 Good morning. My name is Emiliani Politi. I work in the Ulich Research Centre in Germany, and this morning my task is to give you a short introduction on QMM approaches. So far during this week you have seen and used methods based on classical physics. However, if we want to talk about QMM, that is quantum mechanics, molecular mechanics, we need to talk about quantum physics. But I also know that some of you is not so much familiar with quantum physics and quantum mechanics yet. Therefore, in the first part of this lecture, I'm going to revise some basic elements about quantum mechanics and quantum chemistry, essential to understand, let's say, the big picture. But I would like that you keep in mind from the beginning that if you want to use QMM approaches to investigate your systems, what you will see and learn today is absolutely not enough, and you will need to do the effort to go more in depth in the theory of computational quantum chemistry if you really plan to employ it in your investigations. After my lecture, my colleague Dimitri Morzov has prepared for you some tutorials where you will have the chance to watch QMM at work with a practical example and re-calculation by using the CP2K code. This is a rather popular quantum code with many features, some of which I will mention in this lecture, and others will be introduced by Dimitri in the tutorial. Summarizing my intent today is to provide you with a quick overview of the essential concept of quantum mechanics and computational quantum chemistry, and then, shortly introduce QMM in order to make it easier to understand what Dimitri is going to show you in the tutorial. In particular, here is the outline. In the first part we will discuss about when and why quantum mechanics is useful in biology, moreover, as just said, we give you a short recap of the crucial concept in quantum mechanics and computational quantum chemistry. Then we will talk about why we need to introduce hybrid QMM approaches, and in the second part of this lesson, after a break, we will go more in depth about how to build a QMM method and how quantum and classical region can be coupled in a QMM scheme. We will focus in particular on the approaches implemented in the CP2K code. Let's start by answering the question why do we need quantum mechanics in describing biological systems? In fact, in the previous days, you used approaches such as force field-based molecular dynamics, or talking, where the finest resolution is the atom. That is, the atoms are described like point in the space moving according to classical Newtonian equations if the dynamical was of interest. However, there are many phenomena in nature, including in biology, where such level of detail is not sufficient. Here are some examples. First, when chemical reactions are involved. For example, if you want to study enzymatic reactions, secondly, when you want to work with the system containing metal atoms for which no universal classical parameterization is in general available and ad hoc force field parameters have to be tuned in each specific case. Third example is when you want to study a phenomenon that involves proton transport, such as in the aerobic generation of ATP or in photosynthesis. In fact, in hydrogen-bonding solvent like water, protons do not diffuse as they add other common cations. That is, as a random Brownian mass motion due to the thermal fluctuation. Instead, the excess proton diffuses via the so-called grotes mechanism sketch in the picture here at the right. That is, through the formation and the concomitant cleavage of covalent bonds involving neighboring molecules. As last example, quantum mechanics is necessary when we need to perform first principle-based prediction of spectroscopic data, such as absorption and fluorescent spectra or even NMR, because empirical parameterization are unavailable or reliable. What do all these examples have in common? The fact that the dynamical behavior of the electrons inside the atoms is fundamental for a correct description of the phenomenon and cannot be neglected as done, for example, when we use the force field approach. Unfortunately, electrons, as well as the lightest nuclei such as protons, cannot be dynamically described through the classical Newtonian equation and more complex theory is required. That is, quantum mechanics. The fundamental equation of quantum mechanics is the so-called Schrodinger equation, written here at the top of the slide. Calculating the quantum property of the system implies to solve the corresponding Schrodinger equation, which you can consider it the equivalent of the Newtonian equation in the classical world. The unknown variable of the Schrodinger equation is Psi, the so-called wave function of the quantum system. It is a function of all the spatial coordinates of the quantum elements in the system, that is, electrons and nuclei for the molecular system and is a function of the time. Knowing the wave function of a system at a certain time allow us to compute the properties of that system at that time. How? By solving an integral like this, for example. The square modulus of Psi is proportional to the charge density distribution of the system. H in the Schrodinger equation is instead the so-called Hamiltonian, and it represents the physics of the system. It is equivalent to the force field at the classical level, that is, it contains the interaction energy terms of electrons and nuclei. For example, here at the bottom I reported a typical Hamiltonian used to describe molecular systems in quantum chemistry, where now small and capital are represent electronic and nuclear coordinates respectively. From left to right you can recognize the kinetic terms for the nuclei, the kinetic terms for the electrons, the Coulomb interaction terms between the electrons, the one between electrons and nuclei, and finally the Coulomb interaction terms between the nuclei. The Schrodinger equation is mathematically extremely complex to solve. Just to give you an example, with even only one particle, so n equals to 1, and a non-so-complex Hamiltonian, exact analytical solution of Schrodinger equation are not available. Therefore, for the many-body system we usually deal with, it can only be solved approximately by numerical solution generated by a computer, and many approximate approaches have been devised in the year for the same. In particular, if we are interested only in the properties of a system and not to its dynamical behavior, we can prove that it is enough to solve a slightly simpler equation for a time-independent Schrodinger equation, where now the wave function psi is independent from the time. Here at the right I listed some of the many schemes developed to approximately solve this simpler equation, Hartree-Fock theory, couple-cluster density-functional theory, and so on. When instead we are also interested in the dynamics of the quantum system, the full-time-dependent Schrodinger equation needs to be solved, and some different approaches are nowadays available to approximately find the solutions of this equation. These are the so-called ab initio molecular-dynamic schemes, such as the Ehrenfest and the Born-Oppenheimer and the Carparinello molecular-dynamic schemes. All those schemes share the assumption or better, the approximation that the motion of the nuclei, the atomic nuclei and one of the electrons in a molecule can be treated separately, but also, in addition, the assumption that the nuclear motion can be considered as a classical motion. These mostly because the masses of the nuclei are at least three order of magnitude larger than the electronic mass, and thinking classically the nuclei are much slower than the electrons. These assumptions are often collected under the name Born-Oppenheimer approximation, not to be confused with the molecular-dynamic scheme just mentioned. Even if nowadays I have to say this name refers to a more technical aspect of the assumption that I cannot describe here. In the large majority of the molecular systems including the typical large biological systems this approximation is well verified and can be safely employed. I do not have time to enter in much more details about the different ab initio molecular-dynamic schemes, but I want to just briefly mention how the Born-Oppenheimer dynamical scheme works because that is implemented in the CP2K code and we will use it in the tutorials. In these slides I wrote down the two equations that describe the Born-Oppenheimer molecular-dynamic scheme. The first thing to note is the separation between the electronic degrees of freedom and the nuclear degrees of freedom second equation. The second thing to note is that the electronic problem does not evolve in time. That is it is a time-independent Schreininger equation. While the second nuclear while the nuclear degrees of freedom evolve in time as a classical entities that is according to a Newton-like equation. Mass times acceleration the two dots over capital R means second derivative with respect time mass times acceleration equals to minus the gradient of a quantity that represents the potential felt by the nuclei. Schematically, the algorithm associated to this molecular-dynamics approach proceed this way. At each time step a time-independent Schreininger equation involving only the electronic degrees of freedom is solved via some electronic structure method like the ones mentioned before Artifog, density-functional theory and so on. Note that in the H.I. H.I. Hamiltonian in this equation coordinates capital R are not dynamical variables but just parameters. In this approximation the electrons move within a static electric field due to the presence of the nuclei. Then the electronic wave function Psi0 found solving this time independent Schreininger equation is used in the next step of the algorithm to calculate the forces on the nuclei via the right-hand side of the second equation. In fact, the forces are obtained as minus the gradient of the potential that depends on the electronic wave function Psi0. Finally, having obtained the forces the nuclei are moved according to a Newton-like equation. Note that the electronic problem in the first equation is the number of solutions. Each one corresponds to a different energy state. However, among those solutions here we are interested to the wave function corresponding to the state with the smallest energy. That is the ground state as indicated with the subscription 0. Instead, the mean term in the second equation refers to the fact that the electronic problem consisting of solving the time independent Schreininger equation that is of finding Psi0 can be recast in a variational problem that is a problem of finding a minimum. This is the so-called wave function minimization or optimization procedure. And on a computer it is computationally more convenient to implement it than any other algorithm that tries to solve directly the time independent Schreininger equation. We will see an example of this minimization algorithm in a few slides. Let's now focus on the methods to solve the first equation that is the electronic problem. As I mentioned before many methods have been devised in the years to approximately solve the time independent Schreininger equation. Some are very accurate and computational expenses others are computationally less demanding but also limited in accuracy. However almost every quantum code used by computational biophysicist and biochemist implements the density functional theory including CP2K the code that you will use in the tutorial. In fact this relatively recent approach represent probably the best compromise between accuracy and computational cost and it is therefore currently one of the very few approaches that offer the possibility to deal with the system of order of hundreds of atoms with sufficient accuracy. Let's briefly describe this density functional theory of DFT. It is based on the following two theorems proved by the physicist Honenberg and Kohn. First the ground state energy and therefore the ground state properties of a many electron system is a unique functional of the electronic density row. Here functional means a function of another function. In fact the density row is a function of the three special coordinates. In each point of the 3D space you have a value of the electronic density. Second the functional for the ground state energy row is variational in the sense we have mentioned before. The benefit to use this method is that instead to calculate the wave function size 0, which depends on all the electronic coordinates, the properties of the system depend only on row, which in turn depends only on three coordinates, the special coordinates. The drawback is that the functional row is not known and therefore the theory as it is cannot be used in practice. Luckily the physicist Kohn and Schamm in 1965 had the idea to recast the problem in order to make the FT a practical method. The idea is simple given a system of n electrons interacting between each other and with an external potential which could be for example the one of the nuclei in a molecular system. Kohn and Schamm hypothesized that it is always possible to find a fictitious system for by n fictitious non-interacting electrons, that is they do not interact with each other but which interact with an external potential that generates by construction the same electronic density as the one of the real fully interacting system. In this way the problem to find the density row of the real full interacting system that minimize the functional E of row is recast to the problem of solving n single electron equation much easier to solve than a single equation of n electrons. And above all all the terms in the single electron equations are known apart from one the so called exchange correlation function. In these Kohn-Schamm equations the phi i's represent the single electron wave function not to be confused with the psi zero in the previous slide that is the wave function of the entire system for by n electrons and the row can be obtained from the phi i's by using the first equation the first relation in the slide. In practice quantum chemist have proposed many recipes to approximate the unknown exchange correlation e x c for example by approximating to the simplest case homogenous electron gas or by fitting experimental data when you have to specify the level of theory you are going to use to solve the electronic problem with the DFT you need also to state explicitly the exchange correlation functional to be employed. Now having decided which exchange correlation functional to use how do we get the electronic density row in other words how does it work the algorithm that solves the system of n single electron Kohn-Schamm equations the system of Kohn-Schamm equation can be numerically solved through an iterative procedure because those equations are non-linear and that some terms in the equations depend on the electronic density itself that is on the solutions the iterative procedure to solve this equation can be summarized this way first we start with an arbitrary electronic density in order to have a set of equations to solve completely and the question to solve completely defined then we find the PHI eyes that is the single electron wave functions which then can be used to get the new electronic density by using the equality at the bottom and we measure the difference between the new and the previous density if the difference is below some predetermined threshold we consider the new density already converged and we stop the iterations otherwise we take the new density and we go back to the first step this self-consistent approach is frequently employed in quantum chemistry and in general when we have to solve non-linear equations like the Kohn-Schamm equations all these equations all the equations we have seen so far including the Kohn-Schamm equations are continuous equations their solution as for example the PHI eyes in the Kohn-Schamm equation are function defined on the space in this sense are continuous equations however to put such a problem on a computer to solve such equation we need to discretize it how can we discretize the problem of solving the Kohn-Schamm equations this is done by solving by expanding the wave function or a density over a finite set of known function and we refer usually to this set of function as the basis set this way the problem to solve a continuous differential equation is recast to the problem to diagonalize a matrix and find again values and again vector when one wants to specify the type of quantum chemistry calculation is going to perform on a computer they need to specify both the level of theory for example DFT together with the chosen exchange correlation functional and also the employed basis set this defines completed the level of theory that is being used commonly two classes of basis set can be identified the localized basis set such as the atom center gaussian function very suitable to describe the wave function of localized objects like molecules and non-local basis sets such as the plane waves which were originally employed to describe the wave function of periodic system like condensed matter solid state ones both type of basis sets have advantages and disadvantages in the slide I listed the sum of them for your reference since I do not have time to add more details here really the code that you will see in the tutorial CP2K implements also a more sophisticated approach which combines both classes of basis sets this is the so-called hybrid or dual gaussian and plane wave method in short GPW the method uses an atom centered gaussian type basis to describe the wave function but also an auxiliary plane wave basis to describe the density using a plane wave basis set for the charge density means using grids in real space to represent the charge density in fact by a mathematical operation called Fourier transform that computationally can be performed in a very efficient way on a grid with an algorithm called fast Fourier transform one can pass from the representation on the real space grid to the representation of the reciprocal space that is the g space of the plane waves finer grids that is with smaller cells correspond to larger tofs in the reciprocal space what is the advantage of this dual representation that is localized basis set for the wave function and on local basis set for the density the advantage is mainly on performance or better on scaling with the density represented as a sum of plane waves or which is the same on a regular grid the efficiency of the fast Fourier transform algorithm can be exploited to obtain the long range energy terms in a time that scales linearly with the system size thus is convented one of the major bottlenecks of standard gaussian based calculations we will come back on this point in more details later today as we will see as you will see in the tutorial in order to set up a calculation in CP2K that uses the GPW approach you will have to provide information on both the gaussian type and the plane wave basis sets to be used the largest systems investigated so far via full quantum mechanical approaches that is by describing the entire molecular system through quantum mechanics include less than 10,000 atoms in contrast typical sizes of biological system are much larger than 10,000 atoms therefore investigating interesting biological system at full quantum mechanical level is beyond the current state of the hardware and software technology but as we have seen at the beginning there are cases where a quantum chemical solution is required also for large biological system this implies that for these cases at present the only viable way is to resort to multi scale approaches as for example the hybrid quantum mechanical molecular mechanics one in fact in the biological system the regions where the electronic description is necessary is usually especially limited area of the system for example the region where the chemical reaction takes place and this feature makes a QMM approach very suitable for these systems this because in the QMM approach the system is fictitiously separated in two parts that are described at different levels of theory a small part the QM or quantum part usually the chemical active region or in general the region where the electronic degrees of freedom are important which is treated at quantum level by computational demanding electronic structure methods as for example density functional theory and the rest of of the system which contains atoms that for example do not directly participate in the reaction the chemical reaction that is instead described efficiently at a lower level of theory usually by classical force field this part is usually referred as the MMM or classical part a QMM interface is the part of the code or a standalone code that couples in a coherent way the two different resolutions okay we have reached the end of the first part of the lecture in the second part we will go more in depth on how the coupling between the quantum and the classical regions can be done and we will describe different QMM approaches including the one implemented in CP2K in the CP2K code that you will see in the tutorial