12.6.3 Polarization Energy with a Well-Defined Basis Set Limit

The definition of polarization energy lowering in the original ALMO-EDA used the full AO space of each fragment as the variational degrees of freedom. This is based on the assumption that the AO basis functions are fragment-ascribable based on their atomic centers. However, this assumption becomes inappropriate when very large basis sets are used, especially those with diffuse functions (e.g. def2-QZVPPD). In such scenarios, basis functions on a given fragment tend to describe other fragments so that the “absolute localization" constraint becomes weaker and finally gets effectively removed. This is why the original ALMO-EDA scheme does not have a well-defined basis set limit for its polarization energy.

To overcome this problem, Horn and Head-Gordon ⁵⁷³ Horn P. R., Head-Gordon M.
J. Chem. Phys.
(2015), 143, pp. 114111. Link proposed a new definition for the POL term in the ALMO-EDA method based on fragment electrical response functions (FERFs). FERFs on a given fragment are prepared by solving CPSCF equations after its SCF solution is found:

where $𝐇$ is SCF orbital Hessian and ${𝐌}_{\mu }$ is a component ($\mu$) of a multipole matrix with a certain order. The resulting fragment response matrices ($\{{\mathrm{\Delta }}_{\mu }\}$) are a set of $n_{\mathrm{v}}\times n_{\mathrm{o}}$ matrices. Then, a singular value decomposition (SVD) is performed on ${\mathrm{\Delta }}_{\mu }$:

and the left vectors (not including the null vectors) will be used to construct a truncated virtual space, which is used to define the variational degrees of freedom for the SCF-MI problem:

where ${𝐂}_{\mathrm{vir}}$ denotes the original virtual orbitals of the given fragment.

The basic spirit of using FERFs is to obtain a subset of virtuals that is most pertinent to the electrical polarization of a given fragment, while the redundant variational degrees of freedom (which might be CT-like) are excluded. This scheme is shown to give a well-defined basis set limit for the polarization energy that relies on the SCF-MI calculation. The multipole orders dipole (D), quadrupole (Q), and octopole (O)m included on the RHS of Eq. (12.2), decide the span of FERFs on each fragment. Numerical experiments suggest that the inclusion of dipole- and quadrupole-type responses is able to long-range induced electrostatics correctly and also gives a well-defined basis set limit, which is thus recommended as the working basis of the SCF-MI problem. The full span of the polarization subspace of fragment $A$ is thus:

Therefore, each occupied orbital will be paired with eight virtual orbitals (if the employed AO basis is large enough).

The polarization subspaces constructed as in Eq. (12.4) are non-orthogonal between fragments. Therefore, it is named as the “nDQ" model for polarization. There is another version of this method which enforces interfragment orthogonality between the polarization subspaces and it is correspondingly termed as “oDQ" (or with other multipole orders). The preparation of orthogonal FERFs is more complicated (see Ref.  573 Horn P. R., Head-Gordon M.
J. Chem. Phys.
(2015), 143, pp. 114111. Link for the details) and usually gives less favorable polarization energies. For most general cases, we recommend the use of the “nDQ" model. Calculations using FERFs are performed using the generalized SCF-MI procedure introduced in Section 12.6.2.

However, solving the CPSCF equation requires inverting the large orbital Hessian matrix $𝐇$ of dimension $ov\times ov$, which is the time demanding step in solving for FERFs. For example, in Hartree-Fock theory, in the canonical representation, it has matrix elements

Recently, Aldossary and coauthors proposed an uncoupled-FERFs (uFERFs) to improve the conventional FERFs. The idea of uFERFs is simply to adopt the solution for uncoupled CPSCF equation, where the orbital Hessian is approximated with

With this approximation, the orbital Hessian is easily inverted, and the fragment response matrices are simply

Benchmark calculations on the S22, S66 and Ionic43 datasets shows that the POL energy is hardly changed using uFERFs, while a general 5-10 times speedup for FERF creation and 20% average speedup for the whole EDA calculation is achieved. In addition, the importance of monopole uFERFs (response to scaled nuclear charges) for intermolecular interactions is recognized for strong ion-neutral interactions. Therefore, it is recommended to use MDQ-uFERFs for the calculation of POL energy in ALMO-EDA calculations.