3E09
Analysis Tools for Linear Scaling Quantum Chemistry Calculations○William Dawson1, Luigi Genovese2, Takahito Nakajima1
1 RIKEN Center for Computational Science, Japan
2 French Alternative Energies and Atomic Energy Comission, France[Abstract] The development of efficient linear-scaling algorithms alongside steady increases
in computational power have made ab-initio calculations with thousands of atoms more andmore common place. However, compared to calculations on small systems, there still remains
a deficiency in the kinds of tools available to post-process and analyze the results. In this talk,we will discuss new algorithms and computational techniques we have been developing tohelp interpret the results of calculations in the linear-scaling regime. One key approach tounderstanding large systems is the ability to partition a system into smaller building blocks. In
this talk, we will consider how to automatically decompose a system into building blocks
using the information in the one particle density matrix. We will demonstrate the utility ofthese tools by presenting an analysis of various biological systems in realistic environments.
[Introduction] Commonly used computational techniques for computing the properties ofmolecular systems from first-principles are often limited to systems of fewer than one
thousand atoms due to computational costs that grow cubically or worse with the size of thesystem. In order to overcome these bottlenecks, linear-scaling methods both for theconstruction of the Hamiltonian and for computing the density matrix have beendeveloped[1]. However, while these techniques can enable calculations on larger molecules inmore realistic environments, this increase in system size also leads to an increased challenge
in interpreting calculation results. This challenge is made even more difficult by deficiencies
in analysis tools that can efficiently be applied to large systems.
[Methods] One of the main bottlenecks that needs to be avoided when performing linearscaling calculations is the computation of the density matrix through the eigendecompositionof the Hamiltonian. Linear-scaling methods directly compute the density matrix using
methods such as the Fermi Operator Expansion, density matrix minimization, density matrix
purification, etc, which can effectively exploit the sparsity of the density matrix. While thesemethods allow for the efficient calculation of the ground state density, information about theeigenspectrum is lost.
The basis of these diagonalization free methods is the calculation of a projector on to theoccupied subspace of a matrix. By changing the target subspace, these methods can also beemployed to slice up the eigenspectrum into arbitrary patches, which might then berecombined to recover the full eigenspectrum[2]. In many cases, the projectors associatedwith these patches of the spectrum also have significant sparsity, allowing for an efficient
divide and conquer approach to computing the eigenspectrum. Once a local energy envelope
has been chosen, we might also investigate the locality in space using the sparsity of thedensity matrix. Mohr et al.[3] has shown that a good measure for the locality of the region iswhether the sub-density matrix associated with a given region is idempotent.

[Results and Discussion] In Fig. 1, we plot the eigenspectrum of a small protein (1L2Y [4])in a salt water solution computed using the linear scaling version of BigDFT[5]. The density
matrix is moderately sparse, with about 14% of its entries being nonzero. We have also
computed the density matrices associated with only certain subsets of the energy spectrum.
While arbitrary energy regions can lead to significant fill in, if these subset regions are chosenwell, a significant amount of sparsity remains. This locality in space and energy allows for
efficient calculation of the molecular orbitals of this system. This information, combined withthe purity indicator metric, allows for the partitioning of large systems into smaller building
blocks.

Fig. 1. Density matrix sparsity for different energy regions.

[References][1] Bowler, D. R., and Tsuyoshi Miyazaki. "Methods in electronic structure calculations." Reports on Progress inPhysics 75, no. 3 (2012): 036503.
[2] Bai, Zhaojun, and James Demmel. "Using the matrix sign function to compute invariant subspaces." SIAM
Journal on Matrix Analysis and Applications 19, no. 1 (1998): 205-225.
[3] Mohr, Stephan, Michel Masella, Laura E. Ratcliff, and Luigi Genovese. "Complexity Reduction in Large
Quantum Systems: Fragment Identification and Population Analysis via a Local Optimized Minimal Basis."
Journal of chemical theory and computation 13, no. 9 (2017): 4079-4088.
[4] Neidigh, Jonathan W., R. Matthew Fesinmeyer, and Niels H. Andersen. "Designing a 20-residue protein."
Nature Structural and Molecular Biology 9, no. 6 (2002): 425.
[5] Mohr, Stephan, Laura E. Ratcliff, Paul Boulanger, Luigi Genovese, Damien Caliste, Thierry Deutsch, andStefan Goedecker. "Daubechies wavelets for linear scaling density functional theory." The Journal of chemicalphysics 140, no. 20 (2014): 204110.