5.10 Asymptotically Corrected Exchange-Correlation Potentials

No GGA exchange functional can simultaneously produce the correct contribution to the exchange energy density and exchange potential in the asymptotic region of molecular systems,[van Leeuwen and Baerends(1994)] and existing GGA exchange-correlation (XC) potentials decay much faster than the correct $-1/r$ XC potential in the asymptotic region.[Casida and Salahub(2000)] High-lying occupied orbitals and low-lying virtual orbitals are therefore too loosely bound, and $-\varepsilon ^{}_{\rm HOMO}$ becomes far smaller than the ionization energy, despite the exact condition that these should be the same for the exact functional.[van Gisbergen et al.(1996)van Gisbergen, Osinga, Gritsenko, van Leeuwen, Snijders, and Baerends, Wu et al.(2003)Wu, Ayers, and Yang] Moreover, response properties may be poorly predicted from TDDFT calculations with GGA functionals.[van Gisbergen et al.(1996)van Gisbergen, Osinga, Gritsenko, van Leeuwen, Snijders, and Baerends] Long-range corrected hybrid DFT (LRC-DFT), described in Section 5.6, has greatly remedied this situation, but is more expensive that KS-DFT with GGA functionals due to the use of Hatree-Fock exchange. The asymptotic corrections described in this section are designed to remedy the same problems but within the GGA framework.

5.10.1 LB94 Scheme

An asymptotically corrected (AC) exchange potential proposed by van Leeuwen and Baerends is[van Leeuwen and Baerends(1994)]

$\begin{equation} \label{eq:vx_ LB} v_{x}^{\text {LB}} = -\beta \left( \frac{x^2}{1+3 \beta \mbox{sinh}^{-1}(x)}\right) \end{equation}$

(5.64)

where $x = \frac{ \vert \nabla \rho \vert }{\rho ^{4/3}}$ 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:[van Leeuwen and Baerends(1994)]

$\begin{equation} v_{xc}^{\text {LB94}} = v_{xc}^{\text {LDA}} + v_{x}^{\text {LB}} \; . \end{equation}$

(5.65)

The parameter $\beta$ in Eq. eq:vx_LB was determined by fitting to the exact XC potential for Be atom. As mentioned in Refs. Casida:1998 and Hirata:1999b, 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. eq:vx_LB does not come from the functional derivative of an exchange energy functional, the Levy-Perdew virial relation[Levy and Perdew(1985)] is used instead to obtain the exchange energy:

$\begin{equation} E_ x^{\text {LB}} = -\int v_ x^{\text {LB}}([\rho ],\textbf{r})[3\rho (\textbf{r})+\textbf{r}\nabla \rho (\textbf{r})]d\textbf{r} \end{equation}$

(5.66)

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

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 5.60 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
   JOBTYPE     = sp
   EXCHANGE    = lb94
   BASIS       = 6-311(2+,2+)G**
   CIS_N_ROOTS = 30
   RPA         = true
$end

5.10.2 Localized Fermi-Amaldi (LFA) Schemes

Another alternative, proposed by Pan, Fang and Chai,[Pan et al.(2013)Pan, Fang, and Chai] is to use a localized version of Fermi-Amaldi exchange-correlation functional. The resulting exchange density functional, whose functional derivative has the correct $-1/r$ asymptotic behavior, can be directly added to any semi-local density functional. Three variants of this method were proposed in Ref. Chai:2013b. The simplest of these, the strictly-localized Fermi-Amaldi (LFAs) scheme, is implemented in Q-Chem, for molecules consisting of atoms with $Z\le 55$ .

Example 5.61 LFAs-PBE single-point TD-DFT calculation with water molecule

$comment
   Use LFAs-PBE potential for ground-state calculations, followed by
   TDDFT calculations with an adiabatic PBE XC kernel.
$end

$molecule
   0  1
   O
   H1  O  oh
   H2  O  oh  H1  hoh

   oh  =   1.0
   hoh = 110.0
$end.

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

$xc_functional
   X   PBE   1.0
   C   PBE   1.0
   X   LFAs  1.0
$end