What Tools Can We Use?
• Ab Initio (Molecular Orbital) Methods – from the beginning
 full quantum method
 only experimental fundamental constants
 very high accuracy
 complete (all interactions are included)
 systematic improvement possible
 very time consuming (“expensive”)
 only relatively small systems
What is Molecular Orbital Theory?
Molecular orbitals of the hydrogen molecule - constructed from
hydrogen 1s atomic orbitals (basis functions)
s*~ (1sA - 1sB)
1sA


0.735 Å
(H2)
s ~ (1sA + 1sB)
1sB
Ab Initio Methods
Key Steps
Schrödinger Equation
H   E
Space & Time separation
( r ,t )  ( r )( t)  H( r )  E( r )

Non Relativistic Hamiltonian
Separate motion of
electrons and nuclei
2
2
H TV   V
2
Born-Oppenheimer Approximation
Normalisation

( r ,R )  ( r )( R )

2
 c d  1

Antisymmetrisation

(Slater Determinants)
( r1 ,... ri ,r j ,... rn )   ( r1 ,... r j ,ri ,... rn )

Molecular Orbitals
( r )   1 ( r1 ) 2 ( r2 )...  v ( rv )
Ab Initio Methods
Key Steps
Molecular Orbitals

LCAO
 ( r )   1 ( r1 ) 2 ( r2 )...  v ( rv )
N
 i   c i  
 1

Basis Functions

Variation Principle
n m l
2
g(  ,r )  Cx y z exp(  r )
E ( i )  E (  )
For any antisymmetric, normalized
wavefunction, the expectation value for the
energy will always be greater than the energy
of the exact wavefunction.
Assumptions
• Non-Relativistic
– relativistic effects scale naively as Z4/c2 where c = 137
• Born-Oppenheimer approximation
– If nuclear motion is treated as a perturbation, the appropriate parameter
is M–1/4, even for H nuclear motion the effects are expected to be small.
However, there are circumstances where the Born-Oppenheimer
approximation breaks down: crossing electronic states, very high
rotational excitation…
• Finite nuclear size
– ignore
Assumptions
Practical Approximations:
• The Hamiltonian
• The wavefunction is expanded in terms of one-electron functions, the
one-particle space is truncated
• The N-particle basis is not complete, the N-particle space is
truncated
Ab Initio Methods
• Heirarchy of methods to treat electron-electron interactions electron
correlation ie what approximation do we use for H?
-
repulsion
attraction
+
attraction
• Hartree-Fock theory – just consider 1 electron + “average” repulsion
– Need an initial guess of the average repulsion (ie the electron density)
– Iterate until self-consistent
Accuracy of HF theory
• Bond lengths and angles of “normal” organic molecules quite
accurate (within 2%)
• Conformational energies accurate to 1-2 kcal/mol
• Vibrational frequencies for most covalent bonds systematically
too high by 10-12%
• Zero-point vibrational energies ~1-2 kcal/mol
(usually scale by 0.9)
• Protonation/Deprotonation energies ~10kcal/mol (gas phase)
• Reaction barriers may (!!) have very large errors
• Isodesmic reaction energies accurate to 2-5 kcal/mol
(where number of bonds of each type are formally conserved)
Limitations of HF theory
Restricted Hartree-Fock Theory (RHF)
– for an even number of electrons requires double occupancy of
each molecular orbital
– for an odd number of electrons requires that a single (say)
spin up electron is added in an open shell orbital
Consider the dissociation of the H2 molecule:
H+ + H–  H–H  H + H
In RHF H2 can only dissociate to a H+ and H– giving a “dissociation
catastrophe” because we must have doubly occupied orbitals ie a
doubly occupied orbital on H– and no electron on the H+!
Limitations of HF theory
Unrestricted Hartree-Fock Theory (UHF)
– allows molecular orbitals to be singly occupied
– allows the spin up molecular orbitals to be different in energy
(and nature) to the spin down molecular orbitals.
Consider the dissociation of the H2 molecule:
H+ + H–  H–H  H + H
In UHF H2 can dissociate to H + H but at the cost of different
energy spin up and spin down orbitals (what does that mean?) and
the loss of a pure spin states, ie UHF gives an incorrect spin
function which is contaminated by spin states of higher multiplicity!
Basis Sets
• A standard way to solve a differential equation (like the
Schrödinger equation) is to expand the solution in terms of a
set of orthogonal basis functions
• If you choose sensible functions, you might not need too
many…
• We have a wonderful set of sensible orthogonal functions to
choose: atomic basis functions
• Molecular orbitals really do look pretty much like the sum of
the atomic orbitals on the atoms making up the molecule
Atomic Orbitals
• We only really have one-electron orbitals from the H atom,
Slater functions:

STO
(r)  e
 r
• But no closed form for integration…
Gaussian functions


GTO
(r)  e
 r
2
• We can integrate these!


STO
N
GTO
  k n n
n1
Atomic Orbitals
• We can fit multiple Gaussians to Slater functions when we
have to integrate them:
Minimal Basis Sets
• Smallest possible number of atomic orbitals:
– one basis orbital per two inner shell electrons
– one basis orbital for each valence atomic orbital
– For first-row elements there are basis functions resembling
1s, 2s, 2px, 2py, 2pz atomic orbitals.
– The STO's are replaced by n GTO's for the purposes of evaluating
necessary integrals (STO-nG basis set)
These basis set should not be used for any serious calculation!
Minimal (or Single Zeta) basis sets do not work well
because they are not flexible enough to describe how
the atomic orbitals deform in the molecule.
Scaling Basis Sets
• Typically atomic orbitals contract a little in a molecular
environment, we need to model how they get smaller
– We could make the exponent ζ bigger
– But this leads to nasty non-linear optimisation problems…
– We could let the atomic orbital get smaller by linearly combining
it with a smaller orbital (ie with bigger ζ)
– We now have twice as many orbitals but our optimisation is much
easier
Scaling Basis Sets
N-Zeta Basis Sets
• Double-Zeta Basis Sets
– two Slater-type functions for each atomic orbital of the minimal
basis (requiring two exponents, zeta - ), one which is closer to
the nucleus, the other allowing for electron density to move away
from the nucleus
• Triple-Zeta Basis Sets
– three Slater-type functions for each atomic orbital of the minimal
basis (requiring three exponents, zeta – )
• Fit Gaussians to these for the relevant integrals
Split Valence N-Zeta Basis Sets
• Leave the core orbitals as a single Slater function
• Add extra Slater functions only to the valence orbitals
– Valence double-zeta
– Valence triple-zeta
– Valence quadruple-zeta
• Fit Gaussians to these for the relevant integrals, take most
care over the tightest Slater functions, ie the ones closest to
the nucleus
Split Valence N-Zeta Basis Sets
• Valence double-zeta:
– 3 Gaussians per core orbital
– 2 Gaussians per “tight” valence orbital
– 1 Gaussian per “normal” valence orbital
– 6 Gaussians per core orbital
– 3 Gaussians per “tight” valence orbital
– 1 Gaussian per “normal” valence orbital
Pople 3-21G Basis
Pople 6-31G Basis
• Valence triple-zeta:
–
–
–
–
6
3
1
1
Gaussians per core orbital
Gaussians per “tight” valence orbital
Gaussian per “sl smaller” valence orbital
Gaussian per “sl bigger” valence orbital
Pople 6-311G Basis
Split Valence N-Zeta Basis Sets
• Valence double-zeta:
– 3 Gaussians per core orbital
– 2 Gaussians per “tight” valence orbital
– 1 Gaussian per “normal” valence orbital
– 6 Gaussians per core orbital
– 3 Gaussians per “tight” valence orbital
– 1 Gaussian per “normal” valence orbital
Pople 3-21G Basis
Pople 6-31G Basis
• Valence triple-zeta:
–
–
–
–
6
3
1
1
Gaussians per core orbital
Gaussians per “tight” valence orbital
Gaussian per “sl smaller” valence orbital
Gaussian per “sl bigger” valence orbital
Pople 6-311G Basis
Extended Basis Sets
• Orbitals change shape when you make a molecule
• How do you make then “wigglier”
• You add “polarisation” functions
• The next shell up atomic orbitals do exactly what you want!
Polarised Basis Sets
• Add polarisation functions only to non-hydrogen atoms
– H atoms are pretty spherical
• Add one set of extra functions:
– 6-31G(d)/6-31G*
– 6-311G(d)/6-311G*
• Add two sets of polarisation functions
– 6-31G(2df)
– 6-311G(2df)
• Fit each extra polarisation function with a single gaussian if we
have to integrate it
Polarised Basis Sets
• Add polarisation functions only to all atoms
• Add one set of extra functions:
– 6-31G(d,p)/6-31G**
Dunning cc-pVDZ Basis
– 6-311G(d,p)/6-311G** Dunning cc-pVTZ Basis
• Add two sets of polarisation functions
– 6-31G(2df,2pd)
– 6-311G(2df,2pd)
Dunning cc-2pVTZ
• Fit each extra polarisation function with a single gaussian if we
have to integrate it
Polarised Basis Sets
• Add polarisation functions only to all atoms
• Add one set of extra functions:
– 6-31G(d,p)/6-31G**
Dunning cc-pVDZ Basis
– 6-311G(d,p)/6-311G** Dunning cc-pVTZ Basis
• Add two sets of polarisation functions
– 6-31G(2df,2pd)
– 6-311G(2df,2pd)
Dunning cc-2pVTZ
• Fit each extra polarisation function with a single gaussian if we
have to integrate it
In general, polarization functions significantly improve the
description of molecular geometries (bond lengths and angles)
as well as relative energies.
Diffuse Basis Sets
• In some molecules and ions the orbitals expand rather than
contract
– Anions
– Regions of localised negative charge
– Rydberg states…
• Add one set of extra very diffuse (ie very small ) functions
(per shell) model these with a single Gaussian as required
– Add diffuse functions to non-H atoms (H is rarely –ve)
– 6-31+G(d,p)/6-31+G**
Dunning aug’-cc-pVDZ Basis
– 6-311+G(d,p)/6-311+G** Dunning aug’-cc-pVTZ Basis
• Add two sets of diffuse functions
– 6-31++G(d,p)
– 6-311++G(2df,2pd)
Dunning aug’-cc-2pVTZ Basis
Diffuse Basis Sets
• You must use diffuse functions for
–
–
–
–
anions
zwitterions
Rydberg states
weakly bound species
(hydrogen bonds, van der Waals complexes…)
Commonly Used Basis Sets
Basis Set
Description
No. of Basis Functions
H
C,O
H2 O
C6 H 6
STO-3G
Minimal Basis Set. Cheap but
not reliable
1
5
7
36
3-21G
Double-split-valence (or
double-zeta-valence). A small
not always reliable basis set.
2
9
13
66
Split-valence + polarization
basis. A popular mediumsized basis set
2
15
19
102
Another split-valence +
polarization basis with p
functions on H atoms
5
15
25
120
6-31G(d)
6-31G(d,p)
Basis Sets for Heavy Atoms
• Heavy atoms can be problematic
– They have lots of electrons
ignore most of them and replace them with an “effective core
potential” instead
– They have a large +ve nuclear charge so the core electrons
experience huge forces and can travel at relativistic velocities
use “relativistic core potentials” to empirically correct for
relativistic effects
– The energy spacing between the valence and the “core” electrons
gets smaller
use “small core” effective core potentials, or specially designed
“core-valence” basis sets
Basis Set Libraries
https://bse.pnl.gov/bse/portal
The Complete Basis Set (CBS) Limit
• The electronic energy obtained with an infinitely large basis set
• Basis set convergence is depressingly slow
• One-electron basis functions cannot describe the singularities
in the Coulomb potential (or wavefunction) as two electrons
approach each other
Wavefunction
Cusp as two electrons
approach
R12; electron-electron distance
Basis Set Take Home Messages
•
•
•
•
•
•
Convergence is slow!
Need polarization functions (d set on C, N, O.. p set on H…)
For structures and frequencies at least a DZP basis is required to get close to
the HF limit
When mixing different basis sets, ie using different basis sets for different
atoms (eg to model a transition metal complex you may need a specific
transition metal basis and a basis for second row atoms) it is important to
make sure each basis is of the same “quality”, ie do not mix, say, a DZ basis
with a VTZ basis.
A basis set must be physically big enough to span the interactions you are
interested in!
When choosing a basis set, choose the best basis set for your time and
memory limits.