Quantum Chemistry. Molecular Simulation I. The Beginning... Quantum Chemistry. Ψ characterizes the particles motion ( ) ( ) Molecular Orbital Theory

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1 Molecular Smulaton I Quantum Chemstry Quantum Chemstry Ψ H Ψ E = ΨΨ Classcal Mechancs U = E bond +E angle +E torson +E non-bond Molecular Orbtal Theory Based on a wave functon approach Schrödnger equaton Densty Functonal Theory Based on the total electron densty Hohenberg Kohn theorem Jeffry D. Madura Department of Chemstry & Bochemstry Center for Computatonal Scences Duquesne Unversty The Begnnng... Schrödnger equaton Hamltonan operator Wave functon ( ) HΨ= EΨ ħ ħ H = V r V r m = + x y z m Ψ characterzes the partcles moton ( ) ( ) varous propertes of the partcle can be derved Quantum Chemstry Start wth Schrödnger s equaton HΨ= EΨ Make some assumptons Born-Oppenhemer approxmaton Lnear combnaton of atomc orbtals Ψ= c ϕ + c ϕ a a b b Apply the varatonal method E * Ψ HΨdτ * ΨΨ dτ

2 LCAO A practcal and common approach to solvng the Hartree-Fock equatons s to wrte each spn orbtal as a lnear combnaton of sngle electron orbtals (LCAO) K ψ = c ν φ ν ν = the φ v are commonly called bass functons and often correspond to atomc orbtals K bass functons lead to K molecular orbtals the pont at whch the energy s not reduced by the addton of bass functons s known as the Hartree- Fock lmt Bass Sets Slater type orbtals (STO) ( ) ( ζ ) ( ) Gaussan type orbtals (GTO) functonal form n+ / / n r Rnl r = n! r e ζ xyze a a b c - r zeroth-order Gaussan functon 3/4 r α gs ( α, r) = e α π Property of Gaussan functons s that the product of two Gaussans can be expressed as a sngle Gaussan, located along the lne jonng the centers of the two Gaussans a ma n - r mn -a mm r - a nn r a m+ a n -a rc e e = e e STO vs. GTO 0.8 f( x ) 0.6 g( x). f( x ) g( x ) x 6

3 Gaussan expanson the coeffcent the exponent uncontratcted or prmtve and contracted s and p exponents n the same shell are equal Mnmal bass set STO-NG Double zeta bass set lnear combnaton of a contracted functon and a dffuse functon. Splt valence 3-G, 4-3G, 6-3G φ L ( ) = d φ α µ µ µ = Polarzaton to solve the problem of non-sotropc charge dstrbuton. 6-3G*, 6-3G** Dffuse functons fulfll as defcency of the bass sets to descrbe sgnfcant amounts of electron densty away from the nuclear centers. (e.g. anons, lone pars, etc.) 3-+G, 6-3++G Roothaan-Hall equatons The recastng of the ntegro-dfferental equatons nto matrx form. The Hartree-Fock equaton s wrtten as Ths s transformed to gve a Fock matrx (for closedshell systems) K K core Fµν = Hµν + Pλσ ( µν λσ ) ( µλ νσ ) λ= σ = where K ( ) φ ( ) = ε ( ν ν φ ν ν ) F c c ν= ν= K / N λσ = cλcσ = P The energy s K K core E= P H + F µ = ν= the electron densty s expressed as K K ρ( r) P φ ( r) φ ( r) = ( ) µν ν µν µν µ ν µ = ν = In matrx form the Roothaan-Hall equaton s wrtten as FC= SCE 3

4 Solvng the Roothaan-Hall Equaton Common scheme for solvng the Roothaan-Hall equatons s calculate the ntegrals to form the Fock matrx, F calculate the overlap matrx, S dagonalze S form S -/ guess or calculate an ntal densty matrx, P Form the Fock matrx usng the ntegrals and densty matrx solve the secular equaton F -EI =0 to gve the egenvalues E and the egenvectors C by dagonalzng F calculate the molecular orbtal coeffcents, C, from C=S -/ C calculate a new densty matrx, P, from matrx C check for convergence RHF vs. UHF Restrcted Hartree-Fock (RHF) closed-shell molecules Restrcted Open-shell Hartree-Fock (ROHF) combnaton of sngly and doubly occuped molecular orbtals. Unrestrcted Hartree-Fock (UHF) open-shell molecules Pople and Nesbet: one set of molecular orbtals for α spn and another for the β spn. UHF and RHF Dssocaton Curves for H Electron Correlaton The most sgnfcant drawback to HF theory s that t fals to adequately represent electron correlaton. NR HF Ecorr = E E E(H )-E(H) a.u R (a.u.) Confguraton Interactons excted states are ncluded n the descrpton of an electronc state Many Body Perturbaton Theory based upon Raylegh-Schrödnger perturbaton theory 4

5 Confguraton Interacton The CI wavefuncton s wrtten as Ψ= c0ψ 0 + cψ + cψ + where Ψ 0 s the HF sngle determnant where Ψ s the confguraton derved by replacng one of the occuped spn orbtals by a vrtual spn orbtal where Ψ s the confguraton derved by replacng one of the occuped spn orbtals by a vrtual spn orbtal The system energy s mnmzed n order to determne the coeffcents, c 0, c, etc., usng a lnear varatonal approach Many Body Peturbaton Theory Based upon perturbaton concepts The correcton to the energes are ( 0) ( 0) ( 0) = Ψ 0Ψ E H dτ ( ) ( 0) ( 0) = Ψ Ψ E V dτ ( ) ( 0) ( ) = Ψ Ψ E V dτ ( 3) ( 0) ( ) = Ψ Ψ E V dτ Perturbaton methods are sze ndependent these methods are not varatonal H = H0 + V Geometry Optmzaton Dervatves of the energy E( x) ( ) ( ) ( ) ( ) E x E x f = E x + xf x + ( xf x )( xj xj ) + x j j the frst term s set to zero the second term can be shown to be equvalent to the force the thrd term can be shown to be equvalent to the force constant x x Internal coordnate, cartesan coordnate, and redundant coordnate optmzaton choce of coordnate set can determne whether a structure reaches a mnmum/maxmum and the speed of ths convergence. Internal coordnates are defned as bond lengths, bond angles, and torsons. There are 3N-6 (3N-5) such degrees of freedom for each molecule. Chemsts work n ths world. Z-matrx... Cartesan coordnates are the standard x, y, z coordnates. Programs often work n ths world. Redundant coordnates are defned as the number of coordnates larger than 3N-6. 5

6 Frequency Calculaton The second dervatves of the energy wth respect to the dsplacement of coordnate yelds the force constants. These force constants n turn can be used to calculate frequences. All real frequences (postve force constants): local mnmum One magnary frequency (one negatve force constant): saddle pont, a.k.a. transton state. From vbratonal analyss can compute thermodynamc data Charges Mullken Molecular Propertes Löwdn electrostatc ftted (ESP) Bond orders Bondng Natural Bond Analyss Bader s AIM method Molecular orbtals and total electron densty Dpole Moment Energes onzaton and electron affnty Koopman s theorem Energes equatng the energy of an electron n an orbtal to the energy requred to remove the electron to the correspondng on. frozen orbtals lack of electron correlaton effects Dpole Moments The electrc multpole moments of a molecule reflect the dstrbuton of charge. Smplest s the dpole moment nuclear component electronc µ M nuclear = Z ARA A= K K µ = qr ( ) µ electronc = P dτφ r φ µν µ ν µ = ν= 6

7 Molecular Orbtals and Total Electron Densty Electron densty at a pont r ( ) N / ( ) K ( ) ( ) K K ρ r = ψ r = Pµµ φµ r φµ r + Pµν φµ ( r) φν ( r) Number of electrons s N / ( ) K K K N = drψ r = P + P S Molecular orbtals HOMO LUMO = µ = µ = ν = µ + µµ µν µν = µ = µ = ν = µ + Natural Bond Analyss Bondng a way to descrbe N-electron wave functons n terms of localzed orbtals that are closely ted to chemcal concepts. Bader F. W. Bader s theory of atoms n molecules. Ths method provdes an alternatve way to partton the electrons among the atoms n a molecule. Gradent vector path bond crtcal ponts charges are relatvely nvarant to the bass set Wberg Mayer Bond Orders WAB P µν µ onaνonb AB = = ( ) ( ) B PS PS µ onaνonb Bond orders can be computed for ntermedate structures whch can be useful way to descrbe smlarty of the TS to the reactants or to the products. µν νµ Mullken A Löwdn A Charges K K K q = Z P P S µµ µν µν µ = ; ona µ = ; onaν= ; ν µ atomc orbtals are transformed to an orthogonal set, along wth the mo coeffcents φ K ( S ) ' / µ νµ ν ν = q = Z S P A = A φ K / / ( ) µ = ; µ ona µµ 7

8 Electrostatc potentals the electrostatc potental at a pont r, φ(r), s defned as the work done to brng a unt postve charge from nfnty to the pont. the electrostatc nteracton energy between a pont charge q located at r and the molecule equals qφ(r). there s a nuclear part and electronc part φ nucl ( r) = Z M A elec ( r) A= r RA ( r) = nucl( r) + elec ( r) φ φ φ ( r) ' dr ρ φ = ' r r Geometry Water Example Z-MATRIX (ANGSTROMS AND DEGREES) CD Cent Atom N Length/X N Alpha/Y N3 Beta/Z J H O ( ) 3 3 H ( ) 00.08( 3) E(RHF) = A.U. Populaton Analyss ********************************************************************** Populaton analyss usng the SCF densty. ********************************************************************** Alpha occ. egenvalues Alpha vrt. egenvalues Condensed to atoms (all electrons): 3 H O H Energy Total atomc charges: H O H Classcal Mechancs Neglect electrons Treat atoms as spheres and bonds are sprngs Force Feld What s a force feld? A mathematcal expresson that descrbes the dependence of the energy of a molecule on the coordnates of the atoms n the molecule Also ths sometmes used as another term for potental energy functon. What are force felds used for? Structure determnaton Conformatonal energes Rotatonal and Pyramdal nverson barrers Vbratonal frequences Molecular dynamcs 8

9 Force Feld Hstory Pre-970 Harmonc 970 For molecules wth less than 00 atoms one class of force felds went for hgh accuracy to match expermental results The other class of force felds was for macromolecules. Present There are hghly accurate force felds desgned for small molecules and there are force felds for studyng proten and other large molecules Force Feld Frst force felds developed from expermental data X-ray NMR Mcrowave Current force felds have made use of quantum mechancal calculatons CFF MMFF94 There s no sngle best force fled Force Felds MM/3/4: Molecular Mechanc Force feld for small moelcules CHARMM: Chemstry at Harvard Macromolecular Mechancs AMBER: Asssted Model Buldng wth Energy Refnement OPLS : Optmzed Parameters for Lqud Smulaton CFF: Consstent Force Feld CVFF: Valence Consstent Force Feld MMFF94: Merck Molecular Force Feld 94 UFF: Unversal Force Feld Potental Energy Functon The potental energy functon s a mathematcal model whch descrbes the varous nteractons between the atoms of a molecule or system of molecules. In general, the functon s composed of ntramolecular terms (st three terms) and ntermolecular terms (last two terms). U r ( ) = b b ( ) + ( θ θ ) + V [ + ( ) j+ j cos( n jφ δ )] + j 4ε[( σ ) ( σ ) 6] + rj rj qqj 4ε r j 9

10 Bond Stretch l l ( ) 0 E = k l l Angle Bendng 0 E k ( ) θ = θ θ θ Harmonc approxmaton s used k b s the force constant l 0 s the reference bond length Hgher order terms ' '' ( ) ( ) ( ) 3 4 l = l 0 + l 0 + l 0 E k l l k l l k l l Harmonc approxmaton k θ s the bendng force constant θ 0 s the reference angle Other forms nclude ( cos ) Eθ = kθ + θ Torson Interactons ( cos ) ( cos ) ( cos3 ) E V V V ϕ = + ϕ + ϕ ϕ Out-of-plane Bendng ( ) 0 Eω = kω ω ω Represented as a Fourer seres Ths term accounts for the energetcs of twstng the -4 atoms Frst term: mportant for descrbng the conformatonal energes (cs-trans) Second term: mportant for determnng the relatvely large barrer to rotaton about conjugated bonds Thrd term: allows for accurate of the energy barrer for rotaton about bonds where one or both of the atoms n the bond have sp 3 hybrdzaton Harmonc approxmaton k ω s the oop force constant ω 0 s the reference value Dffferent methods n whch to calculate ω MMF: angle between a bond -j and a plane formed by j-k-l and j s the central atom MM3: angle between a bond -j and a pont located n the place formed by -k-j. 0

11 Van Der Waals Interactons E vdw * * 6 Rj Rj = ε R j R j Electrostatcs E elec qq j = DR j Lennard-Jones -6 potental ε s the well depth R j * s the sum of the van der Waals rad (of atoms and j) R j s the dstance between nteractng atoms Coulomb s law q and q j are the charges on atom and j respectvely D s the delectrc constant R j s the dstance between atoms and j Bond ncrement model (used n CFF and MMFF) q q δ = + 0 j j Charge Classes Class I Calculated drectly from experment Class II Extracted from a quantum mechancal wave functon (Mullken analyss) Class III Extracted from a wave functon by analyzng a physcal observable predcted from the wave functon. (Electrostatc fttng) Class IV A parameterzaton procedure to mprove class II and III charges by mappng them to reproduce charge-dependent observables obtaned from experment Cross Terms 0 0 bb = bb ( ) ( ) 0 0 θθ = kθθ ( θ θ) ( θ θ ) 0 0 bθ = bθ ( ) ( θ θ ) E k b b b b E E k b b Bond/bond Needed to get the correct splttng n the vbratonal frequences of the symmetrc and asymmetrc C-H bond stretchng modes Angle/angle Needed to determne correctly the extent of splttng n angular deformaton modes for the cases n whch the bendng modes are centered on a sngle atom Bond/angle Needed to predct the observed bond lengthenng that often occurs when a bond angle s reduced

12 Molecular Mechancs In the molecular mechancs model, a molecule s descrbed as a seres of pont charges (atoms) lnked by sprngs (bonds). A mathematcal functon (the force-feld) descrbes the freedom of bond lengths, bond angles, and torsons to change. The force-feld also contans a descrpton of the van der Waals and electrostatc nteractons between atoms that are not drectly bonded. The force-feld s used to descrbe the potental energy of the molecule or system of nterest. Molecular mechancs s a mathematcal procedure used to explore the potental energy surface of a molecule or system of nterest. Potental Energy Mnmzatons Potental Energy Surface: Has mnma (stable structures) and saddle ponts (transton states). Below: mnma & Saddle Pont. Force F = U Potental Energy Energy Mnmzaton Energy Mnmzaton Methods Gven a functon f whch depends on one or more ndependent varables, x, x,, fnd the values of those varables where f has a mnmum value. f = 0 x f > 0 x Potental Energy Surface Taylor seres expanson about pont x k U( xk) ( x xk) U( xk) U( x) = U( xk) + ( x xk) + + xk xk xj the second term s known as the gradent (force) the thrd term s known as the Hessan (force constant) Algorthms are classfed by order, or the hghest dervatve used n the Taylor seres. Common algorthms ( st Order): Steepest Descent (SD), Conjugated Gradents (CONJ) Non-dervatve Smplex Sequental unvarate method

13 Energy Mnmzaton Methods Dervatve Steepest descents Moves are made n the drecton parallel to the net force Conjugate gradent The gradents and the drecton of successve steps are orthogonal Newton-Raphson Second-order method; both frst and second dervatves are used BFGS Quas-Newton method (a.k.a. varable metrc methods) buld up the nverse Hessan matrx n successve teratons Energy Mnmzaton Methods Truncated Newton-Raphson Intally follow a descent drecton and near the soluton solve more accurately usng a Newton method. Dfferent from QN n that the Hessan s sparse allowng for a faster evaluaton Comparng st Order Algorthms BOTH: terate over the followng equaton n order to perform the mnmzaton: R k = R k- + l k S k Where R k s the new poston at step k, R k- n the poston at the prevous step k-, lk s the sze of the step to be taken at step k and S k s the drecton. SD: At each step the gradent of the potental g k (.e., the frst dervatve n mult-dmensons) s calculated and a dsplacement s added to all the coordnates n a drecton opposte to the gradent. S k = -g k CONJ: In each step, weghs n the prevous gradents to compensate for the lack of curvature nformaton. For all steps k > the drecton of the step s a weghted average of the current gradent and the prevous step drecton,.e., S k = -g k + b k S k- SD versus CONJ SD versus CONJ. Startng from pont A, SD wll follow a path A-B-C. CONJ wll follow A-B-O because t modfes the second drecton to take nto account the prevous gradent along A-B and the current gradent along B-C. 3

14 Convergence Comparson of Methods Small change n energy Small norm of the gradent RMS gradent Number of steps vs. tme du grad = dx grad RMS = n Steepest descents: 500 steps n 4.08 secs (not converged) Conjugate gradent: 7 steps n 5.77 seconds Newton-Raphson: 5 steps n 4.84 seconds Whch Method Should I Use? Must consder Storage: Steepest descents lttle memory needed whle Newton- Raphson methods requre lots. Avalablty of dervatves : Smplex, none are needed, steepest descents, only frst dervatves, Newton-Raphson needs frst and second dervatves. The followng s common practce SD or CG for the ntal rough mnmzaton followed by a few steps of NR. SD s superor to CG when startng structure s far from the mnmum TN method after a few SD and/or CG appears to gve the best overall and fastest convergence Conformatonal Analyss Molecular conformatons term used to descrbe molecular structures that nterconvert under ambent condtons. ths mples several conformatons may be present, n dfferng conc., under ambent condtons. a proper descrpton of the molecular structure, the molecular energy, or the spectrum for a molecule wth several conformatons must comprse a proper weghtng of all of the conformatons. Boltzmann s equaton P = E RT fe E j RT fe j j f s the number of states or conf. of energy E R s.98 cal/mol-k (the deal gas constant) T s the absolute temperature (K) j s the summaton over all the conformatons 4

15 Butane Conformatonal Analyss Conformatonal Analyss Example Usng Boltzmann s equaton We have a populaton of 89.74% at -80 and 0.6% at +/- 60 assumng a relatve energy dfference of.7 kcal/mol. Conformatonal Analyss: A Cautonary Note MM Dredng Term Trans Gauche E Trans Gauche E Stretch Stretch-Bend Bend Torson VDW Total Even though the energetc dfference gven by the two models s smlar, dfferent contrbutons gve rse to those dfferences. Molecular Mechancs Energetcs Sterc energy the energy reported by most molecular mechancs programs energy of structure at 0 K. correct for vbratonal moton by addng the zero pont energy ZPE = ν MM energy s NOT equal to free energy!! MM energy can be equvalent to enthalpy f one assumes the PV term can be gnored 5

16 Molecular Mechancs Energy Stran energy Dfferences n sterc energy are oly vald for dfferent conformatons or confguratons. Stran energy permts the comparson between dfferent molecules. A stranless reference pont must be determned. A partcular reference pont mght be the all trans conformatons of the straght-chan alkanes from methane to hexane (Allnger defnton). These compounds can be used to derve a set of stranless energy parameters for consttuent parts of molecules. Subtractng the stranless energy from the sterc energy Allnger and co-workers concluded that the char cyclohexane has an nherent stran energy due to the presence of,4 van der Waals nteractons between the carbon atom wthn the rng. Molecular Mechancs Energy Interacton energy Ths s the dfference between the energes of two solated speces and the energy of the ntermolecular complex Mathematcally ths s represented as ( ) E = E E + E e ab a b Sterc Energes Usng sterc energes to predct the thermodynamcs of smple tautomerzaton O OH The expermental heat of formaton dfference s approxmately 8 kcal/mol MM sterc energy dfference s.3 kcal/mol The large error s due to error n bond energy terms, I.e. the number/types of bonds broken and made are not precsely balanced. Postonal Isomers In ths case the number and precse types of bonds are retaned. Consder hydracrlc acd vs. lactc acd. HO O OH HO MM energy for hydracrylc acd s 3.3 kcal/mol whle that for lactc acd s.37 kcal/mol yeldng an energy dfference of 0.86 kcal/mol n favor of hydracrylc acd. Expermentally the heat of formaton dfference s 4.4 kcal/mol n favor of lactc acd. O OH 6

17 Conformers and Confguratonal Isomers Molecular mechancs can be employed to predct relatve energes of conformers and confguratonal somers. It should not be used on postonal somers or the relatve energes of dfferent molecules. Molecular mechancs can be used to study the bndng between molecules s ntermolecular nteractons have been approprately parameterzed. Ideal Gas Statstcal Thermodyanmcs The free energy can be wrtten as where G =- kt lnq + PV Q = fe E RT The dfference n free energy can be wrtten as G = RT ln P P Conformatonal Analyss Usng MOE Conformatonal Analyss usng MOE 7

18 Butane Conformatonal Analyss Butane Conformatonal Analyss Start the molecular dynamcs operaton by choosng: MOE Compute Conformatons Dynamcs In the Molecular Dynamcs wndow (prevous slde) Select the Open Database Vewer opton. Ths wll automatcally open the output database (dyn.mdb) n whch the results wll be stored. (Press Tab to advance to the next feld) Change Save Iteratons to 0. Ths s the number of steps to execute between wrtng samples of the trajectory. Under the column labeled Heat Rate, modfy the Iteratons feld to read 00. Ths represents the number of teratons n the heat phase. Change the Tme Stepfrom 0.00 to Clck Okto start the smulaton. Butane Conformatonal Analyss Once the smulaton s complete, you can revew the dynamcs smulaton as a move anmaton: Open the Dynamcs Anmator panel n the Database Vewer: DBV Fle Dynamcs Anmator The MD Anmator anmates the varous conformatons generated by the molecular dynamcs run. (next slde) Butane Conformatonal Analyss Controllng the anmaton: Move the Frame slder by draggng the left mouse button. You can step forwards or backwards through the anmaton usng the >>> and <<< buttons. The Anmaton wheel controls the speed of the anmaton. 8