FMO Capping Method

Note

The information in this section is my interpretation of the cited references. In many cases the descriptions of the algorithms provided in the literature are somewhat vague. Thus what is in this section may not accurately reflect how the underlying implementations actually work. Please feel free to contribute and make this section more accurate.

In FMO calculations caps in the traditional sense are not used.

FMO99 Capping

The initial FMO capping method Kitaura et al.⁴¹, here labeled as FMO99, assigned electrons unevenly to the fragments (i.e., one of the fragments gets both of the bonding electrons). From this assignment, an initial density for fragment $I$ is obtained by:

Defining a reference system (no guidance on how to do this was provided)
Localizing the molecular orbitals of the reference system to the atoms in $I$
- The relevant atoms must have the same geometry in both the reference system and the target system
Discarding contributions to the molecular orbitals from atoms outside $I$
Renormalizing the molecular orbitals
Use the renormalized MOs to form a n-electron density (n depending on how the electrons are assigned)

FMO00 Capping

Note

The language describing the FMO00 method Nakano et al.⁶⁹ uses is: “core” and “valence” orbitals; however, based on subsequent publications, such as Fedorov and Kitaura¹⁷, I believe what they are actually referring to would typically be termed “occupied” and “virtual” orbitals, respectively. The language in this section reflects this assumption.

A new capping method was proposed in a subsequent paper Nakano et al.⁶⁹, here denoted FMO00. In this method, for a bond $a-b$, the bonding electrons are assigned to one of the two atoms in the bond, say $a$. For clarity, we refer to the fragment which gets atom $a$ as $A$, and that which gets $b$ as $B$. $B$’s initial density is computed using the normal AO basis set for $B$, with the usual equations. However, $A$’s initial density is computed using $A$’s usual basis set augmented with the virtual orbitals from $b$. To extract the virtual orbitals of $b$, a modified version of $A$’s Fock operator is used:

\[\widehat{F}'_A = \widehat{F}_A + \sum_{i\in b} \gamma_i\Ket{i}\Bra{i}\]

where $widehat{F}_A$ is the original Fock operator, $i$ runs over the occupied orbitals of $b$, $\gamma_i$ is a large positive coefficient ( default value is between 10$^6$ and 10$^8$), and $\Ket{i}$ is the $i$-th occupied orbital of $b$. In practice orbital $\Ket{i}$ is only decribed by the AOs on the $b$, which is to say, that after localizing $\ket{i}$ only the matrix elements $C_{\mu i}$ where $\mu$ is on $b$ are kept.

FMO04 Capping

In describing a three-body FMO method Fedorov and Kitaura¹⁷, a new capping method was pitched which is effectively a combination of FMO99 Capping and FMO00 Capping. In this study all severed bonds were carbon-carbon bonds. Thus the decision was made to use the occupied orbitals of a methane molecule instead of the actual occupied orbitals. The methane orbitals were split so that one atom, call it $a$, gets one of the $sp^3$ orbitals (specifically the $sp^3$ orbital along the bond) and the other atom, call it $b$, gets the remaining four occupied orbitals. For a given bond, the decision of which atom is $a$ versus which atom is $b$ was made so that $a$ went to the fragment with the smaller ordinal number (presumably the ordinal number was assigned based on user input, but this point was not clarified).