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Lecture 14. Basis Set

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- 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

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)

: A set of L preset basis functions

(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

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

1s

2s

2p

3s

3p

3d

Radial Wave Functions Rnl

node

2 nodes

*Bohr Radius

*Reduced distance

Radial node

(ρ = 4, )

- 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

- * 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

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

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

- 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)

- 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.