n-fall08-week6-7 - J.I Siepmann Chem 8561 46 IDEAL...

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J.I. Siepmann Chem 8561 46 IDEAL POLYATOMIC GAS a polyatomic molecule consists of three or more nuclei and many electrons to derive the canonical partition function for polyatomic molecules, we will follow a similar approach as used for the diatomic molecule use Born-Oppenheimer approximation to separate the motions of nuclei and elec- trons electrons in the field of the n nuclei at fixed conformation of molecule nuclei in the electronic potential u e 1 ( r ) thereafter, we can separate the motion of the nuclei into translations of the center- of-mass, rotations, and intramolecular motions (vibrations) again, the large (continuous) number of translational states allows us to use Boltz- mann statistics, thus Q ( N, V, T ) = ( q trans q rot , vib , elec , nucl ) N N ! (164) each of the n nuclei of the polyatomic molecule requires 3 cartesian coordinates to describe its position, thus the configuration of the polyatomic molecule is specified by 3 n coordinates the position of the COM takes three coordinates the orientation of a given vector (bond of the molecule) in space is specified by 2 angles, thus 2 angles are needed for a linear polyatomic molecule, and 1 additional angle is needed for nonlinear polyatomic molecules to specify the rotation of the molecule around the axis (bond) given by the first 2 angles thus, the remaining 3 n 5 or 3 n 6 coordinates are needed to describe the conformation of the polyatomic molecule (the separations of the n nuclei with respect to each other); all these coordinates are conventionally referred to as vibrational degrees of freedom
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J.I. Siepmann Chem 8561 47 the equivalent of the rigid rotor/harmonic oscillator approximation for diatomic molecules allows us to separate rotational and vibrational degrees of freedom with these approximations all five types of degrees of freedom can be treated as independent, and the canonical partition function can be written as product of individual contributions Q = [ q trans ( V, T ) q rot ( T ) q vib ( T ) q elec ( T ) q nuclear ( T )] N N ! (165) Translational, electronic, and nuclear partition functions the COM is a point particle with mass equal to the sum over all the individual masses of the molecule q trans = bracketleftbigg 2 π ( n i =1 m i ) k B T h 2 bracketrightbigg 3 / 2 V (166) with the conventional choice that the zero of energy is defined by completely sep- arated atoms at rest, the global minimum of the (total) electronic potential has an energy of D e (and the vibrational groundstate, sum of 3 n 6 (nonlinear) or 3 n 5 (linear) modes, has an energy of modes m / 2), then the electronic partition function is q elec = ω e 1 e D e /k B T + · · · (167) and the nuclear partion functions can again be neglected q nucl = 1 (168)
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J.I. Siepmann Chem 8561 48 Rotational partition function in case, the polyatmic molecule is linear, the rotational partition function can be calculated using the equations derived for the diatomic molecule, i.e.
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