Chemistry 324, Spring 2002 Geometry Optimization

Geometry optimization Page 1 of 9 Review Chemistry 324, Spring 2002 Geometry Optimization By now, you have learned several useful things about molecular modeling. First, you have learned from your work with the Spartan tutorial that the first step in developing a molecular model is to draw a picture of your molecule on the computer screen. Drawing the molecule accomplishes several things. First, it tells Spartan about the contents of your molecule. Second, it defines a structure for the model by defining the location of each atom relative to the other atoms in the molecule. Third, it tells Spartan the location of covalent bonds and it tells Spartan whether these bonds are single, double, or triple bonds. You have also learned that many chemical problems can be translated into structure-energy problems. A molecule tends to adopt the structure that makes its potential energy as low as possible. This structure is called the minimum-energy structure or equilibrium geometry. If you want to compute a molecule's properties (geometry, electron distribution, and so on), you must obtain this information from a model of its minimum-energy structure. Minimum-energy structures are also useful for thinking about chemical reactions. For example, the potential energy change that accompanies a reaction, E rxn, equals the difference between the potential energies of the reactants and products, where molecule adopts its minimum-energy structure. Similarly, the potential energy barrier, E, equals the difference between the potential energies of the transition state and reactants, where, once again, the latter adopt their minimumenergy structures. Computational chemists have developed mathematical methods for converting any "trial" structure, such as the structure you build when you draw a molecule with Spartan, into a minimum-energy structure. These methods are called geometry optimization. The following material explains how these methods work, how you can use them to obtain the results you want, and how they sometimes go wrong. Multi-dimensional Potential Energies Surfaces The process of geometry optimization can be visualized by looking at a reaction coordinate diagram. When you build a model, you define its structure (I will call this the trial structure). This structure corresponds to some point on the reaction coordinate diagram, T. Geometry optimization simply changes the structure of this model into a minimum-energy structure, O. E geo The mathematical details of geometry optimization are fairly complicated partly because the mathematical description of molecular structure is complicated. For example, consider a water molecule, H 1 -O-H 2. In order to describe its geometry completely, we must specify the location of every atom in three-dimensional space. This can be accomplished by listing the XYZ coordinates of each atom in order: water geometry = (x H1, y H1, z H1, x O, y O, z O, x H2, y H2, z H2 ) Analytical geometry tells us that we can think of an ordered pair of numbers as a vector in twodimensional space. Since a water molecule's geometry is defined by an ordered list of nine O T

Geometry optimization Page 2 of 9 numbers, its geometry corresponds to a vector in nine-dimensional space. 1 The molecule's potential energy is a function of its geometry, so geometry optimization is really a ten-dimensional problem: nine dimensions define the geometry, a tenth dimension defines the energy at this geometry, and we need to find the geometry in this ten-dimensional space with the lowest energy. When a molecule is as small as water, the problem can be made much simpler by using internal coordinates to define the molecule's geometry. For example, we can completely specify the molecule's geometry using the two HO bond distances and the HOH bond angle, θ. water geometry = (r(h 1 O), r(h 2 O), θ) These three numbers describe the positions of the atoms relative to one another, but they do not tell us where the molecule is located relative to external objects, such as other water molecules. Exactly six numbers are needed to describe the position of any non-linear molecule relative to external objects, so it is always possible to remove six dimensions by working in internal coordinates. 2 This is an enormous simplification when a molecule as small, like water, but it may not matter much for larger molecules. Our new description of water's structure-energy relationship fits into four dimensions (three coordinates for geometry, one coordinate for energy). This is a simple problem for a computer to "visualize", but it is hard for people to draw and look at pictures of four-dimensional functions, so we will take one more step that may not be justified. We will assume that the two HO bond distances are equal to each other (if you have taken inorganic chemistry, then you will recognize that this effectively assumes a geometry of C 2v symmetry). The geometry of "symmetric" water can be expressed using only two numbers: water geometry = (r, θ) A contour graph of potential energy vs. geometry in the vicinity of water's minimum-energy structure might look like the picture shown below. The points that make up a given loop represent geometries of identical energy, but each loop corresponds to a different energy. The minimumenergy structure is located at X and is defined by the ordered pair (r X, θ X ). θ X r HO r X All graphs of energy vs. geometry, regardless of their dimensional complexity, are referred to as potential energy surfaces. The initial trial geomery corresponds to one point on the surface, (r T, θ T ), and the goal of geometry optimization is to locate another point on the surface that corresponds to a minimum-energy geometry, such as (r X, θ X ). θ Mathematical Characteristics of Minimum-Energy Structures Minimum-energy structures have several defining characteristics. These characteristics can be applied to any molecular structure as mathematical tests. If a structure passes all of the tests, it is 1 In other words, the geometry of any molecule containing N atoms can be expressed by a 3N dimension vector consisting of XYZ coordinates of each atom. 2 The geometry of a non-linear molecule containing N atoms can be expressed by a 3N-6 dimension vector consisting of suitably chosen internal coordinates.

Geometry optimization Page 3 of 9 a minimum-energy structure. The first test uses the partial derivative of the potential energy with respect to each of the geometry coordinates. If the structure is located at an energy minimum, all of these derivatives will equal zero because the surface is flat at an energy minimum. It is convenient to collect the partial derivatives in vector form, in which case the vector is called the energy gradient. water energy gradient = ( E/dr, E/dθ) Any point on an energy surface at which the gradient vanishes, that is, at which the gradient equals (0, 0,, 0), is called a stationary point. A minimum-energy structure is always a stationary point, but not all stationary points are minimum-energy structures. Some stationary points are maximum-energy structures, and others are complicated saddle points. Minimum-energy structures can be distinguished from other types of stationary points by examining second derivatives of the energy. First, we calculate second partial derivatives of the energy relative to all possible combinations of the coordinates. Next, we collect these derivatives in a square matrix called the Hessian matrix. Hessian = 2 E/dr 2 2 E/drdθ 2 E/dθdr 2 E/dθ 2 If you have studied linear algebra, you may recall that this kind of matrix can be transformed mathematically into a diagonal form, that is, into a matrix that contains zeros in all of its offdiagonal positions and the eigenvalues of the original Hessian matrix in its diagonal positions. diagonalized Hessian = Eigenvalue #1 0 0 Eigenvalue #2 We will explore this transformation in more detail later on. For now, it suffices to know that the calculation of the Hessian matrix, and its diagonalization, are equivalent to a calculation of the molecule's force constants and vibration frequencies. 3 The distinctive feature of a minimumenergy structure is that all of its force constants, and all of its vibration frequencies, are positive real numbers. The Hessians of other types of stationary points are characterized by one or more negative force constants, and by vibration frequencies that are imaginary numbers. Energy Calculations So far, I have assumed that I can calculate a model's energy reliably, but I have not revealed how this is accomplished. I will not reveal all of these secrets here, but some understanding of energy computations is helpful, even at this early stage. For example, it turns out that the different methods for calculating energy generate different potential energy surfaces. Each type of energy calculation gives a different energy for the same structure, and each calculation generates a unique minimum-energy structures. Consequently, when you describe a model, you must not only mention that a geometry optimization was performed, you must also describe the method used to calculate its energy. Although there are many ways to compute potential energy, all of the methods fall into two broad categories, each with its own advantages and disadvantages. 3 The eigenvalues are the molecule s force constants and are proportional to the square of the vibration frequencies.

Geometry optimization Page 4 of 9 Molecular Mechanics (MM). Molecular mechanics treats a molecule as a collection of bonded atoms. The energy of the molecule depends on perturbations of the chemical bonds and throughspace interactions between nonbonded atoms. Bond perturbations always increase a molecule's energy. Stretching a bond beyond its ideal value, squeezing a bond angle below its ideal value, and twisting a group of atoms into a nonideal conformation, are examples of destabilizing bond perturbations. Interactions between nonbonded atoms, on the other hand, can both destabilize and stabilize a molecule. Therefore, a geometry optimization that is guided by molecular mechanics will produce a structure in which bond perturbations and destabilizing nonbonded interactions are minimized, and stabilizing nonbonded interactions are maximized. Molecular mechanics calculations are useful mainly because they are fast. Computational speed and molecule size are directly related. A calculation that can be performed quickly is also a calculation that can be repeated many times for many different atoms. Therefore, molecular mechanics calculations are often the only tool that can be applied to large molecules, e.g., proteins. The main drawback of molecular mechanics is its reliance on bond perturbations. Before one can identify perturbations and calculate their effects, one must have some idea what bonds are present. Therefore, the outcome of a molecular mechanics calculation depends on your trial structure and your choice of chemical bonds. Different bond patterns give different energies even for the same structure, and they lead to different minimum-energy structures. Also, many molecules, such as resonance hybrids, cannot be described satisfactorily using standard bond types. Molecular mechanics cannot be applied in these cases, regardless of the molecule s size. Quantum Mechanics (QM). Quantum mechanics does not rely on chemical information, like bond patterns. It treats a molecule as a collection of subatomic particles, nuclei and electrons, instead. The energy of the molecule depends on the positions of the nuclei and the distribution of electrons (the latter is estimated using an approximate version of the Schrodinger equation). As a rule, a molecule's energy is lowered when the electrons move in a way that creates bonds between neighboring nuclei. The energy is elevated by destabilizing interactions between pairs of electrons, destabilizing interactions between pairs of nuclei, and antibonding interactions. Quantum mechanical calculations are much slower than molecular mechanics, not only because there are many more subatomic particles in a molecule than there are atoms, but also because the calculation of electron distribution and energy is inherently more time-consuming. Despite this, quantum mechanical calculations are the method of choice for calculating potential energy. The quantum mechanical energy of a structure is a function of structure only. The energy does not depend on how bonds are drawn on the structure, and optimization of a given trial structure will always lead to the same minimum-energy structure no matter how the bonds are drawn. Quantum mechanical models are also completely general. They can be used to study standard molecules, reaction intermediates, excited states, and transition states. A Word about Spartan's Energy Calculations. Spartan energy calculations can be initiated in two different ways. You can set up a variety of molecular mechanics and quantum mechanical energy calculations using the Setup Calculations dialog window. These calculations can give you the energy of the current trial structure (calculate Single Point Energy), the energy and geometry of a minimum-energy structure (calculate Equilibrium Geometry), or the energy and geometry of a transition state. You can also obtain the energy and geometry of a minimum-energy structure by clicking on the Minimize icon. This calculation is always guided by molecular mechanics. If you want to perform a quantum mechanical geometry optimization, you must set it up using the Setup Calculations dialog.

Geometry optimization Page 5 of 9 Geometry Optimization Strategies Computational chemists have developed many different mathematical methods for transforming a trial model into a minimum-energy model. Different computer programs may use different geometry optimization methods, and a given program may even use different methods for different types of energy calculations. For example, a program may use one type of optimization algorithm for molecular mechanics optimizations and a different one for quantum mechanics optimizations. The user may even be able to select the type of geometry optimization method. Some understanding of these methods is useful, but it must be said that computational chemists worry much less about the details of optimization methods than they used to. Today one normally assumes that the methods that have been built into molecular modeling program are efficient and reliable. All geometry optimization methods are essentially trial-and-error procedures. They all begin by calculating the energy, and possibly the gradient and Hessian, of the trial structure. These data are used to predict a new structure that may, or may not, be of lower energy than the original trial structure. All of the calculations are then repeated on the new structure, and the entire process is repeated until a structure is generated that appears to be a minimum-energy structure. The distinguishing characteristics of any given optimization method are the information it requires about each structure and the procedure it uses for predicting a new structure. Together, these characteristics define a strategy for searching the potential energy surface. Search strategies are mathematical procedures, but they are most easily understood using a nonmathematical analogy. Imagine the following scenario: you parachute from an airplane at night over unfamiliar territory and you are told (before you jump) to rendezvous with your comrades at the bottom of the closest valley. The unfamiliar terrain in this scenario corresponds to a potential energy surface in which geographical location (latitude and longitude) corresponds to molecular structure and elevation corresponds to potential energy. Your landing point represents the original trial structure and the rendezvous point (the valley bottom) is the minimum-energy structure. Since it is night, you cannot see distant objects. You must choose your path by feeling the ground surrounding your immediate location, that is, you must rely on information derived from your current location. The simplest way to proceed, but certainly the least efficient, is to head off in a random direction. If a step in a randomly selected direction takes you to a lower elevation, then you repeat the process from your new location. If the step takes you to a higher elevation, then you must return to your previous location and choose a new direction. If, after a certain number of steps, you cannot find a lower elevation to jump to, you can reduce your step size and repeat the random step process. Eventually, by stumbling around the surface at random, you will reach a point where even tiny steps fail to lower your elevation. This point represents a minimum-energy structure. The random search procedure can be improved on by keeping track of your last three locations. If your current location is lower in energy than your two previous locations, you should avoid steps that will carry you backwards! Simplex minimization is a well-known optimization procedure that uses this type of reasoning. It is considerably better than purely random searches, it never fails, it is easy to program, and it can be applied to all kinds of optimization problems, not just geometry optimization. However, it is rarely used in computational chemistry because much more powerful procedures are available. A search strategy can be made more efficient by removing the random elements described above. For example, if we calculate the gradient vector, G, of our trial structure, we won't have to choose our path randomly because the vector G points towards the steepest downhill path.

Geometry optimization Page 6 of 9 Steepest downhill paths (arrows) do not always point towards the valley bottom ( ). r HO θ Unfortunately, as you can see from the drawing above, the steepest downhill path from a given point does not generally point directly towards the minimum energy structure. 4 Therefore, it is still an open question how far we should travel along G before we stop and change directions, i.e., recalculate the gradient. One strategy for using and following gradients is the line search. The computer takes different size steps along the path defined by G until the energy is as low as possible on this line. The gradient is calculated at this new point and a new line search is then initiated, and the process repeats until the gradient vanishes. The largest random element in a line search strategy is step size, how far we should travel along G before we stop to recalculate the gradient. The uncertainty in step size can be substantially reduced, however, by calculating the surface's curvature, i.e., the second derivatives or Hessian matrix, at the trial structure. If a surface has a quadratic shape, the location of the minimumenergy structure is defined completely by the gradient and Hessian. However, even when a surface is only quasi-quadratic, the Hessian can still make a search more efficient by indicating a likely location for the minimum-energy structure. The Newton-Raphson method is a well-known and highly efficient search strategy that uses gradient and curvature information. As you can see, every type of optimization procedure constructs (and then discards) many trial structures before it locates a minimum-energy structure. The computational efficiency of a procedure is determined by the product of two numbers, the number of structures that it examines, and the time it spends calculating the properties (energy, gradient, Hessian) of each structure. Computation time = (# structures) x (time per structure) Simple-minded strategies, like simplex minimization, are very inefficient, because even though they require very little time per structure, they examine a very large number of structures. The relative efficiency of more complicated search strategies is harder to evaluate. Generally speaking, computation time increases: energy << gradient << Hessian matrix. Therefore, a line search may be as efficient as a Newton-Raphson search because the latter requires much more time per structure. Some ideas about how to make searches more efficient are given below. Geometry Optimization Criteria Geometry optimization procedures use several criteria to test each structure. Logically, one expects the gradient, G, to vanish at a minimum-energy structure. This criterion cannot be met in actual practice, however, unless the optimization procedure locates the minimum precisely. Therefore, all optimization procedures allow a little leeway in the size of the gradient. The usual practice is to compute the magnitude of the gradient vector (this is often referred to as the rootmean-square gradient or rmsg), and to insist that this quantity be smaller than some preset number (the latter is called the gradient tolerance). Another criterion that can be applied is to insist that every component in G be smaller than some preset number as well. 4 At any given point, the directions of G and G are perpendicular to the contour line passing through that point.

Geometry optimization Page 7 of 9 Many procedures also use step size as a criterion. Since any geometry can be written as a vector, the difference between the current geometry and the next predicted geometry can be written as a difference vector, D. This quantity is then treated the same way as the gradient vector. That is, the optimization procedure insists that the magnitude of the difference vector, rmsd, become smaller than some preset number (the geometry tolerance). Some procedures also insist that every component in the difference vector be made smaller than some preset number. Some procedures also look at the predicted change in energy. If we assume that energy and structure are linearly related, the difference in energy between the current structure and a new structure is simply the product of the gradient and difference vectors. Procedures that use the energy change criterion insist that the magnitude of the predicted energy change fall below some preset number (the energy tolerance) before halting the optimization. Typically, several criteria are applied simultaneously. For example, a procedure might insist that rmsg be smaller than the gradient tolerance, rmsd be smaller than the geometry tolerance, and the predicted energy change be smaller than the energy tolerance. If any of these criteria are not met, the optimization is continued. Even when a structure satisfies all of the optimization criteria, one needs to approach the structure with a little skepticism. First, there is the possibility that the structure is the wrong kind of stationary point. This can be tested by calculating the Hessian matrix (or, equivalently, by calculating vibration frequencies). Second, there is always some numerical imprecision associated with an optimized structure. This imprecision arises because there is a small region of the potential energy surface that contains structures that will satisfy the optimization criteria, so searches will stop when they enter any part of this region. Fortunately, this region is nearly always quite small, and the numerical imprecision it creates is of no chemical significance. Accelerating (Quantum Mechanics) Geometry Optimizations Nearly all computer programs that perform molecular mechanics geometry optimizations are extremely fast (except for very large molecules) and there is little value in thinking about how to accelerate these calculations (unless you plan on writing your own computer program). High-level quantum mechanics geometry optimizations are much, much slower, however. Time per structure is significant even when only the energy and gradient are calculated. Therefore, it makes sense to look for ways of accelerating these calculations. One way to accelerate a geometry optimization is to use a more efficient optimization procedure, but this is rarely attempted because efficiency is hard to predict and there are other, more reliable ways of accelerating optimizations. The single most effective way to accelerate a geometry optimization is to start with a trial structure that is as close as possible to the minimum-energy structure. The thinking behind this idea is quite simple: the closer your parachute lands you to your destination, the sooner your search will be over. Spartan contains several features that help create good trial geometries. First, as you draw a molecule on the screen, Spartan automatically applies standard bond lengths and angles found in similar molecules. Second, if you click Minimize, Spartan will perform a fast molecular mechanics geometry optimization that will give a much improved estimate of the true minimum-energy structure. Third, when you start a quantum mechanics geometry optimization, Spartan performs a preliminary fast geometry optimization (either molecular mechanics or semi-empirical quantum mechanics) before attempting the requested geometry optimization. You might wonder, If Spartan performs a fast geometry optimization automatically, why bother

Geometry optimization Page 8 of 9 clicking Minimize? The answer to this lies in the fact that both Minimize and the fast preliminary optimizations use less reliable energy calculations. These calculations usually do a good job, but there is a slight chance that they will generate a poor trial structure and cause subsequent geometry optimizations to go awry. You can avoid this kind of bad outcome if you apply Minimize yourself, because you can look at the outcome. If the minimized structure is worse than your trial structure, you can use the Undo command to return to your trial structure, and you can request options in the Setup Calculations dialog window that will turn off the automatic preliminary optimizations. Another effective method for accelerating a geometry optimization is to guess the surface gradient and curvature at the start of an optimization. 5 Gradient and Hessian calculations are very time-consuming, and a fast effective guess can greatly reduce the time needed for the optimization. Many programs use some kind of empirical scheme for guessing gradients and Hessians. Spartan automatically guesses the gradient and Hessian of a trial structure by calculating these quantities using semi-empirical quantum mechanics before attempting the requested geometry optimization. Symmetry. Many molecules have symmetric structures. For example, the two HO bond distances in water are identical. As was shown above, by assuming a symmetric structure for water, the number of internal coordinates that need to be optimized is reduced from three to two. This simplifies the mathematical description of the potential energy surface, and it reduces the number of structures that need to be examined. Symmetry assumptions have an enormous impact on the speed of quantum mechanics calculations. The electron distribution and energy can both be calculated much faster for a symmetric molecule, so the time per structure is greatly reduced, and this leads to a more efficient optimization. The most impressive results are obtained for molecules of high symmetry. For example, benzene contains 12 atoms and its structure is specified by 30 internal coordinates. However, if we assume a symmetric structure for benzene (D 6h point group), only two coordinates need to be specified, the CC and CH distances. The following table shows how symmetry affects Spartan quantum mechanics geometry optimization (HF/3-21G model). Both calculations began with the same symmetrical trial structure for benzene, and both ended with nearly identical optimized structures, but one optimization assumed a symmetric structure at all times and the other did not. The optimization that assumed a symmetric structure was over ten times faster, partly because fewer structures were examined, but also because the time per structure was reduced. Symmetry # internal coord # structures Total opt. time (sec) Time per structure (sec) D 6h 2 3 4.2 1.4 none 30 9 43 4.8 Since symmetry can have such a huge impact, many computer programs, Spartan included, automatically check for symmetry in a trial structure before initiating any calculation. If symmetry is detected, the geometry optimization maintains this symmetry, and the minimum-energy structure has the same (or higher) symmetry as the trial structure. 6 This is an appropriate time to point out that symmetry assumptions are a type of geometry constraint and are not always justified. For example, if one optimizes the geometry of NH 3 by starting with a planar model, programs like Spartan will enforce planar symmetry on the optimization procedure and produce a planar optimized structure. This is incorrect, of course. NH 3 5 Hessian calculations are so time-consuming that most optimization procedures never calculate the true Hessian unless it is requested by the chemist. A pseudo-hessian is used instead. 6 The Setup Calculations dialog window contains a Symmetry checkbox. Spartan checks for symmetry in the trial structure only if this box is checked. If the box is not checked, symmetry in the trial structure is ignored.

Geometry optimization Page 9 of 9 adopts a lower energy pyramidal structure. Unfortunately, the only way to uncover the mistaken symmetry assumption is to calculate the model s vibration frequencies, a potentially timeconsuming process. In this case, one of vibration frequency of planar NH 3 turns out to be an imaginary number, indicating that this stationary point is a transition state and not a true minimum-energy structure. Local vs. Global Minima Virtually every potential energy surface contains multiple energy minima. This is most obviously the case when a molecule is flexible. The potential energy surface for methylcyclohexane contains two energy minima, corresponding to chair conformers (equatorial methyl and axial methyl), and some additional minima, correspond to higher energy conformers. The lowest energy minimum, equatorial chair methylcyclohexane, is called the global minimum, while the other minima are referred to as local minima (these minima are local in the sense that they have lower energies than any other structures in their immediate vicinity). All of the geometry optimization procedures that are described in this chapter tend to locate the local minimum that is closest to the starting trial structure. This result naturally follows from the parachute analogy; after you land, you proceed downhill to the closest rendezvous point. Consequently, if a trial structure of methylcyclohexane begins with the methyl group in an axial position, geometry optimization will almost certainly lead to the axial local minimum. To find the global minimum, we must start the geometry optimization from a different trial structure. Failed Geometry Optimizations Sometimes Spartan returns an error message that reads optimization failed. This message indicates failure only in the sense that an optimized structure has not (yet) been located. That is, a certain number of trial structures were generated, but none of them satisfied all of the optimization criteria. When this occurs, the only corrective action that is needed is to continue the geometry optimization (note: Spartan automatically updates the model s structure, even when an optimization fails, so a second geometry optimization request will begin where the previous search left off). Spartan s behavior is dictated by the fact that geometry optimization is potentially an open-ended process. It is impossible to guess in advance how many trial structures will need to be examined, so Spartan sets a limit on the number that it will check during any given search (the limit is equal to the number of internal coordinates + 20). 7 If this limit is exceeded, Spartan automatically halts the optimization so that you can decide whether to continue. Further Reading A.R. Leach, Molecular Modeling: Principles and Applications, 2/E, Prentice-Hall, Harlow, England, 2001, ISBN 0-582-38210-6, pp. 253-273. 7 You can override Spartan s default limit by typing GeometryCycle = N in the Options box of the Setup Calculations dialog window (N is the limiting number you want to use).