NoteTaking_C1: Computational Organic Chemistry

CHAPTER 1: QUANTUM MECHANICS FOR ORGANIC CHEMISTRY (1)

Basically, this chapter gives a handed summary of the fundamentals of quantum mechanics. I will only cover several key concepts here.

1) Born-Oppenheimer Approximation: 

The total wavefunction is the product of nuclear wavefunction and electronic wavefunction. The approximation is based on the fact that electrons are much lighter than nuclei, and therefore can move much faster. That means electrons can response instantaneously to any changes in the relative positions of nuclei.

2) Hartree-Fock Method

Hartree proposed that the total electronic wavefunction can be separated into a product of one-electron wavefunctions.
Fork suggested using the Slater determinant which is antisymmetric and satisfies Pauli principle as the one-electron wave-function.
Hartree-Fork method is a mean-field method. That is to say each electron is moving in an effective potential produced by the average positions of the remaining electrons. It neglects instantaneous electron-electron interactions.

3) Linear Combination of Atomic Orbitals Approximation (LCAO) 

The molecular orbitals (which are used to construct the Slater determinant) are approximated as an linear combination of the atomic orbitals.

4) Hatree-Fock-Roothaan Procedure

5) RHF, ROHF and UHF

RHF and ROHF: Spin up and spin down electrons share the same spatial description.
UHF: Spin up and spin down electrons do not have the same spatial description.

6) Variational Principle

7) Basis sets

Slater-type orbitals (STOs) vs Gaussian-type orbitals (GTOs)

STOs come from the exact solution of the Shcrodinger equation of hydrogen atom. The integrals of STOs can be only solved using an infinite series and truncation of this infinite series can cause serious errors.

GTO is a gaussian function that mimics the shape of a STO. The integrals can be solved exactly. The trade-off is that GTOs differ in shape of STO: STO has a cusp but GTO is a continuous differentiable.

Single zeta: one basis function for every formally occupied or partially occupied orbitals.
Double zeta: two
Triple zeta: three

The basis functions are usually made up of multiple Gaussian functions.

Polarization functions: a set of functions that mimic the atomic orbitals with angular momentum greater than one.

Example:
carbon: d GTOs; hydorgen: p GTOs;


"*":  adding a set of  polarization functions to all the atoms except hydrogen
"**": adding a set of polarization functions to all the atoms
(2df, 2p): two sets of d functions and one set of f functions are added to nonhydrogen atoms; two sets of p functions are added to hydrogen atoms


Diffuse functions allow the electron density to expand to a larger volume, for example, in the case of long pairs.


Split-valence basis sets (Pople):  6-311G+(2df, p)
The split-valence basis sets were constructed by minimizing the energy of the atom at the HF level with respect to contraction coefficients.

Correlation-consistent basis sets (Dunning): aug-cc-pVNZ, where N is the number of degree to which the valence space is split. As N increases, the numer of polarization functions also increase. "aug" meas the addition of diffuse functions.
The correlation-consistent basis sets were constructed to extract the maximum electron correlation energy for each atom.

Basis sets are built into the common computational chemistry programs. A valu- able web-enabled database for retrieval of basis sets is available from the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory “EMSL Gaussian Basis Set Order Form” (http://www.emsl.pnl.gov/forms/ basisform.html).