Quantum chemistry composite methods

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Quantum chemistry composite methods (also referred to as thermochemical recipes) [1] [2] 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

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. [3]

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)

G4 is a compound method in spirit of the other Gaussian theories and attempts to take the accuracy achieved with G3X one small step further. This involves the introduction of an extrapolation scheme for obtaining basis set limit Hartree-Fock energies, the use of geometries and thermochemical corrections calculated at B3LYP/6-31G(2df,p) level, a highest-level single point calculation at CCSD(T) instead of QCISD(T) level, and addition of extra polarization functions in the largest-basis set MP2 calculations. Thus, Gaussian 4 (G4) theory [4] is an approach for the calculation of energies of molecular species containing first-row, second-row, 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 G4 and the related G4MP2 methods have been extended to cover transition metals. [5] A variant of G4MP2, termed G4(MP2)-6X, has been developed with an aim to improve the accuracy with essentially identical quantum chemistry components. [6] It applies scaling to the energy components in addition to using the HLC. In the G4(MP2)-XK method [7] that is related to G4(MP2)-6X, the Pople-type basis sets [8] are replaced with customized Karlsruhe-type basis sets. [8] In comparison with G4(MP2)-6X, which covers main-group elements up to krypton, G4(MP2)-XK is applicable to main-group elements up to radon.

Feller-Peterson-Dixon approach (FPD)

Unlike fixed-recipe, "model chemistries", the FPD approach [9] [10] [11] [12] [13] consists of a flexible sequence of (up to) 13 components that vary with the nature of the chemical system under study and the desired accuracy in the final results. In most instances, the primary component relies on coupled cluster theory, such as CCSD(T), or configuration interaction theory combined with large Gaussian basis sets (up through aug-cc-pV8Z, in some cases) and extrapolation to the complete basis set limit. As with some other approaches, additive corrections for core/valence, scalar relativistic and higher order correlation effects are usually included. Attention is paid to the uncertainties associated with each of the components so as to permit a crude estimate of the uncertainty in the overall results. Accurate structural parameters and vibrational frequencies are a natural byproduct of the method. While the computed molecular properties can be highly accurate, the computationally intensive nature of the FPD approach limits the size of the chemical system to which it can be applied to roughly 10 or fewer first/second row atoms.

The FPD Approach has been heavily benchmarked against experiment. When applied at the highest possible level, FDP is capable to yielding a root-mean-square (RMS) deviation with respect to experiment of 0.30 kcal/mol (311 comparisons covering atomization energies, ionization potentials, electron affinities and proton affinities). In terms of equilibrium, bottom-of-the-well structures, FPD gives an RMS deviation of 0.0020 Å (114 comparisons not involving hydrogens) and 0.0034 Å (54 comparisons involving hydrogen). Similar good agreement was found for vibrational frequencies.


The calculated T1 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 with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively. T1 vs NIST Expiremental Heat of Formation.png
The calculated T1 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 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, [15] 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 [14] with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively.

Correlation consistent composite approach (ccCA)

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. [16] [17] 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:


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:

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

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

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

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

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

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.

The Correlation Consistent Composite Approach is available as a keyword in NWChem [18] and GAMESS (ccCA-S4 and ccCA-CC(2,3)) [19]

Complete Basis Set methods (CBS)

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. [20] 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(+). [21] The CBS family of methods is available via keywords in the Gaussian 09 suite of programs. [22]

Weizmann-n theories

The Weizmann-n ab initio methods (Wn, n = 1–4) [23] [24] [25] 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, [2] [26] and unprecedented accuracies in prediction of spectroscopic constants. [27] 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.

In an attempt to extend the applicability of the Wnab initio thermochemistry methods, explicitly correlated versions of these theories have been developed: Wn-F12 (n = 1–3) [28] and more recently even a W4-F12 theory. [29] W1-F12 was successfully applied to large hydrocarbons (e.g., dodecahedrane, [30] as well as to systems of biological relevance (e.g., DNA bases). [28] W4-F12 theory has been applied to systems as large as benzene. [29] In a similar manner, the WnX protocols that have been developed independently further reduce the requirements on computational resources by using more efficient basis sets and, for the minor components, electron-correlation methods that are computationally less demanding. [31] [32] [33]

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Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids. It is necessary because, apart from relatively recent results concerning the hydrogen molecular ion, the quantum many-body problem cannot be solved analytically, much less in closed form. While computational results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials.

In computational chemistry and molecular physics, Gaussian orbitals are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.

Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry, but it is also used in nuclear physics. Coupled cluster essentially takes the basic Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation. Some of the most accurate calculations for small to medium-sized molecules use this method.


MOLPRO is a software package used for accurate ab initio quantum chemistry calculations. It is developed by Peter Knowles at Cardiff University and Hans-Joachim Werner at Universität Stuttgart in collaboration with other authors.

Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70. It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16. Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.

Q-Chem is a general-purpose electronic structure package featuring a variety of established and new methods implemented using innovative algorithms that enable fast calculations of large systems on various computer architectures, from laptops and regular lab workstations to midsize clusters and HPCC, using density functional and wave-function based approaches. It offers an integrated graphical interface and input generator; a large selection of functionals and correlation methods, including methods for electronically excited states and open-shell systems; solvation models; and wave-function analysis tools. In addition to serving the computational chemistry community, Q-Chem also provides a versatile code development platform.

Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.

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A basis set in theoretical and computational chemistry is a set of functions that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.

PQS (software)

PQS is a general purpose quantum chemistry program. Its roots go back to the first ab initio gradient program developed in Professor Peter Pulay's group but now it is developed and distributed commercially by Parallel Quantum Solutions. There is a reduction in cost for academic users and a site license. Its strong points are geometry optimization, NMR chemical shift calculations, and large MP2 calculations, and high parallel efficiency on computing clusters. It includes many other capabilities including Density functional theory, the semiempirical methods, MINDO/3, MNDO, AM1 and PM3, Molecular mechanics using the SYBYL 5.0 Force Field, the quantum mechanics/molecular mechanics mixed method using the ONIOM method, natural bond orbital (NBO) analysis and COSMO solvation models. Recently, a highly efficient parallel CCSD(T) code for closed shell systems has been developed. This code includes many other post Hartree–Fock methods: MP2, MP3, MP4, CISD, CEPA, QCISD and so on.

TURBOMOLE Computational chemistry program

TURBOMOLE is an ab initio computational chemistry program that implements various quantum chemistry methods. It was initially developed by the group of Prof. Reinhart Ahlrichs at the University of Karlsruhe. In 2007, TURBOMOLE GmbH, founded by R. Ahlrichs, F. Furche, C. Hättig, W. Klopper, M. Sierka, and F. Weigend, took over the responsibility for the coordination of the scientific development of TURBOMOLE program, for which the company holds all copy and intellectual property rights. In 2018 David P. Tew joined the TURBOMOLE GmbH. Since 1987, this program is one of the useful tools as it involves in many fields of research including heterogeneous and homogeneous catalysis, organic and inorganic chemistry, spectroscopy as well as biochemistry. This can be illustrated by citation records of Ahlrich's 1989 publication which is more than 6700 times as of 18 July 2020. In the year 2014, the second Turbomole article has been published. The number of citations from both papers indicates that the Turbomole's user base is expanding.

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Spartan (chemistry software)

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Pople diagram

A Pople diagram or Pople's Diagram is a diagram which describes the relationship between various calculation methods in computational chemistry. It was initially introduced in January 1965 by Sir John Pople,, during the Symposium of Atomic and Molecular Quantum Theory in Florida. The Pople Diagram can be either 2-dimensional or 3-dimensional, with the axes representing ab inito methods, basis sets and treatment of relativity. The diagram attempts to balance calculations by giving all aspects of a computation equal weight.


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