2/21/2005 CHEM 408 - Sp05L7 - 1SemiSemi--Empirical CalculationsEmpirical Calculations• electronic structure methods can be classified into three categories:ab initio: “from the beginning” -- rigorous (if approximate) solution of the electronic Schrödinger equation (SEQ) without use of empirical informationsemi-empirical: solution of the SEQ wherein some integrals are replaced by empirically derived parametersdensity-functional: analagous to solution a SEQ in which some guesswork and empiricism is needed to define Ĥ2/21/2005 CHEM 408 - Sp05L7 - 2)}|()|{(2111νσµλλσµνλσλσµνµν−+≡∑∑==KKcorePHF)1(||)1( nuclei121211νµµνϕϕ⎭⎬⎫⎩⎨⎧−−∇−≡∑∫∗JJJcoreRrZrdHrrr)1()1(1νµµνϕϕ∗∫≡ rdSr)2()2(1)1()1()|(1221σλνµϕϕϕϕλσµν∗∗∫∫=rrdrdrr∑=≡niiiCCP1 MO2σλλσESCFC =(recall) The HF-SCF Equations:n electrons, K basis fns. {ϕµ}from: J. Sadlej, Semi-Empirical Methods of Quantum Chemistry (Ellis Horwell, 1985)2e integrals, #I ~ K4/8r12e1e2ABCDr12e1e2ABCD2/21/2005 CHEM 408 - Sp05L7 - 3• semi-empirical methods reduce the computational burden using approximations based on the idea ofzero differential overlap (ZDO)0)1()1( =∗νµϕϕif µ ≠ν• all semi-empirical approaches apply this (often crude) approximation in different ways which give rise to their designations as:CNDO - Complete Neglect of Differential OverlapCNDO (Pople ‘65); obsolete except as PPP model for πsystems)INDO - Intermediate Neglect of Differential OverlapINDO (Pople ‘67); MINDO/3 (Dewar ‘75); INDO/S (Zerner ‘73); INDO/S still widely used for spectroscopic calcs., especially of transition metalsNDDO - Neglect of Diatomic Differential OverlapMNDO(Dewar ‘77), AM1(Dewar ‘85), PM3(Stewart ‘89), MNDO/d(Theil‘92)2/21/2005 CHEM 408 - Sp05L7 - 4The CNDO Method•basis set: only valence e using minimal STOsµννµµνδϕϕ=≡∗∫)1()1(1rdSr(in FC=ESC only)λσµνδδλλµµλσµν)|()|( =all 3 & 4 center integrals eliminated• parameterize as:),( if )|( AAA∈=λµγλλµµ) and ( if )()|( BARABAB∈∈=λµγλλµµABABBBAAAAAAcorePPPHFγγγµµµµµµ∑≠+−+= )(21⎪⎩⎪⎨⎧∈∈−∈−=),( if ),( if 2121BAPHAPHFABcoreAAcoreνµγνµγµνµνµνµνµν∑∈≡AAAPPλλλwhere• the Fock matrix elements then become:common to allsemi-empirical methods• assume:µ, λboth on atom A, etc.average repulsion between 2e on atom Aaverage repulsion between e on atom A and e on atom B(valence) epopulation on atom A2/21/2005 CHEM 408 - Sp05L7 - 5• Hcoreelements,AVUHABABcore∈−=∑≠µµµµµfor )1(||)1(121211µµµµϕϕ⎟⎟⎠⎞⎜⎜⎝⎛−−∇−=∗∫AARrZrdUrrr)1(||)1(11 sABBsAABRrZrdVϕϕrrr−=∫- energy of e in ϕµ- obtained from ionization energies of A- attraction between e on A and nucleus B- same for all µ, calc. using s orbital on A and Slater’s rules to determine ζµ⎩⎨⎧∈∈∈=) and (for ),(for 0BASAHABcoreνµβνµµνµν)(21BAABβββ+=AAAAEAIP −≅γ•γvalues estimated from empirical data:)(2)(BBAAABAAAAABABRRγγγγγ+++=and-βABare resonance integrals responsible for A-B bonding- like HMO theory βvalues determined by fitting exptl. datafromhere) (µνµνδ≠Swith2/21/2005 CHEM 408 - Sp05L7 - 6• input parameters {γAA, UsA, UpA, βA} for each atom derived from empirical data • Sµν(RAB) and VAB(RAB) are calculated using STOs & Slater’s rules determine all of the Fµν• iterative solution of FC = EC just like HF theory• huge numbers of integrals eliminated in favor of a small number of parameters; integrals left are all simple ⇒ calculation times negligible compared to equivalent ab initio methods• higher-level semi-empirical models INDO, NDDO, are similar in spirit, but they account for successively more integrals and add more empirical parameters2/21/2005 CHEM 408 - Sp05L7 - 712= optimized+ = from expt.from: J. P. Stewart, J. Computer-Aided Molecular Design 4, 1 (1990).Parameter Sets used by Various Models22115# params:# from exp:11518-2451806-1222215222152/21/2005 CHEM 408 - Sp05L7 - 8Elements ParameterizedTable from: AMPAC 8 User Manual, M. J. S. Dewar et al., Semichem, Inc (2004).2/21/2005 CHEM 408 - Sp05L7 - 9Some Performance TestsTables & other values from F. Jensen, Introduction to Computational Chemistry (Wiley, 1999).err. dipole/D err. bond angles/deg4.3 3.3 3.9err. I.P./eV.78 .61 .59.45 .35 .38(125 molecule test set H, C, N, O, F, Al, Si, P, S, Cl, Br, I)2/21/2005 CHEM 408 - Sp05L7 - 10Some Comments on Semi-Empirical Calculations1,2• energies parameterized against heats of formation (298 K) via:∑∑∆+−=∆atomsatoms)atoms()atoms()molecule()molecule( HEEHfelelfso Eel(molecule) already contains thermal contributions• because energies are parameterized against expt. these SCF calculations effectively contain correlation contributions• weak interactions are poorly modeled• apart from some known failings, the errors observed tend to be statistical rather than systematic (bad for comparisons)1. F. Jensen, Introduction to Computational Chemistry (Wiley, 1999); 2. C. J. Cramer, Essentials of Computational Chemistry (Wiley, 2002).2/21/2005 CHEM 408 - Sp05L7 - 11Density Functional TheoryDensity Functional Theory• focus on electron density rather than wavefunction),...,,(21 nrrrrrrΨ∑∫∫ ∫=−Ψ=niinnrrrrrrdrdrdr122121)(|),...,,(|......)(rrrrrrrrrδρfunction of all(3n) e coordinatesfunction of only 3 coordinates• early models of solids were based on the intuitive idea that the(ground-state) energy of a system should be describable as some functional (function of a function) of the electron density: ][][][][ρρρρeeeNeEETE++= ][][][ρρρKJEee+=electronkinetic energyelectron-nucleus attractionelectron-electron repulsionCoulombexchangerdRrrZEAAAeNrrrr∑∫−−=nuclei||)(][ρρ∫∫′′−′= rdrdrrrrJrrrrrr||)()(21][ρρρ• standard classical Coulomb terms are used for EeNand J:with2/21/2005 CHEM 408 - Sp05L7 - 12∫≅ rdrTerr3/52/32)]([)3(103][ρπρ• use of results from a uniform e gas to approximate the QM kinetic and exchange terms yields the Thomas-Fermi-Dirac model (1920s):∫⎟⎠⎞⎜⎝⎛−= rdrKrr3/43/1)]([343][ρπρ• the TFD model is only useful for semi-quantitative treatment of metals; it is too inaccurate to predict molecular bonding• modern density functional theory (DFT) is based on two theorems established by