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Tutorial on Kinetic and Equilibrium Isotope Effects Ke R. Yang, Benjamin Rudshteyn, and Victor Batista* Email: victor.batista@yale.edu Yale University, Department of Chemistry 225 Prospect Street, New Haven, CT 06520 Written August 13, 2017 Edited January 12, 2018 Table of Contents 1. Background on kinetic and equilibrium isotope effects ............................................................ 1 2. Calculation of KIEs and EIEs from frequencies and free energies ........................................... 2 3. Calculation of 18O-KIEs and 18O-EIEs with a real world example ........................................... 5 4. References ............................................................................................................................ 19 Appendix A. Using “iso=” ........................................................................................................ 20 Appendix B. Computing H/D Kinetic Isotope Effects .............................................................. 20 Appendix C. Aqueous Environments for 16O/18O Kinetic Isotope Effects .................................. 20

1. Background on kinetic and equilibrium isotope effects Chemical reactions happen on free energy surfaces. For a specific element, different isotopes have different masses, thus have different vibrational frequencies, different zero-point energies (ZPEs) and free energies (Figure 1). This slight difference will result in different reaction rates and thermodynamic preferences, which is called the kinetic isotope effect (EIE) and the equilibrium isotope effect (KIE). The isotope effect is most important for H/D since they have very different atomic masses. For other elements, such as C, N, and O, the mass difference between isotopes is relative small and thus the EIEs and KIEs for those elements is much smaller than the H/D isotope effect. However, they can still be measured experimentally and used to study the reaction mechanisms. Computational modeling is very useful to interpret the experimental measured isotope effects in the study of reaction mechanism. In this tutorial, we will discuss how to calculated the EIE and KIE and correlate them with experimental measured KIEs to elucidate the reaction mechanism of complicated catalytic systems.

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Figure 1. Schematic presentation of the original of 18O kinetic isotope effects.

2. Calculation of KIEs and EIEs from frequencies and free energies For a chemical process A à B, where A is the initial state (reactant) and B is the final state (transition state or product), the 18O KIE and EIE are defined as

OKIE18 = 𝑘* O16 ,𝑘* O18 ,

, (1)

and

OEIE18 = 𝐾* O16 ,𝐾* O18 ,

, (2)

where, k(16O) and k(18O) the rate constants of A(16O) à B(16O) and A(18O) à B(18O) for transition state B, and K(16O) and K(18O) the equilibrium constants of A(16O) à B(16O) and A(18O) à B(18O) for product B. The 18O KIEs and EIEs can be calculated from frequencies and free energies. The way to calculate them is discussed in the following section. 2.1 Calculation of 18O-KIEs and 18O-EIEs from frequencies The 18O equilibrium and kinetic isotope effects are calculated by employing the Transition State Theory as formulated by Bigeleisen and Wolfsberg.1-2 For each step of the catalytic mechanism, the vibrational frequencies of reactants and products are analyzed following the Bigeleisen and Goeppert-Mayer approach.1 The Redlich-Teller product rule3 is employed to isotope exchange

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equations (Eq. 3) for the initial (A) and final (B) states using the full set of vibrational frequencies obtained for both light (16O) and heavy (18O) isotopologues via DFT calculations.

A(16O) + B(18O) A(18O) + B(16O) (3)

18O-EIEs for the exchange reactions were obtained using the 3N – 6 vibrational frequencies as 18O-EIE = ZPE × EXC × MMI (4)

where “VP [a.k.a. MMI] is the vibrational product, a classical factor resulting from translational and rotational motion; EXC is a quantum factor, a correction for thermal excitation of vibrations to levels higher than the zero-point levels; and ZPE is an expression of the zero-point energy differences of the isotopic shifts of vibrations in the transition state and reactants.”4 These terms are defined as follows:

𝑍𝑃𝐸 =

∏exp(ℎ𝜐8

9( :;< )/2𝑘𝑇)

exp(ℎ𝜐89( :;A )/2𝑘𝑇)

BCDE8

∏exp(ℎ𝜐F

G( :;< )/2𝑘𝑇)exp(ℎ𝜐F

G( :;A )/2𝑘𝑇)BCDEF

(5)

𝐸𝑋𝐶 =

∏1 − exp(−ℎ𝜐8

9( :;< )/𝑘𝑇)

1 − exp(−ℎ𝜐89( :;A )/𝑘𝑇)

BCDE8

∏1 − exp(−ℎ𝜐F

G( :;< )/𝑘𝑇)1 − exp(−ℎ𝜐F

G( :;A )/𝑘𝑇)BCDEF

(6)

𝑀𝑀𝐼 = 𝑉𝑃 = ∏ (𝜐8

9( :;A )/BCDE8 𝜐8

9( :;< ))

∏ (𝜐FG( :;A )/𝜐F

G( :;< ))BCDEF

(7)

where 𝜐 is the associated vibrational frequency for mode i, h is Planck’s constant, k is Boltzmann constant, and T is the temperature in K. Practically speaking, if you convert the frequency, typically given in cm-1, to kcal/mol, that value is equivalent to hv. The 18O-KIEs associated with located transition state structures were calculated in a similar way,

18O-KIE = 𝜐NOPQ × ZPE × EXC × VP (8)

18O-KIE = 𝜐NOPQ × 18KTS (9)

where 𝜐NOPQ 3, 5 is the ratio of the imaginary frequencies (doesn’t matter if you represent them with minus signs as Gaussian does) of the TSs associated with light (e.g. 16O) and heavy (e.g. 18O) isotopologues and 18KTS is the product of ZPE × EXC × VP with 3N – 6 vibrational frequencies for the reactant and 3N – 7 vibrational frequencies for the TS (Eqs. 10-12).

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𝑍𝑃𝐸 =

∏exp(ℎ𝜐8

9( :;< )/2𝑘𝑇)

exp(ℎ𝜐89( :;A )/2𝑘𝑇)

BCDR8

∏exp(ℎ𝜐F

G( :;< )/2𝑘𝑇)exp(ℎ𝜐F

G( :;A )/2𝑘𝑇)BCDEF

(10)

𝐸𝑋𝐶 =

∏1 − exp(−ℎ𝜐8

9( :;< )/𝑘𝑇)

1 − exp(−ℎ𝜐89( :;A )/𝑘𝑇)

BCDR8

∏1 − exp(−ℎ𝜐F

G( :;< )/𝑘𝑇)1 − exp(−ℎ𝜐F

G( :;A )/𝑘𝑇)BCDEF

(11)

𝑀𝑀𝐼 = 𝑉𝑃 = ∏ (𝜐8

9( :;A )/BCDR8 𝜐8

9( :;< ))

∏ (𝜐FG( :;A )/𝜐F

G( :;< ))BCDEF

(12)

2.2 Calculation of 18O-KIEs and 18O-EIEs from free energies Equation 4 and 8 is very useful to get individual contributions to EIEs and KIEs. However, if we are only interested in the EIE and KIE number, we can use frequencies directly to calculate the free energy and use the free energy difference to calculate KIEs and KIEs. The Gaussian 09 Rev. D.01 program6 can calculate the free energy directly. From Eq. 3, the 18O-KIEs and 18O-EIEs can be calculated directly as

O − KIEor O − EIE1818= exp U−

V𝐺X* Y;< , + 𝐺[* Y;A , − 𝐺X* Y;A , − 𝐺[* Y;< ,\𝑅𝑇

^, (13)

where 𝐺X* Y;A ,, 𝐺X* Y;< ,, 𝐺[* Y;A ,, and 𝐺[* Y;< , are calculated total free energies of A(16O), A(18O), B(16O), and B(18O). They can be found in the standard output files from Gaussian frequency calculations with isotope substitutions. We will use both ways (direct and individual contribution routes) to calculate 18O-KIEs and 18O-EIEs in our next examples and show they give essentially identical results.

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3. Calculation of 18O-KIEs and 18O-EIEs with a real world example We have applied 18O KIE’s to study the catalytic cycle of Mn-Terpy dimer,7 which is biomimetic catalyst to catalyze water oxidation with the sacrificial oxidant oxone (KHOOSO3). Our results suggest the ligand substitution of AcO- by HOOSO3- is the rate-determining step, as summarized in Scheme 1. In this tutorial, we will use the ligand substitution step (I to II) to show to calculate 18O EIEs and KIEs and use them to interpret experimental observed 18O KIE.

Scheme 1. Complete catalytic cycle of O2 evolution from the active catalyst I upon activation by peroxymonosulfate (HOOSO3-) in an acetate buffer under turnover conditions. Intrinsic 18O kinetic isotope effect (KIE) and equilibrium isotope effect (EIE) factors obtained with B3LYP-D2 are indicated in blue. The calculated rate-determining step is labelled with r.d.s. and is highlighted in green.

3.1 Overview of the calculation procedure From Section 2, we can see frequencies and/or free energies of different isotopologues are required inputs to calculation. Thus, one needs to do the following calculations to calculate KIEs and EIEs: a. Optimize structures of stationary points (minimum or saddle point) structures for the chemical process we are interested. b. Perform frequency calculations. c. Perform frequency calculations with isotope substitutions. d. Extract frequencies and/or free energies to calculate EIEs and KIEs.

For the first ligand substitution step, I + HOOSO3- à TSI-II à II + AcOH, we need to optimize the structures of HOOSO3-, AcOH, complex I and II and the transition state connected

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I and II. They are already optimized in our previous study and their structures are shown in Figure 2.

For 18O KIE, • A(16O) = H-16O-O-SO3- • A(18O) = H-18O-O-SO3-,

in Eq. 3 • B(16O) = TSI-II(16O) • B(18O) = TSI-II (18O);

for 18O EIE, • A(16O) = H-16O-O-SO3- • A(18O) = H-18O-O-SO3-,

while • B(16O) = II(16O) • B(18O) = II(18O).

We will show the step-by-step procedure to calculate the 18O KIE and EIE of the ligand exchange step.

We already optimized the structure of reactants (I and HOOSO3-), transitions state structures (TSI-II), and the product structure (II and AcOH). The output files of each optimization are listed in the directory KIE_tutorial/. You can redo the optimization yourself. Regarding the geometry optimization, please refer to the Gaussian User Reference at http://gaussian.com/opt/ and the sample input AcO_MnIII_MnIV_OOSO3_R_Opt_BS1.com for minima and AcO_MnIII_MnIV_HSO5_OAc_TS_Opt_BS1.com for saddle points. Note, you need to use the fragment guess to generate the correct anti-ferromagnetic coupling state in order to start your optimization. You can find how to use fragment guess to prepare anti-ferromagnetic coupling state at http://gaussian.com/afc/ and the example input file AcO_MnIII_MnIV_HSO5_OAc_TS_Frag_BS1.com included with this tutorial.

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Figure 2. Optimized geometries of (a) HOOSO3-, (b) AcOH, complexes (c) II and the (d) transition state connecting I and II (bonds are added in manually). Color code: gray = C, white = H, purple = Mn, blue = N, yellow = S, red = O.

3.2. Calculation of the 18O KIE of ligand exchange step (I à II) We will use the case of HOOSO3- for examples in this tutorial: Step 1. Optimize the geometry of HOOSO3- and perform frequency analysis: Input file: HOOSO3_Opt.com %mem=12000MB %nprocshared=8 %chk=HOOSO3_Opt.chk #p ub3lyp/gen empiricaldispersion=gd2 int=ultrafine scf=(xqc,maxconventionalcycles=120,tight)

a! b!

c! d!

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scrf=(pcm,read,solvent=water) opt freq HOOSO3_Opt, optimization -1 1 S 0.49040900 -0.06969700 -0.00544600 O 0.32491500 -0.93022100 -1.18774800 O 0.53319100 -0.79280700 1.27333800 O 1.44829200 1.03741200 -0.14346800 O -0.93399500 0.87516300 0.04009900 O -2.07320000 -0.02458300 0.13052500 H -2.24015700 -0.20455100 -0.81482400 -H 0 3-21g **** -C 0 3-21g **** -O 0 6-31+g(d) **** -S 0 6-31g(d) **** alpha=1.0 where alpha is the “scaling factor (for all the elements but acidic hydrogens) for the definition of solvent accessible surfaces. In other words, the radius of each atomic sphere is determined by multiplying the van der Waals radius by scale. The default value is 1.2.” from http://www.lct.jussieu.fr/manuels/Gaussian98/00000474.htm The optimization should finish in several (5-6) cycles and generate an output file: HOOSO3_Opt.out. You can get the SCF energy with the “grep” command:

Step 2. Perform frequency calculation with isotope substitution for HOOSO3-.

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Use a text editor to open the output file “HOOSO3_Opt.out”. I used vim editor to open the HOOSO3_Opt.out file.

Below is how it looked like in my terminal:

Search “Thermochemistry” section by typing “/Thermochemistry”, then pressing “return” key. You should see the following window:

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Above the highlighted “Thermochemistry” is information of vibrational modes. You can see the temperature and pressure used in our calculation are 298.150 Kelvin and 1.00000 Atmosphere. The atomic masses of all atoms in HOOSO3- are listed in a. u. The atomic masses in the last column will be used for following frequency calculations. We need to prepare input files to do frequency analysis for H-16O-OSO3- and H-16O-OSO3-: HOOSO3_O16_Freq_BS1.com and HOOSO3_O18_Freq_BS1.com. The input files are shown in the following pages (or see Appendix A). In the frequency calculations of H-16O-OSO3- and H-16O-OSO3-, we can avoid the time time-consuming Hessian evaluation step by using “freq=(readisotopes, readfc)” option to read force constants from previous Opt + Freq calculation. Atomic masses need to be provided in the input file, as well as the temperature and pressure to be used for the evaluation of thermochemical properties. The atomic mass of the sixth atom in “HOOSO3_O16_Freq_BS1.com” is 15.99491, corresponding to 16O; while atomic mass of the sixth atom (the O of the OH group) in “HOOSO3_O18_Freq_BS1.com” is 17.99916, corresponding to 18O. You need to make copies of the HOOSO3_Opt.chk file called HOOSO3_O16_Freq_BS1.chk and HOOSO3_O18_Freq_BS1.chk. You can run frequency analysis for H-16O-OSO3- and H-16O-OSO3- interactively (since the calculation is small) after loading the Gaussian module: [ky254@login-0-0 KIE_tutorial]$ module load Apps/Gaussian/2009-D01 [ky254@login-0-0 KIE_tutorial]$ g09 HOOSO3_O16_Freq_BS1.com HOOSO3_O16_Freq_BS1.out [ky254@login-0-0 KIE_tutorial]$ g09 HOOSO3_O18_Freq_BS1.com HOOSO3_O18_Freq_BS1.out

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Input file for H-16O-OSO3-: HOOSO3_O16_Freq_BS1.com %chk=HOOSO3_O16_Freq_BS1.chk # ub3lyp/gen empiricaldispersion=gd2 guess=read geom=check int=ultrafine scf=(xqc,maxconventionalcycles=120,tight) scrf=(pcm,read,solvent=water) freq=(readisotopes,readfc) input file to perform frequency calculation with different isotopes -1 1 298.15 1.00000 31.97207 15.99491 15.99491 15.99491 15.99491 15.99491 1.00783 C H 0 3-21g **** -N 0 6-31g **** O 0 6-31+g(d) **** -S 0 6-31g(d) **** alpha=1.0

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Input file for H-18O-OSO3-: HOOSO3_O18_Freq_BS1.com %chk=HOOSO3_O18_Freq_BS1.chk # ub3lyp/gen empiricaldispersion=gd2 guess=read geom=check int=ultrafine scf=(xqc,maxconventionalcycles=120,tight) scrf=(pcm,read,solvent=water) freq=(readisotopes,readfc) input file to perform frequency calculation with different isotopes -1 1 298.15 1.00000 31.97207 15.99491 15.99491 15.99491 15.99491 17.99916 1.00783 C H 0 3-21g **** -N 0 6-31g **** O 0 6-31+g(d) **** -S 0 6-31g(d) **** alpha=1.0

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Step 3. Extract the frequencies and/or free energies of HOOSO3- with different isotopes: Use the “Freq.pl” script to extract frequencies of H-16O-OSO3- and H-18O-OSO3- from the output file: [ky254@login-0-0 KIE_tutorial]$ ./Freq.pl HOOSO3_O16_Freq_BS1.out 111.0759 214.4583 272.7586 369.0338 381.2678 488.2048 511.1852 557.4638 658.2161 917.6353 987.8196 1169.9727 1183.5699 1389.6571 3652.0983 [ky254@login-0-0 KIE_tutorial]$ ./Freq.pl HOOSO3_O18_Freq_BS1.out 108.8926 211.4369 269.0138 362.5440 380.9582 487.8238 511.0973 555.8531 657.6055 893.7312 987.7534 1169.9661 1183.4777 1385.6036 3639.7583 Use “grep” command to extract the free energies of H-16O-OSO3- and H-18O-OSO3- from output files: [ky254@login-0-0 KIE_tutorial]$ grep "Free Energies" HOOSO3_O16_Freq_BS1.out Sum of electronic and thermal Free Energies= -774.951720 [ky254@login-0-0 KIE_tutorial]$ grep "Free Energies" HOOSO3_O18_Freq_BS1.out Sum of electronic and thermal Free Energies= -774.951959

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Step 4. Enter the calculated frequencies and free energies of HOOSO3- into the “KIE” tab in the “KIE_tutorial.xlsx” file as shown below:

Step 5. Optimized the geometry of transition state connected complexes I and II and perform frequency analysis (AcO_MnIII_MnIV_HSO5_OAc_TS_Opt_BS1.com). This calculation utilizes the “modifysph” keyword at the end as discussed at the “”PCM Parameters” tab of http://gaussian.com/scrf/. It says the following: “ Alters parameters for one or more spheres. The modified spheres can be indicated in the PCM input in lines following this keyword having the following format: atom radius [alpha] where atom is the atom number or element type. ” In this case, the Mn’s radius is being modified to a value of 2.0. Step 6. Perform frequency calculation with isotope substitution for TSI-II (AcO_MnIII_MnIV_HSO5_OAc_TS). Input files are named AcO_MnIII_MnIV_HSO5_OAc_TS_O16_Freq_BS1.com and AcO_MnIII_MnIV_HSO5_OAc_TS_O18_Freq_BS1.com and are located in KIE_tutorial/. Same as what we have done for H-O-OSO3-, we can perform the frequency calculations of TSI-II with H-16O-OSO3- and H-18O-OSO3-. The isotope is placed in the same place as in Step 2. Similar to what we have done for H-O-OSO3-, we can extract the frequencies and free energies of TSI-II with 16O and 18O isotopes. The results are shown below for reference: $ ./Freq.pl AcO_MnIII_MnIV_HSO5_OAc_TS_O16_Freq_BS1.out -72.7942 11.6768

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14.5561 18.1010 28.9578 33.4290 36.5791 39.9182 41.9481 45.5268 … $ ./Freq.pl AcO_MnIII_MnIV_HSO5_OAc_TS_O18_Freq_BS1.out -71.9581 11.6727 14.5564 18.0944 28.9528 33.4136 36.5444 39.8946 41.9332 45.5085 … The first frequency of TSI-II with 16O and 18O isotopes is labeled with a negative sign, indicating this mode corresponding to an imaginary frequency. The imaginary frequency of with TSI-II with 16O is $ grep "Free Energies" AcO_MnIII_MnIV_HSO5_OAc_TS_O16_Freq_BS1.out Sum of electronic and thermal Free Energies= -3067.651440 $ grep "Free Energies" AcO_MnIII_MnIV_HSO5_OAc_TS_O18_Freq_BS1.out Sum of electronic and thermal Free Energies= -3067.651673 Step 7. Enter the calculated frequencies and free energies of TSI-II with 16O and 18O isotopes into the “KIE” tab in the “KIE_tutorial.xlsx” file.

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After entering the frequencies and free energies of TSI-II with 16O and 18O isotopes, you need to adjust the Column I, J, and K to calculate the EXC, ZPE, and MMI term for all frequencies:

Then, you will find the calculated EXC, ZPE, MMI, and v18R and KIE highlighted in green and their values are 0.972, 0.969, 1.057, 1.012, and 1.007, respectively. The KIE is the one reported in the paper, which is close the experimental observed 1.013 ± 0.003. We can also find the KIE calculated from DDG, which is 1.006, essentially identical to the one calculated directly from frequencies.

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Now, we have shown how to calculate the 18O KIE with reactant and transition state frequencies and/or free energies for 16O and 18O isotopes. Equilibrium isotope effects (EIE) can be calculated in the similar way. We will briefly how to calculate the EIE with reactant and product frequencies and/or free energies for 16O and 18O isotopes. 3.3. Calculation of the 18O KIE of ligand exchange step (I à II) For I + HOOSO3- à II + AcOH, the steps are shown below (steps 1-4 were done already as part of the above procedure): Step 1. Optimize the geometry of HOOSO3- and perform frequency analysis: Step 2. Perform frequency calculation with isotope substitution for HOOSO3-. Step 3. Extract the frequencies and/or free energies of HOOSO3- with different isotopes: Step 4. Enter the calculated frequencies and free energies of HOOSO3- into the “KIE” tab in the “KIE_tutorial.xls” file. Step 5. Optimized the geometry of complex II and perform frequency analysis (AcO_MnIII_MnIV_OOSO3_R_Opt_BS1.com). Step 6. Perform frequency calculation with isotope substitution for complex II (AcO_MnIII_MnIV_OOSO3_R). Input files are named AcO_MnIII_MnIV_OOSO3_R_O16_Freq_BS1.com and AcO_MnIII_MnIV_OOSO3_R_O18_Freq_BS1.com and are located in KIE_tutorial/. Step 7. Enter the calculated frequencies and free energies of complex II with 16O and 18O isotopes into the “EIE” tab in the “KIE_tutorial.xlsx” file.

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We can get the EIE from frequencies to be 1.012 and EIE from free energies to be 1.012. Again, they are essentially identical. Now, we have reproduced the KIE and EIE number of the ligand exchange step in Scheme 1. KIEs and EIEs of other steps and your own system can be calculated by following the same procedure.

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4. References 1. Bigeleisen, J.; Mayer, M. G., Calculation of Equilibrium Constants for Isotopic Exchange Reactions. J. Chem. Phys. 1947, 15, 261-267. 2. Bigeleisen, J.; Wolfsberg, M., Theoretical and Experimental Aspects of Isotope Effects in Chemical Kinetics. Adv. Chem. Phys. 1958, 1, 15-76. 3. Wolfsberg, M.; Hook, W. A. V.; Paneth, P.; Rebelo, L. P. N., Isotope Effects in the Chemical, Geochemical and Biosciences; Springer: New York, 2010. 4. Turnquist, C. R.; Taylor, J. W.; Grimsrud, E. P.; Williams, R. C., Temperature Dependence of Chlorine Kinetic Isotope Effects for Aliphatic Chlorides. J. Am. Chem. Soc. 1973, 95, 4133-4138. 5. Roth, J. P.; Klinman, J. P., Oxygen Kinetic Isotope Effects as Probes of Enzymatic Activation of Molecular Oxygen. In Isotope Effects in Chemistry and Biology, A., K.; Limbach, H.-H., Eds. CRC Press: Boca Raton, FL, 2006; pp 645-670. 6. Frisch, M. J. T., G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision D.01, , Gaussian, Inc.: Wallingford, CT, 2009. 7. Khan, S.; Yang, K. R.; Ertem, M. Z.; Batista, V. S.; Brudvig, G. W., Mechanism of Manganese-Catalyzed Oxygen Evolution from Experimental and Theoretical Analyses of 18o Kinetic Isotope Effects. ACS Catalysis 2015, 5, 7104-7113.

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Appendix A. Using “iso=” If you do not want to use the checkpoint file to obtain the frequencies for the isotopologue, you can use the following notation (the example is water with a different level of theory). Then in GaussView, the isotope on the atom is visible, which is useful for checking. %chk=H2O_18.chk %mem=49GB %nprocshared=12 #p freq uwb97xd/def2svp nosymm int=ultrafine scrf=(smd,solvent=water) for KIE O18 0 1 O(iso=18) 4.32690 4.88403 2.37842 H 4.48957 5.73318 2.81407 H 3.59984 5.07428 1.76823

Appendix B. Computing H/D Kinetic Isotope Effects If you need to compute H/D isotope effects, the procedure is slightly different due to the

relative abundance of D and the ease of D-containing reactants. For example, if you have a KIE from a reaction run in H2O vs. D2O, then use Equation 14 (simplified from the equations earlier in the tutorial). R is the lowest energy minimum before the turnover-limiting/rate-limiting transition state of high energy TS with 1 or 2 indicating the isotopologue.

(14)

The isotope label should then be put on all exchangeable/acidic (i.e. polar) hydrogens. These include, but are not limited to protons bound to oxygens and nitrogens. In Figure 2, these hydrogens would include those of the hydroxyl groups in panels A and B, but none of the H’s in panels C and D (hence the example in the tutorial would have an H/D KIE close to 1).

Appendix C. Aqueous Environments for 16O/18O Kinetic Isotope Effects If the 18O in your reaction can originate from a species that can exchange with water such as OH- (i.e. basic aqueous solution), then simply using the procedure in the tutorial with A = OH- is not enough. We must account for the fractionation of the isotope between H2O and OH- as discussed below.

KIE =kH2OkD2O

= e−(−GR (1)+GTS (1) )/RT

e−(−GR (2 )+GTS (2 ) )/RT

⎛⎝⎜

⎞⎠⎟= e−(GR (2 )+GTS (1)−GR (1)−GTS (2 ) )/RT

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First, we must consider that the experiment measures the quantity in Equation 15, which does not include OH-.

(15)

Therefore, we must replace the denominator as shown in Equation 16 where we must determine the equilibrium constant K.

(16)

We compute the equilibrium constant K, which is for the reaction in Equation 17. (17)

The equilibrium constant expression is given in Equation 18 and is rearranged in Equation 19 to look like Equation 16.

(18)

(19)

If you use def2svp for H, 6-31+G(d,p) for O, and the wb97xd functional, you get these thermal corrections: OH_16.log: Thermal correction to Gibbs Free Energy= -0.007463 H2O_16.log: Thermal correction to Gibbs Free Energy= 0.002986 OH_18.log: Thermal correction to Gibbs Free Energy= -0.007656 H2O_18.log: Thermal correction to Gibbs Free Energy= 0.002758

For equation 17, we get the free energy shown in equation 20.

(20)

That leads to the K shown in equation 21.

(21)

Therefore, equation 16 becomes equation 22 where KIE(18Oapparent) is the KIE that you would normally calculate with A = OH- and B = the transition state of interest

(22)

18KIE = [16O16O] / [16O18O][H2

16O] / [H218O]

18KIE = [16O16O] / [16O18O]K *[16OH − ] / [18OH − ]

16OH − + H2

18O! 18OH − + H216O

K = [H216O] / [H2

18O][16OH − ] / [18OH − ]

= [H216O][18OH − ]

[16OH − ][H218O]

[H216O] / [H2

18O]= K[16OH − ] / [18OH − ]

ΔG = (− − 0.007463− 0.002758 − 0.007656 +  0.002986)au *627.509 kcal /molau

=0.02196kcal /mol

K = e−ΔG/RT = e−(0.02196kcal /mol )/(0.001987204 kcal /(mol*K )*298.15K ) = 0.96361

18KIE = [16O16O] / [16O18O]0.96361*[16OH − ] / [18OH − ]

=KIE(18Oapparent )0.96361