The EDA terms

Second-generation ALMO-EDA

ALMO-EDA (absolutely localized molecular orbital energy decomposition analysis) builds the interaction up in steps, with Kohn–Sham DFT [Horn2016a] [Horn2016b]:

frozen

The fragments’ own orbitals, unchanged, brought together and antisymmetrized. It is divided into three parts:

electrostatic

The permanent electrostatics of the fragments’ densities after an orthogonal decomposition of the frozen wavefunction [Horn2016b] (Q-Chem’s ELEC). It is more attractive than the classical electrostatics below, by as much as 30 kJ/mol for an anion in contact with a solvent molecule.

Pauli

The repulsion from antisymmetrizing the fragments’ occupied orbitals, in the same orthogonal decomposition: frozen − electrostatic − dispersion.

dispersion

Defined against a dispersion-free partner functional (Hartree–Fock exchange). With a functional that adds an empirical correction (-D3, -D4) it also contains that correction, including the D4 three-body (ATM) term.

polarization

The relaxation of each fragment’s orbitals in the presence of the others, kept on its own fragment. The space they relax in is spanned by fragment electric-field response functions (FERF-DQ: dipole and quadrupole fields) [Horn2015], which keeps polarization nearly free of basis-set superposition error.

charge transfer

The rest of the relaxation to the full SCF of the complex, in which electrons may move between fragments.

total

frozen + polarization + charge transfer. These identities hold exactly, and the step checks them.

classical electrostatic

The Coulomb interaction of the isolated fragments’ charge densities and nuclei, including charge penetration (Q-Chem’s CLS ELEC). This is what SAPT calls electrostatics: for Li⁺–PF₆⁻ and PF₆⁻–EC it agrees with SAPT2+/aug-cc-pVDZ to 1.6 kJ/mol. It is exactly pairwise additive, so its many-body part is zero, which is a useful check.

Conventions

• The fragments are at their geometry in the cluster. There is no preparation or distortion term: the energies are relative to the isolated fragments at the same geometry.

• Charge transfer carries the basis-set superposition error; polarization does not.

• Energies are in kJ/mol.

Common terms

To compare schemes, the step also gives a few common terms, used only where the physics really is the same:

Common term	ALMO-EDA	SAPT
electrostatics	classical electrostatic	electrostatics
Pauli	frozen − classical electrostatic − disp.	exchange
dispersion	dispersion	dispersion
frozen	frozen	–
polarization	polarization	–
charge transfer	charge transfer	–
induction	polarization + charge transfer	induction
total	total	total

ALMO-EDA has two electrostatic terms, and only the classical one is comparable with SAPT.

• SAPT’s electrostatics is the Coulomb interaction of the isolated monomers’ densities, which is ALMO’s classical electrostatic term (Q-Chem’s CLS ELEC). The common Pauli term is what is left of the frozen term, frozen − classical electrostatic − dispersion (Q-Chem’s MOD PAULI).

• ALMO’s own electrostatic and Pauli terms come from the orthogonal decomposition of the frozen wavefunction [Horn2016b] (Q-Chem’s ELEC and PAULI). They are not comparable with SAPT.

For PF₆⁻–EC (ωB97M-V/def2-TZVPD against SAPT2+/aug-cc-pVDZ):

• the classical electrostatics is within 1.6 kJ/mol of SAPT’s electrostatics, and the common Pauli term within 1.7 kJ/mol of SAPT’s exchange;

• the orthogonal-decomposition terms are each 36 kJ/mol away.

Both pairs add up to the same thing: frozen = electrostatics + Pauli + dispersion holds in both. The step keeps the native terms under their own names as well.

Because the classical electrostatics is pairwise additive, the common ≥3-body electrostatics is exactly zero. The many-body part of the frozen term, other than dispersion, then appears entirely in the common Pauli term.

SAPT induction contains both polarization and charge transfer, so it corresponds to ALMO’s polarization plus charge transfer, here called induction. Comparing SAPT induction with ALMO polarization alone hides charge transfer, which matters for small cations such as Li⁺.

References

[Horn2016a]

P. R. Horn, Y. Mao and M. Head-Gordon, “Probing non-covalent interactions with a second generation energy decomposition analysis using absolutely localized molecular orbitals”, Phys. Chem. Chem. Phys. 18, 23067–23079 (2016), doi:10.1039/C6CP03784D.

[Horn2016b] (1,2,3)

P. R. Horn, Y. Mao and M. Head-Gordon, “Defining the contributions of permanent electrostatics, Pauli repulsion, and dispersion in density functional theory calculations of intermolecular interaction energies”, J. Chem. Phys. 144, 114107 (2016), doi:10.1063/1.4942921.

[Horn2015]

P. R. Horn and M. Head-Gordon, “Polarization contributions to intermolecular interactions revisited with fragment electric-field response functions”, J. Chem. Phys. 143, 114111 (2015), doi:10.1063/1.4930534.