# Energy Decomposition Analysis (EDA)¶

## Energy Decomposition Analysis of Ammonia Borane¶

This tutorial explains how to perform an energy decomposition analysis (EDA) [1] of Ammonia borane, in which ammonia and borane have a donor-acceptor interaction with each other. Hereby, the user can examine which interactions result in a stable complex.

The bond energy of H_{3}N-BH_{3} is defined as:

in which E_{NH3} and E_{BH3} are the energies of the optimized reactants, and E_{H3N-BH3} is the energy of the optimized complex. The bond energy consists of the preparation energy, also known as the strain energy, and the interaction energy:

The preparation energy is the amount of energy that is required to deform the NH_{3} and BH_{3} from their equilibrium structure to the geometry they have in the complex. The interaction energy is the change in energy when the prepared fragments (NH_{3} and BH_{3} in the complex geometry) are combined to form the complex. A quantitative energy decomposition analysis (EDA) divides the interaction energy in the electrostatic interaction, Pauli repulsion, and attractive orbital interactions:

The electrostatic interaction, which is usually attractive, is the energy between the unperturbed charge distributions of the prepared fragments. The Pauli repulsion is responsible for steric repulsion, it consists of the destablilizing interactions between occupied orbitals of the fragments. The orbital interaction accounts for charge transfer and polarization.

If you want to do an EDA with unrestricted fragments, see the following tutorial instead.

### Geometry Optimization¶

The geometry of the reactants and the product (complex) must be optimized. Note that another basis set and functional can be used as well. Symmetry will be used, which may help in the analysis.

**Perform the following steps for**NH_{3}, BH_{3}and H_{3}N-BH_{3}:**3.**Select**Task**→**Geometry Optimization****4.**Select**XC functional**→**GGA:BP86****5.**Select**Basis set**→**TZP****5.**Select**Numerical quality**→**Good****7.**Run the calculation with**File**→**Run**(give an appropriate name to your calculations)**8.**When the run has finished, click ‘Yes’ to import the optimized coordinates and save it

The bond energy of H_{3}N-BH_{3} can be calculated by subtracting the energies of the reactants, NH_{3} and BH_{3} from the energy of the complex. The energies can be found at the bottom of the logfile or in the output.

The bond energy of H_{3}N-BH_{3} was calculated to be -31.80 kcal/mol with these settings. (-838.47 - (-444.53) - (-362.14) = -31.80)

### EDA¶

Single point calculations of the prepared fragments NH_{3} and BH_{3} are needed to perform an EDA. In the GUI, single points of the fragments will be computed automatically when you follow the next steps for a fragment analysis:

- Perform a new calculation using the geometry of the
**optimized complex**(H_{3}N-BH_{3}):**1.**Select**Task**→**Single Point****2.**Select**XC functional**→**GGA:BP86****3.**Select**Basis set**→**TZP****4.**Select**Numerical quality**→**Good**

**8.**Go in the panel bar to**MultiLevel**→**Fragments****9.**Check the ‘Use fragments’ check box**10.**Run the calculation

When doing calculations on a cluster, single point calculations of NH_{3} and BH_{3} in their prepared geometry must be performed at first. The adf.rkf files of the single point computations are needed for the fragment analyses.

### Analysis¶

The energy decomposition of the interaction energy of H_{3}N-BH_{3} can be found in the output file of the fragment analysis. You can find for instance that H_{3}N-BH_{3} has an interaction energy of -44.63 kcal/mol, which consists of a Pauli repulsion of 108.10 kcal/mol, an electrostatic interaction of -77.53 kcal/mol, and an orbital interaction of -75.19 kcal/mol. This results in an interaction energy of -44.63 kcal/mol.

The orbital interaction is decomposed into contributions from different irreducible representations of the molecular point group. In this case the contributions from A1 (\(\sigma, \sigma^*\)-orbitals) are much more important than those from E1 (\(\pi, \pi^*\)-orbitals).

The bond energy was calculated previously to be -31.80 kcal/mol and accordingly the preparation energy can be calculated to be 12.83 kcal/mol (=-31.80-(-44.63)).
The structure of the fully relaxed NH_{3} is only slightly different from the structure of the NH_{3} fragment in H_{3}N-BH_{3}.
However, the structure of the fully relaxed flat BH_{3} (symmetry D_{3h}) is substantially different than the structure of the BH_{3} trigonal pyramidal fragment in H_{3}N-BH_{3} (symmetry C_{3v}):

The preparation energy for NH_{3} is only 0.11 kcal/mol, the difference between the bond energy of the NH_{3} fragment (-444.42 kcal/mol) and the fully relaxed NH_{3} molecule (-444.53 kcal/mol).
Due to the relatively large structural changes the preparation energy for BH_{3} is much larger, namely 12.72 kcal/mol, the difference between the bond energy of the BH_{3} trigonal pyramidal fragment (-349.42 kcal/mol) and the fully relaxed planar BH_{3} molecule (-362.14 kcal/mol).
Note that the energies of the fragments can be found at the bottom of the logfile or in the output of the fragments.

With **AMSlevels** the molecular orbital diagram can be visualized, in which one can see a donor-acceptor interaction and (repulsive) interactions between occupied orbitals.

**1.****SCM → Levels****2.**In AMSlevels, click on**View → Labels → Show**

By right clicking the BH_{3} 2A1 fragment orbital (LUMO of BH_{3}, acceptor orbital) one can select the corresponding SFO (fragment orbital), which can be visualized with AMSview.
Similar one can select the NH_{3} 2A1 fragment orbital (HOMO of NH_{3}, donor orbital).
After some manipulations, using 50% opacity, one can get the following two AMSview windows that show these fragment orbitals.

## EDA with unrestricted fragments¶

This tutorial explains how to perform an energy decomposition analysis (EDA) of a molecule with unrestricted fragments, for example the CH_{3} groups of ethane. Hereby, the user can examine which interactions result in the stable molecule. To see an explanation of these different interactions or to do an EDA without unrestricted fragments, see the EDA of Ammonia Borane tutorial.

### Geometry Optimization¶

The geometry of the reactants and the product (complex) must be optimized. Note that another basis set and functional can be used as well. . Symmetry will be used, which may help in the analysis.

For ethane:

**3.**Select**Task**→**Geometry Optimization****4.**Select**XC functional**→**GGA:BP86****5.**Select**Basis set**→**TZP****5.**Select**Numerical quality**→**Good****7.**Run the calculation with**File**→**Run**(give an appropriate name to your calculations)**8.**When the run has finished, click ‘Yes’ to import the optimized coordinates and save it

For CH_{3} (methyl) an unrestricted calculation is needed:

**3.**Select**Task**→**Geometry Optimization****4.**Check the**Unrestricted**box.**5.**Enter`1.0`

as Spin polarization**6.**Select**XC functional**→**GGA:BP86****7.**Select**Basis set**→**TZP****8.**Select**Numerical quality**→**Good****9.**Run the calculation with**File**→**Run**(give an appropriate name to your calculations)**10.**When the run has finished, click ‘Yes’ to import the optimized coordinates and save it

The bond energy of C_{2}H_{6} can be calculated by subtracting the energies of the reactants (two CH_{3}) from the energy of the complex. The energies can be found at the bottom of the logfile or in the output.

The bond energy of C_{2}H_{6} was calculated to be -93.53 kcal/mol with these settings. (-920.23 - 2*(-413.35)) = -93.53)

### EDA¶

Single point calculations of the CH_{3} groups in the geometry they have in the product are needed to perform an EDA.
Note that one sometimes need to change the electron configuration of the fragments to make them so called ‘prepared for bonding’ in order to minimize the Pauli repulsion in the electron pair bond. This is not needed in this simple example.

**1.**Perform a new calculation with the optimized molecule, ethane**2.**Select the Task**Single Point****3.**Check the**Unrestricted**box.**4.**Select**XC functional**→**GGA:BP86****5.**Select**Basis set**→**TZP****6.**Select**Numerical quality**→**Good**

**10.**Go in the panel bar to**MultiLevel**→**Fragments****11.**Check the ‘Use fragments’ check box (A warning will popup regarding NOSYM symmetry)**12.**Enter as spin`1`

for one of the fragments and`-1`

for the other fragment

The symmetry has been adjusted to NOSYM. However, we want to use symmetry.
Besides setting the symmetry to AUTO, in this case we also need to symmetrize the coordinates again,
since using fragments will lower the symmetry of ethane that ADF can use from D_{3d} to C_{3v}.
Note that in this case the symmetrization will only reorient the geometry in order to fulfill the molecular orientation requirements in ADF such that ADF can use symmetry.

### Analysis¶

The different energies, where the interaction energy of the CH_{3} groups of ethane consists of, are noted in the output.
It can be noticed that the interaction energy of -111.41 kcal/mol is build out of 180.02 kcal/mol Pauli repulsion, -125.43 kcal/mol electrostatic interaction, and -166.02 kcal/mol orbital interactions.

The orbital interaction is decomposed into contributions from different irreducible representations of the molecular point group. In this case the contributions from A1 (\(\sigma, \sigma^*\)-orbitals) are much more important than those from E1 (\(\pi, \pi^*\)-orbitals).

The bond energy was calculated previously to be -93.53 kcal/mol and accordingly the preparation energy can be calculated to be 17.88 kcal/mol (=-93.53-(-111.41)).
The preparation energy for one CH_{3} fragment is 8.95 kcal/mol, the difference between the bond energy of the CH_{3} trigonal pyramidal fragment (-404.40 kcal/mol) and the fully relaxed planar CH_{3} molecule (-413.35 kcal/mol).
The preparation energy of the other CH_{3} fragment is the same.
Note that the energies of the fragments can be found at the bottom of the logfile or in the output of the fragments.

With AMSlevels the molecular orbital diagram can be visualized, in which one can see an electron-pair bond and (repulsive) interactions between occupied levels.

[1] | F.M. Bickelhaupt, E.J. Baerends J.P., Kohn-Sham Density Functional Theory: Predicting and Understanding Chemistry, Reviews in Computational Chemistry 15, 1 (2000) |