Computational Methods. Chem 561

Computational Methods Chem 561

Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics

Computational Chemistry " Computational methods (a.k.a. molecular modeling) are a set of techniques to investigate chemical problems using computers " Molecular geometry " Activation barriers " Chemical reactivity " IR, UV and NMR spectra " Intermolecular interactions (e.g., enzyme-substrate) " Physical properties (e.g., melting points of polymers)

ab initio methods " from the beginning from first principles " all integrals calculated exactly " approximations " Born -Oppenheimer " the wavefunction is an antisymmetric product of oneelectron spin orbitals " using a finite number of basis set functions to expand each spin orbital (basis set truncation error)

Post-HF Methods " HF SCF does not take into account electron correlation " Electron repulsion is taken into account in an average sense " Two particles are said to be correlated if the pair probability distribution function cannot be factorized into a product of distribution functions for each individual particle " two sources of electron correlation " dynamic (Coulomb) correlation: short-range effect due to electron repulsion " non-dynamic (Fermi) correlation: electrons must obey the Pauli Exclusion Pple., which gives rise to electron exchange. (no two electrons of like spin can occupy the same space) " E el = E HF + E corr " Three approaches to account for correlation energy " Configuration Interaction (CI) " Coupled Clusters Methods " Perturbation Theory

Configuration Interaction " Express the wavefunction as a linear combination of trial wavefunctions (multideterminantal) " Ψ 0 is obtained from solving the HFR equations " Ψ n 0 represent excited state configurations " in practice only double and quadruple excitations are important " To perform a CI calculation " Obtain Ψ 0 using HFR-SCF " Substitute spin orbitals to obtain excited configurations " CIS (all possible single excitations) " CID (double excitations) " CISD (single and double excitations) Φ = c 0 Ψ 0 + c 1 Ψ 1 + c 2 Ψ 2 + " CISDT (single, double, and triple excitations) " Full CI: all possible excitations " Solve for coefficients in linear expansion variationally

Coupled Clusters Method " Not variational " Total correlation energy is the sum of contributions from each pair of occupied orbitals E corr = e ab a<b " where e ab is the correlation energy between the electron pair in spin-orbital χ a and that in spin-orbital χ b " the size of the problem is reduced by ~ a factor of 2

Coupled Clusters Method " The correlation energy is calculated for each pair of electrons by considering a double excitation % H ˆ ' Ψ 0 + & t<u c tu ab Ψ ab tu ( % * = E ab ' Ψ 0 + c tu ab Ψ ab ) & " where the ab pair of electrons has been moved to spin orbitals tu E ab = E 0 + e ab t<u tu ( * ) " The process is repeated for each pair of electrons E corr = a<b e ab

Coupled Clusters Approximation (CCA) " A variation of Coupled Clusters Theory where quadruple excitations are considered as products of double excitations.

Many-Body Perturbation Theory " Electron correlation is taken into account by adding small perturbations to the Hamiltonian H ˆ Φ i = H ˆ 0 + V ˆ ( ) Φ i = E i Φ i " where (0) (1) Φ i = Ψ i + λ Ψ i + λ 2 (2) Ψ i + and E i = E (0) i + λe (1) i + λ 2 E (2) i + " λ is an ordering parameter " The zeroth-order solution is obtained with HF-SCF ˆ H 0 Ψ i (0) = E i (0) Ψ i (0)

Many-Body Perturbation Theory " Not a variational method " Also called Rayleigh-Schrödinger or Møller-Plesset Perturbation Theory " In the Møller-Plesset treatment higher-order corrections to the energy come from excited states where electrons are moved to HF virtual orbitals. " Only second (MP2)and fourth (MP4) order corrections are useful.

Density Functional Theory " It solves the HFR equations in terms of the electron probability density instead of the wavefunction ρ Ψ 2 " The Hohenberg-Kohn Theorem 1 " for a molecule with a nondegenerate ground state, the ground state molecular energy, wavefunction and all other properties are uniquely determined by the groundstate electron probability density, ρ 0 1 Proof offered by Hohenberg and Kohn in 1964

Density Functional Theory " Since the electron probability density is a function of Ψ and Ψ is a function of r, θ, and φ, ρ is a functional (function of a function) " The ground-state electronic energy E 0 is obtained by defining the external potential ( ) = v r and solving the Schrödinger equation i Z α α riα

Density Functional Theory " The ground-state electronic energy is a functional of the electron probability density and is dependent on the external potential E 0 = E v [ ρ 0 ] " The Hohenberg-Kohn Variational Theorem " the true ground state electron density minimizes the energy functional " equivalent to the Variational Theorem

Density Functional Theory " The Kohn-Sham method " find an approximation to the functional that minimizes the energy (variational approach) " start with a reference system of n noninteracting electrons

Density Functional Theory " The ground-state electronic energy can be expanded using the individual terms from the Hamiltonian E 0 = T [ ρ ] 0 + V [ Ne ρ ] 0 + V [ ee ρ ] 0 kinetic energy nuclear-electron attraction electron-electron repulsion " Define the exchange-correlation energy functional E [ xc ρ ] 0 = T [ ρ ] 0 + V [ ee ρ ] 0

Density Functional Theory " V ee includes E [ xc ρ ] 0 = T [ ρ ] 0 + V [ ee ρ ] 0 " a Coulomb energy term (equivalent to J ij ) " an exchange energy term (equivalent to K ij ) " a Coulombic correlation energy term " T[ρ 0 ] and J ij are calculated exactly " K ij and the correlation term are approximated with functionals

Density Functional Theory Types of Functionals " Local Density Approximation (LDA) " E XC is given by local density, approximated as a homogeneous electron gas " Underestimated ionization energies, overestimates bond energies " Generalized Gradient Approximation (GGA) " E XC is improved by also using the gradient of the density (molecules have inhomogeneous electron density) " Accurate atomization energies and reaction barriers, inaccurate van der Waals forces

Density Functional Theory Types of Functionals " Hybrid Density Functional Methods (H-GGA) " A conventional GGA method is combined with some amount of HF exchange " Vast improvement for many molecular properties

Density Functional Theory E xc = E x + E c " Exchange functionals (interaction between electrons of same spin) " Perdew and Wang PW88, PW91 " Becke B88 (a GGA functional) " Correlation functionals (interaction between electrons of opposite spin) " Lee-Yang-Parr LYP (a GGA functional) " Perdew P86 (a GGA functional) " Becke B96 " These parameterized functions cannot be solved analytically and must be integrated numerically

Density Functional Theory Hybrid Functionals ( ) E xc = c 1 K HF + c 2 ρ 4 3 (r)d 3 r + c 3 ε NL x ρ 4 3 (r), ρ(r) d 3 r + c 4 ε L c ρ 4 3 (r) d 3 r + c 5 ε NL c ρ 4 3 (r), ρ(r) d 3 r where K HF is the exact exchange term from HF " B3LYP is a hybrid functional ( ) B E 3LYP xc = a 0 E HF x + ( 1 a 0 )E LSDA B x + a x E 88 VWN x + ( 1 a c )E 5 LYP c + a c E c a 0 = 0.2; a x = 0.72; a c = 0.81 " B3PW91 (Becke s original hybrid functional) ( ) E xc = E LDA xc + a 0 E HF LDA B ( x E x ) + a x ΔE 88 PW 91 x + a c ΔE c a 0 = 0.2; a x = 0.72; a c = 0.81

Comparison of Computational Methods scaling behavior N 3 N 4 N 5 N 6 N 7 N 8 N 9 N 10 method(s) DFT HF MP2 MP3, CISD, CCSD, QCISD MP4, CCSD(T), QCISD(T) MP5, CISDT, CCSDT MP6 MP7, CISDTQ, CCSDTQ From J. Phys. Chem. A, Vol. 111, No. 42, 2007

Semiempirical Methods " Based on SCF " Integrals that are expected to be small are not calculated " Use of parameters from experimental data

Semiempirical Methods " Extended Huckel " used for organic molecules for conjugated π systems " neglect 2-electron terms (H rs instead of F rs ) " use valence electrons only

Semiempirical Methods " NDO methods (developed by John Pople) " CNDO complete neglect of differential overlap ( ) = δ rs δ tu ( rr tt) " set S rs = δ rs and rs tu " ignore many 2-electron integrals " INDO intermediate neglect of differential overlap " exchange integrals are calculated on same atom

Semiempirical Methods " Developed by Dewar " only valence electrons are considered " rely heavily on parametrization " use ionization energies for H ii core " neglect many 2-electron integrals " set S rs = δ rs and rs tu ( ) = δ rs δ tu ( rr tt) " Examples " MNDO " AM1 (Austin model 1 from UT @ Austin) " PM3 (parametric method 3)

Semiempirical Methods " Methods are only as good as their parameters " Usually optimized for a set of molecules " organic compounds with conjugated π systems " amino acids, etc.

Molecular Mechanics Method " not quantum mechanical " atoms are point charges connected by springs V = V str + V bend + V oop + V tors + V VdW + V es bond angle bending torsion around bonds electrostatic interactions between atoms stretching out of plane bending Van der Waals forces

Molecular Mechanics Methods " MM force fields developed by Allinger " MM2 for organic compounds " MM3 for polypeptides and proteins " MM4 for hydrocarbons " AMBER and CHARMM for polypeptides, proteins, and nucleic acids " SYBYL for organic molecules and proteins " used in drug design