5 Density Functional Theory

5.12 Thermally-Assisted-Occupation Density Functional Theory (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).162 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 θ, and existing XC functionals (e.g., LDA or GGAs) can be used in TAO-DFT.163 The computational cost of the method is similar to that of KS-DFT for single-point energy calculations and analytical nuclear gradients, and reduces to the cost of KS-DFT in the absence of strong static correlation effects.

There are several $rem variables that are used for 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
       The parameter m for the value of the fictitious temperature θ=m×10-nEh in TAO-DFT.
TYPE:
       INTEGER
DEFAULT:
       7
OPTIONS:
       m inθ=m×10-nEh, with the other parameter n being set by TAO_DFT_THETA_NDP.
RECOMMENDATION:
       NONE

TAO_DFT_THETA_NDP
       The parameter n for the value of the fictitious temperature θ=m×10-nEh in TAO-DFT.
TYPE:
       INTEGER
DEFAULT:
       3
OPTIONS:
       n θ=m×10-nEh, with the other parameter m being set by TAO_DFT_THETA.
RECOMMENDATION:
       NONE

Note that setting TAO_DFT_THETA = 0 recovers ordinary KS-DFT.162 In addition to the XC functional, a functional Eθ[ρ] is needed in TAO-DFT. Currently available in Q-Chem are an LDA version162 (the ETheta_LDA functional) as well as a version based on the gradient expansion approximation163 (GEA) (ETheta_GEA functional), and the latter may be substituted for the former in the sample jobs below.

Example 5.20  TAO-LDA calculation on Be atom

$molecule
   0 1
   Be
$end

$rem
   JOBTYPE             sp
   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 5.21  TAO-PBE, spin-restricted calculation on stretched N2

$molecule
   0 1
   N1
   N2 N1 4.5
$end

$rem
   JOBTYPE             sp
   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 5.22  TAO-PBE, spin-unrestricted calculation on stretched N2

$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
   GEN_SCFMAN          FALSE
$end

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