1. Structure of Presentation
Theory and Computation
Methods: HF, DFT (B3LYP), MP2
Basis sets: 6-31g*, 6-31g**, cc-pvtz
Molecules of Interest
Methylene (H2C)

2. Theory and Computation
Methods and Basis Sets

3. Quantum Mechanics (and Math) (review)
H = E 
Time-dependent wave equation:
we are most interested in the energy of stationary states and thus can ignore the phase
Kinetic Energy
Potential Energy

4. Quantum Mechanics (and Math) (review)
In molecules we must treat each particle this way! (nucleus + electrons)
Molecular Hamiltonian:
Ke nuclei
Ke e-
Ue-n
Ue-e
Un-n

5. Methods Born-Oppenheimer Approximation: Hartree-Fock Method
Accounts for electron exchange
anti-symmetric exchange or spin wave functions
but not for correlation
Mean field approximation
1st term = kinetic energy of electron
2nd term = nuclear attraction of electron
3rd term = electron repulsion (density)
4th term = exchange-correlation energy
True Correlation
Mean-field

6. Methods HF Moller – Plesset 2nd order (MP2) DFT Etrue= EHF + Ecorr
Perturbation model
Finds solutions by adding a “correction” to HF
DFT
Uses densities instead of wavefunctions
Different functionals treat Exc differently
1st term = kinetic energy of electron
2nd term = nuclear attraction of electron
3rd term = electron repulsion (density)
4th term = exchange-correlation energy
Source: Sousa, S. et al General Performance of Density Functionals . J. Phys. Chem. A 2007, 111,

7. Functionals I have been using B3LYP *most popular in literature*
In B3LYP:
EHF parameter is semi-empirically obtained
a, b, and c are optimized on a set of known molecules
- Thus “Hybrid”
a = 0.2
b = 0.72
c = 0.81
LDA = local density approximation
GGA = generalized gradient approximation
Hybrid = a few empirically defined parameters and a hartree fock term
Meta GGA = uses the second derivative in addition to the gradient and the density
Fully non-local
Exc HF = exact exchange = slater determinant of KS orbitals of non-interacting electrons. The rest can be approximated and optimized
Source: Sousa, S. et al General Performance of Density Functionals . J. Phys. Chem. A 2007, 111,

8. Basis Sets Contracted Gaussian-type orbitals (CGTO)
Controls width of gaussian
normalization
Number of gaussians
L = a + b + c
(angular momentum)
Variationally optimized
Why CGTO?
Easy to compute with gaussians
Gaussians are not accurate enough
Linear combination of enough gaussians can approximate a more accurate slater-type orbital
Good conceptual explanation:

9. Basis sets 6-31g* 6-31g** Treatment of Core Orbitals
6 primitive gaussians per orbital (n = 6)
Treatment of valence orbitals
each valence orbital is comprised of 2 “orbitals”
One composed of 3 primitive gaussians (n=3)
One composed of 1 primitive gaussian (n=1)
*= addition of d-polarization functions above He
6-31g**
** = addition of p-polarization functions on H
Good conceptual explanation:

10. Example Carbon atom orbitals Minimal basis set 6-31g 6-31g* 6-31g**
1s, 2s, 2px, 2py, and 2pz orbitals
A total of 5 orbital basis for carbon
6-31g
6 gaussians to mimic a 1s
3 + 1 gaussians to mimic 2s, 2px, 2py, and 2pz EACH
Adding up gaussians:
4 “valence orbitals” from two CGTOs each
Two CGTO made of 3 GTO and one GTO, respectively
Total of 16 primitive gaussians to describe valence orbitals
Total of 22 primitive gaussians to describe carbon
6-31g*
Adds 6 d-orbital CGTO to allow polarization of non-H atoms
6-31g**
Adds 3 p-orbital CGTO to polarize H-atoms

11. Basis sets
cc-pvtz
Correlation consistent – polarized valence triple zeta
Adds extra functions per atom to describe polarization better
(from Gaussian
website)
Atoms
cc-pVTZ H
3s,2p,1d He
Li-Be
4s,3p,2d,1f
B-Ne
Na-Ar
5s,4p,2d,1f Ca
6s,5p,3d,1f
Sc-Zn
7s,6p,4d,2f,1g
Ga-Kr
Good conceptual explanation:

12. Molecules of Interest

13. Methylene Top-down Side view Singlet ΔE = 9.09 kcal/mol Triplet
Source: Carter, E.; Goddard, W. Electron correlation, basis sets, and the methylene singlet–triplet gap. JCP. 1987, 86,

14. Methylene B3LYP/cc-pvtz Singlet Triplet
SCF E(b3lyp-triplet) = -24,578.5 kcal/mol
SCF E(b3lyp-singlet) = -24,566.7 kcal/mol
ΔEts = 11.8 kcal/mol
kcal/mol

15. Methylene (evaluating accuracy)
Computational Results:
ΔEts(HF) = kcal/mol
ΔEts(b3lyp) = 11.8 kcal/mol
ΔEts(MP2) = kcal/mol
Experimental Results:
ΔE = 9.09 kcal/mol