6.9 Energy Decomposition Analysis (EDA)

The interaction energy between molecules can be calculated with the supermolecular approach: one performs calculations for the supersystem and for the subsystems with size-consistent methods and derive the interaction energy $\Delta E$ by taking the energy difference. The energy decomposition analysis (EDA) allow a partitioning of the Hartree-Fock (HF) or DFT interaction energy in physically meaningful contributions: the classical electrostatic interaction $\Delta E_{\text{ele}}$, the exchange-repulsion $\Delta E_{\text{exrep}}$, the orbital relaxation energy $\Delta E_{\text{orb}}$ and additionally for DFT the correlation interaction $\Delta E_{\text{cor}}$: \[\begin{align} \Delta E_{\text{HF}}& = \Delta E_{\text{ele}} + \Delta E_{\text{exrep}} + \Delta E_{\text{orb}} ~~, \\ \Delta E_{\text{DFT}}& = \Delta E_{\text{ele}} + \Delta E_{\text{exrep}} + \Delta E_{\text{orb}} + \Delta E_{\text{cor}}. \end{align}\](6.21–6.22) Further details and derivations of the different energy contributions can be found in [202].

6.9.1 How to perform

The EDA scheme is implemented in the module ridft and can be done with RI-Hartree-Fock and with all local, gradient corrected, hybrid and meta density functionals (please note that the functionals included in the XCFun library are not supported!).

Calculation of the subsystems:
In HF and hybrid DFT calculations please insert $scfdenapproxl 0 in the control file, at least in a second run. The EDA scheme needs the exchange and Coulomb energies of every system separately. After the subsystem calculation you will find under $subenergy the different energy contribution to the total energy of the system: the one electron energy, Coulomb- and exchange energy, correlation energy in case of DFT calculations, nuclear repulsion energy and optionally the dispersion energy.
Preparation of the supersystem control file:
First run define for the supersystem and take for consistency the same basis set and the same method (i.e. the same functional and the same grid). Please use in case of DFT calculations not the multiple grids m3 to m5, because this would lead to erroneous orbital relaxation energies. If the subsystems are open-shell species the occupation in the EHT submenue of define of the supersystem must be chosen open-shell, too. For open-shell systems the Fermi-smearing is recommended. The sequence of the supersystem coordinates must have the same sequence as the subsystem coordinates. In the case of HF and hybrid DFT calculation use again $scfdenapproxl 0.
Then please insert in the control file :
```
           $subsystems 
             molecule#1 file=sub1/control
             molecule#2 file=sub2/control
        
```
If you use the supermolecular basis set for the calculation of the monomers please insert after $subsystems the option copo:
```
          $subsystems copo
             molecule#1 file=sub1/control
             molecule#2 file=sub2/control
        
```
It is possible to generate orthogonal product wave functions when you use opro instead of copo. But with this choice it is not possible to calculate the different energy contributions of the interaction energy.
You can choose at most ten subsystems.
Generation of the product molecular wave functions:
The module promowa generates RHF and UHF product molecular wave functions. The new (product) start vectors can be found in the files mos for closed-shell systems or in alpha and beta for open-shell systems. Please note that the molecular orbitals of the different subsystems are not orthogonal to each other.
Energy decomposition analysis:
After the supersystem ridft calculation you will find the following output with the different contributions of the interaction energy in Hartree:

                  ---------------------------------------------------- 
                 | * Total Interaction energy  =      -0.0058700698  |
                  ---------------------------------------------------- 
                 : * Electrostatic Interaction =      -0.0134898233  :
                 :             Nuc---Nuc       =      18.2843363825  :
                 :             1-electron      =     -36.5372802833  :
                 :             2-electron      =      18.2394540775  :
                 : * Exchange-Repulsion        =       0.0112325934  :
                 :             Exchange Int.   =      -0.0139477002  :
                 :             Repulsion       =       0.0251802936  :
                 : * Orbital Relaxation        =      -0.0036128399  :
                 .....................................................