Temperature-Dependence of the HCl Spectrum: A Simulation
Contents
1. D gt If you haven t saved your file yet do so This is your last reminder When you ate finished with all 5 functions you are ready to test them by graphing the energy levels of the ground 0 and excited v 1 vibrational states of HCl as a function of rotational quantum number J for 0 lt sJ lt 50 To do this you will need to a make waves that will hold the values of all 51 of the ground and excited state vibrational energies and b assign the appropriate energy values to the waves You have already learned how to make waves in the Igor tutorial For the x axis J values we will take advantage of the wave X which already exists in Igor s memory and contains the integer values 0 1 2 3 such that x O 0 x 1 1 etc To see how this wave is useful in this case click on the command window and type the following text make n 51 VibO Jwave Jwave x VibO RovibEnergy 0 Jwave AppendToTable Vib0 Jwave Do you understand what you just did The first line creates a wave named VibO to hold the values of the ground state vibrational energies and a wave named Jwave to hold the values of J The second line tells Igor to set the waves X and Jwave to be equal so now Jwave 0 0 Jwave 1 1 etc The third line tells Igor to calculate many values of the function RovibEnergy v J using 0 and all of the values in the wave Jwave for the values of J then place the result of the calculation in VibO so now VibO 0 RovibEnergy 0 02. and the reduced mass of the molecule u 1 Jk V I 6 27c Yu where u is given by mm u PELA A 7 m My and m is the mass of an individual atom The constants x and y characterize the degree of anharmonicity in the bond ie the inelasticity of the vibrations due to repulsions and attractions between nuclei and electrons Now let s consider rotational motions The simplest picture of a rotating object is the rigid rotor picture where two masses are connected by a rigid rod whose length is fixed As for the harmonic oscillator the Schrodinger wave equation SWE can be solved exactly for the rigid rotor resulting in the rotational energies Ere J 4BJJ 1 8 where Ep is in units of Joules B is a sort of angular frequency of rotation in Hertz and J 0 1 2 is the rotational quantum number that characterizes the rotational state of the diatomic Le how much rotational energy it has We know however that equation 8 will not be an accurate description of the rotational energies of our diatomic molecule since the rod bond connecting the two masses atoms actually vibrates changes length When a diatomic molecule rotates it acts like a non rigid rotor In fact as it rotates the bond can begin stretching even more due in part to centrifugal forces think of spinning a rock on a slinky This means that the rotational energies must 72 Fall 2004 5 Simulated RoVib Spectrum of HCI Experiment
3. VibO 1 RovibEnergy 0 1 etc The last line tells Igor to display the contents of the waves VibO and Jwave in a table Click on TableO you should see your new wave VibO and the corresponding values for each value of J at v 0 Calculate two values of VibO in your notebook to check that the function RoVibEnergy v J is working properly and that values have been assigned to VibO correctly To plot VibO versus J enter the code 76 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry Display Vib0 vs Jwave Since the energies are quantized this values should not be displayed as a continuous line Modify the symbols to show discrete values and add axes labels Now you must repeat the procedure for the rovib energies when v 1 however you will not need to recreate Jwave it already exists you can just use it as needed Append the Vib1 values to your plot by typing AppendToGraph Vib1 vs Jwave Your graph should look similar to the one in Figure 1 Be sure that both axes start at 0 e e 25000 vibo oo Vib1 e a ea lt 20000 evo g P ao 2 so e amp 15000 e D e e 0 Lu C E 2 10000 oo oo aw on 00 5000 leggpaoeneOO 0 o gt 0000000007 k l 0 10 20 30 40 50 Rotational Quantum Number J Figure 1 The rovibrational energies of HCI as a function of rotational quantum number Print a copy of your graph to hand in with your report
4. rovib Instead we will have Igor calculate values of PQ and RJ using equations 16a b directly TASK 2 Calculation of Peak Positions Transition Energies e Create function PeakPosition to calculate transition energies and check code To calculate the positions of the absorbance peaks along the x axis we will create a function called PeakPosition There are two approaches that we could take to do this The easiest but more inelegant way is to define two separate functions one to calculate the energies of the P branch transitions and one to calculate the energies of the R branch transitions Each of these functions would require the input variables v and J see equations 16a b A more elegant and not much harder approach is to define a single function that has one additional input variable the branch name This approach requires the use of a structure called conditional branching which is used when you want the function to do one thing in one case but something different in another case It is a structure that is common to almost all programming languages The conditional branch in Igor appears as an if else endif structure Our function will gt These selection rules are actually only strictly true for a HO RR system In practice they work quite well for predicting the rovib spectrum of most diatomics at a reasonable resolution 78 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry therefore
5. Ed MIT Press Cambridge MA 1993 On reserve in Murphy Library K P Huber and G Herzberg Molecular Spectra and Molecular Structure IV Constants of Diatomic Molecules Van Nostrand New York 1979 An updated version of the tables in Herzberg pages for H Cl are included in this manual as Appendix B P W Atkins Quanta 2 4 Ed Oxford University Press Oxford 1991 see Angular momentum Branch Vibrational spectroscopy Programming Igor Pro User s Manual 89 Fall 2004
6. Simulated RoVib Spectrum of HCI Experimental Physical Chemistry for HCI obtained from Huber amp Herzberg All constants should be in units of cmt and numbers like 5 3194 x 104 can be entered as 5 3194e 4 These values should also be entered in your notebook with the reference Your global variables are now loaded and are ready to be used in functions To test this go back to the Command window and type Print Be Be Igor should respond in the History area with Be 10 5934 Now that the global variables are loaded it s time to start working on the calculation of the energy levels We will create here a suite of functions that will allow us to calculate the energy of any quantum state v J Our suite will consist of the following functions Function Bv v Calculates B for vibrational level v using Be and ae Function Dv v Calculates D for vibrational level v using De and Be Function Nubar v Calculates v for vibrational level v using V V x and V y Function RotEnergy v J Calculates the rotational energy for state v J Function VibEnergy v Calculates the vibrational energy for state v Function RovibEnergy v J Calculates the total rovib energy for state v J Why are we breaking the calculation up like this One of the cardinal rules in programming is to compartmentalize your code That is break big jobs into small sections This makes debugging easier interpreting someone else s code easie
7. actually require three input variables Branch v and J The variable Branch is a string variable i e a variable that holds a string of text that allows us pass string values of either P or R to our function as the input string values are indicated in Igor by quotation marks The frame of our function looks like this Function PeakPosition Branch v J Calculates P or R transition energies string Branch II Pass values of P or R only variable v J v is the initial state vib quantum number if some expression II if this is a P branch transition do this if the expression is true then calculate the P Branch peak position else if this is an R branch transition do this if the expression is false then calculate the R branch peak position endif End but we need to fill in the squiggly brackets with Igor code We will start with the some expression which needs to be something like Is the string variable Branch equal to PP Although Igor has a number of comparison operators on hand for comparing numbers lt etc they do not work for s rings string comparisons are complicated by questions such as does P equal p or space P etc Igor has a different set of built in functions for comparing strings There is an on line catalog of available functions that is kept in the Function Help menu under the Mise menu Choose String functions from the p
8. both versions of the thermal distribution plot Now return to the FillSpectrum function and modify it to display the true intensities of each line You will need to add the input variable T ambient temperature to parameter list and declare T as a variable in the first set of declaration statements Modify the lines that assign an intensity of 1 to each transition line to now assign an intensity calculated by your function Relativelntensity Run FillSpectrum at room temperature 298 K for v 0 and Jmax 50 Expand the center region You will probably see something that looks like Figure 5 below missing line Intensity N l 0 ao T T T L 2500 2600 2700 2800 2900 3000 3100 3200 Transition Energy cm Figure 5 A slightly flawed spectrum of HCI at room temperature 86 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry If you compare this plot to the experimental HCl spectrum you will see that you are missing P 4 The reason for this is fairly subtle It happens that R 50 2799 2 cm falls within 1 cmt of P 4 2798 97 cm so that both transitions are assigned to the same location in Spectrum0 Since R 50 is calculated after P 4 in the do while loop the value of P 4 is overwritten by the value of R 50 in Spectrum0 The intensity of R 50 at 298 K is so tiny that we end up with a missing line A mote realistic way to handle this is to sum overlapping lines rather t
9. to return the answer to you Function Bv v II Function calculates Bv in cm 1 from global molecular constants variable v return Be AlphaE v 0 5 End 75 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry The double slash is a symbol used by Igor to denote comments text ignored by Igor As you program you must insert comments so that it is easier for other people to understand your code When you feel that you understand how this code works enter it in the Procedure window below the Macro code Test your function by typing in the Command window Print Bv 0 Bv 0 Igor should respond with Bv O 10 4398 Perform this calculation in your notebook using equation 10 to be sure it s right before you go on Now you need to write the code for the other four functions in our suite Use carriage returns between the end of one function or macro and the beginning of the next so that they can be read more easily In creating RotEnergy VibEnergy and RovibEnergy take advantage of the other functions that you have written by using them in your code For example the body of the code for RovibEnergy should look like return VibEnergy v RotEnergy v J Be sure to pay attention to what variables need to be declared for this set of functions it will be some combination of v and J as defined in the function declaration in general it is whatever variables are used by the function
10. total energy of the system the diatomic is given by E vvi Et Exe 1 where E is the energy due to rotations and E is the energy due to vibrations uib Recall that an absorption spectrum is essentially a plot of the amount of light absorbed versus the energy of the light absorbed Therefore in order to simulate the absorption spectrum of a diatomic molecule such as HCl we must be able to calculate the energies of light that are absorbed Since the energies of matter and light are quantized can only have certain values only certain wavelengths or energies of light can be absorbed These were predicted by Neils Bohr based on the simple condition that E rovib 2 photon where Esm is the energy of the absorbed light For most molecules AE is on the order of rovib 10 cmt which puts Eps in the infrared region of the EM spectrum Typically E is larger than E by at least a factor of 10 so that rotational energies are actually small corrections to the vibrational energy spacings Therefore we ll start by examining the expressions for vibrational energy If the bond were a perfect spring i e vibrations were elastic and followed Hooke s law k amp x then the Schrodinger wave equation could be solved exactly and the vibrational energy of the molecule would be given by the expression for the energies of a quantum mechanical harmonic oscillator Epo v bvw 1 2 3a 1 The molecule can al
11. true constants like B TASK 1 Calculation of the Energy levels e Create macro NewConstants to hold molecular constants e Create functions to calculate Enovin v J e Make plots of Erovb 0 J vs J and Enovin 1 J vs J The construction of any molecular spectrum begins with the calculation of the energy levels available to the molecule You will begin by creating a macro called NewConstants that will allow you to enter all of the values for the molecular constants that characterize HCl into Igors memory Although you could use the numerical values of these parameters in every function it is much nicer to be able to change them in one central location Accepted values for all of the necessary molecular parameters can be found in the latest compilation by Huber and Herzberg a portion of 2 K P Huber and G Herzberg Molecular Spectra and Molecular Structure IV Constants of Diatomic Molecules Van Nostrand New York 1979 73 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry which is included as Appendix B note that the values of some of the parameters have been hidden in the footnotes and that you want the data for lowest energy electronic state which is the state labeled X Start up Igor and open the Procedure window the Procedure window is where all the function and macro code is stored The first step in any function or macro is the declaration of variables You will need to declare a va
12. Experiment 5 Temperature Dependence of the HCI Rovib Spectrum A Simulation One of the virtues of spectroscopy is that an enormous amount of information is contained in a single picture the spectrum In this experiment we will assemble the components of a spectrum one piece at a time first the molecular energy levels then the positions of the absorbance peaks and finally the intensities of the peaks As a final part of the process you will make a QuickTime movie of how your spectrum changes as the temperature is changed Numerical simulation is an invaluable tool for designing experiments and interpreting experimental data In this example we can easily change the temperature of the experiment over a range that far surpasses what is practical experimentally Thankfully structured programming in high level data analysis programs such as Igor Pro has become very user friendly You have already seen how to use a few of the programming tools available in Igor Pro in the Igor tutorial For this experiment we will pull together some of these ideas and introduce a few more tools Much of this experiment is taken from one developed and graciously offered by Dr Melissa Hines a professor of chemistry at Cornell University The Rovib Energies of HCI A diatomic molecule can be modeled as two balls atoms connected by a spring bond If we consider the principle motions of the diatomic molecule to be bond vibration and molecular rotation then the
13. HCl Go back to the questions on page 84 and rethink your answers You should feel very comfortable with all of these types of questions by the time you have finished your simulation If 88 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry not go back and play some more with different values of v Jmax and T for your different functions until you think you have a good physical understanding of rovibrational spectroscopy Wrapping it Up In addition to your notebook pages your report should include the following e an abstract e the code that you have written in the Procedure window that is you do not need to print out Graph or Table macros that Igor created e plots of the various figures that you made during the simulation e a labeled diskette containing your Igor experiment free of viruseslll e a short discussion of some of the relevant points raised on page 84 Be sure that your notebook record contains sufficient detail so that someone could reconstruct your simulation without the aid of this manual You may assume that the someone has a working knowledge of Igor that is similar to your understanding before you started this experiment You must include explanations of how the Igor functions that we used x2pnt etc work Don t forget the derivations also General References Spectroscopy J L Steinfeld Molecules and Radiation An Introduction to Modern Molecular Spectroscopy 24
14. Plot the relative intensities of the transitions originating from the ground v 0 and first excited v 1 vibrational states of HCI at room temperature as a function of J for J 0 to 50 You will first need to make the waves IntensityO and Intensity1 to hold the values of these intensities You may want to go back and see how you did this for the Rovib energies earlier Your plot should resemble that in Figure 4 This is because J is not the only quantum number pertaining to rotation there is also the quantum number m which describes the plane of orientation Since m spans the range J to J there are 2J 1 sublevels for each J level In the absence of external fields all of the m sublevels are degenerate they have the same energy and so many m states can be populated with equal probability 85 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry Relative Intensity 0 10 20 30 40 50 Rotational Quantum Number J Figure 4 Relative populations of H CI at room temperature Appearances are deceiving the shape of the v 1 level distribution thermal envelope is actually the same shape as the 0 level distribution but the relative intensities are much smaller Verify the similarity of the v 0 and yv 1 thermal envelopes with a graph generated as follows display IntensityO The wave for v 0 append R Intensity1 Append the wave for v 1 using the right hand axis gt Print out
15. al Physical Chemistry depend not only on the rotational quantum number J but also on the vibrational quantum number v As for the anharmonic oscillator the SWE cannot be solved exactly for such a system However the true rotational energies can be approximated very well using the expression in equation 9 now in wavenumbers having divided by 4c Ew 2 J BJJ 1 DUT YP 9 where B B a v 1 2 10 and D D B v 1 2 11 In much the same way that V x and y are the molecular constants that characterize the vibrations of a particular bond B D and p characterize the rotations of a particular molecule The equilibrium rotational constant B completely characterizes the rotations of a rigid rotor and is related to the equilibrium bond length 7 according to equation 12 b B en 80 cur 12 The centrifugal stretching constant D is related to the equilibrium bond length as well as bond stiffness and the coefficients a and B termed coupling constants characterize the extent of rotational vibrational coupling i e how much the bond stretches in response to rotating If we combine equations 4 and 9 we obtain an expression for the total energy of a diatomic molecule due to rotations and vibrations the rovib energy 13 The subscripts v on V B and D are reminders that the values of these parameters depend on the vibrational quantum number and are not
16. ax to the list of parameters passed to the function in the parentheses next to V include Jmax as a variable in the first variable declaration and change the while J lt 50 statement accordingly Add the ability to change the intensity wave Spectrum0 This is so we can simulate multiple spectra say for transitions starting at v 0 or at v 1 etc We will use a variable named Spectrum to hold the name of the spectrum wave for example Spectrum0 Change all references within the code from Spectrum0 to Spectrum Add Spectrum to the list of parameters passed to the function and declare it as a wave on a separate line between the two other sets of variable declarations Make sure that the line position for P Q is wot marked a J 0 J 1 transition is mot possible since J is never negative While this really should be accomplished within PeakPosition it is acceptable to assign P 0 an intensity of zero in FillSpectrum Be sure to do this outside the body of the loop 83 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry When you have gotten everything to work you should experiment with your function Set your x axis so that it spans the full range of transition energies 0 4000 cm1 Try running FillSpectrum with various different values of Jmax v and Spectrum To change the value of Spectrum you will need to make new Spectrumv waves first i e match the value of V to the name of the spectrum wave so tha
17. be sure that it has a Figure caption or a title Since the wave x contains integer values we could actually use x whenever we need quantum numbers rather than create a new wave to hold quantum numbers You may use either approach but be sure to document your choice in your notebook Positions of Absorption Peaks in the HCI Rovib Spectrum When a molecule absorbs light the energy from the light causes the molecule to undergo a transition from an initial state to a final state According to the Bohr condition the molecule can only absorb the light if the energy of the light matches the difference in energies between the two states of the molecule E pun AE pae However not every one of these photons will be absorbed equally well that is we may not be able to observe a measurable absorbance intensity for each type of photon that matches the Bohr condition This intensity condition is embodied in part in selection rules which are a set of restrictions on the allowed changes in the values of the quantum numbers They are a result of integrals that are required to calculate the intensities of the transitions 77 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry and ultimately arise from fundamental rules such as conservation of angular momentum The selection rules for rovib absorption transitions of a diatomic are Av 1 AJ 1 14 This means that the only rovib transitions that can be obser
18. can simply write Spectrum0 2762 1 to make a peak of unit intensity occur at 2762 cm This turns out to be inelegant because we are limited to 1 cm resolution for absorption peaks However the ease in programming will offset this inconvenience for our purposes Make a wave named Spectrum0 that contains 4001 points we will choose a spectral range of 0 cm to 4000 cm since this covers the spectral range for the IR spectrum of HCl it would be different for a different molecule You now have a wave that will hold the intensities of all the lines in the spectrum that spans the correct spectral range and that is indexed by the resident wave X The initial plan is to set all of the spectral line intensities to be equal to one we will add correct intensities later Realize that only those points in Spectrum0 that correspond to observable transitions should have a non zero intensity all other points in Spectrum0 will have zero intensity What we need to do now is 1 use the function PeakPosition to calculate the transition energies of the observable absorbance peaks and 2 assign an intensity of 1 to each observed transition in Spectrum0 This calls for another Function which we will name FillSpectrum To do this we need to learn another programming structure A loop is useful whenever you want to have the same task repeated In Igor the supported loop structure is a do while loop other programming languages may have slightly different varie
19. double clicking on the Movie icon that you just created the QuickTime viewer called MoviePlayer is already installed on the Macs Mave at least two more movies one with v gt 0 and one with the x axis expanded to cover the region 2800 3500 cm What do you learn from these two new movies Band Head Formation When you watch your movie you should see a band head form as the temperature is increased A band head forms when one branch of the spectrum folds back upon itself so that there is a maximum or minimum transition energy a band head The value of J at which the band folds can be calculated by treating the expression for the transition energy as a continuous function of J Ignoring D terms we have PU Y B B J B B y 20 R J V0 2B 3B By B By 21 where Vgo is the band origin energy due to vibrational energy changes only To find the value of J at which P J reaches a maximum we can differentiate equation 20 with respect to J and set the resulting expression equal to 0 Solving for J we find that 22 where Jp is the rotational quantum number at which the P branch turns around Similarly we find that 23 Derive equations 22 and 23 in your notebook then calculate values of Jp and Jz for the v 0 1 and v 1 2 spectra If the P branch does not fold back then the value of Jp is negative it does not fold This is the case for the ro vib spectrum of
20. eracy g of any rovib state is equal to 2J 1 Furthermore the excess energy above the zero point energy is AE J Erovin y J Exovin 0 0 Therefore the relative intensity of a transition arising from any state vJ is Iv N 2J 1 2 Bed E 00 AT 19 gt Derive equation 19 from equation 18 in your notebook TASK 4 Creating a Thermal Envelope e Create function Relativelntensity to calculate values of relative transition intensities e Plot Relativelntensity vs J for v 0 1 e Modify function FillSpectrum to assign relative values of transition intensities to Spectrum at correct transition energies e Investigate the effects of different vibrational and rotational levels and temperature on the look of the spectrum Using equation 19 write a function called Relativelntensity which has three input variables v J and T Since we are working in units of cm for energy you should use amp 0 69502 cm K as the value of the Boltzmann constant Note that we really should have included amp in our original NewConstants macro You can do this now if you wish You will need to re run the macro once you have changed it gt Show the conversion of k from units of J K to cm K in your notebook Test your function by typing Print Relativelntensity 0 1 298 Relativelntensity 0 1 298 You should get a value of 2 71234 Do this calculation in your notebook by hand to be sure that it s right
21. han overwrite them Go back and modify FillSpectrum so that overlapping lines are summed instead of overwritten use the assignment rather than the assignment Now re run your function The P 4 line should be there Experiment again with FillSpectrum using different values of v Jmax and T Choose a Jmax that allows you to see significant effects on the entire spectrum up to a temperature of 8000 K Go back and look at the questions on page 84 Can you answer these questions better now Print out several representative versions of your spectra at different temperatures v values and Jmax values using illustrative x axis ranges You do not need to print out 50 plots just a few representative versions TASK 6 Making the Movie Create macro RunSimulation to create a movie of the evolution of the spectrum with increasing T e Create movies for v 0 v gt 0 expanded region of spectrum Now you can create your movie showing the evolution of the HCl spectrum as the temperature is raised We will use a temperature range of 1 K to 8000 K To prepare e close all graphs except for the one showing Spectrum0 Igor can only make a movie of the top most window e Resize this graph until it is approximately 7 wide by 4 tall e Autoscale and label the axes e In the upper left hand corner of the graph add annotation which says T 1 R Now you need to write a macro called RunSimulation In its first incarnation you wan
22. indow Print x2pnt Spectrum0 2762 16 Igor should return the value 2762 since point 2762 is the closest point to the number 2762 16 Rather than having actual numbers flying around in the code very clumsy we will use a new variable called PeakPoint to hold the value of x2pnt Spectrum0 2762 16 i e to hold the integer 81 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry value of the peak position Now we can accomplish the task of assigning an intensity of one to the spectral line near point 2762 with the statements PeakPoint x2pnt Spectrum0 2762 16 Spectrum0 PeakPoint 1 Recall that square brackets are used for indexing a wave so the second statement reduces to Spectrum0 2762 1 in this example Of course you don t want the number 2762 16 flying around either Modify this code to replace the value 2762 12 with the code that you use to calculate that peak position Once you have gotten this far it is tempting to just replace set Spectrum0 equal to one at that line position with Spectrum0 PeakPoint 1 and be done with it However it is prudent to put in a line for error checking Suppose for example that we ask Igor to find a point in Spectrum0 that is out of Spectrum0 s range 0 to 4000 cm This might happen if you switch molecules or make a mistake with the input of molecular parameters Clearly there would be a problem In some cases you might just get an error message but i
23. l Physical Chemistry Be sure that you understand how all of these pieces of code work before you go on Also be sure that you are writing in your notebook as you go You may print out your procedure window and paste it into your notebook in the appropriate places if you wish TASK 3 Generating a Stick Spectrum e Make wave Spectrum0 to hold values of transition intensities e Create and modify function FillSpectrum to assign values of transition intensities to Spectrum0 at correct transition energies e Plot FillSpectrum using intensities of 1 for all transition energies e Investigate the effects of different vibrational and rotational levels on the look of the spectrum Now that you have created a function to calculate peak positions you can start to generate a stick spectrum A stick spectrum is a plot of the transition energies peak positions using sticks lines of unit height intensity We will take a rather inelegant approach to creating a spectrum by creating a wave called Spectrum0 that actually holds the values of the intensities of the absorbance peaks i e the y axis in a spectrum rather than the peak positions the x axis The index of Spectrum0 will then represent peak positions in integral numbers of cm1 For example if SpectrumO is indexed from 0 to 4000 our x axis will be allowed to have integer values ranging from 0 to 4000 Consider for example P 1 which occurs for HCl at approximately 2762 cm With our approach we
24. mine the population of each vJ state at a given temperature T According to statistical mechanics this is given by the Boltzmann equation N ABJ AT N j y gn BEC AT 17 vJ where N is the total number of molecules AE v J is the excess rovib energy of the state J above the zero point energy i e the energy needed to access the v J state from the 0 0 state g is the degeneracy of the state and amp is the Boltzmann constant Simply stated the Boltzmann equation says that unless the energy required to populate a state is less than or equal to the thermal energy T the state has little chance of being populated If the state is not populated the intensity of the transition originating from that state will be zero The zero point energy is not included because a molecule can never have an energy less than the zero point energy For the purposes of this simulation only the re ative intensities of spectral lines are important not absolute intensities Therefore only the re ative populations of different states are really 84 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry important The term in parentheses in equation 17 is just a constant at a given temperature and for a given number of molecules Therefore if we consider the populations of all states relative to the 0 0 state then we can write E v AT N j amp t E 0 0 kT N 0 0 80 0 Ta 18 The degen
25. n others you might actually crash the program or even the machine It is good programming technique to try to avoid these sorts of annoyances In this case you will need to make sure that the value of x2pnt Spectrum0 transition energy is greater than the lower bound of Spectrum0 which is 0 and less than the total number of points in Spectrum0 the upper bound The function numpnts can be used to determine the total number of points in Spectrum0 Look up numpnts in the Igor Waves function list The logical operator amp acts like an AND it is called a bitwise AND can be used to compare values of PeakPoint You can now bring all of this information together to construct the body of your do while loop Below is an example of the code you should end up with in place of the old body of the loop do PeakPoint x2pnt Spectrum0 PeakPosition P v J find point closest to transition if PeakPoint gt 0 amp PeakPoint lt numpnts Spectrum0 peak is within range SpectrumO0 PeakPoint 1 endif J 1 while J lt 50 This code will fill in the P branch part of the spectrum Be sure that you understand how it works Ask questions if you don t understand To complete FillSpectrum you need to do the following things e Declare the new variable PeakPoint in the second variable declaration at the beginning of the function Note again that variables that appear in the call statement in the definition of the func
26. ode illustrates several points of style that you should follow in writing code First every line except the first line variable declaration line s and last line are indented by one tab stop This is so that the body of the code the work engine part stands out When we start to work with loop structures the body of the loop should be indented one further tab stop Second the naming of variables in programming code has become standardized over the years when words are strung together to make a variable name the initial letters of each word are capitalized and all other letters are lower case This is to avoid variable names such as inubarexe or INUBAREXP which are almost impossible to read It is a good idea to periodically check your code for errors as you program In Igor Pro you can do this by clicking on the Command window Igor will then automatically compile everything in the Procedure window If your code has errors Igor will give you error messages with some hopefully helpful information on how to fix them If you see a compile button at the bottom of your Procedure window Igor has not yet compiled the code in the window so it will not yet run you can also click on the compile button to compile the code Click on the Command window now to check your code If you do not receive an error message yay look under the Macro menu for your new macro and try to run it Enter the values 74 Fall 2004 5
27. r and additions or modifications easier In this case we are breaking the calculation of equation 13 into five smaller pieces We ll tackle one piece at a time Functions in Igor are defined with the keyword Function instead of Macro Functions and macros are similar but generally functions run faster while macros are more versatile We will use functions when we need a value returned to some other function or macro so that we can use it in another calculation You will be lead through the writing of code for the first function but you must write the others on your own The first function is Bv v which will calculate a value for By using equation 10 The name of the function is Bv while the V in parentheses indicates that the function will need to be given a value of v the vibrational quantum number in order to do the calculation it is an input variable All of the molecular constants in equation 10 have already been declared as global variables and can be used by all functions all of the time The typical syntax for a Function is shown in the sample code below The first line defines the function and the variables that the function needs in order to do the calculation The second line is the variable declaration which lets the program know that you will be using V as a variable in the calculation The meat of the function is in the return statement which simultaneously defines the calculation to be performed and tells the function
28. riable to hold the value of each of the constants that are hidden in equation 13 ie in equations 5 10 and 11 Be Qe De Be V Vx and Vy In Igor the declaration of variables that can be used by the entire program requires two steps declaration of local variables available to the macro alone and declaration of global variables available to the entire Igor experiment The local declaration of variables is accomplished using the keyword variable see line 2 below We will then ask Igor to prompt the user to enter the values of the molecular constants by using the Igor function Prompt lines 3 7 The actual global declaration occurs after the normal variable declaration and is accomplished using the syntax variable G lines 8 9 Finally the macro ends with the command EndMacro line 10 With these structural ideas in mind examine the following code Once you understand it you may type it into your Procedure window Macro NewConstants iBe iAlphaE iDe iBetaE iNuBarE iNuBarExE iNuBarEyE variable iBe iAlphaE iDe iBetaE iNuBarE iNuBarExE iNuBarEyE Prompt iBe Enter Be Prompt iAlphaE Enter AlphaE Prompt iDe Enter De Prompt iBetaE Enter BetaE Prompt iNuBarE Enter NuBarE Prompt iNuBarExE Enter NuBarExE Prompt iNuBarEyE Enter NuBarEyE variable G Be iBe AlphaE iAlphaE De iDe BetaE iBetaE NuBarE iNuBarE variable G NuBarExE iNuBarExE NuBarEyE iNuBarEyE EndMacro Body of code This section of c
29. sitions to 1 It then increments the value of J to J 0 1 1 Since J is still less than 50 the loop starts again with J 1 The calculation and intensity assignment will be done again and again for each incremented J until J 1 51 at which point the loop breaks The variables V and J are declared on two separate lines in this case because V is a value that is passed to the function by the outside user while J is used only internally by the function Note that the maximum allowed value of J here is not physically meaningful it s just some large number Now we need to fill in the body of the loop with Igor code You already have written a function PeakPosition that does the first task in the body of the loop Therefore we can devote our efforts to the second task of assigning an intensity of one to each of these lines It would be nice to just say SpectrumO PeakPosition 1 However this will not actually work because the energies calculated by PeakPosition are non integral for example P 1 2762 16 cm1 but the indices of SpectrumO are integral they are the values in X 0 1 2 3 2762 2763 etc Fortunately Igor has a function that takes any number i e 2762 16 and finds the point number ie index number on a wave that is closest to that value This function is called x2pnt Look up x2pnt in the Igor function list and describe how it works in your notebook To see x2pnt in action type in the following code in the Command w
30. so have translational motion but translational energies are so small that we can neglect them in this analysis 71 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry where Epo is in units of Joules v is the frequency of the vibration in Hertz and v 0 1 2 is the vibrational quantum number that characterizes the vibrational state of the diatomic i e how much vibrational energy it has Since vibrational spectra are typically obtained in units of wavenumbers cm we will convert all energies in Joules to wavenumbers by dividing by Ae Eno v Vo 12 3b Real bonds however do not act like ideal harmonic oscillators because there are attractive forces between electrons on one atom and nuclei on the other and repulsive forces between nuclei and between electrons Real bonds are in fact anharmonic oscillators The Schrodinger wave equation cannot be solved exactly using such real potentials However the true vibrational energy of a diatomic molecule can be fairly accurately modeled as a power series expansion of equation 3 in which each successive term is a smaller correction to the harmonic energy Ep0 V 0 1f2 4 where V V x v 1 2 9 v 1f2y 5 The constants in equation 5 characterize the way that the bond vibrates v is the harmonic vibrational frequency in wavenumbers which depends on both the stiffness of the bond described by the force constant amp
31. t it to repeatedly call FillSpectrum with temperatures increasing from 1 K to 8000 K multiplying by a factor of 1 1 on each pass Use your best value of Jmax I suggest a value larger than Jmax 50 since you will be simulating high temperatures and V 0 and thus also Spectrum0 for a first try You should notice by now that this calls for a loop structure Within your loop structure you will also need to update the annotation on your graph after calculating the spectrum at the new temperature The following code will do this for you NoteString T num2str T K II T is the temp variable Textbox C N text0 NoteString As usual you should not use this code until you understand how it works the second line is Igor s way of making a textbox You need not worry about that part Be sure to declare NoteString as a string variable at the beginning of the macro 87 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry Once you have gotten your simulation running it is time to make the movie In the macro RunSimulation add the command NewMovie on its own line right after your variable declarations then add CloseMovie right before End Within the temperature loop add the command AddMovieFrame afer the annotation update code These commands will slow the macro considerably but it still runs nicely as a QuickTime movie When you run your macro Igor will create the movie It can be viewed by exiting Igor and
32. t when v 0 Spectrum Spectrum0 etc You should get to the point where you are able to answer discuss the following questions e Notice that there is a maximum energy beyond which there are no transitions Why would this be e What happens to the spectrum as JMaXx is increased Is it physically meaningful to change Jmax Is there some value of Jmax that would be physically meaningful How would you calculate it e What does it mean when there is a transition line at 0 cm1 e What happens to the spectrum as you change v Try displaying several spectra say Spectrum0 and Spectrum1 on the same plot use different colors What happens when v is very large Why You should have a good sense of all of the information contained in these plots before you go on gt Print out several representative versions of your stick spectra from your investigations you may have to adjust your x axis You need to do this now because the waves will get modified later and your spectra will look different Take a deep breath this section was the longest one Intensities of the Rovib Transition Lines The reason why your stick spectrum looks nothing like an experimental spectrum is that the intensities are wrong The intensity of a transition is directly proportional to the number of molecules in the znitial state except in the case of very intense light sources Therefore to get the correct the line intensities we must be able to deter
33. ties of loop structures The general form of the do while loop is A more elegant and seemingly logical approach would be to create two waves one for peak position and one for intensity In practice it turns out to be rather complicated to program a versatile way correlate these two waves gt We could have used a Macro but we need the speed of a Function 80 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry do things to be repeated while expression if expression is true go through the loop again In this case you want the value of PeakPosition to be calculated for two branches P and R and for many values of J say J 0 to J 50 You also need to be able to assign a value of 1 to the intensity of each observed transition Therefore the skeleton of the code for our Function looks like this Function FillSpectrum v variable v value of v for the initial rovib state variable J value of J for the initial rovib state J 0 start at J 0 do calculate the P and R absorbance peak positions corresponding to this J set Spectrum0 equal to one at those peak positions J 1 increase J by 1 while J lt 50 check to see if J is still less than or equal to 50 if so loop again End This loop starts by calculating the P and R transition energies for an initial state with the specified value of v that was passed to the function and J 0 then sets the intensities of those tran
34. tion i e must be declared on a separate line from variables that are used only inside of the function i e J and PeakPoint 82 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimental Physical Chemistry At the beginning of the body of the function set the value of all points in Spectrum0 to 0 in case this is not the first time through the calculation You can do this quickly by simply assigning the entire Spectrum0 wave a value of zero Add some code within the loop to calculate the R branch peak positions and set their intensities to equal one Now you can test your new function From the Command line call FillSpectrum with v 0 then plot Spectrum0 type Display Spectrum0 in the command window Igor will choose the x axis automatically Adjust the x axis display to give you a nice picture of the spectrum change the markers to sticks to zero and label the axes Print out a copy of this spectrum sow because it will change later Mine looks like this 1 0 0 85 0 6 4 Intensity 0 4 0 24 0 0 1000 1500 2000 2500 3000 3500 Transition Energy cm Figure 3 A crude HCI stick spectrum Once your function works tweak it as follows e Add the ability to change the maximum value of J used in the calculation in the do loop To do this add Jm
35. ull down menu By clicking on a choice you can view what the function does and how to use it If you surf around the String functions for a few minutes you should find the function cmpstr which will work for our purposes Describe how it works in your notebook We can now replace if some expression with if cmpstr Branch P 0 if this is a P branch transition Be sure that you understand this code before you use it Now you need to complete PeakPosition by filling in the other squiggly brackets with code to actually calculate the P and R branch transition energies Be sure to use a Return statement so that you will have the result of the calculation returned when you call the function Keep your code well commented When you are finished test your code by entering the following Print statement in the command line the response is also shown below the statement Print PeakPosition P 0 10 PeakPosition R 0 10 2651 68 3072 18 Note that you should get exactly these numbers Even small differences mean that you have an error somewhere either in your molecular constants or in the programming gt Calculate the values of E photon A Etot for the P Branch v 0 J 10 v 1 J 9 and R Branch v 0 J 10 v 1 J 11 transitions in your notebook using your values for the molecular constants Does your code work correctly 79 Fall 2004 5 Simulated RoVib Spectrum of HCI Experimenta
36. ved for a diatomic are those that correspond to changes in energy between states with the quantum numbers described in equation 14 Therefore according to the selection rules the allowed rovib absorption Av 1 transition energies for a diatomic are given by E hoton T roil v 1 1 Emi 1 phot s J a J 15 Transitions for which AJ 1 are called P Branch transitions energy Poor and transitions for which AJ 1 are called R Branch transitions energy Rich In a spectrum the x axis represents transition energy frequency wavelength etc Therefore the terms transition energy and peak position i e position along the x axis are used interchangeably P Branch transitions occur at lower transition energies than R Branch transitions which is presumably the source of their names Spectroscopists use the shorthand P J and R J to signify these transition energies Py gt E shorn P Erowid Y Ly mn 1 Emit 16a R J m E om R E pip v 1 J 1 ii E 0 J 1 6b It is important to remember however that P and R J actually correspond to values of Epson and thus to differences in energies of matter E We could write out expression for P J and R J in terms rovib of molecular constants using equations 4 and 9 but since we already have a function that calculates values of E for any value of v and J that would defeat the purpose of compartmentalizing code
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