Talk:Molecular vibration/draft

A molecular vibration occurs when atoms in a molecule are in periodic motion. The frequency of the periodic motion is known as a vibration frequency. Each vibration frequency corresponds to a normal mode of vibration. A molecular vibration is excited when the molecule absorbs a quantum of energy, E, which corresponds to a vibration frequency, ν, according to the well-known relation E=hν, where h is Planck's constant. A fundamental vibration is excited when one quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited and so on to higher overtones.

To a first approximation the motion in a normal vibration can be described as a kind of simple harmonic motion. In this, the harmonic approximation the vibrational energy is a quadratic function with respect to the atomic dispacements and the first overtone would have twice the frequency of the fundamental. In reality vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule as the potential energy of the molecule is more like a Morse potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is infrared spectroscopy because vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. However, Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly.

Vibrational excitation can occur with electronic excitation (vibronic transition) to give vibrational fine structure to electronic transitions, particularly with molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Vibrational coordinates
The coordinate of a normal vibration is a combination of displacements in the positions of atoms in the molecule. When the vibration is excited the amplitude of the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Separation of internal and external coordinates
The separation of vibrations from molecular translation and rotation is acheived by means of the Eckart conditions. In effect these conditions define a set of Cartesian coordinates for the molecule with the origin at the centre of mass of the molecule, and an orientation which is fixed relative to the molecule as a whole. For molecules in the gaseous state, the separation of vibrations from rotations is an approximation as there is interaction via the Coriolis "force". Translational motion does not usually affect vibration frequencies, but it does have an indirect effect on line shapes.

Cartesian coordinates
The displacements of an atom can be decomposed into a set of three atomic Cartesian displacements, each one paralell to one of the Cartesian coordinates of the molecule. In a molecule with N atoms, there are 3N atomic Cartesian displacement coordinates. However three of these correspond to translation of the whole molecule, and three (two in the case of a linear molecule) to rotation. Therefore there are 3N-6(5) vibrations in a molecule with N atoms.

Mass-weighted Cartesian coordinates are useful in quantum mechanical calculations.

Valence coordinates
Valence coordinates are commonly described as follows
 * Stretching: a change in the length of a bond
 * Bending: a change in the angle between two bonds
 * Rocking: a change in angle between a group of atoms and a line through the rest of the molecule.
 * Wagging: a change in angle between the plane of a group of atoms and a plane through the rest of the molecule,
 * Twisting: a change in the dihedral angle between the planes of two groups of atoms
 * Out-of-plane: an atom or group of atoms moves in and out of the molecular plane (applies only to planar molecules).

In a rocking, wagging or twisting coordinate the angles and bond lengths within the groups involved do not change. Rocking may be distinguished from wagging by the fact that the atoms in the group stay in the same plane.

An illustration - ethene
Valence coordinates for the in-plane vibrations of the molecule ethene can be specified in relation to the bond lengths and angles between bonds. here are some examples.
 * Sorry about the quality of the sketch!!


 * C-H stretching: q1 = δr1, q2 = δr2, q3 = δr3, q4 = δr4
 * C-C stretching: q5 = δr5
 * HCH bending: q12 = δβ12, q34 = δβ34
 * CH2 rocking: qr1 = δα15 - δα25, qr2 = δα36 - δα46

In the wagging coordinates a CH2 group moves out of the molecular plane. The twisting coordinate involves a change in the angle between the two CH2 groups.

See infrared spectroscopy for some animated illustrations of valence coordinates.

Symmetry-adapted coordinates
Symmetry-adapted coordinates may be created by applying a projection operator to a set of symmetrically equivalent coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) symmetry-adapted C-H stretching coordinates of the molecule ethene (irreducible representations in brackets) are given by
 * qs1 = q1 + q2 + q3 + q4 (Ag)
 * qs2 = q1 + q2 - q3 - q4 (B3u)
 * qs3 = q1 - q2 + q3 - q4 ((B2u)
 * qs4 = q1 - q2 - q3 + q4 (B1g)
 * q5 is already symmetry-adapted (Ag)

Bending
 * qb1 = q12 + q34 (Ag)
 * qb2 = q12 - q34 (B3u)

Rocking
 * qr1 = qr1 + qr2 (B2u)
 * qr2 = qr1 - qr2 (B3g)

Wagging The two wagging coordinates are combined as a sum or a difference. (B2g and B1u)

Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto.

Normal coordinates
A normal coordinate, Q, may sometimes be constructed directly as a symmetry-adapted coordinate. This is possible when the normal coordinate belongs uniquely to a particular irreducible representation of the molecular point group. In the ethene example there are three such normal coordinates. When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the normal coordinate is a linear combination of the symmetry-adapted coordinates. The mixing coefficients are found by performing a full normal coordinate analysis by means of the Wilson GF method. For example, in ethene there are three vibrations of Ag symmetry, qs1 (symmetric C-H stretching), q5 (C-C stretching) and qb1 (symmetric HCH bending). The mixing in this case is minimal because the vibration frequencies are well-separated. In general, there is more mixing the closer together the vibration frequencies are.
 * Both wagging coordinates, (B2g and B1u)
 * Twisting coordinate (Au)

Selection rules
The classification of molecular vibrations by symmetry is useful because is is easy to derive selection rules for infrared and Raman activity by using information in the character tables. For example, any vibration belonging to the totally symmetric representation of the molecular point group will be active in the Raman spectrum and will be polarised. In ethene there are three such vibrations. Futhermore there are two Raman-active C-H stretching modes, (Ag and B1g); the one of B1g symmetry is depolarised. Thus, if mixing is ignored the Raman-active C-H stretching vibrations at ca. 3000 cm-1 can be assigned unequivocally on the basis of polarisation.

Degeneracy
In molecules with a rotation axis of order 3 or more some vibrations may be doubly degenerate. Degenerate vibrations have the same vibration frequency, but the corresponding normal coordinates are orthogonal to each other. 3-fold degeneracy is possible in molecules of cubic symmetry, primarily tetrahedral or octahedral molecules. 5-fold degeneracy is found in molecules of icosahedral symmetry such as B12H122- and C60.

For example, in methane there are four C-H bonds but only two distinct C-H stretching frequencies, one of which is assigned to a triply degenerate vibration.

Newtonian mechanics
Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics, to calculate the correct vibration frequencies. The basic assumption is that each normal vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, f. The anharmonic oscillator is considered elesewhere.
 * $$Force=- F Q \!$$

By Newton’s second law of motion this force is also equal to a "mass", m, times accelaration.
 * $$ Force = M \frac{d^2Q}{dt^2}$$

Since this is one and the same force the ordinary differential equation follows.
 * $$M \frac{d^2Q}{dt^2} + F Q = 0$$

The solution to this equation of simple harmonic motion is
 * $$Q(t) = A \cos (2 \pi \nu  t) \!$$;  $$  \nu =   {1\over {2 \pi}} \sqrt{F \over M} \!$$

A is the maximum amplitude of the vibration coordinate Q. It remains to define the "mass", M.

In a diatomic molecule, AB, M is the reduced mass, μ given by
 * $$\frac{1}{\mu} = \frac{1}{m_A}+\frac{1}{m_B}$$

In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.
 * $$F=\frac{\partial ^2V}{\partial Q^2}$$

Quantum mechanics
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
 * $$E_n = \left( n + {1 \over 2 } \right) {1\over {2 \pi}} \sqrt{F \over M} \!$$,

where n is a quantum number that can take values of 0, 1, 2 ... The difference in energy when n changes by 1 are therefore equal to the energy derived using classical mechanics. See quantum harmonic oscillator for graphs of the first 5 wave functions. Knowing the wave functions, certain selection rules can be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,
 * $$\Delta n = \pm 1$$

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. Also simultaneous excitation of two vibrations is forbidden in the harmonic approximation but in reality can give rise to sum and difference transitions.

Intensities
In an infrared spectrum the intensity of an absorption band is proportional to the devative of the molecular dipole moment with respect to the normal coordinate. The intensity of Raman bands depends on polarizability. See also transition dipole moment.

Normal coordinate analysis
Normal coordinates are, by definition, linear combinations of internal coordinates. Normal coordinate analysis is concerned with finding the coefficients of the linear transformation. There are currently two approaches. This article is concerned with the second approach. Force constants are defined as
 * 1) Ab Initio quantum mechanical calculations of the force constants as second derivatives with respect to Cartesian coordinates reference needed
 * 2) Derivation of force constants by fitting of experimental vibration frequencies.
 * $$f_{ij}=\frac{\partial ^2V}{\partial q_i \partial q_j}$$

where qi is typically a valence coordinate, such as Each force constant fij is an element of a force constant matrix F of order 3N-6(5). If the molecule has some symmetry the elements of this matrix are not all unique. However, just as symmetry-adapted coordinates are linear combinations of valence coordinates, so a matrix of symmetry-adapted force constants can be constructed by taking linear combinations of valence force constants. gives many examples. For instance, for the water molecule,
 * fr a bond stretching force constant
 * frr an interaction constant connecting two bond-stretching coordinates
 * r2fα a bending force constant (the factor r2 ensures that the bending force constant has the same dimensions as the stretching force constant)
 * etc.

{\mathbf {F}} = \begin{pmatrix}  {f_r+f_{rr}} & {\sqrt{2}rf_{r\alpha}} & 0  \\ {\sqrt{2}rf_{r\alpha}} & {r^2 f_\alpha} & 0  \\ 0 & 0 & {f_r - f_{rr}}  \\ \end{pmatrix} $$ For each force constant element fij there is a corresponding G matrix element gij. See Eckart conditions and FG method for how the G matrix is constructed from the atomic masses and the B matrix, which transforms internal Cartesian coordinates to valence coordinates. The symmetry-adapted G matrix for the water molecule, H2O, is

{\mathbf {G}} = \begin{pmatrix}  {\mu_H+\mu_O(1+cos \alpha)} & {{\sqrt{2} \over {r}} \mu_O sin \alpha} & 0  \\ {{\sqrt{2} \over {r}} \mu_O sin \alpha} & {2 \over r^2}[{\mu_O(1-cos \alpha)]} & 0  \\ 0 & 0 & {\mu_H+\mu_O(1-cos \alpha)}  \\ \end{pmatrix} $$ The vibration frequencies are simply related to the eigenvalues,λ, of GF by λ=4π2ν2. Observed frequencies should, if possible, be adjusted for anharmonicity.
 * GFL=λL

Details concering the determination of the eigenvalues and eigenvectors can be found in. The eigenvectors, columns of L, provide the normal coordinates as linear combinations of the valence coordinates.

Indeterminacy
Examination of the F matrix for the water molecule shows that there are four valence force constants, but only three equations. This illustrates a univeral problem with the the general valence force field (GVFF). For each symmetrised block of F, of order n greater than one, there are more force constants, n(n+1)/2 of them, than the number of observable frequencies. Therefore, additional information is required. The options include
 * 1) use isotopic substition which changes G but not F
 * 2) use Coriolis coupling data
 * 3) simplify the force field, for example, by using the Urey-Bradley force field (UBFF)
 * 4) get data from Ab initio calculations