# 12.9 ALMO-EDA Method for Bonded Interactions

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 DFT575, 573, 574. 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, $\Delta E_{\text{PREP}}$ includes the energy required to distort each radical fragment to the geometry that it adopts in the bonded state $\Delta E_{\text{GEOM}}$, as well as an electronic preparation due to orbital rehybridization $\Delta E_{\text{HYBRID}}$. 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 $sp^{2}$-hybridized with an unpaired electron in a $p$ orbital, while an amine group is often $sp^{3}$- or $sp^{2}$-hybridized with a lone pair in the $p$ orbital in a molecule. Then,

 $\Delta E_{\text{PREP}}=\Delta E_{\text{GEOM}}+\Delta E_{\text{HYBRID}}$

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 $\Delta E_{\text{HYBRID}}$. 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 $sp^{2}$-hybridized planar molecule, while a methyl group in a bond is a pyramidalized $sp^{3}$ 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: $\Delta E_{\text{SC}}$ 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 574, 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 575 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) 573. 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.

BONDED_EDA
Use the bonded ALMO-EDA.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not perform bonded ALMO-EDA. 1 Perform ALMO-EDA with non-orthogonal CI. 2 Perform ALMO-EDA with spin-projected formalism.
RECOMMENDATION:
Set to 2 for all cases where the supersystem is closed shell, only use 1 for cases where the fragments have more than one unpaired spin each.

EDA_CONTRACTION_ANAL
Perform analysis separating orbital contraction from the rest of POL.
TYPE:
BOOLEAN
DEFAULT:
0
OPTIONS:
FALSE Do not perform contraction analysis. TRUE Perform contraction analysis.
RECOMMENDATION:
No recommendation

Example 12.23  Bonded EDA of F${}_{2}$ with MDQ FERFs, frozen analysis in the spin-projected formalism

$molecule 0 3 -- 0 2 F 0.0 0.0 0.0 -- 0 2 F 0.0 0.0 1.382$end

$rem jobtype eda eda2 10 bonded_eda 2 exchange wb97x-d basis aug-cc-pvdz scf_convergence 6 max_scf_cycles 200 roscf true scf_guess fragmo symmetry false sym_ignore true scf_algorithm gdm_ls scfmi_mode 1 scf_print_frgm true child_mp true child_mp_orders 1233 frz_relax true frz_relax_method 2 frz_ortho_decomp 1$end

$rem_frgm scf_convergence 7 scf_algorithm gdm_ls scf_guess sad$end


Example 12.24  Bonded EDA of CH with MDQ FERFS, contraction analysis in the non-orthogonal CI formalism.

$molecule 0 4 -- 0 3 C 0.0000000000 0.0000000000 -0.0525358999 -- 0 2 H 0.0000000000 0.0000000000 1.0525358999$end

$rem jobtype eda eda2 10 bonded_eda 1 exchange hf basis aug-cc-pvdz scf_convergence 6 max_scf_cycles 2000 roscf true scf_guess fragmo scf_algorithm l_bfgs scfmi_mode 1 scf_print_frgm true child_mp true child_mp_orders 1233 frz_relax true frz_relax_method 2 eda_contraction_anal true symmetry false sym_ignore true$end

$rem_frgm scf_convergence 7 scf_algorithm gdm_ls scf_guess sad$end