The %TAE[(T)] diagnostic is an energy-based diagnostic for multi-reference character which is a good a priori predictor for the magnitude of post-CCSD(T) contributions. In particular, there is a high statistical correlation between the %TAE[(T)] diagnostic and the magnitude of connected quadruple and quintuple excitations.[1,2,3] For the 140 molecules in the W4-11 database, a squared correlation coefficient of R^2 = 0.94 is obtained.[2]

The %TAE[(T)] diagnostic is calculated from the atomic and molecular CCSD and CCSD(T) energies. It is defined as:

%TAE[(T)] = 100 × (TAE[CCSD(T)]–TAE[CCSD]) / TAE[CCSD(T)]

where TAE[CCSD] and TAE[CCSD(T)] are the total atomization energies calculated at the CCSD and CCSD(T) levels, respectively.

The %TAE[(T)] diagnostic does not have a steep basis set dependence. Even CCSD(T)/cc-pVDZ energies give fairly useful estimates for post-CCSD(T) contributions, and CCSD(T)/cc-pVTZ energies result in %TAE[(T)] values that are close to the infinite basis set limit.[2]

Here is a worked example of how to calculate the %TAE[(T)] diagnostic for C2H4 at the CCSD(T)/cc-pVDZ level:

1. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ energies for the atoms:

ROHF/cc-pVDZ for H = –0.49980981 Hartree

ROCCSD/cc-pVDZ for C = –37.778529074216 Hartree

ROCCSD(T)/cc-pVDZ for C = –37.780664543434 Hartree

2. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ energies for the molecule:

CCSD/cc-pVDZ for C2H4 = –78.423790315994

CCSD(T)/cc-pVDZ for C2H4 = –78.438791852740

3. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ total atomization energies for the molecule:

TAE[CCSD] = 2×(–37.778529074216) + 4×(–0.49980981) – (–78.423790315994) = 0.86749293 Hartree = 544.4 kcal/mol

TAE[CCSD(T)] = 2×(–37.780664543434) + 4×(–0.49980981) – (–78.438791852740) = 0.87822353 Hartree = 551.1 kcal/mol

4. Calculate the %TAE[(T)] for the molecule:

%TAE[(T)] = 100×(TAE[CCSD(T)]–TAE[CCSD]) / TAE[CCSD(T)] = 100×(551.1–544.4) / 551.1 = 1.2%

This %TAE[(T)] value indicates that the CCSD(T) method can be safely used for C2H4.

1. %TAE[(T)] values smaller than 5% indicate that post-CCSD(T) contributions should not exceed 0.5 kcal/mol.

2. %TAE[(T)] values between 5–10% indicate that post-CCSD(T) contributions should generally not exceed 1.0 kcal/mol. In these cases the CCSD(T) method should be used with caution.

3. %TAE[(T)] values larger than 10% indicate that post-CCSD(T) contributions can certainly exceed 1.0 kcal/mol by significant amounts. In these cases the CCSD(T) method should not be used.

[1] A. Karton, E. Rabinovich, J. M. L. Martin, B. Ruscic. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. Journal of Chemical Physics 125, 144108 (2006). http://dx.doi.org/10.1063/1.2348881

[2] A. Karton, S. Daon, J. M. L. Martin. W4-11: A high-confidence dataset for computational thermochemistry derived from W4 ab initio data. Chemical Physics Letters 510, 165–178 (2011). http://dx.doi.org/10.1016/j.cplett.2011.05.007

[3] A. Karton. A computational chemist’s guide to accurate thermochemistry for organic molecules. Wiley Interdisciplinary Reviews: Computational Molecular Science, 6, 292–310 (2016). http://dx.doi.org/10.1002/wcms.1249

The %TAE[(T)] diagnostic is calculated from the atomic and molecular CCSD and CCSD(T) energies. It is defined as:

%TAE[(T)] = 100 × (TAE[CCSD(T)]–TAE[CCSD]) / TAE[CCSD(T)]

where TAE[CCSD] and TAE[CCSD(T)] are the total atomization energies calculated at the CCSD and CCSD(T) levels, respectively.

The %TAE[(T)] diagnostic does not have a steep basis set dependence. Even CCSD(T)/cc-pVDZ energies give fairly useful estimates for post-CCSD(T) contributions, and CCSD(T)/cc-pVTZ energies result in %TAE[(T)] values that are close to the infinite basis set limit.[2]

**How to calculate %TAE[(T)] values**Here is a worked example of how to calculate the %TAE[(T)] diagnostic for C2H4 at the CCSD(T)/cc-pVDZ level:

1. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ energies for the atoms:

ROHF/cc-pVDZ for H = –0.49980981 Hartree

ROCCSD/cc-pVDZ for C = –37.778529074216 Hartree

ROCCSD(T)/cc-pVDZ for C = –37.780664543434 Hartree

2. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ energies for the molecule:

CCSD/cc-pVDZ for C2H4 = –78.423790315994

CCSD(T)/cc-pVDZ for C2H4 = –78.438791852740

3. Calculate the CCSD/cc-pVDZ and CCSD(T)/cc-pVDZ total atomization energies for the molecule:

TAE[CCSD] = 2×(–37.778529074216) + 4×(–0.49980981) – (–78.423790315994) = 0.86749293 Hartree = 544.4 kcal/mol

TAE[CCSD(T)] = 2×(–37.780664543434) + 4×(–0.49980981) – (–78.438791852740) = 0.87822353 Hartree = 551.1 kcal/mol

4. Calculate the %TAE[(T)] for the molecule:

%TAE[(T)] = 100×(TAE[CCSD(T)]–TAE[CCSD]) / TAE[CCSD(T)] = 100×(551.1–544.4) / 551.1 = 1.2%

This %TAE[(T)] value indicates that the CCSD(T) method can be safely used for C2H4.

**How to interpret %TAE[(T)] values**1. %TAE[(T)] values smaller than 5% indicate that post-CCSD(T) contributions should not exceed 0.5 kcal/mol.

2. %TAE[(T)] values between 5–10% indicate that post-CCSD(T) contributions should generally not exceed 1.0 kcal/mol. In these cases the CCSD(T) method should be used with caution.

3. %TAE[(T)] values larger than 10% indicate that post-CCSD(T) contributions can certainly exceed 1.0 kcal/mol by significant amounts. In these cases the CCSD(T) method should not be used.

**References**[1] A. Karton, E. Rabinovich, J. M. L. Martin, B. Ruscic. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. Journal of Chemical Physics 125, 144108 (2006). http://dx.doi.org/10.1063/1.2348881

[2] A. Karton, S. Daon, J. M. L. Martin. W4-11: A high-confidence dataset for computational thermochemistry derived from W4 ab initio data. Chemical Physics Letters 510, 165–178 (2011). http://dx.doi.org/10.1016/j.cplett.2011.05.007

[3] A. Karton. A computational chemist’s guide to accurate thermochemistry for organic molecules. Wiley Interdisciplinary Reviews: Computational Molecular Science, 6, 292–310 (2016). http://dx.doi.org/10.1002/wcms.1249