Dr. Aniruddha Chakraborty

School of Basic Sciences

Indian Institute of Technology Mandi

Kamand, Himachal Pradesh 175005

India

ph: +(91)1905-237930

achakrab

**Funded Research Projects **

In the following I list a number of research projects ranges from fundamental science to engineering. Some of them are funded or expected to be funded by IIT Mandi & others are externally funded. Few projects are already completed, few are ongoing & few projects are to be submitted for external funding.

- Seed Grant (2012-15): "Dynamical Analysis of Highly Excited Molecular Spectra", IIT Mandi, India, INR. 4,70,000 (P.I.).

Summary: Understanding dynamics of highly excited vibrational motion of small molecules is an interesting contemporary research area. The aim of this research is to extract new dynamical informations encoded in experimental frequency domain spectra. Effective spectroscopic Hamiltonians are very useful to analyze and extract informations from experimental and simulated spectra. The standard approach for building an effective spectroscopic Hamiltonian is well understood for systems far below the dissociation energy of any bond. This project is aimed to extend our method to energies above the dissociation energy of any bond.

*Status: completed.*

- Research Grant (2012-17): "Supramolecular High Energy Compounds: synthesis, Characterization & theoretical Studies", Defence Research & Developement Organization, India, INR. 3600000 (Co-P.I.).

Summary: High energy materials are widely used for several interesting applications. This project is aimed to synthesize a number of new high energetic materials guided by advanced molecular modelling studies. Along with the synthesis, in the theoretical aspect of this project, heat of formation of possible new materials will be estimated using standard density functional theory.

Status: completed.

- Research Grant (proposal - to be submitted): "Understanding complexity of internal dynamics of biological molecules", Department of BioTechnology, India (P.I.).

Summary: The internal dynamics of biological macro-molecules are very complex. One of the major classes of these are proteins. An interesting problem, still far from complete understanding is the connection between sequence of amino acid residues and the very specific native structure. So the first question here is what dictates whether a given sequence has a reliable native structure or not, if it has what that structure would be? or what it is about the sequence that makes it a good folder. Understanding sequence-structure connection is important because it would make it possible to design sequences in the laboratory to fold to a definite structure, which then could be tailored to biological activity. The aim of this project is to know sequence-structure connection and then use that for predicting the kinetic mechanism by which a protein folds to its native structure.

Status: to be submitted.

- Research Grant: "Dynamical Analysis of Highly Excited Molecular Spectra", INR. 27,60,000 (P.I.).

Summary: The classical phase space structure and dynamics of vibrational Hamiltonian with mixed chaotic and regular motions are of great interest. Standard method of studying this system is to use approximate effective Hamiltonians (although it is not correct), which possess a special constant of the motion. This ``total action” with its corresponding quantum number is a very interesting feature of the actual dynamic. The total action is in general called a ``polyad quantum number”. However, useful as an analytical tool, there are limitations in applying the effective Hamiltonian for molecules at higher energies. The limitations are due to the fact that the exact dynamics actually contains in principle, an infinite number of resonance couplings,hence breakdown of the total polyad action. The vibrational energy levels of systems with more than two degrees of freedom that break the polyad action, time-dependent transport, and the representation of them by the use oeffective Hamiltonian are entirely unexplored area of research. This project is aimed to constrcution of an effective polyad-breaking for a system with more than three degrees of freedom & analyze the phase space structure to extruct dynamical information.

Status: to be submitted.

- Research Grant: "Electron solvation by a layer of polar adsorbates - realistic model", Council of Scientific & Industrial Research, India, INR. 3,96,000 (P.I.).

Summary: An electron near a metal surface feels the charge of its image in the metal and therefore it moves under the influence of this attractive potential. Harris et. al., reported an experimental study of the dynamics of electron in image states of a metalsurface having polar adsorbates on it - they find two kinds of states, viz., one localized and the other delocalized. There have been attemps to model the process, but the problem is the nature of the image potential state is not known owing to the lack of detailed knowledge of the geometry of the metal surface. All the theoretical calculations done so far have used flat metal surface. In this project we will consider a model in which we account for non-flateness of the surface.

Status: ongoing.

- Research Grant: "Curve Crossing Problems with Arbitrary Coupling", Council of Scientific & Industrial Research, India, INR. 7,54,400 (P.I.).

*Summary: *Nonadiabatic transition due to potential eenergy curve crossing is an interesting mechanism to induce electronic transitions in collision process. There are examples where the transition is in between two states and also there are cases of involvement of more than two states in the process. There are only few cases where exact analytical solution of two state problem is available but those involve specific shape of potentials and coupling term. The aim of this project is to solve multi-state problems involving both potentials and coupling terms as arbitrary functions of position and time.

Status: ongoing.

- Research Grant (proposal - in preparation): "Thermal Diffusivity in Realistic Solid", (P.I.).

Summary: Temperature and heat flow are the two important quantities in understanding heat conduction. When temperature distribution is not uniform at all points of a solid, then heat energy flows in the direction of less temperature. The Fourier equation is in general used in understanding heat conduction, which is valid only for a stationary homogeneous isotropic solid without any heat source. But in reality we have solids of different shapes, e.g., we can have a cylinder with non-unifrom cylindricity, also there may be real life examples where cylindricity changes with time. The aim of this project is to understand the effect of such types of non-uniformity in shape on thermal diffusivity.

Status: in preparation.

- Research Grant: "Buckled Nano Rod A Two state System and its Quantum Dynamics using exact Hamiltonian model", (P.I.).

Summary: This research project is on theoretical investigation of the dynamics of a mechanical two level system (mechanical equivalent of the 'bit'). The compression of a nano-rod would cause it to buckle. There are now two possible buckled state & the system is interesting as a nano-sized memory device. We have very recently developed a theoretical approach for the calculation of classical rate of transition between two buckled state using an exact model Hamiltonian. Now the plan is to develope a method to calculate quantum rate using the same model Hamiltonian. Then we will apply our method to the case of carbon nanotube. A calculation using system plus reserviour model will also be done.

Status: to be submitted.

- Research Grant: "Understanding the Dynamics of Vibrational excitation in C60 single molecule transistor - An effective Hamiltonian Approach" (P.I.).

Summary: In an interesting experiment, Park et. al., reported a three electrode transistor made using a single C60 molecule. Like standard field effect transistors, the voltage on the "gate" electrode controls the current flowing from the source electrode through the C60 molecule to the drain electrode. The experiment shows that the oscillatory motion of C60 trapped between the two electrodes can be excited by passage electrons through the system. There are attemps to model the process. But this process involve several unknowns, which does not allow to model it appropriately. The nature of potential for centre of mass motion of C60 is not known due to the lack of detailed knowledge of the electrode geometry. The experimental and theoretical work jointly lead to the conclusion that the formation of negatively charged C60 results in a shift of the equilibrium geometry by about 3-4 pm. It was suggested that this shift arises due to the image interaction, though the details of the geometry that would lead to such a shift was never discussed. We plan to consider the simplest possible model, which describe the physics of the problem. The C60 molecule sits under the potential of both electrodes. Adding an extra electron to C60 can change the C60 - metal equilibrium distance due to the image image interaction. When this negatively charged C60 gives that extra electron to the drain electrode, the former equilibrium position is regained and the molecule may be left in an excited state of "center of mass" oscillation. This is reminiscent of two photon processes encountered in resonance Raman scattering. Here we will derive a formula for current, similar to Kramers-Heisenberg-Dirac formula. Using this formula we will estimate the unknown parameters (non-linear experimental fit of the experimental data) which will give us a relatively good understanding of the relative contribution of differnt possible mechanisms for the process.

Status: to be submitted.

- Research Grant: "Effective Hamiltonian for bond dissociation process", INR. 36,00,000 (P.I.).

Summary: A reliable potential energy surface is necessary for analysing the dynamics of a molecule in a particular electronic state. For large polyatomic molecules, a global potential energy surface is out of reach of even the best current computational capabilities. So the only way to make progress is to use analytical functions for potential with several parameters to ﬁt the experimental data. But this method requires one to guess the possible analytical form that defines the potential energy surface. For unknown potential energy surface the effective spectroscopic Hamiltonians are the only method for extracting information from experimental data. This type of Hamiltonian is very useful even when it is too difﬁcult to calculate the spectra starting from the potential energy surface. So far it was not possible to construct an effective Hamiltonian for energies above bond dissociation energy, because of lack of knowledge of how states above dissociation energy (continuum of energy) interact with the states below dissociation energy (discrete of energy). The aim of this project is to propose a method for constructing such effective Hamiltonian and use it for creating new knowledge of bond dissociation process.

Status: to be submitted.

- Research Grant (to be submitted): "Electron-electron correlation in quantum dot", (P.I.).

*Summary: **Understanding the effect of electron correlation in Quantum dot is an interesting and important area of research both theoretically and experimentally. One of the most interesting features of quantum dot is that it is possible to add extra electrons one at a time, step by step. Electrons in a quantum dot feel a central potential due to the semiconductor nanostructure. This is equivalent to an attractive oscillator potential. So the motion of these electrons in these systems is expected to be more like vibrating atoms in real molecules. So one should expect all the effects of an-harmonicity, such as bifurcations, local and normal modes, Fermi resonances, and approximate polyad numbers for electrons in a quantum dot. If these molecule-like effects turn out to have some interesting use in electronic devices, this would become another important area of fundamental research. In this project we will try to explore this option theoretically.*

Status: in preparation.

- Research Grant (to be submitted): "Electron-electron correlation in two electron atoms", (P.I.).

*Summary: **Two electron atom is the most simplest system to study electron correlation. The doubly excited states of atoms with two outer electrons, exhibit molecule-like collective motion. There are even frozen planet state observed in two electron systems, **where the electrons become locked into place on the same side of the nucleus. There are evidence for molecule like collective effects in **atoms with three outer electrons. An open question in this area is the applicability of this type of correlation to motion of electrons in **molecule. A simple effective spectroscopic Hamiltonian was proposed for double excited two electron atom for understanding independent **particle, shielding and correlation effect. The Hamiltonian is constructed by nonlinear least square fit of spectra of two electron systems with both electrons in the n=2 shell. Now my plan of future research is to apply this effective spectroscopic Hamiltonian method for higher shells and for atoms with more than two electrons.*

Status: in preparation.

- Research Grant: "Understanding freezing/melting and glassy behaviour of clusters", (P.I.).

Summary: The clusters are simple systems that exhibit complex behaviour such as freezing/melting, or evben has the possibility of glasy behaviour. The phenomena of internal rearrangement and seeking minima on high energy rugged potential energy surfaces of high dimensionality has been extensively investigated. The dynamical nature of large amplitude motions in these systems has yet to be explored. In this project we plan to identify the essential features of the potential energy surface that has the strongest influence on dynamical behaviour, so one can construct an appropriate reduced dimensional model for further dynamical analysis.

Status: in preparation.

- Research Grant (to be submitted): "Effective Hamiltonian for curve crossing problems", (P.I.).

Summary:Nonadiabetic transitions due to potential energy curve or surface is one of the most important mechanism to efficiently induce electronic transition in collisions. This is very interesting concept and appears in various areas of physics, chemistry and biology. The most typical examples are, of course, a variety of atomic and molecular processes such as atomic and molecular collisions, chemical reactions and molecular spectroscopic processes. Other examples are dynamic processes on solid surfaces, energy relaxation and phase transitions in condensed phase physics, and electron and proton transfer processes in biology. Recently I have reported an exactly solvable model for the curve crossing problems, where the coupling is assumed to be Dirac delta function. In real systems nature of coupling is not very simple in general and so the problem is not in general analytically tractable. So it would be very interesting to construct effective spectroscopic Hamiltonians from experimental or simulated spectra for curve crossing problems.

Status: in preparation.

- Research Grant (to be submitted): "Dynamical Analysis of Internal Motion Large Molecules", (P.I.).

Summary: Larger molecules are formed by linking together several smaller components. So understanding dynamical informations from small molecules may be useful for studying the dynamical nature of internal motion of larger molecule. For understanding dynamics of a hydrocarbon chain CH3-(CH2)n-CH3, one would probably start working with CH3-CH3, in which there is hindered rotational motion of the two CH3 groups. The construction of effective spectroscopic Hamiltonian and the bifurcation analysis of this system can be a very useful starting point for dynamical analysis of hydrocarbon chain larger than CH3-CH3 i.e., n larger than 1. The dynamics become qualitatively very different with the incorporation of more -CH2- group in the hydrocarbon chain due to large amplitude, flexible twisting motions generally seen in short chain molecules. Also there are faster high frequency vibrations, such as those of the individual bonds. So for constructing effective spectroscopic Hamiltonian for larger molecules I plan to use all the required informations obtained from the dynamical studies of smaller molecules.

Status: in preparation.

- Research Grant (to be submitted): "Effective Hamiltonian: Solvent Induced Polyad Breakdown", (P.I.).

Summary: The classical phase space structure with mixed chaotic and regular motions are one of the most interesting areas of research. Standard method of studying this type of systems is to use approximate effective Hamiltonians, which possess a special constant of the motion. This ``total action” with its corresponding quantum number is a very interesting feature of the actual dynamic. The total action is in general called a ``polyad quantum number”. However, useful as an analytical tool, there are limitations in applying the effective Hamiltonian for molecules at higher energies. The limitations are due to the fact that the exact dynamics actually contains in principle, an infinite number of resonance couplings, hence breakdown of the total polyad action. The breakdown of polyad quantum number in gas phase has been studied extensively by effective spectroscopic Hamiltonian method. The goal of this project is to construct the efefctive spectroscopic Hamiltonian for studying vibrational dynamics in condensed phase. The key question is this area are the role of polyad numbers in energy transfer in condensed phase.

Status: in preparation*.*

- Research Grant (to be submitted): "Barrier Crossing Problem of a Long Chain Polymer Molecule ", (P.I.).

Summary: The activated escape of a particle from a metastable well is known as the Kramers problem. This problem was recently applied to the case of polymers. This barrier crossing problem for polymer is important in biology as many biological processes involve the translocation of a chain molecule from one side of the membrane to the other through a pore in the membrane. Theoretical analysis suggests that the escape of a long chain polymer trapped in the less stable well of a double well potential can occur by a kink mechanism. The kink is nothing but the distortion of the polymer caused by the barrier, and it moves with a constant velocity. The barrier crossing rate by kink mechanism is generally faster than by any other mechanisms. To these interesting cases, we plan to apply the recently developed dynamical transition state theory.

Status: in preparation.

- Research Grant (to be submitted): " Understanding replication of DNA Molecule ", (P.I.).

*Summary: *The replication of the DNA molecule is a very interesting problem of current interest. The first step in replication process is the unzipping of the two strands of the DNA molecule. Usually, this is done by enzymes, and mechanical force at the molecular level is involved in the action of these enzymes. There have been many interesting investigations in this area, e.g., the theory of denaturation of DNA under a torque. This process can be thought of equivalent to the problem of pulling a polymer out of a potential well by a force applied to one of its end. The aim of this project is to analyze the dynamics of the pulling out process using dynamical transition state theory.

Status: in preparation.

- Research Grant (to be submitted): "Breaking of a Long Chain Molecule under strain", (P.I.).

Summary*: *The breaking of a polymer chain that is under tension is an interesting research problem. This problem is relevant for understanding material failure under stress, polymer rupture, adhesion, friction, mechanochemistry and biological applications of dynamical force microscopy. Theoretical studies suggest that, simple statistical approaches are likely to fail for this problem. In a very recent work by Ezra et.al., used classical trajectory simulation to investigate this fragmentation kinetics and phase space structure of short tethered atomic chains under constant tensile stress. The aim of this project is to use dynamical transition state theory to understand the problem of breaking of a polymer chain under strain. The plan here is to develope a general framework to study the problem of breaking of a flexible, semi-flexible and ring polymer under starin and apply the same method to study the rupture of carbon nanotube under strain.

*Status: in preparation. *

- Research Grant (to be submitted): "Dynamical Transition State Theory using effective Hamiltonian", (P.I.).

Summary: In 1930s, Eyring and Polanyi studied a chemical reaction, providing the first calculation of the potential energy surface of chemical reaction. This surface contains one minimum associated with the reactant and another minimum for the product. They are separated by a barrier that needs to be crossed for the chemical reaction to occur. Eyring and Polanyi defined the surface's transition state as the path of steepest ascent from the barriers saddle point in coordinate space. Once crossed, this transition state can never be recrossed. The notion of a transition state as a surface of no return defined in coordinate space was soon recognized as fundamentally flawed, because recrossing is possible. So transition state theory provides an upper bound of the exact classical reaction rate. The exact classical reaction rate in general will depend on the choice of dividing surface. This is the key idea behind the variational transition state theory, in which one searches the space between reactants and products for a dividing surface that minimizes the reaction rate. E. Wigner recognized very early that in order to develop a rigorous theory of reaction rates, one must extend the notions above from configuration space to phase space. In the late 1970's Pechukas et. al., shows that, for systems with two degrees of freedom, at energies sufficiently close to the energy at the saddle point, the classical trajectory perpendicular to the reaction path that passes through the saddle point is an unstable periodic orbit. If the orbit of this periodic trajectory is plotted in phase space, it defines a surface known as a periodic orbit dividing surface or PODS - this corresponds to the surface of minimum flux. Quite recently higher dimensional analogue of PODS is established for extracting no return transition states for many degrees of freedom system. The aim of this project is to test and extend theories of reaction rates. A reliable potential energy surface is generally used for studying the reaction rate. In this project we will use the energy level data (simulated or experimental) for constructing the effeective spectroscopic Hamiltonian and that Hamiltonian will be used for rate calculation using dyanmical transition state theory. As the effective Hamiltonian is expressed in terms of coupled oscillatores, so one can estimate the effect of different oscillators as well as different types of coupling on the reaction rate.

Status: in preparation.

**"Fundamental Research - only few can do & fewer can understand."**

Dr. Aniruddha Chakraborty

School of Basic Sciences

Indian Institute of Technology Mandi

Kamand, Himachal Pradesh 175005

India

ph: +(91)1905-237930

achakrab