## Graph theoretic methods for dynamics of electrons and nuclei with post-Hartree-Fock accuracy

The accurate treatment of molecular properties often requires the correlated study of electronic structure with extensive basis sets. But the size of systems that can be considered by standard approaches in electronic structure is greatly limited due to the intrinsic (steeply algebraic) computational scaling of electron correlation methods and the number and quality of basis functions used. Although there has been significant progress through algorithmic improvements for the treatment of electron correlation, the computational expense is still often too high for most systems of chemical interest. This is especially true when dynamics calculations are to be performed with on-the-fly electronic structure or when quantum nuclear effects are to be studied.

In a series of publications (see below) we have shown how graph-theoretic methods may be used to adaptively construct many-body approximations to potential surfaces and AIMD calculations. In the process both extended Lagrangian as well as Born-Oppenheimer based * ab initio * molecular dynamics simulations can be performed at accuracy comparable to CCSD and MP2 levels of theory with DFT-computational cost. Hence for the studies presented below, we presented Car-Parrinello-style dynamics, but with CCSD accuracy. Similarly, we have shown how multiple graphical representations of molecular systems may be simultaneously utilized to construct accurate potential surfaces in agreement with MP2 and CCSD levels of theory, again at DFT cost.

Furthermore we have also shown how condensed-phase simulations on interfaces and liquids may be constructed with hybrid DFT accuracy at gradient-corrected DFT accuracy. The approach utilizes graph-theoretic methods.

### Representative publications

#### 73 Graph theoretic molecular fragmentation methods for electronic potential energy surfaces augmented by machine learning

Xiao Zhu, and Srinivasan S. Iyengar, arXiv.

#### 72 Graph-|Q⟩⟨C|, a Graph-Based Quantum/Classical Algorithm for Efficient Electronic Structure on Hybrid Quantum/Classical Hardware Systems: Improved Quantum Circuit Depth Performance

Juncheng Harry Zhang, and Srinivasan S. Iyengar, * J. Chem. Theory and Comput. * **Article ASAP**, 10.1021/acs.jctc.1c01303 (2022).
**Summary: ** Recently, multiple quantum computing technologies have emerged as potential alternative computational platforms to address complex computational challenges. Additionally, algorithms to
approximate electron correlation problems, for small molecular systems, and quantum nuclear dynamics problems have been implemented on quantum hardware devices. However, application of standard quantum circuit models to treat electronic structure problems leads to a rapid increase in the circuit depth and the number of quantum gates. This contributes greatly to the accumulated error during quantum propagation. This paper outlines a new hybrid quantum+classical algorithm based on a graph-theoretic approach to molecular fragmentation and is geared toward performing electron correlation calculations, potentially on an ensemble of quantum and classical hardware systems. The algorithm studied here is referred to as the “Graph-|Q⟩⟨C|” algorithm since it contains an independent set of classical and quantum algorithmic components inside a single umbrella. That is, the overall computational workload is partitioned, through graph theory based on computational complexity analysis, into (a) classical computing sections that are carried out on traditional classical electronic structure packages, and (b) quantum computing sections that are carried out using quantum circuit models. Furthermore, the Graph-|Q⟩⟨C| algorithm is quantum hardware-agnostic and is developed with the goal to be implemented on all quantum hardware technologies, and, in
fact, is designed to be used on an ensemble of such quantum hardware systems for any given calculation. In essence, our Graph-|Q⟩⟨C| algorithm yields a new approach that reduces the required quantum circuit depth, the number of quantum gates, and the number of CNOT gates (by several orders of magnitude) that contribute to error accumulation, through a graph-theory-based projection operator formalism. Thus, given this
reduction, our algorithm, potentially improves the quantum algorithmic efficiency, provides a new avenue for quantum resource management, and also reduces the accumulation of errors during the demonstrated electronic structure calculations on quantum hardware. Given the limitations of quantum circuit gate fidelities within the gate model, this algorithm, we expect, will become a central piece in the quantum/classical computing of chemical systems.

#### 70 Graph-theory based molecular fragmentation for efficient and accurate potential surface calculations in multiple dimensions

Anup Kumar, Nicole DeGregorio and Srinivasan S. Iyengar, * J. Chem. Theory and Comput. * **Articles ASAP.** (2021). DOI: https://doi.org/10.1021/acs.jctc.1c00065

** Summary: ** The accurate and efficient study of electronic structure and nuclear dynamics is at the core of multiple problems that are critical to materials, biological, and atmospheric research. However, these studies are deeply affected by the steep, polynomial scaling cost of electronic structure, and potentially exponential scaling of quantum nuclear dynamics. In this publication, we present a multi-topology molecular fragmentation approach, based on graph theory, to calculate multi-dimensional potential energy surfaces in agreement with post-Hartree-Fock levels of theory, but at DFT cost.

#### 69 Weighted-Graph-Theoretic Methods for Many-Body Corrections within ONIOM: Smooth AIMD and the Role of High-Order ManyBody Terms

Juncheng Harry Zhang, Timothy C. Ricard, Cody Haycraft, and Srinivasan S. Iyengar, * J. Chem. Theory and Comput. * **17**, 2672 (2021).

**Summary: ** We present a weighted-graph-theoretic approach to adaptively
compute contributions from many-body approximations for smooth and accurate
post-Hartree−Fock (pHF) ab initio molecular dynamics (AIMD) of highly fluxional
chemical systems

#### 68 An efficient and accurate approach to estimate hybrid functional and large basis set contributions to condensed phase systems and molecule-surface interactions

T. C. Ricard and S. S. Iyengar, * J. Chem. Theory and Comput. * **16**, 4790 (2020). (Supporting Information)

**Summary: ** We present an efficient approach to perform accurate hybrid DFT electronic structure calculations for condensed phase systems, at pure DFT cost.

#### 67 Embedded, graph-theoretically defined many-body approximations for wavefunction-in-DFT and DFT-in-DFT: applications to gas- and condensed-phase AIMD, and potential surfaces for quantum nuclear effects

Timothy C. Ricard, Anup Kumar and Srinivasan S. Iyengar, * Int. J. Quant. Chem. * In Press. (2020).

** Summary: ** We discuss a graph theoretic approach to adaptively compute many-body-approximations in an efficient manner to perform (a) accurate post-Hartree-Fock AIMD at DFT cost for medium to large sized molecular clusters, (b) hybrid DFT electronic structure calculations for condensed phase simulations at the cost of pure density functionals, (c) reduced cost on-the-fly basis extrapolation for gas-phase AIMD and condensed phase studies, and (d) similar basis-set extrapolations for condensed phase calculations, and (e) accurate post-Hartree-Fock level potential energy surfaces at DFT cost for quantum nuclear effects.

#### 66 Fragment-based electronic structure for potential energy surfaces using a superposition of fragmentation topologies

A. Kumar and S. S. Iyengar, "Fragment-based electronic structure for potential energy surfaces using a superposition of fragmentation topologies", * J. Chem. Theory and Comput. * **15**, 5769 (2019).

** Summary: ** Molecular fragmentation methods developed by us and by several groups have made it possible to perform accurate electronic structure calculations in large systems. However, problems exist with these ideas when are applied to molecular potential surface calculations where fragments may change as the nuclei move. We develop a Fragment-Based electronic structure method which adaptively utilizes multiple graphical representations for a given molecular system and use these to compute smooth potential energy surfaces in an efficent manner. Graph theory based representations are used to efficiently compute and represent the local interactions between coarse grain units of molecules leading to post-Hartree-Fock accurate results at DFT cost. Benchmarks are provided on protonated waterwire systems ubiquitous in numerous biological ion channels and in material applications.

#### 62 Efficiently capturing weak interactions in ab initio molecular dynamics with on the fly basis set extrapolation

T. C. Ricard and S. S. Iyengar, "Efficiently capturing weak interactions in *ab initio* molecular dynamics with on the fly basis set extrapolation", * J. Chem. Theory and Comput. * **14**, 5535 (2018).

#### 60 Adaptive, geometric networks for efficient coarse-grained ab initio molecular dynamics with post-Hartree-Fock accuracy

T. C. Ricard, C. Haycraft, and S. S. Iyengar, "Adaptive, geometric networks for efficient coarse-grained ab initio molecular dynamics with post-Hartree-Fock accuracy", * J. Chem. Theory and Comput. * **14**, 30 (2018).

#### 56 Efficient, 'On-the-fly', Born-Oppenheimer and Car-Parrinello-type dynamics with coupled cluster (CCSD) accuracy through fragment based electronic structure

C. Haycraft, J. Li and S. S. Iyengar, "Efficient, 'On-the-fly', Born-Oppenheimer and Car-Parrinello-type dynamics with coupled cluster (CCSD) accuracy through fragment based electronic structure" * J. Chem. Theory and Comput. * **13**, 1885 (2017). Supporting Information: Click here

#### 54 Hybrid extended Lagrangian, post-Hartree-Fock Born-Oppenheimer ab initio molecular dynamics with fragment-based electronic structure

J. Li, C. Haycraft and S. S. Iyengar "Hybrid extended Lagrangian, post-Hartree-Fock Born-Oppenheimer ab initio molecular dynamics with fragment-based electronic structure". * J. Chem. Theory and Comput. * **12**, 2493 (2016).

#### 53 Ab initio molecular dynamics using recursive, spatially separated, overlapping model subsystems mixed within an ONIOM based fragmentation energy extrapolation technique

J. Li and S. S. Iyengar "Ab initio molecular dynamics using recursive, spatially separated, overlapping model subsystems mixed within an ONIOM based fragmentation energy extrapolation technique". * J. Chem. Theory and Comput. * **11**, 3978 (2015).

#### 2 Inclusion of molecular flexibility in Partition Coefficient calculations on oligopeptides using stochastic sampling

N. G. J. Richards, P. B. Williams, S. S. Iyengar, *Proceedings of the twenty-seventh Hawaii International Conference on System Sciences* 203 (1994).