# 11.8 Polarizable Embedding Model

The polarizable embedding (PE) model is a fragment-based quantum-classical explicit embedding scheme to model molecular properties in complex heterogeneous environments 708, 709. The theory is explained thorougly in literature 708, 709, 588. In essence, the environment is represented by a multi-center multipole expansion to model electrostatic interactions, whereas polarization is taken into account by dipole-dipole polarizabilities placed at the expansion points. Polarization effects can thus be treated fully self-consistently by mutual polarization of the environment and the quantum region.

A recent tutorial review on how to prepare PE calculations in general (creating embedding potentials) is also available 915. For automated generation of embedding potentials, please refer to the PyFraME tool 1 which is also explained in the aforementioned review.

PE can be used for Hartree-Fock and density-functional theory ground-state SCF methods. In addition, PE has been combined with the algebraic-diagrammatic construction for the polarization propagator (ADC) 847, explained in the subsequent section.

The combined scheme of the PE model and ADC (PE-ADC) 847 is built on top of a PE-HF ground-state calculation and takes into account perturbative corrections of the excitation energies in a density-driven manner. That is, after the Hartree-Fock ground-state calculations, the induced dipole moments in the environment are kept frozen and an ADC calculation is performed as usual. Thereafter, perturbative corrections of the electronic excitation energies are calculated based on i) the transition density (perturbative linear-response-type correction, ptLR), and ii) the difference density (perturbative state-specific correction, ptSS) for each excited state.

## 11.8.0.2 PE Job Control

The PE job control is accomplished in two sections, $rem and$pe. To enabling PE-ADC, specification of the ADC method and other ADC job control parameters (thresholds, max. iterations etc.) should be set in the $rem section. PE-ADC supports the excited state analysis (STATE_ANALYSIS) carried out by the libwfa module. PE Turns PE on. TYPE: BOOLEAN DEFAULT: False OPTIONS: True Perform a PE calculation. False Don’t perform a PE calculation. RECOMMENDATION: Set the$rem variable PE to TRUE to start a PE calculation.

Note:  Turning PE on disables symmetry by setting SYM_IGNORE to TRUE.

Note:  Setting the REM variables USE_LIBQINTS and GEN_SCFMAN to TRUE is required to run PE.

The PE-specific options can be set in the $pe input section. The format of the$pe section requires key and value pairs separated by a space character:

$pe <keyword> <parameter>$end


Note:  The following job control variables belong only in the $pe section. Do not place them in the$rem section.

POTFILE
Path of the potential file.
INPUT SECTION: $pe TYPE: STRING DEFAULT: potfile.pot OPTIONS: Provide the path/name of the potential file. RECOMMENDATION: None DIIS Use DIIS acceleration to obtain induced moments. INPUT SECTION:$pe
TYPE:
BOOLEAN
DEFAULT:
TRUE
OPTIONS:
TRUE Turns DIIS acceleration on. FALSE Turns DIIS acceleration off (normal Jacobi solver is used).
RECOMMENDATION:
TRUE

CONVERGENCE_INDUCED
Threshold for induced moments convergence. Converge induced moments to a residual norm of $10^{-\text{CONVERGENCE\_INDUCED}}$.
INPUT SECTION: $pe TYPE: INTEGER DEFAULT: 8 Corresponding to $10^{-8}$ OPTIONS: $n\leq 12$ Corresponding to $10^{-n}$ RECOMMENDATION: Use the default unless higher accuracy is desired. MAXITER Maximum number of iterations for induced moments. INPUT SECTION:$pe
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
$n\geq 1$
RECOMMENDATION:
Use the default. If more iterations are required to converge the induced moments, there might be an error in the system setup.

BORDER
Activate border redistribution/removal options for sites in proximity to the QM/MM border.
INPUT SECTION: $pe TYPE: BOOLEAN DEFAULT: FALSE OPTIONS: TRUE Enable border options. FALSE Disable border options. RECOMMENDATION: None BORDER_TYPE Remove or redistribute multipole moments/polarizabilities. INPUT SECTION:$pe
TYPE:
STRING
DEFAULT:
REMOVE
OPTIONS:
REMOVE remove multipole moments/polarizabilities. REDIST redistribute multipole moments/polarizabilities.
RECOMMENDATION:
None

BORDER_RMIN
Minimum distance from QM atoms to MM sites to be taken into account for removal/redistribution
INPUT SECTION: $pe TYPE: FLOAT DEFAULT: 2.2 (AU) OPTIONS: $r>0$ (Unit depends on BORDER_RMIN_UNIT) RECOMMENDATION: None BORDER_RMIN_UNIT Unit of BORDER_RMIN, default is atomic units (AU) INPUT SECTION:$pe
TYPE:
STRING
DEFAULT:
AU
OPTIONS:
AU Use atomic units. AA Use Angstrom.
RECOMMENDATION:
None

BORDER_REDIST_ORDER
Order from which on moments are removed. For example, if set to 1 (default), only charges are redistributed and all higher order moments are removed.
INPUT SECTION: $pe TYPE: INTEGER DEFAULT: 1 OPTIONS: $n=0,1,2,3,...$ RECOMMENDATION: None BORDER_N_REDIST Number of neighbor sites to redistribute multipole moments/polarizabilities to. The default (-1) redistributes to all sites which are not in the border region. INPUT SECTION:$pe
TYPE:
INTEGER
DEFAULT:
-1
OPTIONS:
$n=-1,1,2,3,...,\text{number of MM sites}$
RECOMMENDATION:
Use the default value.

BORDER_REDIST_POL
Redistribute polarizabilities? If set to FALSE, polarizabilities are removed.
INPUT SECTION: $pe TYPE: BOOLEAN DEFAULT: FALSE OPTIONS: TRUE Redistribute polarizabilities. FALSE Remove polarizabilities. RECOMMENDATION: None Example 11.33 Input for a PE-HF calculation of 4-Nitroaniline in presence of six water molecules $molecule
0 1
C          8.64800        1.07500       -1.71100
C          9.48200        0.43000       -0.80800
C          9.39600        0.75000        0.53800
C          8.48200        1.71200        0.99500
C          7.65300        2.34500        0.05500
C          7.73200        2.03100       -1.29200
H         10.18300       -0.30900       -1.16400
H         10.04400        0.25200        1.24700
H          6.94200        3.08900        0.38900
H          7.09700        2.51500       -2.01800
N          8.40100        2.02500        2.32500
N          8.73400        0.74100       -3.12900
O          7.98000        1.33100       -3.90100
O          9.55600       -0.11000       -3.46600
H          7.74900        2.71100        2.65200
H          8.99100        1.57500        2.99500
$end$pe
potfile  gen_scfman_pe_potfile.pot
$end$comment
The corresponding potential file \texttt{gen\_scfman\_pe\_potfile.pot} can be
found in the samples folder.
$end$rem
METHOD               HF
BASIS                STO-3G
PE                   TRUE
SYM_IGNORE           TRUE
USE_LIBQINTS         TRUE
\$end


## 11.8.0.3 PE output

After SCF convergence, the PE module prints a summary of PE energy contributions, for example:

----------------------------------------------------------------------
Polarizable Embedding Summary:

Electrostatics:
Electronic:                   0.30901227399981
Nuclear:                      -0.32134940137969
Multipole:                    0.00000000000000
Total:                        -0.01233712737988

Polarization:
Electronic:                   -0.01817189734325
Nuclear:                      0.01717961961137
Multipole:                    -0.02091890381649
Total:                        -0.02191118154837

Total Energy:                    -0.03424830892825
----------------------------------------------------------------------


If a PE-ADC calculation is carried out, the perturbative corrections are printed together with the excitation energies:

  Excited state 1 (singlet, A)                                     [converged]
----------------------------------------------------------------------------
Term symbol:  2 (1) A                                     R^2 =  1.84764e-13

Total energy:                                           -483.3704138865 a.u.
Excitation energy:                                               3.906651 eV
-------------------------------------------
PE ptSS energy correction:                                      -0.001804 eV
Corrected Excitation Energy (ptSS):                              3.904847 eV
-------------------------------------------
PE ptLR energy correction:                                      -0.000096 eV
Corrected Excitation Energy (ptLR):                              3.906554 eV
-------------------------------------------