Lecture 32 : Part 1

Hello and welcome to the course on AI, ML in Materials Engineering. I am Professor Tarun Kumar Kundu from Department of Metallurgical and Materials Engineering, IIT, Kharagpur. So, on request of Professor Bishwas, I will be taking two lectures on Molecular Dynamics and Density Functional Theory based methods ok. So, these two methods have got importance in materials science, computational material science and also designing of new materials even to get the estimation of the properties which cannot we cannot get it from the laboratory setup ok. So, AI, ML are used to get some of the parameters for for for running density functional theory calculation and also molecular dynamics calculation. At the same time the information we get from let us say DFT based calculation or molecular dynamics calculation that can be used as a database to further use this AI, ML method. So, this is a kind of a two way process both these this molecular dynamics and and DFT based simulation add to the database at the same time also AI, ML also help in in in getting better potential for DFT method or even to get the better a functional for potential for for molecular dynamics method or functional for DFT based method ok. So, it is time is very short. So, I will be covering only the main principle behind the molecular dynamics simulation and density functional theory based simulation. So, in the first lecture I will be covering the basic principle behind the molecular dynamics method. So, that you get the the background theory at the same time also when you use AI, ML methods to to get the parameters for for for for MD run. So, you you get to know where it is used and how it is been been used. So, today I will be covering basically introduce with the computational material science different methods suitable for different system size because a a single method you cannot use for different size of the of of the system. Some of the methods are used are useful for very very low scale system or even some some of the methods are are useful are are useful for a kind of the engineering design method or a component of a of a machine and all ok. So, I will be covering also the four paradigms of science where empirical, theoretical, computational and data driven. So, that gives you the connection between the computational and data driven. That means, the computational in this case we can say that the molecular dynamics or the DFT with the with with the machine learning methods ok. So, I will be covering mainly molecular dynamics which is essentially a Newtonian mechanics method, but DFT and the the the ab initio method are essentially we use the quantum chemical methods ok. And also we will see how the the the the atoms are are are modeled in the in the in the molecular dynamics method. So, we will be covering the methods application and limitation of molecular dynamics and inter atomic potential for molecular dynamics, different algorithm used to solve this this Newtonian equation of motion in the in the molecular dynamics methods. We will be covering mainly mainly three mostly used algorithms. And also we will see the what are the the the the the procedure was followed for doing a molecular dynamics calculation. Essentially some of the pre setups, initialization, equilibration, production runs, data generation and property calculation. At the same time also if time permits we will see that what are the criteria in need to see whether the the runs are are valid the MD runs are valid or not ok. So, these are the the the keywords which we can can and can search and also this keywords are essentially described in this lectures ok. So, coming to the position of molecular dynamics in the overall setup for materials computational materials and materials design. As you know that some of the process we can use for very short size may be in the range of let us say few few angstrom, may be in the range of nanometer range we we have to use the the quantum mechanics method which is called the electronic structure method or ab initio methods. And under this ab initio methods are the Hartree-Fock method and density function theory based methods are used. So, Hartree-Fock methods essentially we use the we solve the Schrodinger equation with some of the approximation. In the case of density function theory where instead of wave function we use the the electron density and use the concern equations, but the the Hartree-Fock principles are also have the lineage with the the DFT. So, in the next class when we discuss about the DFT method also we will first discuss the Hartree-Fock method and then we will move to density function theory method ok. So, this electronic method or the ab initio method we cannot we we can use only less may be less than 1000 1000 atoms ok. Then we move to molecular dynamics which is the the main topic for today's lecture. Essentially we can go to little bit larger size may be 10 to the power 10 number of atoms where essentially we solve the the Newtonian law of motion like force equals to mass into accelerations and which is essentially called the atomic scale. So, in this case we cannot deal with the electronic property ok. So, because we do not consider electrons we consider atoms as a kind of a sphere or deformable sphere and then we use the the Newtonian mechanics ok. So, now if you have to go to little bit larger size scale the essentially we need to use some of the statistical based methods. Essentially these are called or meso scale method or micro scale method where we can use the the Monte Carlo simulation and we can use the metropolis Monte Carlo simulation to simulate the thermodynamic example or we can use the kinetic Monte Carlo simulation to stimulate activated process like diffusion and all. And for the larger size for size system essentially we need to go for the finite elemental methods where we divide the system into small elements and calculate the variables like stress,strain,pressure,temperature as a set of algebraic differential equation and or or integral equation and solve by some of the numerical methods. So, in this case we can simulate some of the deformationbehavior or we can simulate the convective transfer of heat and mass and momentum also. And later on we can get also the engineering design. So, essentially it is give you the overall range of the scale and also different method we can use. So, in the in all the cases essentially the the three basic four four among the four basic forcebasic force which is the mainly force we consider only the electromagnetic interactions ok of the among the nucleus and and electrons and and and atom. Sometime the system size may be may be little bit larger, but at the same time also we need to look at the howif a a small portion reacts in that case we need to combine two two methods like quantum mechanics and molecular mechanics which is called thethe multiscale method or it is called QMM method. Similarly, also some of the parameter we can get from the molecular dynamics and those we can use in the in and some of the phenomenological methods or even in the finite element methods which is essentially are the which which are essentially the the multiscale method ok. So, this gives you the overall spectrum of different methods and where we can we can use and the position of molecular dynamics and and DFTDFT in the overalloverall spectrum of the the calculation methods. So, this gives you the different length and timescale for in material science. Here we can see that for thethe atomic range essentially in the range of 10 to the power minus 9 9 millimeter and the timescale in 10 to the power minus 12 seconds we need to use the molecular dynamics or you can and for the case of the electronic structure based property essentially the range in the range of 10 to the power minus 10 minus 11meter and the timescale is also very short in the range of 50 second in that case we have to use the the quantum quantum mechanics ok. So, as we go go down we need to go to the the the quantum mechanics or wave mechanics basebase method at the same time also it it also take more time to to solve it and which also dictates the number of atoms or number of constituent we can we can deal with. Like for in that case in exact sense it can be solved only for one particle system or one electron system, but if we do some of the approximation like in the case of the Hattie-Fock methods we can go up to let us say 50 or 50 or 100100 atoms and the scaling in the range of n to the power 4 where n is the number of up of atom ok. DFT scale we can go to little bit larger size may be in the range of 1000 1000 atoms and and the the the scale generally is range from to n to the power n to n to the power cube ok. Molecular dynamics essentially it is a kind of a linear scaling with respect to the the the number of atoms we we deal with ok. So, essentially this each approach represent a different balance among computational efficiency, tractable system size, accuracy and predictive power. So, essentially as you go to the DFT method or Hattie-Fock method these are more precise compared to let us say molecular dynamics like way. For material science we have to go for molecular dynamics method or DFT method method on the based on the maximum number of atoms that can be handled ok. So, now, fast feed separation method are becoming popular day by day, but molecular dynamics still serves where the quantum effects are negligible. That means, where you do not have to deal with the the electron transfer or electronic interaction, but quantum effects become significant only when the particle whether lambda is compatible with the interatomic distance where may be in the range of let us say in angstrom. Otherwise the use of easier Newtonian equation motion is well justified ok. So, this gives you the perspective on on the molecular dynamics methods ok. So, for paradigm of science because this main course is on the AIML and where does it connect with the molecular dynamics and DFT method. So, essentially first paradigm that means, the earlier time essentially it was kind of the empirical science. So, experimental observation and then also the manipulation of the of the observations and also some of the getting some of the empirical way. Then comes the the model based theoretical system like in the case of let us say laws of thermodynamics and all, but as the time progress the analytical solution of this the the law based system became became difficult by then essentially we need to when at the advent of computers essentially we go for the computational science essentially DFT DFT based methods and and molecular dynamic methods. So, essentially this is in the third third paradigm ok which is today's lecture, but the fourth paradigm comes when the it is of the big data driven science where you have got predictive predictive analysis, clustering, relationship, data mining, anomaly detection and all. So, this now essentially this data driven science require data where from we get the data we get the data from the experiment or we can get the data from the third paradigm or computational science. So, essentially there is a direct linkage between this computational science or thisDFT or MD simulation with the with the AIML. At the same timesome of the parameters required for running DFT or to to to run MD also we can getby using the the get or refine it further by using the the data data data driven science or molecular. So, essentially this is a two way process both these there both of them are contributingand and progressing at the same time. So, now, coming back to the main molecular dynamics methods we need to discuss about the main concept behind behind it essentially we consider the the system is made up of of atoms, but in the case of the atoms essentially we know that it is made up nucleus and electrons, but in the case of the molecular dynamics we omit this electron part and we just club it together as a as a atoms ok, but this essentially we gives thegive the the the mass at the centre of it. So, essentially we consider as a point mass at the same time also we may give some of the charge. So, in MD an atom is basically approximated as a sphere with point mass and electron role is totally neglected at the same. If we neglect the electron mass essentially electronic based property we cannotwe cannot get from the molecular dynamics that means, we do not do not get the reactivity or let us say we cannot get band gaps and all this all this thing ok. So, essentially let us say we atoms now this how this atom behaves in presence of others and under the influence of differentdifferent external parameter like external temperature, force, even stress, even electric field and all ok. So, essentially those field will give thethe force to the to the to the atoms and this atom will move as it moves also it will get different position and and also momentum and we can utilize thosethisthis velocity position and all at the same time also overall energy to get some of thethe thermodynamic property or some of the other process related properties ok. So, a typical procedure of of of MD. So, in this case let us say number 1 atom. So, move from dotted to theto the solid wall position. So, essentially let us say it was initially at the dotted position then it is moved to this due to thedue to the force imparted by the nearbynear nearbynearby atoms and of course, we need to use this step that means, how much it is moving in a in a small small step we need to use the time step at the this time step has to be such that within this time step no such significant process take place. That means, it has to be less than the sum of the whatever the process the high frequency process take place. So, it is gives essentially it is a as as the thethe F equal to m a dictates. So, essentially we need to get the force and this force we get it from thefrom the potential under which the all the atoms are. So, we take the negative gradients of this of the of the the the force field to get the force from the force we can calculate the the acceleration and with this acceleration using this acceleration we can get the new position or new new velocity that is essentially the basic principle from the beyond the molecular dynamics ok. Basic principle is the force calculation and how the the atoms move under thisthe the this force and now we need to deal with let us say thousands eveneven millions of atoms. So, if. So, you need to need to calculate the force on each atom at the same time also how its moves its new position how the velocity changes and all and at the same time also we need to see how it changes with respect to temperature external temperature pressure and all. At the same time also we need to seethe the most of the cases we need to see the molecular dynamics is done at under constant energy. So, in the if its constant energy that means, essentially kinetic energy and potential energy energy are are constant. So, there will be this distribution ok. So, in order to get differentproperties we need to use different condition and this condition are called ensembles ok. We will deal we will see how the what are the different ensembles are used what are the different algorithms are used toto see under the force how the particles moves at the same time also we need to see how we can converge it under the different constraints like that constant temperature constant pressure constant energy and all ok. So, coming back to thethe the main methods molecular dynamics is an integration of a Newtonian equation of motion over time to obtain the time evolution of the atoms and thus the properties of of our interest. Given the initial position and velocities of every atoms and using the provided inter atomic potential the force on each atoms are calculated. We will see what are the differentinter atomic potentials are there. Potential potential is essentially the field as a particle move in in under the gravitational force particles move similar way also thethe the the atoms in molecular dynamics essentially moves due to the the the the the the potential field we can say or potential under the potential energy and this potential energy are given as a kind of the some of thethe analytical forms and all. So, which are parametrics and this . So, AIMLs are mainlyuseful to get these these parameters ok. Most of the cases we get the parameter from the experimental observation or to get the parameter molecular dynamics parameter also we can get it from the ab initial calculation also. So, using this information essentially information means thisthe if you have the initial position and the velocity of every atoms and using thethe potential we can get the get the force. So, using this force and the velocity the the initial position are advanced towards the the lower energy state through a small steps ok which is called the time step and which results a new position new velocity and and so on ok. So, with the new data and inputs the these steps are repeated may be more than 1000 may be million times also until an equilibrium is reach and the system properties do not change with time. That means, we need to first equilibrate it and then also run the system run the the molecular dynamics. So, so during and after equilibration various data like atomic position momenta energy force are stored for each or some of some times time interval which can be used to calculate different property. So, these are the main use of molecular dynamics we can get the property like energies structure and mechanical property expansion or diffusion coefficient melting point, heat capacity, thermal conductivity, free energy, phase diagram and also we can get the structural property like the radial distribution function directly or via the statistical analysis ok. So, this time dependent properties which are collected it and then we can use thethe statistical methods which is called the ergodistic principle if it is satisfy then we canwe can get all this property using thethe statistical analysis. So, what are the other application of molecular dynamics? We can also get the we can get the description for processes and phenomena such as sputtering, vapour deposition, fast plastic flow, crack growth and fast they and fast structure, nano indentation, propagation of soft wave,deterioration, irradiation, ion bombardment, cluster impact and operation of the nano gas. So, these are the few if if application are, but this is not the not not the limited ok. So, the applications are going day by day and people are using anywhere where the the experimental setup are not possible due to either very high temperature requirement or maybevery hazardous atmosphere and all ok. And at the same time also before we do for go for any experiment also people can go for firstsee how may the the materials behave as a first we go for computational methods to see how the materials will behave and then we go for. So, that we can tune number of experiment we need to do and all ok . So, some of the limitation of molecular dynamics compared with the first principle method molecular dynamics is extremely fast and can handle much bigger system which is good news. However, some of the limitation of molecular dynamics include availability of potential is limited especially for multi component system and the accuracy of a potential is always under under underunder questions and length scale is still not macroscopic. So, maybe in the range of some some nano nanometer, but it is not in the macroscopic range and time scale is also limited to nanosecond ok. So, even going to microseconds is also veryvery difficult and time consuming ok . So, no electromagnetic properties we can obtain no electronic property no electromagnetic property magnetic property we can get using the molecular dynamicmethods ok. So, main thing in molecular dynamics is the the the the theforce exerted on the on the on the atom due to the field and this field is represented by the interatomic potential ok. So, we need to discuss more on the interatomic potential. So, when atoms are near enough to fill each other the balance between attraction and repulsion takes place and it is determined by interatomic potential or there is this potential are essentially some of thethe the attraction and and repulsion force between the between the atoms ok . So, atoms can eventually settle down at a minimum potential state at the equilibrium distance following Newton's equation of motion and the sum of all the force acting on the on the atom we can just use F equal to m a and a also we can write it as d v d t or d 2 r d r d 2 r d t d t square or as a kind of the rate of change of the the momentum ok. So, if the total energy is is constant in time which is the case for an isolated system for for MD simulation force is related to the negative gradient of the of the potential ok. So, with respect to the position that means, if you know the potential then if you do thethe the the gradients or that means, the differential with respect to the position and take the negative sign out we can get the force. So, if you know the force essentially and we know the mass. So, essentially we get the acceleration ok. So, if you know the acceleration from there we can calculate thethe the the velocity and momentum ok . So, therefore, if you know the potential of a system as a function of interatomic distance then we can have the forces on atom and thus can solve thisthis this equation for time evolution of the of the of the of the system ok . So, potential for specific system are generated by fitting some certain function with parameters of experimental data like some of the experimental data which is used to get these potentials are essentially potential are not are not, but is the some of the mathematical functions ok. So, this function can be polynomials now a days polynomial function are mostly used for to to get the parameter by using AI and ML method ok and other cases also some of the empirical the relationships also we canwe can use. So, what are the experimental data we can use to fit this potential fit the parameter or to or to get the potential interatomic potentials are equilibrium lattice parameters, cohesive energy, bulk modulus, elastic modulus, vacancy formation energy, thermal expansion coefficient, dielectric constant, vibrational system and also surface energy or or it can be calculated from the first principle methods ok. So, these empirical potential are system specific one should see ensure that this is transferable. Transferable let us say we get the potential for for some let us say two interactingatoms in one environment that that same potential may not be used those two the interaction between two atom in if the if the thethe surrounding atmosphere is different ok. So, this transferability is very importantfor for getting the the empirical potential ok and it should be be careful about using onespecific potential for other systems or conditions ok. So, even under the same system and conditions some potential may result in different dynamics in most of the potential are usually constructed by fitting as to static static properties ok. Significant progress has been made in recent years in generating accurate and reliable potential for many metallic systems semiconductor and some of the other ceramics methods. So, MLAI AI are being increasingly used to generate the inter interactive potential. So, here is the main use of AIMF in molecular dynamics ok . So, we will discuss few important inter types ofinter atomic potentials ok , but this is not the limited people can use nowadays a new types of methodpotential people are people are using ok . So, some of the most used potential are the fair potential essentially this is nothing, but thethe the Lennard-Jones type of potentialdepicting thethe Van der Waals type of forces ok . So, in this . So, if you have n atom systems an atom I interact with two atoms. So, essentially pair atoms we are considering the interaction between 2 2 2 atoms only ok , but overall interaction can be between 2 2 between 2 atoms or it. So, we usually call pair or if the 2 atoms are bonded also we can call it is a bonded interaction ok or it can be interaction between 3 atoms which we call triplet or angular. So, either it can be let us say 3 3 3 3 atoms are not bonded or it can be bonded. So, if the 3 atoms are bonded essentially it form a angle ok . So, we call it is angularangular potential ok or what is one or we we call it thethe the the one thisthis this bending potential ok . And 4 it can be 4 atom systems. So, essentially if this 4we call it quartet or it can be dihedrals. So, if the 4 atoms are bonded. So, essentiallythe the what what changes is the dihedral angle between this ok and so forth and anyway, but in this case we are considering only pair potential then that means, the interaction between 2 atoms only. So, you can. So, this is the overall potential and we are considering only this first first term for the for the pair potential ok. So, considering only the 2 atom pair pair interaction neglecting all others there are n minus 1 interaction per per atom and thus the number of pairs will be in the in order of the of the n square. So, number of pair will be n into n minus 1 by 2. So, it will be it is in the order of n square. So, if you have let us say you have 5 5 atoms. So, this is number of pair interaction the all thisthe the the this this arrows gives you essentially 10 ok, but it is it though it is for a for a small number it isfor for 5 atom it is 10, but essentially if you go larger. So, essentially it will ultimately come down to theas a order of n square ok . So, how we generally dictate? So, if it is bonded essentially we do it is a kind of a bond bond stretching model ok, but if it is not bonded sometime also the bonded also it is bonded at the same time also it can have certain kind of this Van der Waal type induction in that case we need to add both both of this ok. So, if there is no bonding interaction. So, essentially we can like in the case of thethe the let us say noble gas or if some if you have veryhard spherical molecules and all. So, in that case we can which is expressed in terms of the interatomic distance like this potential looks something like this where energy is in the y axis and the separation distance in the in the x axis and it say that it will have a energy corresponding position at which the energy is minimum at the same time also it will have the distance at which the the the energy is 0. So, we consider this this energy well or this epsilon and also thethe separation distance where the potential is 0. So, these two parameter we can use to get get the estimation of thethe repulsive potential and and attractive potential ok. So, attractive potential generally little be in the larger size and if it moves to very close cases essentially we have the the nuclear repulsion repulsion type ok. So, this is the Lennard-Jones potential very famous 4 epsilon sigma by r to the power 12 minus sigma by r to the power 6 and from there we just as I mentioned that the the the the the negative gradients will give you the force. So, the force will be will be 24 epsilon by sigma 2 sigma by r whole to the power 13 minus sigma by r whole to the power of 7. So, this is essentially the Lennard-Jones potential which is used for noble gas at the same time also when the you have got bonded molecules like hydrocarbons and all there also the this is essentially we use the bond bond bond stretching model at the same time also we can add this Lennard-Jones potential in order to give some of the van der Waals types of interaction also. So,the sphere potential may not be used we cannot use it for less for metals where the you which you have got the metallic type of top bonding. So, in that case essentially we go for this embedded atom method or EAM potential. So, in in atoms atoms or ions are in the middles of the the the electron sea and mostly coordinated with bigger number of the the the atoms with which a particular atom is is is kind of the the close or coordinated are quite high may be in the in the range of 8 to 12 ok. And, the coulomb interaction between them are long range one long range means essentially the the not only the nearest nearest neighbor. So, they the the long distance neighbors also have the have have got the intersection that is why it is called the long range one ok. And, this long range order may be covering easily 10 10 10 to 20 atoms ok. So, the embedded atom method consider the effective electron density at a given atomic site as one of the parameters thereby capturing some of the electronic effect while maintaining the simplicity of the of the potential ok. Essentially in metal valenceelectrons are delocalized as an electron clouds like here ok. And, exhibit an additional attraction towards this atom. So, the whatever this the the the delocalizing electrons also have got the attraction with this this this the ionic core ok. And, this EAM potential consider both the contribution from the the the the positive or repulsive pair potential and also the negative or attractive embedding energy ok. So, energy that is the embedding energy nothing, but the energy that is required to embed positively charged atom core into an electron cloud density density that approximate the missing n atom effect in a pair potential. Essentially the electronic or EAM potential you have this the pair potential at the same time also you have got this embedded potential given as a kind of the as a function of this charge density. Essentially this charge density again you need to rely with the the the the the F T base calculation. So, fitting this EAM potential is also a time consuming and if you have different setups let us say different alloys and all. So, it also take times to to get this this EAM potential ok. So, some of the element potential are there for some of the selected systems like say copper aluminum and all ok. So, in the case of the the EAM potential essentially this the second term is the embedding energy function as a and R ij is the scalar distance between the 2 atom i and i and j. And, this first term or the pair potential again this it it is given as the interaction between between between the different different atoms ok. So, 1 2 1 3 like this ok and this is rho rho i. So, essentially this is the the charge density at the at the at a particular core again it is given by it is affected by some of the other charge also or the other atom is or core charge which is given by P i equals to half of the summation of P j R ij where j not equal to i. Now, comes to the Tarsow potential. So, this Tarsow potential are basically you can use for covalent solids ok. And, in covalent solids so, deviation from the scheme of pairwise additives are even even larger since these solids are less closely packed often with a coordination number of only 4 and have strong directional bonds of equivalent strength ok. So, bond angle and bond order are the prime characteristics of these materials and the strength of a bond between 2 atoms depend on the local environment such as the coordination number and also the bond angle. So, somewhere we need to use this coordination number ok. So, Tarsow recognize this geometrical fact and develop a potential in the in this form ok. So, essentially Tarsow potential in the summation of this this the the repulsive repulsive force and also the the attractive force ok. But, in the case of the attractive force essentially we we scale it down using this this bond bond order V ij ok. So, this V ij is the bond order for bond between atom i and j and which is normally a decreasing function of the coordination number. So, this V ij take care of the coordination number and this V ij is essentially inversely proportional to the square root of the number of the coordination number ok. Although this both this term depends upon the distance because you can see that first term and second term attractive term and repulsive term depends upon the distance and this type of potential usually has more than 6 parameters to be fitted. With a good parameterization this potential have worked well for where you have got some of the covalent type of characteristics like silicon carbide semiconductor like silicon, diamond, amorphous carbon, some of the hydrocarbons and and all. Then interacting potentials for for for ionic solids essentially where they more than the covalent force is very less compared to the ionic force or coulombic type of force. So, in that case we need to use this this here this potential essentially we call it Buckingham potential ok. So, in ionic solid ions can be polarized or charged and the coulombic components can spread into a very long range and it generally predominates ok. And to describe potential for this solid one must consider two pair interaction terms short range term at the same time also the the long range potential. So, essentially the ionic charge essentially it have the the short range ok it can be pair pair pair potential or it can be and and also the the long range one ok. So, long range essentially nothing but the coulombic potential coulombic coulombic force ok coulombic potential. And the short range essentially we write it in this form. So, not exactly the Lennard Jones potential essentially, but we we use this form C 1 exponential of minus R ij by rho minus C 2 by R i R ij square where rho C 1 and C 2 are the constant we need to again fit this parameters ok. Z 1 Z 2 are are charges of the this they are this constituting atoms rho C 1 C 2 are the constant in the in the short rangeBuckingham potential. So, this Buckingham potential will be Buckingham potential plus the coulombic potential ok. This coulombic interaction term is the long range type that decay that as you can see it is 1 by R ij. So, for an ionic exchange this term under periodic boundary condition we will see what is the periodic boundary condition and where why you need to use the periodic boundary condition for molecular dynamics QSM. So, thisperiodic condition cannot be treated with the usual energy cut off approach because the potential periodically accumulates and impose a serious divergent voltage. So, instead of energy minimization energy will increase if we treat thisthis 1 by R basis thisthis coulombic potential, but we can effectively dealthis case that are essentially instead of diverging it should go to converging ok. So, essentially for that we use the real some techniques which generates an artificial charge distribution localized around each of the ion twice ok. So, essentially we use some kind of the cancellation process or we need to make this long rangeforce into a short range force ok. So, we need to use some of the reciprocal phase and also distribution of the point point charge into into GaussianGaussian kind of distribution and all ok . So, I am not going very details of this real some technique, but essentially the thing thing is that we need to make it short range and in order to make it short range essentially we need to go for use for the Fourier transformation method in thein the reciprocal phase ok. So, in that case all at the same time also we need to spread out the chargefrom point charge to the kind of a Gaussian kind of distribution ok . So, how we neutralize it essentially how you use this first with the opposite sign essentially which cancel the original charge thus behaves as a short range and next with the same sign the ions expressed in a spherically symmetric a Gaussian type and treated by first Fourier transformation in reciprocal phases. Now, some of the cases in ionic solids some of the the anion mostly the anions are may be also it can deform under the influence of the other other atom . So, you need to have the revenue to to change its charge its its its ion ionic charge under different condition ok. So, in that case we use the the cell cell potentialto represent this highly a polarizable anions. Now, all this all this potential what we have discussed in kind of a atomic gas we we cannot deal with the whatever the the electronic electron transfer or electronic interaction, but people also try whether we can get the some of the flavor of the DFT or or ab initio method in molecular dynamics itself by byby tweaking thethe potential and which is called the reactive force field potential ok. So, this this is also very widely studiedsubjects nowadays and this is the reactive force potentialfield development is also on a big work is going on under AIMLmethods also . So, reactive force methods potential made of tens of tens of parameters that are determined by first principle calculation and these potential are are designated to respond dynamically to the local environment and to describe some of the bond forming or bond breaking in situ. So, bond forming and bond breaking is nothing, but the electronic electronic transfer method ok . Although they are roughly 10 times slower than the thethe earlier mentioned potential and their successful application for hydrocarbons nanotubesthe semiconductor system have made the approach very popular and and promising. At the same time also it is promising for applying the AIML method also . So, after discussing on the potential now we need to see howthis potential dictates the how we get the force from this potential and how this force also dictates how much the particles move ok, how the displacement of the or the displacement of the of the atom ok. So, we need to use some of the algorithm to to or the procedure to to solve thethe the the the the Newton'sNewton's law of motion. Essentially Newton's law of motion is a differential equation ok. So, for differential equation essentially the solution is the integration or analytical solution, but analytical solution of let us say thousands or or let us say millions of atoms is very difficult. So, we need to resort to the numerical method. So, that is why we need to go for numerical algorithm ok . So, the total force of an N atom systems. So, N atom we can represent with for each atom we can represent with x y z or the space coordinate at the same time also we can representwith the the moment of coordinate also in the in phase space ok . So, 3 there. So, it will have 3 N atomic coordinates system with the empirical potential Newton's equation of motion can relate force to the derivative of the potential energy to the change in the position as a function of the time. So, essentially force is nothing, but summation of mass into accelerationacceleration also you can write here d 2 r d d t square ok and this force is nothing, but the thethethe negativegradients of the potential. So, u is the potential and this potential can be different different types of potential as we have discussed ok . In principle the first and second integration of the equation will result in the velocities ok and and the position that means, d r d t will give you the the the velocityvelocity and all ok. So, however, analytical solution because what we are getting we are getting this the the the a ok and a is nothing, but d 2 r d t square. So, if you first integrate you will get the velocity after second integration we get the position ok . So, the analytical solution for a 6 N dimensional system 3 N position and 3 N momenta are simply impossible this is very you cannot do it . So, MD simulation therefore, use the finite difference method in numerical step by step fashion. So, instead of what we do that we convert this differential equation into difference equation ok and we do we do the solution in a step by step way that means, we now we change the the the time we change change then then we change this this this position within a very minutetime which is called the time step ok. So, essentially we use the finite difference method in numericalstep by step fashion and it replace the differential d r d t with finite difference like delta r delta t and thus transform the differential equation into finite difference equation. For example, this position at a later time t t plus delta t is projected forward from the position at time t in the Taylor expression. So, you can use the Taylor expression let us say we let us say we know r t, but we want to get the position at t plus delta t. So, we can use thisthe the the Taylor series. So, the first term is nothing, but d r d t second term half of the del 2 r del t square which is nothing, but the acceleration ok . And we can we can use the higher order terms also, but most of the cases in practical MD run up to third order terms are considered and higher order other terms becomes truncation error expressed as the as the order. So, most of the cases if we use three order r terms essentially the error term will be in the range of t to the delta t to the power of 4 ok . So, now we can manipulate this this Taylor series expansion in different way and we get different different algorithm ok. Because let us say we what we start with in a molecular dynamic simulation that means, at time t equal to 0 what is the position. So, this position generally we can get from the from the structural data from let us say x r d data and all or we can just use a kind of a random distribution of the the the atoms atoms ok . Then we need to get how with times under the influence of differentdifferent external parameter like temperature or pressure ok, then how the position changes ok. So, this position change essentially we need to use different algorithm or different variation or different manipulation of this the the the the the Taylor series expansion to get the the property to to get the their position at later later time at the same time also which is kind of the efficient way. So, that means, which do not diverge at all at the same time also which is this step to do the energy minima or potential energy minimization ok . So, based on this well I will discuss mainly three main types of this algorithm use one of them is called the Valdez algorithm. So, in the Valdez algorithm what essentially we can take let us say we we do that we take the Taylor expansion for forward and backward position. So, this is a forward position that t then we want to get it delta t delta t or if you have let us say also you know the backward step also let us say at t t minus delta t . So, both this case we can write it as a Taylor series expansion now if you add these two . So, essentially we get the position at t plus delta t equals to 2 R t minus R of R at the backward position plus thethe acceleration into delta t square. So, this acceleration we know it from the from the potential ok and if you know the position at time time t and t minus delta t. So, we can get the position at t plus delta t. So, that is essentially the Valdez algorithm ok. So, in this case it requires only one fourth revolution per delta t because a is you get it from the fourth revolution. So, one one fourth, but in the case in this case essentially the error will be in the now if we add it up. So, this delta t this t q term is cancelling out. So, the error will be in the in the in the order of delta t to the power 4 ok. The first time step need the position at at the first time steps need the position at one step backward. So, essentially now you have started 0, but you do not get the minus the position at at minus any any minus time ok backward time. So, what we do . So, for the just when you start it. So, the the first step essentially we can manipulate it in other way. So, in that case we essentially we do the expression like this R at t there then R at delta t. So, essentially t 0 let us say t equal to 0. So, in that case only you use this ok. So, R t plus delta t equal to R t plus V t delta t plus a t by 2 delta t square ok. Of course, in this case let us say when you start it. So, first step we do it like this from second step we can use this this this expression ok, but at the same time let us say in this step essentially you do not have. So, this is the main main main algorithm which we use in the Verlet in the Verlet algorithm case, but in this case there is no expression for velocity which is good, but at the same time also you need to have the velocity to to calculate the kinetic energy or the velocity also relate somewhere with the with temperature and pressure also ok. So, after the we use this the final equation for the second time step t plus delta t velocity which is required to compute the kinetic energy do not appear in the time evolution by the Verlet algorithm. So, there we can calculate it from again using some of the difference equation like velocity you can write the velocity at position at time t plus delta t minus position at t minus delta t divided by 2 dash. So, these are the different different different differential the different what to say that approximation of differential equation to the difference equation ok. So, we need to use this another extra step to calculate the the velocity if we use the the Verlet algorithm ok. So, this is kind of the 2 2 calculation we need to do. So, in order to avoid this another modification is called the velocity Verlet algorithm ok. So, the velocity Verlet algorithm which is most frequently implemented methods in molecular dynamics, where the position velocities and acceleration at time t plus delta t are obtained from the corresponding quantity at time t by advancing t by half step. So, essentially we are advancing the we are getting the velocity at half step advance t plus delta t by 2 again also we are taking the first 2 2 2 2 term v t plus half a t into delta t. At the same time also r we are advancing a one full step not half step essentially r t plus delta t again we are approximate we are writing the taking the 3 3 term ok. So, we are getting r t v t delta t plus half a t delta t square. Now, this v t plus this term is nothing, but v at t plus delta t that is what we are we are writing here at the same time also, but here we are just directly calculating the acceleration from the from from the the the potential potential ok. So, we are taking the differential of this the the the potential with respect to the position and dividing by the mass we get this the the acceleration ok. So, you get the acceleration from from from here ok. So, now, but what we are getting here is the velocity velocity at half half step advance ok. So, the now what you get that the advance of then we need to advance this velocity by full step. So, from a at the previous previous and current time step and an express it using the half step advance velocity and the full step advance a. So, essentially now v at full step we are using the the velocity at the current step plus the we are using the the acceleration at the the current step and the next step. So, these next step acceleration we are getting from the from from the force and we are using this ok. So, essentially this velocity at the the next full step essentially velocity half step plus the half of the acceleration at the next step ok. So, in this case velocities are upgraded only after the new position and acceleration equivalent to new forces are computed ok. So, this is the most used method for numerical solution of the Newtonian equation of motion.