About

The central goal of Nancy Makri's research is to advance the theoretical understanding of quantum mechanical processes in large molecules and the condensed phase. Her work involves developing new theoretical descriptions and simulation methods based on Feynman's path integral formulation of quantum dynamics. These methods are applied to investigate phenomena such as tunneling and coherence, proton and electron transfer reactions in solution and biological systems, the dynamics of quantum fluids, and excitation energy transfer in molecular aggregates. Her research addresses the challenge of solving the Schrödinger equation for complex systems, which is computationally feasible only for small molecules without severe approximations, by creating approaches that enable the study of larger, more complex molecular systems.

Research topics

• Computer Science
• Physics
• Quantum mechanics
• Mathematics
• Pure mathematics
• Statistical physics
• Chemistry
• Mathematical physics
• Materials science
• Mathematical analysis
• Chemical physics
• Molecular physics

Selected publications

• Electronic frustration with quantum dissipation: Entropy, coherence, sign problem, and time evolution

  Physical review. B./Physical review. B · 2026-01-06 · 1 citations

  articleSenior author
  PublisherDOI

• Small matrix path integral propagation for long-time quantum dynamics of multistate systems in one and two dimensions

  The Journal of Chemical Physics · 2026-02-17

  article1st authorCorresponding

  The small matrix path integral (SMatPI) methodology allows fully quantum mechanical calculations on system-bath Hamiltonians under a variety of conditions with minimal array storage and is thus applicable to multistate systems. This work focuses on the SMatPI iteration algorithm, which can be computationally prohibitive in systems with many states. By transforming the SMatPI propagation matrices to the desired initial state, the cost of the iterative multiplication is reduced by a factor of n2, where n is the number of system states. This leads to dramatic savings when n≫10 and allows the simulation of large systems over very long propagation times. Applications are presented on models with up to 50 system states, including exciton transfer dynamics in one-dimensional aggregates and a square lattice.

  PublisherDOI

• Small matrix path integral in imaginary time

  The Journal of Chemical Physics · 2025-09-28 · 4 citations

  articleSenior author

  Thermal equilibrium properties are usually obtained from the imaginary-time path integral representation of the Boltzmann operator in combination with Monte Carlo integration methods. In some situations (identical fermions or frustrated Hamiltonians), the Boltzmann matrix leads to terms of alternating sign, which leads to a sign problem that severely impacts convergence. In this paper, we develop a robust and efficient quadrature-based method suitable for computing the Boltzmann matrix for discrete systems coupled to common or local harmonic baths. By expressing the discretized path integral with the influence functional in terms of a sum of matrix products, we develop a small matrix path integral (SMatPI) decomposition that allows iterative propagation in imaginary time while circumventing the storage of tensors employed in earlier work. The method is illustrated with several examples that involve two- and three-level systems coupled to common or local baths. We show that cyclic tight-binding Hamiltonians with positive coupling parameters give rise to Boltzmann matrix elements with alternating signs, presenting a severe sign problem to Monte Carlo approaches, while the SMatPI algorithm is stable and efficient.

  PublisherDOI

• Frustration Protection of Exciton-Vibration Thermodynamics and Transfer

  The Journal of Physical Chemistry Letters · 2025-12-30 · 2 citations

  articleSenior authorCorresponding

  Recent work reported unusual effects in the time evolution of some cyclic exciton-vibration Hamiltonians, which stems from electronic frustration and the accompanying ground state degeneracy. Here we show that the thermodynamic and dynamical properties of electronically frustrated systems are insensitive to variations of temperature and exciton-vibration coupling strength and that the lack of sensitivity correlates well with a quantitative measure of frustration. Thus, electronic frustration offers significant protection against the effects of large molecular environments.

  PublisherDOI

• Quantum dynamics of dissipative two-level systems and intradimer excitation energy transfer in the presence of static disorder

  The Journal of Chemical Physics · 2025-07-28 · 2 citations

  articleSenior author

  We use the numerically exact, fully quantum mechanical small matrix path integral (SMatPI) methodology to investigate the time evolution of the reduced density matrix (RDM) following photoexcitation of model molecular dimers in the presence or absence of static disorder. The dimer is modeled in terms of a two-level system that represents the excited electronic states of the monomers, which are coupled to a dissipative bath of vibrational modes with an Ohmic spectral density under diverse conditions that correspond to homo- or heterodimers, weak or moderately strong exciton-vibration coupling, high- or low-frequency vibrations, and high or low temperature. Through the equivalence class path integral algorithm, the averaging with respect to static disorder is performed with computational effort comparable to that of a single SMatPI calculation. We find that static disorder alters the dynamics and equilibrium properties of the RDM in significant and often subtle ways, which can mimic effects associated with stronger or weaker dissipation. The impact of disorder is most pronounced at low temperatures, where it tends to suppress coherence and often induces upward shifts in the population of the higher-lying state, while the effects on the off-diagonal RDM element and the eigenstate populations depend nonmonotonically on the asymmetry parameter. At high temperatures, the population shift is weaker and reversed for some parameters.

  PublisherDOI

• Discrete Generalized Quantum Master Equations

  Journal of Chemical Theory and Computation · 2025-05-06 · 4 citations

  article1st authorCorresponding

  Several derivative and integral approximations are explored for discretizing the Nakajima-Zwanzig generalized quantum master equation (NZ-QME or GQME) to obtain discrete quantum master equation (DQME) hierarchies and relations between discrete memory kernel and reduced density matrix (RDM) elements. It is shown that the simplest forward-difference approximation does not allow the reliable determination of the discrete kernel elements, even in the infinitesimal time-step limit, and that discrete kernels obtained in earlier work are flawed, although the procedure can be remedied. The various approximations give rise to DQMEs that differ in structure and in the RDM-kernel relationships. It is shown that the use of a more accurate discretization based on the midpoint derivative and midpoint integral approximations leads to a DQME that exhibits endpoint effects, which reflect the weaker impact of the bath on the RDM during the first time step and which parallel those encountered in the small matrix decomposition of the path integral (SMatPI) with a symmetric factorization of the short-time propagator. The features of the DQME hierarchies and RDM-kernel relations are illustrated through analytical examples involving a simple integrodifferential equation and a scalar GQME model, as well as numerical results for a two-level system (TLS) coupled to a harmonic bath.

  PublisherDOI

• Comment on "Unified framework for open quantum dynamics with memory"

  arXiv (Cornell University) · 2024-10-09 · 2 citations

  preprintOpen access1st authorCorresponding

  A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. We elaborate on these three critiques in this Comment.

  PublisherOA PDFDOI

• Roadmap on methods and software for electronic structure based simulations in chemistry and materials

  Electronic Structure · 2024-05-08 · 17 citations

  articleOpen access

  Abstract This Roadmap article provides a succinct, comprehensive overview of the state of electronic structure (ES) methods and software for molecular and materials simulations. Seventeen distinct sections collect insights by 51 leading scientists in the field. Each contribution addresses the status of a particular area, as well as current challenges and anticipated future advances, with a particular eye towards software related aspects and providing key references for further reading. Foundational sections cover density functional theory and its implementation in real-world simulation frameworks, Green’s function based many-body perturbation theory, wave-function based and stochastic ES approaches, relativistic effects and semiempirical ES theory approaches. Subsequent sections cover nuclear quantum effects, real-time propagation of the ES, challenges for computational spectroscopy simulations, and exploration of complex potential energy surfaces. The final sections summarize practical aspects, including computational workflows for complex simulation tasks, the impact of current and future high-performance computing architectures, software engineering practices, education and training to maintain and broaden the community, as well as the status of and needs for ES based modeling from the vantage point of industry environments. Overall, the field of ES software and method development continues to unlock immense opportunities for future scientific discovery, based on the growing ability of computations to reveal complex phenomena, processes and properties that are determined by the make-up of matter at the atomic scale, with high precision.

  PublisherDOI

• Coherence in Chemistry: Foundations and Frontiers

  Chemical Reviews · 2024-10-23 · 42 citations

  reviewOpen access

  Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

  PublisherOA PDFDOI

• Essential vs Removable Bath Anharmonicity: Path Integral Results with Model Electronic-Vibrational Baths

  The Journal of Physical Chemistry Letters · 2024-09-18 · 7 citations

  article1st authorCorresponding

  Fully quantum mechanical real-time path integral calculations on a two-level system (TLS) interacting with a simple or composite anharmonic bath are presented. The bath consists of a few or many units, each comprising two or three electronic states coupled to secondary harmonic vibrational baths. The primitive harmonic bath approximation leads to qualitatively wrong system dynamics at high temperatures, while the effective harmonic bath mapping becomes exact in the macroscopic bath limit. Significant deviations are observed when the number of bath units is small, including skewing and blue shifts in the population oscillations and enhanced or suppressed coherence. These effects reflect the essential anharmonic character of the bath and cannot be captured by harmonic bath treatments.

  PublisherDOI

Recent grants

• Quantum-Classical Path Integral Methodology

  NSF · $550k · 2014–2017

• Simulation methods for dynamical processes in quantum fluids

  NSF · $440k · 2008–2011

• Forward-backward quantum-semiclassical dynamics in solution

  NSF · $440k · 2005–2009

• Real-time path integral methodology for condensed-phase quantum dynamics

  NSF · $510k · 2020–2024

• Quantum-Classical Path Integral Methodology

  NSF · $475k · 2017–2020

Frequent coauthors

• Sohang Kundu

  University of Illinois Urbana-Champaign

  28 shared

• Jiushu Shao

  Beijing Normal University

  25 shared

• Jonathan Chen

  17 shared

• Graham R. Fleming

  Lawrence Berkeley National Laboratory

  16 shared

• Partha Pratim Roy

  Indian Institute of Technology Roorkee

  15 shared

• Akira Nakayama

  The University of Tokyo

  15 shared

• William H. Miller

  University of Minnesota

  14 shared

• Sambarta Chatterjee

  12 shared

Awards & honors

• Member, National Academy of Sciences, elected 2023
• ACS Award in Theoretical Chemistry, 2023
• Member, American Academy of Arts and Sciences, elected 2021
• ACS Physical Chemistry Division Award in Theoretical Chemist…
• School of Chemical Sciences Teaching Award, University of Il…