Download organic crystals: prediction of crystal structure from molecular structure

the crystal packing of organic molecules
intermolecular energies determine
crystal symmetry and intermolecular geometry
angelo gavezzotti
Dipartimento di Chimica Strutturale
Università di Milano
[email protected]
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Part I. The physical nature
and the computer simulation
of the intermolecular potential
a "modern" molecule is a collection of nuclei
and of charge density points (~20000 significant ones
for a medium-size molecule)
the charge density box
- a molecular orbital calculation - computer programs available
- standard option for ρ(x,y,z) in e- per cubic angstrom
V(i), ρ(i); q(i)= V(i) ρ(i)
• In principle:
The total intermolecular interaction energy
is the expectation value of the Hamiltonian
of a supramolecular system
(just like the molecular energy is the expectation value
of the molecular Hamiltonian)
•In
practice:
for historical and practical reasons,
one usually defines:
- Coulombic terms
- polarization terms
- dispersion terms
- repulsion terms
stabilizing/destabilizing: ENERGIES
< 0, > 0
attractive/repulsive:
FORCES
interactions can be: stabilizing and attractive
stabilizing but repulsive
destabilizing but attractive (activation)
destabilizing and repulsive
5
4
3
2
1
0
-1
-2
3
3.5
4
4.5
dis tanza in angs tro m
5
Coulombic term
++, - - repulsive destabilizing
+ - attractive stabilizing
strongly orientation-dependent
+
+
+
-
+
-
+
+
-
Coulombic potential energy
• Point-charge model: a net charge at each atomic nucleus
In terms of point-charge models, the coulombic energy is
Ecoul =1/(4πε°)
ΣΣ qi,A q j,B /Rij
where do the q's come from?
1) Mulliken population analysis charges (variable with basis set)
2) ESP charges (more consistent)
3) re-scaled Extended Huckel (Gavezzotti and Filippini)
4) other recipes
In terms of central multipoles
The potential due to a charge distribution, at a distance R from
the center of charges, is expanded in a series of multipoles:
V(R) = A°/R +(1/R2) (Ax px+Ay py+Az pz)
+ (1/R3) (Az2 d z2+Axy dxy+Axz dxz+Ayz dyz+ Ax2-y2 dx2-y2)
+ higher terms over 1/R4
, 1/R5 ,...
A coefficients: radial dependence of the potential
p and d functions are the spherical harmonics.
Central multipoles may be appropriate for very small
molecules, but not for large organic molecules
The distributed dipole model
(A.J.Stone)
distributed
multipoles
point charges?
negative nuclei?
• In terms of central or distributed dipoles,
the coulombic energy is a sum of
moment k to moment m terms,
where each moment is a monopole,
dipole, quadrupole moment.
Coulombic potential energy: in terms of full density
Ecoul = ∫ ∫ρA ((r1)) ρB ((r2) )/ |r1 -r2 | d 3r1d 3r2 +
Σk ∫ Zk(A) ρB ((r2) )/ |rk -r2 | d 3r2 +
+ Σm ∫ Zm(B) ρA ((r1) )/ |rm -r1 | d 3r1 + Σk Σm Zk(A) Zm(B) |rm -rk |
requires an analytical form for ρ
Ecoul = Σk Σm qk,A ((r1)) qm,B ((r2) )/ |r1 -r2 |
+ Σk Zk(A) [Σm qm,B ((rm) )/ |rk -rm | ] +
+ Σk Zk(B) [Σm qm,A ((rm) )/ |rk -rm | ] + Σk Σm Zk(A) Zm(B) |rm -rk |
replaces the integrals with a summation, ρ by points
POLARIZATION
attractive stabilizing
+
polarizer
+
+
polarized
Polarization (electrostatic induction) energy
Polarizability is:
a measure of the propensity of a given
electronic environment to yield
under the action of an external electric
field : displacement/restraint
the restraining force is
coulombic attraction between
the displaced charge and the nuclei
polarizability is large when electrons
are at a large distance from a
weakly charged nucleus.
electric field
electron
nucleus
The energy involved in the polarization process
is always stabilizing because the induced dipole µ
always points in the stabilizing direction
(compliance of the polarized medium
to the polarizing field ε)
The linear dipole polarization energy is given by
E pol = - ∫ µ dε = -1/2 α ε2
classical polarization: analogy with
molecular orbitals mixing, including virtual orbitals
sometimes called charge-transfer energy.
the electron game of chemistry..................
bond formation
spin pairing
charge transfer
polarization
Pauli repulsion
avoidance of like spins
NEUTRAL
NEUTRAL
??
DISPERSION:
conceptually similar to polarization
it operates in any chemical system
from NaCl to solid benzene
the "glue of the world"!!!
Dispersion energy
fluctuating dipoles generate induced dipoles in a nearby
electron distribution. The resulting quantum mechanical
dipole-dipole coupling is the effect called dispersion.
stabilization has to do with zero-point oscillator energies,
and dispersion effects are a consequence of the
uncertainty principle, like all zero-point energy effects.
London model for dispersion energy
α molecular polarizability, I ionization potential
at a distance R:
Edisp ≈ [ -I (α)2 ] (R)-6
The dispersion energy is always
stabilizing, like polarization energy is.
Fritz London
REPULSION
Exchange-overlap-Pauli repulsion
Short-range repulsion has no classical analogy.
Pauli principle:no two electrons with same
quantum numbers in the same region of space
mathematical requirement:
proper wavefunctions must be antisymmetrical
Approximate models for the repulsion energy:
SAB = electron density overlap
Erep = A SABγ
Pauli: avoid overlap
reasonable: energy proportional to overlap,
with A and γ parameters
Etot = Ecoul + Epol + Edisp + Erep
chemical interpretations;
for example, coulombic end polarization energies are large
in molecules with a permanent charge separation
the definition of partitioned energies is not unique.
partitioning schemes: Morokuma-Stone IMPT-etc...
One often speaks of "van der Waals" energies
more appropriately one should speak of
Coulomb-London-Pauli energies
Etot = Ecoul + Epol + Edisp + Erep
• full quantum chemical methods
approximate methods:
• the 'atom-atom' potential E = A exp(-BRkm) - C(Rkm)6
E = A (Rkm)-12 - B (Rkm)-6
• semiempirical methods ('PIXEL')
building a model for calculating interaction energies
a molecule at X' = MX°+ t
rotation and translation
the electric field
a molecule at X°
if M and t are space group operations,
calculate lattice energy
inspecting the nature of the
intermolecular chemical bond
exercise
parallel offset benzene dimer
80
Ecoul
Epol
energies, kJ/mol
60
Edisp
40
Erep
20
Etot
0
-20
-40
-60
-80
2
3
4
5
interplanar distance
6
benzoic acid dimer
300
E(coul)
E(pol)
200
energies, kJ/mol
E(disp)
E(rep)
100
E(tot), n=3
0
-100
-200
-300
1
2
3
R(O...H)
4
The energetic glossary
of intermolecular interactions
very strong interactions in charged species
+H NCH CH=CHCOO3
2
, energies in kJ/mol
- +
+
-325
-178
+
+83
-
+
-
• O-H...O=C hydrogen bond
(30-40 kJ mol-1)
• stacking of aromatic rings
with polar substituents
(20-30 kJ mol-1)
• arene-perfluoroarene
(25 kJ mol-1 ring-1)
• very acidic C-H to π-clouds
(10 kJ mol-1)
acetylene-acetone
• very acidic C-H to basic O, N
(<10 kJ mol-1)
benzoquinone
• aliphatic C-H to O,N,π
(<5 kJ mol-1)
Part II
Molecular organization in crystals
Calculation of crystal energies
CRYSTAL:
condensed phase in which matter is organized
with periodic translational symmetry
chemical entities are related by symmetry
that repeats itself periodically in 3D space
the basic chemical unit
an inversion center
translation
inversion
mirror/glide
twofold/screw
P1 or P1space groups
inversion center
plus translation
a screw axis
21
__
b
__
z→
a screw axis plus perpendicular translation:
monoclinic
__
↑
b
↓__
←
c
→
inversion center + screw axis = glide plane
21
→G
→G
exercise: space group P21/c
b
c
basic facts
about molecular crystals
molecular materials
▪ non-linear optics
▪ pigments
▪ explosives
▪ pharmaceutical dosage forms
▪ mesophases (LCD displays)
molecular materials vs. inorganics,
ceramics, etc.
▪ organic chemistry much more versatile
than inorganic chemistry
▪ less weight
▪ more pollution (solvents, etc.)
▪ very little mechanical resistence
sublimation
enthalpy,
kJ/mole...
organic
crystals
sublimation heat, kJ/mol
250
200
150
100
50
0
0
50
100
150
200
250
molecular weight
300
350
400
problem:
a quantitative estimate of the lattice energy
solution:
1) give the computer the co-ordinates
of the asymmetric unit,
the cell dimensions and space group;
these come from X-ray diffraction
2) generate a cluster of molecules and
calculate the lattice sums
alternative solution:
periodicize atomic orbitals (band structure) and calculate
the electronic wavefunction on crystal orbitals (Bloch methods)
Φ(r+R, k) = exp(ik·R) Φ(r,k)
k=a*/2
k=0
Bloch function
160
calculated lattice energy
140
a 10 kJ/mol difference
between obs and calc
is considered a good result
(experimental values
often wrong by much
more than that)
120
100
80
60
40
20
20
40
60
80
100
120
sublimation enthalpy
140
160
Ecoul Epol Edisp Erep Etot
argon
-2
-1
-12
8
-7
DTE
-21
-9
-123
60
-93
SUCROSE
-273 -126 -116 315 -200
Ecoul
NaCl
> -800
Epol Edisp Erep
-108
-29
169
-9
254
forsterite (one ion pair)
[2Mg]4+[SiO4]4- -8156 -1009
weak forces require close contact of molecular
objects: close packing in organic crystals
organic crystals: prediction of crystal structure
from molecular structure
• generate compact structures (geometry)
• compute lattice energies (physics of the interaction)
• ranking generated structures in order of
stability (a problem in polymorphism)
...a largely unsolved problem!
close packing is a necessary
but not a sufficient condition
anhydrous caffeine
computational crystal structures
-40
Coul.
-60
-80
total
-100
-120
-140
1.35
disp.
1.4
1.45
1.5
1.55
1.6
density
many good crystal structures, all close packed...
...and yet no one has ever been able
to have a good crystal of anhydrous caffeine!
for any given molecular structure, many crystal structures
can always be derived by a simple simulation
experimental ∆H's between polymorphs are a few kJ/mol
about 10 generated structures have lattice energies
withing a range of a few kJ mol-1
crystallization is not only a matter of equilibrium
thermodynamics, otherwise one would always crystallize
a Boltzmann distribution of many polymorphs
aspirin, naphthalene and sucrose have been crystallized
in tons without ever observing a second polymorph
"soft matter"
great flexibility in phase behaviour
the good side: many options
the bad side: difficult to control
a slab of crystalline
succinic anhydride
??
liquid
the problem....
kinetics