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{{MOS|article|[[MOS:BBB]]|date=July 2023}}
In  [[quantum chemistry]] and [[physics]], the '''Lieb–Oxford inequality''' provides a lower bound for the indirect part of the [[Coulomb energy]] of a [[Quantum Mechanics|quantum mechanical]] system. It is named after [[Elliott H. Lieb]] and [[Stephen Oxford]].

The inequality is of importance for [[density functional theory]] and plays a role in the proof of [[Lieb–Thirring inequality#The stability of matter|stability of matter]].

==Introduction==

In classical physics, one can calculate the [[Coulomb energy]] of a configuration of charged particles in the following way. First, calculate the [[charge density]] {{math|''ρ''}}, where {{math|''ρ''}} is a function of the coordinates  {{math|''x'' &isin; ℝ{{sup|3}}}}. Second, calculate the Coulomb energy by integrating:

: <math>\frac{1}{2}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{\rho(x)\rho(y)}{|x-y|} \, \mathrm{d}^3 x \, \mathrm{d}^3 y.</math>

In other words, for each pair of points {{math|''x''}} and {{math|''y''}}, this expression calculates the energy related to the fact that the charge at  {{math|''x''}} is attracted to or repelled from the charge at {{math|''y''}}. The factor of {{frac|1|2}} corrects for double-counting the pairs of points.

In quantum mechanics, it is ''also'' possible to calculate a charge density {{math|''ρ''}}, which is a function of  {{math|''x'' &isin; ℝ{{sup|3}}}}. More specifically, {{math|''ρ''}} is defined as the [[expectation value (quantum mechanics)|expectation value]] of charge density at each point. But in this case, the above formula for Coulomb energy is not correct, due to [[exchange interaction|exchange]] and [[electronic correlation|correlation]] effects. The above, classical formula for Coulomb energy is then called the "direct" part of Coulomb energy. To get the ''actual'' Coulomb energy, it is necessary to add a correction term, called the "indirect" part of Coulomb energy. The Lieb–Oxford inequality concerns this indirect part. It is relevant in [[density functional theory]], where the expectation value ρ plays a central role.

==Statement of the inequality==
For a [[quantum mechanics|quantum mechanical]] system of {{math|''N''}} particles, each with charge {{math|''e''}}, the {{math|''N''}}-particle density is denoted by

:<math>P(x_1,\dots,x_N).</math>

The function {{math|''P''}} is only assumed to be non-negative and [[Wave function#Normalized components and probabilities|normalized]]. Thus the following applies to particles with any "statistics". For example, if the system is described by a normalised [[square integrable]] {{math|''N''}}-particle [[wave function]]

:<math>\psi\in L^2(\mathbb{R}^{3N}),</math>

then

:<math>P(x_1,\dots,x_N)=|\psi(x_1,\dots,x_N)|^2.</math>

More generally, in the case of particles with [[Spin (physics)|spin]] having {{math|''q''}} spin states per particle and with corresponding wave function
:<math>\psi(x_1,\sigma_1,\dots,x_N,\sigma_N)</math>
the {{math|''N''}}-particle density is given by 
:<math>P(x_1,\dots,x_N)=\sum_{\sigma_1=1}^q\cdots\sum_{\sigma_N=1}^q|\psi(x_1,\sigma_1,\dots,x_N,\sigma_N)|^2.</math>
Alternatively, if the system is described by a density matrix {{math|''&gamma;''}}, then {{math|''P''}} is the diagonal
:<math>\gamma(x_1, ... , x_N;  x_1, ..., x_N ).</math>

The electrostatic energy of the system is defined as

:<math>I_P=e^2\sum_{1\le i<j\le N}\int_{\mathbb{R}^{3N}}\frac{P(x_1,\dots,x_i,\dots,x_j,\dots,x_N)}{|x_i-x_j|} \, \mathrm{d}^3 x_1\cdots\mathrm{d}^3 x_N.</math>

For {{math|''x'' ∈ ℝ{{sup|3}}}}, the single particle charge density is given by

:<math>\rho(x)=|e|\sum_{i=1}^N\int_{\mathbb{R}^{3(N-1)}}P(x_1,\dots,x_{i-1},x,x_{i+1},\dots,x_N) \, \mathrm{d}^3 x_1\cdots\mathrm{d}^3 x_{i-1} \, \mathrm{d}^3 x_{i+1}\cdots\mathrm{d}^3 x_N</math>

and the direct part of the Coulomb energy of the system of {{math|''N''}} particles is defined as the electrostatic energy associated with the charge density {{math|''ρ''}}, i.e.

: <math>D(\rho)=\frac12\int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{\rho(x)\rho(y)}{|x-y|} \, \mathrm{d}^3 x \, \mathrm{d}^3 y.</math>

The '''Lieb–Oxford inequality''' states that the difference between the true energy {{math|''I''{{sub|''P''}}}} and its semiclassical approximation {{math|''D''(''ρ'')}} is bounded from below as

{{NumBlk|:|<math>
E_P=I_P-D(\rho)\ge -C|e|^\frac23\int_{\mathbb{R}^3}|\rho(x)|^\frac43 \, \mathrm{d}^3 x,
</math>|{{EquationRef|1}}}}
where {{math|''C'' ≤ 1.58}} is a constant independent of the particle number {{math|''N''}}. {{math|''E''{{sub|''P''}}}} is referred to as the indirect part of the Coulomb energy and in density functional theory more commonly as the [[Density functional theory#Derivation and formalism|exchange plus correlation energy]]. A similar bound exists if the particles have different charges {{math|''e''{{sub|1}}, ... , ''e''{{sub|''N''}}}}. No upper bound is possible for {{math|''E''{{sub|''P''}}}}.

==The optimal constant==
While the original proof yielded the constant {{math|1=''C'' = 8.52}},<ref>
{{cite journal
 |last1=Lieb |first1=E. H.
 |year=1979
 |title=A lower bound for Coulomb energies
 |journal=[[Physics Letters A]]
 |volume=70 |issue=5–6 |pages=444–446
 |bibcode=1979PhLA...70..444L
 |doi=10.1016/0375-9601(79)90358-X
}}</ref> Lieb and Oxford managed to refine this result to {{math|1=''C'' = 1.68}}.<ref name="LO1981">
{{cite journal
 |last1=Lieb|first1=E. H.
 |last2=Oxford |first2=S.
 |year=1981
 |title=Improved lower bound on the indirect Coulomb energy
 |journal=[[International Journal of Quantum Chemistry]]
 |volume=19 |issue=3 |pages=427
 |doi=10.1002/qua.560190306
}}</ref> Later, the same method of proof was used to further improve the constant to {{math|1=''C'' = 1.64}}.<ref>
{{cite journal
 |last1=Kin-Lic Chan |first1=G.
 |last2=Handy |first2=N. C.
 |year=1999
 |title=Optimized Lieb-Oxford bound for the exchange-correlation energy
 |journal=[[Physical Review A]]
 |volume=59 |issue=4 |pages=3075
 |bibcode=1999PhRvA..59.3075K
 |doi=10.1103/PhysRevA.59.3075
|url=https://authors.library.caltech.edu/69066/1/PhysRevA.59.3075.pdf
 }}</ref> It is only recently that the constant was decreased to {{math|1=''C'' = 1.58}}.<ref>{{cite journal |last1=Lewin |first1=Mathieu |last2=Lieb |first2=Elliott H. |last3=Seiringer |first3=Robert |title=Improved Lieb–Oxford bound on the indirect and exchange energies |journal=Letters in Mathematical Physics |date=October 2022 |volume=112 |issue=5 |pages=92 |doi=10.1007/s11005-022-01584-5|arxiv=2203.12473 |bibcode=2022LMaPh.112...92L |s2cid=247618886 }}</ref> With these constants the inequality holds for any particle number {{math|''N''}}.

The constant can be further improved if the particle number {{math|''N''}} is restricted. In the case of a single particle {{math|1=''N'' = 1}} the Coulomb energy vanishes, {{math|1=''I''{{sub|''P''}} = 0}}, and the smallest possible constant can be computed explicitly as {{math|1=''C''{{sub|1}} = 1.092}}.<ref name="LO1981"/> The corresponding [[Calculus of variations|variational equation]] for the optimal {{math|''&rho;''}} is the [[Lane–Emden equation]] of order 3. For two particles ({{math|1=''N'' = 2}}) it is known that the smallest possible constant satisfies {{math|''C''{{sub|2}} ≥ 1.234}}.<ref name="LO1981"/> In general it can be proved that the optimal constants {{math|''C''{{sub|''N''}}}} increase with the number of particles, i.e. {{math|''C''{{sub|''N''}} ≤ ''C''{{sub|''N'' + 1}}}},<ref name="LO1981"/> and converge in the limit of large {{math|''N''}} to the best constant {{math|''C''{{sub|''LO''}}}} in the inequality ({{EquationNote|1}}). Any lower bound on the optimal constant for fixed particle number {{math|''N''}} is also a lower bound on the optimal constant {{math|''C''{{sub|''LO''}}}}. The best numerical lower bound was obtained for {{math|1=''N'' = 60}} where {{math|''C''{{sub|60}} ≥ 1.41}}.<ref>{{cite journal|last2=Vuckovic|first2=S.|last3=Gori-Giorgi|first3=P.|year=2016|title=Challenging the Lieb–Oxford bound in a systematic way. Molecular Physics|journal=[[Molecular Physics (journal)|Molecular Physics]]|volume=114|issue=7–8|pages=1076–1085|doi=10.1080/00268976.2015.1136440|last1=Seidl|first1=M.|arxiv=1508.01715|bibcode=2016MolPh.114.1076S|s2cid=100620702}}</ref> This bound has been obtained by considering an exponential density. For the same particle number a uniform density gives {{Math|''C''{{sub|60}} ≥ 1.34}}.

The largest proved lower bound on the best constant is {{math|''C''{{sub|''LO''}} ≥ 1.4442}}, which was first proven by Cotar and Petrache.<ref>{{cite arXiv |last1=Cotar|first1=C.|last2=Petrache|first2=M.|year=2019|title=Equality of the Jellium and Uniform Electron Gas next-order asymptotic terms for Coulomb and Riesz potentials|eprint=1707.07664|class=math-ph}}</ref>  The same lower bound was later obtained in using a uniform electron gas, melted in the neighborhood of its surface, by Lewin, Lieb & Seiringer.<ref name="LLS1905">{{cite journal|last1=Lewin|first1=M.|last2=Lieb|first2=E.H.|last3=Seiringer|first3=R.|year=2019|title=Floating Wigner crystal with no boundary charge fluctuations|journal=Phys. Rev. B|volume=100|issue=3|pages=035127|doi=10.1103/PhysRevB.100.035127|arxiv=1905.09138|bibcode=2019PhRvB.100c5127L|s2cid=162168639}}</ref>  Hence, to summarise, the best known bounds for {{math|''C''}} are {{math|1.44 ≤ ''C'' ≤ 1.58}}.

==The Dirac constant==
Historically, the first approximation of the indirect part {{math|''E''{{sub|''P''}}}} of the Coulomb energy in terms of the single particle charge density was given by [[Paul Dirac]] in 1930 for [[fermions]].<ref>
{{cite journal
 |last1=Dirac |first1=P. A. M.
 |year=2008
 |title=Note on Exchange Phenomena in the Thomas Atom
 |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
 |volume=26 |issue=3 |pages=376–385
 |bibcode=1930PCPS...26..376D
 |doi=10.1017/S0305004100016108
|doi-access=free
 }}</ref> The wave function under consideration is
:<math>\psi(x_1,\sigma_1,\dots,x_N,\sigma_N)= \frac{\det(\varphi_i(x_j,\sigma_j))}{\sqrt{N!}}.</math>
 
With the aim of evoking perturbation theory, one considers the eigenfunctions of the [[Laplacian]] in a large cubic box of volume {{math|{{!}}''&Lambda;''{{!}}}} and sets

: <math>\varphi_{\alpha,k}(x,\sigma) = \frac{\chi_\alpha(\sigma)\mathrm{e}^{2\pi\mathrm{i} k\cdot x}}{\sqrt{|\Lambda|}},</math>

where {{math|''&chi;''{{sub|1}}, ..., ''&chi;''{{sub|''q''}}}} forms an orthonormal basis of {{math|ℂ{{sup|''q''}}}}. The allowed values of {{math|''k'' &isin; ℝ{{sup|3}}}} are {{math|''n''/{{!}}''&Lambda;''{{!}}{{sup|{{frac|1|3}}}}}} with {{math|''n'' &isin; ℤ{{su|p=3|b=+}}}}. For large {{math|''N''}}, {{math|{{!}}''&Lambda;''{{!}}}}, and fixed {{math|1=''&rho;'' = ''N'' {{!}}''e''{{!}}/{{!}}''&Lambda;''{{!}}}}, the indirect part of the Coulomb energy can be computed to be

: <math>E_P(\mathrm{Dirac})=-C |e|^{2/3} q^{-1/3}\rho^{4/3}|\Lambda|,</math>

with {{math|1=''C'' = 0.93}}.

This result can be compared to the lower bound ({{EquationNote|1}}).  In contrast to Dirac's approximation the Lieb–Oxford inequality does not include the number {{math|''q''}} of spin states on the right-hand side. The dependence on {{math|''q''}} in Dirac's formula is a consequence of his specific choice of wave functions and not a general feature.

==Generalisations==
The constant {{math|''C''}} in ({{EquationNote|1}}) can be made smaller at the price of adding another term to the right-hand side. By including a term that involves the [[gradient]] of a power of the single particle charge density {{math|''&rho;''}}, the constant {{math|''C''}} can be improved to {{math|1.45}}.<ref>
{{cite journal
 |last1=Benguria |first1=R. D.
 |last2=Gallegos |first2=P.
 |last3=Tušek |first3=M.
 |year=2012
 |title=A New Estimate on the Two-Dimensional Indirect Coulomb Energy
 |journal=[[Annales Henri Poincaré]]
 |volume=13 |issue=8 |pages=1733
 |bibcode=2012AnHP...13.1733B
 |doi=10.1007/s00023-012-0176-x
|arxiv = 1106.5772 |s2cid=119272701
 }}</ref><ref>
{{cite journal
 |last1=Lewin |first1=Mathieu
 |last2=Lieb |first2=Elliott H.
 |year=2015
 |title=Improved Lieb-Oxford exchange-correlation inequality with a gradient correction
 |journal=[[Physical Review A]]
 |volume=91 |issue=2 |pages=022507
 |bibcode=2015PhRvA..91b2507L
 |doi=10.1103/PhysRevA.91.022507
|arxiv = 1408.3358 |s2cid=119172373
 }}</ref> Thus, for a uniform density system {{math|''C'' ≤ 1.45}}.

==References==
{{reflist}}

==Further reading==
* {{cite book
 |last1=Lieb |first1=E. H.
 |last2=Seiringer |first2=R.
 |year=2010
 |title=The Stability of Matter in Quantum Mechanics
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-19118-0
}}

{{DEFAULTSORT:Lieb-Oxford inequality}}
[[Category:Inequalities]]
[[Category:Density functional theory]]