Basis SetsBasis Sets

Ryan P. A. BettensRyan P. A. Bettens

Department of ChemistryDepartment of Chemistry

National University of SingaporeNational University of Singapore

We’ll look at…We’ll look at…

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

What are basis sets?What are basis sets?

• Simply put, a basis set is a collection (set) Simply put, a basis set is a collection (set) of mathematical functions used to help of mathematical functions used to help solve the Schrödinger equation.solve the Schrödinger equation.

• Each function is centered (has its origin) at Each function is centered (has its origin) at some point in our moleculesome point in our molecule• Usually, but not always, the nuclei are used.Usually, but not always, the nuclei are used.

• Each function is a function of the Each function is a function of the xx,,yy,,zz coordinates of an electron.coordinates of an electron.

An AnalogyAn Analogy1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-10 -5 0 5 10

We desire to reproduce this function

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-10 -5 0 5 10

|x0>,|x2> as basis functions (BF)

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-10 -5 0 5 10

|x0>, |x2>, |x4>, |x6> as BF

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-10 -5 0 5 10

|x0>, |x2>, |x4>, |x6>, |x8>, |x10> as BF

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-10 -5 0 5 10

|x0>, |x2>, |x4>, |x6>, |x8>, |x10>, |x12>, |x14> as BF

c0 = 0.99651 ± 0.000907c2 = -0.16358 ± 0.000302c4 = 0.0078816 ± 2.68e-05c6 = -0.00017293 ± 9.74e-07c8 = 2.0326e-06 ± 1.75e-08c10 = -1.3396e-08 ± 1.64e-10c12 = 4.6961e-11 ± 7.68e-13c14 = -6.8392e-14 ± 1.42e-15

7

0

22

k

kk xcy

Why use basis sets?Why use basis sets?

• We desire one or both of the following.We desire one or both of the following.• The electronic energy of our molecule.The electronic energy of our molecule.• The wavefunction for our molecule so that we The wavefunction for our molecule so that we

may calculate other properties of our may calculate other properties of our molecule. E.g., dipole moment, polarizability, molecule. E.g., dipole moment, polarizability, electron density, spin density, chemical shifts, electron density, spin density, chemical shifts, etc.etc.

• We satisfy our desire by solving the We satisfy our desire by solving the stationary state Schrödinger equation.stationary state Schrödinger equation.

Solving the Stationary State Solving the Stationary State Schrödinger Equation (1)Schrödinger Equation (1)

• We wish to solve: We wish to solve: ĤĤ = = EE• Ĥ Ĥ is the Hamiltonian operator.is the Hamiltonian operator.• is the wavefunction.is the wavefunction.

• ĤĤ is nothing more than a mathematical recipe of is nothing more than a mathematical recipe of operations to be applied to the function operations to be applied to the function such that we such that we obtain a constant times obtain a constant times back again after performing back again after performing the prescribed operations.the prescribed operations.

• The constant will be the energy.The constant will be the energy.• In the Schrödinger equation, the only thing we know In the Schrödinger equation, the only thing we know

before hand is the formula for before hand is the formula for Ĥ.Ĥ.• The formula for The formula for Ĥ involves operations that apply only to Ĥ involves operations that apply only to

the positions (coordinates) of electrons and nuclei in our the positions (coordinates) of electrons and nuclei in our molecule.molecule.

Introducing basis sets…Introducing basis sets…• In order to met our earlier desires we must figure out In order to met our earlier desires we must figure out

what what (the wavefunction) is and with that we will know (the wavefunction) is and with that we will know EE..

• Unfortunately we can only solve the Schrödinger Unfortunately we can only solve the Schrödinger equation to obtain nice formulae for equation to obtain nice formulae for when we have an when we have an hydrogenic atom (H, Hehydrogenic atom (H, He++, Li, Li2+2+, Be, Be3+3+, …) , …)

• If we desire to solve the Schrödinger equation for any If we desire to solve the Schrödinger equation for any system with more than two particles (a nucleus and an system with more than two particles (a nucleus and an electron) then we are forced to make guesses as to what electron) then we are forced to make guesses as to what is. is.

• One guess is to use functions that are similar to the One guess is to use functions that are similar to the formulae obtained already.formulae obtained already.

• That is, functions like That is, functions like ss, , pp, , dd, , ff etc. atomic orbitals (AO’s). etc. atomic orbitals (AO’s).• At this point we might call basis sets, very loosely, as At this point we might call basis sets, very loosely, as

sets of functions like sets of functions like ss, , pp, , dd, , ff, etc. that will be used to , etc. that will be used to describe the behavior of electrons in all systems whether describe the behavior of electrons in all systems whether they be hydrogenic or not.they be hydrogenic or not.

Solving the Stationary State Solving the Stationary State Schrödinger Equation (2)Schrödinger Equation (2)

Ĥ Ĥ Known

Guess

= = EE Only if actually is the wavefunction

EE

If is not the wavefunction

otherwise

== Something elseSomething else

Approximately Solving the Approximately Solving the Stationary State Schrödinger Stationary State Schrödinger

Equation (1)Equation (1)Ĥ Ĥ = = EE Ĥ Ĥ = = E E 22

∫ ∫ Ĥ Ĥ dd = = E E ∫ ∫ 22 ddEE = ∫ = ∫ Ĥ Ĥ dd / ∫ / ∫ 22 dd• If instead we approximate If instead we approximate by by then we then we

can show thatcan show that = ∫ = ∫ Ĥ Ĥ dd / ∫ / ∫ 22 dd • We can always find an energy, We can always find an energy, , this way., this way.• A theorem in QM states that the A theorem in QM states that the ≥ E ≥ E..• If If ≈ ≈ , then , then ≈ ≈ EE..

How we use basis setsHow we use basis sets

• Basis sets are used to approximate Basis sets are used to approximate ..• The bigger and better the basis set the The bigger and better the basis set the

closer we get to closer we get to , and hence , and hence EE..• Nowadays almost everyone utilizes Nowadays almost everyone utilizes

gaussiangaussian functions in basis sets. functions in basis sets.• One or more gaussian-type functions are One or more gaussian-type functions are

used for each AO in each atom in the used for each AO in each atom in the molecule of interest.molecule of interest.

• Let’s look at an example – the H atom, for Let’s look at an example – the H atom, for which we already know what which we already know what should be. should be.

Case Study: H atom (1)Case Study: H atom (1)

• We know that when we solve the Schrödinger We know that when we solve the Schrödinger equation for the H atom we get as possible equation for the H atom we get as possible wavefunctions:wavefunctions:

• = 1s, 2s, 3s, 4s, etc., as well as the p and d… = 1s, 2s, 3s, 4s, etc., as well as the p and d… functions etc.functions etc.

• The lowest energy state is The lowest energy state is = |1s>, with = |1s>, with EE = - = -½ a.u.½ a.u.

• The first excited state is The first excited state is = |2s> = |2s>• Mathematically these functions (in a.u.) look like:Mathematically these functions (in a.u.) look like:

res

2/1

4

121

2

2/1

24

1

2

22

r

ers

Case Study: H atom (2)Case Study: H atom (2)

• Graphically the 1s and 2s orbitals look like.Graphically the 1s and 2s orbitals look like.0.6

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

543210Distance / Angstroms

1s AO 2s AO

Exact Solution to the Schrodinger Equation for the H atom

s Basis Functionss Basis Functions

• Note that the exact s functions are of the form Note that the exact s functions are of the form ee--rr ((i.e.i.e., a Slater), where , a Slater), where is a constant ( is a constant ( = 1 = 1 for H’s 1s, for H’s 1s, = = ½ for H’s 2s).½ for H’s 2s).

• Gaussian basis functions don’t even have the Gaussian basis functions don’t even have the same same formform..

• s basis functions (s basis functions (ggss) are take the form:) are take the form:

Note

24/3

2 rs eg

Where Where is again a constant. is again a constant.Not

Contracted GaussiansContracted Gaussians

• Sometimes a single gaussian function (a single Sometimes a single gaussian function (a single gaussian is termed a gaussian is termed a primitive gaussianprimitive gaussian) can be ) can be improved upon.improved upon.

• A basis function can, in general, be written as a linear A basis function can, in general, be written as a linear combination of primitive gaussians.combination of primitive gaussians.

N

gd1

• Here Here NN is termed the is termed the degree of contractiondegree of contraction..• The The dd are simple numbers called are simple numbers called contraction coefficientscontraction coefficients – they – they

are fixed for the basis set, and do not vary in any calculation.are fixed for the basis set, and do not vary in any calculation.• The The gg are the primitive gaussians, and could be s, p, d, f, etc. are the primitive gaussians, and could be s, p, d, f, etc.

type gaussian functions.type gaussian functions.

Minimal Basis SetsMinimal Basis Sets• Minimal basis sets are constructed such that Minimal basis sets are constructed such that

there in there in only only oneone function per core and valence function per core and valence AOAO..

• For the H and He atoms, we only have one For the H and He atoms, we only have one function, because H and He have no core AO’s function, because H and He have no core AO’s and there is only one valence AO – the 1s AO.and there is only one valence AO – the 1s AO.

• For Li – Ne the electrons in each element will For Li – Ne the electrons in each element will have their behavior represented by 5 functionshave their behavior represented by 5 functions

• 1 function allowing for the electrons in the 1s core AO.1 function allowing for the electrons in the 1s core AO.• 4 functions for the electrons in the 4 functions for the electrons in the nn=2 valence shell, =2 valence shell,

i.e.i.e., 2s (1 function) and the three 2p (3 functions) , 2s (1 function) and the three 2p (3 functions) AO’s.AO’s.

Minimal Basis Set NotationMinimal Basis Set Notation

• A minimal basis set is often represented by the notation A minimal basis set is often represented by the notation STO-STO-nnG, where G, where nn is some non-zero positive integer. is some non-zero positive integer.

• STO stands for “Slater Type Orbital”, with STO stands for “Slater Type Orbital”, with nn primitive primitive gaussians (the “G” above) will be used to approximate it.gaussians (the “G” above) will be used to approximate it.

• nn actually specifies the degree of contraction that will be actually specifies the degree of contraction that will be used to approximate the STO.used to approximate the STO.

• nn is often set to 3, thus a STO-3G basis set is common. is often set to 3, thus a STO-3G basis set is common.• Minimal basis sets represent the simplest (almost the Minimal basis sets represent the simplest (almost the

cheapest and nastiest – there is something else worse!) cheapest and nastiest – there is something else worse!) approximation we can make when we evaluate approximation we can make when we evaluate ..

• To make all this clearer let’s go back to the H atom case To make all this clearer let’s go back to the H atom case study.study.

STO-3GSTO-3G

• NN = 3. = 3.• For the H atom we have the following For the H atom we have the following fixedfixed

constants that will be used to define the one and constants that will be used to define the one and only one basis function H possesses with the only one basis function H possesses with the STO-3G basis set.STO-3G basis set.

• 11 = = dd1111gg1111 + + dd1212gg1212 + + dd1313gg1313

N

gd1

Primitive d1 1 / bohr-2

1 0.154329 3.42525 2 0.535328 0.623913 3 0.444635 0.168856

STO-3G for H (1)STO-3G for H (1)

0.5

0.4

0.3

0.2

0.1

0.0

/

bo

hr-3

543210Distance / Angstroms

Exact and the Three Primitives in the STO-3G Basis Set for H

1s exact d11g11

d12g12

d13g13

These three primitives add together to give the contracted basis function

Largest

Smallest

STO-3G for H (2)STO-3G for H (2)

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

0.6

0.5

0.4

0.3

0.2

0.1

0.0

/

bo

hr-3

543210Distance / Angstroms

Exact (1s AO) and STO-3G basis function for H

1s exact 1

Introducing “Molecular” OrbitalsIntroducing “Molecular” Orbitals

• By analogy with LCAOMO, modern QC By analogy with LCAOMO, modern QC calculations construct MO’s via basis functions.calculations construct MO’s via basis functions.

basis

1

N

ii c

• ii is called an MO, even if the calculation is applied to an atom, in which case they are in actual is called an MO, even if the calculation is applied to an atom, in which case they are in actual

fact AO’s.fact AO’s.

• ccii is called an MO coefficient for MO is called an MO coefficient for MO ii, even thought the coefficient is applied to basis function , even thought the coefficient is applied to basis function

Approximately Solving the Approximately Solving the Stationary State Schrödinger Stationary State Schrödinger

Equation (2)Equation (2)• Recall that we desire to solve,Recall that we desire to solve,

= ∫ = ∫ Ĥ Ĥ dd / ∫ / ∫ 22 dd• We want the lowest We want the lowest possible, because our possible, because our ≥ E ≥ E..• The MO’s are contained within our The MO’s are contained within our function. function.• The only variables we have that we can change in The only variables we have that we can change in

order to get as low an energy as possible is the order to get as low an energy as possible is the MO coefficients, MO coefficients, i.e.i.e., the , the ccii..

• So all the So all the ccii are varied iteratively to minimize the are varied iteratively to minimize the ..

STO-3G for H (3)STO-3G for H (3)

• For our STO-3G on the H atom, we had no For our STO-3G on the H atom, we had no ccii, so nothing could be varied here to , so nothing could be varied here to obtain the lowest obtain the lowest possible. possible.

• The The of the H atom with a STO-3G basis of the H atom with a STO-3G basis set is thus completely fixed at set is thus completely fixed at = -0. = -0.4665819 a.u. = 12.697 eV.4665819 a.u. = 12.697 eV.

• Compare with the exact result of Compare with the exact result of 13.606 eV.13.606 eV.

• This is an error of 87.7 kJ molThis is an error of 87.7 kJ mol-1-1!!

Bigger Basis SetsBigger Basis Sets

• Substantial improvements can be made in Substantial improvements can be made in computing energies and wavefunctions by computing energies and wavefunctions by increasing the number of basis functions.increasing the number of basis functions.

• The next step up from a minimal basis set The next step up from a minimal basis set is a so-called “split valence” basis set.is a so-called “split valence” basis set.

• In split valence basis sets we allow for In split valence basis sets we allow for more than one function per valence AOmore than one function per valence AO..

• We may have 2 or 3 or 4 etc. basis We may have 2 or 3 or 4 etc. basis functions per valence AO.functions per valence AO.

Basis Set Terminology (1)Basis Set Terminology (1)

• 2 basis functions per valence AO is called 2 basis functions per valence AO is called a a valence double zetavalence double zeta basis set. basis set.

• 3 basis functions per valence AO is called 3 basis functions per valence AO is called a a valence triple zetavalence triple zeta basis set. basis set.

• 4 basis functions per valence AO is called 4 basis functions per valence AO is called a a valence quadruple zetavalence quadruple zeta basis set. basis set.

• May have 5, 6, or even higher numbers of May have 5, 6, or even higher numbers of basis function per valence AO.basis function per valence AO.

Basis Set Terminology (2)Basis Set Terminology (2)

• Examples of valence double zeta basis Examples of valence double zeta basis sets are the 3-21G basis set or the 6-31G sets are the 3-21G basis set or the 6-31G basis set.basis set.

• An example of a valence triple zeta basis An example of a valence triple zeta basis set is the 6-311G basis set.set is the 6-311G basis set.

• The above notation is attributed to Pople The above notation is attributed to Pople and co-workers.and co-workers.

Basis Set Terminology (3)Basis Set Terminology (3)

• The Pople general form for basis set The Pople general form for basis set notation is notation is MM--ijk…ijk…G.G.

• MM is the degree of contraction to be used is the degree of contraction to be used for the for the singlesingle basis function per each basis function per each corecore AO.AO.

• The number of digits after the hyphen The number of digits after the hyphen denotes the number of basis functions per denotes the number of basis functions per valence AO.valence AO.

• The value of each digit denotes the degree The value of each digit denotes the degree of contraction to be used for the given of contraction to be used for the given valence basis function.valence basis function.

Basis Set Terminology (4)Basis Set Terminology (4)

• E.g. 3-21G meansE.g. 3-21G means• Each core AO on an atom will be represented Each core AO on an atom will be represented

by a single contracted gaussian basis by a single contracted gaussian basis function. The degree of contraction is 3.function. The degree of contraction is 3.

• This is a valence double zeta basis set as This is a valence double zeta basis set as there are 2 digits after the hyphen.there are 2 digits after the hyphen.

• The first valence basis function will be The first valence basis function will be represented by a contracted gaussian basis represented by a contracted gaussian basis function. The degree of contraction is 2.function. The degree of contraction is 2.

• The second valence basis function will be The second valence basis function will be represented by a primitive gaussian.represented by a primitive gaussian.

Basis Set Terminology (5)Basis Set Terminology (5)

• E.g. 6-311G meansE.g. 6-311G means• Each core AO on an atom will be represented Each core AO on an atom will be represented

by a single contracted gaussian basis by a single contracted gaussian basis function. The degree of contraction is 6.function. The degree of contraction is 6.

• This is a valence triple zeta basis set as there This is a valence triple zeta basis set as there are 3 digits after the hyphen.are 3 digits after the hyphen.

• The first valence basis function will be The first valence basis function will be represented by a contracted gaussian basis represented by a contracted gaussian basis function. The degree of contraction is 3.function. The degree of contraction is 3.

• The second and third valence basis functions The second and third valence basis functions will each be represented by a primitive will each be represented by a primitive gaussian.gaussian.

Calculating the Number of Basis Calculating the Number of Basis FunctionsFunctions

• STO-3GSTO-3G• H and He – 1 basis function.H and He – 1 basis function.• Li – Ne – 1 for the core + 4 for the valence = 5Li – Ne – 1 for the core + 4 for the valence = 5

• 6-31G6-31G• H and He – 2 basis functions.H and He – 2 basis functions.• Li – Ne – 1 for the core + 8 for the valence = 9Li – Ne – 1 for the core + 8 for the valence = 9

• 6-311G6-311G• H and He – 3 basis functions.H and He – 3 basis functions.• Li – Ne – 1 for the core + 12 for the valence = 13Li – Ne – 1 for the core + 12 for the valence = 13

3-21G for H (1)3-21G for H (1)• H has no core AO’s, so there will be two s-H has no core AO’s, so there will be two s-

type basis functions that will be used to type basis functions that will be used to describe the 1s AO of H.describe the 1s AO of H.

• We now have MO coefficients to vary.We now have MO coefficients to vary.• The 1s AO will be represented as a linear The 1s AO will be represented as a linear

combination of the two s-type basis combination of the two s-type basis functions.functions.

• We will also get an “MO” for the 2s AOWe will also get an “MO” for the 2s AO

• 1s1s = = cc1,1s1,1s11 + + cc2,1s2,1s22

• 2s2s = = cc1,2s1,2s11 + + cc2,2s2,2s22

3-21G for H (2)3-21G for H (2) N d / bohr-2

1 2 0.156285 5.44718 0.904691 0.824547

2 1 1 0.183192

0.8

0.6

0.4

0.2

0.0

/

bo

hr-3

543210Distance / Angstroms

1s exact 1

2

Exact and the Two Basis Functions in the 3-21G Basis Set for H

3-21G for H (3)3-21G for H (3)• After minimizing the value of After minimizing the value of we obtain we obtain

= -0.4961986 H = 13.503 eV. = -0.4961986 H = 13.503 eV.• cc1,1s1,1s = 0.37341, = 0.37341, cc2,1s2,1s = 0.71732 = 0.71732• Error = 9.97 kJ molError = 9.97 kJ mol-1-1, a much better result., a much better result.

0.5

0.4

0.3

0.2

0.1

0.0

/

bo

hr-3

543210Distance / Angstroms

Exact (1s AO) and 3-21G 1s "MO" for H

1s exact 1

3-21G for H (4)3-21G for H (4)• We also obtain a solution for the 2s AO of H.We also obtain a solution for the 2s AO of H.

• cc1,2s1,2s = 1.25554, = 1.25554, cc2,2s2,2s = -1.09602 = -1.09602

0.8

0.6

0.4

0.2

0.0

/

bo

hr-3

543210Distance / Angstroms

Exact (2s AO) and 3-21G 2s "MO" for H

2s exact 2

Increasing the Basis SetIncreasing the Basis Set

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

Basis Set No. s functions

/ H Error / kJ mol-1

STO-3G 1 -0.4665819 87.718 3-21G 2 -0.4961986 9.978 6-311G 3 -0.4998098 0.499 cc-pVQZ 4 -0.4999455 0.143 cc-pV5Z 5 -0.4999945 0.014 cc-pV6Z 6 -0.4999992 0.002

BondingBonding• When atoms bond together to form molecules, the When atoms bond together to form molecules, the

electrons that make up the system distribute themselves electrons that make up the system distribute themselves throughout space and between the nuclei to produce the throughout space and between the nuclei to produce the lowest possible overall energy of the system.lowest possible overall energy of the system.

• Certain parts of space have higher densities of electrons, Certain parts of space have higher densities of electrons, while others contain very low densities.while others contain very low densities.

• Basis sets, are functions, which constrain electron Basis sets, are functions, which constrain electron densities to certain regions of space cf. H atom.densities to certain regions of space cf. H atom.

• In order to obtain the correct energy of the system, we In order to obtain the correct energy of the system, we require our basis functions to correctly reflect the real require our basis functions to correctly reflect the real electron density in our system.electron density in our system.

• Thus our basis set should allow for as much flexibility as Thus our basis set should allow for as much flexibility as possible in distributing our electrons around and between possible in distributing our electrons around and between nuclei.nuclei.

• At present, the best way of doing that is by varying MO At present, the best way of doing that is by varying MO coefficients.coefficients.

• Because of this we often need quite a few, and a wide Because of this we often need quite a few, and a wide variety of, fixed functions.variety of, fixed functions.

More Flexibility (1)More Flexibility (1)

• We can increase the number of functions We can increase the number of functions of the same angular type, of the same angular type, e.g.e.g., more s , more s functions.functions.

• E.g. STO-3G E.g. STO-3G → → 3-21G3-21G → → 6-311G6-311G … …• Adding more functions of the same Adding more functions of the same ll type type

(recall (recall ll=0 for s AO) will only allow for =0 for s AO) will only allow for electrons to be further “spread out”, or for electrons to be further “spread out”, or for placing more “nodes” in electron density placing more “nodes” in electron density as we move away from the nucleus.as we move away from the nucleus.

More Flexibility (2)More Flexibility (2)• Here are the 6 s functions used in the Here are the 6 s functions used in the

cc-pV6Z basis (more on this basis set later) for H.cc-pV6Z basis (more on this basis set later) for H.• Electron density is permitted to be more spread out, but Electron density is permitted to be more spread out, but

is spherically symmetric.is spherically symmetric.• There is never any special direction is space that There is never any special direction is space that

electrons prefer to be concentrated.electrons prefer to be concentrated.

1.0

0.8

0.6

0.4

0.2

0.0

/

bohr

-3

543210Distance / Angstroms

1

2

3

4

5

6

6 s functions from the cc-pV6Z basis for H

Introducing PolarizationIntroducing Polarization• We can increase the number of functions of the We can increase the number of functions of the

same angular type, same angular type, e.g.e.g., more s functions., more s functions.• Adding more functions of the same Adding more functions of the same ll type will only type will only

allow for electrons to be further “spread out” or more allow for electrons to be further “spread out” or more nodes to exist.nodes to exist.

• However, it does not allow for a different However, it does not allow for a different directionaldirectional distribution of electron density than what we already distribution of electron density than what we already have.have.

• We can also include higher angular types of We can also include higher angular types of basis functions.basis functions.

• This does allow for different preferred directions in This does allow for different preferred directions in space for electrons to wonder around in.space for electrons to wonder around in.

• For H this would mean allowing p-type functions For H this would mean allowing p-type functions and also d-types, etc., to partake in bonding.and also d-types, etc., to partake in bonding.

• For Li – Ar this would mean including d-type and For Li – Ar this would mean including d-type and also f-type etc.also f-type etc.

Case Study: HCase Study: H22

• Comparing the 6-311G basis with and without Comparing the 6-311G basis with and without polarization functions (p functions) on each H atom polarization functions (p functions) on each H atom in Hin H22, we obtain the following MO coefficients., we obtain the following MO coefficients.

Basis Function

6-311G 6-311G+p-type

s1 (H1) 0.19310 0.18612 s2 (H1) 0.29490 0.28748 s3 (H1) 0.12566 0.13260 pz (H1) - -0.02283 s1 (H2) 0.19310 0.18612 s2 (H2) 0.29490 0.28748 s3 (H2) 0.12566 0.13260 pz (H2) - 0.02283 E / a.u. -1.128038 -1.132491

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

• Each H atom has Each H atom has been “polarized”.been “polarized”.

Basis Set Terminology (6)Basis Set Terminology (6)• Polarization functions are often added separately to atoms other Polarization functions are often added separately to atoms other

than H and He (atoms other than H and He are termed “heavy” than H and He (atoms other than H and He are termed “heavy” atoms).atoms).

• Adding 1 set of polarization functions to heavy atoms is designated Adding 1 set of polarization functions to heavy atoms is designated by a “*” or (d) after the basis set designation.by a “*” or (d) after the basis set designation.

• Adding 1 set of polarization functions to H and He is designated by 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.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 E.g 3-21G* adds a set of d-type functions to all heavy atoms in the molecule.molecule.

• E.g. 3-21G** adds a set of d-type functions to all heavy atoms in the 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 and a set of p-type functions to all H and He atoms in the molecule.molecule.

• 3-21G(d) is synonymous to 3-21G* and 3-21G(d,p) is synonymous 3-21G(d) is synonymous to 3-21G* and 3-21G(d,p) is synonymous to 3-21G**to 3-21G**

• Adding two sets of d-type functions to heavies is denoted by (2d).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 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 heavies, and two sets of p-type and a set of d-type to H and He is designated by (2df,2pd), etc.designated by (2df,2pd), etc.

Diffuse FunctionsDiffuse Functions

• If the problem at hand suggests that electron density If the problem at hand suggests that electron density might be found a long way from the nuclei, then, so-might be found a long way from the nuclei, then, so-called “diffuse” functions can be added.called “diffuse” functions can be added.

• Computing anions is an example were diffuse functions Computing anions is an example were diffuse functions are necessary.are necessary.

• Diffuse functions are of the same type as valence Diffuse functions are of the same type as valence functions (s and p’s for Li – Ar, or just s for H and He).functions (s and p’s for Li – Ar, or just s for H and He).

• Diffuse functions are characterized by small basis set Diffuse functions are characterized by small basis set exponents, exponents, i.e.i.e., small values for the , small values for the ..

• E.g., for the 6-31G basis set with diffuse functions on H, E.g., for the 6-31G basis set with diffuse functions on H, the the ’s are:’s are:(18.7311,2.82539,0.640122); (0.161278); (18.7311,2.82539,0.640122); (0.161278); (0.036)(0.036)

Addition of Diffuse FunctionsAddition of Diffuse Functions• Let’s look at H and HLet’s look at H and H-- with diffuse functions starting with diffuse functions starting

with the 6-311G basis set.with the 6-311G basis set.• ’’s are as follows:s are as follows:

(33.865, 5.09479, 1.15879), (0.32584), (0.102741)(33.865, 5.09479, 1.15879), (0.32584), (0.102741)(0.036)(0.036), , (0.018)(0.018), , (0.009)(0.009), , (0.0045)(0.0045)

• The last three exponents are simply ½ the previous The last three exponents are simply ½ the previous exponent.exponent.

H Atom H- Atom 6-311G +1 diffuse 6-311G +1 diffuse +2 diffuse +3 diffuse +4 diffuse E/H -0.499810 -0.499818 -0. 466672 -0.486963 -0.487714 -0.487741 -0.487741 1 0.23804 0.23760 0.17498 0.15313 0.15465 0.15438 0.15439 2 0.50371 0.50640 0.17469 0.29190 0.27555 0.27772 0.27765 3 0.38357 0.37525 0.76477 0.31512 0.38686 0.37596 0.37635 d1 - 0.00832 - 0.42992 0.21432 0.26041 0.25850 d2 - - - - 0.18420 0.10144 0.10602 d3 - - - - - 0.05208 0.04641 d4 - - - - - - 0.00295

Basis Set Terminology (7)Basis Set Terminology (7)

• In the Pople notation, a single set of diffuse In the Pople notation, a single set of diffuse functions are added to heavy atoms by adding a functions are added to heavy atoms by adding a “+” after the digits representing the number of “+” after the digits representing the number of valence functions.valence functions.

• A second “+” represents a single set of diffuse A second “+” represents a single set of diffuse functions added to H and He atoms.functions added to H and He atoms.

• Thus a 6-31++G basis set has a single set of Thus a 6-31++G basis set has a single set of diffuse functions added to heavy atoms and H diffuse functions added to heavy atoms and H and He atoms.and He atoms.

Example Basis Set DesignationExample Basis Set Designation• 6-311++G(2df,2pd) for benzene.6-311++G(2df,2pd) for benzene.• For CFor C

• A single contracted GTO of degree 6 to mimic the 1s core AO.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, 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.and the remaining two will be made up of a single gaussian each.• A set of diffuse functions will be added, A set of diffuse functions will be added, i.e.i.e., a single diffuse s and a diffuse p, a single diffuse s and a diffuse pxx, p, pyy

and pand pzz..• Two sets of d polarization functions will be added.Two sets of d polarization functions will be added.• A single set of f 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’ 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.part + 4 diffuse + 5 d AO x 2 + 7 f AO = 34.• For HFor H

• Three functions for the 1s AO, the first being a contracted GTO of degree 3, and Three functions for the 1s AO, the first being a contracted GTO of degree 3, and the remaining two are simple primitives.the remaining two are simple primitives.

• A diffuse s function added to them.A diffuse s function added to them.• Two sets of p polarization functions added.Two sets of p polarization functions added.• A single set of d 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 No. basis functions = 1 valence AO x 3 functions for the ‘311’ part + 1 diffuse + 3

p AO x 2 + 5 d AO = 15p AO x 2 + 5 d AO = 15• For CFor C66HH66 we will therefore require a total of 34 x 6 +15 x 6 = 294 basis we will therefore require a total of 34 x 6 +15 x 6 = 294 basis

functions.functions.• This is going to be a fairly big calculation!This is going to be a fairly big calculation!

• Still, an energy calculation on DStill, an energy calculation on D6h6h benzene takes only 5 min on a XP1000 Dec- benzene takes only 5 min on a XP1000 Dec-Alpha.Alpha.

5 d OR 6 d?5 d OR 6 d?

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

• For the d functions we have a 6 possible combinations: xFor the d functions we have a 6 possible combinations: x22, y, y22, z, z22, xy, xz, , xy, xz, yz.yz.

• However, hydrogenic AO’s are actually expressed in-terms of spherical However, hydrogenic AO’s are actually expressed in-terms of spherical polar coordinates, and not Cartesians, so one can take the appropriate polar coordinates, and not Cartesians, so one can take the appropriate linear combinations of the above Cartesians to arrive at 5 functions (2zlinear combinations of the above Cartesians to arrive at 5 functions (2z22 - - xx22 - y - y22, x, x22 – y – y22, xy, xz, yz) instead of 6., xy, xz, yz) instead of 6.

• The missing function actually transforms as an s function, and not a d The missing function actually transforms as an s function, and not a d function (xfunction (x22 + y + y22 + z + z22))

• When using the Pople basis sets it is sometimes necessary to specify 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.whether you wish to use the 5 d set or 6 d set.

2

4/1

3

5128 ry yeg

2

4/1

3

72048 rxy xyeg

Basis sets from other workersBasis sets from other workers• A superb set of basis functions originates from Dunning and co-A superb set of basis functions originates from Dunning and co-

workers.workers.• These authors use a very simple designation scheme.These authors use a very simple designation scheme.• The basis sets are designated as either:The basis sets are designated as either:

• cc-pVcc-pVXXZZ• aug-cc-pVaug-cc-pVXXZ.Z.

• The ‘cc’ means “correlation consistent”.The ‘cc’ means “correlation consistent”.• The ‘p’ means “polarization functions added”.The ‘p’ means “polarization functions added”.• The ‘aug’ means “augmented”, with the functions actually added The ‘aug’ means “augmented”, with the functions actually added

being essentially diffuse functions.being essentially diffuse functions.• The ‘VThe ‘VXXZ’ means “valence-Z’ means “valence-X-X-zeta” where zeta” where XX could be any one of the could be any one of the

followingfollowing• ‘‘D’ for “double”, ‘T’ for “triple”, Q for “quadruple”, or 5 or 6, D’ for “double”, ‘T’ for “triple”, Q for “quadruple”, or 5 or 6, etc.etc.

• Determining the number of basis functions is done by considering Determining the number of basis functions is done by considering the valence space and placing the valence space and placing XX functions down for each valence functions down for each valence AO with the largest value of AO with the largest value of ll..

• We then take one less function as we go up in the We then take one less function as we go up in the ll quantum quantum number, and take an extra function as we go down in number, and take an extra function as we go down in ll quantum quantum number.number.

• If the basis set is an ‘aug’ type, then we add one function across the If the basis set is an ‘aug’ type, then we add one function across the board for each board for each ll-type function we have.-type function we have.

Examples of Dunning’s cc Basis Examples of Dunning’s cc Basis SetsSets

• cc-pVDZ for Li – Necc-pVDZ for Li – Ne• We will have [3s2p1d], which is 3 + 2 x 3 + 1 x 5 = 14 We will have [3s2p1d], which is 3 + 2 x 3 + 1 x 5 = 14

basis functions per atom in this row.basis functions per atom in this row.• Each H and He will have [2sp] = 5 functions.Each H and He will have [2sp] = 5 functions.

• aug-cc-pVDZ for Li - Neaug-cc-pVDZ for Li - Ne• We will have [4s3p2d], which is 4 + 3 x 3 + 2 x 5 = 23 We will have [4s3p2d], which is 4 + 3 x 3 + 2 x 5 = 23

basis functions per atom in this row.basis functions per atom in this row.• Each H and He will have [3s2p] = 9 functions.Each H and He will have [3s2p] = 9 functions.

• cc-pV5Z for Li – Necc-pV5Z for Li – Ne• We will have [6s5p4d3f2gh], which is 6 + 5 x 3 + 4 x 5 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 + 3 x 7 + 2 x 9 + 1 x 11 = 91 basis functions per atom in this row!in this row!

• Each H and He will have [5s4p3d2fg] = 55 functions.Each H and He will have [5s4p3d2fg] = 55 functions.

Wondering what h AO’s look like?Wondering what h AO’s look like?

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

Effective Core PotentialsEffective Core Potentials• Normally applied to third and higher row Normally applied to third and higher row

elements.elements.• A potential replaces the core electrons in a A potential replaces the core electrons in a

calculation with an effective potential.calculation with an effective potential.• Eliminates the need for core basis functions, which Eliminates the need for core basis functions, which

usually require a large number of primitives to usually require a large number of primitives to describe them.describe them.

• May be used to represent relativistic effects, which May be used to represent relativistic effects, which are largely confined to the core. are largely confined to the core.

• Some examples are: CEP-4G, CEP-31G, CEP-Some examples are: CEP-4G, CEP-31G, CEP-121G, LANL2MB (STO-3G 1121G, LANL2MB (STO-3G 1stst row), row),

• LANL2DZ (D95V 1LANL2DZ (D95V 1stst row), SHC (D95V 1 row), SHC (D95V 1stst row) row)

Basis Set QualityBasis Set Quality• ECP minimal basis sets are clearly the worst ECP minimal basis sets are clearly the worst

quality, followed closely by minimal basis sets.quality, followed closely by minimal basis sets.• DZ basis sets are a marked improvement, but DZ basis sets are a marked improvement, but

still generally of low quality.still generally of low quality.• 6-311G6-311G• 6-311G(2df,2pd) ~ cc-pVTZ6-311G(2df,2pd) ~ cc-pVTZ• 6-311++G(2df,2pd)6-311++G(2df,2pd)• aug-cc-pVTZaug-cc-pVTZ• Bigger Dunning’s basis sets now win hands-Bigger Dunning’s basis sets now win hands-

down.down.• A simple comparison can be made by comparing A simple comparison can be made by comparing

the number of s, p, d, f, etc. functions between the number of s, p, d, f, etc. functions between basis sets.basis sets.

One last word…One last word…Unbalanced Basis SetsUnbalanced Basis Sets

• 3-21++G(2df,2pd)3-21++G(2df,2pd)• Only 2 functions per valence AO,Only 2 functions per valence AO,

• but 3 polarization functions and a diffuse?but 3 polarization functions and a diffuse?

• 6-311+G(2df)6-311+G(2df)• 3 functions per valence AO, 3 polarization and 3 functions per valence AO, 3 polarization and

a diffuse on heavies,a diffuse on heavies,• but no polarization nor diffuse on H?but no polarization nor diffuse on H?

• aug-cc-pV5Z on heavies, cc-pVDZ on H.aug-cc-pV5Z on heavies, cc-pVDZ on H.• aug-cc-pV5Z (sp only for H-Ne).aug-cc-pV5Z (sp only for H-Ne).