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12.7 Additional ALMO-EDA Capabilities

12.7.2 ALMO-EDA for Bonded Interactions

(September 1, 2024)

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 765 Levine D. S. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 4812.
Link
, 763 Levine D. S., Head-Gordon M.
J. Phys. Chem. Lett.
(2017), 8, pp. 1967.
Link
, 764 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 with 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 re-hybridization can play such a large role in the energy, ΔEPREP includes the energy required to distort each radical fragment to the geometry that it adopts in the bonded state ΔEGEOM, as well as an electronic preparation due to orbital re-hybridization ΔEHYBRID. 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, NH2, is sp2-hybridized with an unpaired electron in a p orbital, while an amine group is often sp3- or sp2-hybridized with a lone pair in the p orbital in a molecule. Then,

ΔEPREP=ΔEGEOM+ΔEHYBRID (12.22)

This re-hybridized 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 re-separated and the energy difference between the electronically distorted fragments and the ground state fragments is ΔEHYBRID. This corresponds to fixing the α-density and allowing the β-hole to re-optimize in the span of that α-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 re-hybridization 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 sp2-hybridized planar molecule, while a methyl group in a bond is a pyramidalized sp3 fragment; the re-hybridzation cost was already paid by the geometric distortion.

The FRZ energy in the bonded scheme corresponds to the ALMO supersystem formed by combining the RO fragments to form a spin-pure triplet single-determinant wave function without allowing the orbitals to relax further. This term is entirely a non-bonded interaction and will typically be repulsive for a chemical bond because of Pauli repulsion. It includes contributions from inter-fragment 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: ΔESC 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, 764 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, 765 Levine D. S. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 4812.
Link
so 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). 763 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.

BONDED_EDA

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

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 12.28  Bonded EDA of F2 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
   EXCHANGE           wb97x-d
   BASIS              aug-cc-pvdz
   EDA2               10
   BONDED_EDA         2
   SCF_CONVERGENCE    6
   MAX_SCF_CYCLES     200
   ROSCF              true
   SCF_GUESS          fragmo
   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
   INTEGRAL_SYMMETRY    false
   POINT_GROUP_SYMMETRY false
$end

$rem_frgm
   scf_convergence   7
   scf_algorithm     gdm_ls
   scf_guess         sad
$end

Example 12.29  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
   EXCHANGE               hf
   BASIS                  aug-cc-pvdz
   EDA2                   10
   BONDED_EDA             1
   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
   INTEGRAL_SYMMETRY      false
   POINT_GROUP_SYMMETRY   false
$end

$rem_frgm
   scf_convergence   7
   scf_algorithm     gdm_ls
   scf_guess         sad
$end