Quantum Calculations B. Barbiellini Thematics seminar April 21,2005

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

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.

Electronic structure theory H  = E  Ab-initio - from the origins (First-principles) No experimental parameters Few physical constants c, h, m e, q e

min  H|  = E Variational Theorem

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

Climbing Mt. Psi Correlation energy: energy contributions beyond SCF

 = det(  r))det(  r  Independent Particle Model: Hartree-Fock (HF) SCF  is a molecular orbital  is spin up F  =e  F is an effective one-particle hamiltonian which depend on MO’s  Self Consistent Field (SCF).

Linear combination of atomic orbitals termed “basis functions” Basis set – mathematical representation of molecular orbitals Minimal basis set – one basis function for every atomic orbital that is required to describe the free atom H(1s) C(1s,2s,2p) → CH 4 :9 basis functions Larger basis sets are more flexible –better approximation of exact MOs Polarization functions, diffuse functions

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 STOs v. GTOs

Density functional theory Less expensive than post-SCF methods Include some electron correlation E elec = E T + E V + E J + E XC Pure functionals: BP86, BLYP Hybrid HF/DFT: B3LYP Good for geometries, electron affinities Good for large systems Problem: not systematic

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

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.

Distance H-H 

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

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

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

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.