Vibrational spectroscopy of linear molecules
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To determine the vibrational spectroscopy of linear molecules, the rotation and
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
of linear molecules are taken into account to predict which vibrational (normal) modes are active in the
infrared spectrum Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
and the
Raman spectrum Raman spectroscopy () (named after Indian physicist C. V. Raman) is a Spectroscopy, spectroscopic technique typically used to determine vibrational modes of Molecule, molecules, although rotational and other low-frequency modes of systems may als ...
.


Degrees of freedom

The location of a
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
in a 3-dimensional space can be described by the total number of coordinates. Each
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
is assigned a set of ''x'', ''y'', and ''z'' coordinates and can move in all three directions.
Degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
is the total number of variables used to define the motion of a molecule completely. For ''N'' atoms in a molecule moving in 3-D space, there are 3''N'' total motions because each atom has 3''N'' degrees of freedom.Miessler, Gary L., Paul J. Fischer, and Donald A. Tarr. ''Inorganic Chemistry''. Upper Saddle River: Pearson, 2014, 101.


Vibrational modes

''N'' atoms in a molecule have 3''N''
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
which constitute
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, and
vibrations Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
. For
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
molecules, there are 3 degrees of freedom for translational (motion along the x, y, and z directions) and 3 degrees of freedom for rotational motion (rotations in Rx, Ry, and Rz directions) for each atom. Linear molecules are defined as possessing bond angles of 180°, so there are 3 degrees of freedom for translational motion but only 2 degrees of freedom for rotational motion because the rotation about its molecular
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis *Axis (mathematics), a designator for a Cartesian-coordinate ...
leaves the molecule unchanged. When subtracting the translational and rotational degrees of freedom, the degrees of vibrational modes is determined. Number of degrees of vibrational freedom for
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
molecules: 3''N''-6 Number of degrees of vibrational freedom for
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
molecules: 3''N''-5Housecroft, Catherine E., and A. G. Sharpe. ''Inorganic Chemistry''. Upper Saddle River, NJ: Pearson Prentice Hall, 2005, 90.


Symmetry of vibrational modes

All 3''N'' degrees of freedom have
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
relationships consistent with the
irreducible representations In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
of the molecule's
point group In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
. A
linear molecule In chemistry, the linear molecular geometry describes the geometry around a central atom bonded to two other atoms (or ''ligands'') placed at a bond angle of 180°. Linear organic molecules, such as acetylene (), are often described by invoking ...
is characterized as possessing a
bond angle Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
of 180° with either a C∞v or D∞h symmetry point group. Each point group has a
character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters ...
that represents all of the possible symmetry of that molecule. Specifically for linear molecules, the two character tables are shown below: However, these two character tables have infinite number of irreducible representations, so it is necessary to lower the symmetry to a subgroup that has related representations whose characters are the same for the shared operations in the two groups. A property that transforms as one representation in a group will transform as its correlated representation in a subgroup. Therefore, C∞v will be correlated to C2v and D∞h to D2h. The correlation table for each is shown below: Once the point group of the linear molecule is determined and the correlated symmetry is identified, all symmetry element operations associated to that correlated symmetry's point group are performed for each atom to deduce the reducible representation of the 3''N'' Cartsian displacement vectors. From the right side of the character table, the non-vibrational degrees of freedom, rotational (Rx and Ry) and translational (x, y, and z), are subtracted: Γvib = Γ3N - Γrot - Γtrans. This yields the Γvib, which is used to find the correct normal modes from the original symmetry, which is either C∞v or D∞h, using the correlation table above. Then, each vibrational mode can be identified as either IR or Raman active.


Vibrational spectroscopy

A
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
will be active in the IR if there is a change in the dipole moment of the molecule and if it has the same symmetry as one of the x, y, z coordinates. To determine which modes are IR active, the irreducible representation corresponding to x, y, and z are checked with the
reducible representation In mathematics, specifically in the representation theory of group (mathematics), groups and algebra over a field, algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no prop ...
of Γvib.Kunju, A. Salahuddin. ''Group Theory and Its Applications in Chemistry.'' Delhi: Phi Learning, 2015, 83-86. An IR mode is active if the same irreducible representation is present in both. Furthermore, a vibration will be Raman active if there is a change in the
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
of the molecule and if it has the same symmetry as one of the direct products of the x, y, z coordinates. To determine which modes are Raman active, the irreducible representation corresponding to xy, xz, yz, x2, y2, and z2 are checked with the reducible representation of Γvib. A Raman mode is active if the same irreducible representation is present in both.


Example

Carbon Dioxide Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
, CO2 1. Assign point group: D∞h 2. Determine group-subgroup point group: D2h 3. Find the number of normal (vibrational) modes or degrees of freedom using the equation: 3n - 5 = 3(3) - 5 = 4 4. Derive reducible representation Γ3N: 5. Decompose the reducible representation into irreducible components: Γ3N = Ag + B2g + B3g + 2B1u + 2B2u + 2B3u 6. Solve for the irreducible representation corresponding to the normal modes with the subgroup character table: Γ3N = Ag + B2g + B3g + 2B1u + 2B2u + 2B3u Γrot = B2g + B3g Γtrans = B1u + B2u + B3u Γvib = Γ3N - Γrot - Γtrans Γvib = Ag + B1u + B2u + B3u 7. Use the correlation table to find the normal modes for the original point group: ''v1'' = Ag = Σ ''v2'' = B1u = Σ ''v3'' = B2u = Πu ''v4'' = B3u = Πu 8. Label whether the modes are either IR active or Raman active: ''v1'' = Raman active ''v2'' = IR active ''v3'' = IR active ''v4'' = IR active


References

{{Branches of spectroscopy Vibrational spectroscopy