Ab initio

Reactant – Transition State Structure – Product

1. Selection of the theoretical model

2. Geometry optimization

3. Frequency calculation

4. Energy calculation

5. 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)

basis setmethod

Basis Set

Introduction

Schrödinger equation : HΨ=EΨ

The most important result is

the ENERGYENERGY !!! Aim to adequate molecular energy

Problem: - energy is not available

Approximations:

-computer capacity

-CPU time

-size of molecule

Goal: select the most accurate calculation that is computationally feasible for a given molecular system

Model chemistry: theoretical method and basis set

HF MP2 CCSD CCSD(T) CCSDT … Full CI

Minimal …Split-

valence …

Polarized …

Diffuse …High

angular momentum

…

… … … … … … … …

∞ … Exact solution

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)

basis setmethod

Five statements for demythologization:

Nº 1.: The term “orbital” is a synonym for the term “One-Electron” Function (OEF)

Nº 2.: A single centered OEF is synonymous with “Atomic Orbital”. A multi centered OEF is synonymous with “Molecular Orbital”.

Orbital == OEF

AO MO

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)

xexf )(

x f(x)

0.0 1.000

0.1 1.105

0.2 1.221

321033

22

11

00 !3

)0(

!2

)0(

!1

)0(

!0

)0()( x

fx

fx

fx

fxCxCxCxCexf x

33221100 CCCC

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 non-orthogonal AO basis set {} the orthogonal set {} is rotated to the another orthogonal set{}.

1

1

2

1

AO AO MO

orthogonalization SCF

Five statements ….

O

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.

Five statements ….

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

• Slater-type orbitals (J.C. Slater)

– Represent electron density well in valence region and beyond (not so well near nucleus)

– Evaluating these integrals is difficult

• Gaussian-type orbitals (F. Boys)

– Easier to evaluate integrals, but don’t represent electron density well

– Overcome this by using linear combination of GTOs

STOs v. GTOs

g ,r cx n y m z le r 2

d

pg

pp

s( ,r )cx n y m z le r

(Sμ)=

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

• Double-zeta: • Triple-zeta:

36 22Total

42 22Total

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 setsA 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 setsDuring 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 setsBasis 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