5.10 Asymptotically Corrected Exchange-Correlation Potentials

5.10.1 LB94 Scheme

An asymptotically corrected (AC) exchange potential proposed by van Leeuwen and Baerends isvanLeeuwen:1994

vxLB=-β(x21+3βsinh-1(x)) (5.57)

where x=^ρ/ρ4/3 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:vanLeeuwen:1994

vxcLB94=vxcLDA+vxLB. (5.58)

The parameter β in Eq. (5.57) 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. (5.57) does not come from the functional derivative of an exchange energy functional, the Levy-Perdew virial relationLevy:1985 is used instead to obtain the exchange energy:

ExLB=-vxLB[ρ](𝐫)[3ρ(𝐫)+𝐫^ρ(𝐫)]𝑑𝐫 (5.59)

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

       Sets the β parameter for the LB94 XC potential
       n Corresponding to β=n/10000.
       Use the default.

Example 5.21  Applications of LB94 XC potential to N2 molecule.

   TDLDA/LB94 calculation is performed for excitation energies.

   0 1
   N    0.0000    0.0000    0.0000
   N    1.0977    0.0000    0.0000

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