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 that the quality of the calculated heat of reaction depends on both the method used to calculate the energy (E[..]) and the type of reaction.
"A Guide to Molecular Mechanics and Quantum Chemical Calculations" (our methods book), explains that the Quantum Chemical Calculations perform "best" on "isodesmic" reactions. (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 QM methods (SemiEmpirical, Hartree-Fock, DFT, and Moller Plesset).
Sadly, the experimental Heat of Formation 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.
2(CH3CH2NH2) -> 7(H2) + N2 + 4(C-graphite)
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 Heat of 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 HOF calculations. The methods book includes many examples of HF/6-31G* without vibrational corrections performing very 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:
These energies are combined, along with similar calculations for each atomic species, experimental energies of "standard state, atomically pure" systems are subtracted, and the result is a reasonably good approximation of the actual HOF. A brief overview of how these steps are used can be found in our discussion of calculating the exact energy.
Note that each of these steps is relatively time consuming, so G3 calculations are practical only for very small systems.
The reference for G3 can be found at the end of this FAQ.
Return to TopThe quantum mechanical calculations in Spartan assume that molecules are isolated, with a temperature of zero Kelvin, and with stationary nuclei. Experiments are conducted 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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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 the methods book (table 6-17) provides an assessment of the accuracy of this calculation using a variety of computational methods and basis sets for a series of Nitrogen Bases. Regression analysis shows that even simple HF/6-31G* calculations can produce an accuracy of less than 4 kcal/mol on a system covering a 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 few standards you can use these in a regression fit (using the regression tools in Spartan's spreadsheet utility). 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 possible that provides reasonably 'good' predictions. This said, scientists always feel more comfortable with a theory if it reproduces the *exact* answer. In Quantum Chemical calculations, obtaining the "exact" answer is so prohibitively expensive (from a time perspective) that it is 'practically' impossible, however, the "exact" calculation can be described. (see next section).
One definition of the *best* energy is the energy that 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". This allows for the cancellation of systematic errors.)
In order to find what energy is *best* for your needs, start with a related question/reaction for which the answer is known. 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 (scale by .89 for HF or by .975 for DFT). If you need more accuracy, move up to MP2 or DFT energies with larger basis sets. If you need more accuracy, attemp a single point energy calculation using 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. 3-21G* is a popular basis set and is much faster than 6-31G*. Semi-Empirical is the simplest quantum chemical method, and it is an order of magnitude faster than HF/3-21G*. Molecular Mechanics methods, while limited to ground states of common organic chemistry groups, is extremely fast, and in the areas for which it is parameterized, it also fairly accurate.
If you are new to computational chemistry,
Wavefunction's method books
discuss many types of reactions and molecular properties,
quantitatively comparing different theory levels with experimental data..
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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 these terms is currently impossible for any real (non-trivial) molecule. However, a number of methods and recipes have been developed that 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 that attempts to produce accurate results in a "relatively short" period of time. By studying it, one can generalize the approach and imagine a method that should give the *exact* results even if such a recipe is impractical (from a time and/or computational resource perspective).
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 unites 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.
<S2> is the spin operator, and it is relevant in Unrestricted HF 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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