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In the textbooks that I read (namely Ashcroft/Mermin , Marder, etc.) it seems that a distinction is made between the correlations in electron gas and a Couloumb interaction between the electrons. What is exactly meant by the concept of correlations? How is that connected to the interactions in electron gas, and how does the screening enters the picture?

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According to my limited understanding of density functional theory. Coulomb interaction is one of the correlation effects. Besides Coulomb interaction, there are interaction due to Pauli exclusion principle and change of kinetic energy compared with that of non-interacting electron gas.

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It would be useful to have the exact reference in the text where "correlation" is mentioned, but I would argue that correlation is a word which characterizes the collective behavior of electrons in certain circumstances. Such circumstances are varied, but imply that their behavior (and that may be spatial dynamics, but also spin dynamics or other quantities) is not independent. Examples include Coulomb interactions: such interaction mar for instance prevent double occupation of a site in a solid if it is strong enough. The Pauli exclusion principle also imposes constraints on the spin arrangements, thereby inducing correlations between the spins of electrons of a lattice. Other spin spin interactions such may induce ferromagnetism or antiferromagnetism on a lattice, which is another form of correlation.

As a conclusion, there is indeed a difference between correlation, which is a general description of the behavior of electrons, and the interaction or other phenomenon (not necessarily an interaction) which generates it.

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