DENSITY FUNCTIONAL THEORY

From the theory to its practical applications. - Matthew Lane; Professor J. Staunton

THE HARTREE APPROXIMATION

And the problem of the many body wave function.

THE PROBLEM: MANY ELECTRONS

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[1] https://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Electron_shell_002_Helium_-_no_label.svg/2000px-Electron_shell_002_Helium_-_no_label.svg.png

THE PROBLEM: MANY ELECTRONS

unperturbed hydrogen-like Hamiltonian for electron 1 unperturbed hydrogen-like Hamiltonian for electron 2 perturbation part

THE PROBLEM: MANY ELECTRONS

• Perturbation theory for e-e interaction.• Variational methods for Z (shieling.) • Approximate solution.

• Cannot obtain an exact solution.

• Generalise to Z electrons.

Make an informed guess and minimise.

THE PROBLEM: MANY ELECTRONS

• Cannot solved for the ground state. • Electronic effects are too complex.

THE SOLUTION MANY ELECTRONS

• The Hartree Product.• Neglects spin (clearly not antisymmetrised.)

• By intuition, use single electron orbitals from the exact solution of a Hydrogen-like atom.

• Expectation value of Generalised Hamiltonian with Hartree Product:

• Functional of any orbital, not the ground state. • Need to go from every possible function to the optimal ground state orbital.

THE SOLUTION MANY ELECTRONS

THE SOLUTION MANY ELECTRONS

• Need to minimise the functional with respect to the orbitals.

i.e., find he minimum energy by varying the functional form of the orbital.

F[]F[]F[]F[]

THE SOLUTION MANY ELECTRONS

THE SOLUTION MANY ELECTRONS

• Then minimse Functional derivative

THE SOLUTION MANY ELECTRONS

• The Hartree Hamiltonian:

• Minimised energy is not true energy – upper bound. • Does not take into account spin. (Hartree Product)• Ideal potential:

• Each electron ‘feels’ the average of all the others.• Naturally leads to the mean field approximation.

• Iterative process approach true solution - self consistent field approach.

THE SOLUTION MANY ELECTRONS

NEGLECTING ELECTRON CORRELATION EFFECTS

THE HARTREE-FOCK APPROXIMATION The addition of spin.

THE PROBLEM: INDISTINGUISHABILITY AND SPIN

• Electrons are indistinguishable. • Hartree approximation ignored this, no spin.

Improve qualitatively on the Hartree approximation by including spin effects.

THE PROBLEM: INDISTINGUISHABILITY AND SPIN

Symmetric wave function (p=+1)

• Does not go to zero.• Bosons.

Antisymmetric wave function (p=-1)

• Does go to zero. • Fermions.

What happens when you put two particles in the same state? i.e., when state 1 is the same as state 2

[1]Explains the orbital structure of atoms: pairs of electrons, Pauli Exclusion Principle, and s, p etc… shells.

THE SOLUTION INDISTINGUISHABILITY AND SPIN

• Antisymmetrised wave function for the Helium atom – matrix determinant.

• Extend to Z electrons – Slater determinant.

THE SOLUTION INDISTINGUISHABILITY AND SPIN

• Orbital functions spin functions = wave functions.

• Repeat Hartree procedure. single electron Hydrogen

-like Hamiltonian

THE SOLUTION INDISTINGUISHABILITY AND SPIN

• Functional derivative, orthonormality constraint, minimise.

• Coulomb integral:

• Exchange integral:

σ and σ’ are twodifferent spincoordinates

THE SOLUTION INDISTINGUISHABILITY AND SPIN

Fock operator – the Hartree-Fock approximation equivalent of the Hartree operator.

• Hartree-Fock approximation still fails AND requires huge computational power.

KOHEN-HOHENBERG THEOREMS On the road to DFT.

THE PROBLEM: TOO MANY DEGREES OF FREEDOM

• Need to reduce the computational power required to approximately solve.

• Wave function – 3Z coordinates (plus spin degrees of freedom.)• Charge density function – 3 coordinates. • Express the total wave function as a functional of the electron density.

• Hohenberg and Kohn came up with two theorems about the electron charge density and its potential to simplify the problem.

THE PROBLEM: TOO MANY DEGREES OF FREEDOM

The First Theorem:“For a system of interacting particles in some external potential (the potential generated by the nuclei) the electronic potential is uniquely determined by a ground state electron density.”The Second Theorem:“There exists a universal energy functional of the density which is valid for any potential/ any density function. The density function that minimises this energy functional is the ground state density function.”

Both theorems are remarkably easy to prove.

(Electron density determines the state of the system.)

(Electron density which minimises E[n] is the ground state density.)

THE SOLUTION TOO MANY DEGREES OF FREEDOM

(Theorem 1: Electron density determines the state of the system.)

THE SOLUTION TOO MANY DEGREES OF FREEDOM

(Theorem 1: Electron density determines the state of the system.)

Cannot be right.

THE SOLUTION TOO MANY DEGREES OF FREEDOM

(Theorem 1: Electron density determines the state of the system.)

Cannot be right.

THE SOLUTION TOO MANY DEGREES OF FREEDOM

(Theorem 2: Electron density which minimises E[n] is the ground state density.)

THE THOMAS-FERMI MODEL

Introducing the electron density function (semi-Classically).

THE PROBLEM: ELECTRON DENSITY

• From Hohenberg-Kohn theorems:• The wave function is a function of 3Z spacial coordinates, more if spin degrees of freedom are included. • The electron density function is a function of 3 spacial coordinates.

• Want to use semi-Classical considerations to replace wave function with electron density – the Thomas-Fermi model.• i.e., produce an energy functional of the electron density.

• Look for terms describing kinetic + potential + electronic interaction energies.

THE SOLUTION ELECTRON DENSITY

• Kinetic energy term – momentum of ground state electrons. • Fermi statistics: in momentum space, occupied states up to Fermi momentum have volume

• From Heisenberg’s Uncertainty Principle, each h3 of phase space contained an electron pair.

• Equate electron density.

THE SOLUTION ELECTRON DENSITY

• Kinetic energy term – momentum of ground state electrons. • Electrons between p and p+dp

• Classical kinetic energy per unit volume

pF

dp

THE SOLUTION ELECTRON DENSITY

• Kinetic energy term – momentum of ground state electrons. • Integrate over all space.

• Potential energy term – nuclear attraction.

• Electronic interaction energy term – between densities.

THE SOLUTION ELECTRON DENSITY

• Kinetic energy term – momentum of ground state electrons. • Integrate over all space.

• Potential energy term – nuclear attraction.

• Electronic interaction energy term – between densities.

KOHEN-SHAM ANSATZ Further down the road to DFT.

THE PROBLEM: PRACTICALITY

• Repackage the problem to make it easier – obtain the Kohn-Sham equations.

• Assumption 1 : exact ground state density can be replaced by the ground state density of an auxiliary system of non-interacting particles that gives the same result.• Assumption 2 : the auxiliary Hamiltonian uses a spin dependant potential acting on an electron at r with spin σ.

• Trying to find some fictional for a set of non-interacting particles that gives the same ground state density.

THE PROBLEM: PRACTICALITY

• Repackage the problem to make it easier – obtain the Kohn-Sham equations.

• Assumption 1 : exact ground state density can be replaced by the ground state density of an auxiliary system of non-interacting particles that gives the same result.• Assumption 2 : the auxiliary Hamiltonian uses a spin dependant potential acting on an electron at r with spin σ.

• Trying to find some fictional for a set of non-interacting particles that gives the same ground state density.

We know how to deal with single particles in potential wells (set by the Kohn-Sham potential.)

THE SOLUTION PRACTICALITY

Re-write terms from Hartree-Fock in terms of the electron density. •Kinetic energy functional •External (background) potential term (as before)•Hartree energy functional•Exchange-Correlation energy functional•Nuclear interaction energy (as before)

THE SOLUTION PRACTICALITY

• Kohn-Sham energy functional:

• Minimise with respect to electron density function.

(orthonormality constraint, including spin functions)

THE SOLUTION PRACTICALITY

• Three Kohn-Sham equations:• Schrodinger-like equation• Auxiliary Hamiltonian• Kohn-Sham potential• i.e., fictional Kohn-Sham potential of the

auxiliary non-interacting system gives the same ground state electron density.

ε is not the energy – all this Hamiltonian is used for is

obtaining VKS .

THE SOLUTION PRACTICALITY

Now we have the fictional potential of a system of non-interacting particles that has the same ground state electron density as the real

system.

Periodic lattice potentialfrom nuclei.

Mean field potential for electrons

Specific electron correlation effectsand spin interactions.

How do electrons behave in this potential?

A BRIEF INTERLUDE: SOLID STATE Metallic structure and

Tight Binding.

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[2] http://unlcms.unl.edu/cas/physics/tsymbal/teaching/SSP-927/index.shtml

TIGHT BINDING MODEL • Bound electrons still described well by energy levels – wave functions do not extend significantly. • Electron-ion attraction strongest at small separations – conduction electrons excluded from neighbourhood due to Pauli Exclusion Principle. • Conduction electrons move in reduced potential – potential screened by the presence of nearby electrons.

Potential field each electron moves under can be considered a weak periodic potential,AS DESCRIBED BY THE KOHN-SHAM POTENTIAL.

TIGHT BINDING MODEL • Arbitrarily large a; allow a to tend to infinity. • Isolated set of atoms.• Wave functions independently describe localised energy levels specific to each atom.

• a decreasing.• Nearby atoms influence valance electrons. • Atomic wave functions overlap - structures extend throughout crystal. • Degenerate orbitals split – discrete energy levels become narrow energy bands.• Energy bands also overlap.

KOHN-SHAM POTENTIAL PRODCUCES ENERGY BANDS.

[3] http://ecee.colorado.edu/~bart/book/book/chapter2/gif/fig2_2_3.gif

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[4] http://ecee.colorado.edu/~bart/book/book/chapter2/gif/fig2_3_2.gif

DFT AND THERMO-POWER

Extracting information from the density of states and maximising the thermoelectric figure of merit.

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[5] http://0-link.springer.com.pugwash.lib.warwick.ac.uk/article/10.1007%2Fs11664-011-1624-y

THE SEEBECK COEFFICIENT • Free electrons will diffuse from warmer to colder.• From the Drude model:

• More realistically, using the energy dependant electrical conductivity:

Using constant relaxationtime approximation.

Large slope in the DOS at the Fermi

energy.

THE SEEBECK COEFFICIENT • A good thermoelectric material:• High Seebeck coefficient S• High electrical conductivity σ• Low thermal conductivity κ

• Maximise the thermoelectric figure of merit.

THE SEEBECK COEFFICIENT Some Coupling and Requirements:• Only one carrier type.• Low carrier concentration vs conductivity.• High effective mass vs mobility (conductivity.)

• High mobility and low effective mass – semiconductors.• Low mobility and high effective mass – polaron conductors (oxides and chalcogenides).

• Low thermal conductivity (minimise phonon contributions.) Gets large for high conductivity.

Wiedemann–Franz law

THE SEEBECK COEFFICIENT

P. Zwolenski, J. Tobola and S. Kaprzkyi, Journal of Electronic Materials 40, 5, 2011

THE SEEBECK COEFFICIENT

-84μVK-1 -66μVK-1

-151 -69 μVK-1 -49 -30 μVK-1

THE SEEBECK COEFFICIENT -93μVK-1 207μVK-1

-47μVK-1 40 54 μVK-1

A MOTIVATION Optimizing material properties.

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[6] http://www.npl.co.uk/science-technology/engineered-materials/research/nanomaterials/nanostructured-thermoelectrics

THERMO-ELECTRIC MATERIALS• Global energy demand/ waste heat. • Refrigeration (Peltier effect.)

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[8] http://www.meerstetter.ch/images/compendium/Peltier_TEC_Model_Detail_big_6175A.png [7] http://de.bosch-automotive.com/en/parts/parts_and_accessories/motor_and_sytems/starters_alternators_1/spare_parts_for_alternators/spare_parts_for_alternators_1

Electron crystal/ phonon glass materialswith ZT>1 (unit cell complexity and

nanostructures.)

DENSITY FUNCTIONAL THEORY

From the theory to its practical applications. - Matthew Lane; Professor J. Staunton