Bond energy analysis
********************

No special input keys are required. However, if you have open-shell
fragments, see the UNRESTRICTEDFRAGMENTS or FRAGOCCUPATIONS key.

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 bond energy decompositions and applications, see, for
example, Refs. [1] [2] [3] [12] [13] [14] [15] [16] [17] [18]. For a
discussion of forming electron pair bonding between open-shell
molecules, see Ref. [4]. For an ETS-NOCV analysis of the orbital
interaction, see Ref. [5].

In ADF2012 the calculation of the Pauli repulsion for meta-GGAs and
meta-hybrids was implemented. Note that for hybrids this had already
been implemented for closed-shell fragments. In ADF2012, for hybrids,
meta-GGAs, and meta-hybrids the calculation of the Pauli repulsion was
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.

In ADF2026, for calculations with non-atomic fragments, the bond
energy with respect to atomic create runs is printed in the output.
This bond energy is computed as the energy difference between the full
molecule and single atoms (create runs), where in the create runs the
atoms are usually calculated as spherical spin-restricted neutral
atoms.


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 - for example, 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. First, 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}). Second, the interaction
energy \Delta E_\text{int} between the prepared fragments.

   \Delta E = \Delta E_\text{prep} + \Delta E_\text{int} = \Delta
   E_\text{prep,geo} + \Delta E_\text{prep,el} + \Delta E_\text{int}

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

   \Delta E_\text{int} = \Delta V_\text{elst} + \Delta E_\text{Pauli}
   + \Delta E_\text{oi} = \Delta E^0 + \Delta E_\text{oi}

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 to 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. For 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 wave function 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.

   \Delta E_\text{oi} = \sum_\Gamma  \Delta E_{\text{oi},\Gamma}

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 the following reference: 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.

This 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
in an 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 bond 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 with respect to 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 cannot 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 workaround to calculate the total energy of a system:
calculate the total energies of the atomic fragments and add them to
the bonding energy. Because the 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. The
'bonding energy' of such an 'atom' will then be equal to the 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). Meta-GGA functionals,
scalar relativistic (ZORA) calculations, relativistic spin-orbit
calculations, electric fields, and QM/MM are not yet supported.

In particular, the requirements for 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)
===============================

Note:

  The use of the **symmetry keyword** is allowed (however, all the
  points of the grid will be used) and the code has been extended to
  **open-shell systems** (AMS2023). **Scalar relativistic calculations
  (ZORA)** are now allowed (AMS2026). Due to gauge dependence with
  ZORA, only interatomic interactions are calculated. Please note that
  spin-orbit calculations are not allowed with 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 is an energy
decomposition scheme for the **total energy** of the molecular system,
which can be seen as an alternative to the Bickelhaupt-Baerends
scheme, which is a **bond energy** decomposition scheme. A full IQA
decomposition is the default procedure (except for scalar ZORA
calculations), including both intra-atomic and interatomic
interactions.

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). 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 'noncovalent contribution' 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]). This last term also includes
a dispersion correction (if and only if dispersion is requested a
posteriori in the DFT calculation). Note that here, *noncovalent* is
to be understood as the sum of all contributions that are not covalent
(and not only London-type interactions). To summarize, when no
dispersion correction of the Grimme type is added, the *noncovalent*
is equal to the classical electrostatic part (previously coined
*ionic*), while when Grimme's D3 is used, *noncovalent* is the sum of
this ionic part and of the dispersion correction.

As already mentioned, a full IQA decomposition is automatically
performed, including intra-atomic terms. Choosing the **verbose mode**
allows you to get more details about the energy decomposition (all
terms are detailed in this case and described in the output itself).
This is now the default option.

Tip:

  If you need to extract relevant IQA terms along a PES Scan, see also
  the REG/IQA tutorial

You can activate the full IQA analysis 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).

[1] T. Ziegler and A. Rauk, *On the calculation of Bonding Energies by
    the Hartree Fock Slater method. I. The Transition State Method*,
    Theoretica Chimica Acta 46, 1 (1977)

[2] T. Ziegler and A. Rauk, *A theoretical study of the ethylene-metal
    bond in complexes between copper(1+), silver(1+), gold(1+),
    platinum(0) or platinum(2+) and ethylene, based on the Hartree-
    Fock-Slater transition-state method*, Inorganic Chemistry 18, 1558
    (1979)

[3] F.M. Bickelhaupt, N.M. Nibbering, E.M. van Wezenbeek and E.J.
    Baerends, *The Central Bond in the Three CN* Dimers NC_CN, CN-CN,
    and CN-NC: Electron Pair Bonding and Pauli Repulsion Effects*,
    Journal of Physical Chemistry 96, 4864 (1992)

[4] F.M. Bickelhaupt, M. Solà, C. Fonseca Guerra, *Highly polar bonds
    and the meaning of covalency and ionicity -- structure and bonding
    of alkali metal hydride oligomers*, Faraday Discussions 135, 451
    (2007)

[5] M. Mitoraj, A. Michalak and T. Ziegler, *A Combined Charge and
    Energy Decomposition Scheme for Bond Analysis*, Journal of
    Chemical Theory and Computation 5, 962 (2009)

[6] M. Mitoraj, A. Michalak and T. Ziegler, *On the Nature of the
    Agostic Bond between Metal Centers and Beta-Hydrogen Atoms in
    Alkyl Complexes. An Analysis Based on the Extended Transition
    State Method and the Natural Orbitals for Chemical Valence Scheme
    (ETS-NOCV)*, Organometallics 28, 3727 (2009)

[7] M. Mitoraj and A. Michalak, *Natural orbitals for chemical valence
    as descriptors of chemical bonding in transition metal complexes*,
    Journal of Molecular Modeling 13, 347 (2007)

[8] M.A. Blanco, A.M. Pendás and E. Francisco, *Interacting Quantum
    Atoms: a correlated energy decomposition scheme based on the
    quantum theory of atoms in molecules*, Journal of Chemical Theory
    and Computation 1, 1096 (2005).

[9] A.M. Pendás, M.A. Blanco and E. Francisco, *A molecular energy
    decomposition scheme for atoms in molecules*, Journal of Chemical
    Theory and Computation 2, 90 (2006).

[10] V. Tognetti and L. Joubert, *On Atoms-in-Molecules Energies from
     Kohn-Sham calculations*, ChemPhysChem 18, 2675 (2017).

[11] V. Tognetti and L. Joubert, *On the physical role of exchange in
     the formation of an intramolecular bond path between two
     electronegative atoms*, Journal of Chemical Physics 138, 024102
     (2013).

[12] K. Kitaura and K. Morokuma, *A new energy decomposition scheme
     for molecular interactions within the Hartree-Fock
     approximation*, International Journal of Quantum Chemistry 10,
     325 (1976)

[13] T. Ziegler and A. Rauk, *Carbon monoxide, carbon monosulfide,
     molecular nitrogen, phosphorus trifluoride, and methyl isocyanide
     as sigma donors and pi acceptors. A theoretical study by the
     Hartree-Fock-Slater transition-state method*, Inorganic Chemistry
     18, 1755 (1979)

[14] H. Fujimoto, J. Osamura and T. Minato, *Orbital interaction and
     chemical bonds. Exchange repulsion and rehybridization in
     chemical reactions*, Journal of the American Chemical Society
     100, 2954 (1978)

[15] S. Wolfe, D.J. Mitchell and M.-H. Whangbo, *On the role of steric
     effects in the perturbational molecular orbital method of
     conformational analysis*, Journal of the American Chemical
     Society 100, 1936 (1978)

[16] A.J. Stone and R.W. Erskine, *Intermolecular self-consistent-
     field perturbation theory for organic reactions. I. Theory and
     implementation; nucleophilic attack on carbonyl compounds*,
     Journal of the American Chemical Society 102, 7185 (1980)

[17] F. Bernardi, A. Bottoni, A. Mangini and G. Tonachini,
     *Quantitative orbital analysis of ab initio SCF=MO computations :
     Part II. Conformational preferences in H2N---OH and H2N---SH*,
     Journal of Molecular Structure: THEOCHEM 86, 163 (1981)

[18] P.J. van den Hoek and E.J. Baerends, *Chemical bonding at metal-
     semiconductor interfaces*, Applied Surface Science 41/42, 236
     (1989)

[19] J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza and A.M. Pendás,
     *Interacting Quantum Atoms - A review*, Molecules 25, 4028 (2020)
