Development and Applications of Imaginary Time Path Integral Methods

Citation

Korol, Roman (2024) Development and Applications of Imaginary Time Path Integral Methods. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/jy10-rf87. https://resolver.caltech.edu/CaltechTHESIS:08212023-205141057

Abstract

Recent engineering advances have opened up avenues to novel technologies that bridge the gap between the quantum and the classical. In order to understand large-scale quantum systems, a variety of approximate theoretical treatments have been proposed. This thesis focuses on development and applications of path-integral methods, which have enjoyed broad applicability in recent years for exploring nuclear quantum effects in the domains that span physical, bio-, geo-, and materials chemistry.

Feynman's path-integral formulation of quantum statistical mechanics offers powerful and widely used strategies for including nuclear quantum effects in complex chemical systems. These strategies are based on the observation that the quantum Boltzmann statistical mechanics of a quantum system is exactly reproduced by the classical Boltzmann statistical mechanics of an isomorphic ring-polymer system. For the numerically exact calculation of quantum Boltzmann statistical properties, the classical Boltzmann distribution of the ring-polymer system can be sampled using Monte Carlo (i.e., path-integral Monte Carlo, or PIMC) or molecular dynamics (PIMD).

Chapters 1 and 2 of this thesis identify and — with no computational overhead — eliminate the issues in virtually all previous numerical implementations of PIMD that stem from time discretization. The resultant integration scheme requires only a small modification to existing PIMD algorithms and provides accurate statistical and dynamical data in a single-shot simulation with an up to 3-fold increase in the timestep duration.

Chapter 3 transitions from the PIMD method development to the applications of the related PIMC method to understand equilibrium of stable heavy isotopes (D, ¹³C, ¹⁷, and ¹⁸O in small gaseous molecules. We present a collaborative experiment-theory calibration of the temperature dependence of the clumped isotope effect in methane in Chapter 4. We continue in Chapter 5, adding the study of isotopic fractionation between methane, water, and molecular hydrogen. Here we present the first concrete example of the effect of Born-Oppenheimer approximation on PI calculations. Finally, Chapter 6 extends our treatment to ethane and propane. For propane, in addition to multiple clumped isotope effects, there is also a strong site preference for the heavy isotopes to occupy the central (methylene) group.

All the isotopic equilibrium calculations utilize accurate potential energy surfaces and are validated against experimental data in close collaboration with Daniel Stolper's experimental group at Berkeley, representing (to the best of our knowledge) the most accurate reference data available to date.

Thesis Files