Basis Sets

1
Basis Sets

• Ryan P. A. Bettens
• Department of Chemistry
• National University of Singapore

2
Well look at

• what are basis sets.
• why we use basis sets.
• how we use basis sets.
• the physical meaning of basis sets.
• basis set notation.
• the quality of basis sets.

3
What are basis sets?

• Simply put, a basis set is a collection (set) of
  mathematical functions used to help solve the
  Schrödinger equation.
• Each function is centered (has its origin) at
  some point in our molecule
• Usually, but not always, the nuclei are used.
• Each function is a function of the x,y,z
  coordinates of an electron.

4
An Analogy
c0 0.99651 0.000907 c2 -0.16358
0.000302 c4 0.0078816 2.68e-05 c6
-0.00017293 9.74e-07 c8 2.0326e-06
1.75e-08 c10 -1.3396e-08 1.64e-10 c12
4.6961e-11 7.68e-13 c14 -6.8392e-14 1.42e-15
5
Why use basis sets?

• We desire one or both of the following.
• The electronic energy of our molecule.
• The wavefunction for our molecule so that we may
  calculate other properties of our molecule.
  E.g., dipole moment, polarizability, electron
  density, spin density, chemical shifts, etc.
• We satisfy our desire by solving the stationary
  state Schrödinger equation.

6
Solving the Stationary State Schrödinger Equation
(1)

• We wish to solve HY EY
• H is the Hamiltonian operator.
• Y is the wavefunction.
• H is nothing more than a mathematical recipe of
  operations to be applied to the function Y such
  that we obtain a constant times Y back again
  after performing the prescribed operations.
• The constant will be the energy.
• In the Schrödinger equation, the only thing we
  know before hand is the formula for H.
• The formula for H involves operations that apply
  only to the positions (coordinates) of electrons
  and nuclei in our molecule.

7
Introducing basis sets

• In order to met our earlier desires we must
  figure out what Y (the wavefunction) is and with
  that we will know E.
• Unfortunately we can only solve the Schrödinger
  equation to obtain nice formulae for Y when we
  have an hydrogenic atom (H, He, Li2, Be3, )
• If we desire to solve the Schrödinger equation
  for any system with more than two particles (a
  nucleus and an electron) then we are forced to
  make guesses as to what Y is.
• One guess is to use functions that are similar to
  the formulae obtained already.
• That is, functions like s, p, d, f etc. atomic
  orbitals (AOs).
• At this point we might call basis sets, very
  loosely, as sets of functions like s, p, d, f,
  etc. that will be used to describe the behavior
  of electrons in all systems whether they be
  hydrogenic or not.

8
Solving the Stationary State Schrödinger Equation
(2)

• H Y

E Y
Only if Y actually is the wavefunction
otherwise

• E Y

Something else
If Y is not the wavefunction
9
Approximately Solving the Stationary State
Schrödinger Equation (1)

• H Y E Y
• Y H Y E Y2
• ? Y H Y dt E ? Y2 dt
• E ? Y H Y dt / ? Y2 dt
• If instead we approximate Y by y then we can show
  that
• e ? y H y dt / ? y2 dt
• We can always find an energy, e, this way.
• A theorem in QM states that the e E.
• If y Y, then e E.

10
How we use basis sets

• Basis sets are used to approximate Y.
• The bigger and better the basis set the closer we
  get to Y, and hence E.
• Nowadays almost everyone utilizes gaussian
  functions in basis sets.
• One or more gaussian-type functions are used for
  each AO in each atom in the molecule of interest.
• Lets look at an example the H atom, for which
  we already know what Y should be.

11
Case Study H atom (1)

• We know that when we solve the Schrödinger
  equation for the H atom we get as possible
  wavefunctions
• Y 1s, 2s, 3s, 4s, etc., as well as the p and
  d functions etc.
• The lowest energy state is Y0 1sgt, with E -½
  a.u.
• The first excited state is Y1 2sgt
• Mathematically these functions (in a.u.) look
  like

12
Case Study H atom (2)

• Graphically the 1s and 2s orbitals look like.

13
s Basis Functions

• Note that the exact s functions are of the form
  e-ar (i.e., a Slater), where a is a constant (a
  1 for Hs 1s, a ½ for Hs 2s).
• Gaussian basis functions dont even have the same
  form.
• s basis functions (gs) are take the form

Note
14
Contracted Gaussians

• Sometimes a single gaussian function (a single
  gaussian is termed a primitive gaussian) can be
  improved upon.
• A basis function can, in general, be written as a
  linear combination of primitive gaussians.

• Here N is termed the degree of contraction.
• The dmr are simple numbers called contraction
  coefficients they are fixed for the basis set,
  and do not vary in any calculation.
• The gmr are the primitive gaussians, and could be
  s, p, d, f, etc. type gaussian functions.

15
Minimal Basis Sets

• Minimal basis sets are constructed such that
  there in only one function per core and valence
  AO.
• For the H and He atoms, we only have one
  function, because H and He have no core AOs and
  there is only one valence AO the 1s AO.
• For Li Ne the electrons in each element will
  have their behavior represented by 5 functions
• 1 function allowing for the electrons in the 1s
  core AO.
• 4 functions for the electrons in the n2 valence
  shell, i.e., 2s (1 function) and the three 2p (3
  functions) AOs.

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Minimal Basis Set Notation

• A minimal basis set is often represented by the
  notation STO-nG, where n is some non-zero
  positive integer.
• STO stands for Slater Type Orbital, with n
  primitive gaussians (the G above) will be used
  to approximate it.
• n actually specifies the degree of contraction
  that will be used to approximate the STO.
• n is often set to 3, thus a STO-3G basis set is
  common.
• Minimal basis sets represent the simplest (almost
  the cheapest and nastiest there is something
  else worse!) approximation we can make when we
  evaluate y.
• To make all this clearer lets go back to the H
  atom case study.

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STO-3G

• N 3.
• For the H atom we have the following fixed
  constants that will be used to define the one and
  only one basis function H possesses with the
  STO-3G basis set.
• c1 d11g11 d12g12 d13g13

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STO-3G for H (1)
These three primitives add together to give the
contracted basis function
19
STO-3G for H (2)

• There is only 1 basis function for H.
• No flexibility at all in computing e -0.4665819
  a.u.

20
Introducing Molecular Orbitals

• By analogy with LCAOMO, modern QC calculations
  construct MOs via basis functions.

• fi is called an MO, even if the calculation is
  applied to an atom, in which case they are in
  actual fact AOs.
• cmi is called an MO coefficient for MO i, even
  thought the coefficient is applied to basis
  function cm.

21
Approximately Solving the Stationary State
Schrödinger Equation (2)

• Recall that we desire to solve, e ? y H y dt /
  ? y2 dt
• We want the lowest e possible, because our e E.
• The MOs are contained within our y function.
• The only variables we have that we can change in
  order to get as low an energy as possible is the
  MO coefficients, i.e., the cmi.
• So all the cmi are varied iteratively to minimize
  the e.

22
STO-3G for H (3)

• For our STO-3G on the H atom, we had no cmi, so
  nothing could be varied here to obtain the lowest
  e possible.
• The e of the H atom with a STO-3G basis set is
  thus completely fixed at e -0.4665819 a.u.
  12.697 eV.
• Compare with the exact result of 13.606 eV.
• This is an error of 87.7 kJ mol-1!

23
Bigger Basis Sets

• Substantial improvements can be made in computing
  energies and wavefunctions by increasing the
  number of basis functions.
• The next step up from a minimal basis set is a
  so-called split valence basis set.
• In split valence basis sets we allow for more
  than one function per valence AO.
• We may have 2 or 3 or 4 etc. basis functions per
  valence AO.

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Basis Set Terminology (1)

• 2 basis functions per valence AO is called a
  valence double zeta basis set.
• 3 basis functions per valence AO is called a
  valence triple zeta basis set.
• 4 basis functions per valence AO is called a
  valence quadruple zeta basis set.
• May have 5, 6, or even higher numbers of basis
  function per valence AO.

25
Basis Set Terminology (2)

• Examples of valence double zeta basis sets are
  the 3-21G basis set or the 6-31G basis set.
• An example of a valence triple zeta basis set is
  the 6-311G basis set.
• The above notation is attributed to Pople and
  co-workers.

26
Basis Set Terminology (3)

• The Pople general form for basis set notation is
  M-ijkG.
• M is the degree of contraction to be used for the
  single basis function per each core AO.
• The number of digits after the hyphen denotes the
  number of basis functions per valence AO.
• The value of each digit denotes the degree of
  contraction to be used for the given valence
  basis function.

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Basis Set Terminology (4)

• E.g. 3-21G means
• Each core AO on an atom will be represented by a
  single contracted gaussian basis function. The
  degree of contraction is 3.
• This is a valence double zeta basis set as there
  are 2 digits after the hyphen.
• The first valence basis function will be
  represented by a contracted gaussian basis
  function. The degree of contraction is 2.
• The second valence basis function will be
  represented by a primitive gaussian.

28
Basis Set Terminology (5)

• E.g. 6-311G means
• Each core AO on an atom will be represented by a
  single contracted gaussian basis function. The
  degree of contraction is 6.
• This is a valence triple zeta basis set as there
  are 3 digits after the hyphen.
• The first valence basis function will be
  represented by a contracted gaussian basis
  function. The degree of contraction is 3.
• The second and third valence basis functions will
  each be represented by a primitive gaussian.

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Calculating the Number of Basis Functions

• STO-3G
• H and He 1 basis function.
• Li Ne 1 for the core 4 for the valence 5
• 6-31G
• H and He 2 basis functions.
• Li Ne 1 for the core 8 for the valence 9
• 6-311G
• H and He 3 basis functions.
• Li Ne 1 for the core 12 for the valence 13

30
3-21G for H (1)

• H has no core AOs, so there will be two s-type
  basis functions that will be used to describe the
  1s AO of H.
• We now have MO coefficients to vary.
• The 1s AO will be represented as a linear
  combination of the two s-type basis functions.
• We will also get an MO for the 2s AO
• f1s c1,1sc1 c2,1sc2
• f2s c1,2sc1 c2,2sc2

31
3-21G for H (2)
32
3-21G for H (3)

• After minimizing the value of e we obtaine
  -0.4961986 H 13.503 eV.
• c1,1s 0.37341, c2,1s 0.71732
• Error 9.97 kJ mol-1, a much better result.

33
3-21G for H (4)

• We also obtain a solution for the 2s AO of H.
• c1,2s 1.25554, c2,2s -1.09602

34
Increasing the Basis Set

• The table below summarizes the results for
  increasing the number of s basis functions from 1
  (minimal) through 6.

35
Bonding

• When atoms bond together to form molecules, the
  electrons that make up the system distribute
  themselves throughout space and between the
  nuclei to produce the lowest possible overall
  energy of the system.
• Certain parts of space have higher densities of
  electrons, while others contain very low
  densities.
• Basis sets, are functions, which constrain
  electron densities to certain regions of space
  cf. H atom.
• In order to obtain the correct energy of the
  system, we require our basis functions to
  correctly reflect the real electron density in
  our system.
• Thus our basis set should allow for as much
  flexibility as possible in distributing our
  electrons around and between nuclei.
• At present, the best way of doing that is by
  varying MO coefficients.
• Because of this we often need quite a few, and a
  wide variety of, fixed functions.

36
More Flexibility (1)

• We can increase the number of functions of the
  same angular type, e.g., more s functions.
• E.g. STO-3G ? 3-21G ? 6-311G
• Adding more functions of the same l type (recall
  l0 for s AO) will only allow for electrons to be
  further spread out, or for placing more nodes
  in electron density as we move away from the
  nucleus.

37
More Flexibility (2)

• Here are the 6 s functions used in the cc-pV6Z
  basis (more on this basis set later) for H.
• Electron density is permitted to be more spread
  out, but is spherically symmetric.
• There is never any special direction is space
  that electrons prefer to be concentrated.

38
Introducing Polarization

• We can increase the number of functions of the
  same angular type, e.g., more s functions.
• Adding more functions of the same l type will
  only allow for electrons to be further spread
  out or more nodes to exist.
• However, it does not allow for a different
  directional distribution of electron density than
  what we already have.
• We can also include higher angular types of basis
  functions.
• This does allow for different preferred
  directions in space for electrons to wonder
  around in.
• For H this would mean allowing p-type functions
  and also d-types, etc., to partake in bonding.
• For Li Ar this would mean including d-type and
  also f-type etc.

39
Case Study H2

• Comparing the 6-311G basis with and without
  polarization functions (p functions) on each H
  atom in H2, we obtain the following MO
  coefficients.

• Each H atom has directed some electron density
  specifically toward the other H atom.
• Each H atom has been polarized.

40
Basis Set Terminology (6)

• Polarization functions are often added separately
  to atoms other than H and He (atoms other than H
  and He are termed heavy atoms).
• Adding 1 set of polarization functions to heavy
  atoms is designated by a or (d) after the
  basis set designation.
• Adding 1 set of polarization functions to H and
  He is designated by a second or a by (d,p)
  after the basis set designation.
• E.g 3-21G adds a set of d-type functions to all
  heavy atoms in the molecule.
• E.g. 3-21G adds a set of d-type functions to
  all heavy atoms in the molecule and a set of
  p-type functions to all H and He atoms in the
  molecule.
• 3-21G(d) is synonymous to 3-21G and 3-21G(d,p)
  is synonymous to 3-21G
• Adding two sets of d-type functions to heavies is
  denoted by (2d).
• Adding two sets of d-type functions and a set of
  f-type functions to heavies, and two sets of
  p-type and a set of d-type to H and He is
  designated by (2df,2pd), etc.

41
Diffuse Functions

• If the problem at hand suggests that electron
  density might be found a long way from the
  nuclei, then, so-called diffuse functions can
  be added.
• Computing anions is an example were diffuse
  functions are necessary.
• Diffuse functions are of the same type as valence
  functions (s and ps for Li Ar, or just s for H
  and He).
• Diffuse functions are characterized by small
  basis set exponents, i.e., small values for the
  a.
• E.g., for the 6-31G basis set with diffuse
  functions on H, the as are(18.7311,2.82539,0.64
  0122) (0.161278) (0.036)

42
Addition of Diffuse Functions

• Lets look at H and H- with diffuse functions
  starting with the 6-311G basis set.
• as are as follows(33.865, 5.09479, 1.15879),
  (0.32584), (0.102741)(0.036), (0.018), (0.009),
  (0.0045)
• The last three exponents are simply ½ the
  previous exponent.

43
Basis Set Terminology (7)

• In the Pople notation, a single set of diffuse
  functions are added to heavy atoms by adding a
  after the digits representing the number of
  valence functions.
• A second represents a single set of diffuse
  functions added to H and He atoms.
• Thus a 6-31G basis set has a single set of
  diffuse functions added to heavy atoms and H and
  He atoms.

44
Example Basis Set Designation

• 6-311G(2df,2pd) for benzene.
• For C
• A single contracted GTO of degree 6 to mimic the
  1s core AO.
• Three functions per valence AO, the first will be
  a contracted GTO of degree 3, and the remaining
  two will be made up of a single gaussian each.
• A set of diffuse functions will be added, i.e., a
  single diffuse s and a diffuse px, py and pz.
• Two sets of d polarization functions will be
  added.
• A single set of f functions will be added.
• No. basis functions 1 for the core 4 valence
  AO x 3 functions for the 311 part 4 diffuse
  5 d AO x 2 7 f AO 34.
• For H
• Three functions for the 1s AO, the first being a
  contracted GTO of degree 3, and the remaining two
  are simple primitives.
• A diffuse s function added to them.
• Two sets of p polarization functions added.
• A single set of d polarization functions added.
• No. basis functions 1 valence AO x 3 functions
  for the 311 part 1 diffuse 3 p AO x 2 5 d
  AO 15
• For C6H6 we will therefore require a total of 34
  x 6 15 x 6 294 basis functions.
• This is going to be a fairly big calculation!
• Still, an energy calculation on D6h benzene takes
  only 5 min on a XP1000 Dec-Alpha.

45
5 d OR 6 d?

• Because p, d, f, etc. basis functions are
  expressed in terms of Cartesian coordinates like

• For the d functions we have a 6 possible
  combinations x2, y2, z2, xy, xz, yz.
• However, hydrogenic AOs are actually expressed
  in-terms of spherical polar coordinates, and not
  Cartesians, so one can take the appropriate
  linear combinations of the above Cartesians to
  arrive at 5 functions (2z2 - x2 - y2, x2 y2,
  xy, xz, yz) instead of 6.
• The missing function actually transforms as an s
  function, and not a d function (x2 y2 z2)
• When using the Pople basis sets it is sometimes
  necessary to specify whether you wish to use the
  5 d set or 6 d set.

46
Basis sets from other workers

• A superb set of basis functions originates from
  Dunning and co-workers.
• These authors use a very simple designation
  scheme.
• The basis sets are designated as either
• cc-pVXZ
• aug-cc-pVXZ.
• The cc means correlation consistent.
• The p means polarization functions added.
• The aug means augmented, with the functions
  actually added being essentially diffuse
  functions.
• The VXZ means valence-X-zeta where X could be
  any one of the following
• D for double, T for triple, Q for
  quadruple, or 5 or 6, etc.
• Determining the number of basis functions is done
  by considering the valence space and placing X
  functions down for each valence AO with the
  largest value of l.
• We then take one less function as we go up in the
  l quantum number, and take an extra function as
  we go down in l quantum number.
• If the basis set is an aug type, then we add
  one function across the board for each l-type
  function we have.

47
Examples of Dunnings cc Basis Sets

• cc-pVDZ for Li Ne
• We will have 3s2p1d, which is 3 2 x 3 1 x 5
  14 basis functions per atom in this row.
• Each H and He will have 2sp 5 functions.
• aug-cc-pVDZ for Li - Ne
• We will have 4s3p2d, which is 4 3 x 3 2 x 5
  23 basis functions per atom in this row.
• Each H and He will have 3s2p 9 functions.
• cc-pV5Z for Li Ne
• We will have 6s5p4d3f2gh, which is 6 5 x 3
  4 x 5 3 x 7 2 x 9 1 x 11 91 basis
  functions per atom in this row!
• Each H and He will have 5s4p3d2fg 55
  functions.

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Wondering what h AOs look like?

• Check out this site
• http//www.orbitals.com/orb/orbtable.htm

49
Effective Core Potentials

• Normally applied to third and higher row
  elements.
• A potential replaces the core electrons in a
  calculation with an effective potential.
• Eliminates the need for core basis functions,
  which usually require a large number of
  primitives to describe them.
• May be used to represent relativistic effects,
  which are largely confined to the core.
• Some examples are CEP-4G, CEP-31G, CEP-121G,
  LANL2MB (STO-3G 1st row),
• LANL2DZ (D95V 1st row), SHC (D95V 1st row)

50
Basis Set Quality

• ECP minimal basis sets are clearly the worst
  quality, followed closely by minimal basis sets.
• DZ basis sets are a marked improvement, but still
  generally of low quality.
• 6-311G
• 6-311G(2df,2pd) cc-pVTZ
• 6-311G(2df,2pd)
• aug-cc-pVTZ
• Bigger Dunnings basis sets now win hands-down.
• A simple comparison can be made by comparing the
  number of s, p, d, f, etc. functions between
  basis sets.

51
One last wordUnbalanced Basis Sets

• 3-21G(2df,2pd)
• Only 2 functions per valence AO,
• but 3 polarization functions and a diffuse?
• 6-311G(2df)
• 3 functions per valence AO, 3 polarization and a
  diffuse on heavies,
• but no polarization nor diffuse on H?
• aug-cc-pV5Z on heavies, cc-pVDZ on H.
• aug-cc-pV5Z (sp only for H-Ne).