Basis sets

Basis Sets : 

Basis Sets Computational Chemistry 5510 Spring 2006 Hai Lin

Refresh Your Mind: HF Theory : 

Refresh Your Mind: HF Theory Schrödinger Equation Electronic Schrödinger Equation Hartree-Fock Equations Roothann-Hall Equations Best Approximated (Lowest) Energy

What is a Basis Set? : 

What is a Basis Set? A basis set is a combinations of mathematical functions used to represent atomic orbitals. The employed mathematical functions describe the radial and angular distributions of electron density. Based on the LCAO approximation, each one-electron molecular orbital f is approximated by a linear combination of atomic orbitals ci (basis functions) with expansion coefficients ci. f = c1 c1 + c2 c2 + c3 c3 + … s pz py px dz2 dx2-y2 dxy dyz dxz

Slater Type Orbital : 

Slater Type Orbital A Slater Type Orbital (STO) takes the form cz,n,l,m(r,q,f) ? Yl,m(q,f) rn-1e-zr Spherical harmonic function for angular distribution Exponential function for radial distribution r Radial distribution STO describes fairly well the radial electron distribution. However, STO is difficult to handle because integrations can not be calculated analytically. STO is only used in very special cases.

Gaussian Type Orbital : 

Gaussian Type Orbital A Gaussian Type Orbital (GTO) takes the form cz,n,l,m(r,q,f) ? Yl,m(q,f) r2n-2-le-zr2 Spherical harmonic function for angular distribution Exponential function for radial distribution r Radial distribution GTO describes less satisfactorily the radial electron distribution. However, GTO is easy to handle because integrations can be calculated analytically. GTO is used very commonly. Too flat at r ~ 0 Fall off too fast at large r

Approximate STO by GTOs : 

Approximate STO by GTOs STO-nG Slater type orbitals approximated by a linear combination of n primitive Gaussian functions A minimum basis set (only enough functions are used to accommodate all electrons.) Widely used in semi-empirical calculations (especially for n = 3) r Radial distribution r Radial distribution r Radial distribution STO-1G STO-2G STG-3G

Split Basis Sets : 

Split Basis Sets Minimal basis sets (e.g., STO-3G) do not adequately describe anisotropic electron distribution in molecules. Each split basis set has a set of two (or more) functions of different sizes or radial distributions, allowing more flexibility. Split is often made for valence orbitals only, which are chemically important. H C N More diffuse More compact A looser bond (a < b) A tighter bond (a > b) a + b = a + b =

Diffuse Basis Functions : 

Diffuse Basis Functions Normal split valence basis sets describe electron distribution not far away from nuclear centers. Some cases have electron distribution far away from nuclear centers (e.g., anions, molecules with lone pairs of electrons, excited states, transition state) One adds additional s (for H and He) and p (for heavy atoms) functions with very diffuse radial distributions. H H- Cl ? H Cl … H … Br

Polarization Basis Functions : 

Polarization Basis Functions Higher angular momentum functions improves the description for anisotropic electron distribution. Normally p orbitals are added to H and He, d orbitals are added to first-row atoms, f oribtals are added to second-row atoms … H C N Anisotropic distribution that can not be described by s orbitals H C N

Basis Set Contraction : 

Basis Set Contraction Computational cost for HF calculations increases as N4, where N is the number of basis functions. Instead of optimizing all coefficients ci, one predetermines the ratios between some selected coefficients, forming a fixed linear combination of basis functions. For example, f = c1c1 + c2c2 + c3c3 + c4c4 + c5c5 + c6c6 + c7c7 has 7 coefficients (c1 to c7) to be optimized. f = c1(c1 + a2c2 + a3c3 + a4c4 + a5c5) + c6c6 + c7c7 has 3 coefficients (c1, c6, and c7) to be optimized. A loss in accuracy but a gain in efficiency

Basis Set Limit : 

Basis Set Limit The more basis functions, the better representation of the wave function, and thus the lower energy. f = c1c1 + c2c2 + c3c3 + … When the number of basis functions goes infinite, we have the best result corresponding to the complete basis set (CBS) limit. We can not handle infinite number of basis functions, but we may be able to estimate the energy at the CBS limit.

Pople Style Basis Sets : 

Pople Style Basis Sets The basis set notation looks like k-nlm++G** or k-nlm++G(idf,jpd) k primitive GTOs for core electrons n primitive GTOs for inner valence orbitals l primitive GTOs for medium valence orbitals m primitive GTOs for outer valence orbitals + means 1 p diffuse functions added to heavy atoms. ++ means 1 p diffuse functions added to heavy atoms and 1 s diffuse functions added to H atom. * means 1 d polarization functions added to heavy atoms. ** means 1 d polarization functions added to heavy atoms and 1 p polarization functions added to H atom. idf means i d and 1 f polarization functions added to heavy atoms. idf,jpd means i d and 1 f polarization functions added to heavy atoms and j p and 1 d polarization functions added to H atom. E.g., 3-21G, 6-31G, and 6-311G E.g., 6-31+G E.g., 6-31G* E.g., 6-31+G(d,p)

Common Basis Sets : 

Common Basis Sets Pople’s Basis Sets 3-21G 3 primitive GTO for core electrons, 2 for inner and 1 for outer valence orbitals Preliminary geometry optimization; Poor for energy 6-31G 6-31G(d) Common moderate basis set 6-31G(d,p) 6-31+G(d,p) Good for geometry and energy 6-311+G(2df,2p) Good for geometry and accurate energy Dunning’s Correlation-consistent Basis Sets Systematically converge the correlation energy to the basis set limit. Work typically with high-level electron-correlated wave function methods. (aug)-cc-p(C)VXZ, X = D, T, Q, 5, 6, and 7 One can also modify and/or optimize basis sets for a particular task.

Effective Core Potentials : 

Effective Core Potentials Core electrons are chemically unimportant, but require a large number of basis functions for an accurate description of their orbitals. An effective core potential (ECP) is a linear combination of specially designed Gaussian functions that model the core electrons, i.e., the core electrons are represented by a effective potential and one treats only the valence electrons explicitly. Saving computational effort Taking care of relativistic effects partly Important for heavy atoms, e.g., transition metal atoms Core electrons Valence electrons ECP

Select Basis Sets : 

Select Basis Sets Always a compromise between accuracy and computational cost! With the increase of basis set size, calculated energy will converge. We refer such a situation as the complete basis set (CBS) limit. Do you have anything special (anion, transition metal, transtion state)? Use smaller basis sets for preliminary calculations and for heavy duties (e.g., geometry optimizations), and use larger basis sets to refine calculations. Use larger basis sets for critical atoms (e.g., atoms directly involved in bond-breaking/forming), and use smaller basis sets for unimportant atoms (e.g., atoms distant away from active site). Use popular and recommended basis sets. They have been tested a lot and shown to be good for certain types of calculations.

Typical Calculation Procedures : 

Typical Calculation Procedures Read in molecular specification (geometry, charge, multiplicity, ...) Read in method specification (theory, basis set, convergence threshold, ...) Form initial guess for orbitals Compute new orbitals according to Roothaan-Hall equations Differences between new and old orbitals sufficiently small? Additional calculations (gradient, Hessian, dipole, atomic charges, ...) Y N Start End

Summary : 

Summary What is a basis set in the LCAO approximation? Slater & Gaussian type orbitals Split basis sets Diffuse & polarization basis functions Basis set contraction Complete basis set limit Common basis sets Effective core potentials How to select basis sets Typical procedure of calculations

Your Homework : 

Your Homework Read the slides. If you have difficulty in understanding the math in the slides, find a reference. Read textbook §5.1, §5.2, §5.4, §5.4.1, and §5.7. Take notes when you read. Questions: What is a minimum basis set, and what is a triple zeta basis set? What does the basis set notation 6-31+G(d,p) mean? The calculated energy for a given molecule will decrease if the basis set size increases. What will be the changes in energy difference between two molecules (e.g., DE between reactant and product)? Why?