The partition function is so important because everything else one ever wants (and can) know about the macroscopic system can be calculated from it:

Transcription

1 5. Thermochemstry 5.. Molecular partton functon and thermodynamc quanttes One of the man goals of computatonal chemstry s to calculate macroscopc chemcal quanttes that we observe n the laboratory (e.g. equlbrum constants, rate constants etc.). ut what we actually do the calculatons on s a sngle molecule, whch s a mcroscopc system. The connecton between the mcro and macroscopc worlds s made through statstcal mechancs. The fundamental quantty of statstcal mechancs s the partton functon. For a canoncal ensemble, the partton functon Q s wrtten as: Q exp( E T) (5.) In ths equaton, ndex runs over all possble mcroscopc energy states of the system, s oltzmann constant (.3806 x 0-3 JK - ) and T s absolute temperature. The partton functon s so mportant because everythng else one ever wants (and can) now about the macroscopc system can be calculated from t: probablty that the system s found n the state wth energy E (at temperature T): exp( E T) exp( E T) P( E ) (5.) exp( E T) Q nternal energy (at temperature T), whch s the mean (average) value of the total energy (mean s denoted by ): enthalpy entropy U E P( E ) E E exp( E T) exp( E T) ln Q H U pv T pv T Gbbs free energy (from enthalpy and entropy): T ln Q T (5.3) (5.4) ln Q S T ln Q T (5.5) T G H TS (5.6) Partton functon Q establshes the connecton between macroscopc measurable quanttes (thermodynamcs) and mcroscopc energy states of the system. The ey to calculatng all macroscopc measurable ensemble propertes from mcroscopc quanttes s to get Q.

2 However, to calculate Q (and therefore everythng else), we need the energes E of all the possble states of the system. The number of possble states of a general system s enormous and the energy s an excrucatngly complcated functon. It s necessary to mae some smplfyng assumptons. Ideal gas assumpton Snce deal gas molecules do not nteract wth one another, s that we may rewrte the partton functon as: Q N! N! N! N q N! () exp (... ) energy levels exp( g () / exp( / T ) N T ) () N T exp( () / T )... ( N ) exp( ( N ) / T ) (5.7) where the factor of /N! derves from the quantum mechancal ndstngushablty of the partcles, s the total energy of an ndvdual molecule. On gong from the frst to the second lne the exponental of all possble sums of energes was expressed as a product of all possble sums of exponentals of ndvdual energes. That s just math. Gong from the second to the thrd lne, the sum has been changed so that t goes over dscrete energy levels, rather than ndvdual states, and g s the degeneracy of level. The quantty n bracets on the thrd lne defnes the molecular partton functon q. A second consequence of the deal gas assumpton s that pv N T nrt (5.8) where n s the number of moles and R s the unversal gas constant (8.345 J mol K ). Any pv n the above equatons can therefore be replace by NT or nrt. Separaton of energy components We have thus reduced the problem from fndng the ensemble partton functon Q to fndng the molecular partton functon q. In order to mae further progress, we assume that the molecular energy ε can be expressed as a separable sum of electronc, latonal, rotatonal, and vbratonal terms,.e.: (5.9) elect l rot vb

3 (we don t consder nuclear energy as t s generally not mportant n chemstry). Substtutng to (5.7) the molecular partton functon also separates: q q q q q (5.0) elect l rot vb where etc. eelctronc energy levels q elect g exp( elect, / T) (5.) Now we have to fnd expresson for the energes, and by (5.7) or (5.0) molecular partton functons for the ndvdual degrees of freedom. From those, usng (5.7) the canoncal partton functon Q can be obtaned and fnally from (5.) - (5.6) we can get all the thermodynamcs.. Molecular electronc partton functon: The electronc partton functon s usually the smplest to compute. For a typcal, closed shell snglet molecule, the degeneracy of the ground state s unty, and the varous excted states are so hgh n energy that, at least at temperatures below thousands of degrees, they mae no sgnfcant contrbuton to the partton functon. The only one that really matters s the ground electronc state energy: q elect exp( E, / T) (5.) elect ground The electronc component of U usng (5.3) s ndependent of temperature and equal to Eelec. However, f the ground state has multplcty other than one (total spn S greater than zero) a degeneracy factor S+ must be added.. Molecular latonal partton functon: To evaluate q, we assume that the molecule acts as a partcle n a three-dmensonal cubc box of dmenson a 3 where a s the length of one sde of the cube. In realty, the molecule s not confned to any such box, but to get around t, we wll assume that the box s very, very large (a s enormous compared to the sze of the molecule, n fact we ll call t macroscopc). The energy levels for ths elementary quantum mechancal system are gven by: h n x ny nz (5.3) 8Ma where M s the molecular mass, and each energy level has assocated wth t the three unque quantum numbers nx, ny, and nz. ecause the energy levels for the partcle n a box are very, very closely spaced (at least for a box of macroscopc dmensons), the partton functon sum may be replaced by an ndefnte ntegral, and ths ntegral can be evaluated analytcally as: 3

4 q M T h 3 / V (5.4) where volume V replaced a 3. Evaluatng equaton (5.3) for the nternal energy we get: 3 U nrt (5.5) The entropy (equaton 5.5) s slghtly more complcated, ncludng some addtonal terms from the expanson of the logarthm n Q (eqn. 5.7) gvng: S M nrln h T 3/ V N A 5 (5.6) Volume appears n the last expresson. Remember the deal gas assumpton (5.8). 3. Molecular rotatonal partton functon: The rotatonal partton functon can be evaluated from quantum mechancal rotatonal levels of the molecule. We wll not go nto detals here, just summarze the results. We have to dstngush between lnear and non-lnear molecules: lnear molecules the rotatonal nternal energy s: and the entropy U lnear rot nrt (5.7) S lnear rot 3/ I T nrln h (5.8) where I s the molecular moment of nerta and s a symmetry factor, = for asymmetrc molecules and = for symmetrc ones (belongng to Cv and Dh pont groups). non-lnear molecules For non-lnear molecules the rotatonal Schrodnger equaton cannot be n general solved exactly, but t can be suffcently well approxmated. The general formulas are: U non lnear rot n 3 RT (5.9) 4

5 S nonlnear rot nrln I A I I C 8 h T 3 / 3 (5.0) where IA, I, and IC are the prncpal moments of nerta, and σ s agan a symmetry number. 4. Molecular vbratonal partton functon: You mght have notced that for latonal and rotatonal motons (not electronc though), the nternal energy s derectly proportonal to temperature T. That means at zero absolute temperature (0K) the latonal and rotatonal energes are both zero. In other words, the laton and rotatonal motons are completely frozen at 0K. Vbratons are very dfferent. The quantum mechancal expresson for the energy of the harmonc oscllator (3.4) says that even n the ground state (n =0) there s stll a non-zero vbratonal energy: E0 h (5.) Ths s called a ZERO POINT ENERGY (ZPE) and s extremely mportant. A molecule wth N atoms wll have 3N - 6 vbratonal modes (3N -5 f t s lnear) and e ach of them wll contrbute to the ZPE. That means that at 0K the total energy of the molecule wll be ts electronc energy plus the sum of ZPE from all vbratonal modes: U 0 U( T 0K) E elect h (5.) 3N 6 ecause the zero pont energy always has to be counted, t s usually separated from the vbratonal partton functon, n only the hgher energy levels, whose populaton s dependent on temperature, are counted. The formulas for the vbratonal nternal energy and entropy are: U vb R h 3N 6 exp( h / T) (5.3) S vb R 3N 6 h exp( h / T) ln exp( h / T) (5.4) 5

6 5.. Thermochemstry n Gaussan All frequency calculatons nclude thermochemcal analyss of the system. y default, ths analyss s carred out at 98.5 K and atmosphere of pressure, usng the prncpal sotope for each element type. Here s the start of the thermochemstry output for formaldehyde: Thermochemstry Temperature Kelvn. Pressure Atm. Atom has atomc number 6 and mass Atom has atomc number 8 and mass Atom 3 has atomc number and mass Atom 4 has atomc number and mass Molecular mass: amu. Ths secton lsts the parameters used for the thermochemcal analyss: the temperature, pressure, and sotopes. Gaussan predcts varous mportant thermodynamc quanttes at the specfed temperature and pressure, ncludng the thermal energy correcton, heat capacty and entropy. It also gve the zero pont energy (ZPE). These tems are broen down nto ther source components n the output: Zero-pont correcton= (Hartree/Partcle) Thermal correcton to Energy= (E therm = ZPE+E +E rot+e vb) Thermal correcton to Enthalpy= (H therm= E therm+pv) Thermal correcton to Gbbs Free Energy= (G therm= H thermts) Sum of electronc and zero-pont Energes= (E 0=E elect+zpe) Sum of electronc and thermal Energes= (E= E 0+E +E rot+e vb= E elect +E therm) Sum of electronc and thermal Enthalpes= (H= E+pV= E elect+h therm) Sum of electronc and thermal Free Energes= (G= HTS= E elect+g therm) e careful here, because the descrpton s lttle confusng. The thermal correctons and the sums all contan the zero-pont energy, although t s not explctly stated! The Sum of electronc and thermal Energes s the same as the nternal energy we denoted U above. The expressons on the rght also gve you optons for calculatng the thermodynamc quanttes, U, H and G. The entropy s also gven n the output, along wth the heat capacty (Cv) and broen down to ndvdual contrbutons. The Entropy S can be used to calculate the Gbbs free energy from enthalpy. Note though that these are n dfferent unts than the quanttes above: E (Thermal) CV S KCal/Mol Cal/Mol-Kelvn Cal/Mol-Kelvn Total Electronc Translatonal Rotatonal Vbratonal

7 5.3. Scalng factors Raw frequency values computed at the Hartree-Foc level contan nown systematc errors due to the neglect of electron correlaton, resultng n overestmates of about 0%-%. Therefore, t s usual to scale frequences predcted at the Hartree-Foc level by an emprcal factor of Use of ths factor has been demonstrated to produce very good agreement wth experment for a wde range of systems. Our values must be expected to devate even a bt more from experment because of our choce of a medum-szed bass set (by around 5% n all). Frequences computed wth methods other than Hartree-Foc are also scaled to smlarly elmnate nown systematc errors n calculated frequences. The followng table lsts the recommended scale factors for frequences and for zero-pont energes and for use n computng thermal energy correctons (the latter two tems are dscussed later n ths chapter), for several mportant calculaton types (Table from Cramer: Essentals of Computatonal Chemstry: Theores and Models, nd Ed.) 7

8 Sometmes, dfferent scalng factors are used for ZPE and thermal energes than for frequences that are used for smulaton of spectra (Table from Foresman and Frsch: Explorng Chemstry wth Electronc Structure Methods, nd Ed.): Gaussan wll scale frequences for the thermochemstry calculatons, but the actual frequences wll always be unscaled (f you want to scale them you have to do t yourself) Changng thermochemstry parameters y default Gaussan wll carry out frequency and thermochemstry calculaton for the standard temperature and pressure (STP) and most abundant sotopes. y default the frequences are also unscaled. If dfferent condtons, sotopes and/or scalng factors are desred, there are several ways to change them:. An old way, whch does not seem to wor that well anymore s to use ReadIsotopes opton to the Freq eyword n the route secton. Values for all parameters must then be specfed n a separate nput secton followng the molecule specfcaton-and separated from t by a blan lne. #3LYP/6-3G(d) Opt Freq=ReadIsotopes Test Here s the general format for the ReadIsotopes nput secton: temp pressure [scale] sotope for atom sotope for atom. sotope for atom N The scale factor s optonal. Isotopes are specfed as ntegers although the program wll use the actual value. (e.g., 8 specfes 8 O, and Gaussan uses the value ). 8

9 . More drect way, whch now seems to be the only one that wors, s to set the temperature, pressure and scalng factors drectly n the route secton usng, respectvely, eywords Temperature, Pressure and Scale wth the desred values (for temperature n Kelvn, for pressure n atmospheres). For example: #3LYP/6-3G(d) Opt Freq Temperature=73.0 Pressure=0. Scale= Isotopes can be set n the molecule specfcaton secton usng (Iso= ) after the atom symbol. For example: 0 C(Iso=3) H(Iso=) wll mae the frst atom 3 C and the second atom Deuterum. 3. If you already calculated frequences and thermochemstry for one set of parameters and saved the checpont fle, a utlty program freqch can be used to just repeat the frequency calculaton (wthout dong any quantum mechancs) for dfferent condtons, sotopes or wth dfferent scalng factors. just type freqch at the prompt and enter the name of the checpont fle and desred parameters. For example, f I had done a calculaton for water, ths would do t for a heavy water (DO) at 300 K and.0 atmospheres: born 5% freqch Checpont fle? water.ch Wrte Hyperchem fles? n Temperature (K)? [0=>98.5] Pressure (Atm)? [0=> atm].0 Scale factor for frequences durng thermochemstry? [0=>/.].0 Do you want to use the prncpal sotope masses? [Y]: N For each atom, gve the nteger mass number. In each case, the default s the prncpal sotope. Atom number, atomc number 8: [6] 6 Atom number, atomc number : [] Atom number 3, atomc number : [] and the output would follow. You should always mae sure that your parameters are correctly set by checng the output. In ths case I have: Atom has atomc number 8 and mass Atom has atomc number and mass.040 Atom 3 has atomc number and mass.040 Temperature Kelvn. Pressure Atm. 9

10 If you don t use scalng (scale factor =) t wll not be reported. However, f you choose to scale the frequences, for example by , t wll appear n the thermochemstry output: Thermochemstry wll use frequences scaled by The only problem wth usng freqch s that only thermal correctons are reported, but not the total sums that also contan the electronc energy. Whle n the log fle you get: Zero-pont correcton= (Hartree/Partcle) Thermal correcton to Energy= Thermal correcton to Enthalpy= Thermal correcton to Gbbs Free Energy= Sum of electronc and zero-pont Energes= Sum of electronc and thermal Energes= Sum of electronc and thermal Enthalpes= Sum of electronc and thermal Free Energes= from freqch you get only part of t: Zero-pont correcton= (Hartree/Partcle) Thermal correcton to Energy= Thermal correcton to Enthalpy= Thermal correcton to Gbbs Free Energy= To calculate total energes, enthalpes and Gbbs free energes, you need to add the electronc energy to the thermal correctons. The electronc energy s n the log fle from the orgnal job: SCF Done: E(R3LYP) = A.U. after cycles Remember that the thermal correctons already contan the ZPE, so you don t add t agan (t would be there twce). To see that t wors, let s add the Thermal correctons from the freqch output to the electronc energy E(R3LYP) and compare to the summed energes: U = E(R3LYP) + Thermal correcton to Energy = A.U. = A.U. = Sum of electronc and thermal Energes H = E(R3LYP) + Thermal correcton to Enthalpy = A.U. = A.U. = Sum of electronc and thermal Enthalpes G = E(R3LYP) + Thermal correcton to Gbbs Free Energy = A.U. = A.U. = Sum of electronc and thermal Free Energes 0

11 5.5. Example: Enthalpy and Gbbs free energy of a reacton Let's consder the hydraton reacton: H + + HO H3O +. Our goal s to compute the enthalpy change H 98 and Gbbs free energy change G 98 at 98 K and atm pressure for the reacton, at 3LYP/6-3+G(df,p) level of theory. The enthalpy change can be calculated usng these expressons: H 98 = H 98 (products) H 98 (reactants) = H 98 (H3O + ) H 98 (HO) H 98 (H + ) Smlarly: G 98 = G 98 (products) G 98 (reactants) = G 98 (H3O + ) G 98 (HO) G 98 (H + ) We need to do calculatons frequency calculatons (whch also means optmzaton!) on H3O + and HO and get thermochemstry parameters under the specfed condtons (98K, atm happen to be the default). The proton H + not only cannot vbrate, but there s also no electron to even do any quantum calculaton. The only contrbuton to the enthalpy or free energy from the proton s latonal, gven by (for mol): U S H G 3 RT M T Rln h 3 5 RT pv RT H TS 3 / V N A 5 M T Rln h 3 / RT pn A 5 Substtutng the numbers for H + : U = J.mol - = cal.mol - H= J.mol - =.48 cal.mol - S = J.mol -.K - = 6.04 cal.mol -.K - G= 6.54 J.mol - = 6.75 cal.mol - You can chec these numbers because t turns out that Gaussan wll stll calculate thermochemstry, even for atoms and even for a naed proton! Just call Freq and remember that proton has a charge of +. The method does not matter, there are no electrons to calculate. Here s the nput:

12 # Freq Test h plus H and the output: Thermochemstry Temperature Kelvn. Pressure Atm. Atom has atomc number and mass Molecular mass: amu. Zero-pont vbratonal energy Vbratonal temperatures: (Kelvn) 0.0 (Joules/Mol) (Kcal/Mol) Zero-pont correcton= (Hartree/Partcle) Thermal correcton to Energy= Thermal correcton to Enthalpy= Thermal correcton to Gbbs Free Energy= Sum of electronc and zero-pont Energes= Sum of electronc and thermal Energes= Sum of electronc and thermal Enthalpes= Sum of electronc and thermal Free Energes= E (Thermal) CV S KCal/Mol Cal/Mol-Kelvn Cal/Mol-Kelvn Total Electronc Translatonal Rotatonal Vbratonal As you can see the enthalpy (under Sum of electronc and thermal Enthalpes) s A.U., whch s x 67.5 cal.mol - =.48 cal.mol -. The Gbbs free energy (under Sum of electronc and thermal Free Energes) s x 67.5 cal.mol - = 6.75 cal.mol -. Another way to get these numbers s from the E (Thermal), whch s the same as U: H = U + pv = E (Thermal) + RT = ( ) cal.mol - =.48 cal.mol - G = H TS =.48 cal.mol K x 6.04 cal.mol -.K - = 6.75 cal.mol - To get the thermochemstry for HO and H3O + we can buld them n Gabedt (agan remember the charge for the latter s plus one). Here s the nput fle for both jobs lned together (I used Opt Freq snce we are dong full optmzaton anyway):

13 # 3LYP/6-3+g(df,p) Opt Freq test water 0 O H H Ln-- # 3LYP/6-3+g(df,p) Opt Freq test H3O plus O H H H Ths s the useful output for water: Zero-pont correcton= (Hartree/Partcle). Sum of electronc and thermal Enthalpes= Sum of electronc and thermal Free Energes= And ths for the hydronum on: Sum of electronc and thermal Enthalpes= Sum of electronc and thermal Free Energes= Remember that these are the actual enthalpes and free energes, contanng all the contrbutons ncludng ZPE. All that needs to be done s smply convert these numbers to other unts. H 98 (HO) = Hartree = cal.mol - G 98 (HO) = Hartree = cal.mol - H 98 (H3O + ) = Hartree = 48.7 cal.mol - G 98 (H3O + ) = Hartree = 484. cal.mol - Substtutng nto products mnus reactants formulas above, we get the desred enthalpy and free energy changes for the reacton: H 98 = 48.7 (4796.4).5 cal.mol - = 78.0 cal.mol - G 98 = 484. (4797.8) ( 6.3) cal.mol - = 0.0 cal.mol - 3