Vibrational partition function

From Wikipedia, the free encyclopedia

The vibrational partition function traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.

Approximations

Quantum Harmonic Oscillator

The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or vibrational normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degree of freedom of molecules towards thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by:

$E_{{j,i}}=\hbar \omega _{j}(i+{\frac {1}{2}})$

where j is an index representing the vibrational mode, and i is the quantum number for each energy level of the jth vibrational mode. The vibrational partition function is then calculated as:

$Q_{{vib}}=\prod _{j}{\sum _{i}{e^{{-{\frac {E_{{j,i}}}{kT}}}}}}$

v t e Statistical mechanics

Statistical ensembles	Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric Open

Statistical thermodynamics	Characteristic state functions

Partition functions	Translational Vibrational Rotational

Equations of state	Dieterici Van der Waals Ideal gas law Birch–Murnaghan

Entropy	Sackur–Tetrode equation Tsallis entropy Von Neumann entropy

Particle statistics	Maxwell–Boltzmann statistics Fermi–Dirac statistics Fermi-Dirac distribution Bose–Einstein statistics Bose-Einstein distribution

Statistical field theory	Conformal field theory Osterwalder–Schrader axioms

Quantum statistical mechanics	Gibbs canonical ensemble Density matrix Gibbs measure Partition function Weyl quantization Slater determinant

See also	Probability distribution Structureless particles

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.

Vibrational partition function

Approximations

Quantum Harmonic Oscillator

See also