Semi-Empirical Methods CHEM 430

Transcription

1 Semi-Empirical Methods CHEM 430

2 Cost, Hartree%Fock, scales,as,n 4,(N=#, basis,funcfons), Due,to,two% electron, integrals, within,fock, matrix, Semi%empirical,cut, cost,by,reducing, number,of, integrals, Typically,scale, as,n 2, Molecular,detail, 10 ms, thousands of atoms: protein folding, drug binding AtomisFc,MM, Coarse%grain, 100 fs, 10 atoms: photochemistry 1 ms+, 1 million atoms: dynamics of large proteins, cell membranes, viruses Correlated,Method,, (MP2,,CCSD), Density, FuncFonal, Theory, 10 ps, 100 atoms: chemical reactions, 2, ComputaFonal,cost,

3 Semi-Empirical MO Methods the high cost of ab initio MO calculations is largely due to the many integrals that need to be calculated (esp. two electron integrals) semi-empirical MO methods start with the general form of ab initio Hartree-Fock calculations, but make numerous approximations for the various integrals many of the integrals are approximated by functions with empirical parameters these parameters are adjusted to improve the agreement with experiment

4 Semi-Empirical MO Methods core orbitals are not treated by semiempirical methods, since they do not change much during chemical reactions only a minimal set of valence orbitals are considered on each atom (e.g. 2s, 2p x, 2p y, 2p z on carbon) Extended Hückel, Zero Differential Overlap and Neglect of Diatomic Differential Overlap

5 Extended Hückel Theory (Roald Hoffman, 1960 s; implemented in YAeHOMP) H C i = S C i E i! H Hamiltonian Matrix! C i column vector of molecular orbital coefficients! E i orbital energies! S overlap matrix! H ii use valence shell ionization potentials! H ij = K S ij (H ii + H jj )/2 with K = 1.75

6 Extended Hückel Theory (EHT) All core e - are ignored (all modern semiempirical methods use this approximation) Only valence orbitals are considered, represented as STOs (hydrogenic orbitals). Correct radial dependence. Overlap integrals between two STOs as a functions of r are readily computed. (HT assumed that the overlap elements S ij = d ij ). The overlap between STOs on the same atom is zero. Resonance Integrals: For diagonal elements: H µµ = negative of ionization potential (in the appropriate orbital). Valence shell ionization potentials (VSIPs) have been tabulated, but can be treated as an adjustable parameter. For off-diagonal elements: H µn is approximated as (C is an empirical constant, usually 1.75): 1 H = + 2 ( H µµ H nn ) µn µu C µn S RESULT: the secular equation can now be solved to determine MO energies and wave functions. ** Matrix elements do NOT depend on the final MOs (unlike HF) the process is NOT iterative. Therefore, the solution is very fast, even for large molecules.

7 TABLE 10-1 Cartesian Coordinates (in Angstroms) for Atoms of Methane Oriented as Shown in Fig Atom x y z C H a H b H c H d Figure 10-1 Orientation of methane inacartesian axis system.

8 TABLE 10-2 Basis AOs for Methane AO no. Atom Type n a l a m a exp 1 C 2s C 2p z C 2p x 2 1 (1) b C 2p y 2 1 (1) b H a 1s H b 1s H c 1s H d 1s a n, l, m are the quantum numbers described in Chapter 4. b 2p x and 2p y are formed from linear combinations of m=+ 1 and m= 1 STOs, and neither of these AOs can be associated with aparticular value of m.

9 TABLE 10-3 Overlap Matrix for STOs of Table

10 TABLE 10-4 The Extended Hückel Hamiltonian Matrix for CH 4 a a All energies in a.u.

11 TABLE 10-5 Energies for Methane by the Extended Hückel Method MO no. Energy (a.u.) Occ. no

12 Figure 10-2 A drawing of the lowest-energy nondegenerate EHMO for methane. The AOs are drawn as though they do not overlap. This is done only to make the drawing simpler. Actually, the AOs overlap strongly.

13 TABLE 10-7 Coefficients for MOs φ 2,φ 3,φ 4 φ 2 φ 3 φ 4 2s p z p x p y s a s b s c s d

14 Figure 10-3 The three lowest-energy degenerate MOs of methane.

15 Figure 10-7 function of K. Extended Hückel energy difference between staggered and eclipsed ethanes as a

16 TABLE Energy Barriers for Internal Rotation about Single Bonds a Barrier (kcal/mole) b Molecule Calculated Experiment CH 3 CH CH 3 NH CH 3 O H CH 3 CH 2 F CH 3 CHF CH 3 CF CH 3 CH 2 Cl CH 3 CHCH cis CH 3 CHCHCl CH 3 CHO CH 3 NCH a Calculated barriers are for rigid rotation, where no bond length or angle changes occur except for the torsional angle change about the internal axis. b The stable form for the fi t seven molecules has the methyl C H bonds staggered with respect to bonds across the rotor axis. For the last four molecules, the stable form has a C H methyl bond eclipsing the double bond.

17 TABLE 10-8 Mulliken Net AO and Overlap Populations for Methane as Computed by the Extended Hückel Method 2s 2p z 2p x 2p y 1s a 1s b 1s c 1s d 2s p z p x p y s a s b s c s d

18 TABLE Gross AO Populations, Gross Atomic Populations, and Net Atomic Charges for Methane Gross AO Gross atom Net atomic population population charge C 2s C a 2p H a a All 2p AOs and all H AOs have identical values because they are equivalent through symmetry.

19 Really Big Personalities... John Pople Michael Dewar

20 Zero Differential Overlap (ZDO) two electron repulsion integrals are one of the most expensive parts of ab initio MO calculations 1 ( µ% #$ ) = & " µ ( 1 ) " % ( 1 ) " # ( 2 ) " $ ( 2 ) d! 1 d! 2 r 12 neglect integrals if orbitals are not the same ( µ! #$ ) = ( µµ ##) " " µ! where " µ! = 1 if µ =!, " µ! = 0 if µ %! approximate integrals by using s orbitals only CNDO, INDO and MINDO semi-empirical methods #$

21 Neglect of Diatomic Differential Overlap (NDDO) fewer integrals neglected 1 ( µ% #$ ) = & " µ ( 1 ) " % ( 1 ) " # ( 2 ) " $ ( 2 ) d! 1 d! 2 r 12 neglect integrals if µ and! are not on the same atom or! and! are not on the same atom integrals approximations are more accurate and have more adjustable parameters than in ZDO methods parameters are adjusted to fit experimental data and ab initio calculations MNDO, AM1 and PM3 semi-empirical methods recent improvements: PDDG and PM6

22 Neglect,of,Diatomic,DifferenFal, Overlap,(NDDO), No,further,approximaFons,than,ZDO, Integral,approximaFons,are,more,exact,than,ZDO, More,adjustable,parameters,than,ZDO, Introduced,by,John,Pople, 5,

23 Intermediate,Neglect,of,DifferenFal, Overlap,(INDO),, Neglects,all,two%center,two%electron,integrals,which,are,not,of,the, Coulomb,type, Preserves,rotaFonal,invariance, Total,energy,independent,of,rotaFon,of,coordinate,system,!Some,integrals,made,independent,of,orbital,type,(e.g.,,integral, involving,a,p%orbital,must,be,the,same,as,with,an,s%orbital),,!one%electron,integrals,with,two,different,funcfons,on,the, same,atom,and,a,potenfal,energy,operator,from,another,atom, disappear, Happy,medium,between,NDDO,and,CNDO, 6,

24 Complete,Neglect,of,DifferenFal, Overlap,(CNDO), Only,Coulomb,one%center,and,two%center,two%electron,integrals, remain, Integrals,and,approximaFons,the,same,as,in,INDO, CNDO&vs&INDO&vs&NDDO& Only,differ,in,treatment,of,two%electron,integrals, CNDO,&INDO& Reduce,two%electron,integrals,, to,two,parameters,,γ AA,and,γ AB, NDDO& All,one%,and,two%center,, integrals,are,kept, 7,

25 Approximation of 1-Electron Integrals H µµ = U µµ - å B¹ A V AB U µµ from atomic spectra V AB value per atom pair H = 0 µu µ,u on the same atom = b = ( b + ) H b S µu AB µu AB 2 1 b A B One b parameter per element

26 Neglect of 2-Electron Integrals 2-e - integral CNDO INDO NDDO ( µ A µ A lala ) ( µ A µ A lblb ) ( µ A ua lala ) ( µ A ua lblb ) ( µ A ub lala ) ( µ A ua las A ) ( µ A ua lbs B ) ( µ A ua las B ) ( µ A ub las B ) - - -

27 Modified,NDDO,model, Modified,neglect,of,differenFal,overlap,(MNDO), AusFn,Model,1,(AM1), Parametric,Method,Number,3,(PM3), How,to,transform,NDDO,into,working,computaFonal,model, Remaining,integrals,can,be,calculated,from,funcFonal,form,of,the, atomic,orbitals, Remaining,integrals,can,be,made,into,parameters,(and,then,fit,to, experimental,data), Remaining,integrals,can,be,made,into,parameters,(assigned,values, based,on,experimental,data), 8,

28 MNDO, InteracFons,between,O%H,and,N%H,bonds,treated,differently,, " V MNDO ', nn (A, H ) = Z A $ # Core%core,repulsion,treated,differently, V nn MNDO (A, B) = Z A ' Z H s A s H s A s H 1+ e α AR AH Z B ' s A s B s A s B R AH % + e α H R AH ' & 1+ e α AR AB + e α BR ( AB ) Fit,for,elements,H,B,C,N,O,F,Al,Si,P,S,Cl,Zn,Ge,Br,Sn,I,Hg,Pb, Fit,the,α,values, Parameters,taken,from,atomic,spectra,and,others,fit,to,molecular, data, Limita0ons& Sterically,crowded,molecules,,four,membered,rings,,weak, interacfons,,acfvafon,energies,,ethers,,sulfides, 9,

29 AM1, MNDO,repulsion,between,two,atoms,which,are,2%3,Angstroms, apart,is,too,high!acfvafon,energies,too,large,(among,other, errors) due,to,too,repulsive,interacfon,in,the,core%core,potenfal, Modified,core%core,funcFon,from,MNDO,by,adding,Gaussian, funcfons,and,reparameterized, AM1 nn V MINDO nn (A, B) = V & Z A' Z B' # bka ( RAB cka )2 bkb ( RAB ckb )2 (A, B) + + α kb e ( % α ka e RAB $ k ' k k,between,2,and,4,depending,on,the,atom, ak,bk,ck,fit,to,molecular,data,, LimitaFons, Geometries,with,hydrogen,bonds,,alkyl,groups,,nitro,compounds,, peroxide,bonds,,phosphor,compounds,, 10,

30 Some Limitations of AM1 predicts hydrogen bond strengths approximately correct (but geometry often wrong) activation energies much improved over MNDO hypervalent molecules improved over MNDO, but still significant errors alkyl groups systematically too stable by ca 2 kcal/mol per CH 2 group nitro groups too unstable peroxide bonds too short

31 PM3, Same,core%core,repulsion,as,AM1,but,only,two,Gaussian,funcFons, added,to,each,atom, Automated,opFmizaFon,process, All,parameters,opFmized,simultaneously,(MNDO,,AM1,done,by, hand), Significantly,larger,training,set,used, LimitaFons, Hydrogen,bonds,too,short,,gauche,of,ethanol,more,stable,than, trans,,si%x,bonds,too,short,,charge,of,nitrogen,atoms,ohen, incorrect,sign,and/or,magnitude, 11,

32 Some Limitations of PM3 hydrogen bonds are too short by 0.1 A almost all sp 3 nitrogens are pyramidal Si halogen bonds too short structures for NH 2 NH 2, ClF 3 wrong charge on nitrogens unrealistic

33 ParameterizaFon,

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39 Performance,of,, semi%empirical,methods, 13,

40 More,performance,discussion, 14,

41 More,performance,discussion, 15,