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Coupled Cluster theory, Non-linear Dynamics, Machine Learning and Synergetics

Name :  Valay Agarawal

Department: Chemistry

Program:Int B.S. - M.Sc( 05th Year)

Name of supervisor: Prof. Rahul Maitra

Topic of research: Coupled Cluster theory, Non-linear Dynamics, Machine Learning and Synergetics

Description of research work:

Analysing and Accelerating Coupled Cluster Calculations with Nonlinear Dynamics and Synergetics

The iteration scheme associated with single-reference coupled-cluster theory has been analysed using nonlinear dynamics. The solutions to the iterative scheme have been shown to follow the non-linear dynamics principle under perturbation. The period doubling cascade of the solutions, followed by chaotic regime has been recovered, along with recovery of accurate value of the Feigenbaum constant to high accuracy. Further phase space analysis, through multiple recurrence analysis parameters including recurrence rate, determinism, Shannon entropy indicates the presence of a few significant cluster amplitudes, mostly involving valence excitations, that dictate the dynamics, while all other amplitudes are enslaved. This has allowed Starting with a few initial iterations to establish the inter-relationship among the cluster amplitudes, a supervised Machine Learning scheme with polynomial Kernel Ridge Regression model has been employed to express each of the enslaved amplitudes uniquely in terms of the former set of amplitudes. The subsequent coupled cluster iterations are restricted solely to determine those significant excitations, and the enslaved amplitudes are determined through the already established functional mapping. We will show that our hybrid scheme leads to significant reduction in computational time without sacrificing the accuracy. The predicted energy as well as the predicted wave function with our method is highly accurate compared to the exact method. The novelty of the work lies in exploiting the non-linear nature of the Coupled Cluster equations. The work also includes aspects of machine learning, and the saving in computational costs is competitive to Direct Inversion of Iterative Space (DIIS), which is a standard procedure in all commonly used programs. Our work, not only reduces the time costs without sacrificing, but is also customizable to the level of accuracy requirement. This makes the theory more general and applicable. We also do not require any training datasets for our machine learning model, making the theory directly transferable and usable by others without having to face hidden computational and human costs of data generation. 


This works opens up multiple areas of research which will be pursued in the future. Some of them are usage of new machine learning models guided by the internal structure of the iterative scheme for solving the Coupled Cluster equations. This work is the first instance of working at the interface of Non-Linear Dynamics and Quantum Chemistry, and with expertise from other domains of mathematics make it highly liable to future collaborations. The work is also the first demonstration of use of Principle of Synergetics in the area of Quantum Chemistry, opening a gold mine for collaborations in the field of Quantum Chemistry, Quantum Dynamics and so on.  


  1. Stability Analysis of a Doubly Transformed Coupled Cluster Theory: 
    Valay Agarawal, Anish Chakraborty, Rahul Maitra, J. Chem. Phys. 153, 084113 (2020);
  2. Accelerating Coupled-Cluster Calculations with Non-Linear Dynamics and Shallow Machine Learning. Valay Agarawal, Samrendra Roy, Anish Chakraborty, Rahul Maitra, arXiv: 2011.02259, Accepted in J. Chem. Phys.