Many schemes for decomposing quantum chemical calculations of intermolecular
interaction energies into physically meaningful components can be found in the
literature, but the definition of the charge-transfer (CT) contribution has
proven particularly vexing to define in a satisfactory way and typically
depends strongly on the choice of basis
set,^{Ronca:2014, Rezac:2015, Lao:2016b} because as virtual orbitals on
monomer $A$ start to extend significantly over monomer $B$ as the basis set
approaches completeness, the distinction between polarization (excitations
localized on $A$, introduced by the perturbing influence of $B$) and CT
(excitations from $A$ to $B$) becomes blurred.^{Lao:2016b} This
ambiguity renders orbital-dependent definitions of CT highly dependent on the
choice of atomic orbital basis set. On the other hand, constrained density
functional theory (cDFT, Section 5.13),^{Kaduk:2012}
by means of which a CT-free reference
state can be defined based on “promolecule” densities, affords a definition
of CT that is scarcely dependent on the basis set and is in accord with
chemical intuition in simple cases.^{Lao:2016b}

For intermolecular interactions, the cDFT definition of CT can be combined with
a definition of the remaining components of the interaction energy
(electrostatics, induction, Pauli repulsion, and van der Waals interactions)
based on symmetry-adapted perturbation theory (SAPT, Section 12.13).
In traditional SAPT, the CT interaction energy resides within the induction
energy (also known as the polarization energy), which is therefore itself
highly dependent upon the basis set. However, using cDFT to define the CT
component and subtracting this out of the SAPT induction energy, both the CT
and the remaining induction energies are largely independent of basis
set.^{Lao:2016b} SAPT/cDFT therefore provides a stable and
physically-motivated energy decomposition.

While the cDFT definition of CT exhibits only a very mild basis-set dependence, its quantitative details
do depend upon how the charge constraints in cDFT are defined relative to
fragment populations (Section 5.13). For SAPT/cDFT,
both atomic Becke^{Becke:1988a}
and fragment-based Hirshfeld^{Rezac:2015} (FBH) charge partitioning methods are available.
The former involves construction of atomic cell
functions that amount to smoothed Voronoi polyhedra centered about each atom.
A switching function defines the atomic cell of atom $a$, and falls rapidly from
$\approx 1$ near the nucleus for atom $a$, to $\approx 0$ near any other nucleus. Becke^{Becke:1988a}
defined atomic cell functions ${P}_{a}(\text{\mathbf{r}})$ that are products of switching functions and that
can be used to define the cDFT integration weight for monomer $A$ by summing over atoms $a\in A$:

$${w}_{A}^{\mathrm{Becke}}(\text{\mathbf{r}})=\frac{{\sum}_{a\in A}{P}_{a}(\text{\mathbf{r}})}{{\sum}_{b}{P}_{b}(\text{\mathbf{r}})}.$$ | (12.61) |

The sum in the denominator runs over all atoms in both monomers, $A$ and $B$.
Becke populations, however, are rooted in a somewhat arbitrarily-defined topology, based
in part on assumed atomic radii, whereas FBH partitioning
derives physical significance from isolated monomer densities
${\stackrel{~}{\rho}}_{A}(\mathbf{r})$ and ${\stackrel{~}{\rho}}_{B}(\mathbf{r})$.
The cDFT weight function for monomer $A$ is^{Rezac:2015}

$${w}_{A}^{\mathrm{FBH}}(\mathbf{r})=\frac{{\stackrel{~}{\rho}}_{A}(\mathbf{r})}{{\stackrel{~}{\rho}}_{A}(\mathbf{r})+{\stackrel{~}{\rho}}_{B}(\mathbf{r})},$$ | (12.62) |

which is the same “stockholder” scheme used to define atomic Hirshfeld populations (Section 10.2.1),
but applied here to the entire monomer.
In the language of cDFT, the denominator in this expression would be called the *promolecule*
density for the dimer $A+B$. In order to
set a molecular fragment constraint, simply retain the existing syntax in the *$cdft* input
section (as described in Section 5.13) and specify all atoms within a given molecular fragment.

Due to the fact that Becke populations are rooted in a topological scheme based in part on assumed atomic
radii, it is highly recommended that if CDFT_POP is set to BECKE, the rem variable
BECKE_SHIFT should be set to use either the
empirically derived Bragg-Slater radii^{Slater:1964} or ab initio derived radii based on
the universal density criterion^{Pacios:1995} (see Section 10.2.1 for more details).
Using the UNSHIFTED (default) scheme can
lead to highly unphysical results, including a charge-transfer vector that points in the opposite direction.

To perform SAPT/cDFT energy decomposition analysis, the user must request
a normal SAPT or XSAPT calculation (JOBTYPE = XSAPT), and in
addition specify the keyword CDFT-EDA in the *$sapt* input section.
Users of this method are asked to cite Ref. Lao:2016b.

CDFT-EDA

Requests a SAPT/cDFT-based energy decomposition analysis.

INPUT SECTION: *$sapt*

TYPE:

None

DEFAULT:

Not specified

OPTIONS:

The analysis is performed if the keyword is set.

RECOMMENDATION:

None

As shown in the example below, a *$cdft* input section is also required in order to specify the
monomer charges and spins for the cDFT part of the calculation.
The CDFT_POP variable may be set (in the *$rem* section) in order to specify
the electron-counting mechanism for cDFT. The options are either to use atomic Becke populations
(as in traditional cDFT calculations^{Kaduk:2012}), summed up for each monomer,
or else fragment-based Hirshfeld partitioning in which promolecule densities for the monomers are
used to obtain a whole-molecule version of Hirshfeld atomic charges.

CDFT_POP

Sets the charge partitioning scheme for cDFT in SAPT/cDFT

TYPE:

STRING

DEFAULT:

FBH

OPTIONS:

FBH
Fragment-Based Hirshfeld partitioning
BECKE
Atomic Becke partitioning

RECOMMENDATION:

None

$molecule 0 1 -- 0 1 O -0.702196054 -0.056060256 0.009942262 H -1.022193224 0.846775782 -0.011488714 H 0.257521062 0.042121496 0.005218999 -- 0 1 O 2.220871067 0.026716792 0.000620476 H 2.597492682 -0.411663274 0.766744858 H 2.593135384 -0.449496183 -0.744782026 $end $rem JOBTYPE XSAPT EXCHANGE gen BASIS aug-cc-pvdz LRC_DFT true CDFT_POP FBH ! Fragment-Based Hirshfeld charge partitioning $end $xpol embed none print 3 dft-lrc $end $sapt algorithm AO EmpiricalDisp 3 order 2 basis dimer cdft-eda print 3 $end $xc_functional x wPBE 1.0 c PBE 1.0 $end $lrc_omega 500 500 $end $cdft 0 1 1 3 0 1 1 3 s $end