Quantum chemistry composite methods
Encyclopedia
Quantum chemistry composite methods (also referred to as thermochemical recipes) 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.
The various energy changes are assumed to be additive so the combined energy is given by:
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).
's research group, utilizes the correlation consistent basis sets
developed by Dunning and Co-workers. 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:
Core-core and core-valence interactions are accounted for using MP2(FC1)/aug-cc-pCVTZ:
Scalar relativistic effects are also taken into account with a one-particle Douglass Kroll Hess Hamiltonian
and recontracted basis sets:
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.
T1 method as incorporated in Spartan
consists of:
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 with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively.
and zero-point energies, and two added higher level correction parameters. According to the developers, this theory gives significant improvement over G3-theory.
Computational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
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
Basis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...
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
John Pople
Sir John Anthony Pople, KBE, FRS, was a Nobel-Prize winning theoretical chemist. Born in Burnham-on-Sea, Somerset, England, he attended Bristol Grammar School. He won a scholarship to Trinity College, Cambridge in 1943. He received his B. A. in 1946. Between 1945 and 1947 he worked at the Bristol...
. 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:- the molecular geometry is obtained by a MP2Møller-Plesset perturbation theoryMøller–Plesset perturbation theory is one of several quantum chemistry post-Hartree–Fock ab initio methods in the field of computational chemistry...
optimization using the 6-31G(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 interactionQuadratic configuration interactionQuadratic configuration interaction is an extension of Configuration interaction that corrects for size-consistency errors in the all singles and double excitation CI methods ....
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 GAUSSIANGAUSSIANGaussian is a computational chemistry software program 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...
and SPARTANSpartan (software)SPARTAN is a molecular modeling and computational chemistry application from . It contains code for molecular mechanics, semi-empirical methods, ab initio models, density functional models, post-Hartree-Fock models, and thermochemical recipes including T1....
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 6-311G(2df,p) basis set.
- The effect of diffuse functions is assessed using an MP4 calculation with the 6-311+G(d, p) basis set.
- The largest basis set is 6-311+G(3df,2p) used at the MP2 level of theory.
- A Hartree-FockHartree-FockIn computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
geometry optimization with the 6-31G(d) basis set used to give a geometry for: - 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
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
method such as B3LYP
Hybrid functional
Hybrid functionals are a class of approximations to the exchange-correlation energy functional in density functional theory that incorporate a portion of exact exchange from Hartree-Fock theory with exchange and correlation from other sources...
and sometimes the QCISD(T) method in step 1 is replaced by the coupled cluster
Coupled cluster
Coupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry...
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, 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. A Gaussian-4 method has been introduced. An alternative to the Gaussian-n methods is the correlation consistent composite method.The correlation consistent Composite Approach (ccCA)
This approach, developed at the University of North Texas by Angela K. WilsonAngela K. Wilson
Angela K. Wilson is a Professor of computational chemistry at the University of North Texas and co-director of the Center for Advanced Scientific Computing and Modeling .-Career:...
's research group, utilizes the correlation consistent basis sets
Basis set
Basis set can refer to:* Basis * Basis set...
developed by Dunning and Co-workers. 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:
- EccCA = EMP2/CBS + ΔECC + ΔECV + ΔESR + ΔEZPE + ΔESO
The reference energy EMP2/CBS is the MP2/aug-cc-pVnZ (where n=D,T,Q) energies extrapolated at the complete basis set
Basis set
Basis set can refer to:* Basis * 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
Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* 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.
Complete basis set methods (CBS)
These methods by Petersson and coworkers 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. 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
Spartan (software)
SPARTAN is a molecular modeling and computational chemistry application from . It contains code for molecular mechanics, semi-empirical methods, ab initio models, density functional models, post-Hartree-Fock models, and thermochemical recipes including T1....
consists of:
- HFHartree-FockIn computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
/6-31G* optimization. - RI-MP2/6-311+G(2d,p)[6-311G*] single point energy with dual basis set.
- An empirical correction using atom counts, Mulliken bond orders, HFHartree-FockIn computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
/6-31G* and RI-MP2 energies as variables.
T1 follows the G3(MP2) recipe, however, by substituting an HF
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
/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
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
/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 with mean absolute and RMS errors of 8.5 and 11.5 kJ/mol, respectively.
Gaussian-4 (G4)
Gaussian 4 (G4) theory 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 theoryDensity functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
and zero-point energies, and two added higher level correction parameters. According to the developers, this theory gives significant improvement over G3-theory.