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1: procedure QUIRQI ( ω, v ) 2: guess v 3: p = f$_{+}$ ( v ) , q = f$_{−}$ ( v ) 4: while e$_{rel}$ > ϵ and g$_{max}$ > γ and ω < ω old do 5: L [ v ] = [ F , v ] + [ G [ v ] , P ] 6: L [ p ] = f$_{−}$ ( L [ v ]) , L [ q ] = f$_{+}$ ( L [ v ]) 7: filter ( L [ p ] , L [ q ] , τ$_{mtx}$ ) 8: ω p = 〈 p , L [ p ] 〉 2 |〈 p , q 〉| , ω q = 〈 q , L [ q ] 〉 2 |〈 p , q 〉| , ω = ω p + ω q 9: g$_{p}$ = q ω q − L [ p ] , g$_{q}$ = p ω p − L [ q ] 10: e$_{rel}$ = ( ω old − ω ) /ω 11: g$_{max}$ = max i,j { [ g$_{p}$ ] ij , [ g$_{p}$ ] ij } 12: β p = 〈 g$_{p}$ , g$_{p}$ − g old p 〉 〈 g old p , g old p 〉 , β q = 〈 g$_{q}$ , g$_{q}$ − g old q 〉 〈 g old q , g old q 〉 13: ω old ← ω , g old p ← g$_{p}$ , g old q ← g$_{q}$ 14: h$_{p}$ ← g$_{p}$ + β $_{p}$h$_{p}$ , h$_{q}$ ← g$_{q}$ + β $_{q}$h$_{q}$ 15: h$_{v}$ = F ( h$_{p}$ , h$_{q}$ ) , h$_{v}$ ← f a ( h$_{v}$ ) 16: filter ( h$_{p}$ , h$_{q}$ , h$_{v}$ , τ$_{mtx}$ ) 17: L [ h$_{v}$ ] = [ F , h$_{v}$ ] + [ G [ h$_{v}$ ] , P ] 18: L [ h$_{p}$ ] = f$_{−}$ ( L [ h$_{v}$ ]) , L [ h$_{q}$ ] = f$_{+}$ ( L [ h$_{v}$ ]) 19: filter ( L [ h$_{p}$ ] , L [ h$_{q}$ ] , τ$_{mtx}$ ) 20: { λ p , λ q } = argmin { λ$_{p}$,λ$_{q}$ } ω [ p + λ $_{p}$h$_{p}$ , q + λ $_{q}$h$_{q}$ ] 21: p ← p + λ $_{p}$h$_{p}$ , q ← q + λ $_{q}$h$_{q}$ 22: v ← F ( p , q ) , v ← f a ( v ) 23: filter ( p , q , v , τ$_{mtx}$ ) 24: end while 25: end procedure 

ω [ p + λ$_{p}$ h$_{p}$ , q + λ$_{q}$ h$_{q}$ ] involves a two dimensional line- search (line 20) corresponding to minimization of 

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with coupling entering through terms in the denomina- tor such as U$_{pq}$ = 〈 h$_{p}$ , h$_{q}$ 〉 . A minimum in Eq. (3) can be found quickly to high precision by alternately substi- tuting one-dimensional solutions one into the other until self-consistency is reached. This semi-analytic approach starts with a rough guess at the pair { λ$_{p}$, λ$_{q}$ } (eg. found by a coarse scan) followed by iterative substitution, where for example the p -channel update is 

<img src='content_image/414307.jpg'>

with an analogous update for the q -channel obtained by swapping subscripts. As the solution decouples ( S$_{pq}$, T$_{pq}$ and U$_{pq}$ become small) the steps are found independently. 

QUIRQI has been implemented in FreeON [2], which employs the linear scaling Coulomb and Hartree-Fock ex- change kernels QCTC and ONX with cost and accuracy 

controlled by the two-electron screening threshold τ$_{2e}$ [13]. N -scaling solution of the QUIRQI matrix equations is achieved with the sparse approximate matrix-matrix multiply (SpAMM), with cost and accuracy determined by the drop tolerance τ$_{mtx}$ [3, 4, 11]. All calculations were carried out with version 4.3 of the gcc/gfortran compiler under version 8.04 of the Ubuntu Linux distribution and run on a fully loaded 2GHz AMD Quad Opteron 8350. 

For systems studied to date, QUIRQI is found to con- verge monotonically with rates comparable to the TDA as shown in Fig. 1. Based on the comparative perfor- mance presented by TINC, the TDA rate of convergence appears to be a lower bound for RPA solvers. In addition to the convergence rate, performance is strongly deter- mined by the initial guess. The following results have been obtained using the polarization response density along the polymer axis [19], which can be computed in O ( N ) by Perturbed Projection [20]. Also, a relative pre- cision of 4 digits in the excitation energy is targeted with the convergence parameters ϵ = 10 − 4 and γ = 10 − $^{3}$, with exit from the optimization loop on violation of monotonic convergence ( ω > ω old due to precision limitations asso- ciated with linear scaling approximations). 

In Fig. 2, linear scaling and convergence to the bulk limit are demonstrated for a series of polyphenylene vinylene (PPV) oligomers at the RHF/6-31G** level of theory for the threshold combinations { τ$_{mtx}$, τ$_{2e}$ } = { 10 − $^{4}$, 10 − $^{5}$} and { 10 − $^{5}$, 10 − $^{6}$} . Significantly more con- servative thresholds have been used for the Coulomb sums, which incur only minor cost. Convergence is reached in 24 − 25 iterations, with the cost of Coulomb summation via QCTC comparable to the cost of SpAMM( τ$_{mtx}$ = 10 − $^{4}$). In Fig. 3, linear scaling and con- vergence to the bulk limit are demonstrated for a series of (4,3) carbon nanotube segments at the RHF/3-21G level of theory for the same threshold combinations, again with convergence achieved in about 24-25 cycles. In both cases, tightening the pair { τ$_{mtx}$, τ$_{2e}$ } leads to a system- atically improved result. While the { 10 − $^{4}$, 10 − $^{5}$} thresh- olds that work well for PPV lead to a non-monotone be- havior with respect to extent for the nanotube series, dropping one more decade to { 10 − $^{5}$, 10 − $^{6}$} leads to a sharply improved behavior. Dropping thresholds further to { 10 − $^{6}$, 10 − $^{7}$} yields identical results to within the con- vergence criteria ( ∼ four digits) across the series, also scaling with N but at roughly twice the cost. 

These results demonstrate that QUIRQI can achieve both systematic error control and linear scaling in solu- tion of the RPA eigenproblem for systems with extended conjugation. Relative to PPV, the greater numerical sen- sitivity encountered with the nanotube series is consistent with the ground state problem, where a smaller band gap and greater atomic connectivity typically demand tighter thresholds. 

QUIRQI exploits decoupling of the Tsipper functional into nearly independent, pseudo-Hermitian quotients 

