Modern Electronic Structure Theory: Basis Sets

5.61 Physical Chemistry Lecture Notes 1 MODERN ELECTRONIC STRUCTURE THEORY: Basis Sets At this point, we have more or less exhausted the list of electronic structure problems we can solve by hand. If we were limited to solving problems manually, there would be a lot of chemistry we wouldn’t be able to explain! Fortunately, the advent of fast personal computers allows chemists to routinely use more accurate models of molecular electronic structure. These types of calculations typically play a significant role in interpreting experimental results: calculations can be used to assign spectra, evaluate reaction mechanisms and predict structures of molecules. In this way computation is complementary to experiment: when the two agree we have confidence that our interpretation is correct. The basic idea of electronic structure theory is that, within the Born Oppenheimer approximation, we can fix the M nuclei in our molecule at some positions RI. Then, we are left with the Hamiltonian for the electrons moving in the effective field created by the nuclei: Ĥ ≡ − N 1 2 N M ∑ ∇ -∑ ∑ 2 i i=1 i=1 I =1 N ZI 1 +∑ r̂i − R I i< j r̂i − r̂j Eq. 1 Where the first term is the kinetic energy of all N electrons, the second term is the attraction between the electrons and nuclei and the third is the pairwise repulsion between all the electrons. The central aim of electronic structure theory is to find all the eigenfunctions of this Hamiltonian. As we have seen, the eigenvalues we get will depend on our choice of Reaction Unstable the positions of the nuclei – Barrier intermediate Eel(R1,R2,R3,…RM). As was the case with diatomics, these Eel(R1,R2) energies will tell us how stable the molecule is with a given R2 configuration of the nuclei {RI} – Equilibrium if Eel is very low, the molecule will Conformation be very stable, while if Eel is high, R1 the molecule will be unstable in that configuration. The energy Eel(R1,R2,R3,…RM) is called the potential energy surface, and it contains a wealth of information, as illustrated in the picture at above. We can determine the equilibrium configuration of the 5.61 Physical Chemistry Lecture Notes 2 molecule by looking for the minimum energy point on the potential energy surface. We can find metastable intermediate states by looking for local minima – i.e. minima that are not the lowest possible energy states, but which are separated from all other minima by energy barriers. In both of these cases, we are interested in points where ÑEel = 0 . Further, the potential surface can tell us about the activation energies between different minima and the pathways that are required to get from the “reactant” state to the “product” state. Solving the electronic Schrödinger also gives us the electronic wavefunctions Yel(r1,r2,r3,…rN), which allow us to compute all kinds of electronic properties – average positions, momenta, uncertainties, etc – as we have already seen for atoms. We note that while the Hamiltonian above will have many, many eigenstates, in most cases we will only be interested in the lowest eigenstate – the electronic ground state. The basic reason for this is that in stable molecules, the lowest excited states are usually several eV above the ground state and therefore not very important in chemical reactions where the available energy is usually only tenths of an eV. In cases where multiple electronic states are important, the Hamiltonian above will give us separate potential surfaces E1el, E2el, s* potential E3el … and separate wavefunctions surface 1 2 3 Y el, Y el, Y el. The different potential surfaces will tell us about the favored conformations of the s potential molecules in the different electronic surface states. We have already seen this Erxn for H2+. When we solved for the electronic states, we got two eigenstates: s and s*. If we put the electron in the s orbital, the molecule was bound and had a potential surface like the lower surface at right. However, if we put the electron in the s* orbital the molecule was not bound and we got the upper surface. 5.61 Physical Chemistry Lecture Notes 3 So, at the very least our task is clear cut: solve for the eigenstates of Eq. 1. Unfortunately, this task is also impossible in practice, even on a computer. Thus, over the years chemists have developed a vast array of sophisticated techniques that allow us to at least approximate these solutions to within a tolerable degree of accuracy. Learning all the details of these approximations would require a course unto itself: the derivations of the individual approximations are exceedingly complex, and the sheer number of different approximations that can be made is quite impressive. These detailed derivations teach us a lot about what molecules and properties we should expect our approximations to work for and how we should think about improving our calculations in cases where the theory fails. However, the thing that has really brought computational chemistry into the mainstream is the fact that one does not have to understand every nuance of a method in order to know how to use it successfully. It suffices to have a simple, qualitative understanding of how each method works and when it can be applied. Then, coupling that knowledge with a little technical proficiency at using commercial chemistry software packages allows us to run fairly sophisticated calculations on our desktop computer. The next two lectures are intended to prepare us to run these types of calculations. First, we note that nearly all the popular approximations are still based on MO theory – MO theory on steroids in some cases, but MO theory nonetheless. Thus, there are still 5 steps in the calculation 1) Choose an Atomic Orbital Basis 2) Build the Relevant Matrices 3) Solve the Eigenvalue Problem 4) Occupy the orbitals based on a stick diagram 5) Compute the energy In a typical calculation, the computer automatically handles steps 2-4 automatically – we don’t have to tell it anything at all. It is sometimes helpful to know that the computer is doing these things (e.g. The calculation crashed my computer. What was it doing when it crashed? Oh, it was trying to solve the eigenvalue problem.) but we essentially never have to do them ourselves. This leaves two steps (1 and 5) that require some input from us for the calculation to run properly Choosing an Atomic Orbital Basis 5.61 Physical Chemistry Lecture Notes 4 The first point here is that for real electronic structure calculations, you typically use a basis set that is much larger than your intuition might tell you. For example, for H2+ we guessed that we would get a decent result if we wrote: y = c11sA + c2 1sB A basis of this type would be called a minimal basis set, because it is the smallest one that is even close to right. In a more accurate calculation, you might use a basis that looks more like: y = c11s A + c2 1sB + c3 2s A + c4 2sB + c5 2 pxA + c6 2 pxB + c7 2 p yA + c8 2 p yA + c9 2 pzA + c10 2 pzB + c11 3s A + c12 3sB The reason we use such extended basis sets arises from a point that was discussed earlier concerning MO theory. Because our results are variational, a bigger basis always gets us a lower energy, and therefore a longer AO expansion always gets us closer to the ground state energy. In the worst case, the unimportant basis functions will just have coefficients that are very near zero. While such extended basis sets would be a significant problem if we were working things out by hand, computers have no problem dealing with expansions involving even 10,000 basis functions. The second important point about the atomic orbitals we use is that they are not hydrogenic orbitals. The reason for this is that the two-electron integrals involving hydrogenic orbitals cannot all be worked out analytically, making it very difficult to complete Step 2. Instead of hydrogenic orbitals – which decay like e - r – we will use Gaussian orbitals that decay like e -a r . Gaussians do not look very much like hydrogenic orbitals – they don’t have a cusp at r=0 and they decay much too fast at large distances. About the only good thing about them is that they have a mximum at r=0 and decay. These differences between Gaussians and hydogenic orbitals are not a problem, though, because we use extended basis sets as emphasized above. Basically, given enough Gaussians, you can expand anything you like – including a hydrogenic orbital, as shown in the picture at right. Here, we plot a hydrogen 1s orbital in red together with the best approximation 2 5.61 Physical Chemistry Lecture Notes 5 we can get to it with a linear comination of one, two or three Guassians. So while using Gaussians may mean we have to use a few extra AOs, if we use enough of them we should be able to get the same answer. So we plan to use relatively large Gaussian basis sets for our calculations. How exactly do we choose those basis sets? Thankfully, a significant amount of trial-and-error research has distilled the choices down to a few key basis set features. The result is that there exists a number of accurate basis sets available to us. A basis set is just a pre-defined set of AO basis functions for each atom in the periodic table. To use a particular basis set, we just need to specify the name of the basis set and then the computer goes through and picks the correct AOs for every atom in our molecule. Depending on the problem at hand and the accuracy desired we need to consider three important aspects of the AO basis set. Single, Double, Triple, Quadruple Zeta Basis Sets As we have already discussed for MO theory of diatomics, the smallest basis we can think of for describing bonding would include all the valence orbitals of each atom involved. Thus, for H we had 1 s-function, for C there were 2 s-functions and one set of p’s. Similarly, for sulfur we would have needed 3 s-functions and 2 p’s …. A basis of this size is called a minimal or single zeta basis. The term “single zeta” refers to the fact that we have only a single set of the valence functions (Note: single valence might seem like a more appropriate name, but history made a different choice). The most important way to expand the basis is to include more than a single set of valence functions. Thus, in a double zeta (DZ) basis set, one would include 2 sfunctions for H, 3 s- and 2 p-functions for C and 4 s- and 3 p-functions for S. Qualitatively, we think of these basis functions as coming from increasing the n quantum number: the first s function on each atom is something like 1s, the second something like 2s, the third like 3s …. Of course, since we are using Gaussians, they’re not exactly 1s, 2s, 3s … but that’s the basic idea. Going one step further, a triple zeta (TZ) basis would have: H=3s, C=4s3p, S=5s4p. For Quadruple zeta (QZ): H=4s, C=5s4p, S=6s5p and so on for 5Z, 6Z, 7Z. Thus, one has: H,He Li-Ne Na-Ar Names Minimal 1s 2s1p 3s2p STO-3G DZ 2s 3s2p 4s3p 3-21G,6-31G, D95V TZ 3s 4s3p 5s4p 6-311G,TZV 5.61 Physical Chemistry Lecture Notes 6 Unfortunately, the commonly used names for basis sets follow somewhat uneven conventions. The basic problem is that many different folks develop basis sets and each group has their own naming conventions. At the end of the table above, we’ve listed a few names of commonly used SZ,DZ and TZ basis sets. There aren’t any commonly used QZ basis sets, because once your basis is that large, it is best to start including polarization functions (see below). Polarization Basis Functions Note that no matter how high you go in the DZ, TZ, QZ hierarchy, you will never, for example, get a p-function on hydrogen or a d-function on carbon. These functions tend to be important for describing polarization of the electrons; at a qualitative level, the p-functions aren’t as flexible in their angular parts and it’s hard to get them to “point” in as many directions as dfunctions. Thus, particularly when dealing with directional bonding in molecules, it can be important to include some of these higher angular momentum functions in your AO basis. In this situation the basis set is said to contain some “polarization” functions. The general nomenclature of polarization functions is to add the letter “P” to a basis set with a single set of polarization functions, and “2P” to a basis with two sets. Thus, a DZP basis would have: 2s1p on hydrogen, 3s2p1d on carbon and 4s3p1d on sulfur. A TZP basis set would have 3s1p on hydrogen, 4s3p1d on carbon and 5s4p1d on sulfur. DZP TZP H,He 2s1p 3s1p Li-Ne 3s2p1d 4s3p1d Na-Ar 4s3p1d 5s4p1d Names 6-31G(d,p), D95V 6-311G(d,p),TZVP We note that in practice it is possible to mix-and-match different numbers of polarization functions with different levels of zeta basis sets. The nomenclature here is to put (xxx,yyy) after the name of the basis set. “xxx” specifies the number and type of polarization functions to be placed on Hydrogen atoms and “yyy” specifies the number and type of polarization functions to be placed on non-hydrogen atoms. Thus, we would have, for example: H,He Li-Ne Na-Ar 5.61 Physical Chemistry 6-311G(2df,p) 7 Lecture Notes 3s1p 4s3p2d1f 5s4p2d1f Diffuse Functions Occasionally, and particularly when dealing with anions, the SZ/DZ/TZ/… hierarchy converges very slowly. For anions, this is because the extra electron is only very weakly bound, and therefore spends a lot of time far from the nucleus. It is therefore best to include a few basis functions that decay very slowly to describe this extra electron. Functions of this type are called “diffuse” functions. They are still Gaussians ( e -a r ), but the value of a is very, very small causing the atomic orbital to decay slowly. Similar to the situation for polarization functions, diffuse functions can be added in a mix-and-match way to standard basis sets. Here, the notation “+” or “aug-“ is added to a basis set to show that diffuse functions have been added. Thus, we have basis sets like 3-21++G, 6-31+G(d,p), aug-TZP. 2 Aside: Transition Metals Those of you interested in inorganic chemistry will note that no transition metals appear in the tables above. This is not because there aren’t basis sets for transition metals – it is just more complicated to compare different transition metal basis sets. First, we note that many of the basis sets above are defined for transition metals. Thus, for example, a 6-31G(d,p) basis on iron is 5s4p2d1f while a TZV basis for iron is 6s5p3d. The reason we didn’t include this above is that the idea of “valence” for a transition metal is a subject of debate: is the valence and s- and d- function? An s a p and a d? Hence, depending on who put the basis set together, there will be some variation in the number of functions. However, one still expects the same ordering in terms of quality: TZ will be better than DZ, DZP will be better than a minimal basis, etc. Thus, you can freely use the above basis sets for all the elements between K and Kr without significant modification. Extending the above table for specific basis sets gives: 3-21G 6-31G(d,p) 6-311G(d,p) TZV K-Ca 5s4p 5s4p1d 8s7p2d 6s3p Sc-Zn 5s4p2d 5s4p2d1f N/A 6s3p2d Ga-Kr 5s4p1d N/A 8s7p3d 6s5p2d 5.61 Physical Chemistry Lecture Notes 8 Things also become more complicated when dealing with second row transition metals. Here, relativistic effects become important, because the Schrödinger equation predicts that the 1s electrons in these atoms are actually moving at a fair fraction of the speed of light. Under these circumstances, the Schrödinger equation is not strictly correct and we need to start considering corrections for relativistic effects. The most efficient way to incorporate the relativity is to use an effective core potential (ECP). An effective core potential removes the core electrons from the problem and replaces them with an effective potential that the valence electrons feel. This potential reflects the combined interaction with the nucleus and the (relativistic) core electrons. Thus, for an ECP we specify both how many core electrons we want to neglect and how many basis functions we want to use to describe the valence electrons. For example, one popular double zeta ECP is the LANL2DZ basis. As an example, for ruthenium LANL2DZ replaces the 28 core electrons (1s22s22p63s23p63d10=Argon) with an effective potential and uses a 3s3p2d basis to describe the valence orbitals. Thus, for the second transition series we have (using [CoreSize]/ValenceBasisSize as our shorthand): Y-Cd Hf-Hg LANL2DZ [Argon]/3s3p2d N/A 10 SDD [Argon]/8s7p6d [Kr4d 4f14]/8s7p6d As one progresses further up the periodic table, fewer and fewer basis sets are available, simply because less is known about these elements chemistry. This is just a very brief overview of what basis sets are available and how good each one is expected to be. The general idea of using basis sets is to use larger and larger basis sets until the quantity we are computing stops changing. This is based on the idea that we are really using the AO expansion to approximate the exact solution of the Schrödinger equation. If we had an infinite basis, then we would get the exact answer, but with a large and finite basis we should be able to get close enough. Note, however, that the calculation will typically take more time in a larger basis than in a smaller one. Thus, we really want to make the basis just big enough to get the answer right, but no larger. 5.61 Physical Chemistry Lecture Notes 9 It is worth mentioning in closing that there is something of a double whammy when it comes to basis sets for very accurate calculations. In the next lecture we will see that there are two general classes of methods: Selfconsistent field methods – which are less expensive but generally less accurate – and methods that include explicit wavefunction correlation –which are more accurate but more expensive. If we take a look at the convergence of the energy as we increase the size of the AO basis, we will see something like the graph at right. The energy of SCF methods converges exponentially with the size of the basis set: we are pretty close to the infinite basis result with a DZP basis and certainly very close with something like TZ2P. Meanwhile, for the correlated methods the convergence is slower: DZP is nowhere close and in practice something like a polarized QZ basis is necessary to get reasonable results. This is doubly painful for correlated methods, because these approaches were already more expensive than SCF, and now they are even more formidable because we need bigger basis sets. At present, there are two approaches to overcome this: First, one can fit a series of energies from smaller basis sets to try to extrapolate to the infinite basis set limit. An even more sophisticated technique is to augment the one-electron atomic orbital basis with some explicit two electron functions (i.e. functions that depend on r12) With these terms the correlated calculations can often be done with much smaller AO bases, making the calculations much cheaper. The downside is that the equations become even more complicated when r12 is involved, so it takes considerable effort to implement these r12 methods.