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# 11.6.2 Theory

(July 14, 2022)

## 11.6.2.1 Working Equations

For a system whose total density can be represented as $\boldsymbol{\gamma}=\boldsymbol{\gamma}^{A}+\boldsymbol{\gamma}^{B}$, i.e., the molecular orbitals on fragments $A$ and $B$ are orthogonal to each other, its total energy can be expressed as

 \displaystyle\begin{aligned} \displaystyle E_{\text{tot}}[\boldsymbol{\gamma}^% {A}+\boldsymbol{\gamma}^{B}]&\displaystyle=\underbrace{\mathrm{tr}(\boldsymbol% {\gamma}^{A}\mathbf{h})+J[\boldsymbol{\gamma}^{A}]+E_{\text{xc}}[\boldsymbol{% \gamma}^{A}]}_{\text{Energy of A}}+\underbrace{\mathrm{tr}(\boldsymbol{\gamma}% ^{B}\mathbf{h})+J[\boldsymbol{\gamma}^{B}]+E_{\text{xc}}[\boldsymbol{\gamma}^{% B}]}_{\text{Energy of B}}\\ &\displaystyle\qquad+\underbrace{J[\boldsymbol{\gamma}^{A},\boldsymbol{\gamma}% ^{B}]+E_{\text{xc}}[\boldsymbol{\gamma}^{A},\boldsymbol{\gamma}^{B}]}_{\text{% Non-additive terms}}\end{aligned} (11.100)

where $\mathbf{h}$ is the core-Hamiltonian matrix, $J$ and $E_{\text{xc}}$ are the Coulomb and XC energies (exact exchange $K$ will also be present if a hybrid functional is employed). In an embedding calculation where the electron density of $A$ (denoted as $\tilde{\gamma}^{A}$) is optimized at a high-level theory in the presence of $\gamma^{B}$ (which is obtained at a low-level theory and then fixed), the total energy can be expressed as

 $E_{\text{tot}}[\tilde{\gamma}^{A},\gamma^{B}]=E^{\text{high}}[\tilde{\gamma}^{% A}]+E^{\text{low}}[\tilde{\gamma}^{A}+\gamma^{B}]-E^{\text{low}}[\tilde{\gamma% }^{A}].$ (11.101)

Differentiating Eq. (11.101) with respect to $\tilde{\gamma}^{A}$ gives the Fock matrix for the embedding calculation:

 $\mathbf{f}^{\text{A-in-B}}=\frac{\delta}{\delta\tilde{\gamma}^{A}}E_{\text{tot% }}[\tilde{\gamma}^{A},\gamma^{B}]=\mathbf{F}^{\text{high}}[\tilde{\gamma}^{A}]% +\mathbf{v}_{\text{emb}}[\tilde{\gamma}^{A},\gamma^{B}],$ (11.102)

where the embedding potential

 \displaystyle\begin{aligned} \displaystyle\mathbf{v}_{\text{emb}}[\tilde{% \gamma}^{A},\gamma^{B}]&\displaystyle=\frac{\partial}{\partial\tilde{\gamma}^{% A}}\left(E^{\text{low}}[\tilde{\gamma}^{A}+\gamma^{B}]-E^{\text{low}}[\tilde{% \gamma}^{A}]\right)\\ &\displaystyle=\mathbf{F}^{\text{low}}[\tilde{\gamma}^{A}+\gamma^{B}]-\mathbf{% F}^{\text{low}}[\tilde{\gamma}^{A}]\\ &\displaystyle=(\mathbf{J}[\boldsymbol{\gamma}^{A}+\boldsymbol{\gamma}^{B}]-% \mathbf{J}[\boldsymbol{\gamma}^{A}])+(\mathbf{v}^{\text{low}}_{\text{xc}}[% \boldsymbol{\gamma}^{A}+\boldsymbol{\gamma}^{B}]-\mathbf{v}^{\text{low}}_{% \text{xc}}[\boldsymbol{\gamma}^{A}])+\mu\mathbf{P}^{B},\end{aligned} (11.103)

where $\mathbf{P}^{B}=\mathbf{S}\boldsymbol{\gamma}^{B}\mathbf{S}$ is the projector formed by $B$’s orbitals and $\mu$ is a large enough constant (e.g. 10${}^{6}$ a.u.) that effectively elevates the energies of $B$’s orbitals and thereby enforces the orthogonality between MOs on $A$ and $B$. In the Q-Chem implementation (with GEN_SCFMAN_EMBED = TRUE), an alternative approach is employed, where one diagonalizes a modified Fock matrix fromwhich the variational degrees of freedom spanned by $B$’s orbitals are projected out:

 $\tilde{\mathbf{f}}^{\text{A-in-B}}=(\mathbf{I}-\mathbf{S}\boldsymbol{\gamma}^{% B})\mathbf{f}^{\text{A-in-B}}(\mathbf{I}-\boldsymbol{\gamma}^{B}\mathbf{S}),$ (11.104)

where $\mathbf{S}$ is the AO overlap matrix.

One disadvantage of using the energy expression given by Eq. (11.101) is that the embedding potential [Eq. (11.103)] depends on $\tilde{\gamma}^{A}$, which needs to be updated in every iteration of the embedding calculation. Thus this scheme affords no savings in computational cost for DFT-in-DFT calculations. To address this limitation, a linearized approximation was suggested, 696 Lee S. J. R. et al.
Acc. Chem. Res.
(2019), 52, pp. 1359.
which is based on the following expansion (to the first order of $\tilde{\gamma}^{A}-\gamma^{A}$):

 $E^{\text{low}}[\tilde{\gamma}^{A}+\gamma^{B}]-E^{\text{low}}[\tilde{\gamma}^{A% }]\approx E^{\text{low}}[\gamma^{A}+\gamma^{B}]-E^{\text{low}}[\gamma^{A}]+% \mathrm{Tr}\left((\tilde{\gamma}^{A}-\gamma^{A})\mathbf{v}_{\text{emb}}[\gamma% ^{A},\gamma^{B}]\right),$ (11.105)

where

 $\mathbf{v}_{\text{emb}}[\gamma^{A},\gamma^{B}]=\frac{\delta}{\delta\gamma^{A}}% \left(E_{\text{DFT}}[\gamma^{A}+\gamma^{B}]-E_{\text{DFT}}[\gamma^{A}]\right)=% \mathbf{F}[\gamma^{A}+\gamma^{B}]-\mathbf{F}[\gamma^{A}]$ (11.106)

Based on this linearized approximation, the total energy of the entire system is approximately given by

 $E_{\text{tot}}[\tilde{\gamma}^{A};\gamma^{A},\gamma^{B}]=E^{\text{high}}[% \tilde{\gamma}^{A}]+E^{\text{low}}[\gamma^{A}+\gamma^{B}]-E^{\text{low}}[% \gamma^{A}]+\mathrm{Tr}\left((\tilde{\gamma}^{A}-\gamma^{A})\mathbf{v}_{\text{% emb}}[\gamma^{A},\gamma^{B}]\right),$ (11.107)

and the Fock matrix for embedding calculation:

 $\mathbf{f}^{\text{A-in-B}}=\frac{\delta}{\delta\tilde{\gamma}^{A}}E_{\text{tot% }}[\tilde{\gamma}^{A};\gamma^{A},\gamma^{B}]=\mathbf{F}^{\text{high}}[\tilde{% \gamma}^{A}]+\mathbf{v}_{\text{emb}}[\gamma^{A},\gamma^{B}]$ (11.108)

where the embedding potential stays unchanged during the embedding calculation. In Q-Chem 5.4.1 and versions after, this linearized approximation is used by default in projection-based embedding calculations.

For WFT-in-DFT calculations, one can absorb the embedding potential $\mathbf{v}_{\text{emb}}[\gamma^{A},\gamma^{B}]$ into the Hamiltonian that is used for the correlated WFT calculation, converting Eq. (11.107) into

 $E[\Psi^{A};\gamma^{A},\gamma^{B}]=\langle\Psi^{A}|\hat{H}^{\text{A-in-B}}|\Psi% ^{A}\rangle+E^{\text{low}}[\gamma^{A}+\gamma^{B}]-E^{\text{low}}[\gamma^{A}]-% \mathrm{Tr}\left(\gamma^{A}\mathbf{v}_{\text{emb}}[\gamma^{A},\gamma^{B}]% \right).$ (11.109)

## 11.6.2.2 Partition of the Occupied Space

An embedding calculation usually starts from an SCF calculation of the full system at the lower level of theory, which yields canonical MOs. Therefore, it is necessary to partition the occupied space and assign orbitals to fragments $A$ and $B$ (without losing generality, assuming $A$ is the embedded fragment). In the original work by Manby et al., 768 Manby F. R. et al.
J. Chem. Theory Comput.
(2012), 8, pp. 2564.
this was achieved by

• Performing a Pipek-Mezey localization 954 Pipek J., Mezey P. G.
J. Chem. Phys.
(1989), 90, pp. 4916.
of the canonical occupied orbitals;

• Assigning a PM-localized orbital to the “active” fragment $A$ if its Mulliken population on $A$ is greater than 0.4.

f This approach has been adopted as the default occupied space partition method in Q-Chem.

Recently a parameter-free and more robust partition scheme was proposed by Claudino and Mayhall, which is known as the Subsystem Projected AO Decomposition (SPADE) procedure. 219 Claudino D., Mayhall N. J.
J. Chem. Theory Comput.
(2019), 15, pp. 1053.
In this approach, one first transforms the occupied orbitals into the symmetrically orthogonalized AO basis:

 $\bar{\mathbf{C}}_{\text{occ}}=\mathbf{S}^{1/2}\mathbf{C}_{\text{occ}}$ (11.110)

and then denotes the rows in $\bar{\mathbf{C}}_{\text{occ}}$ that correspond to fragment $A$ as $\bar{\mathbf{C}}_{\text{occ}}^{A}$. A singular value decomposition (SVD) is then applied to $\bar{\mathbf{C}}_{\text{occ}}^{A}$: $\bar{\mathbf{C}}_{\text{occ}}^{A}=\mathbf{U}^{A}\Sigma^{A}(\mathbf{V}^{A})^{T}$, and the SPADE orbitals are then obtained by rotating the original $\mathbf{C}_{\text{occ}}$:

 $\mathbf{C}_{\text{occ}}^{\text{SPADE}}=\mathbf{C}_{\text{occ}}\mathbf{V}^{A}.$ (11.111)

The largest gap in the singular value spectrum determines the most appropriate occupied orbital partition under the given fragmentation.

## 11.6.2.3 Truncation of the Virtual Space Using Concentric Localization

A WFT-in-DFT calculation requires not only the occupied orbitals on the “active” fragment but also the virtual orbitals. Unlike the occupied orbitals, the virtual orbitals obtained from a projection-based embedding calculation are not assigned to fragments but stay delocalized. If the full virtual space is used in the post-SCF calculation, the savings on computational cost will be rather limited since only the number of occupied orbitals is reduced. Therefore, it is desirable to further truncate the virtual space so that one can significantly reduce the computational cost of WFT-in-DFT calculations.

Claudino and Mayhall recently proposed a simple and efficient approach to truncate the virtual space based on concentric localization (CL), 220 Claudino D., Mayhall N. J.
J. Chem. Theory Comput.
(2019), 15, pp. 6085.
which shares the same spirit as the SPADE partition scheme for occupied space. As the first step, the original set of delocalized virtual orbitals ($\mathbf{C}_{\text{vir}}$) represented in the working basis (WB) are projected onto the embedded fragment $A$ in a user-specified projection basis (PB):

 ${}^{A}\mathbf{C}_{\text{vir}}^{\prime}=(\mathbf{S}_{\text{PB},A}^{-1})(^{A}% \mathbf{S}_{\text{PB,WB}})\mathbf{C}_{\text{vir}}$ (11.112)

where the superscript “$A$” indicates that only the rows corresponding to fragment $A$’s basis functions are included and $\mathbf{S}_{\text{PB},A}$ denotes the overlap matrix for PB functions on fragment $A$ only. One can choose PB to be the same as WB or even a smaller basis set. A particular set of virtual orbitals denoted as $\mathbf{C}_{0}$ can then be selected by performing an SVD on the overlap between ${}^{A}\mathbf{C}_{\text{vir}}^{\prime}$ and $\mathbf{C}_{\text{vir}}$:

 $(^{A}\mathbf{C}_{\text{vir}}^{\prime})^{T}(^{A}\mathbf{S}_{\text{PB,WB}})% \mathbf{C}_{\text{vir}}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{T}$ (11.113)

with

 $\displaystyle\mathbf{V}$ $\displaystyle=[\mathbf{V}_{\text{span}}|\mathbf{V}_{\text{null}}]$ (11.114a) $\displaystyle\mathbf{C}_{0}$ $\displaystyle=\mathbf{C}_{\text{vir}}\mathbf{V}_{\text{span}}$ (11.114b) $\displaystyle\mathbf{C}_{\text{null},0}$ $\displaystyle=\mathbf{C}_{\text{vir}}\mathbf{V}_{\text{null}}$ (11.114c)

By construction, $\mathbf{C}_{0}$ should consist of the virtual valence shell of the WB. In order to achieve higher accuracy for the embedded correlated method, one can select more virtual orbitals from $\mathbf{C}_{\text{null},0}$ in a stepwise fashion. A recommended way 220 Claudino D., Mayhall N. J.
J. Chem. Theory Comput.
(2019), 15, pp. 6085.
is to singular value decompose the matrix $\mathbf{C}^{T}\mathbf{F}\mathbf{C}_{\text{null},0}$, i.e., the coupling between $\mathbf{C}_{0}$ and $\mathbf{C}_{\text{null},0}$ through the Fock operator

 $\mathbf{C}_{0}^{T}\mathbf{F}\mathbf{C}_{\text{null},0}=\mathbf{U}_{0}% \boldsymbol{\Sigma}_{0}\mathbf{V}_{0}^{T}$ (11.115)

where

 $\displaystyle\mathbf{V}_{0}$ $\displaystyle=[\mathbf{V}_{\text{span},0}|\mathbf{V}_{\text{null},0}]$ (11.116a) $\displaystyle\mathbf{C}_{1}$ $\displaystyle=\mathbf{C}_{\text{null},0}\mathbf{V}_{\text{span},0}$ (11.116b) $\displaystyle\mathbf{C}_{\text{null},1}$ $\displaystyle=\mathbf{C}_{\text{null},0}\mathbf{V}_{\text{null},0}$ (11.116c) $\displaystyle\mathbf{C}_{\text{vir, act}}$ $\displaystyle=[\mathbf{C}_{0}|\mathbf{C}_{1}]$ (11.116d)

As the size of $\mathbf{C}_{1}$ is the same as $\mathbf{C}_{0}$, going through the procedure given by Eq. (11.115) doubles the number of active virtual orbitals. This procedure can be carried on iteratively, rendering the accuracy of this method tunable:

 $\mathbf{C}_{n}^{T}\mathbf{F}\mathbf{C}_{\text{null},n}=\mathbf{U}_{n}% \boldsymbol{\Sigma}_{n}\mathbf{V}_{n}^{T}$ (11.117)

where

 $\displaystyle\mathbf{V}_{n}$ $\displaystyle=[\mathbf{V}_{\text{span},n}|\mathbf{V}_{\text{null},n}]$ (11.118a) $\displaystyle\mathbf{C}_{n+1}$ $\displaystyle=\mathbf{C}_{\text{null},n}\mathbf{V}_{\text{span},n}$ (11.118b) $\displaystyle\mathbf{C}_{\text{null},n+1}$ $\displaystyle=\mathbf{C}_{\text{null},n}\mathbf{V}_{\text{null},n}$ (11.118c) $\displaystyle\mathbf{C}_{\text{vir, act}}$ $\displaystyle=[\mathbf{C}_{0}|\mathbf{C}_{1}|\cdots|\mathbf{C}_{n+1}]$ (11.118d)

The virtual orbitals that span the null space, $\mathbf{C}_{\text{null},n+1}$, will remain inactive in the post-SCF calculations. In practice, one is often able to obtain sub-kcal/mol accuracy by only including $\mathbf{C}_{0}$ and $\mathbf{C}_{1}$, which is known as the “double-$\zeta$” CL shell model.