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Oct 29, 2013 (3 years and 10 months ago)

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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.


Basis sets ad LCAO/MO


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


Think about the H2 molecule:

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:


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.

What about d’s?


We continue, multiplying by 2 pre
-
factors:

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:

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:


“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!



Add more exponents


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?


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:





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.



Sadly: still not good enough!

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

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”
for free electrons.


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.