Transcript 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
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 (complete if
• Larger basis set give higher-quality wave functions.
(but more computationally-demanding)
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H-atom orbitals
Slater type orbitals (STO; Slater)
Gaussian type orbitals (GTO; Boys)
Numerical basis functions
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Hydrogen-Like (1-Electron) Atom Orbitals
Z 2 e 4
En  
32 2 e02  2 n 2
Z2
... atomic unit (hartree)
orwithEn n 1,2,23in
2n
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
1s
node
2 nodes
2s
2p
3s
3p
3d
*Bohr Radius
40  2
a0 
me e 2
*Reduced distance
Radial node
(ρ = 4, r  2a0 / Z )

2 Zr
a0
STO Basis Functions
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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)
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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
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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
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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.