C7790 Introduction to Molecular Modelling -1Petr KulhΓ‘nek kulhanek@chemi.muni.cz National Centre for Biomolecular Research, Faculty of Science Masaryk University, Kamenice 5, CZ-62500 Brno PS/2022 Present Form of Teaching: Rev4 Lesson 6 Thermodynamics & Modelling Statistical Thermodynamics C7790 Introduction to Molecular Modelling TSM Modelling Molecular Structures C9087 Computational Chemistry for Structural Biology C7790 Introduction to Molecular Modelling -2Thermodynamics & Modelling Ξ”πΊπ‘Ÿ 0 = βˆ’π‘…π‘‡ ln 𝐾 𝐾 = 𝐢 π‘Ÿ 𝑐 𝐷 π‘Ÿ 𝑑 𝐴 π‘Ÿ π‘Ž 𝐡 π‘Ÿ 𝑏 Fundamental relation Ξ”πΊπ‘Ÿ 0 = 𝑐Δ𝐺𝑓,𝐢 0 + 𝑑Δ𝐺𝑓,𝐷 0 βˆ’ π‘ŽΞ”πΊπ‘“,𝐴 0 + 𝑏Δ𝐺𝑓,𝐡 0 A B C D We only need to know the properties of individual components involved in the reaction at standard conditions (or at different conditions, which are well defined). What do we need to know? solution at equilibrium We need to know the composition of solution at equilibrium. C7790 Introduction to Molecular Modelling -3Thermodynamics & Modelling Ξ”πΊπ‘Ÿ 0 = βˆ’π‘…π‘‡ ln 𝐾 𝐾 = 𝐢 π‘Ÿ 𝑐 𝐷 π‘Ÿ 𝑑 𝐴 π‘Ÿ π‘Ž 𝐡 π‘Ÿ 𝑏 Fundamental relation Ξ”πΊπ‘Ÿ 0 = 𝑐Δ𝐺𝑓,𝐢 0 + 𝑑Δ𝐺𝑓,𝐷 0 βˆ’ π‘ŽΞ”πΊπ‘“,𝐴 0 + 𝑏Δ𝐺𝑓,𝐡 0 What do we need to know? solution at equilibrium A B C D We only need to know the properties of individual components involved in the reaction at standard conditions (or at different conditions, which are well defined). We need to know the composition of solution at equilibrium. easier for modelling It is hard or impossible to model. A B C D D CD D B C C7790 Introduction to Molecular Modelling -4- Overview C7790 Introduction to Molecular Modelling -5- Overview bridge C7790 Introduction to Molecular Modelling -6- Statistical Thermodynamics Or what you should already know…. C7790 Introduction to Molecular Modelling -7Two approaches - I 1. Phenomenological approach: Thermodynamics examines the interrelationships between quantities that characterize the macroscopic state of the system and changes in these quantities in physical processes. Many of the features of the system can be clarified without a thorough knowledge of its internal structure. It is based on several axiomatically pronounced (and experimentally confirmed) laws, which, in connection with the known properties of the system, served to derive other properties and relationships. The state of the system is described using state functions and equations, which determine the relationships between individual state functions. Level of description: β–ͺ state functions β–ͺ state equations β–ͺ thermodynamic theorems wikipedia.cz, simplified C7790 Introduction to Molecular Modelling -8Two approaches - II 2. Statistical approach: Statistical physics (statistical mechanics) relates two levels of description of physical reality, namely the macroscopic and microscopic levels. In a more traditional sense, it deals with the study of the properties of macroscopic systems or systems, considering the microscopic structure of these systems (statistical thermodynamics). The founders were Ludwig Boltzmann and Josiah Willard Gibbs. Level of description: β–ͺ particles and interactions between them β–ͺ equations of motions wikipedia.cz, simplified C7790 Introduction to Molecular Modelling -9System properties t )(tM Time average: = tott otot dttM t M )( 1 The observable value ( ΰ΄₯𝑀) of the property M can be determined by two approaches: snapshot of the system at time t is called a microstate C7790 Introduction to Molecular Modelling -10System properties t )(tM Time average: Ensemble average: οƒ₯= = K i iiMpM 1 2/6 2/6 1/6 1/6 See later: We can run Monte Carlo simulations to get value of property by molecular modelling. The observable value ( ΰ΄₯𝑀) of the property M can be determined by two approaches: = tott otot dttM t M )( 1 See later: We can run molecular dynamics simulations to get value of property by molecular modelling. snapshot of the system at time t is called a microstate 𝑀𝑖 C7790 Introduction to Molecular Modelling -11Statistical ensemble prototype Statistical ensemble (Gibbs ensemble) is thought construction, in which the ensemble is formed by large number of copies of the system (prototype), whose thermodynamic properties we want to determine. Each replica of the prototype is located in exactly one microstate. Interactions between prototype replicas are very weak (it is practically possible to neglect them), however, sufficient for the ansemble to be located in thermodynamic equilibrium. οƒ₯= = K i ii MpM 1 iM Statistical view: 𝑝𝑖 = 𝑛 𝑖 βˆ— 𝐿 =? L number of copies of the prototype K number of microstates that the prototype can acquire ni number of prototypes in microstate i pi the probability of prototype occurrence in the given microstate i C7790 Introduction to Molecular Modelling -12Types of statistical ensembles The most common types of statistical ensembles include: β–ͺ microcanonical ensemble (NVE) - the prototype contains a constant number of particles, has a constant volume, and energy β–ͺ grand canonical ensemble (mVT) - the prototype has a constant chemical potential, volume, and temperature β–ͺ canonical ensemble (NVT) - the prototype contains a constant number of particles, has a constant volume, and temperature C7790 Introduction to Molecular Modelling -13Canonical ensemble Consider the system (prototype), which has a constant number of molecules, a constant volume, and temperature. NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT NVT prototype 1. The ensemble total energy is equal to the sum of the energies of the subsystems (the copies do not interact with each other) energy conservation: 𝐸 = ෍ 𝑖=1 𝐿 𝐸𝑖 = ෍ 𝑖=1 𝐾 𝑛𝑖 𝐸𝑖 LUUe = (internal energy) (Helmholtz energy) LSSe = (entropy) LFFe = Two constrains apply: 2. The sum of number of prototypes in given microstate must be constant and equal to L (total number of prototype copies): 𝐿 = ෍ 𝑖=1 𝐾 𝑛𝑖 C7790 Introduction to Molecular Modelling -14Entropy of canonical ensemble WkS Be ln= kB - Boltzmann constant kB - is not 1.0 because the definition of absolute temperature It can be shown that the entropy of the statistical ensemble is related to the statistical weight W. The statistical weight W determines number of possible ensemble implementations. = = K i i K n L nnW 1 1 ! ! ),...,( L - number of copies of the prototype K - number of microstates ni - number of prototypes of the given microstate inumber of microstates correction for indistinguishability of individual microstates number of all combinationsStatistical weight C7790 Introduction to Molecular Modelling -15Entropy of canonical ensemble, II Because L is a large number, there is an implementation for which the statistical weight of the distribution W* dominates over others. π‘Šβˆ—(𝑛1 βˆ— , . . . , 𝑛 𝐾 βˆ— ) >> π‘Šπ‘œπ‘‘β„Žπ‘’π‘Ÿπ‘  Then, we search for the ensemble composition, in which its entropy reaches maximum value and all imposed constraints are fulfilled. 𝑆 𝑒 = π‘˜ 𝑏 ln π‘Š (𝑛1 βˆ— , . . . , 𝑛 𝐾 βˆ— ) = π‘˜ 𝑏 ln 𝐿! ς𝑖=1 𝐾 𝑛𝑖! β†’ max! 𝐸 = ෍ 𝑖=1 𝐿 𝐸𝑖 = ෍ 𝑖=1 𝐾 𝑛𝑖 𝐸𝑖 Two constrains apply: 𝐿 = ෍ 𝑖=1 𝐾 𝑛𝑖 C7790 Introduction to Molecular Modelling -16Canonical ensemble - solution kB - Boltzmann constant T - absolute temperature The final result: Q e e e p i j i E K j E E i    βˆ’ = βˆ’ βˆ’ == οƒ₯1 * οƒ₯= βˆ’ = K j E j eQ 1  Canonical partition function: it is a normalization value TkB 1 = Value of observable property: 𝑀 = ෍ 𝑖=1 𝐾 𝑝𝑖 βˆ— 𝑀𝑖 Partition function determines various thermodynamic properties. C7790 Introduction to Molecular Modelling -17Thermodynamic properties Entropy: 𝑆 = π‘ˆ 𝑇 + π‘˜ 𝐡 ln 𝑄 Helmholtz energy F: QTkF B lnβˆ’= TSUF βˆ’= Q eE e eE pEU i j i E K i i K j E E K i i i K i i    βˆ’ = = βˆ’ βˆ’ = = οƒ₯ οƒ₯ οƒ₯ οƒ₯ === 1 1 1* 1 Internal energy: VN B T Q TkU ,ln ln οƒ· οƒΈ οƒΆ    ο‚Ά ο‚Ά = οƒ₯= βˆ’ = K j E j eQ 1  Canonical partition function: Canonical ensemble C7790 Introduction to Molecular Modelling -18Partition function and modelling Helmholtz energy F: QTkF B lnβˆ’=οƒ₯= βˆ’ = K j E j eQ 1  Canonical partition function: Evaluation of all (or important) microstates Molecular dynamics (classical continuous system) 𝑄 = 1 β„Ž3 ΰΆ΅ Ξ© π‘’βˆ’π›½π»(𝒙,𝒑) 𝑑𝒙𝑑𝒑 h - Planck constant H - Hamiltonian (energy of the system) For example: β€’ ideal gas model with contributions from β€’ electronic microstates β€’ vibration microstates β€’ rotation microstates β€’ translation microstates β€’ Monte Carlo simulations E1, E2, E3, … We need to find discrete energies of microstates. We need to solve describe energy evolution in time. C7790 Introduction to Molecular Modelling -19Partition function and modelling Helmholtz energy F: QTkF B lnβˆ’=οƒ₯= βˆ’ = K j E j eQ 1  Canonical partition function: Consider only the most important microstate approximation E1, E2, E3, … The most important microstate is the microstate with the lowest energy. 𝐹 = 𝐸1 Very often used for qualitative consideration or when computationally demanding methods are employed (typically quantum chemical calculations). C7790 Introduction to Molecular Modelling -20- Summary bridge