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Quantum chemistry composite methods

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Quantum chemistry composite methods

Quantum chemistry composite methods (also referred to as thermochemical recipes)[1] 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-n Theories 1
    • Gaussian-2 (G2) 1.1
    • Gaussian-3 (G3) 1.2
    • Gaussian-4 (G4) 1.3
  • T1 2
  • Correlation Consistent Composite Approach (ccCA) 3
  • Complete Basis Set methods (CBS) 4
  • Weizmann-n Theories 5
  • References 6

Gaussian-n Theories

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 and Spartan programs also give 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 2 is replaced by the coupled cluster method CCSD(T).

The G2(+) variant, where the "+" symbol refers to added diffuse functions, better describes anions than conventional G2 theory. The 6-31+G(d) basis set is used in place of the 6-31G(d) basis set for both the initial geometry optimization, as well as the second geometry optimization and frequency calculation. Additionally, the frozen-core approximation is made for the initial MP2 optimization, whereas G2 usually uses the full calculation.[2]

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, additionally a spin-orbit correction term and an empirical correction for valence electrons are introduced. This gives some core correlation contributions to the final energy. The HLC takes the same form but with different empirical parameters.

Gaussian-4 (G4)

Gaussian 4 (G4) theory [3] is an approach for the calculation of energies of molecular species containing first-row (Li–F), second-row (Na–Cl), and third row main group elements. G4 theory is an improved modification of the earlier approach G3 theory. The modifications to G3- theory are the change in an estimate of the Hartree–Fock energy limit, an expanded polarization set for the large basis set calculation, use of CCSD(T) energies, use of geometries from density functional theory and zero-point energies, and two added higher level correction parameters. According to the developers, this theory gives significant improvement over G3-theory.


The calculated T1[1] heat of formation (y axis) compared to the experimental heat of formation (x axis) for a set of >1800 diverse organic molecules from the NIST thermochemical database[4] with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively.

The T1 method.[1] is an efficient computational approach developed for calculating accurate heats of formation of uncharged, closed-shell molecules comprising H, C, N, O, F, Si, P, 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,[5] HF/6-31G* and RI-MP2 energies as variables.

T1 follows the G3(MP2) recipe, however, by substituting an

  • Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley and Sons. pp. 224–228.  
  • Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 164–169.  
  1. ^ a b c Ohlinger, William S.; Philip E. Klunzinger; Bernard J. Deppmeier; Warren J. Hehre (January 2009). "Efficient Calculation of Heats of Formation". The Journal of Physical Chemistry A (ACS Publications) 113 (10): 2165–2175.  
  2. ^ Mikhail N. Glukhovtsev; Addy Pross; Leo Radom (1996). "Gas-Phase Non-Identity SN2 Reactions of Halide Anions with Methyl Halides:  A High-Level Computational Study". J. Am. Chem. Soc. 118 (26): 6273–6284.  
  3. ^ Curtiss, Larry A.; Paul C. Redfern; Krishan Raghavachari (2007). "Gaussian-4 theory". The Journal of Chemical Physics (AIP Publications) 126 (8): 084108.  
  4. ^ a b [1] NIST Chemistry WebBook
  5. ^ Mulliken, R. S. (1955). "Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I". The Journal of Chemical Physics 23 (10): 1833–1831.  
  6. ^ Deyonker, Nathan J.; Cundari, Thomas R.; Wilson, Angela K. (2006). "The correlation consistent composite approach (ccCA): An alternative to the Gaussian-n methods". J. Chem. Phys. 124 (11): 114104.  
  7. ^ Fabian, Walter M. F. (2008). "Accurate thermochemistry from quantum chemical calculations?". Monats. Chem. 139 (4): 309.  
  8. ^ "Correlation consistent Composite Approach (ccCA)". NWChem. 
  9. ^ Petersson, G. (2002). "Complete Basis Set Models for Chemical Reactivity: from the Helium Atom to Enzyme Kinetics". In Cioslowski, J. Quantum-Mechanical Prediction of Thermochemical Data 22. Springer Netherlands. pp. 99–130.  
  10. ^ Srinivasan Parthiban; Glênisson de Oliveira; Jan M. L. Martin (2001). "Benchmark ab Initio Energy Profiles for the Gas-Phase SN2 Reactions Y- + CH3X → CH3Y + X- (X,Y = F,Cl,Br). Validation of Hybrid DFT Methods". J. Phys. Chem. A 105 (5): 895–904.  
  11. ^ "CBS Methods". Gaussian 09 User's Reference. Gaussian, Inc. 
  12. ^ J. M. L. Martin and G. de Oliveira (1999). "Towards standard methods for benchmark quality ab initio thermochemistry—W1 and W2 theory.". Journal of Chemical Physics.  
  13. ^ A. D. Boese, M. Oren, O. Atasoylu, J. M. L. Martin, M. Kállay, and J. Gauss (2004). "W3 theory: Robust computational thermochemistry in the kJ/mol accuracy range.". Journal of Chemical Physics.  
  14. ^ A. Karton, E. Rabinovich, J. M. L. Martin and B. Ruscic (2006). "W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions.". Journal of Chemical Physics.  
  15. ^ A. Karton, S. Daon and J. M. L. Martin (2011). "W4-11: A high-confidence dataset for computational thermochemistry derived from W4 ab initio data.". Chemical Physics Letters.  
  16. ^ A. Karton and J. M. L. Martin (2010). "Performance of W4 theory for spectroscopic constants and electrical properties of small molecules.". Journal of Chemical Physics.  
  17. ^ A. Karton, and J. M. L. Martin (2012). "Explicitly correlated Wn theory: W1–F12 and W2–F12.". Journal of Chemical Physics.  


In an attempt to extend the applicability of the W1 and W2 ab initio thermochemistry methods, explicitly correlated versions of these theories have been developed (W1–F12 and W2–F12).[17] W1–F12 was successfully applied to large aromatic systems (e.g., tetracene) as well as to systems of biological relevance (e.g., DNA bases).

The Weizmann-n ab initio methods (Wn, n = 1–4)[12][13][14] are highly accurate composite theories devoid of empirical parameters. These theories are capable of sub-kJ/mol accuracies in prediction of fundamental thermochemical quantities such as heats of formation and atomization energies,[15] and unprecedented accuracies in prediction of spectroscopic constants.[16] The ability of these theories to successfully reproduce the CCSD(T)/CBS (W1 and W2), CCSDT(Q)/CBS (W3), and CCSDTQ5/CBS (W4) energies relies on judicious combination of very large Gaussian basis sets with basis-set extrapolation techniques. Thus, the high accuracy of Wn theories comes with the price of a significant computational cost. In practice, for systems consisting of more than ~9 non-hydrogen atoms (with C1 symmetry), even the computationally more economical W1 theory becomes prohibitively expensive with current mainstream server hardware.

Weizmann-n Theories

The Complete Basis Set (CBS) methods are a family of composite methods, the members of which are: CBS-4M, CBS-QB3, and CBS-APNO, in increasing order of accuracy. These methods offer errors of 2.5, 1.1, and 0.7 kcal/mol when tested against the G2 test set. The CBS methods were developed by George Petersson and coworkers, and they make extrapolate several single-point energies to the "exact" energy.[9] In comparison, the Gaussian-n methods perform their approximation using additive corrections. Similar to the modified G2(+) method, CBS-QB3 has been modified by the inclusion of diffuse functions in the geometry optimization step to give CBS-QB3(+).[10] The CBS family of methods is available via keywords in the Gaussian 09 suite of programs.[11]

Complete Basis Set methods (CBS)

The Correlation Consistent Composite Approach is available as a keyword in NWChem.[8]

The last two terms are Zero Point Energy corrections scaled with a factor of 0.989 to account for deficiencies in the harmonic approximation and spin-orbit corrections considered only for atoms.

ΔESR = EMP2-DK/cc-pVTZ-DK - EMP2/cc-pVTZ

Scalar relativistic effects are also taken into account with a one-particle Douglass Kroll Hess Hamiltonian and recontracted basis sets:

ΔECV= EMP2(FC1)/aug-cc-pCVTZ - EMP2/aug-cc-pVTZ

Core-core and core-valence interactions are accounted for using MP2(FC1)/aug-cc-pCVTZ:

ΔECC = ECCSD(T)/cc-pVTZ - EMP2/cc-pVTZ

The reference energy EMP2/CBS is the MP2/aug-cc-pVnZ (where n=D,T,Q) energies extrapolated at the complete basis set limit by the Peterson mixed gaussian exponential extrapolation scheme. CCSD(T)/cc-pVTZ is used to account for correlation beyond the MP2 theory:


This approach, developed at the University of North Texas by Angela K. Wilson's research group, utilizes the correlation consistent basis sets developed by Dunning and co-workers.[6][7] Unlike the Gaussian-n methods, ccCA does not contain any empirically fitted term. The B3LYP density functional method with the cc-pVTZ basis set, and cc-pV(T+d)Z for third row elements (Na - Ar), are used to determine the equilibrium geometry. Single point calculations are then used to find the reference energy and additional contributions to the energy. The total ccCA energy for main group is calculated by:

Correlation Consistent Composite Approach (ccCA)

with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively. [4]

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