Local-density approximation: Difference between revisions

Revision as of 05:53, 16 October 2018

Revision as of 03:36, 17 February 2019

Line 15:

== Applications ==

Local density approximations, as with Generalised Gradient Approximations (GGA) are employed extensively by [[solid-state physics|solid state physicists]] in ab-initio DFT studies to interpret electronic and magnetic interactions in semiconductor materials including semiconducting oxides and [[Spintronics]]. The importance of these computational studies stems from the system complexities which bring about high sensitivity to synthesis parameters necessitating first-principles based analysis. The prediction of [[Fermi level]] and band structure in doped semiconducting oxides is often carried out using LDA incorporated into simulation packages such as CASTEP and DMol3.<ref>{{cite journal| last1=Segall| first1=M.D.| last2=Lindan| first2=P.J | title= First-principles simulation: ideas, illustrations and the CASTEP code | journal= Journal of Physics: Condensed Matter | year= 2002| volume=14| issue=11| pages=2717|bibcode = 2002JPCM...14.2717S |doi = 10.1088/0953-8984/14/11/301 }}</ref> However an underestimation in [[Band gap]] values often associated with LDA and [[Density functional theory#Approximations .28exchange-correlation functionals.29|GGA]] approximations may lead to false predictions of impurity mediated conductivity and/or carrier mediated magnetism in such systems.<ref>{{cite journal| last1=Assadi| first1=M.H.N| title= Theoretical study on copper's energetics and magnetism in TiO₂ polymorphs| journal= Journal of Applied Physics | year=2013| volume=113| issue=23| pages= 233913| url=https://arxiv.org/ftp/arxiv/papers/1304/1304.1854.pdf233913–233913–5| doi=10.1063/1.4811539|arxiv = 1304.1854 |bibcode = 2013JAP...113w3913A |display-authors=etal}}</ref>

== Homogeneous electron gas ==

Line 37:

:<math>\epsilon_{c} = \frac{1}{2}\left(\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + \dots\right)\ ,</math>

where the Wigner-Seitz parameter <math>r_{s}</math> is dimensionless.<ref name="Murray Gell-Mann and Keith A. Brueckner 1957 364">{{cite journal | title = Correlation Energy of an Electron Gas at High Density | author = Murray Gell-Mann and Keith A. Brueckner | journal = Phys. Rev. | volume = 106 | pages = 364364–368 | year = 1957 | doi = 10.1103/PhysRev.106.364 | issue = 2| bibcode = 1957PhRv..106..364G }}</ref> It is defined as the radius of a sphere which encompasses exactly one electron, divided by the Bohr radius. The Wigner-Seitz parameter <math>r_s</math> is related to the density as

:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}\ .</math>

Line 46:

:<math>\epsilon_{c} = a \ln \left( 1 + \frac{b}{r_s} + \frac{b}{r_s^2} \right) .</math> <ref>{{cite journal | title = Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities | author = Teepanis Chachiyo | journal = J. Chem. Phys. | volume = 145 | pages = 021101 | year = 2016 | doi = 10.1063/1.4958669 | issue = 2| bibcode = 2016JChPh.145b1101C }}</ref>

The parameters <math>a</math> and <math>b</math> ''are not'' from empirical fitting to the Monte Carlo data, but from the theoretical constraint that the functional approaches high-density limit. The Chachiyo's formula is more accurate than the standard VWN fit function.<ref>{{cite journal | title = A simpler ingredient for a complex calculation | author = Richard J. Fitzgerald | journal = Physics Today | volume = 69 | pages = 20 | year = 2016 | doi = 10.1063/PT.3.3288 | issue = 9| bibcode = 2016PhT....69i..20F }}</ref> In the [[atomic units|atomic unit]], <math> a = \frac{ \ln(2) - 1 } {2 \pi^2} </math>. The closed-form expression for <math> b </math> does exist; but it is more convenient to use the numerical value: <math> b = 20.4562557 = \exp(\text{C}/2a) </math>. Here, <math>\text{C}</math> has been evaluated exactly using a closed-form integral and a zeta function (Eq. 21, G.Hoffman 1992).<ref>{{cite journal | title = Correlation energy of a spin-polarized electron gas at high density | author = Gary G. Hoffman | journal = Phys. Rev. B | volume = 45 | pages = 83708730–8733 | year = 1992 | doi = 10.1103/PhysRevB.45.8730 | issue = 15| bibcode = 1992PhRvB..45.8730H }}</ref> <math>\text{C} = \tfrac{\ln(2)}{3} - \tfrac{3}{2\pi^2} \left [ \zeta(3) + \tfrac{22}{9} -\tfrac{\pi^2}{3} + \tfrac{32\ln(2)}{9} - \tfrac{8\ln^2(2)}{3} \right ] + \tfrac{2(1-\ln 2)}{\pi^2} \left [ \ln(\tfrac{4}{\alpha \pi}) + \left \langle \ln R_0 \right \rangle _{\text{av}} - \tfrac{1}{2} \right ].</math> Keeping the same functional form,<ref>{{cite journal | title = Comment on “Communication"Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities”densities" [J. Chem. Phys. 145, 021101 (2016)] | author = Valentin V. Karasiev | journal = J. Chem. Phys. | volume = 145 | pages = 157101 | year = 2016 | doi = 10.1063/1.4964758 | issue = 2| arxiv = 1609.05408 | bibcode = 2016JChPh.145o7101K }}</ref> the parameter <math>b</math> has also been fitted to the Monte Carlo simulation, providing a better agreement. Also in this case, the <math>r_{s}</math> must either be in the atomic unit or be divided by the Bohr radius, making it a dimensionless parameter.<ref name="Murray Gell-Mann and Keith A. Brueckner 1957 364"/>

As such, the Chachiyo formula is a simple (also accurate) first-principle correlation functional for DFT (uniform electron density). Tests on phonon dispersion curves <ref>{{cite journal | title = Study of the first-principles correlation functional in the calculation of silicon phonon dispersion curves | author = Ukrit Jitropas and Chung-Hao Hsu| journal = Japanese Journal of Applied Physics | volume = 56 | issue = 7| pages = 070313 | year = 2017 | doi = 10.7567/JJAP.56.070313 | bibcode = 2017JaJAP..56g0313J }}</ref> yield sufficient accuracy compared to the experimental data. Its simplicity is also suitable for introductory density functional theory courses.<ref>{{cite book|last=Boudreau|first=Joseph|author2=Swanson, Eric |title=Applied Computational Physics|publisher=Oxford University Press|year=2017|isbn=978-0-198-70863-6|page=829}}</ref><ref>{{cite web |url=https://compphys.go.ro/dft-for-a-quantum-dot/ |title=DFT for a Quantum Dot |last=Roman |first=Adrian |date=November 26, 2017 |website=Computational Physics Blog |access-date=December 7, 2017}}</ref>

[[File:Correlation funtionals comparison.gif|thumb|Comparison between several LDA correlation energy functionals and the quantum Monte Carlo simulation]]

Line 54:

Accurate [[quantum Monte Carlo]] simulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566–569 | year = 1980 | doi = 10.1103/PhysRevLett.45.566 | bibcode=1980PhRvL..45..566C | issue = 7}}</ref> The most popular LDA's to the correlation energy density interpolate these accurate values obtained from simulation while reproducing the exactly known limiting behavior. Various approaches, using different analytic forms for ''ε''_c, have generated several LDA's for the correlation functional, including

* Vosko-Wilk-Nusair (VWN) <ref name="vwn">{{cite journal | title = Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis | author = S. H. Vosko, L. Wilk and M. Nusair | journal = Can. J. Phys. | volume = 58 | pages = 12001200–1211 | year = 1980 | doi = 10.1139/p80-159 |bibcode = 1980CaJPh..58.1200V | issue = 8 }}</ref>

* Perdew-Zunger (PZ81) <ref name="pz81">{{cite journal | title = Self-interaction correction to density-functional approximations for many-electron systems | author = J. P. Perdew and A. Zunger | journal = Phys. Rev. B | volume = 23 | pages = 50485048–5079 | year = 1981 | doi = 10.1103/PhysRevB.23.5048 |bibcode = 1981PhRvB..23.5048P | issue = 10 }}</ref>

* Cole-Perdew (CP) <ref>{{cite journal | title = Calculated electron affinities of the elements | author = L. A. Cole and J. P. Perdew | journal = Phys. Rev. A | volume = 25 | pages = 12651265–1271 | year = 1982 | doi = 10.1103/PhysRevA.25.1265 |bibcode = 1982PhRvA..25.1265C | issue = 3 }}</ref>

* Perdew-Wang (PW92) <ref name=pw92>{{cite journal | title = Accurate and simple analytic representation of the electron-gas correlation energy | author = John P. Perdew and Yue Wang | journal = Phys. Rev. B | volume = 45 | pages = 13244–13249 | year = 1992 | doi = 10.1103/PhysRevB.45.13244 |bibcode = 1992PhRvB..4513244P | issue = 23 }}</ref>

Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained [[Møller–Plesset perturbation theory#Rayleigh-Schr.C3.B6dinger perturbation theory|perturbatively]] from the HEG model.<ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages = 1002–1011 | year = 1934 | url = http://link.aps.org/abstract/PR/v46/p1002 | doi = 10.1103/PhysRev.46.1002 | format = abstract |bibcode = 1934PhRv...46.1002W | issue = 11 }}</ref>

== Spin polarization ==