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Number of vibrational modes

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For a molecule with N atoms, the positions of all N nuclei depend on a total of 3 N coordinates, so that the molecule has 3 N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates. A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates. An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third r...
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Vibrational coordinates

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The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν , the frequency of the vibration. Internal coordinates edit Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene, Stretching: a change in the length of a bond, such as C–H or C–C Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule. Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule, Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups. Out–of–plane: a change in the angle between any one of the C–H bonds and the plane defi...
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Quantum mechanics

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In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by E n = h ( n + 1 2 ) ν = h ( n + 1 2 ) 1 2 π k m {\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}}\!} , where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v . The difference in energy when n (or v ) changes by 1 is therefore equal to h ν {\displaystyle h\nu } , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to...
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