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 Gaussian1 (G1) introduced by John Pople. This was quickly replaced by the Gaussian2 (G2) which has been used extensively. The Gaussian3 (G3) was introduced later.
Contents

Gaussiann Theories 1

Gaussian2 (G2) 1.1

Gaussian3 (G3) 1.2

Gaussian4 (G4) 1.3

T1 2

Correlation Consistent Composite Approach (ccCA) 3

Complete Basis Set methods (CBS) 4

Weizmannn Theories 5

References 6
Gaussiann Theories
Gaussian2 (G2)
The G2 uses seven calculations:

the molecular geometry is obtained by a MP2 optimization using the 631G(d) basis set and all electrons included in the perturbation. This geometry is used for all subsequent calculations.

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 6311G(d) basis set. Such a calculation in the Gaussian and Spartan programs also give the MP2 and MP4 energies which are also used.

The effect of polarization functions is assessed using an MP4 calculation with the 6311G(2df,p) basis set.

The effect of diffuse functions is assessed using an MP4 calculation with the 6311+G(d, p) basis set.

The largest basis set is 6311+G(3df,2p) used at the MP2 level of theory.

A Hartree–Fock geometry optimization with the 631G(d) basis set used to give a geometry for:

A frequency calculation with the 631G(d) basis set to obtain the zeropoint 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 631+G(d) basis set is used in place of the 631G(d) basis set for both the initial geometry optimization, as well as the second geometry optimization and frequency calculation. Additionally, the frozencore approximation is made for the initial MP2 optimization, whereas G2 usually uses the full calculation.^{[2]}
Gaussian3 (G3)
The G3 is very similar to G2 but learns from the experience with G2 theory. The 6311G basis set is replaced by the smaller 631G 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 spinorbit 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.
Gaussian4 (G4)
Gaussian 4 (G4) theory ^{[3]} is an approach for the calculation of energies of molecular species containing firstrow (Li–F), secondrow (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 zeropoint energies, and two added higher level correction parameters. According to the developers, this theory gives significant improvement over G3theory.
T1
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, closedshell 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:

HF/631G* optimization.

RIMP2/6311+G(2d,p)[6311G*] single point energy with dual basis set.

An empirical correction using atom counts, Mulliken bond orders,^{[5]} HF/631G* and RIMP2 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.

^ ^{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.

^ Mikhail N. Glukhovtsev; Addy Pross; Leo Radom (1996). "GasPhase NonIdentity SN2 Reactions of Halide Anions with Methyl Halides: A HighLevel Computational Study". J. Am. Chem. Soc. 118 (26): 6273–6284.

^ Curtiss, Larry A.; Paul C. Redfern; Krishan Raghavachari (2007). "Gaussian4 theory". The Journal of Chemical Physics (AIP Publications) 126 (8): 084108.

^ ^{a} ^{b} [1] NIST Chemistry WebBook

^ Mulliken, R. S. (1955). "Electronic Population Analysis on LCAOMO Molecular Wave Functions. I". The Journal of Chemical Physics 23 (10): 1833–1831.

^ Deyonker, Nathan J.; Cundari, Thomas R.; Wilson, Angela K. (2006). "The correlation consistent composite approach (ccCA): An alternative to the Gaussiann methods". J. Chem. Phys. 124 (11): 114104.

^ Fabian, Walter M. F. (2008). "Accurate thermochemistry from quantum chemical calculations?". Monats. Chem. 139 (4): 309.

^ "Correlation consistent Composite Approach (ccCA)". NWChem.

^ Petersson, G. (2002). "Complete Basis Set Models for Chemical Reactivity: from the Helium Atom to Enzyme Kinetics". In Cioslowski, J. QuantumMechanical Prediction of Thermochemical Data 22. Springer Netherlands. pp. 99–130.

^ Srinivasan Parthiban; Glênisson de Oliveira; Jan M. L. Martin (2001). "Benchmark ab Initio Energy Profiles for the GasPhase SN2 Reactions Y + CH3X → CH3Y + X (X,Y = F,Cl,Br). Validation of Hybrid DFT Methods". J. Phys. Chem. A 105 (5): 895–904.

^ "CBS Methods". Gaussian 09 User's Reference. Gaussian, Inc.

^ 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.

^ 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.

^ A. Karton, E. Rabinovich, J. M. L. Martin and B. Ruscic (2006). "W4 theory for computational thermochemistry: In pursuit of confident subkJ/mol predictions.". Journal of Chemical Physics.

^ A. Karton, S. Daon and J. M. L. Martin (2011). "W411: A highconfidence dataset for computational thermochemistry derived from W4 ab initio data.". Chemical Physics Letters.

^ 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.

^ A. Karton, and J. M. L. Martin (2012). "Explicitly correlated Wn theory: W1–F12 and W2–F12.". Journal of Chemical Physics.
References
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 Weizmannn ab initio methods (Wn, n = 1–4)^{[12]}^{[13]}^{[14]} are highly accurate composite theories devoid of empirical parameters. These theories are capable of subkJ/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 basisset 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 nonhydrogen atoms (with C1 symmetry), even the computationally more economical W1 theory becomes prohibitively expensive with current mainstream server hardware.
Weizmannn Theories
The Complete Basis Set (CBS) methods are a family of composite methods, the members of which are: CBS4M, CBSQB3, and CBSAPNO, 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 singlepoint energies to the "exact" energy.^{[9]} In comparison, the Gaussiann methods perform their approximation using additive corrections. Similar to the modified G2(+) method, CBSQB3 has been modified by the inclusion of diffuse functions in the geometry optimization step to give CBSQB3(+).^{[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 spinorbit corrections considered only for atoms.

ΔE_{SR} = E_{MP2DK/ccpVTZDK}  E_{MP2/ccpVTZ}
Scalar relativistic effects are also taken into account with a oneparticle Douglass Kroll Hess Hamiltonian and recontracted basis sets:

ΔE_{CV}= E_{MP2(FC1)/augccpCVTZ}  E_{MP2/augccpVTZ}
Corecore and corevalence interactions are accounted for using MP2(FC1)/augccpCVTZ:

ΔE_{CC} = E_{CCSD(T)/ccpVTZ}  E_{MP2/ccpVTZ}
The reference energy E_{MP2/CBS} is the MP2/augccpVnZ (where n=D,T,Q) energies extrapolated at the complete basis set limit by the Peterson mixed gaussian exponential extrapolation scheme. CCSD(T)/ccpVTZ is used to account for correlation beyond the MP2 theory:

E_{ccCA} = E_{MP2/CBS} + ΔE_{CC} + ΔE_{CV} + ΔE_{SR} + ΔE_{ZPE} + ΔE_{SO}
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 coworkers.^{[6]}^{[7]} Unlike the Gaussiann methods, ccCA does not contain any empirically fitted term. The B3LYP density functional method with the ccpVTZ basis set, and ccpV(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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