Quantum chemical methods can be used to calculate heats of reactions. The Heat of formation (HOF) is a special type of reaction.
For example, to calculate the bond separation energy of
Experience has shown us that the quality of the calculated heats of reaction depends on both the method used to calculate the energy (E[..]) and the type of reaction.
Our methods book explains that the "best" type of reaction is an "isodesmic" reaction. (An Isodesmic reaction is one in which the number and type of bonds remain the same during the reaction. For an example see the discussion on basicities.) The book also lists other types of reactions and gives quantitative examples of the accuracy one can expect for different theories and basis sets.
Sadly, the experimental HOF consists of some of the most difficult reactions to calculate: breaking all bonds and creating atomically pure compounds such as H2 for hydrogen and graphite for carbon.
Thus, good HOF calculations usually require methods better than MP6 with very large basis sets. These can be extremely time consuming, even for molecules of only a few heavy atoms! The G3 method is a good place to start if you are interested in accurate Heats of Formation.
Chances are that if you think you want the HOF formation,
you may only need to compare the relative differences in energy
of certain reactions, and can use a less time-consuming approach than
the methods required for Heat of formation calculations. The
includes many examples of HF/6-31G* without vibrational
corrections performing well (against experiment). The discussion on
finding the *best* energy also covers
this and other common theory levels.
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The G3 method (the most used of the Gx series) is a time consuming recipe for calculating accurate "Heats of Formation" (HOF) from first principles. G3 consists of 10 calculations:
All these energies are combined, along with similar calculations done for each atomic species. The result is then subtracted by experimental energies of the "standard state atomically pure" systems to arrive at a reasonably good approximation for the actual HOF. A quick overview of how these steps are used can be found in our discussion of calculating the exact energy.
Note that each of these steps are relatively time consuming and practical only for very small systems. The G3(mp2) method is often more than twice as fast, but still prohibitively expensive for all but small molecules.
The reference to the G3 papers can be found at the end of this FAQ.
The T1 method is a simpler recipie to calculate the absolute heat of formation. It is similar in concept to the G3 method but replaces the post MP2 calculations with empirical relations based on mulliken bond order and implements faster MP2 approximations. Thus it is much faster than G3 or G3(MP2) and can be applied to much larger molecules. The steps of the calculation are:
The quantum mechanical calculations in Spartan assume that molecules are isolated, with a temperature of zero Kelvin, and with stationary nuclei. Real experiments are carried out with vibrating molecules at finite temperature (often 298.15 K). In order to correct for these differences Spartan can use calculated frequency data to determine a set of normal-mode vibrational frequencies (vi). (These are the same data used to generate IR spectra.)
Using this approximation, the energy corrections are fairly straightforward. Splitting the Enthalpy correction into four (4) parts to include the 'Zero point energy' (Zp), the temperature correction (Hv), enthalpy due to translation(Ht) , and enthalpy due to rotation (Hr).
Note that for small molecules this is a very good approximation. However as molecules get larger the 'normal-mode' approximation becomes less valid. For example:
These problems are magnified if one is examining the entropy or free energies of larger molecules. For these reasons low frequency vibrations are often systematically ignored.
The entropy in Spartan is calculated from the same data as vibrational enthalpy (Hv). The equation for entropy (S) can be written as follows:
The constants ui are proportional to the
vibrational frequencies and vA are inversely
proportional to the moment of inertia of the rigid body. It is
important to note that these equations ignore any entropy due to
conformational flexibility, which may dominate for
larger molecules. Also note that low frequency vibrations will
dominate the calculation of entropy. It is these
modes which break the normal mode assumption, thus making
entropy calculations of larger molecules suspect.
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You can use the isodesmic proton transfer reaction relative to a standard to calculate the relative basicities. For example if we choose the ammonia molecule as a standard we would write:
where B is the compound of interest. Then the relative proton affinity is:
Relative proton affinity is a measure of the basicity which can easily be converted to pH. In our methods book (table 6-17) we show the accuracy of this calculation for a number of methods and basis sets for series of Nitrogen Bases. A regression analysis shows that even simple HF/6-31G* methods can produce an accuracy of less than 4 kcal/mol on a system covering 125 kcal/mol range. A sample Spartan file demonstrating this can be found on our website.
There is also an interesting example of predicting pKa's from the graphical map of the elctrostatic potential. This can be found on page 478 of A Guide to Molecular Mechanics and Quantum Chemical Calculations.
The pH of a compound should be related the energy (enthalpy) of the reaction described above
If you know the pH of a couple standards you can use these in a regression fit (using regression tools in Spartans spreadsheet). This will result in a simple linear relation correlating 'energy' with 'pH'.
Science is a series of approximations. The trick is to use the largest simplification one can do and still get reasonably 'good' predictions. This said, we always feel more comfortable with a theory if a recipe exists to get the *exact* answer. In QM calculations this is so prohibitively expensive as to be impossible, yet it can be described. We describe this 'exact' calculation in the next section.
One definition of the *best* energy is the energy which answers one's questions with enough accuracy, yet consumes the minimum amount of time and resources. As mentioned in the discussion on calculating heats of reactions, this is often achieved by writing a near-isodesmic reaction and calculating an energy difference. (The key here is the 'energy difference'. It allows cancellation of systematic errors.)
In order to find what is *best* for your question, I would suggest starting with a question/reaction in which you know the answer but which is similar to your reaction of interest. Try this reaction with HF/6-31G* or HF/3-21G*. If that isn't accurate enough then add the zero-point energy (ZPE) corrections. (scaled by .90 for HF or by .99 for DFT) If that fails, try MP2 or DFT energies with larger basis sets. If that fails attemp single point CCSD with ZPE corrections. Finally try the G3(MP2) or G3 methods.
Conversely, if HF/6-31G* is good enough, you may want to try faster methods to see if they are also acceptable. HF/3-21G* is a popular basis set and much faster than 6-31G*. Semi-empirical is the simplest method using quantum mechanics approaches while requiring an order of magnitude fewer resources and time. Mechanics methods, while limited to ground states of common chemistry groups, is extremely quick and in the areas in which it is parameterized can be fairly accurate.
The complete Enthalpy (H) and the Gibb's Free Energy (G) of a given conformer of a molecule in the dilute gas phase can be written
Calculating all of these terms is impossible for the foreseeable future for any real/non-trivial molecule. However a number of methods and recipes have been proposed to attempt to approach this limit. I will describe such an algorithm using the G3 method. The G3 method is a specific recipe available in Spartan which attempts to produce accurate results in a relatively short amount of time. By studying it we can generalize it to imagine a method which should give the *exact* results even if such a recipe is impractical.
Below is the results of a G3 run on a water molecule which I will use as reference:
Energy HF/6-31G(d) -76.009809 1 Energy MP2/6-31G(d) -76.196848 2 Energy MP4/6-31G(d) -76.207326 3 perturbation correction -0.000565 4 polarization correction -0.074488 5 diffuse correction -0.012979 6 basis set correction -0.081653 7 spin-orbit correction 0.000000 8 electron pair HLC -0.025544 9 ---------------------- ------------------ G3 energy (Ee) -76.402556 HF/6-31G(d) zero point 0.020515 11 G3 energy (Eo) -76.382040 HF/6-31G(d) Vibrational 0.020518 12 HF/6-31G(d) Translation 0.001416 13 HF/6-31G(d) Rotational 0.001416 RT constant 0.000944 G3 kT energy (H298) -76.378260 Atomic Energy -76.032990 14 dH(298) -0.091381 dH(298) -57.34 kCal/mol
Here I will describe successive improvements to the energy. The details of each step are not discussed here, as the main point is to provide an overview of what is required for an *exact* answer.
HF methods ignore electron correlation energy. The goal of DFT and other post HF methods is to account for this energy.
Another weakness in approach described so far is the basis set. There are a number of possible improvements that can be made to the basis set:
If we carry all the above corrections correctly to their infinite limit we should have the *exact* electronic energy. We label this energy Ee. But this would still not be good enough. We've made the assumption that the nuclei do not move. Quantum mechanics tells us that everything is moving, even at absolute zero; the so called "zero point" energy (Zo). By doing a normal mode analysis we can calculate this zero point energy.
The above energy would be correct at zero Kelvin, but chances are that the system we are studying exists at a non-zero temperature. We can make normal mode corrections to the energy, which we now call enthalpy and label it H298. (298.15 K being the most common temperature of interest.)
At this point we notice that the energy is in some strange units called Hartrees. In order to compare with other results it is useful to do a conversion.
This final conversion step is not always necessary. What is necessary in QM calculations is to ensure that when comparing energies, the energies are not only in the same units, but define the same zero. (Typically that means the same basis set and theory level, and molecules with the exact same constituents and number of electrons.)
Often, when examining reactions we require even more; the Gibbs free energy (G) which is calculated by subtracting entropy: G = H - TS + PV.
Combining all of these corrections terms "correctly" (for which there is currently no 'good' systematic procedure) we should arrive at the "exact" answer. At least for a completely isolated system in a vacuum. Adding a real environment (such as a solvent) is vastly more difficult.
Thankfully, a careful examination of the types of chemistry questions one typically asks shows that one does not need to do all this work. (See our methods book for a thorough discussion of this.) Surprisingly, in many cases, HF/6-31G(d) is a good approximation to reality. This happy accident makes quantum chemistry possible on today's computers.
Spartan gives users the ability to choose from a number of advanced correlated methods, and implements the G3 recipe, which is a standard approach for improving the accuracy of calculated energies.
Spartan also provides the flexibility to work with a wide variety of theoretical models:.
All three approaches have a different definition of zero, and use different units (by convention).
Given this definition one cannot compare molecular mechanics energies of different molecules. This restriction holds true even for different isomers. One can compare energies of different conformations. In fact this is one of the most popular and accurate uses of molecular mechanics.
By convention abinitio results are returned in units of Hartrees. See the units section to convert from Hartrees to more common units
Spartan automatically calculates the solvation energy (whenever possible) using the SM54 solvation method:
The "energy(aq)" displayed in the properties dialogue is the sum of the base energy and this Energy of Solvation:
The SM54 solvation method is a modification to the semi-empirical methods used to model a molecules interaction with water. There is a parameterization for both AM1 (the default) and PM3. These parameterizations are labeled SM54A and SM54P respectively.
BSSE is an acronym for "Basis Set Superposition Error". BSSE can occur when calculating reaction energies. For example in the A + B = AB calculation, the energy of AB could be lower because B's basis sets may lower the A part (on the right hand side) and of course will not change the energy of A on the left hand side.
If one worries about this error, the solution in Spartan is to do all three calculation (A, B, and AB) at a higher basis set. Typically this BSSE correction energy is small for large basis sets. Often smaller than other neglected terms such as incomplete basis, neglect of electron correlation, infinite gas phase, approximate geometries and zero-point energy.
Another well known way of dealing with BSSE is the 'counter poise' method. In this approach the calculations on the left hand side'A' and 'B' have included in them the basis functions of 'B' and 'A' added respectively. Unfortunately this method is geometry dependent and assumes the geometry of A and B do not change much when going from A and B to AB.
Spartan's preferred way of dealing with BSSE is to do a single point energy with a large basis set using the "dual-basis" approximation. This is typically more accurate than the usual small basis-set counter poise' method, includes a correction to finite basis set size, and is much quicker than the larger basis set calculation. As an example of the affect of different basis sets we summarize the results for a HF dimer of water. The dimer was optimized at for each basis set.
|T1 + 4RT||-12.62||includes Trans/Rot/PV|
|G3(MP2)Ee + 4RT||-10.66||includes thermodynamics|
|G3||-14.70||better electron correlation & basis|
"Quantum Chemistry: |
Fundamentals to Applications",
1999, pg 240; Vespremi, Feher
Using this data to assign a rough aproximation of the errors involved in a HF/6-31G* calculation of the interactio energy of two water molecules (1 hydrogen bond) we get
Not suprisingly G3 and G3(MP2) address all these errors. For reactions where the number of bonds stays the same, we would expect the T1 theory to address these errors to some extent. In this non-isodesmic reaction (a hydrogen bond is created/broken) one would expect a dual-basis RI-MP2 calculation with a frequency correction to do well. This calculation is summarized below.
|-152.0304565||-23.61||Use this geometry|
basis set and
|" " + 4RT||-15.55|
|" " + Hv[6-31G*]||-18.13||includes HF vibrations|
basis set and
|" " + 4RT||-12.52|
|" " + Hv[6-31G*]||-15.10||includes HF vibrations|
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In Spartan we have a shortcut that can be used to calculate this 'Interaction Energy'. Note that this is available only as a single point energy. (Any geometry optimization must be done prior to this calculation.) Freeze one of the fragments with the frozen atoms symbol and type in the keyword INTERACTIONENERGY. This keyword will calculate all three parts of the 'reaction', for a total of three energy calculations. To calculate the "Counter-Poise" correction one may type INTERACTIONENERGY=CP. If you want to calculate/show the the intermediate BSSE energies one may type INTERACTIONENERGY=BSSE which will calculate a total of 5 energies. Example output for INTERACTIONENERGY=BSSE is shown below:
Combined Energy -152.06977558 (hartrees) Parts -152.06243373 [ -76.03122486 + -76.03120887 ] Interaction -0.00734185 = -19.27602192 kJ/mol BSSE coorection -0.00082962 [ -0.00062318 + -0.00020644 ] Interaction (CP) -0.00651223 = -17.09786380 kJ/mol
<S2> is the spin operator, and it is relevant in UHF calculations. While UHF (or ROHF) is required for open shell systems and to get certain bond separation energies correct, it suffers from the disadvantage that it's wavefunctions are not (exact) eigenfunctions of the total spin operator. This is because the UHF ground state can be contaminated with functions corresponding to states of higher spin multiplicity.
<S2> is a measure of spin contamination and if often used as a test of how good the UHF wavefunction is. Singlet states should have a value of 0.0, doublets 0.75, and triplets 2.0. If <S2> is within +- .02 of these values the wavefunction is usually considered acceptable.
The <S2> is printed out when the PROPPRINTLEV=1 keyword is used, and is represented in the output file as <S**2>
1 kcal/mol = 6.948 e-21 J = 4.184 kJ/mol 1 au (hartree)= me*e^4/h-bar^2 = 4.3597482(26) 10^-18 J * = 4.35974381(34)10^-18 J (1998 CODATA) = 2625.5000 kJ/mol = 627.510 kcal/mol 627.5095602 kcal/mol * 627.50947093 kcal/mol (1998 CODATA [new Na]) = 27.212 ev = 27.2113961(81) ev * 1 ev = 1.60217733(49) 10^-19 J * 4.184 J = 1 Calorie (a constant) 1 kT (T=300K) = 0.595 kcal*In places where multiple values are listed for a given conversion, the first is the approximation used in Spartan, the second is the 'exact' value (as of 1973, 1986 or 1998).
Entropy: 1 e.u. = 4.184 J/mol*K = 1 cal/mol*K Pressure: 1 kbar = 10^8 Pa = 986.923267 atm 1 atm = 101.325 k Pa (exact) * Length: 1 A = 10^-10 m = 1.8897269 au (old value) = 1.889725988579 au 1 au (Bohr) = h-bar^2/(me*e^2) = 0.529177249(24) A * = 0.5291772083(19) A (new CODATA 1998) Mass: 1 AMU = 1.6605402(10) 10^-27 Kg (Atomic Mass Unit) = 1.66053873(13) 10^-27 Kg (new CODATA 1998) Mass C12 = 12.0 AMU = 12.0 g/mol/Na 1 mn = 1.67492716(13) 10^-27 Kg (Mass of neutron) 1 mp = 1.67262158(13) 10^-27 Kg (Mass of proton) 1.007276470(12) AMU 1 me = 9.1034897(54) 10^-31 Kg (Mass of electron) 9.10938188(72)10^-31 Kg 0.5109906(15) Mev Wavenumber: 1 cm^-1 = 2.9979 10^-10 s^-1 = 0.29979 THz 2.19474.7 cm-1= 1 Hartree^-1/2 Bohr^-1 AMU^-1/2 Wavelength: (for light = 1/Wavenumber) = h*c/Energy (for light) 1 nm = 1239.837/ev (ie. homo-lumo gap) = 1.9166 10^-4/kJ (Na in energy) Charge: 1 au = 1 e = 1.602 10^-19 C = 2.452 10^-18 esu*cm Dipole moment: 1 debye(D) = 3.336e-30 C*m = 0.20824 e*A 1 au = 8.479e-30 C*m = 2.542e-18 esu*cm = 2.542 D Polarizability: 1 au = 14.83e-30 m^3 = 14.83 A^3 Moment-of-Inertia: I cm^-1 = 60.1997601/I[ AMU*bhors^2 ] I cm^-1 = 16.8576522/I[ AMU*A^2 ]
Speed of Light : c : 2.99792458 10^10 cm/s * (exact) Avogadro's Num. : Na : 6.0221367(36) 10^23 * Na : 6.02214199(47)10^23 (1998 CODATA) Gas Constant : R : 8.314510(70) J/K/mol * R : 8.314472(15) J/K/mol (1998 CODATA) Boltzmann const : k : 1.380658(12) 10^-23 J/K * 1.3806503(24) 10^-23 J/K (1998 CODATA) Planck const. : h : 6.626075(40) 10^-34 J s * 6.62606876(52)10^-34 J s (1998 CODATA) fine-structure : alpha: 1/137.0359895(61) 7.297352533(27) 10^-3 (1998 CODATA)*In places where multiple values are listed for a given conversion the first is the approximation used in Spartan, the second is the 'exact' value (as of 1973, 1986 or 1998).
No. Data sets using the older constants have been generated
for more than 15 years. To make sure newer versions maintain
backward compatibility we continue to use the older values for
these fundamental constants and conversion factors. Even though
each new digit is an important scientific achievement,
the increased precision is well underneath the noise present in
the chemical measurements Spartan deals with.
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