12.6 Projector-Based Density Embedding

Conventional super-molecular KS-DFT calculations contain several non-additive terms consisting of kinetic energies and exchange-correlation effects. These non-additive terms are eliminated by applying a level-shift projection operator to maintain orthogonality between fragments when performing a 1-in-2 energy calculation.

The process begins by performing a KS-DFT calculation on the full system, or super-molecule. The KS-DFT calculation is then repeated for fragment 1-in-2, which is fragment 1 in the presence of the localized MOs of fragment 2 taken from the super-molecular full calculation. The Fock Matrix for this calculation is constructed as follows,

$${𝐟}^{(1)}=𝐡+𝐉[{\gamma }^{(1)}+{\gamma }^{(2)}]-{\nu }_{xc}[{\gamma }^{(1)}+{\gamma }^{(2)}]+\mu {𝐏}^{(2)}$$

(12.79)

where P is the level-shift projection operator constructed as:

$${𝐏}_{\alpha \beta }^{(2)}={[𝐒{\gamma }^{(2)}𝐒]}_{\alpha \beta }$$

(12.80)

where ${\gamma }^{(2)}$ is the localized density of fragment 2, and S is the AO overlap matrix. Upon convergence, an energy correction term is added to the final energy to account for the level-shift projection operator contribution to the Fock Matrix energy. The correction term is calculated as the following:

$$E_{\mathrm{correction}}=\mu *tr({\gamma }^{(1)}{𝐏}^{(2)})$$

(12.81)

Once the KS-DFT energy of fragment 1-in-2 is computed, a post KS method can be applied to this converged density to obtain the high-level QM additive energy of fragment 1. The same procedure can be repeated for fragment 2-in-1, without continuing to a post-DFT method to yield the low-level QM additive energy of the fragment 2. These energies are then summed to yield the total energy.

To use density embedding on a system, one must split the super-molecular system into two fragments indexed 1 and 2, and set EMBEDMAN to 1. This is done through the standard Q-Chem fragment input syntax. Two separate jobs must be run to find the total energy of fragment 1-in-2 at a high level QM theory, and fragment 2-in-1 at a low level QM theory. The order of the fragments in the $molecule section determines which fragment will undergo the high level QM. The user must submit a separate job for the 2-in-1 low-level QM calculation, with the order of the fragments reversed and EMBED_THEORY set to 0, which is the default value. The user must then add the final energies of the calculations to determine the total QM/QM embedded energy.

For the current Q-Chem implementation of density embedding, it is necessary to specify the basis as MIXED, which requires to define the basis for each individual atom. When using CCSD(T), one should specify CCMAN2 as TRUE, for Q-Chem’s most updated coupled-cluster code. The current implementation of density embedding only works in combination with the following settings: SCF_ALGORITHM = DIIS, INCFOCK = 0, and PURECART = 222. It is also recommended that users disable symmetry for calculations with SYMMETRY = FALSE, and SYM_IGNORE = TRUE. Refer to the sample input for correct job settings.

$molecule
   0 1
--
   0 1
   He      -4.63032        2.10289       -1.62399
--
   0 1
   F       -8.00612        1.74605       -1.25581
   H       -7.40964        1.84462       -0.47756
$end

$rem
   EXCHANGE        PBE
   CORRELATION     PBE
   BASIS           mixed  ! Must specify basis sets per atom in $basis section below
   PURECART        222
   MAX_SCF_CYCLES  100
   SCF_ALGORITHM   DIIS
   INCFOCK         0
   SYMMETRY        false
   SYM_IGNORE      true
   CC_SYMMETRY     false
   EMBEDMAN        1      ! Turning density embedding on
   EMBED_THEORY    2      ! Running MP2 on fragment 1 in 2, or He in FH
   EMBED_MU        7      ! Default value
   EMBED_THRESH    500    ! Default value for assigning MOs to fragments
$end

$basis
  he 1
  6-31G
  ****
  f 2
  6-31G
  ****
  h 3
  6-31G
  ****
$end

@@@

$molecule
0 1
--
0 1
   F       -8.00612        1.74605       -1.25581
   H       -7.40964        1.84462       -0.47756
--
0 1
   He      -4.63032        2.10289       -1.62399
$end

$rem
  EXCHANGE        PBE
  CORRELATION     PBE
  BASIS           mixed
  PURECART        222
  MAXSCF          100
  SCF_ALGORITHM   DIIS
  INCFOCK         0
  SYMMETRY        false
  SYM_IGNORE      true
  EMBEDMAN        1
  EMBED_THEORY    0      ! There will be no post-DFT calculations for HF-in-He
$end

$basis
  f 1
  6-31G
  ****
  h 2
  6-31G
  ****
  he 3
  6-31G
  ****
$end