Pertemuan Ke-2 Molekuler Mekanik

Mekanika

Molekuler

Pendahuluan

• Mekanika molekuler (Molecular Mechanics) adalah pendekatan modeling berdasarkan mekanika klasik

• Terminologi yang sama dengan pendekatan ini adalah Force Field Method

• Satuan penyusun (building blocks) dalam metode mekanika molekuler adalah atom, elektron tidak dianggap sebagai partikel individual

• Konsekuensinya ikatan antar atom tidak dilihat sebagai hasil penyelesaian persamaan Schrödinger untuk elektron

• Informasi tentang ikatan dinyatakan secara eksplisit yang

berarti diillustrasikan secara fisik bukan sebagai hasil interaksi elektron valensi

• Mekanika molekuler telah terbukti bermanfaat untuk

Menguraikan Sistem

•

Deskripsi sistem

: apa unit dasar

(partikel) yang dipilih, ada berapa

banyak?

•

Kondisi awal

: Dimana posisi partikel

dan bagaimana kecepatannya

•

_Interaksi

_{: Apa bentuk persamaan}

matematika untuk gaya yang bekerja

antar partikel tersebut

•

_{Persamaan dinamik}

_{: Apa bentuk}

Hierarki Satuan

Penyusun untuk

Mengurai Sistem Kimia

Elektron

Quarks

Protons Neutrons

Nuklei

Atoms Molekul

Pemilihan Satuan

Penyusun (

Building

Blocks)

• _{Jika memilih inti atom dan elektron sebagai}

partikel penyusun, kita bisa mengurai atom dan

molekul namun tidak bisa mengurai struktur internal inti atom

• Jika memilih atom sebagai partikel penyusun, kita bisa mengurai struktur molekul namun tidak bisa mengurai distribusi elektron

• Jika memilih molekul (asam amino) sebagai

partikel penyusun, kita bisa mengurai struktur overall makromolekul (protein) namun tidak bisa mengurai pergerakan atom-atom dalam

Pemilihan Kondisi Awal

• _{Posisi ruang yang lengkap (complete phase space)}

dari suatu sistem adalah sesuatu yang sangat besar: Mencakup semua nilai yang mungkin dari posisi dan kecepatan satu partikel

• Kita hanya bisa mengurai sebagian kecil saja dari kondisi ini

• _{Misalnya suatu isomer (struktural atau}

konformasional) dan reaksi kimia ingin coba diuraikan

• _{Senyawa C6H6 memiliki banyak kemungkinan}

struktur dan konformasi, namun jika kita spesifik

Interaksi Partikel dan

Persamaan Dinamik

•

Pada level atomik, interaksi dasar yang

bekerja hanya interaksi elektromagnetik

•

Pada pendekatan Mekanika Molekuler,

interaksi antar partikel disusun dalam

bentuk parameter yaitu interaksi stretching,

bending, torsional, van der Waals dll.

•

Persamaan dinamik menjelaskan bagaimana

suatu sistem berubah dengan perubahan

waktu, Misal: dengan menggunakan GLBB

kita bisa menjelaskan posisi sistem setelah

waktu tertentu

�=�0� �

1 2 � �

2

Interaksi Fundamental

Name Particles Rang_{e (m)} Relative Strengt h

Strong

Interaction Quarks  10

-15

100

Weak

Interaction Quarks, leptons  10

-15

0.001

Electromagneti

c Charged particles

 ₁

Gravitational Mass

particles

Keterangan

• Strong interaction adalah gaya yang menahan inti atom agar tetap utuh walaupun ada tolak menolak antar proton didalam

• Weak interaction gaya yang bertanggung jawab pada peluruhan inti atom dengan mengkoversi neutron menjadi proton (-decay)

• Keduanya adalah gaya yang bekerja short range dan hanya signifikan within the atomic nucleus

• Interaksi elektromagnetik dan gravitasional berbanding terbalik dengan jarak partikel

• Interaksi elektromagnetik terjadi antara partikel bermuatan

Pendekatan didalam

Force

Field

a.k.a

Mekanika Molekuler

• Force field menggunakan pendekatan mekanika klasik seperti persamaan Newton untuk mendeskripsikan sistem

• Aspek kuantum dan energi elektron ditiadakan/tidak diperhitungkan

• Dengan pendekatan klasik, permasalahan direduksi menjadi menentukan energi sistem pada struktur geometri tertentu

• Seringkali juga digunakan untuk menentukan geometri untuk molekul yang paling stabil atau konformasi terbaik yang melibatkan interkonversi antar konformasi

• Untuk keperluan ini perhitungkan diarahkan pada

Potential Energy

Definisi

(Flash info)

• Atom-atom dalam molekul disatukan bersama oleh ikatan kimia

• Saat atom terdistorsi, ikatan akan meregang atau menekuk/ mengkerut yang menyebabkan energi potensial sistem

meningkat

• Setelah susunan geometri atom-atom yang baru terbentuk, molekul berada dalam kondisi stasioner. Pada posisi ini

energi sistem tidak dipengaruhi oleh energi kinetik tetapi oleh posisi atom-atom (potensial)

• Energi dari molekul merupakan fungsi dari posisi inti, saat inti bergerak, elektron secara cepat akan menyesuaikan 

Terminologi dalam

Mekanika Molekuler

• Molekul dalam MM diilustrasikan sebagai ball and spring dimana atom digambarkan memiliki ukuran

dan kelembutan tertentu sedangkan ikatan

digambarkan memiliki panjang dan kekakuan

tertentu

• _{Dasar dari pendekatan FF/MM ini adalah bahwa} molekul tersusun atas unit dengan struktur yang serupa hanya berada dalam molekul yang berbeda

• _{Misalnya semua ikatan ini sama pada molekul apa pun}

CH memiliki panjang 1,06 sd 1,10 Å

Vibrasi regang CH 2900 sd 3300 cm-1

Tipe Atom dalam MM

• Penggambaran molekul yang tersusun atas unit struktural (gugus fungsi) serupa dengan bentuk molekul yang berbeda pada Kimia Organik

• Kimiawan organik biasanya

menggunakan ball n stick atau huruf nama atom dan garis ikatan untuk menggambarkan molekul

• FF method mirip dengan pendekatan ini dengan penambahan atom dan ikatan tidak memiliki satu ukuran dan panjang yang fixed

• Unit struktural yang serupa pada molekul yang berbeda ini

diimplementasikan dalam FF dengan istilah tipe atom

• Tipe atom tergantung pada nomor atom dan jenis ikatan kimia yang terlibat

Energi dalam

Force

Field

Method

• Energi dalam Force Field ditulis sebagai jumlah dari semua suku

• Masing-masing suku menguraikan energi yang

dibutuhkan untuk mendistorsi molekul dalam arah tertentu

EFF = Estr + Ebend + Etors + Evdw + Eel + Ecross

• Dimana Estr adalah energi stretching ikatan antara

2 atom, Ebend energi yang dibutuhkan untuk

membengkokkan sudut ikatan, Etors energi untuk

proses rotasi memutar disekitar ikatan, Evdw dan Eel

The Stretch Energy

• Estr adalah fungsi energi untuk meregangkan ikatan

antara 2 tipe atom A dan B

• Dalam bentuknya yang paling sederhana, Estr

dituliskan sebagai deret Taylor disekitar Panjang ikatan “natural” atau “kesetimbangan” R0.

• Parameter R0 bukan Panjang ikatan kesetimbangan

sembarang molekul,

• Ia adalah parameter yang saat digunakan untuk

The Bending Energy

• _{Ebend adalan energi yang dibutuhkan untuk}

membengkokan sudut yang dibentuk oleh 3 atom ABC, dimana ada ikatan yang terbentuk

antara A dan B dan antara B dan C

• Bentuk persamaannya juga merupakan deret Taylor disekitar sudut ikatan “natural” yang berakhir pada orde kedua dan memberikan pendekatan harmonik

�_����=

(

� ���− �₀���

)

=����

(

���� −�₀���

)

2

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}

• 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

• Eoop may be written as a harmonic term in the angle

 (the equilibrium angle for a planar structure is

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

• The torsional energy is fundamentally different from Estr

and Ebend in three aspects:

1. 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.

2. The torsional energy function must be periodic in the angle

: if the bond is rotated 360° the energy should return to

the same value.

3. 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.

�

_����

(



)

=

∑

�=1

�

_�

���

(

�



)

• 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 Vn constants determine

the size of the barrier for rotation around the B—C bond.

• Depending on the situation, some of these Vn constants

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 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

• _{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

• 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).

�_���( _���)=�_{���������}(���)− �

��

(���)6

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.

• _{The interaction between point charges is given}

by the Coulomb potential, with  being a

dielectric constant.

• 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

���(���)=

��_��  ���

 

���(���)=

 � �

 ₍���₎3 (cos −3��� ����� ��)

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

• _{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) of the bonded terms.

• _{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)_�

���/����=�

���

(���� −�₀���)

[

(���−�₀��)−₍���− �₀��)

]

Aplikasi MM: Padatan

Ionik

• Aplikasi mekanika molekuler pada padatan ionik serupa dengan kalkulasi energi kisi

• Bahkan metode MM memang bisa digunakan untuk menghitung energi kisi, juga efek

adanya cacat pada senyawa ionik dan sifat kristal

• Pertanyaan awal untuk mengurai energi kisi:

• Apa gaya yang menahan ion-ion berkumpul membentuk kristal pada lattice site –nya

Aspek Energi dalam Ikatan

Ionik:

Energi Kisi

•

_{Misalkan ada suatu reaksi antara unsur}

logam yang reaktif (Li) dan mudah melepas

elektron dengan gas halogen (F) yang

cenderung menarik elektron:

Li(g)



Li

+

(g) + e

-

IE

1

= 520 kJ

F(g) + e

-



F

-

(g) EA = -328 kJ

•

_{Reaksi total:}

Li(g) + F(g)



Li

+

(g) + F

-

(g) IE

•

Energi total yang dibutuhkan reaksi ini bahkan

lebih besar karena kita harus mengkonversi Li

dan F kedalam bentuk gas

•

Akan tetapi eksperimen menunjukkan enthalpi

pembentukan padatan LiF (∆H

0

f

) = -617 kJ

•

Jika kedua unsur dalam bentuk gas:

•

Li

+

(g) + F

-

(g)

_

LiF(g) ∆H

0

= -755 kJ

•

Energi kisi adalah perubahan enthalpi yang

menyertai ion-ion gas yang bergabung

membentuk padatan ionik:

•

_Li

+

(g) + F

-

(g)



LiF(s) ∆H

0

kisi

LiF = energi kisi

Nilai Energi Born-Haber

•



H

_oatom

Li = 161 kJ

•

_{BE F}

₂

_{= 159 kJ}

•

IE

1

(Li) = 520 kJ

•

EA (F) = -328 kJ

•



H

_oLattice

(LiF) = -1050 kJ

•



H

_of

LiF = -617 kJ

•

Total Energi :



Hof LiF = H_oatom Li + ½ BE F₂ + IE₁ (Li) + EA (F) +



Pendekatan Mekanika

Molekuler

• Ion-ion diasumsikan berada pada situs kisi masing-masing sesuai dengan muatan formalnya, sehingga NaCl misalnya membentuk 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 obtain an accurate value.

• To calculate lattice energies, this summation be achieved for simple lattice structures by introducing the Madelung constant.

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}

• 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).

•

Nearest Neighbors (first shell):

This

reference sodium ion has two negative

chloride ions as its neighbors on either side at

r

0

so 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 2r

0

is

=

 

•

Next Next Nearest Neighbors (third

shell)

: The attractive Coulomb energy

due to the next two chloride ions

neighbors at a distance 3r

₀

is

•

_{and so on. Thus the total energy due to}

• 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 would naturally differ.

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.

•

_{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 often ignored.

•

Salts such as magnesium oxide can be

thought of as close-packed arrays of anions

with cations occupying the

octahedral

holes

.

•

_{Because the cations are held apart by the}

• _{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

• 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.

• _{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}

Crystal Defects in Silver Chloride

• Silver halides are used in photography to capture light and form an image.

• The action of light on the halide produces silver which forms the black areas of the negative (Figure 2.2).

• Two most common point defects in crystals are Frenkel defects and Schottky defects. What are these?

• _{In Schottky defects, equal numbers of cations and anions are}

missing (for 1 : 1 structures such as AgCl).

• In Frenkel defects, an ion is displaced from its lattice site to an interstitial site; for example, a small cation in a crystal with the NaCl structure can move to a tetrahedral hole from the

octahedral hole normally occupied.

• _{We can use molecular mechanics to estimate the energies of}

these defects in silver halides.

• In a Frenkel defect, there is a vacancy where an ion should be and an ion in a more crowded interstitial position. Would you expect the ions in the vicinity of the defect to stay on their lattice positions?

• It would be reasonable to suppose that the ions would adjust

their positions to allow the interstitial atom more room, and to take up the space left by the vacancy.

• When calculating the energy of formation of the defect the

nearest atoms are allowed to adjust their position to obtain the lowest energy for the crystal including the defect.

• Figure 2.3 (overleaf) shows how the chloride ions move when

(a) Perfect AgCl. (b) A Frenkel defect in

AgCl

• For an estimate of the actual numbers of defects we need to know the Gibbs energy of formation, but the major

contribution comes from the internal energy.

• Calculated values for the energy of formation of cation

Frenkel defects in NaCl and AgCl are 308 kJ mol-1 and 154 kJ mol-l, respectively.

(a

)

Zeolite

• Zeolites* have frameworks of silicon, aluminium and oxygen atoms which form channels and cages, e.g. Figure 2.4.

• They form a wide variety of structures but all are based on silicon

tetrahedrally bound to oxygen.

• Differing numbers of silicon atoms are replaced by aluminium. Other cations, notably those of Groups I, II and the lanthanides, are present in the

structures to balance the charge.

• _{Silicon is not normally thought of as forming Si}4+

ions; indeed silica, SO2, and silicates do contain

silicon covalently bonded to oxygen

• We do have to make some allowance for the

covalency of the Si-0 bonds.

• _{The most successful way of doing this is to add a}

term that represents the resistance of OSiO

and OAlO bond angles to deviation from the

tetrahedral angle.

• The covalency of zeolites and related compounds

• For one form of silica, SO2, for example, a calculated

lattice energy of -12416.977 kJ mol-1 had contributions of

-16029.976 kJ mol-1 from electrostatic interactions,

+3553.796 kJ mol-1 from intermolecular force terms and

the core-shell spring term, and 1.913 kJ mol-1 from those

OSiO bond angles that were not tetrahedral.

• Here the intermolecular force terms are about 20% of the electrostatic interaction.

• The energy due to the term keeping the angles

tetrahedral is small, but without this term the zeolite structure is lost.

• With this addition, the structures of a wide variety of

Modelling Organic

Molecules

• The power of this method for organic molecules lies in the

adoption of a relatively small set of parameters that can be transferred to any molecule you want.

• But what sort of parameters might be needed? Can we

simply use electrostatic and intermolecular forces? How do we allow for bonds and different conformations

• Let us start by looking at a very simple molecule - ethane.

Ethane is H3C-CH3.

• As for solids, we do need to include an electrostatic

interaction, but what charge are we going to give carbon and hydrogen atoms?

• Obviously +4 or -4 on C and +I or -1 on H are unrealistic and

would not even give a neutral molecule.

• _{When two atoms form a covalent bond, they share}

electrons. If the atoms are unalike then one atom has a larger share than the other, resulting in a

positive charge on one atom and a negative charge on the other.

• But the charge transferred is less than one electron. For diatomic molecules, the charge on each atom can be obtained experimentally.

• In the molecule HCI, for example, the hydrogen atom has a charge of +O. 18 and the chlorine atom a

charge of -0.18.

• _{The fractional charges are known as partial}

• _{A convenient way of setting up a set of transferable}

partial charges is to give each atom a contribution to the partial charge from each type of bond that it is involved in.

• _{For example, in chloroethane, CH3CH2Cl, we need to}

consider contributions for the carbon atoms for carbon bound to carbon, carbon bound to hydrogen and carbon bound to chlorine.

• _{Carbon bound to carbon is given a value of zero.}

• _{For elements such as oxygen, which can be singly or}

doubly bonded (C-O or C=O), we need different partial charge contributions for each type of bond

• _{In one available computer program carbon bonded to}

hydrogen gives a contribution of +0.053

• As well as the electrostatic interaction arising from the partial charges, we also need intermolecular forces.

Cocaine analogue -

Ecgonine