Density Functional Theory (DFT)

Limitations to HF-based calculations:

1. HF model breaks down in the limit of the ideal metal

2. Using one particle wave functions expanded variationally in a basis set, the computational requirements grow in powers anywhere from 3-7 of the number of basis functions. These kinds of limitations drastically limit the study of problems such as the chemistry of transition metal surfaces, interfaces, bulk compounds, and large molecular systems.

Basic ideas behind DFT:

* A knowledge of the density is all that is necessary for a complete determination of all molecular properties.

* E. Bright Wilson, 1965

* If one knew the exact electron density, [[rho]](r), then the cusps of this density would occur at the positions of the nuclei.

* Furthermore, he argued that a knowledge of |[[rho]](r)| at the nuclei would give their nuclear charges.

* Thus, he argued, that the full Schrodinger Hamiltonian was known because it is completely defined once the position and charge of the nuclei are given.

==>In principle, the wavefunction and E are known, and thus everything is known.

Uniform Electron Gas: The Local Density Approximation

* Exc[[[rho]]] No straightforward, systematic way in which the exchange-correlation functional can be systematically improved, unlike traditional HF-based methods.

* The way forward in DFT is to start from a model for which there is an exact solution: The Uniform Electron Gas Model (Parr and Yang).

* Original ideas: The Thomas-Fermi Model

Thomas 1927; Fermi 1927, 1928, 1975

->Realized that statistical considerations can be used to approximate the distribution of electrons in an atom.

->Assumptions of Thomas:

1. "Electrons are distributed uniformly in the 6-D phase space for the motion of an electron at the rate of 2 for each h3 of volume."

2. There is an effective potential field that "is itself determined by the nuclear charge and this distribution of electrons.

-> Thomas-Fermi Kinetic Energy Functional

*Consider electrons distributed randomly in a 3D cell

*Energy levels given by 3D PIB

*Density of states derivable from assumption that, for high quantum numbers, the number of distinct E levels for E<e, can be approximated by the volume of 1 octant of a sphere with radius R in the space (nx,ny,nz).

*Total energy of a a cube of N electrons requires the Fermi-Dirac distribution summed over all energy levels.

*Thus, one can relate the total K.E. and the electron density. Summing over contributions from all cells gives the functional:

* Local Density Approximation

* One of the most important ideas in modern density functional theory.

* Electronic properties are determined as functionals of the electron density by applying locally relations appropriate for a homogeneous electronic system.

* TTF: approximation of the electronic K.E. in terms of the density E: the rigorous energy formula gives the K.E. in terms of the first order density matrix. Thus, taking into consideration the classical electrostatic energies of electron-nuclear attraction and electron-electron repulsion:

neglecting exchange and correlation terms.

Using F[[[rho]]], we get an energy formula for an atom in terms of electron density alone (for molecules, the second term is modified):

*Now, assume that for the ground state of an atom of interest that the electron density minimizes the energy functional ETF[[[rho]](r)] under the constraint (incorporated using Lagrange Multipliers):

N: total # electrons

*The ground state electron density must satisfy the Variational Principle:

yielding the Euler-Lagrange equation:

where, , is the electrostatic potential at point r due to the nucleus and entire electron distribution,

This is solved for the electron density, which is then inserted into to give the total density.

* This is the Thomas-Fermi Theory of the atom

*Very simple model which has been modified and improved over the years.

*Breaks down when applied to molecules

->No molecular binding is predicted

->Accuracy for atoms is not high

=>Viewed as an oversimplified model of not much importance for predictions in atomic, molecular, or solid state physics.

=>Very important for development of later theories.

Hohenberg and Kohn, 1964

* Landmark paper

* Provided fundamental theorems showing that, for ground states, that the TF model may be regarded as an approximation to an exact theory, the density functional theory:

1. There exists and exact energy functional E[r], and

2. There exists an exact Variational Approach of the form as just seen.

* Recall, for an electronic system described by

both the ground state energy and wave function are determined by the minimization of the energy functional:

But, for an N-electron system, the external potential v(r) completely fixes H; so N and v(r) determine all properties for the ground state. (Not surprising, since v(r) defines the whole nuclear frame for a molecule, which, together with the total number of electrons, determines all of the electronic properties).

* In place of N and v(r), the first H-K theorem (Hohenberg * Kohn, 1964) legitimizes the use of electron density as the basic variable. It states: The external potential v(r) is determined within a trivial additive constant, by the electron density [[rho]](r). Since [[rho]](r) determines the number of electrons, [[rho]](r) also determines the ground state wave function and all other electronic properties of the system.

* Hohenberg-Kohn Theorem I:

The electron density [[rho]](r) determines the external potential. Proof by contradiction.

We may represent the ground state energy as a functional of the density as

nonclassical terms (exchange and correlation energy)

For a trial density, , such that and where

This provides justification for the variational theorem in TF in that is an approximation to .

* Variational Principle: Hohenberg-Kohn Theorem II, 1964

Allows us to write down the condition that the energy is stationary w.r.t. changes in the density, subject to the constraint:

for which the Euler-Lagrange equation is, in terms of functional derivatives,

This equation is exact for [[rho]](r), only if we know the functional forms of T[[[rho]]] and Vee[[[rho]]]. Note that T[[[rho]]] and Vee[[[rho]]] are independent of v(r), i.e., the sum is a universal function of [[rho]](r). Once an explicit form (approximate or accurate), is obtained, the method can be applied to any system. The above is the basic working equation for density functional theory (DFT).

* Kohn and Sham, 1965

* We now convert this equation into a set of working equations.

* Kohn and Sham introduced the idea of considering the determinental wavefunction for N noninteracting electrons in N orbitals, ji. For such a system, the K.E. and the electron density are exactly given by:

The orbitals obey,

and the energy of the system is given by,

Considering the problem of interacting electrons, we write the energy in different ways:

The exchange-correlation potential is related to the functional derivative as

Comparing these last two expressions with

we deduce that the problem has been recast into one moving noninteracting electrons in N orbitals which obey

Kohn-Sham Equations for the K-S orbitals . The key property of these orbitals is that they give the exact density through,

once the exact exchange -correlation functional has been determined.

              Hartree Fock                            Kohn Sham                           
orbitals      Satisfy a single-particle equation      Potential is local and orbital      
              with a mathematically non-local         dependent. Includes correlation     
              (i.e., orbital dependent) exchange                                          
              potential                                                                   
sum of        Does not equal the total interacting    Equals the total interacting        
orbital       electron density                        density by definition.              
densities                                                                                 

DFT as an alternative to HF-based methods

*In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. The Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinental wave function.

*In density functional theory, the fundamental role is taken over by an observable quantity, the electron density. An important theorem of DFT states that the correct ground state density determines rigorously all electronic properties of the system, in particular its total energy. The total energy of a system can be expressed as a functional of the density and this functional is minimized by the ground state density.

*Practical implementation of DFT leads to effective 1-electron equations, the Kohn-Sham equations, where the orbital dependent exchange operator of the HF equations are formally replaced by an exchange-correlation operator that depends only on electron density (and spin density in spin-polarized calculations).

==>Simpler matrix elements

One particle wave functions represented as

Gaussian-type orbitals

Slater-type orbitals

Numerical functions

Plane Waves

Augmented plane waves

Inherently' includes electron correlation

*Does not lead to the scaling problems of HF-based correlative methods

*Potentially good choice for accurate description of electronic and structural properties of solids, surfaces and interfaces.

*Relatively little known about the systematic performance of DFT applied to typical organic molecules.

1. Analytic gradients which are critical for geometry optimizations have only recently been developed.

2. Besides the expansion of molecular orbitals in finite basis sets, there are issues of adequate representations of the electron density and the exchange correlation potential.

3. More than 1 explicit form for the exchange-correlation potential has been proposed.