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BASIS SET
The atom-centeredfunctions used to describe the atomic orbitals are known as basis
functions and collectively form a basis set.
The basis set is an approximate representation of the atomic orbitals (AOS)-They help
us to calculate molecular orbitals(MOS) using the Linear Combination of Atomic
Orbitals(LCAO)approximation.
They are a Set of One-particle Functions used to build Molecular Orbitals.
They are intended to convert the partial differential equations of the model into
algebraic equations suitable for efficient implementation on a computer.
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Larger basissets give a better approximation to the atomic orbitals
as they place fewer restrictions on the wavefunction.
Larger basis sets attract a higher computational cost.
Basis sets are carefully designed to give the best description for the
lowest cost
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A set offunctions is called the basis set.
A basis set is a set of basis functions that are centred on a specific atom.
e density is large in core orbitals, bonds, lone pairs.
e density is smaller in orbitals that are far away from nuclei.
The basis function should describe something.
Basis Sets and Basis Functions
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Consider spherical harmonicsas an example. It is a function
which can describe the angular properties of hydrogen atom.
But it fails to describe hydrogen atom without the radial
function.
By considering both radial function and spherical harmonics,
H atom can be solved
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Basis Sets inQuantum Chemistry
LCAO-MO approximation: MO's built from AO’s
An"orbital"is a one-electron function
AO's represented by atom-centred Gaussians in most quantum
chemistry programs
Some older programs used "Slater functions"(STO's)
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N is anormalization constant
a, b, c control angular momentum, L= a + b + c
ς (zeta)controls the width of the orbital(Large ς gives
tight function,
Small ς gives diffuse function)
These are H-atom-like functions, at least for 1s;
however, they lack radial nodes and are not pure
spherical harmonics but possess correct short-range and
long-range behaviour.
SLATER TYPE ORBITAL
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Gaussian Type Orbitals
Again,a,b, c control angular momentum,
L = a+b+c
Again ς controls width of orbital
No longer H-atom-like, even for 1s
Much easier to compute
Almost universally used by quantum
chemists
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STO & GTO
STOSrequire more calculating, which takes tremendous amounts of
time, however their calculations have been found to be more accurate
than GTOS.
On the other hand, GTOs, although less accurate, are much faster to
calculate than STOS.
By adding several GTOS, we can mimic the STOs accuracy.
As the number of GTOs used increased, the better they were able to
model the STO equation.
When using GTOs to model STOS, the new equations are given a new
name.
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They are identifiedas STO-nG equations where is n constant that
represents the number of GTOS used.
For instance, two common equations are the STO-3G and the STO-6G in
which 3 and 6 GTOs are used respectively.
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Contracted Gaussian typeorbitals (CGTO’s)
STO's are more accurate, but it takes longer to compute integrals using
them.
So we use a linear combination of enough GTO's to mimic an STOis
called an STO-nG even though it is made of contracted GTO’s
Reduces the number of basis functions
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Types of Basisset
• Minimal basis sets contain the minimum number of basis
functions all of the electrons in the atom
For example:
1. H a single function (1s)
2. C has 5 functions, (1s, 2s,2px,2py,2pz)
3. Al has 9 functions,(1s,2s,2px,2py,2pz,3s,3px,3py,3pz)
• Functions are always added in shells
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1)Minimal basis set:
•One basis function (STO, GTO,or CGTO)for each atomic orbital in the
atom
• The STO-3G basis set is a minimal basis set where each atomic orbital is
made up of 3 Gaussians.
• Minimal basis sets are not well suited to model the anisotropic effects
of bonding
• the exponents do not vary, the orbitals have a fixed size and therefore
cannot expand or contract
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2)Double-zeta basis set:Two basis functions for each AO. It allows the
treatment of spatially different bonds at the same atom.
• Zeta value accounts for the size of the AO
ς
3)Triple-zeta basis sets- Three basis functions for each AO and etc.
• Having different-sized functions allows the orbital to get bigger or
smaller when other atoms approach it.
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Benefits of MultipleZeta Basis Sets:
• Each contracted Gaussian function gets a variational coefficient the
definition of molecular orbitals
• More coefficients means more variational flexibility to get a lower
energy wavefunction
• More basis functions gives more flexibility in describing bonding
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Split Valence Basisset
• "Split-Valence“ basis uses only one basis function for each core AO, and a larger basis
for the valence AO’s.
• In the Split-valence basis set each valence orbital are modeled by two or more basis
functions that have different exponents.
• They allow for size variations that occur in bonding.
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Polarization Techniques
• Asother atoms approach, an atom's orbitals might want to shift to one
side or the other(polarization).
• An s orbital can polarize in one direction if it's mixed with a p orbital.
• p orbitals can polarize if mixed with d orbitals.
• In general, to polarize a basis function with angular momentum l, mix
it with basis functions of angular momentum l +1
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• Polarization functionshave higher angular momentum
• They allow for anisotropic variations that occur in bonding and help
model the inter-electronic cusp.
• This gives "polarized double-zeta", or "double-zeta plus polarization" basis sets,
etc.
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Diffuse Functions
Diffuse functionshave small exponents; this means the electron is
held far away from the nucleus.
Necessary for anions, Rydberg states, very electronegative atoms
(fluorine) with a lot of electron density.
Necessary for accurate polarizabilities or binding energies of van der
Waals complexes.
It is very bad to do computations on anions without using diffuse
functions; results could change completely.
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Poples Basis Set
Developedby the late Nobel Laureate, John Pople, and popularized by
the Gaussian set of programs
STO-3G is a minimal basis set in which each AO is represented by 3
Gaussian, chosen to mimic the behavior of an STO
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.
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• 6-31G* [or6-31G(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 added with diffuse functions for non- hydrogen atoms.
• 6-31 ++ G has diffuse functions for hydrogen also
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STO-3G – eachoccupied atomic orbital is constructed from three gaussian
functions.
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Basis set DescriptionNo of functions for
H Water Benzene
STO-3G Minimal 1 7 36
6-31G Double zeta split valence 2 13 66
6-31G(d) Double zeta split valence with d polarisation 2 9 102
6-31G(d,p) Double zeta split valence with d & p
polarisation
5 25 120
6-311+G(d,p) Triple zeta split valence with d & p
polarisation and diffuse functions for non
hydrogen atoms
6 34 168
Most read
Most read
Most read![21
• 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 added with diffuse functions for non- hydrogen atoms.
• 6-31 ++ G has diffuse functions for hydrogen also](/proxy?url=https%3A%2F%2Fimage.slidesharecdn.com%2Fbasissets-250313160805-ae2a2d42%2F85%2FBasis-sets-for-learning-research-methodoloy-21-320.jpg)
