# Basis sets - Staff.ncl.ac.uk

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29 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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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 Schr
ö
dinger equation:

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:

The function
Ψ

is then “stored” as a number of
coefficients:

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

Try to imagine what the true wavefunction will
be like:

V

ψ

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:

Choose the coefficients to minimise the energy.

How many basis functions?

The more the better (i.e. the more accurate).

Energy always greater than true energy, but
approaches it from above.

The more you use, the slower the calculation!

In fact time depends on number
-
cubed!

The better they are, the fewer you need.

There is a close relationship between chemistry
ideas and basis sets.

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”.

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:

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[
-
b
r]

Our function will decay more quickly: exp[
-
b
r
2
]

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:

Alongside the s orbital that will give 4 independent
functions for the exponent.

We continue, multiplying by 2 pre
-
factors:

This introduces 6 further functions

i.e. giving 10 including the s and p’s

Of these 6 functions, 5 are the d
-
orbitals

-
type orbital:

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?

-
functions:

“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:

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?

3

or 343 times!

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:

We will now have 8 functions per atom.

(8/4)
3

or 8 times slower than C4G!

(28/8)
3

or 43 times faster than ddpp.

The C44G* basis

Main shortcoming: change of shape of s/p
functions when solid is formed.

Need d
-
type functions.

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

E
tot
/at

(Ha)

E
rel
/at

(eV)

a
0

(au)

B
0

(GPa)

E
g

(eV)

Time (s)

Expt

10.263

97.9

1.17

216

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”

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.