Lecture 11. Basis Functions & 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

(J. C. Slater & V. Fock, 1930) (Review) Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli) (J. C. Slater & V. Fock, 1930) (Review) Each  has variational parameters (to be changed to minimize E) including the effective nuclear charge  (instead of the formal nuclear charge Z) Variational condition Variation with respect to the one-electron orbitals {i}, which are orthonormal or its combination for lower E = ij

Hartree-Fock equation (One-electron equation) spherically symmetric Veff includes spherically symmetric & - Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way. - Any one electron sees only the spatially averaged position of all other electrons. - VHF(i) is spherically symmetric. - (Instantaneous) electron correlation is ignored. Spherical harmonics (s, p, d, …) are valid angular-part eigenfunction (as for H-like atoms). - Radial-part eigenfunction of H-like atoms are not valid any more. optimized

Solution of HF-SCF equation gives

Basis set to expand atomic orbitals in the one-electron Hartree-Fock equations : a set of L preset basis functions (complete if ) Larger basis set give higher-quality wave functions and lower energies (but they are more computationally-demanding). H-atom orbitals Slater type orbitals (STO; Slater) Gaussian type orbitals (GTO; Boys) Numerical basis functions

Basis set (a set of basis functions) Slater type (STO) Gaussian-type (GTO) or r2 larger basis set lower energy

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.

Hydrogen-Like (1-Electron) Atom Orbitals

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

STO Basis Functions GTO 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.)

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

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

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.

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

Basis Set Comparison

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

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

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

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)

Polarization Functions. Good for Geometries

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 wave function

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

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)

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.

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)