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**Basis Sets Ryan P. A. Bettens Department of Chemistry**

National University of Singapore

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**We’ll 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.

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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.

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**An Analogy We desire to reproduce this function**

|x0>,|x2> as basis functions (BF) |x0>, |x2>, |x4>, |x6> as BF |x0>, |x2>, |x4>, |x6>, |x8>, |x10> as BF |x0>, |x2>, |x4>, |x6>, |x8>, |x10>, |x12>, |x14> as BF c0 = ± c2 = ± c4 = ± 2.68e-05 c6 = ± 9.74e-07 c8 = e-06 ± 1.75e-08 c10 = e-08 ± 1.64e-10 c12 = e-11 ± 7.68e-13 c14 = e-14 ± 1.42e-15

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**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.

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

We wish to solve: ĤY = EY Ĥ is the Hamiltonian operator. Y is the wavefunction. Ĥ 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 Ĥ. The formula for Ĥ involves operations that apply only to the positions (coordinates) of electrons and nuclei in our molecule.

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**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 (AO’s). 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.

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

= E Y Known Guess Only if Y actually is the wavefunction otherwise E Y = Something else If Y is not the wavefunction

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

Ĥ Y = E Y Y Ĥ Y = E Y2 ∫ Y Ĥ Y dt = E ∫ Y2 dt E = ∫ Y Ĥ Y dt / ∫ Y2 dt If instead we approximate Y by y then we can show that e = ∫ y Ĥ 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.

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

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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 = |1s>, with E = -½ a.u. The first excited state is Y1 = |2s> Mathematically these functions (in a.u.) look like:

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Case Study: H atom (2) Graphically the 1s and 2s orbitals look like.

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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 H’s 1s, a = ½ for H’s 2s). Gaussian basis functions don’t even have the same form. s basis functions (gs) are take the form: Not Where a is again a constant. Note

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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.

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

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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 let’s 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) Largest a Smallest a These three primitives add together to give the contracted basis function

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**STO-3G for H (2) There is only 1 basis function for H.**

No flexibility at all in computing e = a.u.

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**Introducing “Molecular” Orbitals**

By analogy with LCAOMO, modern QC calculations construct MO’s 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 AO’s. cmi is called an MO coefficient for MO i, even thought the coefficient is applied to basis function cm.

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

Recall that we desire to solve, e = ∫ y Ĥ y dt / ∫ y2 dt We want the lowest e possible, because our e ≥ E. The MO’s 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.

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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 = a.u. = eV. Compare with the exact result of eV. This is an error of 87.7 kJ mol-1!

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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.

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**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.

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**Basis Set Terminology (3)**

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

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**Basis Set Terminology (5)**

E.g G 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

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3-21G for H (1) H has no core AO’s, 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

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3-21G for H (2)

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3-21G for H (3) After minimizing the value of e we obtain e = H = eV. c1,1s = , c2,1s = Error = 9.97 kJ mol-1, a much better result.

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

c1,2s = , c2,2s =

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**Increasing the Basis Set**

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

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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.

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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 l=0 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.

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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.

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**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.

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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”.

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**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.

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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 p’s 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 a’s are: ( , , ); ( ); (0.036)

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**Addition of Diffuse Functions**

Let’s look at H and H- with diffuse functions starting with the 6-311G basis set. a’s are as follows: (33.865, , ), ( ), ( ) (0.036), (0.018), (0.009), (0.0045) The last three exponents are simply ½ the previous exponent.

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**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 G basis set has a single set of diffuse functions added to heavy atoms and H and He atoms.

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**Example Basis Set Designation**

G(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 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 d AO = 15 For C6H6 we will therefore require a total of 34 x 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.

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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 AO’s 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.

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**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.

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**Examples of Dunning’s cc Basis Sets**

cc-pVDZ for Li – Ne We will have [3s2p1d], which is x 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 x 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 x x x x 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 AO’s look like?**

Check out this site:

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**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)

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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 G(2df,2pd) aug-cc-pVTZ Bigger Dunning’s 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.

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**One last word… Unbalanced Basis Sets**

3-21++G(2df,2pd) Only 2 functions per valence AO, but 3 polarization functions and a diffuse? 6-311+G(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).

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