NoteTaking_C2: Computational Organic Chemistry

CHAPTER 1: QUANTUM MECHANICS FOR ORGANIC CHEMISTRY (2)

1. Electron Correlation

The difference between exact energy and the energy at the HF limit is called correlation energy.

2. Configuration Interaction

The total wavefunction is a linear combination of configurations.

How to reduce the computational expense required to calculate the matrix elements of the CI hamiltonian ?

Brillouin's Theorem: the matrix element between HF configuration and any singlet excited configuration is zero.

Condon-Slater rules: Configurations that differ by three or more electron occupancies will be zero.

The CI expansion need only contain configurations that are of the spin and symmetry of interest. 

Two approaches to  reduce the size of matrix:

1) Delete some subset of virtual MOs from being occupied or frozen some core electrons. 

2) Truncate the expansion at some level of excitation. 

CID

CISD

Note: By Brillouin's theorem, the singlet excited configurations will not mix with HF reference, but it can overlap with the double excited configurations.

Size consistency: truncated CI is not size-consistent.

3. Moller-Plesset Perturbation theory

The full Hamiltonian is divided into HF Hamiltonian and a perturbation component that is essentially the instantaneous electron-electron correlation.
MP theory is computationally more efficient than CI, but it is not variational.
Including higher-order corrections is not guaranteed to converge the energy.
MP theory is size-consistent.

4. Coupled-Cluster theory (Cizek)

The total wave-function is the result of an operator on HF wavefunction.
The operator T is an expansion of operators Ti, where the Ti operator generates all of the configurations with i electron excitations.

CCD coupled-cluster doubles
CCSD coupled-cluster singles and doubles
CCSD(T) the effect of triples contributions is incorporated in a perturbative way

5. Multiconfiguration SCF (MCSCF) theory and Complet Active Space SCF (CASCF) theory

The underlying assumption to the CI expansion is that the single-configuration reference, the HF wavefunction, is a reasonable description of the molecule.  In some cases, such as cyclobutadiene with D_4h geometry, the HF wavefunction does not capture the inherent multi-configurational nature of the electron distribution. To capture this non-dynamic correlation, we must determine the set of MOs that best describe each of the configuration. This is equal to optimizing the LCAO coefficients of the MOs of each configuration and the coefficient of each configuration self-consistently. 
CASSCF(n.m)
Complet Active Space SCF (CASSCF) procedure indicates that all configurations involving a set of MOs and a given number of electrons comprise the set of configurations to be used in the MCSCF procedure. 

6. Composite Energy Methods