12.5.1 Energy Decomposition Analysis Based on Absolutely Localized Molecular Orbitals

The strength of intermolecular binding is inextricably connected to the fundamental nature of interactions between the molecules. Intermolecular complexes can be stabilized through weak dispersive forces, electrostatic effects (e.g., charge–charge, charge–dipole, and charge–induced dipole interactions) and donor-acceptor type orbital interactions such as forward and back-donation of electron density between the molecules. Depending on the extent of these interactions, the intermolecular binding could vary in strength from just several kJ/mol (van der Waals complexes) to several hundred kJ/mol (metal–ligand bonds in metal complexes). Understanding the contributions of various interaction modes enables one to tune the strength of the intermolecular binding to the ideal range by designing materials that promote desirable effects. One of the most powerful techniques that modern first principles electronic structure methods provide to study and analyze the nature of intermolecular interactions is the decomposition of the total molecular binding energy into the physically meaningful components such as dispersion, electrostatic, polarization, charge transfer, and geometry relaxation terms.

Energy decomposition analysis based on absolutely-localized molecular orbitals (ALMO-EDA) is implemented in Q-Chem, ⁵⁹⁸ Khaliullin R. Z. et al.
J. Phys. Chem. A
(2007), 111, pp. 8753. Link including the open shell generalization. ⁵¹⁶ Horn P. R. et al.
J. Chem. Phys.
(2013), 138, pp. 134119. Link In ALMO-EDA, the total intermolecular binding energy is decomposed into the “frozen density” component (FRZ), the polarization (POL) term, and the charge-transfer (CT) term. The “frozen density” term is defined as the energy change that corresponds to bringing infinitely separated fragments together without any relaxation of their MOs. The FRZ term is calculated as a difference between the FRAGMO guess energy and the sum of the converged SCF energies on isolated fragments. The polarization (POL) energy term is defined as the energy lowering due to the intrafragment relaxation of the frozen occupied MOs on the fragments. The POL term is calculated as a difference between the converged SCF MI energy and the FRAGMO guess energy. Finally, the charge-transfer (CT) energy term is due to further interfragment relaxation of the MOs. It is calculated as a difference between the fully converged SCF energy and the converged SCF MI energy.

The total charge-transfer term includes the energy lowering due to electron transfer from the occupied orbitals on one molecule (more precisely, occupied in the converged SCF MI state) to the virtual orbitals of another molecule as well as the further energy change caused by induction that accompanies such an occupied/virtual mixing. The energy lowering of the occupied-virtual electron transfer can be described with a single non-iterative Roothaan-step correction starting from the converged SCF MI solution. Most importantly, the mathematical form of the SCF MI(RS) energy expression allows one to decompose the occupied-virtual mixing term into bonding and back-bonding components for each pair of molecules in the complex. The remaining charge-transfer energy term (i.e., the difference between SCF MI(RS) energy and the full SCF energy) includes all induction effects that accompany occupied-virtual charge transfer and is generally small. This last term is called higher order (HO) relaxation. Unlike the RS contribution, the higher order term cannot be divided naturally into forward and back-donation terms. The BSSE associated with each charge-transfer term (forward donation, back-bonding, and higher order effects) can be corrected individually.

To perform energy decomposition analysis, specify fragments in the $molecule section and set JOBTYPE to EDA. For a complete EDA job, Q-Chem

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  performs the SCF on isolated fragments (use the $rem_frgm section if convergence issues arise but make sure that keywords in this section do not affect the final energies of the fragments),

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  generates the FRAGMO guess to obtain the FRZ term,

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  converges the SCF MI equations to evaluate the POL term,

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  performs evaluation of the perturbative (RS or ARS) variational correction to calculate the forward donation and back-bonding components of the CT term for each pair of molecules in the system,

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  converges the full SCF procedure to evaluate the higher order relaxation component of the CT term.

The FRGM_LPCORR keyword controls evaluation of the CT term in an EDA job. To evaluate all of the CT components mentioned above set this keyword to RS_EXACT_SCF or ARS_EXACT_SCF. If the HO term in not important then the final step (i.e., the SCF calculation) can be skipped by setting FRGM_LPCORR to RS or ARS. If only the total CT term is required then set FRGM_LPCORR to EXACT_SCF.

ALMO charge transfer analysis (ALMO-CTA) is performed together with ALMO EDA. ⁵⁹⁶ Khaliullin R. Z., Bell A. T., Head-Gordon M.
J. Chem. Phys.
(2008), 128, pp. 184112. Link The ALMO charge transfer scale, $\mathrm{\Delta }Q$, provides a measure of the distortion of the electronic clouds upon formation of an intermolecular bond and is such that all CT terms (i.e., forward-donation, back-donation, and higher order relaxation) have well defined energetic effects (i.e., ALMO-CTA is consistent with ALMO-EDA).

To remove the BSSE from the CT term (both on the energy and charge scales), set EDA_BSSE to TRUE. Q-Chem generates an input file for each fragment with MIXED basis set to perform the BSSE correction. As with all jobs with MIXED basis set and d or higher angular momentum basis functions on atoms, the PURECART keyword needs to be initiated. If EDA_BSSE = TRUE then general basis sets (BASIS = GEN) cannot be used in the current implementation.

Please note that the energy of the geometric distortion of the fragments is not included into the total binding energy calculated in an EDA job. The geometry optimization of isolated fragments must be performed to account for this term.

In the 5.2 release and after, the “EDA2" driver (see Section 12.7) will be employed by default when JOBTYPE = EDA is set, which covers almost all features of the first-generation ALMO-EDA/CTA while including many new features (such as further decomposition of the frozen term). The original implementation of the first-generation ALMO-EDA is still accessible by setting EDA2 = FALSE.