Lecture 12. Basis Set

1. Lecture 12. Basis Set • Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 9.1-9.6 • Essentials of Computational Chemistry. Theories and Models, C. J. Cramer, • (2nd Ed. Wiley, 2004) Ch. 6 • Molecular Modeling, A. R. Leach (2nd ed. Prentice Hall, 2001) Ch. 2 • Introduction to Computational Chemistry, F. Jensen (2nd ed. 2006) Ch. 3 • Computational chemistry: Introduction to the theory and applications of • molecular and quantum mechanics, E. Lewars (Kluwer, 2004) Ch. 5 • LCAO-MO: Hartree-Fock-Roothaan-Hall equation, • C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) • EMSL Basis Set Exchange https://bse.pnl.gov/bse/portal • Basis Sets Lab Activity • http://www.shodor.org/chemviz/basis/teachers/background.html

2. Solving One-Electron Hartree-Fock Equations LCAO-MO Approximation Linear Combination of Atomic Orbitals for Molecular Orbital • Roothaan and Hall (1951) Rev. Mod. Phys. 23, 69 • Makes the one-electron HF equations computationally accessible • Non-linear  Linear problem (The coefficients { } are the variables)

3. : A set of L preset basis functions Basis Set to Expand Molecular Orbitals (complete if ) • Larger basis set give higher-quality wave functions. • (but more computationally-demanding) • H-atom orbitals • Slater type orbitals (STO; Slater) • Gaussian type orbitals (GTO; Boys) • Numerical basis functions

4. Hydrogen-Like (1-Electron) Atom Orbitals or in atomic unit (hartree) Ground state Each state is designated by four (3+1) quantum numbers n, l, ml, and ms.

5. Hydrogen-Like (1-Electron) Atom Orbitals

6. 1s 2s 2p 3s 3p 3d Radial Wave Functions Rnl node 2 nodes *Bohr Radius *Reduced distance Radial node (ρ = 4, )

7. STO Basis Functions • Correct cusp behavior (finite derivative) at r  0 • Desired exponential decay at r  • Correctly mimic the H atom orbitals • Would be more natural choice • No analytic method to evaluate the coulomb and XC (or exchange) integrals GTO Basis Functions • Wrong cusp behavior (zero slope) at r  0 • Wrong decay behavior (too rapid) at r  • Analytic evaluation of the coulomb and XC (or exchange) integrals • (The product of the gaussian "primitives" is another gaussian.)

9. (not orthogonal but normalized)   or  above Smaller for Bigger shell (1s<2sp<3spd)

10. Contracted Gaussian Functions (CGF) • The product of the gaussian "primitives" is another gaussian. • Integrals are easily calculated. Computational advantage • The price we pay is loss of accuracy. • To compensate for this loss, we combine GTOs. • By adding several GTOs, you get a good approximation of the STO. • The more GTOs we combine, the more accurate the result. • STO-nG (n: the number of GTOs combined to approximate the STO) STO GTO primitive Minimal CGF basis set

11. Extended Basis Set: Split Valence • * minimal basis sets (STO-3G) • A single CGF for each AO up to valence electrons • Double-Zeta (: STO exponent) Basis Sets (DZ) • Inert core orbitals: with a single CGF (STO-3G, STO-6G, etc) • Valence orbitals: with a double set of CGFs • Pople’s 3-21G, 6-31G, etc. • Triple-Zeta Basis Sets (TZ) • Inert core orbitals: with a single CGF • Valence orbitals: with a triple set of CGFs • Pople’s 6-311G, etc.

12. Double-Zeta Basis Set: Carbon 2s Example 3 for 1s (core) 21 for 2sp (valence)

13. Basis Set Comparison

14. Double-Zeta Basis Set: Example 3 for 1s (core) 21 for 2sp (valence) Not so good agreement

15. Triple-Zeta Basis Set: Example 6 for 1s (core) 311 for 2sp (valence) better agreement

16. Extended Basis Set: Polarization Function • Functions of higher angular momentum than those occupied in the atom • p-functions for H-He, • d-functions for Li-Ca • f-functions for transition metal elements

17. Extended Basis Set: Polarization Function • The orbitals can distort and adapt better to the molecular environment. • (Example) Double-Zeta Polarization (DZP) or Split-Valence Polarization (SVP) • 6-31G(d,p) = 6-31G**, 6-31G(d) = 6-31G* (Pople)

18. Polarization Functions. Good for Geometries

19. wave function Extended Basis Set: Diffuse Function • Core electrons and electrons engaged in bonding are tightly bound. •  Basis sets usually concentrate on the inner shell electrons. • (The tail of wave function is not really a factor in calculations.) • In anions and in excited states, loosely bond electrons become important. • (The tail of wave function is now important.) •  We supplement with diffuse functions • (which has very small exponents to represent the tail). • + when added to H • ++ when added to others

20. Dunning’s Correlation-Consistent Basis Set • Augmented with functions with even higher angular momentum • cc-pVDZ (correlation-consistent polarized valence double zeta) • cc-pVTZ (triple zeta) • cc-pVQZ (quadruple zeta) • cc-pV5Z (quintuple zeta) (14s8p4d3f2g1h)/[6s5p4d3f2g1h] Basis Set Sizes

21. Effective Core Potentials (ECP) or Pseudo-potentials • From about the third row of the periodic table (K-) • Large number of electrons slows down the calculation. • Extra electrons are mostly core electrons. • A minimal representation will be adequate. • Replace the core electrons with analytic functions • (added to the Fock operator) representing • the combined nuclear-electronic core to the valence electrons. • Relativistic effect (the masses of the inner electrons of heavy atoms are • significantly greater than the electron rest mass) is taken into account by • relativistic ECP. • Hay and Wadt (ECP and optimized basis set) from Los Alamos (LANL)

26. ab initio or DFT Quantum Chemistry Software • Gaussian • Jaguar (http://www.schrodinger.com): Manuals on website • Turbomole • DGauss • DeMon • GAMESS • ADF (STO basis sets) • DMol (Numerical basis sets) • VASP (periodic, solid state, Plane wave basis sets) • PWSCF (periodic, solid state, Plane wave basis sets) • CASTEP (periodic, solid state, Plane wave basis sets) • SIESTA (periodic, solid state, gaussian basis sets) • CRYSTAL (periodic, solid state, gaussian basis sets) • etc.