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    Real Molecules: Normal Mode and Degrees of freedom

    Normal modes are used to describe the different vibrational motions in molecules. Each modecan be characterized by a different type of motion and each mode has a certain symmetry

    associated with it. Group theory is a useful tool in order to determine what symmetries the

    normal modes contain and predict if these modes are IR and/or Raman active. Consequently, IRand Raman spectroscopy is often used for vibrational spectra.

    Overview of Normal Modes

    In general, a normal mode is an independent motion of atoms in a molecule that occurs without

    causing movement to any of the other modes. Normal modes, as implied by their name, areorthogonal to each other. In order to discuss the quantum-mechanical equations that govern

    molecular vibrations it is convenient to convert Cartesian coordinates into so called normal

    coordinates. Vibrations in polyatomic molecules are represented by these normal coordinates.

    There exists an important fact about normal coordinates. Each of these coordinates belongs to an

    irreducible representation of the point the molecule under investigation. Vibrationalwavefunctions associated with vibrational energy levels share this property as well. The normal

    coordinates and the vibration wavefunction can be categorized further according to the point

    group they belong to. From the character table predictions can be made for which symmetriescan exist. The irreducible representation offers insight into the IR and/or Raman activity of themolecule in question.

    Degrees of freedom

    3N where N represents the number of nuclei present in the molecule is the total number of

    coordinates needed to describe the location of a molecule in 3D-space. 3N is most often referred

    to as the total number of degrees of freedom of the molecule being investigated.

    3N, the total number of degrees of freedom, can be divided into:

    3 coordinates to describe the translational motion around the center of mass; these coordinates

    are called the translational degrees of freedom

    3 coordinates to describe the rotational motion in non-linear molecules; for linear molecules only

    2 coordinates are required; these coordinates are called the rotational degrees of freedom

    the remaining coordinates are used to describe vibrational motion; a non-linear molecule has 3N- 6 vibrational degrees of freedom whereas a linear molecule has 3N -5 degrees of freedom.

    Examples: C2H6and CO2

    1) In C2H6N=8 --> 3N=24: ethane has 24 degrees of freedom, 3 translational and 3 rotational,the remaining 18 degrees of freedom are vibrational which is again consistent with 3N -6 =3(8)-

    6=18

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    2) In CO2N=3 --> 3N=9: carbon dioxide has 9 degrees of freedom, 3 translation and 2 rotational,

    the remaining 4 degrees are vibrational degrees of freedom, also consistent with 3N - 5 = 3(3)-5

    = 4

    Overview of degrees of freedom

    Linear Nonlinear

    Translational degrees of freedom 3 3

    Rotational degrees of freedom 2 3

    Vibrational degrees of freedom 3N - 5 3N -6

    Possible 3 Rotational Modes and 3 Translational Modes

    Vibrations are of two types.One which causes change in bond length called as

    Stretching.Other whcich causes change in bond angle called as bending .

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