Quantum chemistry composite methods

Quantum chemistry composite methods are computational chemistry methods that aim for high accuracy by combining the results of several calculations. They combine methods with a high level of theory and a small basis set with methods that employ lower levels of theory with larger basis sets. They are commonly used to calculate thermodynamic quantities such as enthalpies of formation, atomization energies, ionization energies and electron affinities. They aim for chemical accuracy which is usually defined as within 1 kcal/mol of the experimental value. The first systematic model chemistry of this type with broad applicability was called Gaussian-1 (G1) introduced by John Pople. This was quickly replaced by the Gaussian-2 (G2) which has been used extensively. The Gaussian-3 (G3) was introduced later.

Gaussian-2 (G2)

The G2 uses seven calculations:

1. the molecular geometry is obtained by a MP2 optimization using the 6-31G(d) basis set and all electrons included in the perturbation. This geometry is used for all subsequent calculations.
2. The highest level of theory is a quadratic configuration interaction calculation with single and double excitations and a triples excitation contribution (QCISD(T)) with the 6-311G(d) basis set. Such a calculation in the GAUSSIAN program also gives the MP2 and MP4 energies which are also used.
3. The effect of polarization functions is assessed using an MP4 calculation with the 6-311G(2df,p) basis set.
4. The effect of diffuse functions is assessed using an MP4 calculation with the 6-311+G(d, p) basis set.
5. The largest basis set is 6-311+G(3df,2p) used at the MP2 level of theory.
6. A Hartree-Fock geometry optimization with the 6-31G(d) basis set used to give a geometry for:
7. A frequency calculation with the 6-31G(d) basis set to obtain the zero-point vibrational energy (ZPVE)

The various energy changes are assumed to be additive so the combined energy is given by:

EQCISD(T) from 2 + [EMP4 from 3 - EMP4 from 2] + [EMP4 from 4 - EMP4 from 2] + [EMP2 from 5 + EMP2 from 2 - EMP2 from 3 - EMP2 from 4]

The second term corrects for the effect of adding the polarization functions. The third term corrects for the diffuse functions. The final term corrects for the larger basis set with the terms from steps 2, 3 and 4 preventing contributions from being counted twice. Two final corrections are made to this energy. The ZPVE is scaled by 0.8929. An empirical correction is then added to account for factors not considered above. This is called the higher level correction (HC) and is given by -0.00481 x (number of valence electrons -0.00019 x (number of unpaired valence electrons). The two numbers are obtained calibrating the results against the experimental results for a set of molecules. The scaled ZPVE and the HLC are added to give the final energy. For some molecules containing one of the third row elements Ga - Xe, a further term is added to account for spin orbit coupling.

Several variants of this procedure have been used. Removing steps 3 and 4 and relying only on the MP2 result from step 5 is significantly cheaper and only slightly less accurate. This is the G2MP2 method. Sometimes the geometry is obtained using a density functional theory method such as B3LYP and sometimes the QCISD(T) method in step 1 is replaced by the coupled cluster method CCSD(T).

Gaussian-3 (G3)

The G3 is very similar to G2 but learns from the experience with G2 theory. The 6-311G basis set is replaced by the smaller 6-31G basis. The final MP2 calculations use a larger basis set, generally just called G3large, and correlating all the electrons not just the valence electrons as in G2 theory. This gives some core correlation contributions to the final energy. The HLC takes the same form but with different empirical parameters. A Gaussian-4 method has been introduced.[1] An alternative to the Gaussian-n methods is the correlation consistent composite method.[2]
Complete basis set methods (CBS)

These methods by Petersson and coworkers[3] have some similarity to G2 and G3 but contain an MP2 extrapolation to the complete basis set limit as one step.
T1

The T1 method.[4] is an efficient computational approach developed for calculating accurate heats of formation of uncharged, closed-shell molecules comprising H, C, N, O, F, S, Cl and Br, within experimental error. It is practical for molecules up to molecular weight ~ 500 a.m.u.

T1 method as incorporated in Spartan consists of:

1. HF/6-31G* optimization.
2. RI-MP2/6-311+G(2d,p)[6-311G*] single point energy with dual basis set.
3. An empirical correction using atom counts, Mulliken bond orders, HF/6-31G* and RI-MP2 energies as variables.

T1 follows the G3(MP2) recipe, however, by substituting an HF/6-31G* for the MP2/6-31G* geometry, eliminating both the HF/6-31G* frequency and QCISD(T)/6-31G* energy and approximating the MP2/G3MP2large energy using dual basis set RI-MP2 techniques, the T1 method reduces computation time by up to 3 orders of magnitude. Atom counts, Mulliken bond orders and HF/6-31G* and RI-MP2 energies are introduced as variables in a linear regression fit to a set of 1126 G3(MP2) heats of formation. The T1 procedure reproduces these values with mean absolute and RMS errors of 1.8 and 2.5 kJ/mol, respectively. T1 reproduces experimental heats of formation for a set of 1805 diverse organic molecules from the NIST thermochemical database[5] with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively.
Other methods

References

1. ^ Curtiss, Larry A.; Paul C. Redfern and Krishnan Raghavachari (2007). "Gaussian-4 Theory". The Journal of Chemical Physics (American Institute of Physics) 126: 084108. doi:10.1063/1.2436888.
2. ^ DeYonker, Nathan J.; Thomas R. Cundari and Angela K. Wilson (2006). "The correlation consistent composite method (ccCA): An alternative to the Gaussian-n methods". The Journal of Chemical Physics (American Institute of Physics) 124: 114104. doi:10.1063/1.2173988.
3. ^ http://www.wesleyan.edu/chem/faculty/petersson/
4. ^ Ohlinger, William Sean; Phlip E. Klunzinger, Bernard J. Deppmeier, and Warren J. Hehre (2009). "Efficient Calculation of Heats of Formation". The Journal of Physical Chemistry A (ACS Publications) 113 (10): 2165-2175. doi:10.1021/jp810144q.
5. ^ NIST Chemistry WebBook

* Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley and Sons. pp. 224–228. ISBN 0 471-48552 7.
* Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 164–169. ISBN 0470011874.