Introduction to Computational ChemistryFor the Beginner…Focus on…PowerPoint PresentationSetting up the problem…But, electrons and nuclei are in constant motion…Time-Independent Schrödinger EquationHamiltonian for a system with N-particlesBorn-Oppenheimer ApproximationBorn-Oppenheimer Approx. cont.Slide 11Self-consistent Field (SCF) TheorySCF cont.Slide 14Slide 15Hartree-Fock EquationsSlide 17Slide 18Slide 19Employ the method of Langrange Multipliers:Define the Fock Operator, FiAfter a unitary transformation, lij0 and liiei.Koopman’s TheoremBasis Set ApproximationMO ExpansionTotal Energy in MO basisGeneral SCF ProcedureComputational EffortRestricted and Unrestricted Hartree-FockComparison of RHF and UHFAb Initio (latin, “from the beginning”) Quantum Chemistry Summary of approximationsIntroduction to Computational ChemistryNSF Computational Nanotechnology and Molecular Engineering Pan-American Advanced Studies Institutes (PASI) Workshop January 5-16, 2004California Institute of Technology, Pasadena, CAAndrew S. IchimuraFor the Beginner…There are three main problems:1. Deciphering the language.2. Technical implementation.3. Quality assessment.Focus on…Calculating molecular structures and relative energies.1. Hartree-Fock (Self-Consistent Field)2. Electron Correlation3. Basis sets and performanceAb initio electronic structure theoryHartree-Fock (HF)Electron Correlation (MP2, CI, CC, etc.)Molecular propertiesGeometry predictionBenchmarks for parameterizationTransition StatesReaction coords.SpectroscopicobservablesProddingExperimentalistsGoal: Insight into chemical phenomena.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 Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena.€ Vij= V (rij) =qiqj4πε0rij=qiqjrijrijqiqjBut, 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 = maF = forcea = accelerationr = position vectorm = particle mass€ −dVdr= md2rdt2In Quantum Mechanics, particle behavior is described in terms of a wavefunction, . € ˆ H Ψ = ih∂Ψ∂tHamiltonian Operator€ ˆ H Time-dependent Schrödinger Equation € i = −1;h = h 2π( )Time-Independent Schrödinger EquationIf H is time-independent, the time-dependence of  may be separated out as a simple phase factor. € ˆ H (r,t) =ˆ H (r)Ψ (r,t) = Ψ (r)e−iEt / h€ ˆ H (r)Ψ (r) = EΨ (r)Time-Independent Schrödinger EquationDescribes the particle-wave duality of electrons.Hamiltonian for a system with N-particlesSum of kinetic (T) and potential (V) energy€ ˆ H =ˆ T +ˆ V € ˆ T =ˆ T i= −h22mii=1N∑i=1N∑∇i2= −h22mii=1N∑∂2∂xi2+∂2∂yi2+∂2∂zi2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟€ ∇i2=∂2∂xi2+∂2∂yi2+∂2∂zi2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Laplacian operatorKinetic energy€ ˆ V = Vijj>1N∑i=1N∑=qiqjrijj>1N∑i=1N∑Potential energyWhen 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.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 wavefunction for a fixed nuclear geometry.–The second describes the nuclear wavefunction, where the electronic energy plays the role of a potential energy.• So far, the Hamiltonian contains the following terms:€ ˆ H =ˆ T n+ˆ T e+ˆ V ne++ˆ V ee+ˆ V nn€ ˆ T nKinetic energy of nuclei, n€ ˆ T eKinetic energy of electrons, e€ ˆ V neElectron-nuclear attraction€ ˆ V eeElectron-electron repulsion€ ˆ V nnInternuclear repulsionBorn-Oppenheimer Approx. cont.•In other words, the kinetic energy of the nuclei can be treated separately. This is the Born-Oppenheimer approximation. As a result, the electronic wavefunction 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.•Solution of the nuclear wavefunction leads to physically meaningful quantities such as molecular vibrations and rotations.0EHHH. + H.Limitations of the Born-Oppenheimer approximation•The total wavefunction 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” wavefunction - a product of nuclear and electronic wavefunctions - must be used.•In writing the Hamiltonian as a sum of electron kinetic and potential energy terms, relativistic effects have been ignored. These are normally negligible for lighter elements (Z<36), but not for the 4th period or higher. •By neglecting relativistic effects, electron spin must be introduced in an ad hoc fashion. Spin-dependent terms, e.g., spin-orbit or spin-spin coupling may be calculated as corrections after the electronic Schrödinger equation has been solved.The electronic Hamiltonian becomes,€ ˆ H =ˆ T e+ˆ V ne ++ˆ V ee+ˆ V nnB.O. approx.; fixed nuclear coord.Self-consistent Field (SCF) TheoryGOAL: Solve the electronic Schrödinger equation, He=E.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 wavefunction is used in He=E, then the energy must be greater than or equal to the exact energy. The equality holds when  is the exact wavefunction.In practice: Generate the “best” trial function that has a number of adjustable parameters. The energy is minimized as a function of these parameters.SCF cont.The energy is calculated as an expectation value of the Hamiltonian operator:€ E =Ψ∗∫ˆ H eΨ dτΨ∗Ψ dτ∫Introduce “bra-ket” notation,€ Ψ∗∫ˆ H eΨdτ