5.12.3 Thermally-Assisted-Occupation (TAO) DFT‣ 5.12 Unconventional DFT Methods ‣ Chapter 5 Density Functional Theory ‣ Q-Chem 6.0 User’s Manual

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). ¹⁹⁴ 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 ¹⁹⁴ Chai J.-D.
J. Chem. Phys.
(2012), 136, pp. 154104. Link , GGAs ¹⁹⁵ Chai J.-D.
J. Chem. Phys.
(2014), 140, pp. 18A521. Link or global hybrid GGAs ¹⁹⁶ Chai J.-D.
J. Chem. Phys.
(2017), 146, pp. 044102. 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.

TAO_DFT

TAO_DFT
       Controls whether to use TAO-DFT.
TYPE:
       Boolean
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not use TAO-DFT TRUE Use TAO-DFT
RECOMMENDATION:
       NONE

TAO_DFT_THETA

TAO_DFT_THETA
       The parameter $m$ (the mantissa) for the value of the fictitious temperature $\theta=m\times 10^{-n}\ E_{h}$ in TAO-DFT.
TYPE:
       INTEGER
DEFAULT:
       7
OPTIONS:
       $m$ Customize the mantissa for the fictitious temperature.
RECOMMENDATION:
       NONE

TAO_DFT_THETA_NDP

TAO_DFT_THETA_NDP
       The parameter $n$ (the exponent) for the value of the fictitious temperature $\theta=m\times 10^{-n}\ E_{h}$ in TAO-DFT.
TYPE:
       INTEGER
DEFAULT:
       3
OPTIONS:
       $n$ Customize the exponential power for the fictitious temperature.
RECOMMENDATION:
       NONE

Note that setting TAO_DFT_THETA = 0 recovers ordinary Kohn-Sham DFT. ¹⁹⁴ 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 ¹⁹⁴ 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 ¹⁹⁵ 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 GGAs. Currently available in Q-CHEM is an LDA version (the EThetaX_LDA functional). ¹⁹⁶ Chai J.-D.
J. Chem. Phys.
(2017), 146, pp. 044102. Link

Example 5.28 TAO-LDA calculation on Be atom

$molecule
   0 1
   Be
$end

$rem
   BASIS               6-31G*
   EXCHANGE            gen
   TAO_DFT             true
   TAO_DFT_THETA       7    ! default, theta=7 mhartree
   TAO_DFT_THETA_NDP   3    ! default
$end

$xc_functional
   X   S             1.0
   C   PW92          1.0
   X   ETheta_LDA    1.0
$end

Example 5.29 TAO-PBE, spin-restricted calculation on stretched N ${}_{2}$

$molecule
   0 1
   N1
   N2 N1 4.5
$end

$rem
   BASIS               6-31G*
   EXCHANGE            gen
   TAO_DFT             true
   TAO_DFT_THETA       40  ! theta = 40 mhartree
   TAO_DFT_THETA_NDP   3
$end

$xc_functional
   X PBE         1.0
   C PBE         1.0
   X ETheta_LDA  1.0
$end

Example 5.30 TAO-PBE, spin-unrestricted calculation on stretched N ${}_{2}$

$molecule
   0 1
   N1
   N2 N1 5.0
$end

$rem
   JOBTYPE             opt
   UNRESTRICTED        true
   BASIS               6-31G*
   EXCHANGE            gen
   TAO_DFT             true
   TAO_DFT_THETA       40  ! theta = 40 mhartrees
   TAO_DFT_THETA_NDP   3   ! can omit this line
   SCF_GUESS           gwh
   SCF_GUESS_MIX       3   ! mix in 30% LUMO in alpha to break symmetry
$end

$xc_functional
   X PBE         1.0
   C PBE         1.0
   X ETheta_LDA  1.0
$end

Example 5.31 TAO-PBE0 calculation on H ${}_{2}$ molecule

$molecule
   0 1
   H1
   H2 H1 1.00
$end

$rem
   JOBTYPE             opt
   UNRESTRICTED        true
   BASIS               6-31G*
   EXCHANGE            gen
   TAO_DFT             true
   TAO_DFT_THETA       20  ! theta = 20 mhartrees
   TAO_DFT_THETA_NDP   3   ! can omit this line
   SCF_GUESS           gwh
   SCF_GUESS_MIX       3   ! mix in 30% LUMO in alpha to break symmetry
$end

$xc_functional
   X ETheta_LDA    1.00
   X EThetaX_LDA   0.25
   X HF            0.25
   X PBE           0.75
   C PBE           1.00
$end

Example 5.32 TAO-B3LYP calculation on H ${}_{2}$ molecule

$molecule
   0 1
   H1
   H2 H1 1.00
$end

$rem
   JOBTYPE             opt
   UNRESTRICTED        true
   BASIS               6-31G*
   EXCHANGE            gen
   TAO_DFT             true
   TAO_DFT_THETA       174 ! theta = 17.4 mhartrees
   TAO_DFT_THETA_NDP   4
   SCF_GUESS           gwh
   SCF_GUESS_MIX       3   ! mix in 30% LUMO in alpha to break symmetry
$end

$xc_functional
   X ETheta_LDA    1.00
   X EThetaX_LDA   0.20
   X HF            0.20
   X Slater        0.08
   X Becke88       0.72
   C LYP           0.81
   C VWN1RPA       0.19
$end

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5.12.3 Thermally-Assisted-Occupation (TAO) DFT