12.7 Frozen-Density Embedding Theory based methods 12.7 Frozen-Density Embedding Theory based methods 12.8 Polarizable Embedding Model

12.7.1 FDE-ADC

FDE-ADC⁷⁵⁵ is a density embedding method based on the combination of the Algebraic Diagrammatic Construction scheme for the polarization propagator (ADC, Section 7.9) and Frozen-Density Embedding Theory (FDET). In this particular variant the subsystem A is represented by a wave function whereas subsystem B is described by a density. The FDE-ADC method uses the linearized FDET approximation.¹⁰⁴³

12.7.1.1 FDE-ADC Job Control

The FDE-ADC job control is accomplished in two sections, $rem and $fde. Enabling FDE-ADC, specification of the ADC method and other ADC job control parameters (thresholds, max. iterations etc.) should be set in the $rem section. FDE-ADC also supports the excited state analysis (STATE_ANALYSIS) carried out by the libwfa module.

The fragments are specified via the fragment descriptors (see Section 13) in the $molecule section, whereas the first fragment corresponds to the embedded species (A) while the second fragment represents the environment (B).

Note: The current implementation allows only for closed shell fragments.

FDE
       Turns density embedding on.
TYPE:
       BOOLEAN
DEFAULT:
       False
OPTIONS:
       True Perform an FDE-ADC calculation. False Don’t perform FDE-ADC calculation.
RECOMMENDATION:
       Set the $rem variable FDE to TRUE to start a FDE-ADC calculation.

METHOD
       Determines which FDE-ADC method should be used if FDE = True.
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       adc(2) Perform an FDE-ADC(2)-s calculation. adc(2)-x Perform an FDE-ADC(2)-x calculation. adc(3) Perform an FDE-ADC(3) calculation (potential constructed with MP(2) density). cvs-adc(2) Perform an FDE-ADC(2)-s calculation of core excitations. cvs-adc(2)-x Perform an FDE-ADC(2)-x calculation of core excitations. cvs-adc(3) Perform an FDE-ADC(3) calculation of core excitations.
RECOMMENDATION:
       None

The FDE-ADC job control with respect to embedding parameters is accomplished via options in the $fde input section. The format of the $fde section requires key and value pairs separated by a space character:

$fde
   <Keyword>  <parameter>
$end

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

The super-molecular expansion (SE) uses the full basis set of the super-system for calculations on each fragment. Because of the computational cost this option should only be used for small to medium sized super-systems. Note that for visualization of orbitals or densities SE only supports the generation of volumetric data via MAKE_CUBE_FILES (MolDen files are not supported, i.e. MOLDEN_FORMAT should be avoided).

The reassembling of density matrix⁷⁵⁵ (RADM) option allows for calculations on larger systems by only including the basis functions of the embedded species for the ADC calculation. RADM introduces an approximation for the construction of the embedding potential by using an artificially (but cheaply) constructed density matrix for subsystem A. With RADM, all regular options for visualization are supported (MAKE_CUBE_FILES and MOLDEN_FORMAT). The RADM option is the recommended choice for an FDE-ADC calculation.

Analogous to a regular DFT calculation in Q-Chem(by using METHOD) the exchange-correlation functional combination can either be selected with one keyword XC_Func,or by defining X_Func and C_Func (similar to EXCHANGE and CORRELATION).

T_Func
       Kinetic energy functional used for the construction of the embedding potential.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       TF Use Thomas-Fermi kinetic energy functional.
RECOMMENDATION:
       None

XC_Func
       Exchange-Correlation functional used for the construction of the embedding potential.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA exchange-correlation functionals available in Q-Chem.
RECOMMENDATION:
       Only use LDA or GGA-type functionals.

X_Func
       Exchange functional used for the construction of the embedding potential.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA exchange functionals available in Q-Chem.
RECOMMENDATION:
       Only use LDA or GGA-type functionals. XC_Func and X_Func are mutually exclusive.

C_Func
       Exchange-Correlation functional used for the construction of the embedding potential.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA correlation functionals available in Q-Chem.
RECOMMENDATION:
       Only use LDA or GGA-type functionals. XC_Func and C_Func are mutually exclusive.

Expansion
       Specifies which basis set expansion should be used.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       SE/super/supermolecular Supermolecular basis is used for both System A and B. RADM Use RADM approximation (see above).
RECOMMENDATION:
       SE should be used for testing purposes only since it is very expensive for large systems. Use the RADM approximation for larger systems.

rhoB_method
       Method to calculate the environment density (B).
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       HF Use Hartree-Fock method. DFT Use Density Functional Theory.
RECOMMENDATION:
       If DFT is specified, the respective exchange-correlation functional has to defined using the keyword XC_FUNC_B or X_FUNC_B and C_FUNC_B.

XC_Func_B
       Exchange-Correlation functional used for the environment DFT calculation.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA/global-hybrid-GGA exchange-correlation functionals available in Q-Chem.
RECOMMENDATION:
       None

X_Func_B
       Exchange functional used for the environment DFT calculation.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA exchange functionals available in Q-Chem.
RECOMMENDATION:
       XC_Func_B and X_Func_B are mutually exclusive.

C_Func_B
       Correlation functional used for the environment DFT calculation.
INPUT SECTION: $fde
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       All LDA/GGA correlation functionals available in Q-Chem.
RECOMMENDATION:
       XC_Func_B and C_Func_B are mutually exclusive.

PrintLevel
       Print level for FDE-ADC output.
INPUT SECTION: $fde
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 minimum print level 1 extended print level 2 maximum print level 3 max. print level and additional text files (densities, etc.)
RECOMMENDATION:
       None

12.7.1.2 Examples

Example 12.29 Input for a FDE-ADC(2)/cc-pVDZ calculation in supermolecular expansion on CO embedded in one water molecule.

$rem
   SYM_IGNORE           = true
   METHOD               = adc(2)
   EE_STATES            = 2
   BASIS                = cc-pvdz
   FDE                  = true
   MEM_STATIC           = 1024
   MEM_TOTAL            = 16000
   ADC_DAVIDSON_MAXITER = 100
   ADC_DAVIDSON_CONV    = 5
$end

$molecule
   0 1
--
   0 1
   C    -3.618090    1.376803   -0.020795
   O    -4.735683    1.525556    0.115023
--
   0 1
   O    -7.956372    1.485406    0.116792
   H    -6.992316    1.421133    0.177470
   H    -8.105846    2.442220    0.111599
$end

$fde
   T_Func       TF
   XC_Func      PBE
   expansion    super
   rhoB_method  HF
$end

12.7.1.3 FDE-ADC output

In general the FDE-ADC output indicates all important stages of the FDE-ADC calculation, which are:

1.

Generation of $\rho_{A}^{\mathrm{ref}}$ ,
2.

Generation of $\rho_{B}$ ,
3.

Construction of the embedding potential,
4.

Start of FDE-ADC calculation and
5.

Final FDE-ADC summary.

In the following table definitions of the terms printed to the output are collected. These quantities are printed for every state, i.e. for every $\rho_{A}^{I}(\mathbf{r})$ . In addition, the non-electrostatic interactions with respect to the reference density $\rho_{A}^{\mathrm{ref}}(\mathbf{r})$ are printed at the top of the FDE-ADC summary.

Subsystem Energies
Embedded system (A)	$\langle\Psi_{A}^{I}\|\hat{H}_{A}+v_{\mathrm{emb}}^{\mathrm{lin}}\|\Psi_{A}^{I}\rangle$
Environment (B)	$E_{B}=E_{v_{B}}^{\mathrm{HK}}[\rho_{B}]\;\text{or}\;E_{\mathrm{HF}}$
Electrostatic Interactions
rho_A <-> rho_B	$J_{\mathrm{int}}[\rho_{A},\rho_{B}]=\int\int\frac{\rho_{A}(\mathbf{r})\rho_{B}% (\mathbf{r}^{\prime})}{\|\mathbf{r}-\mathbf{r}^{\prime}\|}\mathrm{d}\mathbf{r}% \mathrm{d}\mathbf{r}^{\prime}$
rho_A <-> Nuc_B	$V_{B}^{\mathrm{nuc}}[\rho_{A}]=\int\rho_{A}(\mathbf{r})v_{B}(\mathbf{r})% \mathrm{d}\mathbf{r}$
rho_B <-> Nuc_A	$V_{A}^{\mathrm{nuc}}[\rho_{B}]=\int\rho_{B}(\mathbf{r})v_{A}(\mathbf{r})% \mathrm{d}\mathbf{r}$
Nuc_A <-> Nuc_B	$V_{N_{A}N_{B}}=\sum_{i}\sum_{j}\frac{Z_{i}Z_{j}}{\|R_{i}-R_{j}\|}$
Non-Electrostatic Interactions
non-additive E_xc	$E_{xc}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$
non-additive T_s	$T_{s}^{\mathrm{nad}}[\rho_{A}^{I},\rho_{B}]$
integrated v_xc nad	$\int\rho_{A}^{I}(\mathbf{r})v_{xc}^{\mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf% {r}$
integrated v_T nad	$\int\rho_{A}^{I}(\mathbf{r})v_{T}^{\mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf{r}$
Final FDE-ADC energies
Delta_Lin	$\int(\rho_{A}^{I}(\mathbf{r})-\rho_{A}^{\mathrm{ref}}(\mathbf{r}))v_{xc,T}^{% \mathrm{nad}}(\mathbf{r})\mathrm{d}\mathbf{r}$
Final Energy (A)	$E_{A}^{\mathrm{emb}}[\Psi_{A}^{I},\rho_{B}]=\langle\Psi_{A}^{I}\|\hat{H}_{A}\|% \Psi_{A}^{I}\rangle+J_{\mathrm{int}}[\rho_{A}^{I},\rho_{B}]+V_{B}^{\mathrm{nuc% }}[\rho_{A}^{I}]$
	$+E_{xc,T}^{\mathrm{nad}}[\rho_{A}^{\mathrm{ref}},\rho_{B}]+\Delta^{\mathrm{lin% }}[\rho_{A}^{I},\rho_{A}^{\mathrm{ref}},\rho_{B}]+V_{A}^{\mathrm{nuc}}[\rho_{B% }]+V_{N_{A}N_{B}}$
Final Energy (A+B)	$E_{A}^{\mathrm{emb}}[\Psi_{A}^{I},\rho_{B}]+E_{B}$

Table 12.9: Definition of output terms.

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12.7.1 FDE-ADC

12.7.1.1 FDE-ADC Job Control

12.7.1.2 Examples

12.7.1.3 FDE-ADC output