Ab initio
Reactant – Transition State Structure – Product
1.
2.
3.
4.
5.
Selection of the theoretical model
Geometry optimization
Frequency calculation
Energy calculation
Refining the theoretical model
Description of the theoretical model
QCISD(T)/6-311+G(3df,2p)//MP2/6-311G(d,p)
Energy calculation
Geometry optimisation
MP2/6-311G(d,p)
method
basis set
Basis Set
Introduction
Schrödinger equation : HΨ=EΨ
The most important result is
the
ENERGY !!!
Aim to adequate molecular energy
Problem:
- energy is not available
Approximations:
-computer capacity
-CPU time
-size of molecule
Model chemistry: theoretical method and basis set
HF
MP2
CCSD
CCSD(T) CCSDT
…
Minimal
…
Splitvalence
…
Polarized
…
Diffuse
…
High
angular
momentum
…
…
∞
…
…
…
…
…
Full CI
…
…
…
Exact
solution
Goal: select the most accurate calculation that is
computationally feasible for a given molecular system
Model Chemistries - three areas
of consideration
• Basis sets
• Theoretical methods
QCISD(T)/6-311+G(3df,2p)//MP2/6-311G(d,p)
Energy calculation
Geometry optimisation
MP2/6-311G(d,p)
method
basis set
Five statements for
demythologization:
Nº 1.: The term “orbital” is a synonym for the term “OneElectron” Function (OEF)
Nº 2.: A single centered OEF is synonymous with “Atomic
Orbital”. A multi centered OEF is synonymous with
“Molecular Orbital”.
AO
MO
Orbital == OEF
Five statements ….
Nº 3.: 3 ways to express a mathematical function:
• Explicitly in analytical form
(hydrogen-like AOs)
• As a table of numbers
(Hartree-Fock type AOs for numerous atoms)
• In the form of an expansion
(expression of an MO in terms of a set of AO)
f ( x)  e x  C 0 x 0  C1 x1  C 2 x 2  C3 x 3   
f ( x)  e
x
x
f(x)
0.0
1.000
0.1
1.105
0.2
1.221
f (0) 0 f (0) 1 f (0) 2 f (0) 3
x 
x 
x 
x  
0!
1!
2!
3!
  C00  C11  C2 2  C33  
Five statements ….
Nº 4.: The generation of MOs (-s) from AOs (-s) is equivalent to the
transformation of an N-dimensional vector space where {}is the original
set of non-orthogonal functions. After orthogonalization of the nonorthogonal AO basis set {} the orthogonal set {c} is rotated to the another
orthogonal set{}.
c1
1
1
2
c

AO
orthogonalization
c
O AO
SCF

MO

Five statements ….
Nº 5.: There are certain differences between the shape of numerical
Hartree-Fock atomic orbitals (HF-AO), the analytic Slater type orbitals
(STO) and the analytic Gaussian type functions (GTF).
However , these differences are irrelevant to the final results as the MO
can be expanded in terms of any of these complete sets of functions to any
desired degree of accuracy.
Atomic orbital basis sets
Basis set
• Basis functions approximate orbitals of
atoms in molecule
• Linear combination of basis functions
approximates total electronic
wavefunction
• Basis functions are linear combinations
of gaussian functions
– Contracted gaussians
– Primitive gaussians
STOs v. GTOs
• Slater-type orbitals (J.C. Slater)
s( ,r )  cx y z e
n
m
l
- r
– Represent electron density well in valence
region and beyond (not so well near nucleus)
– Evaluating these integrals is difficult

• Gaussian-type orbitals (F. Boys)
g ,r   cx y z e
– Easier to evaluate integrals, but don’t
represent electron density well
– Overcome this by using linear combination of

GTOs
(S )= c   d g
n
μ
p
m
p
l
p
-r 2
Minimal basis set
• One basis function for every atomic
orbital required to describe the free
atom
• Most-common: STO-3G
• Linear combination of 3 Gaussian-type
orbitals fitted to one Slater-type orbital
• CH4: H(1s); C(1s,2s,2px,2py,2pz)
More basis functions per atom
• Split valence basis sets
• Double-zeta: 2 “sizes” of basis functions
for each valence atomic orbital
– 3-21G CH4: H(1s,1s'),
C(1s,2s,2s',2px,2py,2pz,2px',2py',2pz')
• Triple-zeta: 3 “sizes” of basis functions
for each valence atomic orbital
– 6-311G CH4: H(1s,1s',1s''),
C(1s,2s,2s',2s'',2px,2py,2pz, 2px',2py',2pz',2px'',2py'',2pz'')
More basis functions per atom
• Split valence basis sets
• Triple-zeta:
• Double-zeta:
Total
36
22
Total
42
22
Ways to increase a basis set
• Add more basis functions per atom
– allow orbitals to “change size”
• Add polarization functions
– allow orbitals to “change shape”
• Add diffuse functions for electrons with
large radial extent
• Add high angular momentum functions
Add polarization functions
• Allow orbitals to change shape
– Add p orbitals to H
– Add d orbitals to 2nd row atoms
– Add f orbitals to transition metals
• 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**
Add diffuse functions
• “Large” s and p orbitals for “diffuse electrons”
– Lone pairs, anions, excited states, etc.
• 6-31+G - diffuse functions per heavy atom
• 6-31++G - diffuse functions both per heavy
atom and per H atom
High angular momentum functions
• “Custom-made” basis sets
• 6-31G(2d) - 2d functions per heavy atom
• 6-311++G(3df,3pd)
– Triple-zeta valence
– Diffuse functions on heavy atoms, H atoms
– 3d, 1f functions per heavy atom; 3p, 1d
functions per H atom
Minimal basis sets
A common naming convention for minimal basis sets is STO-XG, where X is an integer.
This X value represents the number of Gaussian primitive functions comprising a single basis function.
In these basis sets, the same number of Gaussian primitives comprise core and valence orbitals.
Minimal basis sets typically give rough results that are insufficient for research-quality publication,
but are much cheaper than their larger counterparts. Here is a list of commonly used minimal basis sets:
STO-2G
STO-3G
STO-6G
STO-3G* - Polarized version of STO-3G
Split-valence basis sets
During most molecular bonding, it is the valence electrons which principally take part in the bonding. In recognition of this fact,
it is common to represent valence orbitals by more than one basis function, (each of which can in turn be composed of a fixed
linear combination of primitive Gaussian functions). The notation for these split-valence basis sets is typically X-YZg.
In this case, X represents the number primitive Gaussians comprising each core atomic orbital basis function.
The Y and Z indicate that the valence orbitals are composed of two basis functions each
Here is a list of commonly used split-valence basis sets:
3-21g
3-21g* - Polarized
3-21+g - Diffuse functions
3-21+g* - With polarization and diffuse functions
6-31g
6-31g*
6-31+g*
6-31g(3df, 3pd)
6-311g
6-311g*
6-311+g*
SV(P)
SVP
Double, triple, quadruple zeta basis sets
Basis sets in which there are multiple basis functions corresponding to each atomic orbital,
including both valence orbitals and core orbitals or just the valence orbitals, are called double, triple,
or quadruple-zeta basis sets. Here is a list of commonly used multiple zeta basis sets:
multiple zeta basis sets:
cc-pVDZ - Double-zeta
cc-pVTZ - Triple-zeta
cc-pVQZ - Quadruple-zeta
cc-pV5Z - Quintuple-zeta, etc.
aug-cc-pVDZ, etc. - Augmented versions of the
preceding basis sets with added diffuse functions
TZVPP - Triple-zeta
QZVPP - Quadruple-zeta