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• 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)
Hartree-Fock Self-Consistent-Field Methodbased 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
Hartree-Fock equation (One-electron equation)
- 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 validangular-part eigenfunction (as for H-like atoms).
- Radial-part eigenfunction of H-like atoms are not valid any more. optimized
spherically symmetricVeff includes
&
Solution of HF-SCF equation gives
Basis set to expand atomic orbitalsin the one-electron Hartree-Fock equations
• 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
: a set of L preset basis functions(complete if )
Basis set (a set of basis functions)
or r2
Slater type (STO)
Gaussian-type (GTO)
larger basis set
lower energy
...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
Hydrogen-Like (1-Electron) Atom Orbitals
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
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.)
(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)
Minimal CGF basis set
STO
GTO primitive
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-Caf-functions for transition metal elements
Extended Basis Set: Polarization Function
• The orbitals can distort and adapt better to the molecular environment.
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)
ab initio or DFT Quantum Chemistry Software
• Gaussian• Jaguar (http://www.schrodinger.com): Manuals on