Chem 7040 Statistical Thermodynamics Problem Set #6 - Computational Due 1 Oct at beginning of class

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1 Chem 7040 Statistical Thermodynamics Problem Set #6 - Computational Due 1 Oct at beginning of class Good morning, gang! I was coerced into giving a last-minute Chem Dept recruiting talk in balmy Minnesota this afternoon. You are on your own for today s class period, but hopefully this class will be a chance for you to work together. As advertised, you will be writing your own Monte Carlo program using Excel. The spreadsheet that you generate (including plots) will be ed to me as an additional homework assignment. You will work in pairs; only one spreadsheet needs to be ed to me per pair. (I recommend that you use the better-equipped computer of the pair.) The pairs are Kitt Erica Maggie Carlos Nick Danielle Bryce Arpa Alex JD Steve Tao Yuqing Adam Cady Michael Abhinaw Dave Bin Robert The notes below will guide you through the process. In this problem, you will be performing a classical Monte Carlo simulation at T=298K on a Morse oscillator, which is representative of the vibrational (internal) motion in the NaCl molecule. Of course, a simulation of this one-dimensional system with Monte Carlo is not necessary. Since we have the potential function (see below), we could obtain ensemble averages on a simple grid. But the purpose is two-fold: (1) to give you experience with Monte Carlo without a complicated, many-particle program, and (2) to allow you to compare to the exact results. A page of Excel tips & tricks is provided, including basic functions for those that need a refresher, as well as more advanced functions that will be needed for this exercise. Excel Basics & Functions to Use Basics: Manipulation of cells in Excel is handled by cell addresses. The column (letter) and row (number) define a cell, such as B12. These cells can be referenced for arithmetic. For example, to multiply the two shown cells, I would enter =B3*C3 in the highlighted cell (D3). After pressing Enter, the product will appear in the new cell. Importantly, any later change to B3 or C3 is automatically reflected in the product; no re-entry of the formula is required. Page 1 of 6

2 A very handy feature of Excel is that formulae may be filled down. For example, if columns B and C contained many numbers and you wished to perform the same multiplication as the example above for each pair, you can put your formula in D3. Then drag the box in the lower-right corner of the cell highlight down for as many rows as you wish. The formula will fill each cell and adjust the contents accordingly. After filling down, the last cell shown would contain =B13*C13 as its formula and 28 as its value. In some instances, you may not want Excel to adjust the formula as you fill down. (For example, if you reference fixed parameters at each cell, you always want it to reference the same cell.) A cell s address can be fixed with the $ symbol. To fix the whole cell B3, you would use $B$3. To fix just its column, use $B3. To fix just its row, use B$3. When filling down, fixed addresses are not changed. Special functions for this exercise: The function rand() generates a uniformly distributed random number between 0 and 1. Entering =rand() into a cell will call this function. One (mildly annoying) aspect of this function is that any update to the spreadsheet in other cells will cause this function to be called anew. Therefore, a sheet with many instances of this function will be constantly changing values. This behavior will not be a problem, but don t be alarmed if your numbers change every time you enter something new! Excel can also accommodate conditional statements. The =if() function is the one that you will use here. The arguments are if(test, do if true, do if false). For example, consider the case where you want to use the original value if the value of another cell (A5) is positive but just use -1 if it s negative. You would enter =if(a5>0,a5,-1) An advanced trick that will be useful to you is nested conditionals. Take the previous example, but add the caveat that if the number is less than -100, use -2 instead of -1. The entry would be =if(a5>0,a5,if(a5<-100,-2,-1)) In this example, the do if false command is yet another if() statement. This syntax will be helpful in the Metropolis testing of your Monte Carlo routine. The sum function is also very handy. Entering =sum(a1:a30) will sum the first 30 elements of column A. Similar expressions exist for averages and standard deviations. The countif function will count the number of cells in a listed range that satisfy some criteria. For example, the number of cells in the first 30 cells of column A that are positive would be computed as =countif(a1:a30, >0 ) (Note the use of quotes.) One tricky caveat to this syntax is that it cannot directly be used to compare to cell values in the conditional. To check if these cells are equal to another cell (B1, for example), use =countif(a1:a30, = &B1) Page 2 of 6

3 Classical Averages Within the classical approximation, canonical ensemble averages may be obtained as ( ) ( ) where the distribution function is ( ) ( ) and the classical partition function is ( ) Morse Oscillator Potential A Morse potential is a commonly used bond function in chemistry. Unlike the harmonic oscillator, a Morse oscillator properly dissociates at long bond lengths. It also includes anharmonicity. As a function of the internuclear distance, the Morse potential is ( ) ( ( ) ) Here, is the equilibrium bond length, and is the bond dissociation energy at. The parameter describes the width of the potential. For the NaCl molecule the Morse parameters are Note that all parameters have been listed in atomic units. The atomic unit for distance is the Bohr radius. ( ) The atomic unit for energy is the Hartree. ( ) In atomic units,. In the spreadsheet that you will create, I strongly recommend that all of your work be performed in atomic units to avoid any conversion fiascos. To compute in atomic units, multiply your temperature (in Kelvin) by Page 3 of 6

4 1. Your first Excel-based task is to view the potential function. On a uniformly spaced grid of 100 points in Excel, plot the Morse potential between. For best visualization, plot the potential up to twice the dissociation energy on the y axis. Put this data and the corresponding plot in the first sheet of your workbook. In Excel, sheets are labeled by tabs at the bottom left corner of the workbook. Name this one Potential (double-click to rename). As step-by-step hints (a) Determine the grid spacing. (b) Place the potential parameters at the top of the spreadsheet so that you can reference them throughout. (c) Create a column of bond length (R) values. (d) Create a matched column of corresponding potential values, using the parameters from (b). Using the $ syntax will be useful here. (e) Insert a line plot and associate it with your data. Please put the plot near the top of your sheet. 2. The next step is to generate a sheet with reference (exact) values for comparison. Switching to Sheet2 (and renaming it Exacts ), compute the following quantities on a grid of 100 points over the range. (a) Average bond length. Compare this length to the equilibrium (bottom of the well) value. Hint: You may use the rectangle rule to integrate as needed. Use the formulae provided earlier for computing averages. (b) Average potential energy (c) Normalized position (bond length) distribution plot this one. Hint: The normalization constant is just the partition function, which is the integral over your distribution. Again, please put these items near the top of your Exacts sheet. Page 4 of 6

5 3. Your Sheet3 (renamed MonteCarlo ) will now finally be your Monte Carlo simulation. (a) Place all potential energy parameters near the top of your sheet, to be referenced throughout. (b) Create columns for Monte Carlo step #, R_trial, V(R_trial), delta V, R_saved, V(R_saved) (c) Use the Excel functions described above and the Metropolis algorithm described in class yesterday to generate a Metropolis Monte Carlo simulation of your Morse oscillator. This step will likely require the most thought and time of this exercise. (Note: Your first step may start at any R. Choosing R=Re is convenient but not necessary. Your first Monte Carlo step s formulae may look different from those of the remaining rows.) You should set up your sheet to perform 10,000 steps. The virtues of the fill down feature will quickly become apparent! Hint: The lone sampling parameter in Metropolis MC is the maximum displacement in a single step. You should establish this parameter as a controllable knob at the top of your sheet. Check the acceptance rate of your simulation at the end. It should be somewhere in the 30-50% range for optimal efficiency. (In the limit of infinite sampling, this rate doesn t matter, but we do not have infinite resources.) If your acceptance rate is too large, increase the maximum displacement. If the acceptance rate is too small, decrease the maximum displacement. (d) Compute the average bond length of your oscillator. (e) Compute the average potential energy of your oscillator. (f) For the average bond length, determine the standard deviation. (Note! This value is a real, physical quantity, which describes the width of the distribution. It is not an error bar.) (g) With the standard deviation available, determine your sampling error bar in the bond length as. (h) Add a new column to generate the running bond length average. This value is the average bond length over the steps performed so far. At the first MC step, it s just the starting value. At the next, it is the average over two values, etc. Compute this value over the course of your simulation and plot the results as a function of MC step #. What do you notice? (i) Finally, generate the binned bond length distribution. This distribution is equivalent to the one you plotted on a grid previously (#2 use the same grid), but here it involves the results of the Monte Carlo simulation. It is essentially just a histogram of R values. You will need to create some new columns. First, generate your binning grid. The spacing should be (max-min)/(# points). There are two ways to determine in which bin an R value fits. The first method (bad idea) is to loop over the grid points and find which one the value fits. This method is very slow and tedious. The other option is to use the round function. For example, if my R value is in cell A1, my minimum bin R value is in cell A2, and my bin spacing is in cell A3 then =round((a1-a2)/a3,0) Page 5 of 6

6 will provide the bin number into which the R value falls. (The last 0 is the number of digits to round toward.) Perform this check for each MC step. Then use countif() to determine the number of sampling points in each bin. Finally, normalize and plot your distribution. For comparison, plot the exact result (from #2) on the same graph. When your spreadsheet is complete, send it to ryan.steele@utah.edu. If you do not complete the spreadsheet during class time, you may work on it over the weekend and it to me by class time on Monday. (No classes are scheduled after ours in room 2010 on Fridays, so you may stay as long as you wish.) I will be back in SLC first thing Saturday morning, so please feel free to ask questions via if you get stuck. Page 6 of 6