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Lecture 14. Basis Set. Molecular Quantum Mechanics, Atkins & Friedman (4 th ed. 2005), Ch. 9.1-9.6 Essentials of Computational Chemistry. Theories and Models, C. J. Cramer, (2 nd Ed. Wiley, 2004) Ch. 6 Molecular Modeling, A. R. Leach (2 nd ed. Prentice Hall, 2001) Ch. 2

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lecture 14 basis set
Lecture 14. 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 http://gnode2.pnl.gov/bse/portal
  • Basis Sets Lab Activity
  • http://www.shodor.org/chemviz/basis/teachers/background.html
slide2

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)
basis set to expand molecular orbitals

: 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
slide4

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.

slide6

1s

2s

2p

3s

3p

3d

Radial Wave Functions Rnl

node

2 nodes

*Bohr Radius

*Reduced distance

Radial node

(ρ = 4, )

sto basis functions
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.)
slide9

(not orthogonal but normalized)

  or  above

Smaller for Bigger shell (1s<2sp<3spd)

slide10

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
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.
slide12

Double-Zeta Basis Set: Carbon 2s Example

3 for

1s (core)

21 for

2sp (valence)

slide14

Double-Zeta Basis Set: Example

3 for 1s (core)

21 for 2sp (valence)

Not so good agreement

slide15

Triple-Zeta Basis Set: Example

6 for 1s (core)

311 for 2sp (valence)

better agreement

slide16

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
slide17

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)
slide19

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
slide20

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
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
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.
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