The role of electron localization in density

functional

QCM Journal Club

Debabrata Pramanik

30th Oct. 2014

Outlines of my talk…

1. Introduction to density functional theory (DFT)

2. Existing exchange-correlation (xc) functional

3. Introduction to new functional

4. Comparing results with exact Kohn-Sham (KS) potential

5. Conclusions

Basic Introduction to DFT

Any property of a system of many interacting particles can be viewed

as a functional of the ground state density ( )

In principle determines all information in many-body wave

function for ground and all excited states

Existence proofs of such functional is given by the work of

Hohenberg & Kohn and also by Mermin

But they provide no guidance for constructing functionals

Also, no exact functional are known for system more than

one electron

DFT would remain a minor curiosity today if it were not

for the ansatz made by Kohn & Sham

It provided a useful way to provide useful, approximate

ground state functionals for real system of many electrons

Kohn - Sham ansatz replaces the many body interacting

problem with an auxiliary independent particle problem

with all many body effects included in an exchange-

correlation functionals

LDA, GGA are widely used approximations for the

exchange-correlation functionals

Many-body Hamiltonian:

Using Born-Oppenheimer approximation KE term of the nucleus can be

neglected.

Ignoring KE, the fundamental Hamiltonian would be

Adopting Hartree atomic units,

Classical interaction of nuclei with one another

Time dependent Schrodinger equation

Expectation value of any operator will be

Density operator

Total energy will be the expectation value of the Hamiltonian

Classical Coulomb energies can be written

Thus the total energy can be written as

Thus all long range interactions cancel in the difference, so that effects of exchange and correlation are short ranged.

So, the Many- body Hamiltonian is difficult to solve.

In DFT we try to express the whole Many-body equation in terms of density of system particles n(r).

Thomas- Fermi – Dirac approximation: Example of a Functional

First term is the local approximation to the KE,

Third term is the local exchange term,

Thus ground state density and energy can be found by minimizing the functional E[n] for all possible n(r) with the constraint on the total number of electrons.

Using the method of Lagrange multipliers, we can get the solution by minimizing the energy functional.

However, Thomas- Fermi type approach starts with approximations that are too crude, missing essential physics and chemistry, such as shell structures of atoms and binding of molecules.

The Hohenberg – Kohn (HK) theorems : The approach of HK was to formulate DFT as an exact theory of many-body systems.

The formulation applies to any system of interacting particles in an external potential where the Hamiltonian can be written as,

Schematic representation of HK theorems:

Intricacies of exact DFT: The theorems given by HK were in terms of unknown functionals of the density.

The Kohn- Sham (KS) ansatz:

Schematic representation of KS ansatz:

New Functional

Success of DFT depends upon the approximations made in the exchange-correlation (xc) part of the Kohn-Sham functional.

However, these approximations become less secure when there is strong correlation and/or current flow present.

Particular attention has been given here to improve the time-dependent xc potential.

Here, they demonstrate that electron localization, driven by Coulomb interaction and Pauli principle, can form a powerful ingredient in approximations in the KS potential.

Electron localization function (ELF) provides a useful indicator of localization

=

Where C is HF Fermi Hole Curvature

can be interpreted as prob. density for finding one particle at with spin projection

and simultaneously a second particle at with spin projection

L =

Here, L=1 complete localization L=0.5 homogeneous electron gas Here, L ranges from 0 to 1.

Starting point is the KS potential, originally derived for a system of two spin full electrons in their spin-zero ground state.

For a one electron system, out of all regions of space where the electron density is dominated by any one KS orbital.

For such a region the KS equation may be approximated by

For the dominant orbital where,

in the region, yielding the ground – state KS potential, which they termed as Single orbital approximation (SOA),

New Functional

This equation is the approximation to the universal KS functional.

The authors have seen that not only SOA works very well for strongly localized orbital regions but also accounts for non-local features and corrects self-interaction in the KS potential in the region of low localization.

Model System (double well):

They have studied a ground state system where the electrons are highly localized

Two spinless electrons

Subject to an external potential

The ext. potential consisting of two identically separated wells together with a potential step between them.

First, compared SOA with exact KS pot.

Then, extend by combining SOA with a potential suited to delocalized systems, in proportions depending on strength of localization.

A useful approximate functional must give accurate densities when applied self-consistently without prior knowledge of the exact density. The SOA as it is unanchored to the external potential, is not suited to this. By mixing SOA with a suitable reference pot. we can better approximate KS potential.

Conclusions

This single orbital approximation (SOA) functional reproduces important non-local features of the xc potential for systems where there is electron localization.

The crucial self-interaction correction is also described very well.

MLP (mixed localization potential), a weighted mixture of the SOA and a reference potential, extends the good performance to regions of reduced localization. The method can be applied self-consistently for a variety of challenging

ground-state and time-dependent situations to obtain accurate results.

The MLP uses a measure of localization, f, for which simple approximations already give successful results for a system with various degrees of localization.

References:

1. The role of electron localization in density functionals, M. J. P. Hodgson, J. D. Ramsden, T. R. Durrant, and R. W. Godby arXiv:1409.5666v1

2. Richard M Martin, Electronic Structure Basic Theory and Practical Methods

3. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964) 4. J. F. Dobson, J. Chem. Phys. 94, 4328 (1991).

Acknowledgments:

1. Prof. Prabal K Maiti 2. Prof. Manish Jain 3. Prof. Diptiman Sen 4. Ranjan 5. Manisha 6. Tathagata 7. Khokan Roy 8. All my Labmates

Thank You