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Thermodynamic Potentials

Thermodynamic Potentials

Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...

Potential-Energy Criterion for Equilibrium

Potential-Energy Criterion for Equilibrium

Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the...

The Nernst Equation

The Nernst Equation

Nonstandard Reaction Conditions
The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eºcell = +0.291 V and ΔG° = −56.2 kJ is spontaneous.

Effects of Temperature on Free Energy

Effects of Temperature on Free Energy

The spontaneity of a process depends upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or the other depending upon the temperature of the substance in question. Likewise, some chemical reactions can also exhibit temperature-dependent spontaneities. To illustrate this concept, the equation relating free energy change to the enthalpy and entropy changes for the process is considered:

Fermi Level

Fermi Level

The Fermi-Dirac function is represented by an S-shaped curve indicating the probability of an energy state being occupied by an electron at a given temperature. The Fermi level is the energy level at which there is a fifty percent chance of finding an electron, and it is positioned between the lower-energy valence band and the higher-energy conduction band.
At absolute zero temperature, electrons fill all energy states up to the Fermi level, leaving upper states empty. As the temperature rises,...

Atomic Nuclei: Nuclear Spin State Population Distribution

Atomic Nuclei: Nuclear Spin State Population Distribution

Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.

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Updated: May 31, 2026

Characterization of Thermal Transport in One-dimensional Solid Materials

Characterization of Thermal Transport in One-dimensional Solid Materials

Published on: January 26, 2014

The optimized effective potential with finite temperature.

R A Lippert¹, N A Modine, A F Wright

¹MIT Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA.

Journal of Physics. Condensed Matter : an Institute of Physics Journal

|June 22, 2011

Summary

This summary is machine-generated.

We present a new method for calculating electronic structures using the optimized effective potential (OEP) method. This approach extends OEP calculations to finite temperatures, improving density functional methods.

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Area of Science:

Quantum Chemistry
Computational Physics
Materials Science

Background:

The optimized effective potential (OEP) method offers enhanced accuracy in electronic structure calculations by incorporating exact exchange energy.
Traditional density functional approximations like the local density approximation have limitations that OEP can potentially overcome.

Purpose of the Study:

To derive a density-matrix-based expression for the gradient of the Kohn-Sham energy with respect to the effective potential.
To generalize the OEP method to the finite temperature regime, extending previous zero-temperature work.

Main Methods:

A novel density-matrix-based derivation of the Kohn-Sham energy gradient was performed.
The derived gradient was utilized for iterative energy minimization to determine the OEP.
The methodology was extended from the zero-temperature limit to finite temperatures.

Main Results:

A finite-temperature version of the optimized effective potential (OEP) equation was derived by setting the calculated gradient to zero.
The new derivation provides a pathway for more accurate electronic structure computations at non-zero temperatures.

Conclusions:

The developed finite-temperature OEP equation represents a significant advancement for density functional theory.
This work enables more precise electronic structure calculations, particularly in systems where temperature effects are crucial.