Random phase approximation in projected oscillator orbitals

Abstract

The projected oscillator orbitals (pOOs) are localized virtual orbitals constructed by multiplying localized occupied orbitals by harmonics. Following a recent paper by Mussard and Ángyán (Theor Chem Acc 134:1, 2015), further developments of projected oscillator orbitals are shown, notably the equations for pOOs of general order as well as their overlaps are derived. The performance of these localized virtual orbitals is demonstrated up to third order. It is found that a good fraction of the aug-cc-pVQZ RPA correlation energy is recovered by use of a smaller number of pOOs. This is especially true where considering only the long-range correlation energy, which is important for the description of London dispersion forces.

Appendix

1.1 Relations between pOOs

The natural expression for a pOO of order n is given by the binomial theorem:

$$\begin{aligned} \vert i_{\alpha \dots \eta } \rangle&={\hat{P}}\, ({\hat{r}}_\alpha -D^i_\alpha )\dots ({\hat{r}}_\eta -D^i_\eta )\vert i \rangle = \sum _{k=1}^{n} A^n_k \end{aligned}$$

(11)

where $A^n_k$ is defined for convenience:

$$\begin{aligned} A^n_k =(-1)^{n-k}\underbrace{(D^i\dots D^i)}_{n-k} {\hat{P}}\, \underbrace{\phantom {()}{\hat{r}}\dots {\hat{r}}}_{k}\vert i \rangle \end{aligned}$$

(12)

and actually contains ${n \atopwithdelims ()k}$ repetitions with different assignations of the indices $\alpha \dots \eta$. For each of those repetitions, a different set of k indices among $\alpha \dots \eta$ is assigned to the k position operators and the remaining indices are given to the vector components $D^i$. For example, consider a pOO $\vert i_{\alpha \beta \gamma } \rangle$ of order 3. The term $A^3_1$ will be:

$$\begin{aligned} A^3_1{}^{\phantom {\prime \prime }}&=(D^i_\alpha D^i_\beta ) {\hat{P}}\,{\hat{r}}_\gamma \vert i \rangle \\ A^3_1{}^{\prime \phantom {\prime }}&=(D^i_\alpha D^i_\gamma ) {\hat{P}}\,{\hat{r}}_\beta \vert i \rangle \\ A^3_1{}^{\prime \prime }&=(D^i_\beta D^i_\gamma ) {\hat{P}}\,{\hat{r}}_\alpha \vert i \rangle \end{aligned}$$

i.e. it contains ${3 \atopwithdelims ()1}=3$ repetitions with different assignations of the indices $\alpha$, $\beta$ and $\gamma$.

We seek to prove Eq. (5), which amounts with the current notations to:

$$\begin{aligned} \vert i_{\alpha \dots \eta } \rangle =A^n_n -\sum _{k=1}^{n-1} \underbrace{\phantom {\vert nk \rangle }D^i \dots D^i}_{n-k} \underbrace{\vert K \rangle }_{\begin{array}{c} \text {pOO of}\\ {\text {order }}k \end{array}} \end{aligned}$$

(13)

where the same repetitions with different assignations of the indices $\alpha \dots \eta$ are found now for each terms of the sum. We begin by inserting in Eq. (13) the binomial expression of Eq. (11) for the pOO $\vert K \rangle$:

$$\begin{aligned} \vert i_{\alpha \dots \eta } \rangle&=A^n_n -\sum _{k=1}^{n-1} \underbrace{(D^i\dots D^i)}_{n-k} \sum _{m=1}^k A_m^k \end{aligned}$$

(14)

where we can rewrite the term in the sums using:

$$\begin{aligned}&\underbrace{(D^i\dots D^i)}_{n-k} A_m^k \nonumber \\&\quad = (-1)^{k-m} \underbrace{(D^i\dots D^i)}_{n-k} \underbrace{(D^i\dots D^i)}_{k-m} {\hat{P}}\, \underbrace{\phantom {(}{\hat{r}}\dots {\hat{r}}}_{m}\vert i \rangle \nonumber \\&\quad =(-1)^{k+n}A^n_m \end{aligned}$$

(15)

This yields:

$$\begin{aligned} \vert i_{\alpha \dots \eta } \rangle&=A^n_n +\sum _{k=1}^{n-1} \sum _{m=1}^k (-1)^{k+(n-1)} A^n_m \nonumber \\&=A^n_n +\sum _{m=1}^{n-1} \left[ \sum _{k=m}^{n-1} (-1)^{k+(n-1)}\right] A^n_m \end{aligned}$$

(16)

where in the last line the double summation was simply rearranged (think of summations over lines versus over columns of a matrix whose columns are composed of $A^n_m$).

In terms of assignations of the indices $\alpha \dots \eta$, in Eq. (15) there is ${n \atopwithdelims ()k}$ ways to assign indices to the first string of $n-k$ “$D^i$” and ${k \atopwithdelims ()m}$ ways to assign indices to the second string of $k-m$ “$D^i$”. This hence becomes a combinatorial problem, and recovering Eq. (11) from Eq. (16) boils down to proving that:

$$\begin{aligned} \sum _{k=m}^{n-1} (-1)^{k+(n-1)} {n\atopwithdelims ()k}{k\atopwithdelims ()m}={n\atopwithdelims ()m} \end{aligned}$$

(17)

which should be the number of repeated occurrences of the $A^n_m$ term with different indices in Eq. (16). This amounts to prove that:

$$\begin{aligned} \sum _{k=m}^{n-1} (-1)^{k+(n-1)} \frac{{n\atopwithdelims ()k}{k\atopwithdelims ()m}}{{n\atopwithdelims ()m}}=1 \end{aligned}$$

(18)

where, manipulating the fraction, the expression to study is:

$$\begin{aligned}&\sum _{k=m}^{n-1} (-1)^{k+(n-1)} {n-m\atopwithdelims ()k-m} \nonumber \\&\quad =\sum _{k=m}^{n} (-1)^{k+(n-1)} {n-m\atopwithdelims ()k-m} - (-1)^{-1} {n-m\atopwithdelims ()n-m} \nonumber \\&\quad =\left\{ \sum _{k^\prime =0}^{n-m} (-1)^{k^\prime } {n-m\atopwithdelims ()k^\prime }\right\} (-1)^{n-m+1} + 1 \nonumber \\&\quad =1 \end{aligned}$$

(19)

where the first step consists simply in adding and subtracting the n-th element of the sum, the second step is a change of the dummy index of the sum and the final step is a realization that the expression in curvy brackets is the binomial theorem for $(1+(-1))^{n-m}=0$. This proves Eq. (5).

1.2 Workflow

Although in our original paper [1] relations to construct all ingredients for the pOOs using solely data from the occupied space are presented (this is the purpose of the projected schemes), for the clarity of the proof-of-concept calculations in this paper, the $N_{\text {pOO}}\times N_{\text {vir}}$ matrix $\mathbf{V}$ to transform virtual orbitals to pOOs is introduced:

$$\begin{aligned} \vert i_{\alpha \dots \eta } \rangle =\sum _a^{N_{\text {vir}}} V_{i_{\alpha \dots \eta } a} \vert a \rangle \end{aligned}$$

(20)

We assume here that the pOOs can be expanded in the virtual basis, i.e. that the pOO space is a subspace of the virtual space. This should be more rigourously checked in later work. For example, the $\mathbf{V}$ matrix corresponding to the first-order pOO reads:

$$\begin{aligned} V^{(1)}_{i_\alpha a}=\langle i \vert {\hat{r}}_\alpha \vert a \rangle \end{aligned}$$

(21)

and, given Eq. (5), repeated in Appendix as Eq. (13), the $\mathbf{V}$ matrix corresponding to a N-th order pOO simply reads:

$$\begin{aligned} V^{(N)}_{i_{\alpha \dots \eta } a}=\langle i \vert {\hat{r}}_\alpha \dots {\hat{r}}_\eta \vert a \rangle -\sum _{k=1}^{n-1} \underbrace{\phantom {V_a}D^i\dots D^i}_{n-k} \underbrace{V^{(K)}_{\dots a}}_{\begin{array}{c} {\text {matrix for}}\\ \hbox { order}\ k\\ \text {pOO} \end{array}} \end{aligned}$$

(22)

The overlap between pOOs is $\mathbf{S}=\mathbf{V}\mathbf{V}^\dagger$ and the pseudo-canonical basis is obtained by solving the generalized eigenvalue equation

$$\begin{aligned} \mathbf{f}\mathbf{X}=\mathbf{S}\mathbf{X}\pmb \epsilon \end{aligned}$$

(23)

Because of linear dependencies built in the construction of the pOOs, the overlap matrix $\mathbf{S}$ is in general only positive semidefinite instead of positive definite. Hence the pseudo-canonical basis of effective size $N_{\text {eff}}\le N_{\text {pOO}}$ can alternatively by found with the $N_{\text {pOO}}\times N_{\text {eff}}$ matrix $\mathbf{X}$ that simultaneously diagonalizes $\mathbf{f}$ and $\mathbf{S}$.

From these relations, one can then construct from the $N_{\text {AO}}\times N_{\text {vir}}$ transformation matrix $\mathbf{C}$ the following sequential transformation matrices:

$$\begin{aligned} \underbrace{\mathbf{C}}_{\text {AO}\rightarrow {\text {vir}}} \rightarrow \underbrace{\mathbf{C}\mathbf{V}^\dagger }_{\text {AO}\rightarrow {\text {pOO}}} \rightarrow \underbrace{\mathbf{C}\mathbf{V}^\dagger \mathbf{X}}_{\text {AO}\rightarrow \text {pseudo-cano}} \end{aligned}$$

(24)

to easily either try out the pOOs, draw them, or calculate energies.