Density functional theory (DFT) is a computational quantum mechanics modeling method used in physics and chemistry to investigate the electronic structure of molecules and condensed phases. DFT was awarded the 1998 Nobel Prize in Chemistry. DFT approximates the complex quantum many-body problem by considering electron density as a basic variable instead of wave functions. Common approximations include the local density approximation (LDA) and generalized gradient approximation (GGA), which include additional information about the density gradient. DFT is widely used today due to its good accuracy and scaling better than other computational methods.

1. Density
Functional
Theory
1998 Nobel Chemistry
Walter Kohn + John A. Pople
Presenter: Wei-Ting Chen

2. Partial ordering by Klein1

3. History
• Atomic chemistry 1800s
• Rutherford-Bohr model for quantum physical
interpretation with electron shells.
• Molecular orbital theory in 1933
• Early 1950’s computer development fueled
computational QM

4. Motivation
• Electron is the main factor for a lot of the properties of
matter
• Nanoscience, our device are scaling down
• DFT application saves experimental probing
• DFT was used in solid state physics since 1970s, but
has great potentials if applicable for chemistry
• Exact wave function solution is unsolvable for many
electron systems

5. Mathematical Model
• SE: Ĥψ= [T+V+U]ψ= Eψ
• T=e kinetic Energy
• V=e-N attraction
• Electron-electron interaction (U) prevents separation into
single particle wave function for exact solution
• (HK)DFT: E=Ê[ρ(→v)]
• Ê is a single unique functional, but probably extremely
complex
• Approximated functionals

6. Model cont’d
• Kohn-Sham DFT(KS-DFT)
• Estimating E with Hartree-Fock(HF) kinetic w/ orbital, e-
N, e-e Coulomb, and Exchange-Correlation
• Correlation is e-e interaction
• Exchange is electron exchange interaction
• Most DFT are similar in the first 3 major terms, usually
only Exchange-Correlation is case-specific
• Sometimes orbitals are avoided (usually not in Chem.)

7. Local Density Approximation
(LDA)
• Exchange functional: 𝐸 𝑥
𝐿𝐷𝐴 𝜌 = −𝐶 𝜌(𝑟)4/3 𝑑𝑟
• Several different type of Correlation terms
• Exchange functional (S) is derived from Homogeneous
Electron Gas (HEG), analytical
• S-VWN, S-PZ81…etc
• Low computation cost, low accuracy by itself
• Integration localized to individual electron density
• Some errors cancel out

8. Graphite imaging with Imaging PS vs LDA. by Heske et al3

9. Ground state of C60 calculated with DFT (DFT2)

10. Generalized Gradient Appx.
(GGA)
• Works like LDA with weighted consideration for
derivative of the point density with regards to distance
• Becke 88 exchange
• 𝐸 𝑥
𝐵88
= 𝐸 𝑥
𝐿𝐷𝐴
− (𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛)
• Lee-Yang-Parr correlation
• B-LYP, PW-LYP…etc

11. Comparison
• DFT doesn’t have hierarchy since it is not variational
• DFT also doesn’t have single convergence
• HF and Coupled Cluster Theory are single convergence
• Coupled Cluster works by perturbation, can always go to
higher accuracy
• Vanilla DFT=n^3, HF=n^4, CC=n^7

12. Limitations
• DFT not very accurate for dynamics due to activation
energy being a bit off (a few kcal/mol)
• Modification to KS-DFT can be made to accommodate
for far out electrons using ad hoc diffuse integrals and
asymptotic correction.
• Often have to be used with other methods (usually no
longer ab initio)

13. Current DFT
• Since KS-DFT is partially HF dependent most HF
programs adopted DFT add-ons (i.e. Gaussian)
• Figuring out optical properties of Chromophores
• Structure-property relationships
• Design to property parameters

14. Sources
1. ”Is chemistry ‘The Central Science’? How are different sciences related? Co-
citations, reductionism, emergence, and posets” Alexandru T. Balaban, Douglas J.
Klein Scientometrics 2006, 69, 615-637.
2. Density Functional Theory. (2015, September 22). Retrieved October 18, 2015, from
https://en.wikipedia.org/wiki/Density_functional_theory
3. Heske, C., Treusch, R., Himpsel, F., Kakar, S., Terminello, L., Weyer, H., & Shirley, E.
(1999). Band widening in graphite. Phys. Rev. B Physical Review B, 4680-4684.