lecture 14. basis set
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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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)
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Basis Set to Expand Molecular Orbitals
• 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
: A set of L preset basis functions(complete if )
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...3,2,1 with 32 222
02
42
nne
eZEn
2
2
2n
ZEn or in atomic unit (hartree)
Each state is designated by four (3+1) quantum numbers n, l, ml, and ms.
Ground state
Hydrogen-Like (1-Electron) Atom Orbitals
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Hydrogen-Like (1-Electron) Atom Orbitals
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Radial Wave Functions Rnl
1s
2s
2p
3s
3p
3d
*Reduced distance
*Bohr Radius
2
20
0
4
ema
e
0
2
a
ZrRadial node
(ρ = 4, ) Zar /2 0
2 nodesnode
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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.)
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(not orthogonal but normalized)
or above
Smaller for Bigger shell (1s<2sp<3spd)
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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)
Minimal CGF basis set
STO
GTO primitive
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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.
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Double-Zeta Basis Set: Carbon 2s Example
3 for 1s (core)
21 for 2sp
(valence)
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Basis Set Comparison
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Double-Zeta Basis Set: Example
3 for 1s (core)
21 for 2sp (valence)
Not so good agreement
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Triple-Zeta Basis Set: Example
6 for 1s (core)
311 for 2sp (valence)
better agreement
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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-Caf-functions for transition metal elements
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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)
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Polarization Functions. Good for Geometries
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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
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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
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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 aresignificantly greater than the electron rest mass) is taken into account byrelativistic ECP.
• Hay and Wadt (ECP and optimized basis set) from Los Alamos (LANL)
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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.