basis Functions</a><br>7.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-symmetry-properties-rotational-wave-functions-and-direction-cosines#7.2">Direction cosines</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-application-continuous-three-dimensional-rotation-reflection-group"><strong>Application of the Continuous Three-Dimensional Rotation-Reflection Group to Symmetric Top Rotational Functions</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-nuclear-spin-functions"><strong>Symmetry Properties of Laboratory-Fixed Nuclear Spin Functions, Nuclear Spin Statistics, and Parities</strong></a><br>9.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-nuclear-spin-functions">Symmetry properties of the nuclear spin functions</a><br>9.2<a href="https://www.nist.gov/pml/methane-symmetry-operations-nuclear-spin-stats"> Nuclear spin statistics and overall parities</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators"><strong>Symmetry Properties of Selected Quantum Mechanical Operators</strong></a><br>10.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators">The Hamiltonian</a><br>10.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators#10.2">The electric dipole-moment operator</a><br>10.3 <a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators#10.3">The total angular momentum operator</a><br>10.4 <a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators#10.4">The proton spin operators</a><br>10.5 <a href="https://www.nist.gov/pml/methane-symmetry-operations-quantum-mechanical-operators#10.5">The vibrational angular momentum operator</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-td-symmetry-species"><strong>Applications of the <em>T<sub>d</sub></em> Symmetry Species</strong></a><br>11.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-td-symmetry-species">Dipole transition selection rules</a><br>11.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-td-symmetry-species#11.2">Perturbations and perturbation operators</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-comparison-present-treatment"><strong>Comparison of the Present Treatment with Those of Jahn and Moret-Bailly</strong></a><br>12.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-comparison-present-treatment#12.1">Jahn's treatment</a><br>12.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-comparison-present-treatment#12.2">Moret-Bailly's treatment</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions"><strong>Application of the Continuous Three-Dimensional Rotation-Reflection Group to Triply Degenerate Vibrational Wave Functions</strong></a><br>13.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions">Spherical polar vibrational coordinates and angular momentum operators</a><br>13.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions#13.2">Comparison of vibrational and rotational angular momentum eigenfunctions</a><br>13.3 <a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions#13.3">Transformation properties of the vibrational angular momentum eigenfunctions</a><br>13.4 <a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions#13.4">Van Vleck's reversed angular momenta</a><br>13.5 <a href="https://www.nist.gov/pml/methane-symmetry-operations-triply-degenerate-vibrational-wave-functions#13.5">The molecule-fixed inversion operation <em>i</em></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-coupling-j-and-ls"><strong>Coupling of <em>J</em> and <em>L</em> ~<sub><em>s</em></sub> in υ<sub>s</sub>&nbsp;=&nbsp;1 States</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-laboratory-fixed-rotations"><strong>Laboratory-Fixed Rotations of CH<sub>4</sub> Considered as an Arbitrary Three-Dimensional Body</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-irreducible-tensorial-sets"><strong>Irreducible Tensorial Sets</strong></a><br>16.1 <a href="https://www.nist.gov/pml/methane-symmetry-operations-irreducible-tensorial-sets#16.1">Vector coupling coefficients</a><br>16.2 <a href="https://www.nist.gov/pml/methane-symmetry-operations-coupling-vibrational-and-rotational-basis-functions">Coupling of vibrational and rotational basis functions</a><br>16.3 <a href="https://www.nist.gov/pml/methane-symmetry-operations-coupling-vibrational-and-rotational-basis-functions#16.3">Coupling of nuclear spin basis functions</a><br>16.4 <a href="https://www.nist.gov/pml/methane-symmetry-operations-vibration-rotation-operators">Vibration-rotation operators in the transformed Hamiltonian</a><br>16.5 <a href="https://www.nist.gov/pml/methane-symmetry-operations-vibration-rotation-operators#16.5">Operators involving the nuclear spin angular momentum</a><br>16.6 <a href="https://www.nist.gov/pml/methane-symmetry-operations-vibration-rotation-operators#16.6">Reduced matrix elements: the Wigner-Eckart theorem</a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-time-reversal-hermitian-conjugation"><strong>The Compound Operation Time-Reversal x Hermitian-Conjugation</strong></a></li><li><a href="https://www.nist.gov/pml/methane-symmetry-operations-acknowledgements-and-references"><strong>References</strong></a></li></ol>
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Created July 8, 2009, Updated June 2, 2021
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