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Understanding weak interactions. “Symmetry E70 (Butterflies)” by M.C. Escher - 1948. “Symmetry E72 (Fish and Boats)” by M.C. Escher - 1949. Understanding the origins and magnitudes of weak interactions.

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Understanding weak interactions

“Symmetry E70 (Butterflies)”

by M.C. Escher - 1948

“Symmetry E72

(Fish and Boats)”

by M.C. Escher - 1949


Understanding the origins and magnitudes of weak interactions

  • Molecular crystals being made up of molecules are affected by intramolecular and intermolecular forces:

    • Intramolecular forces affect the physical properties and shape of molecules which are important in crystal packing

    • The intermolecular forces are generally much weaker and short-range in their effect

  • This combination of strong and weak introduces diversity in the properties of molecular crystals

  • This is in contrast to simple ionic crystals which are dominated by strong long-range columbic forces


Understanding the origins and magnitudes of weak interactions contd.

  • Understanding the origins and magnitudes of intermolecular forces is therefore necessary to understanding the properties of molecular crystals

  • In particular it is necessary to understand their dependence on the following:

    • molecular properties

    • intermolecular separation

    • intermolecular orientation

  • The understanding of intermolecular interactions in the context of crystal packing and the utilization of such understanding in the design of new solids with desired physical and chemical properties is in fact the focus of two closely related fields: crystal engineering and supramolecular chemistry


Crystal engineering
Crystal Engineering interactions

  • The field of crystal engineering aims to predict and control crystal assembly, and hence structure, solid state properties and reactivity.

  • Has a role to play in many fields: materials design, binding of dyes onto clothing, the understanding of bone growth, the formation of clatharate hydrates which block pipes in the oil industry, melting point suppression, etc.

  • Has found application in the field of “green” chemistry where reactions are carried out without the use of solvents, and where the reactions observed often do not have solution phase counterparts.



Van Der Waals Forces interactions Dipole-dipoleDipole-induced dipoleDispersion/London forces



Interaction between dipolar molecules repulsive forces

  • The electric field produced by a dipole μ along its own direction at a distance r from its centre is 2μ/r3

  • For two dipoles aligned head to tail at a distance r apart, the interaction energy U between them is given by:U = -2μ1μ2/r3


Interaction between dipolar molecules repulsive forcescontd.

  • However, in crystals the dipoles will not be necessarily be well aligned with each other.

  • By considering:

    • the random orientation of the dipoles,

    • defining their relative orientations using polar coordinates,

    • and including attractive and repulsive interaction components, the following equation may be obtained:

      U = -(μ1μ2/r3){2cos θ1cos θ2 - sin θ1sin θ2cos(φ1 - φ2)}

Attractive component

Repulsive component


Interaction between dipolar molecules repulsive forcescontd.

U = -(μ1μ2/r3){2cos θ1cos θ2 - sin θ1sin θ2cos(φ1 - φ2)}

  • The above equation shows that the interaction energy between two dipoles in a crystal will be inversely proportional to r3

  • The equivalent equation for an interaction between two dipoles in solution has a dependence proportional to r6

  • The dipole-dipole interaction in the solid-state therefore has a longer effective range

  • Lastly, and most importantly the above equation has repulsive and attractive components so depending on the relative orientations of the dipoles this interaction can be attractive or repulsive


Dipole-induced dipole interactions repulsive forces

  • The electric field of one dipole can induce a dipole on a second polarizable molecule

  • This interaction is dependent on the component μ1 of the electric field of the dipole along the line joining it and the molecule centered at distance r

  • It is also dependent on the polarizability α2 of the second molecule

  • The interaction between the dipole and induced dipole is given by the following equation: U = -4α2μ12/r6

  • Note the dependence proportional to r6 indicating that this is a very short range interaction


Dipole-induced dipole interactions repulsive forcescontd.

  • In contrast to dipole-dipole interactions, this interaction is always attractive as only the magnitude of the interaction depends on the relative orientations of the two molecules

  • Since polar molecules can also be polarized, this interaction also contributes to the total interaction energy between the two dipolar molecules, i.e. Utotal = Udipole-dipole + ΣUdipole-induced dipole


Dispersion/London forces repulsive forces

  • Non-polar molecules can interact with each other even though they do not have permanent dipoles

  • A dipole in one molecule can be created from small instantaneous and transient changes in the positions of electrons (charge displacements = r1 = x1y1z1) in the molecule leading to a transient dipole

  • This transient dipole can then induce a dipole in the second molecule with the appropriate charge displacements

  • These two dipoles attract each other leading to a decrease in the potential energy of the system

  • The attraction between the molecules depends strongly on their polarizability


Dispersion/London forces repulsive forcescontd.

  • The total potential energy is therefore made up from the energy needed to produce the dipoles (I) and the energy of interaction between the two resulting dipoles (II): U = e2r12/2α + e2r22/2α + (e2/r3)(x1x2 + y1y2 - 2z1z2)

  • Allowing for notation Term II is the same as the equation for a dipole-dipole interaction that was shown earlier

    U = -(μ1μ2/r3){2cos θ1cos θ2 - sin θ1sin θ2cos(φ1 - φ2)}

  • Note that the repulsion term is present. This is because transient dipoles can also be repulsive. However, because of induction the interaction is on average attractive and not zero as one would expect. (See Atkins)

(1)

I

II


Dispersion/London forces repulsive forcescontd.

  • An alternative and more powerful form of equation (1) can be obtained by expressing the transient dipole as an oscillator: E0 = 3hv0 - (3/4)hv0α2r-6 + .......

  • The 3hv0 term is the zero pointenergy of the two isolated species (depends on the ionization energy of each species) while the second term is the attractive dispersion energy

  • v0 is the characteristicfrequency of one of the species (how often it “flickers”)

  • The attraction between the molecules depends strongly on their polarizability

(2)


Dispersion/London forces repulsive forcescontd.

  • In practice, more complex charge displacements can occur between molecules leading to quadrupoles and higher multipoles

  • The true dispersion force experienced by a molecule is therefore better described by the following equation: Udispersion = c6r-6 + c8r-8 + c10r-10 + ........ where c6,c8 etc. are constants.

  • Quadropoles and higher terms have often been neglected but can be very important. In CO2 the heat of sublimation at 0 K (27 kJ/mol) has been found to be 45% due to quadropole-quadropole interactions and 55% due to simple dispersion interactions.


Final comments on VDW forces repulsive forces

  • Dipole-dipole, dipole-induced dipole and dispersion forces are often collectively referred to as van der Waals forces

  • The above expressions show that VDW forces should be largest between polar or very polarizable molecules

  • In general polarizability is known to increase with increasedmolecular volume (larger molecules) and increased numbers of π bonds



Repulsive forces repulsive forces

  • The closest distance molecules can reach each other is the point where attractive and repulsive forces are exactly in balance

  • The repulsive forces start becoming significant when the electron clouds of two molecules begin to overlap

  • There are two sources for this force:

    • The Pauli exclusion principle – maximum of 2 electrons per orbital with opposite spins

    • Decreased electron density in areas where overlap has occurred hence less effective shielding and hence greater coulombic repulsion between the nuclei on the two molecules


Repulsive forces repulsive forcescontd.

  • Repulsive forces are very difficult to calculate since they depend on the shapes and nature of the molecule of interest

  • Instead, empirical potentials are used and these tend to have the form ar-n (n equal to 12 usually) or be-cr with a, b, c andn being empirically determined constants for individual atoms types

  • Since repulsion forces are inversely dependent on r12, they are only important at very close range

  • Also, the above equations assume isotropic repulsions between atoms. In practice anisotropic versions of these equations are used as these better describe the behavior of real crystal structures.


Summing attractive and repulsive components together – atom-atom potentials

U = (qiqj)/(Drij) + A/rij12 - C/rij6

  • Where qiand qjare the fractional charges on the atoms,

  • D is the effective dielectric constant,

  • Akl is the repulsive coefficient,

  • and Ckl the attractive coefficient.

  • If one removes the electrostatic component (to make the above equation more general) then one gets left with the 6:n or Lennard-Jones’ form: Ukl(rij) = Akl/rij12 - Ckl/rij6



What about H-bonding? typical lattice energy program

U = (qiqj)/(Drij) + A/rij12 - C/rij6

1 kcal = 4.184 kJ


H-bonding typical lattice energy programcontd.


H-bonding typical lattice energy programcontd.

  • Strong H-bonds (ionic hydrogen bonds) are formed by groups containing an electron density deficiency on the donor group, i.e. OH+, NH+, or an excess of electron density in the acceptor group, i.e., F-, OH-, C-O-, P-O- etc.

  • Moderate H-bonds are generally formed by neutral donor and acceptor groups, i.e., OH, NH, and –O-, C=O, NAr in which donor atoms are electronegative to the H-atom and acceptor atoms have unshared lone-pair electrons.

  • Weak H-bonds are formed when the H-atom is covalently bonded to a slightly more electronegative atom, as in C-H, Si-H, or when the acceptor has no lone pairs but has π electrons.


H-bonding typical lattice energy programcontd.


H-bonding typical lattice energy programcontd.

  • H-bonds have group-pair properties, e.g., P-OH, H-O-H and C-OH have different donor and acceptor abilities resulting in different H-bonds with different D…A distances and angles.

  • Covalent interactions on the other hand are more atom-atom pair in nature. This means that C-C, C=C, C-N etc bonds can be more easily classified into typical bond lengths and angles, VDW and covalent radii etc…


Three- and four-centre H-bonding typical lattice energy program

  • One of the consequences of H-bonding being mostly electrostatic in nature is its ability to form bifurcated (3 centre) and trifurcated (4 centre) bonds, i.e., it is possible for an H-atom to interact with more than one acceptor.

  • These commonly occur when the number of acceptors exceed the numbers of donors in a crystal, e.g. in carbohydrates, nucleosides and nucleotides where a lot of ether (R-O-R) or carbonyl groups are present.

2- and 3-centre H-bonds typically found in amino acid structures


Three- and four-centre H-bonding typical lattice energy programcontd.

What are the criteria for picking out trifurcated H-bonds?

  • Since the interaction is an attractive one the H atom should lie with 0.2 Å from the plane defined by the donor and two acceptor atoms.

  • Alternatively, the 3 angles around the H atom, a + b + c ≈ 360°.

a

c

b


Resonance assisted hydrogen bonding (RAHB) typical lattice energy program

  • H-bonding can often be assisted by the presence of conjugated π bonds. In effect the π bonds act to cooperatively assist the H-bond making the interaction more covalent in character and as a consequence stronger.

  • This phenomenon is referred to as Resonance-Assisted Hydrogen bonding (RAHB).


Resonance assisted hydrogen bonding (RAHB) typical lattice energy programcontd.

  • The effect of RAHB on H-bonding has been illustrated by examining molecules of the following type, in which the degree of delocalization/conjugation (Q) was determined from the molecular bond lengths, and plotted against the H-bond O…O distance. Increased delocalization (lower Q values) leads to shorter O…O lengths.

Q = d1 – d4 + d3 – d2


Principles of Crystal Packing typical lattice energy programTrends Followed by Molecules when Forming a Crystal


Rule 1 typical lattice energy program

  • Maximize density and minimize free volume

  • This is the primary packing rule in all kinds of crystals and is referred to as Kitaigorodsky’s principle of close packing

  • It is especially important in polymorphic systems for crystals that are stable at T = 0 K where no energy is available to support more open structures

  • At higher temperatures, entropy (TS) plays a large role in allowing polymorphs with more molecular freedom to occur


Rule 2 typical lattice energy program

  • Satisfy hydrogen bond donors and acceptors and any other special kind of intermolecular interactions

  • H-bonds are very strong in comparison with VDW forces. As a consequence their presence is structure determining in molecular crystals.

  • Also, in a structure, the strongest hydrogen bond donors always connect to the strongest hydrogen acceptors


Rule 3 typical lattice energy program

  • Minimize electrostatic energy

  • Like - like repulsive interactions must be minimized in favor of like - unlike attractive interactions


Lets look now at the observed space group frequencies typical lattice energy program

  • Though there are 230 space groups more than 75% of all organic molecules crystallize in only 10 of these – probably 99% of all molecules crystallize in just 30 space groups

  • The top ten are: P21/c, P-1, P212121, P21, C2/c, Pbca, Pnma, Pna21, Pbcn and P1

  • Now that we have some understanding of the nature of weak interactions can we rationalize why these space groups are so popular?

    • Are some symmetries more favorable than others?


Principles of Crystal Packing typical lattice energy programClassification of Symmetries into Favorable and Unfavorable


Inversion centers are favorable typical lattice energy program

  • In centrosymmetric space groups molecules are related to each across a centre of symmetry

  • Inversion centers are especially favorable for crystal packing since they diminish like-like interactions and are compatible with translation.

  • They are unique in that they change the direction but not the orientation of intermolecular vectors, i.e. minimize the repulsive component of dipole-dipole interactions

  • Molecules related by inversion centers are often connected to each other by hydrogen bonding which is an energy lowering interaction


Inversion centers are favorable typical lattice energy programcontd.


Mirror planes are always occupied, usually by mirror-symmetric molecules

  • Unoccupied mirror planes are especially unfavorable because they require like-like interactions between adjacent molecules

  • This symmetry maximizes the repulsive component of the dipole-dipole interaction

  • If unoccupied, they produce a sheet of empty space in the crystal which has serious consequences in terms of packing density.


Mirror planes are always occupied, usually by mirror-symmetric molecules contd.

Mirror symmetric molecules

Note relatively empty space


Groups with 3-, 4- and 6-fold rotation axes do not usually occur unless the axes are located within molecules of appropriate symmetry

  • The reason for this is that it is difficult to fill the space around these rotation axes unless the molecules have the shape of the symmetry. The problem is especially acute for 4- and 6-fold axes. How many organic molecules containing a 4-fold axis do you know?

  • If unoccupied, they create an infinite rod of empty space with a diameter of 3-3.5 Å in the crystal, hence lowering the packing density of the structure

  • To compensate for this, areas where these occur are usually filled with disordered solvent leading to the formation of a solvate


Groups with 3-, 4- and 6-fold rotation axes occur unless the axes are located within molecules of appropriate symmetrycontd.

Benzene


Twofold axes are sometimes occupied and sometimes not occur unless the axes are located within molecules of appropriate symmetry

  • There seems to be no energy gain or loss associated with the use of this symmetry

  • As mentioned for rotation axes above, if they are unoccupied they can lead to lower packing densities


Finally occur unless the axes are located within molecules of appropriate symmetry

  • 21 screw axes are more favorable than glide planes, which are comparable to pure translations.

  • Translations are however still very favorable otherwise crystals would not exist.

  • Translations tend to occur together with other symmetry elements, i.e. P1 is not that popular

  • Having gone through the last few slides have a look at the top 10 space groups again:

    • P21/c, P-1, P212121, P21, C2/c, Pbca, Pnma, Pna21, Pbcn and P1


Supplementary – examples of weak interactions occur unless the axes are located within molecules of appropriate symmetry


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