4 leading to aggressive convergence rates equivalent to the fully Hermitian TDA, while remaining variational with respect to matrix truncation ( τ$_{mtx}$ ). However, QUIRQI is not variational with respect to the screening parame- ter τ$_{2e}$ . It can be systematically improved by tightening τ$_{2e}$ though, due to rigorous error bounds based on the Schwartz inequality [13]. These properties present op- portunities for more precise error control as suggested by Rubensson, Rudberg and Salek [12]. Further, these properties are expected to hold even for the most general SCF models, with the only difference being an increas- ingly localized transition density matrix and larger cost prefactor with an increasing DFT component. Finally, a variational approach allows considerable flexibility in the path to solution, as errors due to approximation can be overcome by optimization, offering opportunities for single precision GPU acceleration, variable thresholding, incremental Fock builds as well as extrapolation tech- niques. This work was supported by the U.S. Department of Energy and Los Alamos LDRD funds. Los Alamos Na- tional Laboratory was operated by the Los Alamos Na- tional Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396. Special acknowl- edgments go the International Ten Bar Café for scientific vlibations, and to Sergei Tretiak and Nicolas Bock for valuable input. Figure 3: Approach to the bulk limit of the first excited state of the (4,3) carbon nanotube segment at the 3-21G/RPA level of theory, with inset showing linear scaling cost for HF ex- change (ONX), sparse linear algebra (SpAMM) and Coulomb sums (QCTC). Figure 2: Approach to the bulk limit of the PPV first excited state at the 6-31G**/RPA level of theory, with inset showing linear scaling cost for HF exchange (ONX) and sparse lin- ear algebra (SpAMM). The cost of Coulomb sums with much tighter thresholds are comparable to those for the SpAMM. K. Nemeth, A. M. N. Niklasson, A. Odell, E. Schwe- gler, C. J. Tymczak, and V. Weber, 2009, FreeON ( Sep-18-2009 ), Los Alamos National Laboratory (LA- CC-04-086, Copyright University of California)., URL http://www.nongnu.org/freeon/ .  2.45  2.5  2.55  2.6  2.65  2.7  2.75  2.8  4  8  12  16  20  24 RPA Excitation Energy (eV)  nm ( $_{mtx}$=1D-4, $_{2e}$=1D-5) ( $_{mtx}$=1D-5, $_{2e}$=1D-6)  0  20  40  60  80  100  120  140  160  0  500  1000 1500 Total Time (hours) Atoms ONX( $_{2e}$=1D-6) SpAMM( $_{mtx}$=1D-5) QCTC  3.1  3.15  3.2  3.25  3.3  3.35  3.4  3.45  0  10  20  30  40  50  60 RPA Excitation Energy (eV)  nm ( $_{mtx}$=1D-4, $_{2e}$=1D-5) ( $_{mtx}$=1D-5, $_{2e}$=1D-6)  0  10  20  30  40  50  60  70  80  90  0 500 1000 Total Time (hours) Atoms ONX( $_{2e}$=1D-5) ONX( $_{2e}$=1D-6) SpAMM( $_{mtx}$=1D-4) SpAMM( $_{mtx}$=1D-5) [3] Challacombe, M., 1999, J. Chem. Phys. 110 , 2332. [4] Challacombe, M., 2000, Comp. Phys. Comm. 128 , 93. [5] Furche, F., 2008, J. Chem. Phys. 129 (11), 114105. [6] Harl, J., and G. Kresse, 2009, Phys. Rev. Lett. 103 (5), 056401. [7] Igumenshchev, K. I., S. Tretiak, and V. Y. Chernyak, 2007, J. Chem. Phys. 127 (11), 114902. [8] Lu, D., Y. Li, D. Rocca, and G. Galli, 2009, Phys. Rev. Lett. 102 (20), 206411. [9] Magyar, R. J., and S. Tretiak, 2007, J. Chem. Theor. Comp. 3 , 976. [10] Narita, S., and T. Shibuya, 1992, Can. J. Chem. 70 , 296. [11] Niklasson, A. M. N., C. J. Tymczak, and M. Challa- combe, 2003, J. Chem. Phys. 118 , 8611. [12] Rubensson, E. H., E. Rudberg, and P. Salek, 2008, J. Math. Phys. 49 (3), 032103. [13] Schwegler, E., M. Challacombe, and M. Head-Gordon, 1997, J. Chem. Phys. 106 , 9708. [14] Simoncini, V., 2005, SIAM J. Num. Anal. 43 (3), 1155 . [15] Simoncini, V., and D. B. Szyld, 2007, Num. Lin. Alg. App. 14 (1), 1 . [16] Song, J.-W., M. A. Watson, and K. Hirao, 2009, J. Chem. Phys. 131 (14), 144108. [17] Tamura, T., and T. Udagawa, 1964, Nuc. Phys. 53 , 33. [18] Toulouse, J., I. C. Gerber, G. Jansen, A. Savin, and J. G. Ángyán, 2009, Phys. Rev. Lett. 102 (9), 096404. [19] Tretiak, S., C. M. Isborn, A. M. N. Niklasson, and M. Challacombe, 2009, J. Chem. Phys. 130 (5), 054111. [20] Weber, V., A. M. N. Niklasson, and M. Challacombe, 2004, Phys. Rev. Lett. 92 , 193002. [21] Yam, C. Y., S. Yokojima, and G. Chen, 2003, J. Chem. Phys. 119 (17), 8794. [22] Yokojima, S., and G. H. Chen, 1999, Chem. Phys. Lett. 300 (5-6), 540. [1] Autschbach, J., 2009, Chem. Phys. Chem. 10 , 1757. [2] Bock, N., M. Challacombe, C. K. Gan, G. Henkelman, ∗ Electronic address: mchalla@lanl.gov