The exact density embedding method adapted from the method of Manby, Miller,
and coworkers^{597} allows embedding calculations to extend beyond
electrostatic embedding. This embedding scheme allows for the fragmentation of
a system into two interacting subsystems, which can be treated at two different
levels of quantum mechanics (QM/QM), for example coupled cluster embedded in
DFT. This type of embedding fully accounts for polarization as well as quantum
mechanical exchange, as calculated from the super-molecular embedding density
and the exchange correlation functional used. The goal of this embedding
theory is to perform, say, a DFT calculation on fragment 1 in the presence of
fragment 2.

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,

$${\mathbf{f}}^{(1)}=\mathbf{h}+\mathbf{J}[{\gamma}^{(1)}+{\gamma}^{(2)}]-{\nu}_{xc}[{\gamma}^{(1)}+{\gamma}^{(2)}]+\mu {\mathbf{P}}^{(2)}$$ | (12.79) |

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

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

$${E}_{\mathrm{correction}}=\mu *tr({\gamma}^{(1)}{\mathbf{P}}^{(2)})$$ | (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.

EMBEDMAN

Turns density embedding on.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
Do not use density embedding.
1
Turn on density embedding.

RECOMMENDATION:

Use EMBEDMAN for QM/QM density embedded calculations.

EMBED_THEORY

Specifies post-DFT method performed on fragment one.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
No post HF method, only DFT on fragment one.
1
Perform CCSD(T) calculation on fragment one.
2
Perform MP2 calculation on fragment one.

RECOMMENDATION:

This should be 1 or 2 for the high-level QM calculation of fragment
1-in-2, and 0 for fragment 2-in-1 low-level QM calculation.

EMBED_MU

Specifies exponent value of projection operator scaling factor, $\mu $ [Eq.
(12.79) and (12.81)].

TYPE:

INTEGER

DEFAULT:

7

OPTIONS:

n
$\mu ={10}^{n}$.

RECOMMENDATION:

Values of 2 - 7 are recommended. A higher value of $\mu $ leads to better
orthogonality of the fragment MOs but $\mu >{10}^{7}$ introduces numerical
noise. $$ results in non-additive terms becoming too large. Energy
corrections are fairly insensitive to changes in $\mu $ within the range of
${10}^{2}-{10}^{7}$.

EMBED_THRESH

Specifies threshold cutoff for AO contribution used to determine which MOs
belong to which fragments

TYPE:

INTEGER

DEFAULT:

500

OPTIONS:

n
Threshold $=n/1000$

RECOMMENDATION:

Acceptable values range from 0 to 1000. Should only need to be tuned for
non-highly localized MOs

$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