computational

1. Basis Set
In many electron systems, the unknown MOs are expressed in terms of a known set of functions - a basis set.
They are set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method
Basis sets are Linear combination of basis functions approximates total electronic wavefunction.
For polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)




 
 c
’s are called basis functions
Slater Type Orbitals (STOs)
STOs are linear combinations of one electron wavefunctions and are represented as
STOs represent electron density well in valence region and beyond (not so well near nucleus)
They give exact solution for hydrogen atom and are used for atomic calculations. STOs have correct nuclear cusp condition. But STOs are computationally expensive.
Gaussian Type Orbitals (GTOs)
GTOs have general form
GTOs allow easier evaluation of integrals but shows poor performance near the nucleus and at long distances. A solution for this is to use a linear combination of
enough GTOs to mimic an STO – Contracted gaussian type orbitals (CGTO). (GTOs are not orbitals but functions). Functions which are used to represent orbitals are
known as Gaussian primitives. For the complete description of orbitals, a linear combination of gaussian primitives to be used as basis function which is contraction.

2. Eg: STO-3G basis set - Representation of the Slater Type Orbital (STO) as a linear combination of 3 Gaussian primitive functions.
STO-NG designates "N" Gaussian primitives that are used to simulate the STO basis function.
Types of Basis sets
1. Minimal basis sets
In minimal basis sets, one basis function (one CGTO) for every atomic orbital required to describe the free atom
e.g. 1s, 2s, 2px, 2py, 2pz for carbon – 5 basis functions
Most-common: STO-3G
Linear combination of 3 primitive Gaussians fitted to one Slater-type orbital
Also known as single zeta basis set (zeta, , is the exponent used in Slater-type orbitals)
Double Zeta (DZ) Basis Set
Each function in a minimal basis set is doubled (Two basis function for each orbital)
One set is tighter (closer to the nucleus, larger exponents), the other set is looser (further from the nucleus, smaller exponents)
Tripple Zeta (TZ) Basis Set
Three basis functions for each AO.
Split Valence Basis Set
Only the valence part of the basis set is doubled (fewer basis functions means less work and faster calculations and the core orbitals are represented by a minimal basis,
since they are nearly the same in atoms an molecules
Eg: 3-21G (3 gaussians for 1s, 2 gaussians for the inner 2s,2p, 1 gaussian for the outer 2s,2p)
6-31G (6 gaussians for 1s, 3 gaussians for the inner 2s,2p, 1 gaussian for the outer 2s,2p)

3. • Triple-zeta:
• Double-zeta:

4. Polarization Functions
Higher angular momentum functions added to a basis set to allow for angular flexibility
They allow orbitals to change shape
Add p orbitals to H
Add d orbitals to 2nd row atoms
Add f orbitals to transition metals
Egs:
6-31G(d) - d functions per heavy atoms; also denoted: 6-31G*
6-31G(d,p) - d functions per heavy atoms and p functions to H atoms: also deonoted: 6-31G**
Diffuse Functions
“Large” s and p orbitals for “diffuse electrons” (the electron is held far away from the nucleus)
ie, lone pairs, anions, excited states, etc.
Here functions with very small exponents added to a basis set
Egs: 6-31+G – one set of diffuse s and p functions on heavy atoms
6-31++G – a diffuse s function on hydrogen as well as one set of diffuse s and p functions on heavy atoms.

5. Pople Basis Sets
 Contracted basis sets developed by the late Nobel Laureate, John Pople, and popularized by the Gaussian set of programs
 STO-nG are minimal basis sets in which each AO is represented by n Gaussians, chosen to mimic the behavior of a STO
 3-21G is a split valence basis set where core orbitals are a contraction on 3 GTO’s, the inner part of the valence AOs is a contraction of 2 GTO’s and the outer
part is given by 1 GTO.
 Pople’s split-valence double-zeta basis set is called 6-31G; the core orbital is a CGTO made of 6 Gaussians, and the valence is described by two orbitals - one
CGTO made of 3 Gaussians, and one single Gaussian.
 6-31G* [or 6-31G(d)] is 6-31G with added d polarization functions on non-hydrogen atoms; 6-31G** [or 6-31G(d,p)] is 6-31G* plus p polarization functions for
hydrogen
 6-311G is a split-valence triple-zeta basis; it adds one GTO to 6-31G
 6-31+G is 6-31G plus diffuse s and p functions for non hydrogen atoms; 6-31++G has diffuse functions for hydrogen also.
Pseudopotentials, Effective Core Potentials (ECP)
Core orbitals do not change much during chemical interactions and only valence orbitals feel the electrostatic potential of the nuclei and of the core electrons.
Therefore, a pseudopotential or ECP is used to replace the electrostatic potential of the nuclei and of the core electrons. ECP reduces the size of the basis set needed to
represent the atom (but introduces additional approximations).

6. Calculations using Gaussian Software
General form of an input for the calculation in Gaussian
%mem=500mb (specify memory) – not always necessary
#P HF/6-31G(d) (theory and basis set- model chemistry) opt (keyword for calculation)
Blank line
test1 HF/6-31G(d) geom opt formaldehyde (Title section)
Blank line
0 1 (charge and spin multiplicity)
Molecule specification (cartesian coordinates or Z-matrix)

7. Z-Matrix
It is a convenient way to specify the geometry of molecules. It works by identifying each atom in a molecule by a bond
distance, bond angle and dihedral angle (internal coordinates) in relation to other atoms in the molecule.
Building a Z-Matrix:
To construct a Z-matrix, one should follow these steps:
1. Draw the molecule.
2. Assign one atom to be #1.
3. Starting with atom #1, assign all other atoms a sequential number. List the atoms you numbered, in order, down your paper,
one right under the other.
4. Place the atom designated as #1 at the origin of your coordinate system. The first atom does not have any defining
measurements since it is at the origin.
5. To identify the second atom, you must only define its bond length to the first atom.
6. For the third atom, you must define a bond length to atom #1 and a bond angle between atom #3 and atoms #1 and #2. (Bond
angles are the angles between three atoms.)

8. 7. Remember that you can only use previously defined atoms when defining your current atom. This means that you cannot
reference atom #7 when defining atom #5.
8. To identify atom #4 and all other atoms, you must include a bond length, bond angle and a dihedral angle. (Dihedral angles are
the angles between an atom and the plane created by three other atoms.) This is done by using neighboring atoms to the atom you
are describing.