Density Functional Theory and Thermochemistry using Gaussian.

In this lab we will consider two topics: Density functional calculations and the first principles calculation of thermochemical data. Density Functional Calculations are only slightly more expensive than Hartree-Fock calculations and they are generally quite accurate. DFT calculations are used widely in calculations of geometries and vibrational frequencies. They also provide quite accurate values for reaction energetics and atomization energies, provided suitably large basis sets are used. The only caveat with DFT calculations is that they can be somewhat erratic, and they cannot be systematically improved. Typically things are OK, even for difficult systems, if restricted and unrestricted DFT calculations yield the same results. Such a test is only required if you are dealing with an unusual molecule or exotic transition state. For organic molecules the most often used functional is B3LYP, while also PBE1PBE provides good results. For transition metal compounds the BP86 and also PW91 functionals are cited to give most reliable results. DFT calculations are as easy to use as Hartree-Fock calculations. The only additional choice is the energy functional. We will run geometry and frequency calculations using DFT to examine thermochemical data. In order to calculate reaction enthalpies one needs to calculate accurate geometries for reactions and products, corresponding vibrational frequencies, and very accurate electronic energies. From the geometries one obtains rotational constants and these are used to calculate the rotational contribution to the so-called partition function. From the vibrational frequencies one can calculate the zero-point frequencies, and the vibrational contributions to the partition function. In Gaussian this information is calculated whenever vibrational frequencies are calculated, and a table is printed giving the translational / rotational / vibrational (TRV) corrections to the enthalpy at standard temperature (298.15 K) and pressure (1 atm.). You will see Thermal correction to Enthalpy These corrections (in Hartree) can be added to the electronic energy, calculated at the same or a different level of accuracy. Doing this for reactants and products and subtracting one obtains the enthalpy of the reaction. Alternatively you might be interested in calculating the free energy, using

G H T SΔ = Δ − Δ . This can also be calculated from the partition function. In Gaussian you will see: Thermal correction to Gibbs Free Energy The Gibbs free energy is particularly relevant to calculate activation energies. In both cases (free energies and enthalpies) it is very important to calculate accurate electronic energies, which are ‘converged’ with respect to the basis set. Let us go through the procedure by example. 1. Calculate the reaction enthalpy for the reaction N2 + 3 H2 → 2 NH3. a. Optimize the geometry and calculate frequencies for all molecules at the B3LYP level and a cc-PVTZ basis set (to be written on the keyword line). For H2 you might alternatively use CCSD.

b. Add the electronic energy to the “Thermal correction to Enthalpy” from the vibrational frequency calculation for 1 mol of each species, and calculate the reaction enthalpy at the B3LYP // cc-PVTZ level of accuracy in kcal / mol, using 1H = 627.5095 kcal/mol. In quantum chemistry kcal / mol is still a widely used unit, although kJ / mol is the correct SI unit, of course. The experimental value is -10.98 kcal/mol. 2. An often used technique is to calculate single point electronic energies at the ab initio level, in particular MP2 or CCSD(T), while DFT is used to obtain geometries and “TRV” statmech contributions. Use the same geometries as in exercise 1 and calculate the CCSD(T) correlation energy EΔ (the difference between the CCSD(T) total energy and the Hartree-Fock energy) using both the cc-PVTZ and the cc-PVQZ basis sets. In Gaussview you would select energy under jobtype, and CCSD & include triples under method. In the accompanying excel sheet you can see where you can find this piece of data in the output file. From this calculate the (third-order) extrapolated (TQ) correlation electronic energy, given by

3 3

3 3

4 ( ) 3 ( ) 64 ( ) 27 ( )( )4 3 37

E cc PVQZ E cc PVTZ E QZ E TZE TQ Δ − − Δ − Δ − ΔΔ = =

−

The number 43 is used because of the quadruple zeta basis set, while 33 corresponds to triple zeta. As an aside for the assignment, one can also calculate (DT) extrapolated energies, based on a cc-PVTZ and cc-PVDZ calculation, using 33 and 23 respectively. The precise formula reads

3 3

3 3

3 ( ) 2 ( ) 27 ( ) 8 ( )( )3 2 19

E cc PVTZ E cc PVDZ E TZ E DZE DT Δ − − Δ − Δ − ΔΔ = =

−

This might be useful for larger molecules, as CCSD(T) calculations are very expensive. Such small basis set calculations are really not sufficiently accurate, and I would recommend using DFT results instead. In later labs we will consider so-call explicit r12 calculations to do thermochemistry using Molpro, and the PNO-pCCSD or CEPA approaches in the ORCA program, which is a highly efficient method applicable to larger molecules. Also calculate the extrapolated Hartree-Fock value using the (fifth order) extrapolation formula

5 5

5 5

4 ( ) 3 ( ) 1024 ( ) 243 ( )( )4 3 781

HF HF HF HFHF

E cc PVQZ E cc PVTZ E QZ E TZE TQ − − − −= =

−

The same (fifth order) extrapolation formula can be used for the DFT(TQ) total energy calculated in the cc-PVTZ and cc-PVQZ basis sets. From the CCSD(T) calculation we can obtain the HF, MP2, CCSD and CCSD(T) value for the energy and the correlation energy. These additional energies are calculated on the way towards convergence of the iterative CCSD equations. In the excel file it is indicated where you can pick up the various quantities from the Gaussian output file. Use this to

calculate the reaction enthalpy at these various levels of approximation. You can calculate the result for three different basis sets: T, Q and (TQ) and monitor the convergence of results. In the excel file listed on my website I have collected the results for the calculations thus far. I also performed the basis set extrapolations, and calculated the reaction energies. The excel file contains all the numbers so you can run the example yourself and check if it all matches. The energies may not be precisely the same since your geometry may possibly be slightly different. The numbers can be expected to be very close though. In the above we calculated reaction enthalpies. Of further interest are heats of formation and atomization energies. We cannot calculate heats of formation directly as the elemental forms are often in the solid state. Atomization energies can be calculated, but it is hard to obtain high accuracy, as atoms are often open-shell, and very large basis sets are required to maintain a proper balance between singlets (for most molecules) and high spin wave functions (for most atoms, following Hund’s rule). In the literature 5Z and 6Z basis sets have been used! Moreover, the atomic calculations involve empirical corrections to account for spin-orbit coupling. This is discussed in the book by Cramer, and gives rise to annoying complications. To obtain heats of formation and/or atomization energies I advocate using “hydrogenation” energies. This involves reacting a molecule with H2 gas to saturated atom-hydrides like CH4, NH3, H2O, HF, PH3, NaH and so forth. In contrast to atomization energies the Hydrogenation energies are easier to calculate, and one can obtain balanced results by using the same basis set and method for both molecule and atom-hydride. Let us write the hydrogenation reactions for a hypothetical AB molecule as

2( ) ( ) ( ) ( )2 n m

n mAB g H g AH g BH g++ → +

The hydrogenation reaction enthalpy and Gibbs free energy can be calculated for the molecule of interest, (for any method and basis set). We used precisely this kind of reaction to study reaction enthalpies above. Next we can use a thermodynamic cycle to obtain the heat of formation, employing some experimental data:

2

2

( ) ( ) ( ) ( ) ( )2

( ) ( )

( ) ( ) ( )2

n m

formation formation

n mAB g H g H hydrogenation reaction AH g BH g

H molecule AB H atom hydrides

n mA ele B ele H g

++ Δ − +

Δ Δ −

++ +

or

0 0 0298 298 298( ) ( ) ( )formation Hydrogenation reaction formationH M H M H atom hydrides−Δ = Δ −Δ − .

The heats of formation for the atom-hydrides are obtained from experimental data, provided on the NIST web site. To calculate atomization energies we would use the following cycle

2

22 1 2 1 2

( ) ( ) ( ) ( ) ( )2

( ) ( )

( )

n m

atomization atomizationS S

n mAB g H g H hydrogenation reaction AH g BH g

H molecule AB H H atom hydrides

A B n m H+ +

++ Δ − +

Δ + Δ −

+ + + Hence:

0 0298 298

0 0298 298 2

( ) ( )

( )( ) ( )2

atom hydrogenation reaction

atom atom

H molecule H molecule

n mH atom hydrides H H

−Δ = Δ

+−Δ − − Δ

Using similar cycles we can calculate G H T SΔ = Δ − Δ , and SΔ itself. All of the data we calculate in this lab can be compared directly to experiment. A very useful website is http://srdata.nist.gov/cccbdb/, which contain accurate experimental data. Moreover it contains a large computational data base as well. It is not unlikely that you find the numbers you are calculating on this web site! There is further documentation on how calculations of thermochemical data are carried out, and this may provide useful reference material for you. The following table contains data from the NIST web site that can be used to calculate heats of formation and atomization energies using the intermediate step of calculating atom-hydrides. Table 1. Experimental heats of formation, heats of atomization and molar entropies of atom-hydrides (in kJ/mol) at 298 K (From http://srdata.nist.gov/cccbdb/) Atom-Hydride

∆H˚formation,298 (kJ/mol)

S˚298 (J/K mol)

∆H˚atomization,298 (kJ/mol)

H2 0 130.7 293.1±0LiH 140.62±0.04 170.91 236.7±0BeH2 125.52 173.1 634.5±?BH3 88±10 171.9 1131.0±10CH4 -74.60±0.30 186.4 1663.3±0.3NH3 -45.94±0.35 192.8 1645.3±0.3H2O -241.81±0.03 118.8 927.0±0HF -273.30±0.70 173.8 570.7±0.7NaH 124.26±19.20 188.4 201.2±19.2MgH 229.79±6.10 193.2 135.3±6.1AlH 249.25±3.50 187.9 298.7±3.5SiH4 34.70±1.50 204.2 1287.3±1.5PH3 5.4±1.7 210.2 1281.6±1.7H2S -20.60±0.50 205.8 733.8±0.5HCl -92.31±0.10 186.9 431.6±0.1

DFT calculations often provide reasonable thermochemical data, but reliable high accuracy may require CCSD(T) calculations in a very large basis set. Such calculations are prohibitively expensive for many molecules. Recent progress allows one to use explicitly correlated, or so-called r12 methods, which greatly reduce the size of the basis set needed. Another development is the use of local correlation methods. Local r12-CCSD(T) methods are available in the MOLPRO package, while ORCA provides a pno-CCSD approach (no triples yet), and we recently implemented (Lee Huntington et al.) the pno-PCCSD approach. The development of accurate and reliable methods to obtain thermochemical data is an active area of research. 6. In this lab you are asked to do a number of calculations, which only require slight modifications in the input. It is hard to keep track of this using Gaussview. Instead I find it easier to copy the .com file of a calculation previously prepared in Gaussview, and edit the file (using nano, emacs or vi) to do the next calculation. Finally you can write a little “script” that manually runs each calculation on either abacus or thooft. By editing the script it is easy to keep track of things. Let us go through the procedure. Take one of the molecules (let’s say C2H2 for definiteness) of your assignment and create the directory C2H2. Then a. Prepare the acetylene molecule in Gaussview and run a B3LYP/cc-PVTZ optimization and frequency, input file b3lyp_freqopt.com b. Create a single energy point B3LYP cc-PVQZ calculation, inputfile b3lyp_qz.com (do not run), starting from the optimized geometry. c. Copy b3lyp_qz.com to cc_qz.com and modify cc_qz.com to do a CCSD(T)/cc-PVQZ calculation. d. Copy cc_qz.com to cc_tz.com and modify to do a CCSD(T)/cc-PVTZ calc. e. Copy cc-qz.com to cc-dz.com and modify to do a CCSD(T)/cc-PVDZ calc. f. Transfer the files to thooft.uwaterloo.ca. or abacus.uwaterloo.ca, and put them in a suitable directory. These machines have the G09 executable running. You can also edit the files on these machines of course, as usual. e. Create a script joball containing the lines: jobgauss b3lyp_qz jobgauss cc_dz jobgauss cc_tz jobgauss cc_qz f. copy in the submit script as usual, changing the submit file to reflect your working directory and submit to the queue. g. Edit the output files copy/past the results in an excel file and process the data using the above formulas, or following the example in the excel file. This way you can create a table of all required energies, without errors due to typos or wrong calculations. You only have to make sure that the formulas in the excel sheet are correct.

h. Follow similar steps for the CH4 molecule and for H2 (You might CCSD to get the geometry and frequencies for the latter molecule ). i. From the above set of calculations and the data in the Table you can get the heats of formation and atomization for acetylene, and the reaction enthalpy for the atom-hydrogenation reaction for acetylene, at various levels of methodology. Assignment. Below I listed an extensive set of molecules (partitioned into subsets). Out of this set pick two molecules that react to a third (A+B → C) and repeat the exercises as we did them in the lab (as explained above). In addition I would like you to calculate the cc-PVDZ results and do the (DT) extrapolation in addition. You can build on the excel file that I put on the web site. I would like you to calculate the optimal B3LYP/cc-PVTZ geometry, zeropoint frequencies and thermal enthalpies, as well as the cc-PVDZ, cc-PVTZ, and cc-PVQZ basis set results for the B3LYP, SCF, MP2, CCSD and CCSD(T) methods. In an excel sheet list the results from the Gaussian calculation and calculate the heats of formation for each species at the (TQ) and (DT ) extrapolated basis set limit. Find the data you are calculating on the NIST website, listed above, and discuss your results. In adition calculate the heats of formation and atomization energies for the species in your reaction. Comment on the convergence of the various results with respect to the basis set, and the level of correlation. It will be clear that it is very hard to solve the Schrödinger equation to sufficient accuracy in a balanced way for different types of molecules and atoms. Set a) Carbon and Hydrogen containing Molecules. H2, C2H2, C2H4, C2H6, CH4, CH2 Set b) Exotic Carbon –Hydrogen containing compounds: cyclopropane (C3H6), CH2C, allene (CH2=C=CH2), C3 Set c) Oxygen containing molecules: CO, CH2O, H2O, H2O2, CO2, CH2CO Set d) Nitrogen containing molecules: N2, NH2, HCN, HNO, HNO2, N2O (linear), HNO2 Set e) Remaining species: F2, HF, F2O, BH, BH2, HOF