Basic theoretical concepts behind DFT-LDA-based ab-initio ...

Welcome to the coffee seminar:

Basic theoretical concepts behind DFT-LDA-based ab-initio calculations Jan-Martin Wagner 30 April 2008

(DFT: density-functional theory, LDA: local-density approximation)

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Overview •

General aim; starting point

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Basic approximations (solid-state theory): – Spin degeneracy, periodic boundary conditions, adiabatic approximation

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Density-functional theory (DFT) and local density approximation (LDA) – Hohenberg–Kohn theorem – The universal F functional – Variational principle – Kohn–Sham equation – Local-density approximation – XC functionals beyond LDA

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Further approximations (for numerical treatment): – Frozen-core approximation, pseudopotentials, k-point summation

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Excursion: How to overcome the gap problem

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Summary

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General aim; basic approximations •

General aim: Calculate all materials! properties on fully quantummechanical grounds (i.e., without use of experimental parameters)

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Starting point: Schrödinger equation for a solid, containing approx. 1023 particles (number of variables of the wavefunction ! ) ! unsolvable

Let!s try to tackle the most simple case, described in the following. •

Basic approximations (solid-state theory): – Spin degeneracy: electrons always occur pairwise, occupying both spin states ! no magnetism, no currents – Periodic boundary conditions: calculate representative part of an infinite bulk system ! no surfaces – Perfect crystal: no defects – Adiabatic approximation: separate treatment of nuclei and electrons ! “frozen-lattice” calculation

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Adiabatic approximation •

Hamilton operator: ________________________

! Frozen-lattice eigenvalue equation: •

Expansion of the total wave function using the frozen-lattice functions (X: all nuclear coordinates, x: all electronic coordinates):

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Adiabatic approximation: keep only one term ! Schrödinger equation of the crystal becomes equation for the nuclei:

! Efrozen determines their equilibrium positions R:

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Density-functional theory (DFT) •

Search for the frozen-lattice ground-state! ! Zero temperature without zero-point motion

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Separate the electronic contributions:

! Ground-state energy: •

, where

Electronic problem to be solved (ground-state only): , where V ext depends on the nuclear coordinates:

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DFT: Hohenberg–Kohn theorem •

Hohenberg–Kohn theorem: Local external potentials which differ by more than a constant lead to different ground-state densities. Vice versa, the external potential is determined by the ground-state density (up to a c.).

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Proof via reductio ad absurdum [Phys. Rev. 136, B864 (1964)]: Let V with ground-state "0 lead to the ground-state density n. Assume V ! ! V + const with ground-state "0! leads to the same density n. Being the primed gound-state means:

E0" = #0" H "#"0 < #0 H "#0 = # 0 [H + V " $ V ]# 0 E0" < E0 + # 0 [V " $ V ]#0 = E0 + % [V "(r) $ V (r)] n(r)dr In the same way for the unprimed ground-state:

!

!

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E0 < E0" + # "0 [V $ V "]#0" = E0" + % [V(r) $ V "(r)] n"(r)dr Adding the inequalities and using n’ = n leads to a contradiction (q. e. d.):

E0" + E0 < E0 + E0"

DFT: Hohenberg–Kohn theorem •

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What does the Hohenberg–Kohn theorem mean? Hamilton operator

External potential

H=T+W+V

H

V ext

V’

H"=#" H–K theorem

"0 Ground-state

n = |"|2

n0

"0 ’

Ground-state density

Just this: The wavefunction is determined by the density! ! Use the density as basic quantity; it depends only on one variable r! But: How shall this work? Information contained only in " will be lost! ! Well, we!ll see …

DFT: Variational principle •

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Towards the variational principle: The universal F functional Remember the eigenvalue equation to be solved:

For a certain type of electron–electron interaction, this depends only on the external potential ! functional notation (for given V ext):

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How to transfer this into a functional of n? Let "[n] be the gound-state wavefunction determined by n (via HK), then … – doesn!t work: The potential is determined only up to a constant! Instead, define a universal functional (for given n):

DFT: Variational principle •

The variational principle: How to find the ground-state density? Since F does not depend on V ext, one can construct an energy functional where n and V ext can be chosen independently:

ext

Since E[n,V ] = "[n ] H "[n ] , one finds for a given V ext the corresponding ground-state density from the following variation:

!

! Use this instead of the eigenvalue problem of the frozen-lattice Hamiltonian for the ground-state determination!

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But: No explicit expression is known for F!

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And: How to vary n in a systematic way? ! Kohn–Sham approach!

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DFT: Kohn–Sham approach

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Basic idea: Apply the variational principle to a formally non-interacting, auxiliary many-electron system to obtain an effective external potential leading to the same ground-state density and energy as the true system.

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Main advantage: The many-particle wavefunction can be composed of single-particle ones; the problem is separable into single-particle equations [Phys. Rev. 140, A1133 (1965)] ! Index “s”

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What happens to F? Remember:

! For vanishing W e-e it reduces to Ts: •

How to link this to the true, interacting system? First consider the Hartree (classical Coulomb) interaction as functional of n:

Use this to define the exchange and correlation (XC) functional:

DFT: Kohn–Sham approach •

What does “exchange and correlation functional” mean? – Exchange energy: non-classical energy contribution due to the electrons being Fermions ! antisymmetric wavefunctions; but here: density ! explicit correction needed!

– Correlation energy: Electrons aren!t independent classical particles but can show interference effects (! phase of the common wavefunction) ! “correlation”: all nonclassical behaviour beyond exchange effects ! The XC functional takes care of all nonclassical effects which are lost due to the use of a density instead of a wavefunction! •

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Define the corresponding XC potential:

In this way, XC effects enter the formally non-interacting auxiliary system, described by the …

DFT: Kohn–Sham equation •

Kohn–Sham equation: single-particle equation of the non-interacting auxiliary system (i.e., it has no physical meaning of its own!), …

… which contains an effective potential (the KS potential) taking care of the many-particle interactions (Hartree and XC interaction), …

… and which needs to be solved iteratively, since everything depends on the density (N is assumed even; factor 2 due to spin degeneracy):

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Good news: This self-consistent scheme is mathematically exact, it provides the ground-state energy and density of the interacting system.

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Bad news: No explicit expression is known for the exact XC functional.

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Local-density approximation (LDA)

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Define the Kohn–Sham XC energy density as an integral kernel:

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Homogeneous electron gas: The XC energy density is spatially constant, depends just on the electron density n. It is known from quantum Monte Carlo calculations [Ceperley & Alder, Phys. Rev. Lett. 45, 566 (1980)].

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Local-density approximation (LDA): Replace the true XC energy density by that of the homogeneous electron gas, taken at the local density:

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Advantage: LDA works surprisingly well, also for systems with significantly varying density; reason: (V˜ XC + V H ) fulfills the same sum rule as the exact expression ! a partial cancellation of errors can be expected.

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Drawback: LDA overestimates the bonding ! too small lattice constants, too high bulk moduli ! and cohesive energies.

XC functionals beyond LDA

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Corrections to LDA (minor improvements; increased effort worth only in certain cases, mainly when exchange splittings are underestimated): – Self-interaction correction (SIC): In LDA, the Coulomb self-interaction is not properly cancelled. – Exact exchange: Calculate the exchange contribution from the KS wavefunctions, not from the density.

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Concepts beyond LDA (mainly tailored for specific problems): – Generalized gradient approximation (GGA): includes dependence on the gradient of the density ! semilocal functional; improves the cohesive energy, but worsenes the lattice constants; various construction schemes known (PW86, BP, LYP, PW91, PBE, …). – Extensions of GGA: include the second derivative of the density, include the kinetic energy density; motivation: modifications needed to correctly describe the |r| ! " limit of XC energy and XC potential simultaneously. – Hybrid functionals: mixing exact exchange and GGA parts; e.g. B3LYP:

Further approximations

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Frozen-core approximation: Independent treatment of core and valence electrons; possible if their densities don!t overlap (and no overlap between cores of different atoms) ! KS equation for the valence electron density only ! significantly reduced numerical effort

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Pseudopotential approximation: Further reduce numerical effort for valence electrons in the core region by avoiding the oscillations of their wavefunctions; acheived by replacing the singular atomic Coulomb potential with a finite potential, roughly like this: Wavefunction Potential true

pseudo

– Outside the “cutoff radius”, pseudopotential and pseudo wavefunction are identical to the true potential/wavefunction. – The generation of ab-initio pseudopotentials (from atomic all-electron calculations) is a “science of its own”; different schemes (NC, US, PAW).

Numerical treatment

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Choice of basis functions: Represent all quantities by vectors/matrices, obtained as coefficients relative to a certain basis; most popular: plane waves, Gauß-like functions. – Advantage of localized functions: “natural choice” for molecules. – Advantage of plane waves: Fourier expansion, systematic convergence; choice of cutoff energy determines number of Fourier coefficients (k: inside first Brillouin zone, G: reciprocal lattice):

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Brillouin zone summation: KS energy functional, to be minimized with respect to the density n in order to obtain E0el:

– For a solid: sum over KS eigenvalues ! band-structure energy

Numerical treatment – Band-structure energy: sum over all occupied states in the first Brillouin zone:

– Periodic boundary conditions ! dense mesh of “allowed” k points; too many for numerical calculations ! use special k points (SP), adapted to the crystal symmetry. Then: !

, with

• “Specifications” of DFT-LDA ab-initio calculations:

– name of the code (SIESTA, VASP, PWSCF, ABINIT, CASTEP, …) – type of pseudopotentials – numerical parameters (k-points: type and set size, cutoff energy, XC functional)

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Excursion: The gap problem •

What is “the gap problem”? DFT calculations based on the KS equation do not reproduce the band-gap energy of semiconductors – because DFT is a ground-state theory, and the eigenvalues of the KS equation have no physical meaning! Therefore, it is a wrong expectation if one wants to get the correct gap here. (Note: Even with the correct XC functional, the KS equation cannot produce the correct gap ! it!s not due to the LDA!)

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How to calculate a correct gap? Use Green!s functions (Dyson equation, GW approximation) or timedependent DFT (TD-DFT)! Then it looks like this:

Excursion: The gap problem

From: Aulbur et al., “Quasiparticle calculations …”, Solid State Physics, Vol. 54 (2000), p. 1

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Summary (what to take along)

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DFT (HK theorem, variational principle) is an exact theory for the ground-state of an arbitrary many-electron system.

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The KS equation is an exact effective single-particle equation for an auxiliary system, its iterative solution provides the ground-state energy and density of the real system. The KS eigenvalues and wavefunctions have no physical meaning!

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No excited states can be obtained from DFT (Hohenberg–Kohn–Sham theory), therefore also the band-gap energy cannot be obtained. It is a misuse of HKS theory to take the eigenvalue difference for the gap.

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To obtain the gap, use a GW or a TD-DFT calculation.

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Only using ab-initio pseudopotentials makes a DFT calculation a true ab-initio calculation.

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Literature: – R. M. Dreizler, E. K. U. Groß: “Density Functional Theory”, Springer 1990 – H. Eschrig, “The Fundamentals of Density Functional Theory”, Teubner 1996 – J.-M. Wagner, Ph. D. thesis, 2004 (www.db-thueringen.de/servlets/DocumentServlet?id=9461)