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Understanding Crystal Packing and Polymorphism of Organic Crystals with Density Functional Theory

1. RESEARCH DESCRIPTION

Organic molecules including most of pharmaceutical compounds are prone to polymorphic

formation in the solid state. Because of the variation in crystallization environment (e.g.,

solvent, temperature, using of additives, and concentration), the same molecules can pack

differently and form different crystal lattices or polymorphs. As a result, the physical,

chemical and mechanical properties of the crystals can be dramatically affected.

Among various types of polymorphism, conformational polymorphism is commonly

encountered, particularly for the pharmaceutical compounds which often have aromatic

moieties. Conformational polymorphism exhibits different conformations of the same

molecule in different crystal lattices. There is apparently a tight connection between the

crystal packing and the molecular conformation. From the viewpoint of energy, the lattice

energy of the crystals and the molecular energy are mutually controlled by each other.

We propose to examine the conformational polymorphism with density functional theory

(DFT). By assuming energy is a functional of electron density, DFT has established a realm

of methods and principles in exploring electronic structures of various types of molecular

systems including organic crystals. Pushing current studies of polymorphism of organic

molecules that have been focused mainly on the geometric packing and skeleton structures of

molecules, we aim to evaluate the electronic structures of crystal lattice, particularly using a

DFT-derived concept named nuclear Fukui function, for understanding how crystal packing

affects molecular conformation in conformational polymorphs. With DFT, not only can we

study the energetic properties of whole molecules, but we can reveal the electronic behaviors

of each atom. We have obtained very interesting preliminary data, and we plan to carry out a

systematic study and to understand how the electronic structure plays the role in establishing

the connection between the crystal structure and the conformations of molecules. We expect

results of this study will lead to the prediction of polymorphism of organic molecules, and

shed light on the rational design of crystal structures.

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2. RESEARCH METHODOLOGY

To study the effect of crystal packing on electronic structures of different polymorphs, we

will calculate and compare nuclear Fukui functions of atoms in various crystal lattices.

Nuclear Fukui function has been developed according to the conceptual DFT. DFT claims

that the electron density is the fundamental quantity for describing atomic and molecular

ground states, and energy is a functional of electron density 1-3. As a molecular system

changes from a ground state to another because of perturbations in electron population or the

number of electrons, dN, as well as the external potential that is defined by atomic positions

and nuclear charges, δv(r), the expansion of the system energy change to second order may

be expressed as 4, 5:

Eq. 1

where r is the position vector, µ the electronic chemical potential (the opposite of the

electronegativity 6), characterizing electron’s escape tendency from the equilibrium, ρ(r) the

electron density, η the hardness, f(r) the Fukui function, and β(r, r’) the linear response

function. The Fukui function may be capable of describing the sensitivity of a molecular

system to electronic and nuclear perturbations 7, 8. The hardness is related to Klopman’s

frontier molecular orbital theory 9, calculated by the energy gap between ionization potential,

I, and electron affinity, A 10:

Eq. 2

The inverse of hardness is softness, S 11. It is believed that hardness indicates a resistance to

charge transfer, while softness measures ease of transfer and is associated with polarizability 2. Consequently, the dependence of hardness on molecular deformation is called the nuclear

stiffness 12, 13:

Eq. 3

where Qi = Ri – Ri,0 is the displacement vector of atom i from its equilibrium position, Ri,0.

∫∫∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+⎥

⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛

∂∂

= ')'()()'()(2

1)()(

)(21)(

)(

222

2

2rrrr

rrrr

rrr

rddvddv

vvEdNddv

NvEdN

NEddv

vEdN

NEdE

NvNv δδδ

δδ

δδ

)',()()( rrrr βηρµ f

221

21

2

2 AINN

E

vv

−≅⎟

⎠⎞

⎜⎝⎛

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=µη

Nii ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=Q

G η

3

Similarly, the nuclear reactivity index is defined as the derivative of electronic chemical

potential with respect to the displacement of an atom 14:

Eq. 4

Their contributions to the global hardness and chemical potential can be expressed as:

Eq. 5

where η0 and µ0 are the hardness and chemical potential of un-deformed molecules at

equilibrium. Should there be a conformational change or even a bond breaking, contributions

of Gi and Φi to the decrease of hardness and chemical potential may be revealed from their

scalar products with atomic displacements (i.e., Gi⋅Qi and Φi ⋅Qi), likely to predict how much

each atom is involved in the conformational change, which atoms are involved in the

reaction, whether the molecule accepts or donates electron(s), and/or whether the reacting

bond is shortened or stretched 12. It has been shown that large absolute values of Gi and Φi

can be used to identify those atoms and bonds that are involved in a chemical reaction 13.

Furthermore, the nuclear stiffness and nuclear reaction index can be calculated by atomic or

Hellmann-Feynman forces 14:

Eq. 6

where Fi+ and Fi

- are forces acting on the same atom i when the number of its electrons has

increased (+) or decreased (-), respectively. Thus, from their relationship with electronic

forces on ionized species, Gi and Φi may be able to reveal how much an atom participates in

a reaction. A large force on an atom indicates a large displacement, resulting in a bond

breaking, shortening or conformational change. As a first derivative of the system energy

with respect to the number of electrons (Eq. 1), the nuclear reactivity index may be better for

describing the reactivity than the nuclear stiffness. In fact, the concept of nuclear reactivity

index has been extended into so-called nuclear Fukui function 14:

Eq. 7

Nii ⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂−=

QΦ

µ

∑∑

∑∑

⋅−=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=

⋅+=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=

iii

ii

Ni

iii

ii

Ni

QΦQQ

QGQQ

00

00

µµµµ

ηηηη

)(

)(

21

21

−+

−+

−=

+−=

iii

iii

FFΦ

FF G

−−

++

−=

−=

iii

iii

FFΦ

FFΦ0

0

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to characterize a nucleophilic or electrophilic attack, respectively. Since a chemical reaction

is driven by the change in system energy, and is accompanied by the electron transfer and

atomic displacement, the nuclear Fukui function, a local function to describe system

sensitivity to a simultaneous perturbation in the number of electrons, N, and the nuclear

position, R, may be useful for characterizing the reactivity of crystals with respect to crystal

packing. For a molecule in equilibrium, Fi0 is close to zero; Fi

+ or Fi

- alone may be able to

study the reactivity of respective atoms in a nucleophilic or electrophilic reaction 15. For the

purpose of this study, nuclear Fukui function can also provide further insights.

Because nuclear Fukui function describes the tension applied to each atom due to electronic

perturbation in a molecular system, we hypothesize that the influence of crystal packing on

the conformation of molecules in a crystal can be better manifested by nuclear Fukui

functions. For the conformational polymorphism, the molecule has different conformations

in different polymorphs due to the constraints of each crystal lattice. The most stable a

packing motif, the smaller nuclear Fukui functions are. As conventional methods to study

each conformer as a whole, details of how different chemical moieties in a conformation are

affected are overlooked. Thus, we believe using nuclear Fukui function and other DFT

concepts will lead to further fundamental understandings

of polymorphic formation.

3. PRELIMINARY RESULTS

We have studied indomethacin polymorphs with DFT.

Two polymorphs have been identified. Lattice

parameters of the monoclinic α-form (mp 152 – 154 °C,

P21) are a = 5.462, b =25.310, c = 18.152 Å, β = 94.38°,

and Z = 6 16; lattice parameters of the triclinic γ-form (mp

160 – 161 °C, P_1 ) are a = 9.295, b = 10.969, c = 9.742 Å,

α = 69.38, β = 110.79, γ = 92.78°, and Z = 2 17. The

crystal structures of two forms, as shown in Fig. 1, were

optimized with the lattice parameters fixed prior to the

calculations of electronic structures and properties. A

periodic ab initio program, Crystal 03 18, was used for the

Figure 1. Crystal structures of the α-form (a), and γ-form (b) of indomethacin. Six molecules in each unit cell of the α-form are divided into three symmetrically different pairs as marked

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optimization and single-point electronic calculation.

Hartree-Fock (HF) and DFT with B3LYP exchange-

correlation functional 19, 20 were used for the

structural optimization and electronic calculation,

respectively. The Pople’s 6-21G basis sets were

used for each calculation method. In addition,

single molecule of indomethacin was calculated

with the Gaussian 03 code package (Gaussian, Inc.,

Wallingford, CT).

In this preliminary study, Φi+ was calculated of the

two polymorphs of indomethacin. (Results are not

shown, but Φi- values give similar conclusions.) Fi

0

and Fi+ of each atom were obtained from

calculations of the neural and anionic species of a

crystal structure, respectively. The molecular

structure of the anionic species was kept the same as

its neural counterpart while an extra electron was

introduced to the unit cell during the calculation.

Fig. 2 illustrates the scales and directions of nuclear

Fukui functions of each conformer of the two

polymorphs. Values of the functions are not shown

due to the length limitation of this proposal. It can

be seen that the largest nuclear Fukui functions are

associated with C9, O1, and C10 as well as their

neighbor atoms (C11, N1, C15, and C1). We

believe the large physical stress on these atoms due

to the electronic perturbation may stem from

energy-unfavorable conformations. As shown in

Fig. 2, N1, C9, O1, and C10 bridge two aromatic

rings of indomethacin molecule, the indole and

phenyl rings. Since the carbonyl group between the

Figure 2. Nuclear Fukui functions of three symmetrically different molecules of the α-form, #1 (a), #2 (b) and #3 (c), and the molecule of the γ-form (d) of indomethacin represented as arrows originated from each atom. Arrows are color-coded from red to green to blue indicating values from the largest to the smallest of each molecule.

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two rings also provides p-orbitals, the ideal conformation

for the molecule would be delocalization of all p-orbitals

(excluding those of the –COOH), forming one single

aromatic plane by the indole, carbonyl and benzoyl groups.

The current conformations of indomethacin molecules in

the α- and γ-forms are likely due to the steric repulsion

between the chlorobenzoyl and the methyl of the indole

ring. It is apparently illustrated that the two rings are on

two different planes, forcing the separation of two local

aromatic systems. To support this argument, molecular

orbitals of isolated indomethacin molecule were calculated

with Gaussian 03. As shown in Fig. 3, the HOMO is

mainly delocalized on the indole ring, not extending to the

phenyl ring at all. On the other hand, the LUMO spreads

over the chlorobenzoyl part of the molecule. The separation of the aromatic structures is

likely to be the result of their conformations. Therefore, the balance between the steric

repulsion and the tendency to form one aromatic system may be metastable and sensitive to

any electronic perturbation, causing the two rings to re-align their positions and/or to change

their conformations. The large values of the nuclear Fukui functions of the atoms that

connect the two ring structures may indicate the localization of the two aromatic systems,

characterizing the tension between the steric

repulsion and the inclination to form a single

sharing of the p-orbitals.

The conformations of the molecules in the two

polymorphs are significantly different. The major

difference stems from the relatively position of

the two aromatic rings. There are two dihedral

angles involved between the two rings, C1-N1-

C9-O1 (-158.58, -24.48, 29.49 and -32.42 for the

three asymmetric molecules in the α- form and

the molecule in the γ-form respectively) and O1-

Figure 3. Molecular orbitals of indomethacin single molecule calculated with Gaussian 03 by B3LYP/6-311G**//B3LYP/6-311**, highest occupied molecular orbital (a), and lowest unoccupied molecular orbital (b).

Figure 4. Energy map of single molecule as a function of the two dihedral angles. Positions of conformations in the crystals are marked. Original is the global minimum.

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C9-C10-C15 (51.80, -41.51, 43.32, and -23.25, respectively). The energy map of single

indomethacin molecule as a function of two dihedral angles is shown in Fig. 4. The contour

map is produced by calculating energies of the molecule at different angles while keeping

other bond angles and lengths fixed. The molecule in the γ-form and one of molecules in the

α- form have conformations close to the global minimum. Another two molecules in the α-

form have conformations in different valleys. More polymorphs seem to exist because of

variations of the two dihedral angels. Therefore, we may conclude that the crystal packing

(i.e., the α- and γ-forms) determines the conformations of the indomethacin molecules.

Conversely, due to the unique feature of the molecule as illustrated above, each conformation

is a result of a delicate balance between the steric hindrance and the propensity to delocalize

all possible p-orbitals. Clearly, the nuclear Fukui function illustrates how exactly the crystal

packing can affect the conformation of molecules in a crystal, and it can provide fundamental

understandings regarding the structure-energy relationship. By controlling the relative

positions of the indole and phenyl rings, we may be able to design new polymorphs.

4. RESEARCH PLAN

We plan to examine a few crystal systems in this project with the

similar approaches. We want to focus on molecular systems that

have two aromatic systems separated by a bridging group with p-

orbitals. The bonds between each aromatic system and the

bridging group need to be single bonds so that the two aromatic

systems can rotate freely. One possible system is 5-methyl-2-[(2-

nitrophenyl)amino]-3-thiophenecarbonitrile, as show in Fig. 5. This system has exhibited

more than six polymorphs. It is also called ROY due to its red, orange and yellow colors in

various polymorphs.

We will use the Cambridge Structural Database 21 for searching for model systems for the

research. Currently, the Database has more than 330,000 entries of various organic crystals,

including almost all the polymorphs that have been reported. The Database comes with tools

that allow the database search based on various criteria including the match of a pre-defined

chemical substructure. Our substructures will include two phenyl rings, one phenyl ring and

one indole ring, and one phenyl ring and a 5-member ring linked by either a carboxyl or an

SN

HNO2

N

Figure 5. The ROY structure.

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amine group. We will find related organic crystals that have polymorphs, and select three or

four interesting molecules for this study.

As we illustrate in the Preliminary Results, we will use DFT to calculate the electronic

structures of crystal structures for each selected molecule. We will then derive nuclear Fukui

functions of each molecule in the crystal lattice and correlate to the relative position of the

two aromatic rings. The energy map of each single molecule will be calculated with

Gaussian. Through these calculations, we expect that 1) we will understand what structural

features lead to the polymorphism; 2) we will identify whether more polymorphs are

possible; 3) we will identify the relative physical and chemical stability among the

polymorphs; and 4) we will gain the insight into the conformational polymorphism with

regard to the structure-energy relationship.

Including ROY, we will investigate three or four systems depending on the computational

resources that we can secure. Currently, we have one Linux cluster with 28 nodes, and we

will use the University’s Superdome when possible. The plan of the project is summarized

below:

5. FUTURE DIRECTION

Our ultimate goal is to develop new methods for polymorph prediction based upon electronic

structures of molecules. This study will provide fundamental understandings of the crystal

packing and molecular structure as well as their connections. The generated results will

allow us to identify how the molecular structure affects its conformations and packing motifs

in crystals. DFT is the right computational tool for accomplishing the goals.

Time Milestone

Months 1 – 2 Search the Cambridge Structural Database; Identify model systems that

have required structural features and also have polymorphs.

Months 3 – 10 For each crystal system including ROY, we will use Crystal 03 and

Gaussian 03 to calculate electronic structures of both crystals and single

molecules, and to derive nuclear Fukui functions and energy maps.

Months 11 – 12 Analyze data and write manuscripts.

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6. REFERENCES

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B864-B871.

2) Kohn, W.; Becke, A. D.; Parr, R. G., Density functional theory of electronic structure.

Journal of Physical Chemistry 1996, 100, 12974-12980.

3) Parr, R. G.; Yang, W. T., Density-functional theory of the electronic-structure of

molecules. Annual Review of Physical Chemistry 1995, 46, 701-728.

4) Nalewajski, R. F., Coupling relations between molecular electronic and geometrical

degrees of freedom in density functional theory and charge sensitivity analysis.

Computers & Chemistry 2000, 24, 243-257.

5) Parr, R. G.; Yang, W., Density-functional theory of atoms and molecules. Oxford

University Press: New York, NY, 1989.

6) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E., Electronegativity - density

functional viewpoint. Journal of Chemical Physics 1978, 68, 3801-3807.

7) Ayers, P. W.; Parr, R. G., Variational principles for describing chemical reactions.

Reactivity indices based on the external potential. Journal of the American Chemical

Society 2001, 123, 2007-2017.

8) Ayers, P. W.; Parr, R. G., Variational principles for describing chemical reactions:

The fukui function and chemical hardness revisited. Journal of the American

Chemical Society 2000, 122, 2010-2018.

9) Klopman, G., Chemical reactivity and concept of charge- and frontier-controlled

reactions. Journal of the American Chemical Society 1968, 90, 223-234.

10) Parr, R. G.; Pearson, R. G., Absolute hardness - companion parameter to absolute

electronegativity. Journal of the American Chemical Society 1983, 105, 7512-7516.

11) Yang, W. T.; Parr, R. G., Hardness, softness, and the fukui function in the electronic

theory of metals and catalysis. Proceedings of the National Academy of Sciences of

the United States of America 1985, 82, 6723-6726.

12) Ordon, P.; Komorowski, L., Nuclear reactivity and nuclear stiffness in density

functional theory. Chemical Physics Letters 1998, 292, 22-27.

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13) Luty, T.; Ordon, P.; Eckhardt, C. J., A model for mechanochemical transformations:

Applications to molecular hardness, instabilities, and shock initiation of reaction.

Journal of Chemical Physics 2002, 117, 1775-1785.

14) Cohen, M. H.; Gandugliapirovano, M. V.; Kudrnovsky, J., Electronic and nuclear-

chemical reactivity. Journal of Chemical Physics 1994, 101, 8988-8997.

15) De Proft, F.; Liu, S. B.; Geerlings, P., Calculation of the nuclear fukui function and

new relations for nuclear softness and hardness kernels. Journal of Chemical Physics

1998, 108, 7549-7554.

16) Chen, X. M.; Morris, K. R.; Griesser, U. J.; Byrn, S. R.; Stowell, J. G., Reactivity

differences of indomethacin solid forms with ammonia gas. Journal of the American

Chemical Society 2002, 124, 15012-15019.

17) Kistenmacher, T. J.; Marsh, R. E., Crystal and molecular structure of an

antiinflammatory agent, indomethacin, 1-(p-chlorobenzoyl)-5-methoxy-2-

methylindole-3-acetic acid. Journal of the American Chemical Society 1972, 94,

1340-1345.

18) Doll, K.; Saunders, V. R.; Harrison, N. M., Analytical hartree-fock gradients for

periodic systems. International Journal of Quantum Chemistry 2001, 82, 1-13.

19) Becke, A. D., Density-functional exchange-energy approximation with correct

asymptotic-behavior. Physical Review A 1988, 38, 3098-3100.

20) Lee, C. T.; Yang, W. T.; Parr, R. G., Development of the colle-salvetti correlation-

energy formula into a functional of the electron-density. Physical Review B 1988, 37,

785-789.

21) Allen, F. H.; Davies, J. E.; Galloy, J. J.; Johnson, O.; Kennard, O.; Macrae, C. F.;

Mitchell, E. M.; Mitchell, G. F.; Smith, J. M.; Watson, D. G., The development of

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Information and Computer Sciences 1991, 31, 187-204.

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7. CURRENT SUPPORT

“Development of Hybrid Nanocrystals for Simultaneously Targeted Delivery of Therapeutic

and Bioimaging Agents”, DOD Medical Research Program #BC050287, PI, 03/01/06 –

02/28/09, $434,506

“CAREER: Towards Fundamental Understanding and Rational Control of Crystal Growth”,

NSF #DMR0449633, Sole-PI, 03/01/05 – 02/29/10, $496,399

8. CURRENT FUNDS FOR THE RESEARCH

None. This is a new research initiative, not yet being funded by any source. This research is

different from our NSF CAREER project in which solvent-solid interactions are examined

with DFT and electron-based concepts such as hardness and softness.

9. SUPPORT REQUESTED

$28,000 is requested for the stipend and tuition for Ms. Clare Aubrey-Medendorp.

Principal Investigator/Program Director (Last, first, middle):

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