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Computational Chemistry Computational Chemistry 2018 Lecturer: Dos. Vesa Hänninen [email protected] Room B432 Contents of the course Lectures • Principles of quantum chemistry • Hartree Fock theory • Wave function methods • Density functional methods • Semiempirical methods • Graphical models • Molecular dynamics Computer exercises • Introduction to supercomputer environment (CSC) • Introduction to computation chemistry programs • Calculations of molecular properties • Computer simulations of molecular structures, vibrations, chemical reactions,… What is computational chemistry? • Relies on results of theoretical chemistry and computer science o Theoretical chemistry is defined as mathematical description of chemistry. It is development of algorithms and computer programs to predict chemical properties. o Computational chemistry is application of existing computer programs and methodologies to specific chemical problems. • Practice of efficient computer calculations to obtain chemical and physical properties of molecules, liquids and solids + = There are two different aspects to computational chemistry: 1. Computational studies can be carried out to find a starting point for a laboratory synthesis, or to assist in understanding experimental data, such as the position and source of spectroscopic peaks. 2. Computational studies can be used to predict the possibility of so far entirely unknown molecules or to explore reaction mechanisms that are not readily studied by experimental means. Thus, computational chemistry can assist the experimental chemist or it can challenge the experimental chemist to find entirely new chemical objects. Importance of computational chemistry Computational methods are considered important in scientific community, see: The top 100 papers in science Top cited papers in physics of last 110 years The Most Cited Chemistry Papers Published, 2004–14 The most cited chemistry articles, 2005 Also, computational chemistry has featured in a number of Nobel Prize awards, most notably in 1998 and 2013. Some applications Drug design Nanotechnological modelling and simulations Atmospheric science Computational chemistry programs There are many self-sufficient software packages used by computational chemists. Some include many methods covering a wide range, while others concentrating on a very specific range or even a single method. Details of most of them can be found in: •Biomolecular modelling programs: proteins, nucleic acid. •Molecular mechanics programs. •Quantum chemistry and solid state physics software supporting several methods. •Molecular design software •Semi-empirical programs. •Valence bond programs. Challenges and limitations • Theory level: Is the precision high enough for the application? • The computer programs: Efficiency and implementation • Computer resources: Is there enough power for the task? About CSC . CSC IT Center for Science Ltd is administered by the Ministry of Education, Science and Culture. CSC maintains and develops the state-owned centralised IT infrastructure and uses it to provide nationwide IT services for research, libraries, archives, museums and culture as well as information, education and research management. Researchers can use it’s large collection of scientific software and databases. CSC has offices in Espoo's Keilaniemi and in the Renforsin Ranta business park in Kajaani. Methods of computational chemistry • Ab initio that use rigorous quantum mechanics + accurate – computationally expensive good results for small systems ~102 atoms • Semi empirical that use approximate quantum mechanics – relies on empirical or ab initio parameters + affordable and in some cases accurate limited to well defined systems with ~104 atoms • Molecular mechanics that use classical mechanics – relies on empirical force fields without accounting for electronic properties (no bond breaking or forming) + very affordable and used as a virtual experiment can handle extremely large systems ~109 atoms Principles of quantum chemistry • The state of the system is specified by a (normalized) wave function 휓 • For every measurable property (observable) of a system such as energy 퐸 for example, there exist a corresponding operator (퐻 for 퐸) • Observables satisfy the eigenvalue equation. For example 퐻휓 = 퐸휓. • The expectation value of the observable, for example 퐸, is given by ∞ 퐸 = 휓∗퐻휓 푑휏 −∞ or using Dirac’s notation as 퐸 = 휓 퐻 휓 Majority of computational chemistry revolves around finding a solution to the static Schrödinger equation 퐻휓 = 퐸휓 The list of closed-form analytic solutions is VERY short. The list of famous chemical problems includes the H atom, the harmonic oscillator, the rigid rotor, the Morse potential, and the ESR/NMR problem. Hydrogen atom atomic orbitals can be used as a basis for the molecular orbitals. Harmonic oscillator basis for the molecular vibrational motion . Morse oscillator basis for the molecular stretching vibration . Rigid rotor basis for molecular rotational motion Variational method Yields approximate solution for the Schrödinger equation. Variational principle states that the expectation value of the Hamiltonian for trial wavefunction 휙 must be greater than or equal to the actual ground state energy 휙 퐻 휙 = 퐸[휙] ≥ 퐸 휙 휙 0 Example: Trial function expanded as a linear combination eigenfunctions of the hydrogen ground state 휓0 and the first exited state 휓1 휙 = 푐0 휓0 + 푐1 휓1 where 푐0 and 푐1 are unknown coefficients, so called “variational parameters”. The hydrogenic energy corresponding to this trial function is 2 2 2 2 푐0 휓0 퐻ℎ 휓0 + 푐1 휓1 퐻ℎ 휓1 + 푐0푐1 휓0 퐻ℎ 휓1 + 푐1푐0 휓1 퐻ℎ 휓0 푐0 퐸0 + 푐1 퐸1 퐸 휙 = 2 2 = 2 2 ≥ 퐸0 푐0 + 푐1 푐0 + 푐1 Any variations in the trial function 휙 = 푐0 휓0 + 푐1 휓1 which lower the energy expectation value are bringing the approximate energy closer to the exact value. The best solution can be obtained via optimization, i. e. searching the values of variational parameters 푐푖 which minimize the energy. 휕퐸 = 0 휕푐푖 푐0 퐸 푐1 For hydrogen atom, the optimal solution is obviously found when 푐0 = 1 and 푐1 = 0. In reality we don’t know the eigenfunctions. Instead we use basis functions which are physically relevant for the problem in hand. 휙 = 푐0 휑0 + 푐1 휑1 The hydrogenic energy corresponding to this trial function is 2 2 푐0 휑0 퐻ℎ 휑0 + 푐1 휑1 퐻ℎ 휑1 + 푐0푐1 휑0 퐻ℎ 휑1 + 푐1푐0 휑1 퐻ℎ 휑0 퐸 휙 = 2 2 푐0 + 푐1 휕퐸 2푐0 휑0 퐻ℎ 휑0 + 2푐1 휑0 퐻ℎ 휑1 2푐0퐸 = 2 2 − 2 2 = 0 휕푐0 푐0 + 푐1 푐0 + 푐1 → 휑0 퐻ℎ 휑0 − 퐸 푐0 + 휑0 퐻ℎ 휑1 푐1 = 0 휕퐸 = 0 → 휑0 퐻ℎ 휑1 푐0 + ( 휑1 퐻ℎ 휑1 − 퐸)푐1 = 0 휕푐1 We have a system of linear equations. According to linear algebra the energy eigenvalues can be obtained by finding solutions of a characteristic equation 휑 퐻 휑 − 퐸 휑 퐻 휑 퐻 − 퐸 퐻 0 ℎ 0 0 ℎ 1 = 0,0 0,1 = 0 휑0 퐻ℎ 휑1 휑1 퐻ℎ 휑1 − 퐸 퐻0,1 퐻1,1 − 퐸 The energies can be obtained by diagonalizing the Hamiltonian matrix 휑 퐻 휑 휑 퐻 휑 퐻 퐻 푑푖푎 퐸 0 퐻 = 0 ℎ 0 0 ℎ 1 = 0,0 0,1 0 휑0 퐻ℎ 휑1 휑1 퐻ℎ 휑1 퐻0,1 퐻1,1 0 퐸1 In general when we have N basis functions determinant of a characteristic polynomial becomes 퐻0,0 − 퐸 ⋯ 퐻0,푁 ⋮ ⋱ ⋮ = 0 퐻0,푁 … 퐻푁,푁 − 퐸 The secular determinant for N basis functions gives an N-th order polynomial in which is solved for N different roots, each of which approximates a different eigenvalue. Example: For a helium atom we can choose the trial function as follows: 2 휙 푟1, 푟2 = 퐶 1 + 푝푟12 + 푞 푟1 − 푟2 exp −훼 푟1 + 푟2 where 퐶 is normalization constant and 푝, 푞, and 훼 are variational parameters. After optimization: 푝 = 0.30, 푞 = 0.13, and 훼 = 1.816 퐸 = −2.9024 a.u. (Three parameters) 퐸 = −2.9037462 a.u. (1024 parameters) 퐸 = −2.9037843 a.u. (Experimental value) Using one parameter trial function 휙 = 퐶∙exp −훼 푟1 + 푟2 the minimum energy is 퐸 = −2.848 a.u. Born-Oppenheimer approximation Hamilton operator includes the kinetic and potential energy parts of the electrons and nuclei 퐻 = 푇 + 푉 The kinetic energy operator for electrons and nuclei can be written as 푛 2 푁 2 ℏ ∇ ℏ ∇푗 푇 = − 푖 − 2 푚e 2 푚n 푖 푗 2 2 2 2 휕 휕 휕 where ∇푖 = 2 + 2 + 2, 푛 and 푁 are numbers, and 푚e and 푚n are masses of electrons and nuclei, 휕푥푖 휕푦푖 휕푧푖 respectively. The potential energy operator includes the electron-electron, nuclei-nuclei, and electron-nuclei parts 푛 푛 푁 푁 푛 푁 2 2 2 1 푒 1 푍푗푍푗′푒 푍푗푒 푉 = + − 2 4휋휀0 퐫푖 − 퐫푖′ 2 4휋휀 퐑 − 퐑 ′ 4휋휀 퐫 − 퐑 푖 푖′≠푖 푗 푗′≠푗 0 푗 푗 푖 푗 0 푖 푗 Where 퐫푖 and 퐑푗 are positions of electrons and nuclei and 푍푗 is the atomic number of nuclei 푗. In practice, it is impossible to solve the Scrödinger equation for the total wavefunction Ψ(퐫푖, 퐑푗) exactly. Practical solution: Let’s approximate the wavefunction in a form, where it is factorized in electronic motion and nuclear motion parts 휓(퐫푖, 퐑푗) ≈ 휓el(퐫푖; 퐑푗)휓n (퐑푗) where function 휓(퐫푖; 퐑푗) describes electronic motion (depending parametrically on the positions of nuclei) and function 휓(퐑푗) describes the nuclear motions (vibrations and rotations). With this assumptions, the problem can be reformulated to two separate Scrödinger equations: 퐻el휓el 퐫푖; 퐑푗 = 푉(퐑푗)휓el 퐫푖; 퐑푗 퐻n휓n 퐑푗 = 퐸n휓n 퐑푗 The former equation is for the electronic problem, considering the nuclei to be fixed. The eigenvalue 푉(퐑푗) can be called as interatomic potential, which is then used as a potential energy for the latter equation for the nuclear motion problem. This procedure, the so called Born-Oppenheimer approximation, is justified because electron is lighter than the proton by the factor 2000, the electron quickly rearranges in response to the slower motion of the nuclei. + Example: For the fixed nuclear positions the H2 -ion the total energy operator can be written as We can write the above Hamiltonian further simplified by using atomic units a.u.
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