Linear Response at the Basis Set Limit Bryan Sundahl, Robert Harrison Institute for Advanced Computational Science, Stony Brook University Introduction Methods ?@A ? ? + P Response theory allows for the modeling of a molecular system’s The standard method of computing response states and excitation 6. = B! C@E [−2 2IH − KJ + L!?MN 6. − 2 1 − $# ΓO ] response to external stimuli, and accurate ways of modeling this energies is to project a trial wave function into a basis , , /?@A = B!? [−2 2IH − KJ + L! /? − 2 1 − $#+ ΓP ] response are paramount to validating theory and simulations. Response and solve the corresponding matrix eigenvalue equation[4]. . CDE ?MN . O Because the basis inside MADNESS is not band-limited and can eDfg to external stimuli is exactly what experimentalists measure in their labs. B! = j = −2 (- ± 5) Thus response theory is the concrete link between theory and introduce noise of any frequency, we rearrange the response C±E 4 i 0 ?@A th 6. : p excitation state at iteration n+1 IH: Coulomb operator equations into a fixed-point integral equation. The resulting ?@A th experiment, where direct comparison between theoretical results and /. : p de-excitation state at iteration n+1 KJ : Exchange Operator ? B! C@E: Greens function for excitation L!?MN : Nuclear potential operator experimental results can be realized. equation is then iterated to solution. This scheme is applied to a ? + B! CDE: Green’s function for de-excitation $# : projection operator onto ground batch of small molecules. state Response Equations Results The time-dependent Hartree-Fock equation, an approximation to the Excitation Energies (Hartree) time-dependent Schrödinger equation, can be written in its density Molecule Water Benzene Methane Nitrogen ( State NWChem MADNESS NWChem MADNESS NWChem MADNESS NWChem MADNESS Figure 6: The energies reported here are singlet excitation energies matrix form[1] as: " ! $# − $ # " ! = ' $ # . This equation can be transformed in the Tam-Danchoff approximation of time-dependent Hartree-Fock () 1 0.315436647 0.31534032 0.219247962 - 0.407768570 0.40805661 - 0.36071202 theory. The states reported are the 5 lowest energy states found by into the coupled linear response equations via perturbation theory. 2 0.376880764 0.37684455 0.225086842 - 0.408049785 0.40835542 0.379966838 - the MADNESS solver or the NWChem solver. Dashes indicate that state was not found by that solver. NWChem calculations were done + + + 3 0.400931459 0.39877028 0.240334266 0.23905479 0.408101968 0.40839831 0.379966840 - 1 − $# "! − -. 6. 0 + Γ. 0 = 56.(0) with the aug-cc-pvqz basis set. 4 - 0.40915514 - 0.23973037 0.455846241 0.45412497 - 0.38852576 + , + + , , 1 − $# "! − -. /. 0 + Γ. (0) = −5/.(0) 5 - 0.42049066 0.255872262 0.25460790 0.455972834 0.45423702 - 0.38852576

78# + , = , = Γ. 0 = $ × : 6; 0 <; 0 + : <; 0 /; (0 ) <.(0) 7$ ; ; $#+: projection op. onto ground state electron density Figure 8: A timing profile of a single water molecule with 60 excited "!+: ground state Fock operator states requested. ”Creating gamma” is the application of the perturbed + -.: ground state energy of molecular orbital p two electron operators. “V0 * x” is the application of the potential to the 5: excitation energy response functions. “Create resp. matrix” is the evaluation of matrix 6 0 : excitation function / 0 : de-excitation function inner products. “Diag. resp. matrix” involves diagonalization as well as < 0 : ground state molecular orbital truncation of MADNESS functions. “Aug. resp. matrix” adding in the previous iterations response functions and evaluating inner products (done to help with convergence.) MADNESS The MADNESS software stack allows for representation of functions in an adaptive basis set of Legendre polynomials and/or multiwavelets, thus limiting basis set error to a user-specified precision [2][3]. Operators have separated representations for efficient application in multiple Figure 7 : Current log-log plot of number of states dimensions. Fast, unitary transformation between function requested of single water molecule versus time to completion. Each calculation ran for 40 iterations, 20 at a representations allows for a choice of the best space (real or lower accuracy followed by 20 at a higher accuracy. multiwavelet) in which to compute for a given problem. Conclusions Figures 1-5 (Left to right, Through the application of the Green’s function operator, accurate excited states for top to bottom): Plots of the some small molecules was observed. The timing of these calculations was reported. scaling Legendre Future work will add in full TDHF and TDDFT theories, as well as improving on the polynomials (top) up to scaling with both system size and number of excited states requested. order 2 with one level of References

refinement, and plots of [1] P. A. M. , Proc. Cambridge Philos. Soc., 1931, 27, 240. corresponding [2] R. Harrison, G. Fann, T. Yanai, Z. Gan, and G. Beylkin. Multiresolution : Basic theory and initial applications. Journal of Chemical Physics, 2004. multiwavelet functions [3] B. Alpert, G. Beylkin, D. Gines, and L. Vozovoi. Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics, 2002. (bottom) [4] M. E. Casida. Time-dependent density functional response theory for molecules. In D. E. Chong, editor, Recent Advances in Density Functional Methods. World Scientific, 1995