1. Lecture 11. Basis Functions & Basis Set
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch
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
Basis Sets Lab Activity

2. (J. C. Slater & V. Fock, 1930) (Review)
Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli)
(J. C. Slater & V. Fock, 1930) (Review)
Each  has variational parameters (to be changed to minimize E) including the effective nuclear charge  (instead of the formal nuclear charge Z)
Variational condition
Variation with respect to the one-electron orbitals {i}, which are orthonormal
or its combination for lower E
= ij

3. Hartree-Fock equation (One-electron equation) spherically symmetric
Veff includes
spherically symmetric
&
- Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way.
- Any one electron sees only the spatially averaged
position of all other electrons.
- VHF(i) is spherically symmetric.
- (Instantaneous) electron correlation
is ignored.
Spherical harmonics (s, p, d, …) are valid
angular-part eigenfunction (as for H-like atoms).
- Radial-part eigenfunction of H-like atoms are not valid any more.
optimized

4. Solution of HF-SCF equation gives

5. Basis set to expand atomic orbitals
in the one-electron Hartree-Fock equations
: a set of L preset basis functions
(complete if )
Larger basis set give higher-quality wave functions and lower energies
(but they are more computationally-demanding).
H-atom orbitals
Slater type orbitals (STO; Slater)
Gaussian type orbitals (GTO; Boys)
Numerical basis functions

6. Basis set (a set of basis functions)
Slater type
(STO)
Gaussian-type
(GTO)
or r2
larger
basis set
lower
energy

7. 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.

8. Hydrogen-Like (1-Electron) Atom Orbitals

9. Radial Wave Functions Rnl2p 3s 3p 3d
node
2 nodes
*Bohr Radius
*Reduced distance
Radial node
(ρ = 4, )

10. STO Basis Functions GTO 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.)

12. (not orthogonal but normalized)
  or  above
Smaller for Bigger shell (1s<2sp<3spd)

13. 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

14. 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.

15. Double-Zeta Basis Set: Carbon 2s Example
3 for
1s (core)
21 for
2sp (valence)

16. Basis Set Comparison

17. Double-Zeta Basis Set: Example
3 for 1s (core)
21 for 2sp (valence)
Not so good agreement

18. Triple-Zeta Basis Set: Example
6 for 1s (core)
311 for 2sp (valence)
better agreement

19. 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

20. 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)

21. Polarization Functions. Good for Geometries

22. 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

23. 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

24. 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)

29. ab initio or DFT Quantum Chemistry Software
Gaussian
Jaguar ( 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.

30. 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)