Local random phase approximation with projected oscillator orbitals

Abstract

An approximation to the many-body London dispersion energy in molecular systems is expressed as a functional of the occupied orbitals only. The method is based on the local-RPA theory. The occupied orbitals are localized molecular orbitals, and the virtual space is described by projected oscillator orbitals, i.e., functions obtained by multiplying occupied localized orbitals with solid spherical harmonic polynomials having their origin at the orbital centroids. Since we are interested in the long-range part of the correlation energy, responsible for dispersion forces, the electron repulsion is approximated by its multipolar expansion. This procedure leads to a fully non-empirical long-range correlation energy expression. Molecular dispersion coefficients calculated from determinant wave functions obtained by a range-separated hybrid method reproduce experimental values with <15 % error.

Appendices

Appendix 1: Dipolar oscillator orbitals in local frame

Let ${\varvec{\mathcal{R}}}^i$ be the rotation matrix which transforms an arbitrary vector ${\mathbf{v}}$ from the laboratory frame to the vector ${\mathbf{v}}^{\mathrm{loc}}$ in the local frame defined as the principal axes of the second moment tensor of the charge distribution associated with a given localized occupied orbital i, ${\varvec{\mathcal{R}}}^i\cdot {\mathbf{v}} = {\mathbf{v}}^{\mathrm{loc}}.$ The expression in the local frame of a POO $i_\alpha$ constructed from the LMO i is then:

$$\begin{aligned} \vert i_\alpha ^{\mathrm{loc}} \rangle&= \left( {\hat{I}}-\sum _m^{\mathrm{occ}}\vert m \rangle \langle m \vert \right) \left({\varvec{\mathcal{R}}}^i\cdot \left({\mathbf{r}}-{\mathbf{D}}^i\right) \right) _\alpha \vert i \rangle \nonumber \\&= \left({\varvec{\mathcal{R}}}^i\cdot {\mathbf{r}}\right) _\alpha \vert i \rangle - \sum _{m}^{\mathrm{occ}} \left({\varvec{\mathcal{R}}}^i\cdot \langle m \vert {\mathbf{r}}\vert i \rangle \right) _\alpha \vert m \rangle \nonumber \\&= \sum _\beta {\mathcal{R}}^i_{\alpha \beta } r_\beta \vert i \rangle - \sum _{m}^{\mathrm{occ}} \sum _\beta {\mathcal{R}}^i_{\alpha \beta } \langle m \vert r_\beta \vert i \rangle \vert m \rangle \end{aligned}$$

(39)

Appendix 2: Riccati equations in POO basis

The first-order wave function $\varPsi ^{(1)}$ can be written in terms of Slater determinants $\vert \ldots ab\ldots \vert$ formed with LMOs and canonical virtual orbitals a and b on the one hand, and on the other hand in terms of Slater determinants $\vert \ldots m_\alpha n_\beta \ldots \vert$ formed with LMOs and POOs $m_\alpha$ and $n_\beta$. That is to say that:

$$\varPsi ^{(1)} = \sum _{ij}^{\mathrm{occ}} \sum _{ab}^{\mathrm{virt}} T_{ab}^{ij} \vert \ldots ab \ldots \vert \approx \sum _{ij}^{\mathrm{occ}} \sum _{m_\alpha n_\beta }^\mathrm{POO} T_{m_\alpha n_\beta }^{ij} \vert \ldots m_\alpha n_\beta \ldots \vert .$$

(40)

The canonical virtual orbitals and the POOs in question are related by (see Eq. 6):

$$\vert m_\alpha \rangle =\sum _a^{\mathrm{virt}} \vert a \rangle V_{a m_\alpha } \quad \text {and}\quad \vert n_\beta \rangle =\sum _b^{\mathrm{virt}} \vert b \rangle V_{bn_\beta }.$$

(41)

This allows us to write:

$$\begin{aligned} \varPsi ^{(1)}&= \sum _{ij}^{\mathrm{occ}} \sum _{ab}^{\mathrm{virt}} T_{ab}^{ij} \vert \ldots ab \ldots \vert \nonumber \\&\approx \sum _{ij}^{\mathrm{occ}} \sum _{m_\alpha n_\beta }^\mathrm{POO} T_{m_\alpha n_\beta }^{ij} \vert \ldots m_\alpha n_\beta \ldots \vert \nonumber \\&= \sum _{ij}^{\mathrm{occ}} \sum _{m_\alpha n_\beta }^{\mathrm{POO}} \sum _{ab}^{\mathrm{virt}} V^{}_{am_\alpha } T_{m_\alpha n_\beta }^{ij} V^\dagger _{n_\beta b} \vert \ldots ab \ldots \vert , \end{aligned}$$

(42)

and leads to the transformation rule between the amplitudes in the VMO and in the POO basis:

$${\mathbf{T}}^{ij} = {\mathbf{V}} {\mathbf{T}}_{\mathrm{POO}}^{ij} {\mathbf{V}}^\dagger.$$

(43)

Multiplication of the Riccati equations of Eq. 9 by ${\mathbf{V}}^\dagger$ and ${\mathbf{V}}$ from the left and from the right, respectively, and expressing the amplitudes in POOs using Eq. 43 leads to:

$$\begin{aligned}{\mathbf{V}}^\dagger {\mathbf{R}}^{ij}{\mathbf{V}}&={\mathbf{V}}^\dagger {\mathbf{B}}^{ij}{\mathbf{V}} +{\mathbf{V}}^\dagger \left({\varvec{\epsilon}}+{\mathbf{A}}\right) ^{im}\left({\mathbf{V}} {\mathbf{T}}_{\mathrm{POO}}^{mj}{\mathbf{V}}^\dagger \right){\mathbf{V}} \nonumber \\&\quad +{\mathbf{V}}^\dagger \left({\mathbf{V}} {\mathbf{T}}_{\mathrm{POO}}^{im}{\mathbf{V}}^\dagger \right) \left({\varvec{\epsilon}}+{\mathbf{A}}\right) ^{mj}{\mathbf{V}} \nonumber \\&\quad +{\mathbf{V}}^\dagger \left({\mathbf{V}} {\mathbf{T}}_{\mathrm{POO}}^{im}{\mathbf{V}}^\dagger \right) {\mathbf{B}}^{mn}\left({\mathbf{V}}{\mathbf{T}}_{\mathrm{POO}}^{nj}{\mathbf{V}}^\dagger \right){\mathbf{V}}={\mathbf{0}} , \end{aligned}$$

(44)

where implicit summation conventions are supposed on m and n. Recognizing the expression for the overlap matrix, ${\mathbf{S}}_{\mathrm{POO}}={\mathbf{V}}^\dagger{\mathbf{V}},$ we obtain:

$$\begin{aligned} {\mathbf{R}}^{ij}_{\mathrm{POO}}&= {\mathbf{B}}^{ij}_{\mathrm{POO}} + \left({\varvec{\epsilon}}_{\mathrm{POO}}+{\mathbf{A}}_{\mathrm{POO}}\right) ^{im} {\mathbf{T}}_{\mathrm{POO}}^{mj} {\mathbf{S}}_{\mathrm{POO}} \nonumber \\&\quad + {\mathbf{S}}_{\mathrm{POO}} {\mathbf{T}}_{\mathrm{POO}}^{im} \left({\varvec{\epsilon}}_{\mathrm{POO}}+{\mathbf{A}}_{\mathrm{POO}}\right) ^{mj} \nonumber \\&\quad + {\mathbf{S}}_{\mathrm{POO}} {\mathbf{T}}_{\mathrm{POO}}^{im} {\mathbf{B}}^{mn}_{\mathrm{POO}} {\mathbf{T}}_{\mathrm{POO}}^{nj} {\mathbf{S}}_{\mathrm{POO}}={\mathbf{0}} ,\end{aligned}$$

(45)

which defines ${\mathbf{R}}^{ij}_{\mathrm{POO}},$ ${\varvec{\epsilon}}^{ij}_{\mathrm{POO}}$, ${\mathbf{A}}^{ij}_{\mathrm{POO}}$ and ${\mathbf{B}}^{ij}_{\mathrm{POO}}$ and which are the Riccati equations seen in Eq. 12.

Appendix 3: Solution of the Riccati equations in POO basis

To derive the iterative resolution of the Riccati equations seen in Eq. 12, we write explicitly the fock matrix contributions hidden in the matrix ${\varvec{\epsilon}}$. The matrix elements in canonical virtual orbitals, $\epsilon ^{ij}_{ab},$ read:

$$\epsilon ^{ij}_{ab}=\delta _{ij}\,f_{ab}-\delta _{ab}\,f_{ij},$$

(46)

so that the matrix element in POOs is (we omit the “POO” indices):

$$\epsilon ^{ij}_{m_\alpha n_\beta } =V^\dagger _{m_\alpha a}\epsilon ^{ij}_{ab}V_{b n_\beta } =\delta _{ij}\,f_{m_\alpha n_\beta } - S_{m_\alpha n_\beta }\,f_{ij}.$$

(47)

The terms in the Riccati equations containing the matrix ${\varvec{\epsilon}}$ then read (we use implicit summations over m and n):

$${\varvec{\epsilon}}^{im} {\mathbf{T}}^{mj} {\mathbf{S}} = {\mathbf{f}}{\mathbf{T}}^{ij}\,{\mathbf{S}} -f_{im}\, {\mathbf{S}} {\mathbf{T}}^{mj}\,{\mathbf{S}}$$

(48)

$${\mathbf{S}}{\mathbf{T}}^{im} {\varvec{\epsilon}}^{mj}= {\mathbf{S}} {\mathbf{T}}^{ij}\,{\mathbf{f}} -{\mathbf{S}}{\mathbf{T}}^{im}\,{\mathbf{S}}\, f_{mj}.$$

(49)

We this in mind, the Riccati equations of Eq. 12 yield:

$$\begin{aligned} {\mathbf{R}}^{ij}&= {\mathbf{B}}^{ij} + \left({\mathbf{f}} -f_{ii}\, {\mathbf{S}}\right) {\mathbf{T}}^{ij}\,{\mathbf{S}} + {\mathbf{S}} {\mathbf{T}}^{ij}\left({\mathbf{f}} -{\mathbf{S}}\,f_{jj}\right) \nonumber \\&\quad - \sum _{m\ne i} f_{im}\,{\mathbf{S}}{\mathbf{T}}^{mj}\,{\mathbf{S}} - \sum _{m\ne j} {\mathbf{S}}\, {\mathbf{T}}^{im} {\mathbf{S}}\, f_{mj} \nonumber \\&\quad + {\mathbf{A}}^{im}\, {\mathbf{T}}^{mj}\,{\mathbf{S}} + {\mathbf{S}}{\mathbf{T}}^{im}{\mathbf{A}}^{mj} + {\mathbf{S}}\, {\mathbf{T}}^{im}{\mathbf{B}}^{mn}{\mathbf{T}}^{nj}\,{\mathbf{S}}={\mathbf{0}}. \end{aligned}$$

(50)

Remember that the matrices are of dimension $N_{\mathrm{POO}} \times N_{\mathrm{POO}}$. Due to the nonorthogonality of the POOs and the non diagonal structure of the fock matrix, the usual simple updating scheme for the solution of the Riccati equations should be modified in a similar fashion as in the local coupled cluster theory [29]. The fock matrix in the basis of the POOs will be diagonalized by the matrix ${\mathbf{X}}$ obtained from the solution of the generalized eigenvalue problem:

$${\mathbf{f}}{\mathbf{X}} = {\mathbf{S}}\, {\mathbf{X}}\, {\varvec{\epsilon}}.$$

(51)

Note that the transformation ${\mathbf{X}}^\dagger {\mathbf{f}}\, {\mathbf{X}}$ does not brings us back to the canonical virtual orbitals. We can write the transformation by the orthogonal matrix ${\mathbf{X}}$ as $X^\dagger _{{\overline{a}} i_\alpha }\, f_{i_\alpha j_\beta }X_{j_\beta {\overline{b}}}=\delta _{{\overline{a}}{\overline{b}}}\,\varepsilon _{\overline{b}}$, where ${\overline{a}}$ and ${\overline{b}}$ are pseudo-canonical virtual orbitals that diagonalize the fock matrix expressed in POOs. The Riccati equations of Eq. 50 are transformed separately for each pair [ij] in the basis of the pseudo-canonical virtual orbitals that diagonalize ${\mathbf{f}}_{\mathrm{POO}}$:

$$\begin{aligned} {\mathbf{X}}^\dagger {\mathbf{R}}^{ij}\, {\mathbf{X}}&={\mathbf{X}}^\dagger {\mathbf{B}}^{ij} {\mathbf{X}} +\left( {\mathbf{X}}^\dagger {\mathbf{f}} - f_{ii} \,{\mathbf{X}}^\dagger {\mathbf{S}}\right) {\mathbf{T}}^{ij} {\mathbf{S}}{\mathbf{X}} +{\mathbf{X}}^\dagger {\mathbf{S}} \,{\mathbf{T}}^{ij}\left( {\mathbf{f}}{\mathbf{X}} - {\mathbf{S}}\, {\mathbf{X}} f_{jj}\right) \nonumber \\&\quad -\sum _{m\ne i} f_{im} {\mathbf{X}}^\dagger {\mathbf{S}} {\mathbf{T}}^{mj} {\mathbf{S}}\, {\mathbf{X}} -\sum _{m\ne j} {\mathbf{X}}^\dagger {\mathbf{S}} {\mathbf{T}}^{im} {\mathbf{S}} {\mathbf{X}} f_{mj} \nonumber \\&\quad +{\mathbf{X}}^\dagger {\mathbf{A}}^{im}\, {\mathbf{T}}^{mj} {\mathbf{S}}\, {\mathbf{X}} +{\mathbf{X}}^\dagger {\mathbf{S}} {\mathbf{T}}^{im}{\mathbf{A}}^{mj} {\mathbf{X}} \nonumber \\&\quad +{\mathbf{X}}^\dagger {\mathbf{S}}{\mathbf{T}}^{im} {\mathbf{B}}^{mn}\, {\mathbf{T}}^{nj} {\mathbf{S}}{\mathbf{X}}={\mathbf{0}} , \end{aligned}$$

(52)

which can be simplified by the application of the generalized eigenvalue equation Eq. 51 and the use of the relationships ${\mathbf{I}} = {\mathbf{S}} {\mathbf{X}} {\mathbf{X}}^\dagger = {\mathbf{X}} {\mathbf{X}}^\dagger {\mathbf{S}}$:

$$\begin{aligned}&{\overline{{\mathbf{R}}}}^{ij} ={\overline{{\mathbf{B}}}}^{ij} + \left({\varvec{\epsilon}} - f_{ii} {\mathbf{I}}\right) {\overline{{\mathbf{T}}}}^{ij} +{\overline{{\mathbf{T}}}}^{ij}\left({\varvec{\epsilon}}- f_{jj} {\mathbf{I}}\right) \nonumber \\&\quad - \sum _{m\ne i} f_{im} {\overline{{\mathbf{T}}}}^{mj} - \sum _{m\ne j} {\overline{{\mathbf{T}}}}^{im}f_{mj} \nonumber \\&\quad + {\overline{{\mathbf{A}}}}^{im} {\overline{{\mathbf{T}}}}^{mj} + {\overline{{\mathbf{T}}}}^{im} {\overline{{\mathbf{A}}}}^{mj} + {\overline{{\mathbf{T}}}}^{im} {\overline{{\mathbf{B}}}}^{mn}{\overline{{\mathbf{T}}}}^{nj}={\mathbf{0}} ,\end{aligned}$$

(53)

with the notations:

$$\begin{aligned}&{\overline{{\mathbf{R}}}}^{ij} = {\mathbf{X}}^\dagger {\mathbf{R}}^{ij} {\mathbf{X}} \nonumber \\&{\overline{{\mathbf{A}}}}^{ij} = {\mathbf{X}}^\dagger {\mathbf{A}}^{ij} {\mathbf{X}} \nonumber \\&{\overline{{\mathbf{B}}}}^{ij} = {\mathbf{X}}^\dagger {\mathbf{B}}^{ij} {\mathbf{X}} \nonumber \\&{\overline{{\mathbf{T}}}}^{ij} = {\mathbf{X}}^\dagger {\mathbf{S}}\,{\mathbf{T}}^{ij} {\mathbf{S}}\,{\mathbf{X}} .\end{aligned}$$

The new Riccati equations of Eq. 53 can be solved by the iteration formula:

$$\begin{aligned} {\overline{T}}^{ij(n)}_{{\overline{a}}{\overline{b}}} = - \frac{{\overline{B}}^{ij}_{{\overline{a}}{\overline{b}}} + \Delta {\overline{R}}^{ij}_{{\overline{a}}{\overline{b}}}\left( {\overline{{\mathbf{T}}}}^{(n-1)}\right) }{\varepsilon _{\overline{a}} - f_{ii} +\varepsilon _{\overline{b}} - f_{jj}}, \end{aligned}$$

(54)

where $\Delta {\overline{{\mathbf{R}}}}^{ij}({\overline{{\mathbf{T}}}})$ is

$$\begin{aligned} \Delta {\overline{{\mathbf{R}}}}^{ij}\left( {\overline{{\mathbf{T}}}}\right)&=-\sum _{m\ne i} f_{im} {\overline{{\mathbf{T}}}}^{mj} -\sum _{m\ne j} {\overline{{\mathbf{T}}}}^{im}f_{mj}\nonumber \\&\quad +{\overline{{\mathbf{A}}}}^{im} {\overline{{\mathbf{T}}}}^{mj} +{\overline{{\mathbf{T}}}}^{im} {\overline{{\mathbf{A}}}}^{mj} +{\overline{{\mathbf{T}}}}^{im} {\overline{{\mathbf{B}}}}^{mn} {\overline{{\mathbf{T}}}}^{nj}. \end{aligned}$$

(55)

As presented here, the update of the “non-diagonal” part of the residue is done in the pseudo-canonical basis. After convergence, we could transform the amplitudes back to the original POO basis according to:

$$\begin{aligned} {\mathbf{T}}^{ij}&= \left( {\mathbf{X}}^\dagger {\mathbf{S}}\right) ^{-1} {\overline{{\mathbf{T}}}}^{ij} \left( {\mathbf{S}} {\mathbf{X}}\right) ^{-1} \nonumber \\&= {\mathbf{S}}^{-1}\left( {\mathbf{X}}^\dagger \right) ^{-1} {\overline{{\mathbf{T}}}}^{ij}{\mathbf{X}}^{-1} {\mathbf{S}}^{-1} \nonumber \\&= {\mathbf{X}} {\mathbf{X}}^\dagger \left( {\mathbf{X}}^\dagger \right) ^{-1} {\overline{{\mathbf{T}}}}^{ij}{\mathbf{X}}^{-1}{\mathbf{X}} {\mathbf{X}}^\dagger \nonumber \\&= {\mathbf{X}} {\overline{{\mathbf{T}}}}^{ij}{\mathbf{X}}^\dagger . \end{aligned}$$

(56)

However, this back-transformation is not necessary since the correlation energy can be obtained directly in the pseudo-canonical basis, as:

$$\begin{aligned} \sum _{ij}^{\mathrm{occ}} \text {tr}\left\{ {\overline{{\mathbf{B}}}}^{ij} {\overline{{\mathbf{T}}}}^{ij}\right\}&= \sum _{ij}^{\mathrm{occ}} \text {tr}\left\{ {\mathbf{X}}^\dagger {\mathbf{B}}^{ij} {\mathbf{X}} {\mathbf{X}}^\dagger {\mathbf{S}} {\mathbf{T}}^{ij} {\mathbf{S}} {\mathbf{X}}\right\} \nonumber \\&= \sum _{ij}^{\mathrm{occ}} \text {tr}\left\{ {\mathbf{B}}^{ij} {\mathbf{T}}^{ij} {\mathbf{S}} {\mathbf{X}} {\mathbf{X}}^\dagger \right\} =\sum _{ij}^{\mathrm{occ}} \text {tr}\left\{ {\mathbf{B}}^{ij}{\mathbf{T}}^{ij}\right\} . \end{aligned}$$

(57)

Appendix 4: Riccati equations in the local excitation approximation

The local excitation approximation imposes that in the matrices ${\mathbf{R}}$, ${\varvec{\epsilon}}$, ${\mathbf{A}}$ and ${\mathbf{B}}$, the excitations remain on the same localized orbitals. In this approximation the Riccati equations of Eq. 12, with explicit virtual indexes, read:

$$\begin{aligned} R^{ij}_{i_\alpha j_\beta }&= B^{ij}_{i_\alpha j_\beta } + (\epsilon +A)^{im}_{i_\alpha m_\gamma } T^{mj}_{m_\gamma p_\delta }S_{p_\delta j_\beta } \nonumber \\&\quad + S_{i_\alpha p_\gamma } T^{im}_{p_\gamma m_\delta } (\epsilon +A)^{mj}_{m_\delta j_\beta } \nonumber \\&\quad + S_{i_\alpha p_\gamma } T^{im}_{p_\gamma m_\delta }B^{mn}_{m_\delta n_\tau } T^{nj}_{n_\tau q_\zeta }S^{~}_{q_\zeta j_\beta }=0. \end{aligned}$$

(58)

In this context, the terms containing the matrix ${\varvec{\epsilon}}$ are (with explicit POO indexes):

$$\begin{aligned} \epsilon ^{im}_{i_\alpha m_\gamma } T^{mj}_{m_\gamma p_\delta } S _{p_\delta j_\beta } = f_{i_\alpha i_\gamma } T^{ij}_{i_\gamma p_\delta } S_{p_\delta j_\beta } -f_{im} S_{i_\alpha m_\gamma } T^{mj}_{m_\gamma p_\delta } S_{p_\delta j_\beta } \end{aligned}$$

(59)

$$\begin{aligned} S _{i_\alpha p_\gamma } T^{im}_{p_\gamma m_\delta } \epsilon ^{mj}_{m_\delta j_\beta } = S_{i_\alpha p_\gamma } T^{ij}_{p_\gamma j_\delta } f_{j_\delta j_\beta } -S_{i_\alpha p_\gamma } T^{im}_{p_\gamma m_\delta } S_{m_\delta j_\beta }f_{mj}. \end{aligned}$$

(60)

Inserting this in Eq. 58 and using the shorthand notation $R^{ij}_{i_\alpha j_\beta }\equiv R^{ij}_{\alpha \beta }$, $B^{ij}_{i_\alpha j_\beta }\equiv B^{ij}_{\alpha \beta }$, $f _{i_\alpha j_\beta }\equiv f^{ij}_{\alpha \beta }$ and $S _{i_\alpha j_\beta }\equiv S^{ij}_{\alpha \beta },$ (note the matrix elements $T^{ij}_{i_\gamma p_\delta }$ cannot yet be translated to the shorthand notation) one obtains:

$$\begin{aligned} R^{ij}_{\alpha \beta }&= B^{ij}_{\alpha \beta } +f^{ii}_{\alpha \gamma } T^{ij}_{i_\gamma p_\delta } S^{pj}_{\delta \beta } -f_{im} S^{im}_{\alpha \gamma } T^{mj}_{m_\gamma p_\delta } S^{pj}_{\delta \beta } +A^{im}_{\alpha \gamma } T^{mj}_{m_\gamma p_\delta } S^{pj}_{\delta \beta } \nonumber \\&\quad +S^{ip}_{\alpha \gamma } T^{ij}_{p_\gamma j_\delta } f^{jj}_{\delta \beta } -S^{ip}_{\alpha \gamma }T^{im}_{p_\gamma m_\delta } S^{mj}_{\delta \beta } f_{mj} +S^{ip}_{\alpha \gamma } T^{im}_{p_\gamma m_\delta }A^{mj}_{\delta \beta } \nonumber \\&\quad +S^{ip}_{\alpha \gamma } T^{im}_{p_\gamma m_\delta }B^{mn}_{\delta \tau } T^{nj}_{n_\tau q_\zeta }S^{qj}_{\zeta \beta }=0. \end{aligned}$$

(61)

It is then a further approximation to tell that the POOs coming from different LMOs have a negligible overlap, i.e., that $S^{ij}_{\alpha \beta }=\delta _{ij} S^{ii}_{\alpha \beta }.$ The Riccati equations become:

$$\begin{aligned} R^{ij}_{\alpha \beta }&= B^{ij}_{\alpha \beta } +f^{ii}_{\alpha \gamma } T^{ij}_{i_\gamma j_\delta } S^{jj}_{\delta \beta } -f_{ii} S^{ii}_{\alpha \gamma } T^{ij}_{i_\gamma j_\delta } S^{jj}_{\delta \beta } +A^{im}_{\alpha \gamma } T^{mj}_{m_\gamma j_\delta } S^{jj}_{\delta \beta } \nonumber \\&\quad +S^{ii}_{\alpha \gamma } T^{ij}_{i_\gamma j_\delta } f^{jj}_{\delta \beta } -S^{ii}_{\alpha \gamma } T^{ij}_{i_\gamma j_\delta } S^{jj}_{\delta \beta } f_{jj} +S^{ii}_{\alpha \gamma }T^{im}_{i_\gamma m_\delta } A^{mj}_{\delta \beta } \nonumber \\&\quad +S^{ii}_{\alpha \gamma } T^{im}_{i_\gamma m_\delta } B^{mn}_{\delta \tau } T^{nj}_{n_\tau j_\zeta } S^{jj}_{\zeta \beta }=0, \end{aligned}$$

(62)

which, in turn, allows us to use the shorthand notation $T_{i_\alpha j_\beta }\equiv T^{ij}_{\alpha \beta }$ to arrive at:

$$\begin{aligned} {\mathbf{R}}^{ij}&= {\mathbf{B}}^{ij} +{\mathbf{f}}^{ii}{\mathbf{T}}^{ij}{\mathbf{S}}^{jj} -f_{ii} {\mathbf{S}}^{ii}{\mathbf{T}}^{ij}{\mathbf{S}}^{jj} +{\mathbf{A}}^{im}{\mathbf{T}}^{mj}{\mathbf{S}}^{jj} \nonumber \\&\quad +{\mathbf{S}}^{ii}{\mathbf{T}}^{ij}{\mathbf{f}}^{jj} -{\mathbf{S}}^{ii}{\mathbf{T}}^{ij}{\mathbf{S}}^{jj} f_{jj} +{\mathbf{S}}^{ii}{\mathbf{T}}^{im}{\mathbf{A}}^{mj} \nonumber \\&\quad +{\mathbf{S}}^{ii}{\mathbf{T}}^{im}{\mathbf{B}}^{mn}{\mathbf{T}}^{nj}{\mathbf{S}}^{jj}=0 ,\end{aligned}$$

(63)

Appendix 5: Screened dipole interaction tensor

Any interaction $L({\mathbf{r}})$ can be expanded in multipole series using a double Taylor expansion around appropriately selected centers, here ${\mathbf{D}}^i$ and ${\mathbf{D}}^j$, such that, with ${\mathbf{r}}={\mathbf{r}}^i-{\mathbf{r}}^j=({\mathbf{r}}^i-{\mathbf{D}}^i)+{\mathbf{D}}^{ij}-({\mathbf{r}}^j-{\mathbf{D}}^j)$ where ${\mathbf{D}}^{ij}={\mathbf{D}}^i-{\mathbf{D}}^j$:

$$\begin{aligned} L({\mathbf{r}})&= L^{ij}\left( {\mathbf{D}}^{ij}\right) +\sum _\alpha \left( {\hat{r}}^i_\alpha -D^i_\alpha \right) L^{ij}_\alpha \left( {\mathbf{D}}^{ij}\right) +\sum _\alpha \left( {\hat{r}}^j_\alpha -D^j_\alpha \right) L^{ij}_\alpha \left( {\mathbf{D}}^{ij}\right) \nonumber \\&\quad +\sum _{\alpha \beta }\left( {\hat{r}}^i_\alpha -D^i_\alpha \right) \left( {\hat{r}}^j_\beta -D^j_\beta \right) L^{ij}_{\alpha \beta }\left( {\mathbf{D}}^{ij}\right) +\cdots , \end{aligned}$$

(64)

where the definitions of $L^{ij}_\alpha ({\mathbf{D}}^{ij})$, $L^{ij}_{\alpha \beta }({\mathbf{D}}^{ij})$ are obvious. For example, in the case of the long-range interaction, $L({\mathbf{r}})$ will be defined according to the RSH theory as:

$$L({\mathbf{r}}) =\frac{\text {erf}(\mu r)}{r},$$

(65)

with $r=|{\mathbf{r}}|.$

The multipolar expansion of the long-range interaction leads to the following first and second-order interaction tensors:

$$L^{ij}_\alpha \left( {\mathbf{D}}^{ij}\right)= - \frac{D^{ij}_\alpha }{{D^{ij}}^3} \left( 1-\frac{2}{\sqrt{\pi }} {D^{ij}} \mu \hbox{e}^{-\mu ^2{D^{ij}}^2}-\text{erf}\left( \mu {D^{ij}}\right) \right)$$

(66)

$$\begin{aligned} L^{ij}_{\alpha \beta }\left( {\mathbf{D}}^{ij}\right)&= \frac{3D^{ij}_\alpha D^{ij}_\beta }{{D^{ij}}^5} \left( \text{erf}\left( \mu {D^{ij}}\right) - \frac{2}{3\sqrt{\pi }} {D^{ij}}\mu \text{e}^{-\mu ^2{D^{ij}}^2} \left( 3+2 {D^{ij}}^2\mu ^2\right) \right) \nonumber \\&\quad -\frac{\delta _{\alpha \beta }{D^{ij}}^2}{{D^{ij}}^5} \left( \text{erf}\left( \mu {D^{ij}}\right) - \frac{2}{\sqrt{\pi }} D^{ij} \mu \text{e}^{-\mu ^2{D^{ij}}^2}\right) . \end{aligned}$$

(67)

Remembering that the full-range Coulomb interaction tensor reads $T^{ij}_{\alpha \beta }({\mathbf{D}}^{ij})=3 D^{ij}_\alpha D^{ij}_\beta -\delta _{\alpha \beta }{D^{ij}}^2){D^{ij}}^{-5},$ the long-range interaction tensor can be written in an alternate form which clearly shows the damped dipole–dipole interaction contribution:

$$\begin{aligned} L^{ij}_{\alpha \beta }\left( {\mathbf{D}}^{ij}\right)&= T^{ij}_{\alpha \beta }\left( {\mathbf{D}}^{ij}\right) \left( \text{erf}\left( \mu {D^{ij}}\right) - \frac{2}{3 \sqrt{\pi }}{D^{ij}} \mu \text{e}^{-\mu ^2 {D^{ij}}^2} \left( 3+ 2 {D^{ij}}^2 \mu ^2 \right) \right) \nonumber \\ {}&\quad - \delta _{\alpha \beta } \hbox {e}^{-\mu ^2 {D^{ij}}^2} \frac{4 \mu ^3}{3 \sqrt{\pi }}. \end{aligned}$$

(68)

The trace of the tensor product (used for the spherically averaged $\hbox {C}_6$) then reads:

$$\begin{aligned} \sum _{\alpha \beta } L^{ij}_{\alpha \beta }L^{ij}_{\alpha \beta }&=\frac{6}{{D^{ij}}^6}\left( 4 \text{e}^{-2 {D^{ij}}^2 \mu ^2} {D^{ij}} \mu \left( \frac{ {D^{ij}} \mu \left( 3+4 {D^{ij}}^2 \mu ^2+2 {D^{ij}}^4 \mu ^4\right) }{3 \pi }\right. \right. \nonumber \\&\left. \left. \quad -\frac{ \left( 3+2 {D^{ij}}^2 \mu ^2\right) \text{erf}\left( {D^{ij}} \mu \right) }{3 \sqrt{\pi }} \right) +\text{erf}\left( {D^{ij}} \mu \right) ^2\right) \nonumber \\ {}&=\frac{6}{{D^{ij}}^6} F_{\mathrm{damp}}^\mu \left( {D^{ij}}\right) .\end{aligned}$$

(69)

Appendix 6: Fock matrix element in POO basis

The occupied–occupied block of the fock matrix, $f_{ij},$ is known. The POOs are orthogonal to the occupied subspace of the original basis set, they satisfy the local Brillouin theorem, i.e., the occupied-virtual block is zero. As a result, in the local excitation approximation, we need only to deal with the fock matrix elements $f^{ii}_{\alpha \beta }$:

$$f^{ii}_{\alpha \beta }=\langle i_\alpha \vert \,{\hat{f}}\vert i_\beta \rangle =\langle i \vert {\hat{r}}_\alpha {\hat{Q}}\,{\hat{f}}{\hat{Q}}{\hat{r}}_\beta \vert i \rangle,$$

(70)

from which we directly derive the quantity $f^i_[{\mathrm{M}]}$ of Eq. 36:

$$f^i_{\mathrm{[M]}}=\sum _\alpha f^{ii}_{\alpha \alpha } =\sum _{ab}^{\mathrm{virt}} \langle i \vert {\hat{r}}_\alpha \vert a \rangle f_{ab} \langle b \vert {\hat{r}}_\alpha \vert i \rangle .$$

(71)

Since we would like to express everything in occupied orbitals, we expand the projector ${\hat{Q}}$ and use that the occupied-virtual block of the fock matrix is zero to obtain the following expression:

$$f^{ii}_{\alpha \beta } = \langle i \vert \,{\hat{r}}_\alpha \,{\hat{f}} {\hat{r}}_\beta \vert i \rangle - \sum _{mn}^{\mathrm{occ}} \langle i \vert {\hat{r}}_\alpha \vert m \rangle\, f_{mn} \langle n \vert {\hat{r}}_\beta \vert i \rangle .$$

(72)

In order to transform the triple operator product, ${\hat{r}}_\alpha\, {\hat{f}} {\hat{r}}_\beta ,$ let us consider the following double commutator:

$$\left[ {\hat{r}}_\alpha ,\left[ {\hat{r}}_\beta ,{\hat{f}}\right] \right] = - \delta _{\alpha \beta } .$$

(73)

Note that this holds provided that the fockian contains only local potential terms, which commute with the coordinate operator: see later for the more general case. In this special case, the double commutator can be written as

$$\left[ {\hat{r}}_\alpha ,\left[ {\hat{r}}_\beta ,{\hat{f}}\right] \right] = {\hat{r}}_\alpha {\hat{r}}_\beta {\hat{f}} -{\hat{r}}_\alpha {\hat{f}} {\hat{r}}_\beta -{\hat{r}}_\beta {\hat{f}} {\hat{r}}_\alpha +{\hat{f}} {\hat{r}}_\beta {\hat{r}}_\alpha =-\delta _{\alpha \beta },$$

(74)

which allows us to express the two triple products:

$${\hat{r}}_\alpha {\hat{f}} {\hat{r}}_\beta +{\hat{r}}_\beta {\hat{f}} {\hat{r}}_\alpha = \delta _{\alpha \beta } +{\hat{r}}_\alpha {\hat{r}}_\beta {\hat{f}} +{\hat{f}} {\hat{r}}_\beta {\hat{r}}_\alpha .$$

(75)

The diagonal matrix element of the triple operator product is then:

$$\langle i \vert {\hat{r}}_\alpha {\hat{f}} {\hat{r}}_\beta \vert i \rangle = \tfrac{1}{2}\delta _{\alpha \beta } +\tfrac{1}{2}\left( \langle i \vert {\hat{r}}_\alpha {\hat{r}}_\beta {\hat{f}} \vert i \rangle + \langle i \vert {\hat{f}}{\hat{r}}_\beta {\hat{r}}_\alpha \vert i \rangle \right)$$

(76)

Since the localized orbitals satisfy local Brillouin theorem, we finally obtain for the matrix elements of the fock operator with multiplicative potential (typically Kohn-Sham operator with local or semi-local functionals) between two oscillator orbitals:

$$\begin{aligned} f^{ii}_{\alpha \beta }&= \tfrac{1}{2}\delta _{\alpha \beta } + \tfrac{1}{2} \sum _m^{\mathrm{occ}} \left( \langle i\, \vert {\hat{r}}_\alpha {\hat{r}}_\beta \vert m \rangle\, f_{mi} +f_{im}\langle m \vert {\hat{r}}_\alpha {\hat{r}}_\beta \vert i \rangle \right) \nonumber \\&\quad - \sum _{mn}^{\mathrm{occ}} \langle i\, \vert {\hat{r}}_\alpha \vert m \rangle \, f_{mn} \langle n \vert {\hat{r}}_\beta \vert i \rangle .\end{aligned}$$

(77)

From this, we obtain the quantity $f^i_{\mathrm{[O]}}$ of Eq. 38:

$$\begin{aligned} f^i_{\mathrm{[O]}}=\sum _\alpha f^{ii}_{\alpha \alpha }&=\tfrac{3}{2} +\tfrac{1}{2} \sum _m^{\mathrm{occ}} \left( \langle i \,\vert \hat{{\mathbf{r}}}^2\vert m \rangle\, f_{mi} +f_{im}\langle m \vert \hat{{\mathbf{r}}}^2\vert i \rangle \right) \nonumber \\&\quad - \sum _{mn}^{\mathrm{occ}} \sum _\alpha \langle i\, \vert {\hat{r}}_\alpha \vert m \rangle\, f_{mn} \langle n \vert {\hat{r}}_\alpha \vert i \rangle .\end{aligned}$$

(78)

In the more general case, i.e., when the fockian contains a nonlocal exchange operator, like in hybrid DFT and in Hartree–Fock calculations, the relation seen Eq. 73 does not hold any more and the commutator of the position operator with the fockian contains an exchange contribution [76, 77], which gives rise to an additional term:

$$\left\langle {i}\vert \left[ {\hat{r}}_\alpha , \left[ {\hat{r}}_\beta ,\hat{K}\right] \right] \vert {i}\right\rangle = \sum _m^{\mathrm{occ}} \left\langle {i m}\vert \left( {\hat{r}}_\alpha -{\hat{r}}_\alpha ^\prime \right) w\left( {\mathbf{r}},{\mathbf{r}}^\prime \right) \left( {\hat{r}}_\beta -{\hat{r}}_\beta ^\prime \right) \vert {m i}\right\rangle ,$$

(79)

where the nonlocal exchange operator is defined as

$$\hat{K} = \sum _m^{\mathrm{occ}} \int \hbox {d}{\mathbf{r}}^\prime \phi _m^\dagger \left( {\mathbf{r}}^\prime \right) w\left( {\mathbf{r}},{\mathbf{r}}^\prime \right) \hat{P}_{{\mathbf{r}} {\mathbf{r}}^\prime } \phi _m\left( {\mathbf{r}}^\prime \right) ,$$

(80)

where $\hat{P}_{{\mathbf{r}} {\mathbf{r}}^\prime }$ is the permutation operator that changes the coordinates ${\mathbf{r}}^\prime$ appearing after $\hat{K}$ to ${\mathbf{r}},$ and we recall that $w({\mathbf{r}},{\mathbf{r}}^\prime )$ is the two-electron interaction. Hence, the diagonal blocks of the POO fockian in the general case can be written as:

$$\begin{aligned} \langle i_\alpha \vert {\hat{f}}\vert i_\beta \rangle&=\tfrac{1}{2} \delta _{\alpha \beta } - \tfrac{1}{2} \sum _m^{\mathrm{occ}} \langle i m \vert \left( {\hat{r}}_\alpha -{\hat{r}}_\alpha ^\prime \right) w\left( {\mathbf{r}},{\mathbf{r}}^\prime \right) \left( {\hat{r}}_\beta -{\hat{r}}_\beta ^\prime \right) \vert m i \rangle \nonumber \\&\quad + \tfrac{1}{2} \sum _m^{\mathrm{occ}} \left( \langle i \vert {\hat{r}}_\alpha {\hat{r}}_\beta \vert m \rangle f_{mi} +f_{im}\langle m \vert {\hat{r}}_\alpha {\hat{r}}_\beta \vert i \rangle \right) \nonumber \\&\quad - \sum _{mn}^{\mathrm{occ}} \langle i \vert {\hat{r}}_\alpha \vert m \rangle f_{mn} \langle n \vert {\hat{r}}_\beta \vert i \rangle . \end{aligned}$$

(81)

In the present work, the exchange contribution, which is present only in the case of Hartree–Fock or hybrid density functional fockians and which would give rise to non-conventional two-electron integrals, is not treated explicitly. Possible approximate solutions for this problem will be discussed in forthcoming works. Although we do not use the elements of the off-diagonal ($i \ne j$) blocks of the POO fock operator, for the sake of completeness we give its expression:

$$\begin{aligned} \langle i_\alpha \vert {\hat{f}}\vert j_\beta \rangle&= - \langle i\, \vert {\hat{r}}_\alpha \hat{\nabla }_\beta \vert j \rangle + \sum _m^{\mathrm{occ}} \langle i m \vert {\hat{r}}_\alpha w\left( {\mathbf{r}},{\mathbf{r}}^\prime \right) \left( {\hat{r}}_\beta -{\hat{r}}_\beta ^\prime \right) \vert m j \rangle \nonumber \\&\quad + \sum _m^{\mathrm{occ}} \langle i \,\vert {\hat{r}}_\alpha {\hat{r}}_\beta \vert m \rangle\, f_{mi} \nonumber \\&\quad - \sum _{mn}^{\mathrm{occ}} \langle i \,\vert {\hat{r}}_\alpha \vert m \rangle\, f_{mn} \langle n \vert {\hat{r}}_\beta \vert i \rangle . \end{aligned}$$

(82)

To derive this, instead of the double commutator of Eq. 73, one needs to consider the product of the commutator with a coordinate operator

$${\hat{r}}_\alpha \left[ {\hat{r}}_\beta ,{\hat{f}}\right] = {\hat{r}}_\alpha \left[ {\hat{r}}_\beta ,\hat{T}\right] - {\hat{r}}_\alpha \left[ {\hat{r}}_\beta ,\hat{K}\right] ,$$

(83)

where $\hat{T}$ is the kinetic energy operator, and $[{\hat{r}}_\beta ,\hat{T}]={\hat{\nabla}}_\beta$.