Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 06 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

2. Density Functional Theory HF optimize the e- wavefunction the wavefunction is essentially uninterpretable, lack of intuition e- correlation is only accounted for using post-HF methods DFT optimize the e- density Increased in popularity within last 2 decades. Hamiltonian depends ONLY on the positions and atomic number of the nuclei and the number of e-. Given a known e- density  form the H operator  solve the Schrödinger Eq.  determine the wavefunctions and energy eigenvalues. Hohenberg and Kohn – proved ground-state E is uniquely defined by the e- density. E is a unique functional of r(r). Functional example:

3. Significance (1). The wave function Y of an N-electron system includes 3N variables, while the density, r no matter how large the system is, has only three variables x, y, and z. Moving from E[Y] to E[r] in computational chemistry significantly reduces the computational effort needed to understand electronic properties of atoms, molecules, and solids. (2). Formulation along this line provides the possibility of the linear scaling algorithm currently in fashion, whose computational complexity goes like O(NlogN), essentially linear in N when N is very large. (3). The other advantage of DFT is that it provides some chemically important concepts, such as electronegativity (chemical potential), hardness (softness), Fukui function, response function, etc..

4. Density Functional Theory • ‘Local’ functional: • ‘Non-local’ or ‘gradient-corrected’: • As with MO theory, the density (in exact DFT) obeys a variational principle – the lower E is more accurate. • In DFT, the E functional is written as: • First term: interaction with external potential (nuclei) • Second term: KE(e-) + e-/e- interactions

5. Density Functional Theory • Solution: optimize e- density until E is minimized. • Constraints on e- density? • How do we include this constraint? Lagrange multipliers (m): • This is the DFT equivalent of the Schrödinger Eq. Vext indicates constant external potential (nuclear positions). ** Central crux of DFT: What is the function, F[r(r)]?

6. Density Functional Theory • Kohn and Sham split F[r(r)] into three terms: F[r(r)] = EKE[r(r)] + EH[r(r)] + EcC[r(r)] • EKE[r(r)] = e- kinetic energy • EH[r(r)] = e-/e- Coulombic interaction • EcC[r(r)] = e- exchange/correlation + KE correction + E(self-interaction) • One-electron Kohn-Sham equations: • Solution (SCF approach): • guess density • derive orbitals • calculate new density from orbitals • repeat

7. Density Functional Theory • The solution hinges on VCC[r(r)]: • We must find the functional: ECC[r(r)]. Unfortunately, there is no way to solve for this functional, but we can attempt to find expressions that work well. • Since we must invoke approximations for this term, the implementation of DFT is no longer variational (unlike HF). • DFT remains size consistent (despite losing variational behavior). • There are two basic implementations (approximations) of DFT: • Local-density approximation (LDA) • Generalized gradient approximation (GGA) • LDA • The value of the exchange energy depends only on the local density. • The e- density may vary as a function of r, but r is single-valued, and the fluctuations in r away from r do not affect the value of ECC at r. • LSDA: variation of LDA accounting for spin polarization (open-shell systems), similar to UHF method, which splits solutions in to a and b spins. • ECC is based on the uniform electron-gas model, which is known accurately, and can be cast into an analytical form. • Functionals: VWN, VWN5 (Vosko-Wilk-Nusair)

8. Density Functional Theory • GGA • The value of the exchange energy depends on the local density AND on the gradient of the density. Overcomes LDA tendency to overbind. Adds ~20% to compute time. • Exchange and Correlation contributions usually calculated separately. • BLYP – popular functional including exchange contribution from (B)ecke and correlation contribution from (L)ee, (Y)ang, and (P)arr. • Functionals: BLYP, BP86, BPW91 • Hybrid Functionals • Incorporate HF exchange contribution into the DFT functional • Exact exchange for a non-interacting system can be calculated using HF (using KS orbitals). • Very popular • Functionals: B3LYP, B3PW91, B1PW91, PBE1PBE • Periodic Systems • DFT often used • Periodic plane waves • Car-Parrinello MD • Ab initio MD • Chemical reactions • On-the-fly potentials

9. Density Functional Theory • Similarities with HF: • A basis set is still needed, but can be more flexible (numerical basis functions) • Solution of secular equation • SCF procedure is still used • Differences with HF: • e- correlation is implicitly included • The solution of the secular equation is computationally more efficient – formally scales as N3 as opposed to N4. • Sometimes empirical parameters are included • Some properties are easier to extract from HF than from DFT • DFT has challenge of systematic improvability – difficult to predict performance of 2 different functionals. HF is more predictable, with full CI (with an infinite basis set) as the ultimate goal.

10. Density Functional Theory • PERFORMANCE: • Formally scales as N3, but improvements are possible • Convergence w.r.t. basis set size is more rapid • DFT SCF is sometimes more problematic, thus HF orbitals can be used as an initial guess for KS orbitals • Not capable of describing London dispersion forces – dispersion not included in functionals. This can artificially arise from BSSE. • H-bonded systems: heavy-atom/heavy-atom distances typically too short by 0.1 angstrom, but energetics o.k. (need diffuse functions in basis set). • Complexes with charge-transfer interactions, DFT overpredicts the interactions. • DFT sometime overstabilizes systems, increasing symmetry. • Increasing basis set size does not always improve the accuracy. • Hybrid functionals typically outperform pure functionals. • In view of exceptions, DFT usually performs at level of MP2 theory or better, but not as consistent. DFT does a much better job with transition metals than MO theory. • In general, literature provides guidance with regards to performance.