12.9.1 Overview

EDA schemes have been very successful at elucidating the nature of noncovalent interactions. On the other hand, these methods are often inadequate for the analysis of covalent bonds. In fragment-based approaches, the key difficulty arises from the need to correctly spin-couple two open-shell radical fragments into a closed-shell bond in a spin-pure way. The ALMO-EDA methodology was extended by Levine to accomplish this within HF and KS DFT ⁷¹³ Levine D. S. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 4812. Link ^{, 711} Levine D. S., Head-Gordon M.
J. Phys. Chem. Lett.
(2017), 8, pp. 1967. Link ^{, 712} Levine D. S., Head-Gordon M.
Proc. Natl. Acad. Sci. USA
(2017), 114, pp. 12649. Link . If HF is used, the final wave function whose interaction energy is being decomposed is the CAS(2,2)/1-pair perfect-pairing/TCSCF wave function. At present, only a single bond may be analyzed by these schemes.

The method begins wtih two doublet radical fragments, each of which is described by a restricted open-shell (RO) Hartree-Fock (HF) or Kohn-Sham DFT single determinant. In the bonded EDA scheme, because orbital rehybridization can play such a large role in the energy, $\mathrm{\Delta }{E}_{\text{PREP}}$ includes the energy required to distort each radical fragment to the geometry that it adopts in the bonded state $\mathrm{\Delta }{E}_{\text{GEOM}}$, as well as an electronic preparation due to orbital rehybridization $\mathrm{\Delta }{E}_{\text{HYBRID}}$. For example, an F atom has an unpaired electron in a $p$ orbital, while an F atom in a bond will be $sp$-hybridized. The amine radical, NH${}_2$, is $s{p}^2$-hybridized with an unpaired electron in a $p$ orbital, while an amine group is often $s{p}^3$- or $s{p}^2$-hybridized with a lone pair in the $p$ orbital in a molecule. Then,

This rehybridized state is obtained by relaxing the ALMO supersystem obtained from the fragments, permitting only on-fragment doubly-occupied–singly-occupied orbital rotations (which are well-defined due to the RO nature of the fragments). The so-optimized fragments are then reseparated and the energy difference between the electronically distorted fragments and the ground state fragments is $\mathrm{\Delta }{E}_{\text{HYBRID}}$. This corresponds to fixing the $\alpha$-density and allowing the $\beta$-hole to reoptimize in the span of that $\alpha$-density. This is a kind of polarization, although we draw a distinction from the electronic polarization step which appears later in both the bonded and non-bonded ALMO schemes. Another reason why it makes sense to place the rehybridization energy here is that it is already partially accounted for by the fact that the geometry of the radical fragment is fixed to be that of the interacting fragment. For instance, free methyl radical is an $s{p}^2$-hybridized planar molecule, while a methyl group in a bond is a pyramidalized $s{p}^3$ fragment; the rehybridzation cost was already paid by the geometric distortion.

The FRZ energy in the bonded scheme corresponds to the ALMO supersystem formed by combineing the RO fragments to form a spin-pure triplet single-determinant wave function without allowing the orbitals to relax futher. This term is entirely a nonbonded interaction and will typically be repulsive for a chemical bond because of Pauli repulsion. It includes contributions from interfragment electrostatics, Pauli repulsion, exchange-correlation, and dispersion. The EDA2 frozen decomposition scheme may be applied to this term in the spin-projected formalism (i.e. BONDED_EDA = 2).

A new term is introduced for the bonded EDA scheme: $\mathrm{\Delta }{E}_{\text{SC}}$ of the spin-coupling energy. This energy difference is caused by electron pairing and loosely corresponds to the idea of covalency. Like FRZ, SC will be evaluated with frozen orbitals, but while FRZ is typically strongly repulsive (dominated by Pauli repulsion), SC is typically strongly attractive in the overlapping regime associated with covalent bond formation. For this reason and because we are primarily interested in the singlet surface (as opposed to the triplet surface of the initial supersystem), FRZ and SC may be grouped together into a total frozen orbital term (FRZ + SC).

In the KS DFT scheme  ⁷¹² Levine D. S., Head-Gordon M.
Proc. Natl. Acad. Sci. USA
(2017), 114, pp. 12649. Link , this spin-coupled wave function is formed by forming the broken-symmetry DFT determinant and spin-projecting out the triplet contaminant. Since the wave function is constructed from RO fragment, spin-contamination can only occur within the half-occupied space and hence the triplet contaminant is the only possible contaminant. We therefore obtain and exact singlet wave function. For HF, this is exactly equivalent to the scheme based on nonorthogonal CI ⁷¹³ Levine D. S. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 4812. Link as long as there are only two unpaired spins among the fragments (i.e. the supersystem is closed shell). This is usually the case and so, as the schemes are equivalent, we advocate only using the spin-projected formalism as it is much more efficient.

The POL and CT terms are similarly defined as in the non-bonded ALMO scheme. FERFs may be used to define polarization but monopole FERFs, which describe the expansion or contraction of orbitals (which occurs in some cases on bond formation should be included) ⁷¹¹ Levine D. S., Head-Gordon M.
J. Phys. Chem. Lett.
(2017), 8, pp. 1967. Link . The CT term gives an indication of the level of ionic character in the bond. Taken together the various terms describe a fingerprint for the bond being studied. Further details for how to analyze the results may be found in the referenced literature.

Considerations for using the bonded ALMO-EDA:

• •

  SCFMI_MODE = 1 is required.

• •

  ROSCF = TRUE must be set.

• •

  There are no presets of the bonded ALMO-EDA. Therefore, set EDA2 = 10.

• •

  DIIS may not be used for SCF_ALGORITHM. Use GDM_LS for BONDED_EDA = 2 and L_BFGS for BONDED_EDA = 1.