Bond energy analysis¶

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). In a symmetric molecule 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 alternative, which can also be used in asymmetric molecules, is the ETS-NOCV analysis, which decomposes the bonding interactions in the context of natural orbitals for chemical valence, see Refs [5] [6] [7].

The pure orbital interaction effect of forming electron pair bonding between open shell molecules can approximately be calculated with a bond energy analysis in which all virtual orbitals are removed from the fragments, see Ref. [4].

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 Journal of Computational Chemistry 22, 931 (2001).

Notes on double-hybrid functionals and MP2¶

Since ADF2019.3, double-hybrid density functionals have been implemented in ADF. These include contributions from virtual orbitals from a MP2 calculation. The evaluation of MP2 energies requires orbital energies as input, which, however, are not available for orthogonal fragments. Thus, whenever a double-hybrid calculation is carried out, the Bonding energy with respect to the individual fragments is calculated and printed out. However, all terms which rely on the orthogonal fragment density (e.g. Pauli repulsion terms) do not include the MP2 part of the double-hybrid.

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.

Interacting Quantum Atoms (IQA)¶

The interacting quantum atoms approach (IQA) has been developed by Angel Martín Pendás and coworkers in the framework of real-space partitions of the molecular space [8] [9] [19]. This can be seen as an alternative to the Bickelhaupt-Baerends’ bond energy decomposition scheme but for the total energy of the molecular system. In this new version, a full IQA decomposition is now proposed as the default procedure, including both intra-atomic and inter-atomic interactions (see below).

In a wavefunction context, the IQA QTAIM partition of the molecular energy leads exactly to:

\(E=\sum _A\left(T^A+E_{Ne}^{AA}+E_{eeCl}^{AA}+E_{eeXC}^{AA}\right)+\frac 1 2\sum _{A{\neq}B}\left(E_{NN}^{AB}+E_{eN}^{AB}+E_{Ne}^{AB}+E_{eeCl}^{AB}+E_{eeXC}^{AB}\right)\)

The first four terms of the right-hand side in the equation above correspond to intra-atomic (also coined ‘self’) terms for each atom ‘A’: the atomic kinetic energy (T^A), the interaction energy between the nucleus and the electron density inside basin A \((E_{Ne}^{AA})\), and the repulsion energy between electrons in A, decomposed into classical electrostatic (\(E_{eeCl}^{AA}\)) and exchange-correlation contributions (\(E_{eeXC}^{AA}\)).

The last sum in the equation corresponds to the interactions between each atom pair ‘AB’ (bonded or not bonded by a bond path): the repulsion energy between nuclei in A and B (\(E_{NN}^{AB}\)), the attraction of the electrons in A by the nucleus in B (\(E_{eN}^{AB}\)) the attraction of the electrons in B by the nucleus in A (\(E_{Ne}^{AB}\)), and the repulsion energy between electrons in A with those in B, which can be split into a classical electrostatic contribution (\(E_{eeCl}^{AB}\)) and an exchange-correlation one (\(E_{eeXC}^{AB}\)).

Within a Kohn-Sham DFT framework, the implementation of this partition is not straightforward (see ref. [10] for more details). In ADF2018 a partial energy decomposition limited to interatomic interactions was proposed. For any atom-atom pair ‘AB’, we evaluate:

\(E_{inter}^{AB}=E_{NN}^{AB}+E_{eN}^{AB}\left[\rho \right]+E_{Ne}^{AB}\left[\rho \right]+E_{eeCl}^{AB}\left[\rho \right]+E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\)

where \(E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\) corresponds to the full exact exchange (‘HF-like’) interacting energy between the two atoms. Tognetti and Joubert have shown that despite underlying approximations, this expression can be safely used at least for semi-quantitative purposes [11].

Three energetic terms are printed for each atom pair. The first one corresponds to the total interaction energy between the two atoms ( \(E_{inter}^{AB}\)). This energy is then split into two contributions: the ‘covalent part’ corresponds to the exchange energy between the two atomic basins ( \(E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\)), while the ‘ionic one’ is the sum of all remaining electrostatic contributions ( \(E_{NN}^{AB}+E_{eN}^{AB}+E_{Ne}^{AB}\left[\rho \right]+E_{eeCl}^{AB}\left[\rho \right]\)).

As already mentioned, a full IQA decomposition is now automatically performed, including intra-atomic terms (a detailed description of the implementation will be soon available). Choosing the verbose mode allows to give more details about the energy decomposition (all terms are detailed in this case and described in the output itself).

You can activate these atom-atom interactions via the IQA input block:

IQA
   Enabled Yes/No
   Print [normal | verbose]
   AtomsToDo integer_list
End

IQA

Type:	Block
Description:	Total energy decomposition based on the interacting quantum atoms (IQA) approach and using QTAIM real-space partition.

Enabled

Type:	Bool
Default value:	No
GUI name:	Calculate: Interacting Quantum Atoms (IQA)
Description:	Calculate the total energy decomposition using the interacting quantum atoms (IQA) approach and the QTAIM real-space partitioning.

Print

Type:	Multiple Choice
Default value:	normal
Options:	[normal, verbose]
Description:	Minimal output (default) or verbose mode (detailed energy decomposition)

AtomsToDo

Type:	Integer List
GUI name:	Include atoms
Description:	Define a subset of atoms for which the IQA atom-atom interactions are calculated (no intra-atomic terms). If left empty, all atoms will be included (full IQA).

References