First principles electronic structure: density functional theory

*Electronic Schroedinger equationAtomic units, classical nuclei, Born-Oppenheimer approximation.

Describes 99% of condensed matter, materials physics, chemistry, biology, psychology, sociology, ....

3 basic parameters: electronic mass, proton mass, electric charge

*Prediction: in principle, just need to know what atoms are, can predict all properties of quantum system to very high accuracyNo parameters, no garbage in, garbage out

*Solving the Schroedinger EquationFor example, consider that we have n electrons populating a 3D space. Lets divide 3D space into NxNxN=2x2x2 grid points. To reconstruct , how many points must we keep track of?

n = # electrons (N3n) (N3)188101098100109081000109008

*Two solution philosophies (approximate)

Density functional theoryVariational wave functions (CI, CC, VMC, DMC)Basic objectDensityMany-body wave functionBasic equationE=F[p]H=EEfficiency strategyDensity is only a 3D functionEfficient parameterization of the wave functionAccuracy limitationDont know the functionalRun out of computer time to add parametersArgument for accuracyArgue functional is accurate, compare to experimentVariational theorem: lower energy is closer

BEFORE density functional theoryThomas-Fermi (TF) theory (1927)Find expression for energy as a function of electron densityKinetic energy is hard

Approximate as non-interacting gas

Energy functional

Thomas-Fermi: what is it missing?Kinetic energyExternal fieldCharge density interactionModel: fluid of charge with additional resistance to high densityMissing: Exclusion principleElectron correlationProper kinetic energyCannot describe chemical bonding!

Density functional theoryHohenberg and Kohn (Phys. Rev. 1964) introduced two theorems:Electron density the external potential + constantTotal energy of any density is an upper bound to the ground state energy, if we know the functional

But mapping is unknownVery clear and detailed proofs of these theorems can be found in Electronic Structure, Richard M. Martin (Cambridge University Press 2004 ,ISBN 0 521 78285 6).

First theoremFor each external potential, there is a unique ground state electronic densityproof outline:Suppose there are two different potentialsSuppose they have the same ground state densityShow that this leads to an inconsistency

Hohenberg-Kohn I- The external potential corresponds to a unique ground state electron density. - A given ground state electron density corresponds to a unique external potential- In particular, there is a one to one correspondence between the external potential and the ground state electron density

Second theoremVariational principle with respect to the densityinstead of finding an unknown function of 1023 variables, we need only find an unknown function of 3, the charge density. incredible simplification:Still need to find the functional!

Hohenberg-Kohn IIThere exists a universal functional of the density F[(r)] such that the ground state energy E

is minimized at the true ground state density. Note how very useful this is. We now have a variational theorem to obtain the ground state density (and, correspondingly, the energy)A functional is a mapping from a function (the electron density) to a number (the ground state energy).The equation that we need to solve comes from taking a functional derivative

KS IIwhile these theorems are important, they dont solve the problemwe simply dont know, a priori, the explicit form of the functional:

this aspect of the problem was addressed by Kohn and Sham in 1965Walter Kohn was awarded the 1998 Nobel Prize in Chemistry for his development of density functional theoryWalter Kohns Nobel Lecture can be found athttp://nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html (compared to:

What might F[(r)] look like?UNIVERSAL!From simple inspection:Naively, we might expect the functional to contain terms that resemble the kinetic energy of the electrons and the coulomb interaction of the electrons

Density Functional Theory Completely rigorous approach to any interacting problem in which we can map, exactly, the interacting problem to a non-interacting one.

Ingredients of Density Functional TheoryIngredients: Note that what differs from one electronic system to another is the external potential of the ionsHohenberg-Kohn I: one to one correpondence between the external potential and a ground state densityHohenberg-Kohn II: Existence of a universal functional such that the ground state energy is minimized at the true ground state density

The universality is important. This functional is exactly the same for any electron problem. If I evaluate F for a given trial orbital, it will always be the same for that orbital - regardless of the system of particles.Kohn-Sham: a way to approximate the functional F

Euler-Lagrange SystemThe Hohenberg-Kohn theorems give us a variational statement about the ground state density:the exact density makes the functional derivative of F exactly equal to the negative of the external potential (to within a constant)If we knew how to evaluate F, we could solve all Coulombic problems exactly.However, we do not know how to do this. We must, instead, approximate this functional. This is where Kohn-Sham comes in.

Kohn-Sham ApproachKohn and Sham said:Separate kinetic energy coulomb energy, and other

Kinetic energy of the system of non-interacting electrons at the same density.

Coulomb is the electrostatic term (Hartree)

Exchange-correlation is everything else

Kohn-Sham ApproachThe next step to solving for the energy is to introduce a set of one-electron orthonormal orbitals.Now the variational condition can be applied, and one obtains the one-electron Kohn-Sham equations.Where VXC is the exchange correlation functional, related to the xc energy as:

Comparison with Hartree-FockNaturally, youre remembering the Hartree-Fock equations and realizing that this equation is in fact quite similar:So, just as with Hartree-Fock, the approach to solving the Kohn-Sham equations is a self-consistent approach.

That is, an initial guess of the density is fed into the equation, from which a set of orbitals can be derived. These orbitals lead to an improved value for the density, which is then taken in the next iteration to recompute better orbitals. And so on.

The Exchange-Correlation FunctionalThe exchange-correlation functional is clearly the key to success of DFT.

One of the great appealing aspects of DFT is that even relatively simple approximations to VXC can give quite accurate results.

The local density approximation (LDA) is by far the simplest and used to be the most widely used functional. Approximate as the xc energy of homogeneous electron gas

Ceperley and Alder* performed accurate quantum Monte Carlo calculations for the electron gas

Fit energies(e.g., Perdew-Zunger) to get f(p)

Solving the Kohn-Sham SystemTo solve the Kohn-Sham equations, a number of different methods exist.

Basis set expansion of the orbitals -Localized orbitals : molecules, etc -Plane waves: solids, metals, liquids

Meaning of the orbitals: -Kohn-Sham: In principle meaningless, only representation of the density -Hartree-Fock: Electron distributions in the non-interacting approximation

There is some ad-hoc justification for using K-S orbitals as approximate quasiparticle distributions, but its qualitative at best (see Fermi liquid theory).

like Hartree, LDA-DFT equations must be solved self-consistentlygreat effort to develop efficient and scalable algorithmsremarkably successfulwidely availablecan download computer codes that perform these calculations self-consistent approach

Local Density ApproximationIn the original Kohn-Sham paper, the authors themselves cast doubt on its accuracy for many properties. We do not expect an accurate description of chemical bonding.

And yet, not until at least 10 years later (the 70s), time and time again it was shown that LDA provided remarkably accurate results.

LDA was shown to give excellent agreement with experiment for, e.g., lattice constants, bulk moduli, vibrational spectra, structure factors, and much more.

One of the reasons for its huge success is that, in the end, only a very small part of the energy is approximated.

Local Density ApproximationLDA also works well because errors in the approximation of exchange and correlation tend to cancel.

For example, in a typical LDA atom, theres a ~10% underestimate in the exchange energy.

This error in exchange is compensated by a ~100-200% overestimate of the correlation energy.

Local Density ApproximationIn theory, the LDA method should work best for systems with slowly varying densities (i.e., as close to a homogeneous electron gas as possible).

However, it is interesting that even for many systems where the density varies considerably, the LDA approach performs well!

Good & Bad - Local Density ApproximationGround state densities well-representedCohesive energies are pretty good; LDA tends to overbind a system (whereas HF tends to underbind)Bond lengths are good, tend to be underestimated by 1-2% Good for geometries, vibrations, etc.Structures of highly correlated systems (transition metals, FeO, NiO, predicts the non-magnetic phase of iron to be ground state)Doesnt describe weak interactions well.Makes hydrogen bonds stronger than they should be.Band gaps (shape and position is pretty good, but will underestimate gaps by roughly a factor of two; will predict metallic structure for some semiconductors)Total energies

Local Density ApproximationOne place where LDA performs poorly is in the calculation of excited states.

*Beyond LDA

DescriptionLDAValue of the densityGGA (PBE, BLYP, PW91)Dependence on gradientsmeta-GGA (TPSS)LaplacianDFT+UAd-hoc correction for localized orbitalsHybrid DFT (B3LYP, HSE)Hartree-Fock overestimates gaps. Mix in ~20% of HF, get gaps about right.

*There are many different density functionals Each works best in different situationsDifficult to know if it worked for the right reasons

Practically, very good accuracy/computational cost