Optimization of Geometries by Energy Minimization

Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997. All rights reserve. You are welcome to use this ocument in your own classes but commercial use is not allowe without the permission of the author. Goal: To unerstan the theory unerpinning energy minimization an optimization of the geometry of a molecule. Performance Objectives At the en of this exercise stuents will be able to: 1. explain what geometry optimization is;. outline how the Newton-Raphson metho works; 3. istinguish between harmonic an anharmonic potentials an how these affect convergence; 4. explain the significance of coupling between internal coorinates 5. apply the metho to another small molecule an implement this using Mathca or some other software. Introuction: Optimization of molecular geometries by minimizing the energy is a common research tool in theoretical chemistry. Energy minimization can be performe using quantum methos, semi-empirical methos, or by molecular mechanics. Molecular mechanics is the basis of most commercial software packages that rely on the extremely rapi evaluation of energies an forces as their basis. Molecular mechanics are force fiel methos, which use empirical force constants (secon erivatives of the energy with respect to changes in geometrical coorinates such as bon istances an angles). This ocument will use force fiels to illustrate how geometries are optimize using the Newton-Raphson metho. We will also examine the epenence of convergence on the quaratic nature of the potential energy surface. The Newton-Raphson algorithm also applies to quantum mechanical methos, where exact secon erivatives are rarely compute because of computational emans. Therefore we will also explore the use of approximate guesses for the secon erivatives. Last Moifie Sept. 3, 1997 1

Part I. Derivation of the Newton-Raphson Metho for a Diatomic Molecule. Consier the potential energy curve for a typical bon (using the Morse potential): r e 1.000 i 0.. 600 r i i 100 V i 1. 1 e 1. r r i e 4 V i The ocument Exploring the Morse Potential contains etails of preparing plots like this one using appropriate units. This ocument can be obtaine from the New Traitions Mathca ocument Web site: http://www.niagara.eu/~tjz/mathca/mathinx.htm 0 0 5 r i Notice the shape of the curve. the minimum is clearly visible so that we can pick off the value of r at the minimum easily. Picking off the minimum in a potential function becomes increasingly more ifficult as the number of atoms in a molecule increases. We must also consier how the change in one bon length or angle affects the potential for another. The implication is that we must make several passes to fin the minimum for all lengths an angles in a molecule. this means that we shoul use a computer to o the job an we nee an algorithm (proceure) to implement a computer search for the minimum for the whole molecule (each bon an angle at an optimum value). We will use basic calculus concepts. the overall process escribe here is calle the Newton Rahpson metho. So let's start. Last Moifie Sept. 3, 1997

First, let us approximate the energy at the istance r as a Taylor's series expansion about a first guess for the equilibrium bon istance, r 1. E r E r. 1 1 E r 1 r r.. 1 E r 1 r r 1 r r graient force constant plus higher orer terms which we will neglect (this will give a parabolic plot of E vs. r). The graient an force constant are evaluate at r 1, an r is the variable. To reiterate, r1 is the first guess for the best r but of course we may not know the best r but by using our chemical knowlege we can make a goo guess. The graient g 1 is the graient for the first guess an H 1 is the first guess Hessian. E r E r. 1 1 g 1 r r.. 1 H 1 r r 1 So far we have an approximate potential energy function for the system. Let's solve for the energy minimum for this potential. Take the erivative of the above equation an set the left han sie equal to 0. Solving the resulting equation for r will give the istance at which the graient, g 1, in the approximate potential vanishes. 0 r E r 1 g. 1 1 r r.. 1 H 1 r r 1 The first term in the square bracket is a constant, so its erivative is zero. The erivative of the secon term in the square bracket is simply the graient evaluate at r 1. The erivative of the thir term in the square bracket is simply H 1 times (r-r 1 ). 0 0 g 1. H 1 r r 1 Instea of solving for r, let us solve for the correction to r 1, r 1 = r-r 1, to give the new guess for r e :. H 1 r r 1 g 1 r r 1 g 1 H 1 r 1 g 1 H 1 Last Moifie Sept. 3, 1997 3

When ae to r 1, r 1 gives an improve estimate for the equilibrium bon istance, which we will call r in recognition of the fact that it is very unlikely that this new r is r e because of the approximations that have been mae. r will be the exact equilibrium bon istance if the potential energy surface is quaratic (the Taylor's series expansion to secon orer only is not an approximation for a quaratic function), an if g 1 an H 1 are the exact graient an Hessian at r = r 1. Otherwise, an iterative proceure is require, with the above equations repeate for r, g an H, etc. until convergence is achieve. Reasonable convergence criteria are that the isplacement an graient for the current step are smaller than some small tolerance, since they are zero at the exact r e. What we are oing here is following the same steps use by molecular moeling software. Fin r, correct the initial guess for the coorinate an cycle through again. How o these equations appear if there is more than one coorinate, with coupling between the coorinates? For n coorinates g, r an r become vectors, length n, an H becomes the Hessian matrix, nxn. The Hessian is also the force constant matrix, an coupling between coorinates appears as off-iagonal elements (Beginners shoul review force constants an Hooke's Law). The Hessian is exactly the same force constant matrix that is use in normal moe analysis (Wilson FG or GF metho for example), provie that the force fiel an coorinate efinitions are the same. We will rename the coorinate vector from r to the more general q.. H 1 q q 1 g 1 A new wrinkle is that ivision by a matrix is not efine, so we must left-multiply both sies of the equation by the inverse of the Hessian: 1 H.. 1 1 H 1 q q. 1 H 1 g 1 We min our p's an q's in physics ( momenta an coorinates) q q 1. H 1 1 g 1 q 1. H 1 1 g 1 Part II. Example: the CH molecule with anharmonic potentials - no coupling. In molecular mechanics, only the potential energy is calculate. In the methylene molecule, we will only consier the bon stretching coorinate (istance), an the bening coorinate (angle). Again, we see that the same type of coorinates are use for vibrations an geometry. The simplicity of force fiels is that they use simple analytic forms for the potential, with constants that are etermine empirically. With simple analytic forms, first an secon erivatives are even simpler analytic forms. Draw the CH molecule an ientify the coorinates. Last Moifie Sept. 3, 1997 4

The only requirements for us to get starte are an initial guess for the geometry (q), an the expression for the energy. To moify this example for another molecule you change the coorinates an energy expressions using the material in this ocument as a moel. A typical force fiel expression is that from the MM program: Stretching: E s ( r ) 71.94. k. s r r... 3 0 71.94 k s r r 0 where r 0 is the "equilibrium bon istance" for that type of bon, an k s is the force constant for that type of bon. The potential is a harmonic oscillator function plus an anharmonicity correction. k s an r 0 are tabulate as part of the force fiel in commercial software. MM parameters for the C-H single bon are: k s 4.6 r 0 1.113 The units here are milliynes per Angstrom. Stuent Exercise: Plot the stretching energy vs. r. Does the plot resemble the shape of the Morse iatomic curve given above? What problem o you see with the stretch in this force fiel? What are the potential consequences of this with respect to energy minimization. Place your graph an answers in the space below. A suggeste range is 0.80 to.0 angstrom. Bening: Similarly, the ben in MM is treate as a harmonic oscillator, with a unusual term thrown in to get [1,1,1]tricyclopentane correct. This term shoul be insignificant in most cases. Draw the [1,1,1]tricyclopentane molecule an explain why this might be important for angle optimization. E b ( θ ) 0.0568. k. b θ θ.. 0 0.0568 k.. b 7.0 10 8 6 θ θ 0 k b 0.3 θ 0 109.4 where θ 0 is the "equilibrium bon angle" for that type of bon, an k b is the force constant for that type of bon. The bon istances are in Angstroms, angles are in egrees an energies are in kcal/mole. Last Moifie Sept. 3, 1997 5

Stuent Exercise: Plot the bening energy vs. θ. Also plot the harmonic oscillator function using the same force constant. How ifferent are the graphs of the two functions? What is the appropriate range for a bon angle? Place your graph an answers in the space below. Try plots with range 0 to 00 an 50 to 150 egrees. Combine Bening an Stretching - No Coupling First, efine the starting geometry. r 1.000 θ 100.0 Make the coorinate vector. These are guesses use to start the search for an optimum geometry, one with a minimum energy. i 0.. q 0 r q 1 r q θ Stuent Exercise: Have Mathca print the coorinate vector q in the space below. Here we are just observing how Mathca hanles the ata Evaluate the energy with respect to q 0, q 1 an q : E b ( θ ) 0.0568. k. b θ θ.. 0 0.0568 k.. b 7.0 10 8 6 θ θ 0 E s ( r ) 74.19. k. s r r... 3 0 74.19 k s r r 0 E bs ( θ, r) 0.0 E E s q 0 E s q 1 E b q Stuent Exercise: Have Mathca evaluate the energy in the space below. More observation Last Moifie Sept. 3, 1997 6

Make a graient vector: g b θ E b ( θ) g s r E s ( r) g 0 g s g 1 g s g g b Stuent Exercise: Print the graient vector. If one of the components of the graient is negative, oes that inicate that the coorinate is less than, greater than, or equal to the equilibrium value? Why? Place your answers in the space below. (Draw potential energy curves to help you answer the question.) Now make the Hessian: H bb θ E b ( θ) H ss r E s ( r) H bs θr E bs ( θ, r) The coupling between stretches an bens is zero in this example. j 0.. H 0, 0 H ss H 1, 1 H ss H, H bb Stuent Exercise: Print the Hessian matrix. What oes the fact that all of the off-iagonal elements are zero inicate about the coupling between coorinates? Are the iagonal elements positive or negative? Is the force constant larger for the stretching or bening? Explain. The answer to the last question hols true for almost all cases. Place your answers in the space below. Last Moifie Sept. 3, 1997 7

Now calculate the isplacement vector an a to the original vector: q. H 1 g qnew q q Stuent Exercise: Print the isplacement vector an new coorinate vector. When the isplacements are ae to the coorinate vector, are the new coorinates closer to the equilibrium values? Sketch potential energy graphs (by han) an inicate the changes in the coorinates. Place your answers in the space below. Stuent Exercise: Note that the new coorinates are not equal to the equilibrium values. Why woul the angle be much closer to its equilibrium value than the bon istance is to r 0? Place your answer in the space below. Helpful Hint: The geometry preicte in one section will be able to be checke against the starting point of the next section for most parts. We can iterate by setting q=qnew, an repeating. r 1.09017 θ 109.4616 These coorinates shoul have the values obtaine immeiately above. q qnew g b θ E b ( θ) g s r E s ( r) g 0 g s g 1 g s g g b H bb θ E b ( θ) H ss r E s ( r) H bs θ r E bs ( θ, r) H 0, 0 H ss H 1, 1 H ss H, H bb Last Moifie Sept. 3, 1997 8

q. H 1 g qnew q q Stuent Exercise: Print the coorinates, isplacement, graient, Hessian an new coorinates in the space below. Question: How o the istances an angle compare to the equilibrium values? How oes q in the secon iteration compare to q in the first iteration? How o the Hessian an graient compare to the Hessian an graient from the first step? Place your answers in the space below. Stuent Exercise: Repeat the optimization step again in the space below. r 1.11163 θ 109.4 q qnew g b θ E b ( θ) g s r E s ( r) g 0 g s g 1 g s g g b H bb θ E b ( θ) H bs θ r E bs ( θ, r) H ss r E s ( r) H 0, 0 H ss H 1, 1 H ss H, H bb Last Moifie Sept. 3, 1997 9

q. H 1 g qnew q q Stuent Exercise: Print the isplacement, new geometry coorinates, an the graient. Does the geometry come within 0.0001 angstrom an 0.01 egrees of the equilibrium values? (In a case with no coupling the final optimum geometry will simply be the same as the r 0 an θ 0.) Woul a proceure that uses a criterion that all elements of q < 0.0001 stop or go one aitional step in the example being stuie here? What is the relationship between the bening element of g an q? Place your answers in the space below. Stuent Exercise: Go back through Part II, an a lines to print the energy for each step in the optimization. Is there a tren in the energies? Hint: What is the title of this ocument? Last Moifie Sept. 3, 1997 10

Part III. Example: the CH molecule with harmonic potentials - no coupling. Let us start with the same starting geometry as in part II: r 1.000 θ 100.0 We now reefine the E s (r), E b (q), an E bs (q,r) functions in Part II, in the space below with the anharmonic oscillator terms excise. E b ( θ ) 0.0568. k. b θ θ 0 E s ( r ) 71.94. k. s r r 0 E bs ( θ, r) 0.0 Stuent Exercise: Here you be the computational chemist. Print the coorinate vector q. Why oes it have the values in it that it oes? If it is not correct, what o you nee to o to make it correct? Put your answers in the space below. Last Moifie Sept. 3, 1997 11

Stuent Exercise: Calculate g, H, q, an qnew in the space below, an print them in the space below. Stuent Exercise: Does the Newton-Raphson optimization converge in one step for quaratic potentials as avertise? Explain. Last Moifie Sept. 3, 1997 1

Part IV. Example: the CH molecule with harmonic potentials - with coupling. In this part we will see the effect of coupling between the stretch an ben. First reefine the stretch-ben interaction to be nonzero, yet harmonic: E bs ( θ, r ) 0.003. r r. 0 θ θ 0 Why is this harmonic? We cannot use the same formulas for the graients as above, because the erivative of the energy with respect to one of the coorinates is no longer simply the erivative of one term only. A more general an flexible way to calculate E, g an H follows: r 1.000 θ 100.0 E r 1, r, θ E s r 1 E s r E b ( θ) E bs r 1, θ E bs r, θ r 1 r r r g 0 E r 1, r, θ g r 1 E r 1, r, θ g 1 r E r 1, r, θ θ H 0, 0 r 1 E r 1, r, θ H, 0 1 E r 1, r, θ H r 1 r, 0 E r 1, r, θ r 1 θ H 1, 1 E r 1, r, θ r H 1, E r 1, r, θ r θ Notice how each matrix element is written explicitly. In other wors we are builing up the whole Hessian piece by piece. H, θ E r 1, r, θ Last Moifie Sept. 3, 1997 13

Stuent Exercise: Fill in the other Hessian values below so that the Hessian matrix is symmetric. Why is the Hessian always symmetric, i.e. what property of partial erivatives allows us to say H 1,0 = H 0,1? Print the Hessian an graient below also. Now we will compute the isplacement: q. H 1 g qnew q q Stuent Exercise: Print q an qnew. What is the significance of the q an q new? Stuent Exercise: Is the geometry converge? Do another iteration below an see if the graient an isplacement are zero. Recompute g an H, an print below. q qnew r 1 1.1131 r 1.1131 θ 154.3839 This θ is very large. It was chosen for its ramatic effect on the calculation. Last Moifie Sept. 3, 1997 14

Now we will compute the isplacement: q. H 1 g qnew q q Stuent Exercise: Is the geometry converge? That is, oes Newton-Raphson converge in one step for this case? What is the source of numerical error in q? Are the final values of r an q the same as r 0 an q 0? Why? Last Moifie Sept. 3, 1997 15

Part V. Example: the CH molecule with anharmonic potentials - with coupling. Stuent Exercise: There shoul be no nee for the stuent to o calculations to answer the following two questions. Will Newton-Raphson converge in one step for this case? Will the final values of r an θ be the same as r 0 an q 0? Part VI. Example: the CH molecule with approximate Hessians. Using approximate Hessians is not necessary in molecular mechanics, since they are trivial to compute. For many computational methos, however, computing the secon erivatives is very time consuming or even impossible at present. Optimizations using Hessians that are compute at less accurate levels of theory are far superior to an a priori guess by a human. Experience researchers can make goo guesses for the iagonal force constants from experience, but not the off iagonal terms, which can sometimes be quite large. In this part, the stuent will be aske to make guesses for the Hessian, an see the effects on optimization behavior. r 1.000 θ 100.0 E r 1, r, θ E s r 1 E s r E b ( θ) E bs r 1, θ E bs r, θ r 1 r r r g 0 g 1 E r 1, r, θ r 1 E r 1, r, θ r Here we set up the problem by efining the terms to be use for an initial guess structure g E r 1, r, θ θ Last Moifie Sept. 3, 1997 16

Stuent exercise: Create a iagonal approximate Hessian with values for the iagonal force constants that are at least smaller than half of the values in Part IV. Print the q, g an H to make sure they are correct. Compute the isplacements an print below. Are the isplacements to small or too large compare to the first step in Part IV? Continue with the new coorinates an approximate Hessian an etermine how many steps lea t the optimum structure. What o you o next? Last Moifie Sept. 3, 1997 17

Stuent exercise: Create a iagonal approximate Hessian with values for the iagonal force constants that are at least larger than twice the values in Part IV. Print the q, g an H to make sure they are correct. Compute the isplacements an print below. Are the isplacements to small or too large compare to the first step in Part IV? Does the minimization process converge for your guess Hessian? Closure Question: If you nee to choose approximate force constants is it better to go too large or too small. Why? Last Moifie Sept. 3, 1997 18

What is the next natural topic to iscuss if one wishes to unerstan how moern software packages optimize structures? Most molecular coorinates are actually store as Cartesian coorinates (x,y,z), whereas all of the coorinates in this ocuments have been using internal coorinates (r an θ, inepenent of translation an rotation). This will be emonstrate in a future ocument which explains the transformation from Cartesian to internal an vice versa. Also inclue in this ocument-to-be will be how to calculate other types of internal coorinates such as torsional an out of plane angles, an how to preict the stock market exactly (just checking to see if you rea this far). Mastery Exercise: Repeat the Newton Raphson energy minimization for H O using r = 1.00 an θ = 100.0 You may guess force constants or fin them in the literature. The author acknowleges the National Science Founation for support of the 1997 NSF-UFE Workshop on "Numerical Methos in the Unergrauate Curriculum Using the Mathca Software" an the organizers (Jeffrey Maura, Anrej Wierzbicki, an Siney Young, University of South Alabama). The author also acknowleges the Henry an Camille Dreyfus Founation (SG-96-17) an the NSF-ILI program (DUE-965033) for equipment support to evelop the use of computers in the unergrauate chemistry curriculum at the University of Alabama at Birmingham. Last Moifie Sept. 3, 1997 19