5.12.3 Thermally-Assisted-Occupation DFT (TAO-DFT)

Aiming to study the ground-state properties of large, strongly correlated systems with minimum computational complexity, Prof. Jeng-Da Chai recently developed thermally-assisted-occupation density functional theory (TAO-DFT). ²²² Chai J.-D.
J. Chem. Phys.
(2012), 136, pp. 154104. Link Unlike conventional multi-reference methods, the computational complexity of TAO-DFT increases very insignificantly with the size of the active space (i.e., an active space restriction is not needed for TAO-DFT calculations), and TAO-DFT appears to be very promising for the study of large poly-radical systems. TAO-DFT is a DFT scheme with fractional orbital occupations produced by the Fermi-Dirac distribution, controlled by a fictitious temperature $\theta$, and existing XC functionals (e.g., LDA ²²² Chai J.-D.
J. Chem. Phys.
(2012), 136, pp. 154104. Link , GGAs ²²³ Chai J.-D.
J. Chem. Phys.
(2014), 140, pp. 18A521. Link , global hybrid GGAs ²²⁴ Chai J.-D.
J. Chem. Phys.
(2017), 146, pp. 044102. Link or range-separated hybrid GGAs ²²⁴ Chai J.-D.
J. Chem. Phys.
(2017), 146, pp. 044102. Link ^{, 794} Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link ) can be used in TAO-DFT. The computational cost of the method is similar to that of Kohn-Sham DFT for single-point energy calculations and analytical nuclear gradients, and reduces to the cost of Kohn-Sham DFT in the absence of strong static correlation effects.

There are several $rem variables that are used for TAO-DFT.

Note that setting TAO_DFT_THETA = 0 recovers ordinary Kohn-Sham DFT. ²²² Chai J.-D.
J. Chem. Phys.
(2012), 136, pp. 154104. Link In addition to the XC functional, a functional $E_{\theta }[\rho ]$ is needed in TAO-DFT. Currently available in Q-Chem are an LDA version ²²² Chai J.-D.
J. Chem. Phys.
(2012), 136, pp. 154104. Link (the ETheta_LDA functional) as well as a version based on the gradient expansion approximation ²²³ Chai J.-D.
J. Chem. Phys.
(2014), 140, pp. 18A521. Link (GEA) (the ETheta_GEA functional), and the latter may be substituted for the former in the sample jobs below. Furthermore, a functional $E_{x,\theta }[\rho ]$ is also needed in TAO-DFT for global hybrid (GH) GGAs. Currently available in Q-CHEM is an LDA version (the EThetaX_LDA functional) ²²⁴ Chai J.-D.
J. Chem. Phys.
(2017), 146, pp. 044102. Link . Moreover, two functionals $E_{x,\theta }^{\text{SR}}[\rho ]$ and $E_{x,\theta }^{\text{LR}}[\rho ]$ are needed in TAO-DFT for range-separated hybrid (RSH) GGAs, and their LDA versions (the SR_ETHETAX_LDA and LR_ETHETAX_LDA functionals) are now available in Q-CHEM ⁷⁹⁴ Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link . Note that both $E_{x,\theta }^{\text{SR}}[\rho ]$ and $E_{x,\theta }^{\text{LR}}[\rho ]$ depend on the range-separation parameter $\omega$, and $E_{x,\theta }^{\text{SR}}[\rho ]$ reduces to $E_{x,\theta }[\rho ]$ when $\omega =0$. Therefore, the SR_ETHETAX_LDA functional with $\omega =0$ is another LDA version of $E_{x,\theta }[\rho ]$, denoted as ETHETAX19_LDA.

To improve the performance of TAO-DFT functionals, the B97-type functionals with the semi-empirical dispersion correction based on the D4 model are optimized in TAO-DFT using the system-independent fictitious temperatures, resulting in TAO-B97-D4, TAO-B97X-D4 and TAO-$\omega$B97X-D4 ⁷⁹⁴ Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link . With the constraint $\theta =0$, the B97-type RSH functional is also optimized in KS-DFT, resulting in KS-$\omega$B97X-D4 ⁷⁹⁴ Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link (see Example 5.6.3 for the usage).