# Bond energy analysis¶

No special input keys are required, except if one has open shell fragments, see key FRAGOCCUPATIONS.

ADF calculates various chemically meaningful terms that add up to the bond energy, with an adaptation of Morokuma’s bond energy decomposition to the Kohn-Sham MO method. The individual terms are chemically intuitive quantities such as electrostatic energy, Pauli repulsion, and orbital interactions. The latter are symmetry decomposed according to the Ziegler transition state method. For a discussion of bonding energy decompositions and applications see e.g. [3, 110, 112, 130-136]

In ADF2012 the calculation of the Pauli repulsion for metaGGA’s and metahybrids is implemented. Note that for hybrids this was already implemented before in case of closed shell fragments. In ADF2012 for hybrids, metaGGA’s, and metahybrids the calculation of the Pauli repulsion is also implemented if one is simulating an unrestricted fragment with the key FRAGOCCUPATIONS.

In ADF2012 for hybrids the exact exchange contribution to the Pauli term is isolated and the contributions to the orbital term are divided among orbital symmetries.

## Bond energy details¶

In the framework of Kohn-Sham MO theory and in conjunction with the fragment approach, one can decompose the bond energy between the fragments of a molecular system - say, a base and a substrate for E2 elimination - into contributions associated with the various orbital and electrostatic interactions. In ADF, we follow a Morokuma-type energy decomposition method. The overall bond energy \(\Delta E\) is divided into two major components. In the first place, the preparation energy \(\Delta E_\text{prep}\) corresponding to the amount of energy required to deform the separated fragments, A and B say, from their equilibrium structure to the geometry they acquire in the overall molecule ( \(\Delta E_\text{prep,geo}\) ), and to excite them to their valence electronic configuration ( \(\Delta E_\text{prep el}\) ). In the second place, the interaction energy \(\Delta E_\text{int}\) between the prepared fragments.

In the following step, the interaction energy \(\Delta E_\text{int}\) is further decomposed into three physically meaningful terms, which are printed in the ADF output file.

The term \(\Delta V_\text{elst}\) corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the prepared fragments as they are brought together at their final positions, giving rise to an overall density that is simply a superposition of fragment densities \(\rho_A + \rho_B\). (Note that we use the convention that energy terms containing potential energy only, kinetic energy only, or both kinetic and potential energy are indicated by V, T, and E, respectively.) For neutral fragments, \(\Delta V_\text{elst}\) is usually attractive. The Pauli repulsion \(\Delta E_\text{Pauli}\) arises as the energy change associated with going from \(\rho_A + \rho_B\) the wave function \(\Psi^0 = NA[\Psi_A \Psi_B]\) that properly obeys the Pauli principle through explicit antisymmetrization (*A* operator) and renormalization (*N* constant) of the product of fragment wave functions. It comprises the destabilizing interactions between occupied orbitals, and is responsible for any steric repulsion. In case of neutral fragments, it can be useful to combine \(\Delta V_\text{elst}\) and \(\Delta E_\text{Pauli}\) in a term \(\Delta E_0\) which, in the past, has been conceived as the steric interaction. However, we prefer to reserve the designation steric interaction or repulsion for \(\Delta E_\text{Pauli}\) because that is, as already mentioned, the only source of net repulsive interactions between molecular fragments. Finally, the wavefunction is allowed to relax from \(\Psi^0\) to the fully converged wave function \(\Psi\). The associated orbital interaction energy \(\Delta E_\text{oi}\) accounts for electron pair bonding, charge transfer (e.g., HOMO-LUMO interactions) and polarization (empty/occupied orbital mixing on one fragment due to the presence of another fragment). This can be further decomposed into the contributions from the distinct irreducible representations \(\Gamma\) of the interacting system using the extended transition state method. In systems with a clear \(\sigma\)/\(\pi\) separation, this symmetry partitioning proves to be very informative.

An extensive discussion of the physical meaning of all the terms in the energy decomposition is given in F.M. Bickelhaupt and E.J. Baerends, *Kohn-Sham Density Functional Theory: Predicting and Understanding Chemistry*, In: Rev. Comput. Chem.; Lipkowitz, K. B. and Boyd, D. B., Eds.; Wiley-VCH: New York, 2000, Vol. 15, 1-86.

Text is mostly taken from: Chemistry with ADF, G. te Velde, F.M. Bickelhaupt, E.J. Baerends, C. Fonseca Guerra, S.J.A. van Gisbergen, J.G. Snijders, T. Ziegler J. Comp. Chem. **22** (2001) 931.

## Total energy evaluation¶

ADF normally does not calculate the total energy of a system (the energy wrt bare nuclei and free electrons). However, ADF calculates the energy of the system with respect to fragment energies. By default, these fragments are the spherical spin-restricted neutral atoms, but one can also use larger fragments. For this reason total energies from other programs could not be compared to ADF directly. Note, however, that only energy difference comparisons are meaningful. These are the only energies that play a role in chemistry of course, and for this one does not need total energies.

If you really want to calculate the total energies, there are two options in ADF

**Total energy by adding the binding energy of the atoms**

There is a work-around to calculate the total energy of a system: calculate the total energies of the atomic fragments and add them to the bonding energy. Because total energy of an atom is, by definition, the energy difference between the atom and the (nucleus+free electrons) system one can calculate it by calculating a single atom with the charge equal to the number of electrons. ‘Bonding energy’ of such an ‘atom’ will then be equal to negative of the total energy of the atomic fragment. Care should be taken to apply this procedure to frozen-core fragments. In this case, it only makes sense to remove the valence electrons and leave the frozen core.

**TOTALENERGY keyword**

The total energies have not been tested extensively and should therefore be used with caution. In addition to bond energies it is now possible to compute total energies with ADF by including the keyword TOTALENERGY in the input. This work is in progress.

```
TOTALENERGY
```

The total energy will be computed for the chosen XC functional (LDA, GGA, hybrid functionals, or Hartree-Fock). MetaGGA functionals, (ZORA) scalar relativistic and relativistic spin-orbit calculations, electric fields and QM/MM are not supported yet.

In particular the requirements to the integration accuracy are somewhat higher than for bond energies. It is recommended to use an integration grid (BeckeGrid) of quality “Good”. If in doubt, a convergence test with respect to the integration accuracy is recommended.