1. Computational Chemistry • Molecular Mechanics/Dynamics F = Ma • Quantum Chemistry SchrÖdinger Equation H = E

2. Density-Functional Theory Hy = Ey SchrÖdinger Equation Wavefunction Hamiltonian H = - (h2/2me)ii2+ iV(ri) + ije2/rij Energy Text Book: Density-Functional Theory for Atoms and Molecules by Robert Parr & Weitao Yang

3. Hohenberg-Kohn Theorems 1st Hohenberg-Kohn Theorem: The external potential V(r) is determined, within a trivial additive constant, by the electron density r(r). Implication: electron density determines every thing.

4. 2nd Hohenberg-Kohn Theorem: For a trial density r(r), such that r (r) 0 and , Implication: Variation approach to determine ground state energy and density.

5. 2nd Hohenberg-Kohn Theorem: Application MinimizeEν[ρ] by varying ρ(r): under constraint: (N is number of electrons) Then, construct Euler-Langrage equation: Minimize this Euler-Langrage equation: (chemical potential or Fermi energy)

6. Thomas-Fermi Theory

7. Ground state energy Constraint: number of electrons

8. Using :

9. Kohn-Sham Equations In analogy with the Hohenberg-Kohn definition of the universal function FHK[ρ], Kohn and Sham invoked a corresponding noninteracting reference system, with the Hamiltonian in which there are no electron-electron repulsion terms, and for which the ground-state electron density is exactly ρ. For this system there will be an exact determinantal ground-state wave function The kinetic energy is Ts[ρ]: /2

10. For the real system, the energy functional /2

11. νeff(r) is the effective potential: νxc(r) is exchange-correlation potential:

12. Density Matrix One-electron density matrix: Two-electron density matrix:

14. Thomas-Fermi-Dirac Theory

18. where, rs is the radius of a sphere whose volume is the effective volume of an electron;

19. The correlation energy: At high density limit: At low density limit: where, rs is the radius of a sphere whose volume is the effective volume of an electron. In general:

20. Xα method If the correlation energy is neglected: we arrive at Xα equation: Finally:

21. Further improvements General Gradient Approximation (GGA): Exchange-correlation potential is viewed as the functional of density and the gradient of density: Meta-GGA: Exchange-correlation potential is viewed as the functional of density and the gradient of density and the second derivative of the density: Hyper-GGA: further improvement

22. The hybrid B3LYP method The exchange-correlation functional is expressed as: where, ,

24. Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003) B3LYP/6-311+G(d,p) B3LYP/6-311+G(3df,2p) RMS=21.4 kcal/mol RMS=12.0 kcal/mol RMS=3.1 kcal/mol RMS=3.3 kcal/mol B3LYP/6-311+G(d,p)-NEURON & B3LYP/6-311+G(d,p)-NEURON: same accuracy

27. First-Principles Methods Usage: interpret experimental results numerical experiments Goal: predictive tools Inherent Numerical Errors caused by Finite basis set Electron-electron correlation Exchange-correlation functional How to achieve chemical accuracy: 1~2 kcal/mol?

28. In Principle: DFT is exact for ground state TDDFT is exact for excited states To find: Accurate / Exact Exchange-Correlation Functionals Too Many Approximated Exchange-Correlation Functionals System-dependency ofXC functional ???

29. Existing Approx. XC functional When the exact XC functional is projected onto an existing XC functional, it should be system-dependent

30. EXC[r] is system-dependent functional of r Any hybrid exchange-correlation functional is system-dependent

31. Neural-Networks-based DFT exchange-correlation functional Exp. Database XC Functional Neural Networks Descriptors must be functionals of electron density

33. v- and N-representability We can minimize E[ρ] by varying density ρ, however, the variation can not be arbitrary because this ρ is not guaranteed to be ground state density. This is called the v-representable problem. A density ρ (r) is said to be v-representable if ρ (r) is associated with the ground state wave function of Homiltonian Ĥ with some external potential ν(r).

34. v- and N-representability For more information about N-representable density, please refer to the following papers. ①. E.H. Lieb, Int. J. Quantum Chem. (1983), 24(3), p 243-277. ②. J. E. Hariman, Phys. Rev. A (1988), 24(2), p 680-682.

39. c c c c

40. Basis set of GTFs STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G** ------------------------------------------------------------------------------------- complexity & accuracy Minimal basis set: one STO for each atomic orbital (AO) STO-3G: 3 GTFs for each atomic orbital 3-21G: 3 GTFs for each inner shell AO 2 CGTFs (w/ 2 & 1 GTFs) for each valence AO 6-31G: 6 GTFs for each inner shell AO 2 CGTFs (w/ 3 & 1 GTFs) for each valence AO 6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows 6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen Polarization Function

41. Diffuse/Polarization Basis Sets: For excited states and in anions where electronic density is more spread out, additional basis functions are needed. Polarization functions to 6-31G basis set as follows: 6-31G* - adds a set of polarized d orbitals to atoms in 2nd & 3rd rows (Li - Cl). 6-31G** - adds a set of polarization d orbitals to atoms in 2nd & 3rd rows (Li- Cl) and a set of p functions to H Diffuse functions + polarization functions: 6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets. Double-zeta (DZ) basis set: two STO for each AO

42. 6-31G for a carbon atom: (10s12p)  [3s6p] 1s 2s 2pi (i=x,y,z) 6GTFs 3GTFs 1GTF 3GTFs 1GTF 1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)

45. Time-Dependent Density-Functional Theory (TDDFT) Runge-Gross Extension: Phys. Rev. Lett. 52, 997 (1984) Time-dependent system (r,t) Properties P (e.g. absorption) TDDFT equation: exact for excited states