Mekanika Molekuler

1. MekanikaMolekuler Pertemuan ke-2

2. Pendahuluan • Mekanikamolekuler (Molecular Mechanics) adalahpendekatan modeling berdasarkanmekanikaklasik • Terminologi yang samadenganpendekataniniadalahForce FieldMethod • Satuanpenyusun (building blocks) dalammetodemekanikamolekuleradalah atom, elektrontidakdianggapsebagaipartikel individual • Konsekuensinyaikatanantar atom tidakdilihatsebagaihasilpenyelesaianpersamaan Schrödinger untukelektron • Informasitentangikatandinyatakansecaraeksplisit yang berartidiillustrasikansecarafisikbukansebagaihasilinteraksielektronvalensi • Mekanikamolekulertelahterbuktibermanfaatuntukmenjelaskansistemdenganmolekulbesarataupadatankristalin

3. MenguraikanSistem • Deskripsisistem: apa unit dasar (partikel) yang dipilih, adaberapabanyak? • Kondisiawal: Dimanaposisipartikel dan bagaimanakecepatannya • Interaksi: Apabentukpersamaanmatematikauntukgaya yang bekerjaantarpartikeltersebut • Persamaandinamik: Apabentukpersamaanmatematika yang sistem yang berubahdenganwaktu?

4. HierarkiSatuanPenyusununtukMenguraiSistem Kimia Elektron Molekul Atoms Nuklei Protons Neutrons Makromolekul Quarks

5. PemilihanSatuanPenyusun (Building Blocks) • Jikamemilihintiatom dan elektronsebagaipartikelpenyusun, kitabisamenguraiatom dan molekulnamuntidakbisamenguraistruktur internal inti atom • Jikamemilihatomsebagaipartikelpenyusun, kitabisamenguraistrukturmolekulnamuntidakbisamenguraidistribusielektron • Jikamemilihmolekul (asam amino) sebagaipartikelpenyusun, kitabisamenguraistruktur overall makromolekul (protein)namuntidakbisamenguraipergerakan atom-atom dalammolekul.

6. PemilihanKondisiAwal • Posisiruang yang lengkap (complete phase space) darisuatusistemadalahsesuatu yang sangatbesar: Mencakupsemuanilai yang mungkindariposisi dan kecepatansatupartikel • Kita hanyabisamenguraisebagiankecilsajadarikondisiini • Misalnyasuatu isomer (strukturalataukonformasional) dan reaksikimiaingincobadiuraikan • Senyawa C6H6memilikibanyakkemungkinanstruktur dan konformasi, namunjikakitaspesifik pada senyawabenzene, makakondisiawalsistemmenjadilebihsederhana

8. InteraksiPartikel dan PersamaanDinamik • Pada level atomik, interaksidasar yang bekerjahanyainteraksielektromagnetik • Pada pendekatanMekanikaMolekuler, interaksiantarpartikeldisusundalambentuk parameter yaituinteraksi stretching, bending, torsional, van der Waals dll. • Persamaandinamikmenjelaskanbagaimanasuatusistemberubahdenganperubahanwaktu, Misal: denganmenggunakan GLBB kitabisamenjelaskanposisisistemsetelahwaktutertentu

9. Interaksi Fundamental

10. Keterangan • Strong interactionadalahgaya yang menahan inti atom agar tetaputuhwalaupunadatolakmenolakantar proton didalam • Weak interactiongaya yang bertanggungjawab pada peluruhan inti atom denganmengkoversi neutron menjadi proton (-decay) • Keduanyaadalahgaya yang bekerjashort range dan hanyasignifikanwithin the atomic nucleus • Interaksielektromagnetik dan gravitasionalberbandingterbalikdenganjarakpartikel • Interaksielektromagnetikterjadiantarapartikelbermuatan • Interaksigravitasionalterjadiantarapartikel yang memiliki masa

11. PendekatandidalamForceFielda.k.aMekanikaMolekuler • Force field menggunakanpendekatanmekanikaklasiksepertipersamaan Newton untukmendeskripsikansistem • Aspekkuantum dan energielektronditiadakan/tidakdiperhitungkan • Denganpendekatanklasik, permasalahandireduksimenjadimenentukanenergisistem pada strukturgeometritertentu • Seringkali juga digunakanuntukmenentukangeometriuntukmolekul yang paling stabilataukonformasiterbaik yang melibatkaninterkonversiantarkonformasi • Untukkeperluaniniperhitungkandiarahkan pada penentuanenergi minima pada potential energy surface

12. Potential Energy Surface (Flash info)

13. Definisi(Flash info) • Atom-atom dalammolekuldisatukanbersama oleh ikatankimia • Saat atom terdistorsi, ikatanakanmeregangataumenekuk/mengkerut yang menyebabkanenergipotensialsistemmeningkat • Setelah susunangeometri atom-atom yang baruterbentuk, molekulberadadalamkondisistasioner. Pada posisiinienergisistemtidakdipengaruhi oleh energikinetiktetapi oleh posisi atom-atom (potensial) • Energidarimolekulmerupakanfungsidariposisi inti, saat inti bergerak, elektronsecaracepatakanmenyesuaikan • Hubunganantaraenergimolekuler dan geometrimolekulerdapatdipetakanmenjadisebuahpotential energy surface

14. TerminologidalamMekanikaMolekuler • Molekuldalam MM diilustrasikansebagaiball and spring dimanaatomdigambarkanmemilikiukuran dan kelembutantertentusedangkanikatandigambarkanmemilikipanjang dan kekakuantertentu • Dasar daripendekatan FF/MM iniadalahbahwamolekultersusunatas unit denganstruktur yang serupahanyaberadadalammolekul yang berbeda • Misalnyasemuaikataninisama pada molekulapa pun • CH memilikipanjang 1,06 sd 1,10 Å • Vibrasiregang CH 2900 sd 3300 cm-1 • CH dapatdikembangkanlagimenjadi CH yang terikat padaikatantunggal, gandaataurangkap 3

15. Tipe Atom dalam MM • Penggambaranmolekul yang tersusunatas unit struktural (gugusfungsi) serupadenganbentukmolekul yang berbeda pada Kimia Organik • Kimiawanorganikbiasanyamenggunakan ball n stick atauhurufnama atom dan garisikatanuntukmenggambarkanmolekul • FF method miripdenganpendekataninidenganpenambahan atom dan ikatantidakmemilikisatuukuran dan panjang yang fixed • Unit struktural yang serupa pada molekul yang berbedainidiimplementasikandalam FF denganistilahtipe atom • Tipe atom tergantung pada nomor atom dan jenisikatankimia yang terlibat • Dalam MM2 ada 71 tipe atom yang berbeda

16. EnergidalamForceFieldMethod • Energidalam Force Field ditulissebagaijumlahdarisemuasuku • Masing-masingsukumenguraikanenergi yang dibutuhkanuntukmendistorsimolekuldalamarahtertentu EFF = Estr + Ebend + Etors + Evdw + Eel + Ecross • DimanaEstradalahenergi stretching ikatanantara 2 atom, Ebendenergi yang dibutuhkanuntukmembengkokkansudutikatan, Etorsenergiuntuk proses rotasimemutardisekitarikatan, Evdw dan Eelmenguraikaninteraksi atom-atom non-ikatan dan Ecrossmenguraikankoplingantar 3 sukuawaldiatas

17. Energidalam Force Field

18. The Stretch Energy • Estradalahfungsienergiuntukmeregangkanikatanantara 2 tipe atom A dan B • Dalambentuknya yang paling sederhana, Estrdituliskansebagaideret Taylor disekitar Panjang ikatan “natural” atau “kesetimbangan” R0. • Parameter R0bukan Panjang ikatankesetimbangansembarangmolekul, • Iaadalah parameter yang saatdigunakanuntukmenghitungstrukturenergi minimum suatumolekulakanmenghasilkangeometridengan Panjang ikatankesetimbanganberdasarkaneksperiment

19. The Bending Energy • Ebendadalanenergi yang dibutuhkanuntukmembengkokansudut yang dibentuk oleh 3 atom ABC, dimanaadaikatan yang terbentukantara A dan B dan antara B dan C • Bentukpersamaannya juga merupakanderet Taylor disekitarsudutikatan “natural” yang berakhir pada ordekedua dan memberikanpendekatanharmonik

20. The Out-of-Plane Bending Energy • If the central B atom in the angle ABC is sp2-hybridized, there is a significant energy penalty associated with making the centre pyramidal, since the four atoms prefer to be located in a plane. If the four atoms are exactly in a plane, the sum of the three angles with B as the central atom should be exactly 360°, however, a quite large pyramidalization may be achieved without seriously distorting any of these three angles. • Taking the bond distances to 1.5Å, and moving the central atom 0.2 Å out of the plane, only reduces the angle sum to 354.8 (i.e. only a 1.7° decrease per angle). • The corresponding out-of-plane angle, , is 7.7 for this case. • Very large force constants must be used if the ABC, ABD and CBD angle distortions are to reflect the energy cost associated with the pyramidalization.

21. This would have the consequence that the in-plane angle deformations for a planar structure would become unrealistically stiff. • Thus a special out-of-plane energy bend term (Eoop) is usually added, while the in-plane angles (ABC, ABD and CBD) are treated as in the general case above • Eoopmaybewritten as a harmonic term in the angle  (the equilibrium angle for a planar structure is zero) or as a quadratic function in the distance d, as given in equation below atau

22. The Out-of-Plane Bending Energy

23. The Torsional Energy • Etors describes part of the energy change associated with rotation around a B—C bond in a four-atom sequence A—B—C—D, where A—B, B—C and C—D are bonded • Looking down the B—C bond, the torsional angle is defined as the angle formed by the A—B and C—D bonds as shown in Figure. The angle  may be taken to be in the range [0°,360°] or [−180°,180°].

24. The torsional energy is fundamentally different from Estr and Ebend in three aspects: • A rotational barrier has contributions from both the non-bonded (van der Waals and electrostatic) terms, as well as the torsional energy, and the torsional parameters are therefore intimately coupled to the non-bonded parameters. • The torsional energy function must be periodic in the angle : if the bond is rotated 360° the energy should return to the same value. • The cost in energy for distorting a molecule by rotation around a bond is often low, i.e. large deviations from the minimum energy structure may occur, and a Taylor expansion in  is therefore not a good idea. • To encompass the periodicity, Etors is written as a Fourier series.

25. The n = 1 term describes a rotation that is periodic by 360°, the n = 2 term is periodic by 180°, the n = 3 term is periodic by 120°, and so on. The Vnconstants determine the size of the barrier for rotation around the B—C bond. • Depending on the situation, some of these Vnconstants may be zero. In ethane, for example, the most stable conformation is one where the hydrogens are staggered relative to each other, while the eclipsed conformation represents an energy maximum. • As the three hydrogens at each end are identical, it is clear that there are three energetically equivalent staggered, and three equivalent eclipsed, conformations. • The rotational energy profile must therefore have three minima and three maxima.

26. Energi Torsional Etana

27. The Van der Waals Energy • Evdw is the van der Waals energy describing the repulsion or attraction between atoms that are not directly bonded. • Together with the electrostatic term Eel, it describes the non-bonded energy. • Evdw may be interpreted as the non-polar part of the interaction not related to electrostatic energy due to (atomic) charges. • This may for example be the interaction between two methane molecules, or two methyl groups at different ends of the same molecule.

28. Evdw is zero at large interatomic distances and becomes very repulsive for short distances. • In quantum mechanical terms, the latter is due to the overlap of the electron clouds of the two atoms, as the negatively charged electrons repel each other. • At intermediate distances, however, there is a slight attraction between two such electron clouds from induced dipole–dipole interactions, physically due to electron correlation • Even if the molecule (or part of a molecule) has no permanent dipole moment, the motion of the electrons will create a slightly uneven distribution at a given time.

29. This dipole moment will induce a charge polarization in the neighbor molecule (or another part of the same molecule), creating an attraction, and it can be derived theoretically that this attraction varies as the inverse sixth power of the distance between the two fragments. • Evdw is very positive at small distances, has a minimum that is slightly negative at a distance corresponding to the two atoms just “touching” each other, and approaches zero as the distance becomes large. • A general functional form that fits these conditions is given in eq. (2.11).

30. The Electrostatic Energy: Charges and Dipoles • The other part of the non-bonded interaction is due to internal (re)distribution of the electrons, creating positive and negative parts of the molecule. • A carbonyl group, for example, has a negatively charged oxygen and a positively charged carbon. • At the lowest approximation, this can be modelled by assigning (partial) charges to each atom. • Alternatively, the bond may be assigned a bond dipole moment. These two descriptions give similar (but not identical) results. • Only in the long distance limit of interaction between such molecules do the two descriptions give identical results.

31. The interaction between point charges is given by the Coulomb potential, with being a dielectricconstant. • The atomic charges can be assigned by empirical rules, but are more commonly assigned by fitting to the electrostatic potential calculated by electronic structure methods

32. Cross Terms • The first five terms in the general energy expression, eq. (2.1), are common to all force fields. The last term, Ecross, covers coupling between these fundamental, or diagonal, terms. Consider for example a molecule such as H2O. • It has an equilibrium angle of 104.5° and an O—H distance of 0.958 Å. If the angle is compressed to say 90, and the optimal bond length is determined by electronic structure calculations, the equilibrium distance becomes 0.968Å, i.e. slightly longer. • Similarly, if the angle is widened, the lowest energy bond length becomes shorter than 0.958Å. This may qualitatively be understood by noting that the hydrogens come closer together if the angle is reduced. • This leads to an increased repulsion between the hydrogens, which can be partly alleviated by making the bonds longer. If only the first five terms in the force field energy are included, this coupling between bond distance and angle cannot be modelled.

33. It may be taken into account by including a term that depends on both bond length and angle. Ecross may in general include a whole series of terms that couple two (or more) ofthebondedterms. • The components in Ecross are usually written as products of first-order Taylor expansions in the individual coordinates. • The most important of these is the stretch/bend term, which for an A—B—C sequence may be written as in eq. (2.31)

34. Aplikasi MM: PadatanIonik • Aplikasimekanikamolekuler pada padatanionikserupadengankalkulasienergikisi • Bahkanmetode MM memangbisadigunakanuntukmenghitungenergikisi, juga efekadanyacacat pada senyawaionik dan sifatkristal • Pertanyaanawaluntukmenguraienergikisi: • Apagaya yang menahan ion-ion berkumpulmembentukkristal pada lattice site –nya • Jawabannyaadalahgayatarikelektrostatikantaramuatanpositif dan muatannegatif

35. AspekEnergidalamIkatanIonik: Energi Kisi • Misalkanadasuatureaksiantaraunsurlogam yang reaktif (Li) dan mudahmelepaselektrondengan gas halogen (F) yang cenderungmenarikelektron: Li(g)  Li+(g) + e- IE1 = 520 kJ F(g) + e-  F-(g) EA = -328 kJ • Reaksi total: Li(g) + F(g)  Li+(g) + F-(g) IE1 + EA = 192 kJ

36. Energi total yang dibutuhkanreaksiinibahkanlebihbesarkarenakitaharusmengkonversi Li dan F kedalambentuk gas • Akan tetapieksperimenmenunjukkanenthalpipembentukanpadatanLiF (∆H0f) = -617 kJ • Jikakeduaunsurdalambentuk gas: • Li+(g) + F-(g)  LiF(g) ∆H0 = -755 kJ • Energikisiadalahperubahanenthalpi yang menyertai ion-ion gas yang bergabungmembentukpadatanionik: • Li+(g) + F-(g)  LiF(s) ∆H0kisiLiF = energikisi = -1050 kJ

37. Daur Born-Haber

38. Nilai Energi Born-Haber • Hoatom Li = 161 kJ • BE F2 = 159 kJ • IE1 (Li) = 520 kJ • EA (F) = -328 kJ • HoLattice (LiF) = -1050 kJ • HofLiF = -617 kJ • Total Energi : HofLiF = Hoatom Li + ½ BE F2 + IE1 (Li) + EA (F) + HoLattice

39. PendekatanMekanikaMolekuler • Ion-ion diasumsikanberada pada situs kisimasing-masingsesuaidenganmuatanformalnya, sehingga NaCl misalnyamembentuk array of Na+ and Cl- ions. • The net interaction can be obtained by summing the interactions over all the pairs of ions, including not only the attraction between Na+ and Cl- but also the repulsion between ions of the same sign. • The net interaction decreases with distance but slowly so that it is difficult to obtainanaccuratevalue. • To calculate lattice energies, this summation be achieved for simple lattice structures by introducing the Madelungconstant. • However, for layer structures with low symmetry this approach is not feasible, as a single Madelung constant will not suffice.

40. Madelung Constants • There are many factors to be considered such as covalent character and electron-electron interactions in ionic solids. • But for simplicity, let us consider the ionic solids as a collection of positive and negative ions. In this simple view, appropriate number of cations and anions come together to form a solid. • The positive ions experience both attraction and repulsion from ions of opposite charge and ions of the same charge. • The Madelung constant is a property of the crystal structure and depends on the lattice parameters, anion-cation distances, and molecular volume of the crystal

41. Before considering a three-dimensional crystal lattice, we shall discuss the calculation of the energetics of a linear chain of ions of alternate signs • Let us select the positive sodium ion in the middle (at x = 0) as a reference and let r0 be the shortest distance between adjacent ions (the sum of ionic radii). • The Coulomb energy of the other ions in this 1D lattice on this sodium atom can be decomposed by proximity (or "shells").

42. Nearest Neighbors (first shell): This reference sodium ion has two negative chloride ions as its neighbors on either side at r0so the Coulombic energy of these interactions is • Next Nearest Neighbors (second shell): Similarly the repulsive energy due to the next two positive sodium ions at a distance of 2r0is = =

43. Next Next Nearest Neighbors (third shell): The attractive Coulomb energy due to the next two chloride ions neighbors at a distance3r0is • and so on. Thus the total energy due to all the ions in the linear array is = -

44. We can use the following Maclaurin expansion • to simplify the sum in the parenthesis of Equation before as to obtain • The first factor of Equation is the Coulomb energy for a single pair of sodium and chloride ions, while (2 ln2) the factor is the Madelung constant (M1.38 ) per molecule. • The Madelung constant is named after Erwin Medelung and is a geometrical factor that depends on the arrangement of ions in the solid. If the lattice were different (when considering 2D or 3D crystals), then this constant wouldnaturallydiffer.

45. MM Approach … • Since the computer programs in use are set up to be of general application, they employ methods that give a good approximation to the sum over an infinite lattice for any unit cell. • However, electrostatic interaction is not that has to be considered. We know, for example, that ions are not just point charges but have a size; the shell of electrons around each nucleus prevents too close an approach by other ions. • We therefore include a term to allow for the interaction between shells on the different ions. It would be possible to give each ion a fixed size and insist that the ions cannot be closer than their combined radii. • However, most programs use a different approach by including terms representing intermolecular forces.

46. The intermolecular forces act between cations, and between cations and anions, as well as between anions. • For oxides in particular, however, the cation-cation term is oftenignored. • Salts such as magnesium oxide can be thought of as close-packed arrays of anions with cations occupying the octahedralholes. • Because the cations are held apart by the anions, the cation-cation interaction is un-important.

47. Octahedral Hole

48. The final thing we need to take into account is the polarizability of the ions. This is a measure of how easily the ions are deformed from their normal spherical shape. • In a perfect crystal, the ions are in very symmetrical environments and can be thought of as spherical. If one ion moves to an interstitial site, leaving its original position vacant, then the environment may not be so symmetrical and it may be deformed by the surrounding ions. • A very simple way to model this is to divide the ionic charge between a core that stays fixed at the position of the ion and a surrounding shell that can move off-center. The distribution of the charge is obtained by adjustment to fit the properties of a crystal containing that ion.

49. The shell behaves as though it were attached to the core by springs. Take a chloride ion, for example. If the surrounding ions move so that there is a greater positive charge in one direction, then the shell will move so that the total charge on the ion is distributed over two centers producing a dipole. • Opposing this will be the pull of the springs that attach it to the core.

50. For ionic solids, the most important term for lattice energies is the electrostatic term; for sodium chloride, for example, the total lattice energy in a typical calculation is -762.073 kJ mol-1, of which -861.135 kJ mol-1 is due to the electrostatic interaction while the intermolecular force and shell terms contribute +99.062 kJ mol-1. • Thus the contributions of the intermolecular force and shell terms are about 10% of the electrostatic interactions. • These other terms may have a greater relevance in the study ofdefects.