Electronic Structure Calculations In Quantum Chemistry
Electronic Structure Calculations in QuantumChemistryAlexander B. PachecoUser Services ConsultantLSU HPC & LONIsys-help@loni.orgHPC TrainingLouisiana State UniversityBaton RougeNov 16, 2011Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20111 / 55
Outline1Introduction2Ab Initio Methods3Density Functional Theory4Semi-empirical Methods5Basis Sets6Molecular Mechanics7Quantum Mechanics/Molecular Mechanics (QM/MM)8Computational Chemistry Programs9Exercises10Tips for Quantum Chemical CalculationsElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20112 / 55
IntroductionWhat is Computational Chemistry?Computational Chemistry is a branch of chemistry that usescomputer science to assist in solving chemical problems.Incorporates the results of theoretical chemistry into efficient computerprograms.Application to single molecule, groups of molecules, liquids or solids.Calculates the structure and properties of interest.Computational Chemistry Methods range from123Highly accurate (Ab-initio,DFT) feasible for small systemsLess accurate (semi-empirical)Very Approximate (Molecular Mechanics) large systemsElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20113 / 55
IntroductionTheoretical Chemistry can be broadly divided into twomain categories1Static Methods Time-Independent Schrödinger EquationĤΨ EΨ Quantum Chemical/Ab Initio /Electronic Structure Methods Molecular Mechanics2Dynamical Methods Time-Dependent Schrödinger Equationı Ψ ĤΨ t Classical Molecular Dynamics Semi-classical and Ab-Initio Molecular DynamicsElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20114 / 55
Tutorial GoalsProvide a brief introduction to Electronic Structure Calculations inQuantum Chemistry.12345Overview of Quantum Chemical methods.What kind of calculations can we carry out?What experimental properties can we study/understand?How to create input files?Tips and Tricks to run calculations?Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20115 / 55
Ab Initio MethodsAb Initio, meaning "from first principles", methods solve the Schrödingerequation and does not rely on empirical or experimental data.Begining with fundamental and physical properties, calculate howelectrons and nuclei interact.The Schrödinger equation can be solved exactly only for a few systems Particle in a BoxRigid RotorHarmonic OscillatorHydrogen AtomFor complex systems, Ab Initio methods make assumptions to obtainapproximate solutions to the Schrödinger equations and solve itnumerically."Computational Cost" of calculations increases with the accuracy of thecalculation and size of the system.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20116 / 55
What can we predict with Ab Initio methods?Molecular Geometry: Equilibrium and Transition StateDipole and Quadrupole Moments and polarizabilitiesThermochemical data like Free Energy, Energy of reaction.Potential Energy surfaces, Barrier heightsReaction Rates and cross sectionsIonization potentials (photoelectron and X-ray spectra) and ElectronaffinitiesFrank-Condon factors (transition probabilities, vibronic intensities)Vibrational Frequencies, IR and Raman Spectra and IntensitiesRotational spectraNMR SpectraElectronic excitations and UV-VIS spectraElectron density maps and population analysesThermodynamic quantities like partition functionElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20117 / 55
Ab Initio TheoryBorn-Oppenheimer Approximation: Nuclei are heavier than electronsand can be considered stationary with respect to electrons. Also knowas "clamped nuclei" approximations and leads to idea of potentialsurfaceSlater Determinants: Expand the many electron wave function in termsof Slater determinants.Basis Sets: Represent Slater determinants by molecular orbitals, whichare linear combination of atomic-like-orbital functions i.e. basis setsElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20118 / 55
Born-Oppenheimer ApproximationSolve time-independent Schrödinger equationĤΨ EΨFor many electron system:X e2 ZαX e2 2 X 2α 2 X 2 X e2 Zα ZβĤ i 2 α Mα2me i4π 0 Rαβ4π R4π 0 rij0 αiα,ii jα β {z } {z} {z} {z }{z} T̂nT̂eV̂enV̂nnV̂ee{z }V̂The wave function Ψ(R, r) of the many electron molecule is a function ofnuclear (R) and electronic (r) coordinates.Motion of nuclei and electrons are coupled.However, since nuclei are much heavier than electrons, the nucleiappear fixed or stationary.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 20119 / 55
Born-Oppenheimer ApproximationBorn-Oppenheimer Approximation: Separate electronic and nuclearmotion:Ψ(R, r) ψe (r; R)ψn (R)Solve electronic part of Schrödinger equationĤe ψe (r; R) Ee ψe (r; R)BO approximation leads tothe concept of potentialenergy surfaceThe electronic potential is afunction of nuclearcoordinates.V(kcal/mol)V(R) Ee Vnn3002001000De 100 2001In Molecular Dynamics, thenuclei move along this energysurface obeying Newton’sLaws of Motion.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduRe23RHO H(a0)45Nov 11, 201110 / 55
Potential Energy SurfacesThe potential energy surface (PES) is multi-dimensional (3N 6 fornon-linear molecule and 3N 5 for linear molecule)The PES contains multiple minima and maxima.Geometry optimization search aims to find the global minimum of thepotential surface.Transition state or saddle point search aims to find the maximum of thispotential surface, usually along the reaction coordinate of interest.Picture taken from Bernard Schlegel’s course slide at http://www.chem.wayne.edu/ hbs/chm6440/Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201111 / 55
Geometry OptimizationsGeometry optimization is used to find minima on the potential energy surface, with theseminimum energy structures representing equilibrium structures.Optimization also is used to locate transition structures, which are represented by saddlepoints on the potential energy surface.Optimization to minima is also referred to as energy minimization.During minimization, the energy of molecules is reduced by adjusting atomic coordinates.Energy minimization is done when using either molecular mechanics or quantummechanics methods, and it must precede any computational analyses in which thesemethods are applied.For example, geometry optimization can be used to123characterize a potential energy surfaceobtain a structure for a single-point quantum mechanical calculation, which providesa large set of structural and electronic propertiesprepare a structure for molecular dynamics simulation - if the forces on atoms aretoo large, the integration algorithm may fail.These energies apply to molecules in a hypothetical motionless state at 0 K. Additionalinformation is needed to calculate enthalpies (e.g., thermal energies of translation,vibration, and rotation) and free energies (i.e., entropy).Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201112 / 55
Wavefunction MethodsThe electronic Hamiltonian (in atomic units, , me , 4π 0 , e 1) to besolved isX 1X Zα Zβ1 X 2 X ZαĤe i 2 iRiαrijRαβα,ii jα βCalculate electronic wave function and energyEe hψe Ĥe ψe ihψe ψe iThe total electronic wave function is written as a Slater Determinant ofthe one electron functions, i.e. molecular orbitals, MO’s1ψe N!φ1 (1)φ1 (2)···φ1 (N)φ2 (1)φ2 (2)···φ2 (N)············Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduφN (1)φN (2)···φN (N)Nov 11, 201113 / 55
MO’s are written as a linear combination of one electron atomicfunctions or atomic orbitals (AO’s)φi NXcµi χµµ 1cµi MO coefficientsχµ atomic basis functions.Obtain coefficients by minimizing the energy via Variational Theorem.Variational Theorem: Expectation value of the energy of a trialwavefunction is always greater than or equal to the true energyEe hψe Ĥe ψe i ε0Increasing N Higher quality of wavefunction Higher computationalcostElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201114 / 55
Ab Initio MethodsThe most popular classes of ab initio electronic structure methods:Hartree-Fock methods Hartree-Fock (HF) Restricted Hartree-Fock (RHF): singlets Unrestricted Hartree-Fock (UHF): higher multiplicities Restricted open-shell Hartree-Fock (ROHF)Post Hartree-Fock methods Møller-Plesset perturbation theory (MPn) Configuration interaction (CI) Coupled cluster (CC)Multi-reference methods Multi-configurational self-consistent field (MCSCF)Multi-reference configuration interaction (MRCI)n-electron valence state perturbation theory (NEVPT)Complete active space perturbation theory (CASPTn)Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201115 / 55
Hartree-Fock1Wavefunction is written as a single determinantΨ det(φ1 , φ2 , · · · φN )2The electronic Hamiltonian can be written asXXĤ h(i) v(i, j)ii jX Zα11where h(i) 2i and v(i, j) 2rriαiji,α3The electronic energy of the system is given by:E hΨ Ĥ Ψi4The resulting HF equations from minimization of energy by applying ofvariational theorem:f̂ (x1 )φi (x1 ) εi φi (x1 )where εi is the energy of orbital χi and the Fock operator f , is defined asX f̂ (x1 ) ĥ(x1 ) Ĵj (x1 ) K̂j (x1 )jElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201116 / 55
Hartree-Fock1Ĵj Coulomb operator average potential at x due to chargedistribution from electron in orbital φi defined as Z φj (x2 )φj (x2 )Ĵj (x1 )φi (x1 ) dx2 φi (x1 )r122K̂j Exchange operator Energy associated with exchange ofelectrons No classical interpretation for this term. Z φj (x2 )φi (x2 )dx2 φj (x1 )K̂j (x1 )φi (x1 ) r123The Hartree-Fock equation are solved numerically or in a spacespanned by a set of basis functions (Hartree-Fock-Roothan equations)ZKXφi Cµi φ̃µSµν dx1 φ̃ µ (x1 )φ̃ν (x1 )µ 1ZXXFµν dx1 φ̃ µ (x1 )f̂ (x1 )φ̃ν (x1 )Fµν Cνi εiSµν CνiννFC SCεElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201117 / 55
Hartree-Fock1The Hartree-Fock-Roothan equation is a pseudo-eigenvalue equation2C’s are the expansion coefficients for each orbital expressed as a linear combination of thebasis function.3Note: C depends on F which depends on C need to solve self-consistently.4Starting with an initial guess orbitals, the HF equations are solved iteratively or selfconsistently (Hence HF procedure is also known as self-consistent field or SCF approach)obtaining the best possible orbitals that minimize the energy.SCF procedure1Specify molecule, basis functions and electronic state of interest2Form overlap matrix S3Guess initial MO coefficients C4Form Fock Matrix F5Solve FC SCε6Use new MO coefficients C to build new Fock Matrix F7Repeat steps 5 and 6 until C no longer changes from one iteration to the next.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201118 / 55
SCF Flow ChartForm overlap matrixSInput Coordinates,Basis sets etcGuess InitialMO CoefficientsCForm Fock MatrixFUpdate CC C0SolveFC0 SC0 εCalculate PropertiesENDyesnoSCF Converged? C-C0 tolElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201119 / 55
Post Hartree-Fock Methods Methods that improve the Hartree-Fock results by accounting for the correlation energyare known as Post Hartree-Fock methods The starting point for most Post HF methods is the Slater Determinant obtain fromHartree-Fock Methods. Configuration Interaction (CI) methods: Express the wavefunction as a linearcombination of Slater Determinants with the coeffcients obtained variationallyP Ψi i ci Ψi i Many Body Perturbation Theory: Treat the HF determinant as the zeroth order solutionwith the correlation energy as a pertubation to the HF equation.Ĥ Ĥ0 λĤ 0(0)εi Ei Ψi i (0) Ψi i(1) λEi (2) λ2 E i(1)λ Ψi i ···(2)λ2 Ψi i · · · Coupled Cluster Theory:The wavefunction is written as an exponential ansatz Ψi eT̂ Ψ0 iwhere Ψ0 i is a Slater determinant obtained from HF calculations and T̂ is an excitationoperator which when acting on Ψ0 i produces a linear combination of excited Slaterdeterminants.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201120 / 55
ScalingScaling MP5,CISDT,CCSDT9N!Full CIN Number of Basis FunctionsElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201121 / 55
Density Functional TheoryDensity Functional Theory (DFT) is an alternative to wavefunction based electronicstructure methods of many-body systems such as Hartree-Fock and Post Hartree-Fock.In DFT, the ground state energy is expressed in terms of the total electron density.ρ0 (r) hΨ0 ρ̂ Ψ0 iWe again start with Born-Oppenheimer approximation and write the electronic HamiltonianasĤ F̂ V̂extwhere F̂ is the sum of the kinetic energy of electrons and the electron-electron interactionand V̂ext is some external potential.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201122 / 55
DFTModern DFT methods result from the Hohenberg-Kohn theorem1The external potential Vext , and hence total energy is a unique functional of theelectron density ρ(r)hΨ Ĥ Ψi E[ρ]hΨ ΨiEnergy 2The ground state energy can be obtained variationally, the density that minimizesthe total energy is the exact ground state densityE[ρ] E[ρ0 ], if ρ 6 ρ0If density is known, then the total energy is:E[ρ] T[ρ] Vne [ρ] J[ρ] Enn Exc [ρ]whereEnn [ρ] ZX ZA ZBRABA BJ[ρ] Vne [ρ] 12Zρ(r1 )ρ(r2 )dr1 dr2r12Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduρ(r)Vext (r)drNov 11, 201123 / 55
DFTIf the density is known, the two unknowns in the energy expression are the kinetic energyfunctional T[ρ] and the exchange-correlation functional Exc [ρ]To calculate T[ρ], Kohn and Sham introduced the concept of Kohn-Sham orbitals which areeigenvectors of the Kohn-Sham equation 1 2 veff (r) φi (r) εi φi (r)2Here, εi is the orbital energy of the corresponding Kohn-Sham orbital, φi , and the densityfor an ”N”-particle system isρ(r) NX φi (r) 2iThe total energy of a system isZE[ρ] Ts [ρ] dr vext (r)ρ(r) VH [ρ] Exc [ρ]Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201124 / 55
DFTs is the Kohn-Sham kinetic energy which is expressed in terms of the Kohn-Sham orbitalsasTs [ρ] N ZXi 1 1dr φ i (r) 2 φi (r)2vext is the external potential acting on the interacting system (at minimum, for a molecularsystem, the electron-nuclei interaction), VH is the Hartree (or Coulomb) energy,Z1ρ(r)ρ(r0 )VH drdr02 r r0 and Exc is the exchange-correlation energy.The Kohn-Sham equations are found by varying the total energy expression with respectto a set of orbitals to yield the Kohn-Sham potential asZρ(r0 )δExc [ρ]veff (r) vext (r) dr0 r r0 δρ(r)where the last term vxc (r) δExc [ρ]is the exchange-correlation potential.δρ(r)Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201125 / 55
DFTThe exchange-correlation potential, and the corresponding energy expression, are theonly unknowns in the Kohn-Sham approach to density functional theory.There are many ways to approximate this functional Exc , generally divided into twoseparate termsExc [ρ] Ex [ρ] Ec [ρ]where the first term is the exchange functional while the second term is the correlationfunctional.Quite a few research groups have developed the exchange and correlation functionalswhich are fit to empirical data or data from explicity correlated methods.Popular DFT functionals (according to a recent poll) PBE0 (PBEPBE), B3LYP, PBE, BP86, M06-2X, B2PLYP, B3PW91, B97-D, M06-L,CAM-B3LYP http://www.marcelswart.eu/dft-poll/index.html http://www.ccl.net/cgi-bin/ccl/message-new?2011 02 16 009Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201126 / 55
DFT Flow ChartSelectP initial(n)2ρ(n) (r) Ni φi (r) (n)ĥKSContruct KohnSham Operator1(n) 2 veff (r)2Solve(n 1) (n 1)(r) εiφi(r)P(n 1)N(n 1)2ρ(r) i φi(r) (n)Setρ(n) ρ(n)n n 1(n 1)ĥKS φiCalculatePropertiesENDyesDensity Converged? ρ(n 1) ρ(n) tolElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.edunoNov 11, 201127 / 55
Semi-empirical MethodsSemi-empirical quantum methods: Represents a middle road between the mostly qualitative results frommolecular mechanics and the highly computationally demandingquantitative results from ab initio methods. Address limitations of the Hartree-Fock claculations, such as speed andlow accuracy, by omitting or parametrizing certain integralsintegrals are either determined directly from experimental data or calculated fromanalytical formula with ab initio methods or from suitable parametric expressions.Integral approximations: Complete Neglect of Differential Overlap (CNDO) Intermediate Neglect of Differential Overlap (INDO) Neglect of Diatomic Differential Overlap (NDDO) ( Used by PM3, AM1, .)Semi-empirical methods are fast, very accurate when applied to molecules that aresimilar to those used for parametrization and are applicable to very large molecularsystems.Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201128 / 55
Heirarchy of MethodsSemi-empirical methodsCNDO,INDO,AM1,PM3Hartree FockHF-SCFExcitation HierarchyCIS,CISD,CISDTCCS,CCSD,CCSDTPertubation HierarchyMP2,MP3,MP4Multiconfigurational PT3Full CIElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201129 / 55
Basis SetsSlater type orbital (STO) or Gaussian type orbital (GTO) to describe theAO’sχSTO (r) xl ym zn e ζrχGTO (r) xl ym zn e ξr2where L l m n is the total angular momentun and ζ, ξ are orbitalexponents.Amplitude1STOGTO0.50012345r (a0)Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201130 / 55
STO or GTOWhy STOWhy GTOCorrect cups at r 0Desired decay at r Correctly mimics H orbitalsNatural Choice for orbitalsComputationally expensive tocompute integrals andderivatives.Wrong behavior at r 0 andr Gaussian Gaussian GaussianAnalytical solutions for mostintegrals and derivatives.Computationally less expensivethan STO’sElectronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201131 / 55
Pople family basis set1Minimal Basis: STO-nG Each atom optimized STO is fit with n GTO’s Minimum number of AO’s needed2Split Valence Basis: 3-21G,4-31G, 6-31G Contracted GTO’s optimized per atom. Valence AO’s represented by 2 contracted GTO’s3Polarization: Add AO’s with higher angular momentum (L) 3-21G* or 3-21G(d),6-31G* or 6-31G(d),6-31G** or6-31G(d,p)4Diffuse function: Add AO with very small exponents for systems withdiffuse electron densities 6-31 G*, 6-311 G(d,p)Electronic Structure Calculations in Quantum ChemistryHPC@LSU - http://www.hpc.lsu.eduNov 11, 201132 / 55
Correlation consistent basis set Family of basis sets of increasing sizes. Can be used to extrapolate basis set limit. cc-pVDZ: Double Zeta(DZ) with d’s on heavy ato
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