1. science
2. chemistry

Basis Sets
Patrick Briddon
Contents


What is a basis set? Why do we need them?
Gaussian basis sets
Uncontracted
 Contracted



Accuracy: a case study
Some concluding thoughts
What is a basis set?
Solutions to the Schrödinger equation:
1d

 V  E
2
2 dx
2
are continuous functions, ψ(x).
→ not good for a modern computer (discrete)
Why a basis set?


Idea:
write the solution in terms of a series of functions:
 x    cii x 
i

The function Ψ is then “stored” as a number of
coefficients:
c1 , c2 , c3 , 
A few questions …



What shall I choose for the functions?
How many of them do I need?
How do I work out what the correct coefficients
are?
Choosing Basis functions

V
ψ
Try to imagine what the true wavefunction will
be like:
Choosing Basis functions
ψ
Basis states
The coefficients


These are determined by using the variational
principle of quantum mechanics.
If we have a trial wave-function:
 x    cii x 
i

Choose the coefficients to minimise the energy.
How many basis functions?

The more the better (i.e. the more accurate).


The more you use, the slower the calculation!


Energy always greater than true energy, but
approaches it from above.
In fact time depends on number-cubed!
The better they are, the fewer you need.
Basis sets ad LCAO/MO


There is a close relationship between chemistry
ideas and basis sets.
Think about the H2 molecule:
   1s H1   1s H 2 
   1s H1   1s H 2 
*
Basis sets and LCAO




Physicists call this LCAO (“linear combination
of atomic orbitals”)
The basis functions are the atomic orbitals
Chemists call this “molecular orbital theory”
There is a big difference though:
In LCAO/MO the number of basis functions is
equal to the number of MOs.
 There is no “variational freedom”.

What about our basis functions?

Atomic orbitals are fine, but they are:
Not well defined – you can’t push a button on a
calculator and get one!
 Cumbersome to use on a computer


AIMPRO used Gaussian orbitals

It is called a “Gaussian Orbital” code.
Gaussian Orbitals

The idea:

 r    ci exp   i r  R i 2

i

There are thus three ingredients:



An “exponent”,  – controls the width of the Gaussian.
A “centre” R – controls the location
A coefficient – varied to minimise the energy
The Exponents

Typically vary between 0.1 and 10





Si: 0.12 up to 4;
F: 0.25 up to 10
These are harder to find than coefficients.
Small or large exponents are dangerous
Fixed in a typical AIMPRO run:




determined for atom or reference solid.
i.e. vary exponents to get the lowest energy for bulk Si;
Put into “hgh-pots”
then keep them fixed when we look at other defect systems.
The Positions/Coefficients

Positions: we put functions on all atoms
In the past we put them on bond centres too
 Abandoned – what if a bond disappears during a
run?
 You cannot put two identical functions on the same
atom – the functions must all be different.
 That is why small exponents are dangerous.


Coefficients: AIMPRO does that for you!
How good are Gaussians?

Problems near the nucleus?
True AE wave function was a cusp
 … but the pseudo wave function does not!

How good are Gaussians?

Problems at large distance?
True wave function decays exponentially: exp[-br]
 Our function will decay more quickly: exp[-br2]



Not ideal, but is not usually important for
chemical bonding.
Could be important for VdW forces


But DFT doesn’t get them right anyway
Only ever likely to be an issue for surfaces or
molecules (our solution: ghost orbitals)
AIMPRO basis set


We do not only use s-orbitals of course.
Modify Gaussians to form Cartesian Gaussian
functions:


  p    y  R exp   r  R  
  p   z  R  exp   r  R  
  p x   x  Rix  exp   i r  R i 2
2
y
iy
i
i
2
z

iz
i
i
Alongside the s orbital that will give 4 independent
functions for the exponent.
What about d’s?

We continue, multiplying by 2 pre-factors:


 y    y  R  exp   r  R  
 z   z  R  exp   r  R  
 xy  x  R   y  R exp   r  R  
 xz  x  R z  R  exp   r  R  
  yz    y  R z  R  exp   r  R  
 x 2   x  Rix 2 exp   i r  R i 2
2
2
2
iy
i
i
2
2
2
iz
i
i
2
ix
iy
i
i
2
ix
iz
i
i
2
iy
iz
i
i
What about d’s?

This introduces 6 further functions



i.e. giving 10 including the s and p’s
Of these 6 functions, 5 are the d-orbitals
One is an additional s-type orbital:

 
 x 2    y 2    z 2   x  Rix 2   y  Riy 2  z  Riz 2 exp   i r  R i 2

 r  R i  exp   i r  R i 
2
2


ddpp and all that







We often label basis sets as “ddpp”.
What does this mean?
4 letters means 4 different exponents.
The first (smallest) has s/p/d functions (10)
The next also has s/p/d functions (10)
The last two (largest exponents) have s/p (4 each)
Total of 28 functions
Can we do better?

Add more d-functions:




Add more exponents



“dddd” with 40 functions per atom
this can be important if states high in the conduction
band are needed (EELS).
Clearly crucial for elements like Fe!
ddppp
Pddppp
Put functions in extra places (bond centres)

Not recommended
How good is the energy?





We can get the energy of an atom to 1 meV when
the basis fitted.
BUT: larger errors encountered when transferring
that basis set to a defect.
The energy is not well converged.
But energy differences can be converged.
So:
ONLY SUBTRACT ENERGIES CALCULATED
WITH THE SAME BASIS SET!
Other properties





Structure converges fastest with basis set
Energy differences converge next fastest
Conduction band converges more slowly
Vibrational frequencies also require care.
Important to be sure, the basis set you are using
is good enough for the property that you are
calculating!
Contracted basis sets


A way to reduce the number of functions whilst
maintaining accuracy.
Combine all four s-functions together to create a single
combination:
 s   0.1e




0.1r 2
 0.2e
0.5r 2
 0.7e
1.4 r 2
 0.3e
3.5r 2
The 0.1, 0.2, etc. are chosen to do the best for bulk Si.
They are then frozen – kept the same for large runs.
Do the same for the p-orbitals.
This gives 4 contracted orbitals
The C4G basis


These 4 orbitals provide a very small basis set.
How much faster than ddpp?




Answer: (28/7)3 or 343 times!
Sadly: not good enough!
You will probably never hear this spoken of!
Chemistry equivalent: “STO-3G”

Also regarded as rubbish!
The C44G basis

Next step up: choose two different s/p combinations:
 s1   0.1e
0.1r 2
 s2   0.4e

 0.2e
 0.5 r 2
 0.7e
1.4 r 2
 0.5e
1.4 r 2
We will now have 8 functions per atom.



 0.1r 2
 0.2e
0.5 r 2
(8/4)3 or 8 times slower than C4G!
(28/8)3 or 43 times faster than ddpp.
Sadly: still not good enough!
 0.3e
3.5 r 2
 0.4e
3.5 r 2
The C44G* basis






Main shortcoming: change of shape of s/p
functions when solid is formed.
Need d-type functions.
Add 5 of these.
Gives 13 functions
What we call C44G* (again “PRB speak”)
Similar to chemists 6-31G*
The C44G* basis




13 functions still (28/13)3 times faster than ddpp
Diamond generally very good
Si: conduction band not converged – various
approaches (Jon’s article on Wiki)
Chemists use 6-31G* for much routine work.
Results for Si (JPG)
Basis
Num
Expt
Etot/at
(Ha)
Erel/at
(eV)
a0
(au)
B0
(GPa)
Eg
(eV)
10.263
97.9
1.17
216
Time (s)
512
dddd
40
-3.96667 0.000 10.175
95.7
0.47
25339
ddpp
28
-3.96431 0.064 10.195
96.9
0.52
8348
27173
C44G*
13
-3.96350 0.086 10.192
98.5
0.74
1149
4085
Si-C4G
4
-3.94271 0.652 10.390
92.1
2.28
107
411
The way forwards?





13 functions still (28/13)3 times faster than ddpp
4 functions was (28/4)3 times faster.
Idea at Nantes: form combinations not just of
functions on one atom.
Be very careful how you do this.
Accuracy can be “as good as” ddpp.
Plane Waves



Another common basis set is the set of plane waves
– recall the nearly free electron model.
We can form simple ideas about the band structure
of solids by considering free electrons.
Plane waves are the equivalent to “atomic orbitals”
for free electrons.
 r    cG e
G
iG r
Gaussians vs Plane Waves

Number of Gaussians is very small
Gaussians: 20/atom
 Plane Waves: 1000/atom



Well written Gaussian codes are therefore faster.
Plane waves are systematic: no assumption as to
true wave function
Assumptions are dangerous (they can be wrong!)
 … but they enable more work if they are faster

Gaussians vs Plane Waves

Plane waves can be increased until energy converges



In reality it is not possible for large systems.
Number of Gaussians cannot be increased indefinitely
Gaussians good when we have a single “difficult atom”




Carbon needs a lot of pane waves → SLOW!
1 C atom in 512 atom Si cell as slow as diamond
True for 2p elements (C, N, O, F) and 3d metals.
Gaussians codes are much faster for these.
In conclusion





Basis set is fundamental to what we do.
A quick look at the mysterious “hgh-pots”.
Uncontracted and contracted Gaussian bases.
Rate of convergence depends on property.
A good publication will demonstrate that results
are converged with respect to basis.