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

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).

NoteTaking_B2: Crystal & Crystal Structures

CH2 LATTICES, PLANES AND DIRECTIONS

1. 14 Bravais Lattices

{cP, cF cI, tP, tI}, {oP, oI, oC, oF},{hR, hP}, {mP, mB, aP}
c: cubic
o: orthorhombic
h: hexagonal
m: monoclinic
a: triclinic
P: primitive
F: face-centered
I: body-centered
B/C: base-centered

2. Lattice planes and Miller indices

1) Miller indices, (hkl),  represent a set of identical parallel lattice planes.

2) "The values of h, k and l are the reciprocals of the fractions of a unit cell edge, a, b and c respectively intersected by an appropriate plane. "
3) "Negative intersections are written with a negative sign over the index (bar)"

How to determine Miller indices?

Trave along the axes in turn and then counting the number of spaces between planes encountered from one lattice point to the next.

4) Curly brakets, {hkl}, designate identical planes by virtue of the symmetry of the crystal.

5) Miller indices for hexagonal lattices : (hkil), where i=-(h+k)

3. Directions

Directions are generated written as [uvw]. The direction of [uvw] is simply the vector pointing from the origin to the lattice point with coordinates u, v,w.

4. Zone

A zone is a set of planes, all of which are parallel to a single direction. 

5. Useful Formula 

NoteTaking_B1: Crsytal & Crystal Structure

CH1. CRYSTALS AND CRYSTAL STRUCTURES

keywords: axes, inter-axial angles (alpha for b and c, beta for c and a, and gamma for a and b)

1. Seven crystal systems

Cubic, Tetragonal, Orthorhombic, Monoclinic, Triclinic, Hexagonal, Trigonal(Rhombohedra).
Rhombohedra crystal is often more conveniently described in terms of hexagonal crystal.

PbTiO3, tetragonal---> cubic
BaTiO3,  rhombohedra ---> orthorhombic ---> tetragonal---> cubic
BiFeO3,  Rhombohderal (Hexagonal)

2.  Strukturbericht symbols for simple structure types

A1: cubic-close packed, face-centered cubic
Copper, Cu

A2: body-centered cubic
Tungsten, W

A3: hexagonal close-packed
(0,0,0), (1/3, 2/3, 1/2)
Magnesium, Mg

Halite: MX
Sodium chloride, NaCl

Rutile, MX2
titanium oxide, TiO2
Ti: 0, 0, 0; 0.5, 0.5, 0.5,
O: 0.3, 0.3, 0.0
      0.8, 0.2, 0.5
      0.7, 0.7, 0.0
      0.2, 0.8, 0.5

Fluorite, MX2
calcium fluroite, CaF2,