Lecture 14 basis set
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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

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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


Lecture 14 basis set

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


Lecture 14 basis set

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.


Lecture 14 basis set

Hydrogen-Like (1-Electron) Atom Orbitals


Lecture 14 basis set

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.)


Lecture 14 basis set

(not orthogonal but normalized)

  or  above

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


Lecture 14 basis set

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.


Lecture 14 basis set

Double-Zeta Basis Set: Carbon 2s Example

3 for

1s (core)

21 for

2sp (valence)


Lecture 14 basis set

Basis Set Comparison


Lecture 14 basis set

Double-Zeta Basis Set: Example

3 for 1s (core)

21 for 2sp (valence)

Not so good agreement


Lecture 14 basis set

Triple-Zeta Basis Set: Example

6 for 1s (core)

311 for 2sp (valence)

better agreement


Lecture 14 basis set

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


Lecture 14 basis set

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)


Lecture 14 basis set

Polarization Functions. Good for Geometries


Lecture 14 basis set

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


Lecture 14 basis set

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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