E. Kwan Lecture 9: Introduction to Computational Chemistry Chem 117 Introduction to Computational Chemistry Key Questions (1) What kinds of questions can computational chemistry Eugene E. Kwan help us answer? February 22, 2010. - Mechanistic: What are the intermediates and transition states along the reaction coordinate? What factors are responsible for selectivity? Scope of Lecture the As molecules pass from reactants to products, do they the Odyssey PES stay along the minimum energy path? Cluster optimization getting started - Physical: molecular What is the equilibrium geometry of a molecule? with Gaussian introduction to mechanics DFT computational chemistry What will the spectra of a molecule look like (IR, UV-vis conformational NMR, etc.)? What do the lines represent? electron searching correlation basis sets HF methods - Conceptual: and orbitals Where are the charges in a molecule? What do its orbitals look like? Where are the electrons? Key References 1. Molecular Modeling Basics Jensen, J.H. CRC Press, 2009. Why are some molecules more stable than others? Is a molecule aromatic? What are the important 2. Computational Organic Chemistry Bachrach, S.M. Wiley, hyperconjugative interactions in a molecule? 2007. (2) How are potential energy surfaces (PESs) studied? 3. Essentials of Computational Chemistry: Theories and Models How do I locate ground states and transition states? (2nd ed.) Cramer, C.J. Wiley, 2004. (3) How is energy evaluated? 4. Calculation of NMR and EPR Parameters: Theory and What are the differences between HF, DFT, MP2, etc? Applications Kaupp, M.; Buhl, M.; Malkin, V.G., eds. Wiley- When is each method applicable? VCH, 2004. (4) How do I perform calculations here at Harvard? How do I use Gaussian? I thank Professor Jensen (Copenhagen) for providing some What do I need to do to get started on the Odyssey Cluster? useful material for this lecture. E. Kwan Lecture 9: Introduction to Computational Chemistry Chem 117 The Potential Energy Surface (PES) the energy is a function of all the nuclear coordinates: In every introductory organic textbook, you see diagrams like: Eq Ex 1112,,,,, y z x y 22 z energy transition states Now, if you're sharp, you can point out that we don't really care where the center of mass is, but what we really care about in 0 terms of chemistry, is simply the bond length r. So in some reactants sense, the PES is single-dimensional. The typical appearance products of such potentials for bonds is something like: C H + Cl -10 RCH + HCl H -21 RCH + HCl R -31 RCR + HCl R "reaction coordinate" Here are some important questions that you should ask: - What is a "reaction coordinate," exactly? - What does a transition state look like? - Where do the numbers for these energies come from? I can't answer all of these questions here, but the primary Wikipedia purpose of computational chemistry is to connect experimental results with a theoretical potential energy Note that at the bottom of the well, the PES is described well surface so we can understand nature. One can ask static by a harmonic oscillator--a quadratic function. This is an questions like "What is the equilibrium geometry for this important approximation that is made in many calculations. molecule?" or dynamic ones like "What is the mechanism of this reaction?". But, wait! Where are the electrons? Why aren't we giving them coordinates? It is a basic assumption of standard What exactly is a potential energy surface (PES)? Consider computational techniques that because the nuclei are much dihydrogen: heavier than the electrons, the nuclei are essentially "frozen" r from the point of view of the electrons. This is the Born- HH Oppenheimer approximation. From quantum mechanics, we know that the electrons are described by wave functions, How many variables do I need to describe the geometry of rather than specific position and momentum coordinates as in dihydrogen? Six. In Cartesian coordinates, I could say that classical mechanics. E. Kwan Lecture 9: Introduction to Computational Chemistry Chem 117 The Potential Energy Surface (PES) solid = function; dashed = derivative Now, a more complicated molecule will necessarily require more coordinates. For example, you can think of water as needing 6 two O-H bond lengths and the H-O-H angle: 4 H r 2 2 OH -1.5 -1.0 -0.5 0.5 1.0 1.5 r1 If you want to characterize the entire PES, and you want to do -2 10 points per coordinate, that would make a thousand points total. Since evaluating the energy as a function of geometry -4 requires minutes if not hours per point, this is clearly going to be impractical for all but the exceptionally patient (and we're not). -6 If we want to know whether a particular stationary point is a To summarize, the PES is a multidimensional function. It takes local minimum or maximum, we compute the second derivative, the molecular geometry as input, and gives the energy as which is positive for minima, 0 for asymptotes, and negative for output. It has a lot of dimensions and is very, very complicated. maxima. For multidimensional functions, the analog of the second derivative is called the Hessian: Fortunately, transition state theory tells us that we only need to characterize a few stationary points on the PES, rather than the entire thing. A stationary point is anywhere the gradient is zero. For dihydrogen: EEE 0 xxy121 This is the multidimensional analog of the derivative. For example, consider this function: yx5 51 x In chemistry, we are particularly interested in energy minima. From high school calculus, you know that the derivative is: A stationary point is a local minimum if the Hessian is postive- dy 4 definite there. To evaluate this, one transforms from Cartesian 55x to normal coordinates, which amounts to diagonalizing the dx Hessian or "performing a frequency analysis." The "lack of The function has a local maximum and a local minimum, which any imaginary frequencies" means that the matrix is positive occur when dy/dx is zero: definite. E. Kwan Lecture 9: Introduction to Computational Chemistry Chem 117 The Potential Energy Surface (PES) The minimum energy path, reaction coordinate, or intrinsic Can one have a stationary point that is neither a local maximum reaction coordinate connect the starting materials and nor a local minimum? Certainly: saddle points. Consider y=x3: products. Thus, to understand a reaction, we will need to 2 locate a minimum of three stationary points: one each for the starting material, transition state, and product. 1 energy transition states 0 -1.0 -0.5 0.5 1.0 reactants products -1 C H + Cl -10 RCH + HCl H -21 -2 RCH + HCl starting materials and products correspond to local minima R (stationary points with zero imaginary frequencies) while -31 RCR + HCl transition states correspond to first-order saddle points R (stationary points with exactly one imaginary frequency). "reaction coordinate" The energy difference between the reactant and transition state will tell us about the rate of the reaction; the difference between the reactant and product will tell us about how exothermic the reaction is. We can also examine the geometry of the molecules at all the stationary points, and then draw some conclusions about reactivity, selectivity, or other trends. Q: How do I know if the answers that the computer gives actually mean something real? It is very important to be considering this question at all times when dealing with computations. To fit into the scientific method, a computation must make a tangible prediction that can be verified by experiment. One can now predict geometries (X-ray), reaction barriers (kinetics), spectroscopic properties (IR and NMR), etc. As it turns out, in many cases, the agreement between theory and experiment is now very good. http://www.chem.wayne.edu/~hbs/chm6440/PES.html E. Kwan Lecture 9: Introduction to Computational Chemistry Chem 117 Energy Minimization 11dE Q: How do we locate the stationary points? RReg R g F kdR k Rg The general approach is: (1) Make a guess for the geometry of a molecule at its stationary point; (2) Use a computer program Here, Rg is some guess for the value of Req. F is the force, or to move the atoms in such a way that the gradient drops to negative gradient. That means that if we know what the spring zero; and (3) Perform a frequency analysis to verify the nature of constant k is, we can get to the stationary point in one step. the stationary point is correct. However, we usually don't know this, so we take small scaled steps in the direction of the gradient until it falls below some For now, let us consider step (2); the optimization process. value. This is the method of steepest descent. Imagine we have a one-dimensional, quadratic PES with coordinate R (discussion borrowed from Jensen, pp. 8-12): Energy 1.4 1.2 1.0 0.8 Clearly, we want to be taking as few steps as possible. For a 0.6 harmonic oscillator, k is the derivative of the gradient: R 2 0.4 g dE dE RReg2 R 0.2 dR dR R eq Rg g R -1.0 -0.5 0.0 0.5 1.0 This works great if we are near a quadratic portion of the PES. In reality, many PESs have very flat regions where this kind of The energy E of this harmonic oscillator potential is given by: 1 Newton-Raphson step will be too large. (An excellent EkRR()2 animation of how this works can be found on Wikipedia at 2 eq http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif.) Thus, most programs will scale back quadratic steps when their size Taking the derivative, we have: exceeds some pre-determined threshold.