Quantum Calculations

1. Quantum Calculations B. Barbiellini bba@neu.edu Thematics seminar April 21,2005

2. Goal: Solve the Schrödinger equation Application: Description of chemical bonds

3. Outline • Independent Particle Approximation (IPM) and Hartree Fock (HF) SCF: Basis sets. • Other theoretical methods: DFT and QMC. • Illustrative example: Study of Hydrogen bond in ice and water.

5. Electronic structure theoryH = E Ab-initio - from the origins (First-principles) No experimental parameters Few physical constants c, h, me, qe

6. VariationalTheorem min<| H|> = E

7. Theoretical Methods • SCF & post-SCF methods (CI) • Density functional theory (DFT) • Stochastic methods: Quantum Monte Carlo (QMC)

8. Climbing Mt. Psi Correlation energy: energy contributions beyond SCF

9. Independent Particle Model: Hartree-Fock (HF) SCF  = det(j(a,r))det(j(b,r)) • is a molecular orbital a is spin up F j =e j F is an effective one-particle hamiltonian which depend on MO’s  Self Consistent Field (SCF).

10. Larger basis sets are more flexible • better approximation of exact MOs • Polarization functions, diffuse functions Basis set – mathematical representation of molecular orbitals • Linear combination of atomic orbitals termed “basis functions” • Minimal basis set – one basis function for every atomic orbital that is required to describe the free atom H(1s) C(1s,2s,2p) → CH4: 9 basis functions

11. STOs v. GTOs • Slater-type orbitals (J.C. Slater) • Represent electron density well in valence region and beyond (not so well near nucleus) • Evaluating these integrals is difficult • Gaussian-type orbitals (F. Boys) • Easier to evaluate integrals, but do not represent electron density well • Overcome this by using linear combination of GTOs

12. Density functional theory • Less expensive than post-SCF methods • Include some electron correlation • Eelec = ET + EV + EJ + EXC • Pure functionals: BP86, BLYP • Hybrid HF/DFT: B3LYP • Good for geometries, electron affinities • Good for large systems • Problem: not systematic

13. Example:Gaussian Input basis set method key words } route section blank line blank line charge, multiplicity } molecular structure section atomic symbols (or numbers) xyz coordinates (or z-matrix) blank line #RHF/6-31G(d) Pop=Full Test RHF/6-31G(d) formaldehyde single point 0,1 C 0.0 0.0 0.0 O 0.0 1.22 0.0 H 0.94 -0.54 0.0 H -0.94 -0.54 0.0 } title section

14. Quantum Monte Carlo • Deals with the many body wave-function. • Include electron correlation (Jastrow terms). • Variation QMC --- Stochastic Gradient Approximation (SGA). • Diffusion QMC (almost exact, fixed node approximation) --- computational expensive.

18. b Distance H-H

22. Scattered x rays in ice Isaacs et al., PRL 82 (1999) 600

23. Compton Profile Anisotropy Wavelike fringes corresponding to interference between the electrons on neighboring sigma and hydrogen bonding sites

26. B(r) Fourier transform CP: MO orbital autocorrelation function

27. Conclusion Quantum calculations are of interest because they can deal with electronic effects, electron de-localization, charge-transfer, and other phenomena, which are otherwise difficult or impossible to treat at the level of classical mechanics.