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.
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.
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.
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 grid points are used), and the code supports open-shell systems (AMS2023), as well as both scalar relativistic (ZORA, AMS2026) and spin-orbit (SO, AMS 2027) calculations. ZORA IQA calculations are restricted to interatomic IQA terms only because of gauge dependence. For reliable IQA energies, use at least Good numerical quality.
The interacting quantum atoms approach (IQA) was developed by Angel Martín Pendás and coworkers within the framework of real-space partitions of molecular space [8] [9] [19]. It is an energy decomposition scheme for the total energy of a molecular system and can be viewed as an alternative to the Bickelhaupt-Baerends 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 to the following exact expression:
\(E=\sum _A\left(T^A+E_{Ne}^{AA}+E_{eeCl}^{AA}+E_{eeXC}^{AA}\right)+\frac 1 2\sum _{A{\neq}B}\left(E_{NN}^{AB}+E_{eN}^{AB}+E_{Ne}^{AB}+E_{eeCl}^{AB}+E_{eeXC}^{AB}\right)\)
The first four terms on the right-hand side are intra-atomic (also called self) terms for each atom \(A\): the atomic kinetic energy (TA), the interaction energy between the nucleus and the electron density inside basin \(A\) (\(E_{Ne}^{AA}\)), and the electron-electron repulsion within \(A\), decomposed into classical electrostatic (\(E_{eeCl}^{AA}\)) and exchange-correlation contributions (\(E_{eeXC}^{AA}\)).
The last sum describes the interactions for each atom pair \(AB\), whether or not the atoms are connected by a bond path. It contains the repulsion between the nuclei in \(A\) and \(B\) (\(E_{NN}^{AB}\)), the attraction between the electrons in each basin and the nucleus of the other basin (\(E_{eN}^{AB}\) and \(E_{Ne}^{AB}\)), and the interbasin electron-electron repulsion. The latter is split into classical electrostatic (\(E_{eeCl}^{AB}\)) and exchange-correlation (\(E_{eeXC}^{AB}\)) contributions.
Within a Kohn-Sham DFT framework, the implementation of this partition is not straightforward (see Ref. [10] for more details). For any atom pair \(AB\), we evaluate:
\(E_{inter}^{AB}=E_{NN}^{AB}+E_{eN}^{AB}\left[\rho \right]+E_{Ne}^{AB}\left[\rho \right]+E_{eeCl}^{AB}\left[\rho \right]+E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\)
Here, \(E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\) is the full exact-exchange (HF-like) interaction energy between the two atoms. Tognetti and Joubert have shown that, despite the underlying approximations, this expression can be safely used at least for semi-quantitative purposes [11].
When an a posteriori dispersion correction is applied, an additional term is included in the interatomic interaction energy:
\(E_{inter}^{AB}(\mathrm{with\ dispersion})=E_{inter}^{AB}+E_{disp}^{AB}\)
The form of \(E_{disp}^{AB}\) depends on the selected dispersion correction. For Grimme D3 and D4, it is composed of pairwise two-body contributions and, when enabled, three-body Axilrod-Teller-Muto (ATM) contributions:
\(E_{disp}^{AB}=E_{disp,2}^{AB}+E_{disp,3}^{AB}\)
The D3 and D4 implementations distribute each three-body contribution over the six ordered atom pairs belonging to the corresponding atom triplet. The resulting pair contributions are then included in the IQA pair decomposition, so that their sum reproduces the total dispersion energy. In D4, the two-body dispersion coefficients additionally depend on atomic partial charges obtained from the electronegativity equilibration model.
Eventually, ADF reports three energy terms for each atom pair. The first is the total interaction energy, \(E_{inter}^{AB}\). The second is the covalent part, given by the exchange energy between the two atomic basins, \(E_{eeX}^{AB}\left[\left\{\psi _i^{KS}\right\}\right]\). The third is the noncovalent contribution, which contains the remaining electrostatic terms, \(E_{NN}^{AB}+E_{eN}^{AB}+E_{Ne}^{AB}\left[\rho \right]+E_{eeCl}^{AB}\left[\rho \right]\). If dispersion is requested, the third term also includes the a posteriori dispersion contribution, \(E_{disp}^{AB}\).
Please note that all dispersion terms available in ADF can be included in the IQA energy decomposition (the D4 correction is now available).
By default, ADF performs the full IQA decomposition, including the intra-atomic terms (except ZORA calculations), and uses verbose mode to report all contributions.
Spin-orbit exchange¶
For SO exchange integrals, occupied one-electron states are two-component spinors \(\psi_p(\mathbf r)=\left(\psi_{p,1}(\mathbf r),\psi_{p,2}(\mathbf r)\right)^T\). For two points \(\mathbf r \in \Omega_A\) and \(\mathbf r' \in \Omega_B\), IQA constructs the one-particle density kernel in spin space,
\(\Gamma_{ss'}(\mathbf r,\mathbf r')=\sum_{p\in\mathrm{occ}} w_p\,\psi_{p,s}(\mathbf r)\,\psi_{p,s'}^*(\mathbf r')\).
Here, \(p\) labels a spinor and \(\mathrm{occ}\) denotes the set of occupied spinors. The indices \(s,s'\in\{1,2\}\) label their two components, and the asterisk denotes complex conjugation.
The occupation weight in the density kernel is
\(w_p=\frac{f_p}{n_{rd}}\),
where \(f_p\) is the total, possibly fractional, occupation associated with the spinor multiplet containing \(p\), and \(n_{rd}\) is the number of members of that multiplet (the dimension of the corresponding double-group representation). When time-reversal symmetry is preserved, two degenerate spinors related by time reversal form a Kramers pair. Each partner can mix both spin components, and the pair has \(n_{rd}=2\). In the implementation, each member is scaled by \(\sqrt{f_p/n_{rd}}\); the product of its values entering \(\Gamma\) therefore has weight \(w_p\). Thus, occupations, including fractional occupations, are already included in \(\Gamma\).
The \(2\times2\) matrix \(\Gamma\) is the SO analogue of the scalar one-particle density matrix. Its diagonal elements couple equal spinor components, whereas its off-diagonal elements account for spin mixing. The exchange pair-density factor is the squared Frobenius norm of this matrix,
\(K(\mathbf r,\mathbf r')=\sum_{s,s'=1}^2\left|\Gamma_{ss'}(\mathbf r,\mathbf r')\right|^2=\mathrm{Tr}\!\left[\Gamma(\mathbf r,\mathbf r')\Gamma^{\dagger}(\mathbf r,\mathbf r')\right]\).
Here, \(|\Gamma_{ss'}|^2\) is the squared modulus of a complex matrix element, \(\Gamma^{\dagger}\) is the conjugate transpose of \(\Gamma\), and \(\mathrm{Tr}\) denotes the matrix trace. Thus, \(K\) includes both the same-component and mixed-component contributions. The interatomic exchange energy is then
\(E_{eeX}^{AB}= -\int_{\Omega_A}\!\int_{\Omega_B}\frac{K(\mathbf r,\mathbf r')}{\left|\mathbf r-\mathbf r'\right|}\,d\mathbf r\,d\mathbf r'\).
No additional factor of \(1/2\) appears because \(E_{eeX}^{AB}\) is defined for a pair \(AB\) counted only once. In the total IQA energy expression above, the factor \(1/2\) instead compensates for the sum containing both \(AB\) and \(BA\). For sufficiently distant atom pairs, ADF evaluates this exchange contribution using a multipole expansion or the asymptotic \(-DI/(2R_{AB})\) approximation, where \(DI\) is the delocalization index and \(R_{AB}\) is the internuclear distance. This construction is the SO analogue of scalar-MO exchange, with explicit summation over spinor components and complex conjugation.
IQA input¶
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:
verbose
- Options:
[normal, verbose]
- Description:
Normal output (minimal) or verbose output (default; 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