5.10.2 LB94 Scheme‣ 5.10 Asymptotically Corrected Exchange-Correlation Potentials ‣ Chapter 5 Density Functional Theory ‣ Q-Chem 6.0 User’s Manual

An asymptotically corrected (AC) exchange potential proposed by van Leeuwen and Baerends is ¹²¹² van Leeuwen R., Baerends E. J.
Phys. Rev. A
(1994), 49, pp. 2421. Link

v_{x}^{\text{LB}}=-\beta\left(\frac{x^{2}}{1+3\beta\mbox{sinh}^{-1}(x)}\right)

(5.64)

where $x=\|\hat{\bm{\nabla}}\rho\|/\rho^{4/3}$ is the reduced density gradient. For an exponentially-decaying density, this potential reduces to $-1/r$ in the asymptotic region of molecular systems. The LB94 xc potential is formed by a linear combination of LDA XC potential and the LB exchange potential: ¹²¹² van Leeuwen R., Baerends E. J.
Phys. Rev. A
(1994), 49, pp. 2421. Link

v_{xc}^{\text{LB94}}=v_{xc}^{\text{LDA}}+v_{x}^{\text{LB}}\;.

(5.65)

The parameter $\beta$ in Eq. (5.64) was determined by fitting to the exact XC potential for Be atom. As mentioned in Refs. 180 Casida M. E. et al.
J. Chem. Phys.
(1998), 108, pp. 4439. Link and 499 Hirata S., Head-Gordon M.
Chem. Phys. Lett.
(1999), 314, pp. 291. Link , for TDDFT calculations, it is sufficient to include the AC XC potential for ground-state calculations followed by TDDFT calculations with an adiabatic LDA XC kernel. The implementation of the LB94 XC potential in Q-Chem takes this approach, using the LB94 XC potential for the ground state calculations, followed by a TDDFT calculation with an adiabatic LDA XC kernel. This TDLDA/LB94 approach has been widely applied to study excited-state properties of large molecules.

Since the LB exchange potential in Eq. (5.64) does not come from the functional derivative of an exchange energy functional, the Levy-Perdew virial relation ⁷¹⁴ Levy M., Perdew J. P.
Phys. Rev. A
(1985), 32, pp. 2010. Link is used instead to obtain the exchange energy:

E_{x}^{\text{LB}}=-\int v_{x}^{\text{LB}}[\rho](\textbf{r})\bigl{[}3\rho(% \mathbf{r})+\mathbf{r}\bm{\cdot}\hat{\bm{\nabla}}\rho(\mathbf{r})\bigr{]}\;d% \mathbf{r}

(5.66)

An LB94 calculation is requested by setting EXCHANGE = LB94 in the $rem section. Additional job control and examples appear below.

LB94_BETA

LB94_BETA
       Sets the $\beta$ parameter for the LB94 XC potential
TYPE:
       INTEGER
DEFAULT:
       500
OPTIONS:
       $n$ Corresponding to $\beta=n/10000$ .
RECOMMENDATION:
       Use the default.

Example 5.21 Applications of LB94 XC potential to N ${}_{2}$ molecule.

$comment
   TDLDA/LB94 calculation is performed for excitation energies.
$end

$molecule
   0 1
   N    0.0000    0.0000    0.0000
   N    1.0977    0.0000    0.0000
$end

$rem
   EXCHANGE      lb94
   BASIS         6-311(2+,2+)G**
   CIS_N_ROOTS   30
   RPA           true
$end

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5.10.2 LB94 Scheme