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An Overview of Computational Chemistry

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An Overview of Computational Chemistry Chem. 141 Dr. Mack An Overview of Computational Chemistry 1 Theoretical Chemistry: The mathematical description of chemistry Computational Chemistry: A mathematical method that is sufficiently well developed that it can be automated for implementation on a computer. Chemical Problems Computer Programs Physical Models Math formulas 2 1 Note that the words exact and perfect do not appear in these definitions. •Computational chemistry is based on a approximations and assumptions. •Only a real experimental measurement can approach the limits of exactness! 3 What does Computational Chemistry Calculate? Energy, Structure, and Properties • Molecular Geometries: • What is the energy for a given geometry? • How does energy vary when geometry changes? • Which geometries are stable? • How does energy change w/r extenal perturbation? 4 2 What else can be computed: •Enthalpies of formation •Dipole moment •Orbital energy levels (HOMO, LUMO, others) •Ionization energy (HOMO energy) •Electron affinity (LUMO energy) •Electron distribution (electron density) •Electrostatic potential •Vibrational frequencies and normal modes (IR spectra) •Electronic excitation energy (UV-Vis spectra) •NMR chemical shifts and coupling constants •Reaction path and barrier height •Reaction rate 5 What can one learn from computational calculations? • Molecular Visualization (Graphic Representation) • Molecular Mechanics (Classical Newtonian Physics) • Semi-empirical Molecular Orbital Theory Quantum • Ab Initio Molecular Orbital Theory Mechanical • Density Functional Theory Methods • Geometry Optimization • Molecular Dynamics 6 3 Types of Calculations: Molecular Mechanics (MM) Empirical energy functions parameterized against experimental dataFast, simple, generally not very accurate (> 104 atoms) Semi-empirical Molecular Orbital (MO) Methods Can treat moderate sized molecules (> 102 atoms). Accuracy depends on parameterization. Ab-initio Molecular Orbital Methods Computationally demanding (> 10 atoms) I will focus on Accuracy can be systematically improved. these topics Density Function Theory (DFT) More efficient than ab-initio calculations (> 10 atoms) Accuracy varies, however, there is no systematical way to improve the accuracy. 7 Types of Computational Calculations: Ab Initio: The term "Ab Initio" is latin for "from the beginning". •Computations of this type are derived directly from theoretical principles, with no inclusion of experimental data. •Mathematical approximations are usually a simple functional form for a an approximate solution to a differential equation. A wave function! (The Shrödinger Equation) The most common type of ab initio calculation is called a Hartree-Fock calculation. The ? of complicated system (molecule) is generated from a linear combination of simple functions (basis set). The parameters for the SE are varied until the solution (energy of the system) is optimized. 8 4 Molecular Transition States properties Reaction coords. Ab initio electronic structure theory Hartree-Fock (HF) Spectroscopic Electron Correlation (MP2, CI, CC, etc.) observables Geometry Prodding prediction Benchmarks for Experimentalists parameterization Goal: Insight into chemical phenomena. 9 Setting up the problem… What is a molecule? A molecule is “composed” of atoms, or, more generally as a collection of charged particles, positive nuclei and negative electrons. The interaction between charged particles is described by; Coulomb Potential qj rij qiq j qiq j Vij = V (rij) = = 4pe0rij rij qi Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena. 10 5 But, electrons and nuclei are in constant motion… In Classical Mechanics, the dynamics of a system (i.e. how the system evolves in time) is described by Newton’s 2nd Law: F = force F = ma a = acceleration dV d2r r = position vector - = m 2 dr dt m = particle mass In Quantum Mechanics, particle behavior is described in terms of a wavefunction, Y. ¶Y Hˆ Y = ih Time-dependent Schrödinger Equation t ¶ i = -1;h = h 2p ( ) Hˆ Hamiltonian Operator 11 Time-Independent Schrödinger Equation Hˆ (r,t) = Hˆ (r) If H is time-independent, the time- -iEt / h dependence of Y may be separated out Y (r,t) = Y(r)e as a simple phase factor. Hˆ (r)Y(r) = EY(r) Time-Independent Schrödinger Equation Describes the particle-wave duality of electrons. 12 6 Hamiltonian for a system with N-particles Hˆ = Tˆ + Vˆ Sum of kinetic (T) and potential (V) energy N N h2 N h2 æ ¶ 2 ¶ 2 ¶ 2 ö Tˆ = Tˆ = - Ñ 2 = - ç + + ÷ Kinetic energy å i å 2m i å2m ¶x 2 ¶y 2 ¶z2 i=1 i=1 i i=1 i è i i i ø æ 2 2 2 ö 2 ¶ ¶ ¶ Ñi = ç 2 + 2 + 2 ÷ Laplacian operator è ¶xi ¶yi ¶zi ø N N N N q q ˆ i j V = ååVij = åå Potential energy r i=1 j>1 i=1 j>1 ij (Coulombic) When these expressions are used in the time-independent Schrodinger Equation, the dynamics of all electrons and nuclei in a molecule or atom are taken into account. 13 Born-Oppenheimer Approximation • Since nuclei are much heavier than electrons, their velocities are much smaller. To a good approximation, the Schrödinger equation can be separated into two parts: – One part describes the electronic wave function for a fixed nuclear geometry. – The second describes the nuclear wave function, where the electronic energy plays the role of a potential energy. + ˆ ˆ ˆ ˆ ˆ ˆ H = Tn + Te + Vne + Vee +Vnn n = nuclear e = electronic ne = nucleus-electron ee = electron-electron nn = nucleus-nucleus 14 7 • Since the motions of the electrons and nuclei are on different time scales, the kinetic energy of the nuclei can be treated separately. This is the Born-Oppenheimer approximation. As a result, the electronic wave function depends only on the positions of the nuclei. • Physically, this implies that the nuclei move on a potential energy surface (PES), which are solutions to the electronic Schrödinger equation. Under the BO approx., the PES is independent of the nuclear masses; that is, it is the same for isotopic molecules. E H. + H. 0 H H • Solution of the nuclear wave function leads to physically meaningful quantities such as molecular vibrations and rotations. 15 Limitations of the Born-Oppenheimer approximation •The total wave function is limited to one electronic surface, i.e. a particular electronic state. •The BO approx. is usually very good, but breaks down when two (or more) electronic states are close in energy at particular nuclear geometries. •In such situations, a “ non-adiabatic” wave function - a product of nuclear and electronic wave functions - must be used. The electronic Hamiltonian becomes, ˆ ˆ ˆ ˆ ˆ H = Te +Vne + + Vee +Vnn B.O. approx.; fixed nuclear coordinates 16 8 Hartree-Fock Self-consistent Field (SCF) Theory GOAL: Solve the electronic Schrödinger equation, HeY=E Y. PROBLEM: + – Exact solutions can only be found for one-electron systems, e.g., H2 . SOLUTION: – Use the variational principle to generate approximate solutions. Variational principle – If an approximate wave function is used in He Y =E Y, then the energy must be greater than or equal to the exact energy. – The equality holds when Y is the exact wave function. In practice: – Generate the “best” trial function that has a number of adjustable parameters. – The energy is minimized as a function of these parameters. 17 The energy is calculated as an expectation value of the Hamiltonian operator: ˆ Y | H e | Y E = Y | Y If the wave functions are orthogonal and normalized (orthonormal), dij = 1 Yi | Yj = dij (Kroenecker delta) dij = 0 Then, ˆ E = Y | H e |Y ˆ ˆ ˆ ˆ ˆ H = Te +Vne + + Vee +Vnn 18 9 Hartree Approximation • assume that a many electron wave function can be written as a product of one electron functions Y(r1,r2 ,r3 ,L) = f(r1 )f(r2 )f(r3 )L • Use the variational method energy, solving the many electron Schrödinger equation is reduced to solving a series of one electron Schrödinger equations • In this approximation, each electron interacts with the average distribution of the other electrons 19 Variational Theorem • the expectation value of the Hamiltonian is the variational energy ò Y*Hˆ Ydt = Evar ³ Eexact ò Y*Ydt • the variational energy is an upper bound to the lowest energy of the system • any approximate wave function will yield an energy higher than the ground state energy • parameters in an approximate wave function can be varied to minimize the Evar • this yields a better estimate of the ground state energy and a better approximation to the wave function 20 10 Since electrons are fermions, S = 1/2, the total electronic wave function must be antisymmetric (change sign) with respect to the interchange of any two electron coordinates. (Pauli principle - no two electrons can have the same set of quantum numbers.) Each electron resides in a spin-orbital, a product of spatial and spin functions. F(1,2) = f1a(1)f2b(2) - f1a(2)f2b(1) jI – spin-orbit wave function a = spin up state and b = spin down state (for electrons 1 & 2) (Spin functions are orthonormal: a | a = b | b =1; a | b = b | a = 0) Interchange the coordinates of the two electrons, F(2,1) = f1a(2)f2b(1) - f1a(1)f2b(2) F(2,1) = -F(1,2) 21 A more general way to represent antisymmetric electronic wave functions is in the form of a determinant. For the two-electron case, f1a(1) f2b(1) F(1,2) = = f1a(1)f2b(2) - f1a (2)f2b(1) f1a(2) f2b(2) For an N-electron N-spinorbital wave function, f1(1) f2(1) L fN (1) f (2) f (2) L f (2) F = 1 2 N , f |f = d SD L L L L i j ij f N f (N) L f (N) 1( ) 2 N A Slater Determinant (SD) satisfies the antisymmetry requirement. Columns are one-electron wave functions, molecular orbitals. Rows contain the electron coordinates. One more approximation: The trial wave function will consist of a single SD.
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