Chinese Journal of Chemical Physics  2017, Vol. 30 Issue (1): 50-62

The article information

C. Anzline, S. Israel, R. Niranjana Devi , R. A. J. R. Sheeba , P. Richard Rajkumar
Anzline C., Israel S., R. Niranjana Devi, R. A. J. R. Sheeba, P. Richard Rajkumar
High Resolution Synchrotron Diffraction Study on Charge Density Distribution of Ampicillin Trihydrate
利用高分辨同步辐射衍射研究氨苄青霉素三水合物的电荷密度分布:密度泛函理论和多极模型之间的相关性
Chinese Journal of Chemical Physics, 2017, 30(1): 50-62
化学物理学报, 2017, 30(1): 50-62
http://dx.doi.org/10.1063/1674-0068/30/cjcp1607143

Article history

Received on: July 14, 2016
Accepted on: October 1, 2016
High Resolution Synchrotron Diffraction Study on Charge Density Distribution of Ampicillin Trihydrate
C. Anzlineb, S. Israela, R. Niranjana Devi a, R. A. J. R. Sheeba c, P. Richard Rajkumar a     
Dated: Received on July 14, 2016; Accepted on October 1, 2016
a. Research and Postgraduate Department of Physics, The American college, Madurai 625002, Tamil Nadu, India;
b. Mother Teresa Women's University, Kodaikanal 624102, Tamil Nadu, India;
c. Research and Postgraduate Department of Physics, Madura College, Madurai 625001, Tamil Nadu, India
*Author to whom correspondence should be addressed. S. Israel,E-mail:israel.samuel@gmail.com,Tel.:+91-986538-4403
Abstract: Charge density distribution in ampicillin trihydrate was investigated experimentally. Results were compared with the quantum calculations using density functional theory. The charge derived properties including Mulliken atomic charges, dipole moment, and molecular electrostatic potential were calculated. The multipole analysis was done for the refinement of experimental population parameters. The structure factors obtained from multipole treatment were used for the construction of Fourier maps. Topological properties of the charge distribution were discussed and the characteristics of (3,-1) critical points were analyzed.
Key words: Charge density    Multipole analysis    Density functional theory    Electrophilicity index    
I. INTRODUCTION

Ultra high resolution crystallography and charge derived density has become attractive tools to investigate the bonding and electronic structure of crystals and molecules in ground state [1, 2] and excited state [3]. During the past two decades, a number of exciting advances have occurred in the field of X-ray charge density studies, including the development of reliable helium based cooling devices, and the availability of intense short wavelength synchrotron radiation [4] that gives accurate X-ray information on the chosen samples. After a slow start, the number of synchrotron charge density studies increases rapidly. The use of synchrotron radiation allows measurements on very small crystal samples at short wavelengths and thereby drastically decreases sources of uncertainty such as absorption of the X-ray beam in the crystal and rescattering of the incident and diffracted beams [5]. Advances in computers, softwares, and X-ray sources and detectors have made charge density more commonly available and more readily applicable to molecules of increasing size [6]. Calculation of charge densities in molecules has been a preoccupation [7] of theoretical chemists for some time and several experimental and theoretical charge density investigations for the crystalline solids and small molecules have been reported [8-14]. Since X-rays are mostly scattered by electrons, the X-ray diffraction technique can be used for charge density mapping, identification of intermolecular interactions, and charge transfer in co-crystals with different molecular components [15, 16].

A fundamental work on molecular electron density distributions by Bader laid the foundations for the theory of atoms in molecules (AIM). In Bader's words: "the charge density provides a description of the distribution of charge throughout real space and is the bridge between the concept of state functions in Hilbert space and the physical model of matter in real space" [17]. Bader's early studies of molecular electron density distributions coincided with the ground breaking formulation of modern density functional theory [18]. The experimentally obtained distribution can be compared directly with the theoretical results obtained by density functional theory (DFT). Such comparisons would reveal the deficiencies in both theory and experiment. Now it has reached a stage where it is possible to derive properties like net charges, molecular dipole moments and electrostatic potentials and other important quantum chemical parameters [19]. The extraction of these one electron topological properties has contributed to an understanding of the electron density distribution in molecular solids [20].

The field of crystal engineering has experienced a significant development from the beginning of 21st century. More attention is now being paid to the impact of material properties in drug discovery [21]. Crystal engineering principles are now being actively considered for application to pharmaceuticals to modulate the properties of these valuable materials [22]. Because the physical properties that influence the performance of pharmaceutical solids are reasonably well appreciated, there is a unique opportunity to apply crystal engineering techniques for appropriate biopharmaceutical performance of oral drugs. The interaction of pharmaceutical compounds with biomolecules is done by electrostatic interaction. In order to understand the chemical interaction of pharmaceutical compounds, it is desired to have a complete mapping of localized charges in a molecule. Research challenges in biomedical and medical research are fascinating to many researchers [23]. More recently, both small molecule and macro molecular crystallography are applied together in research relevant to medicinal chemistry [24, 25]. The electron density is a quantum mechanical observable that can be measured directly through scattering experiments. The most adopted method to reconstruct the electron density is the multipolar model. The multipolar electron density can be compared to the electron density computed by quantum chemical methods that use various degrees of approximation to solve the Schrodinger equation [26]. Charge densities with its topology and its derived electrostatic properties are essential for the understanding of inter atomic or inter molecular interactions [27-29].

In this work, the electron density described by the Hansen and Coppens [30] formalism using experimental XRD data was estimated and compared with the charge density estimated theoretically using DFT for ampicillin trihydrate. Ampicillin trihydrate is a broad-spectrum semi-synthetic penicillin derivative that inhibits bacterial cell-wall synthesis by inactivating transpeptidases on the inner surface of the bacterial cell membrane. It is effective in the treatment of gram-positive and gram-negative bacterial infections produced by Streptococcus, Bacillus anthracis, Haemophilus influenzae, Neisseria gonorrhoeae, and Escherichia coli. [31, 32]. This antibiotic is used in the treatment of upper respiratory tract infections, genital and urinary tract infections but it does not work for colds, flu or other viral infections. The title compound has been used as an antibiotic since 1961. The crystal structure was reported in 1968 [33] but no atomic coordinates were given in the work. Boles et al. in 1978 reported the crystal structure of amoxicillin trihydrate and compared with the structure of ampicillin [34]. Burley et al. have reported the crystal structure from the synchrotron X-ray powder diffraction and the details of O-H…O and N-H…O bonds were discussed [35]. In this work, the theoretical and experimental charge densities of the title compound were analyzed. While analyzing, the water molecule was left out deliberately as it did not have any appreciable effect on the charge density.

II. Theoretical charge density using DFT

Chemical and physical properties of micro or macro molecular system can be evaluated by the theoretical methods on high-speed computers. The multi electron wave function used to describe a many electron system employing one electron functions (orbitals) incorporates the Pauli exclusion principle and the resulting differential equation is referred to as Hartree-Fock (HF) equation. This equation is solved through an iterative process and the convergence is achieved once a self-consistent field is obtained. The use of molecular orbitals to describe a molecule containing n electrons involves the linear combinations of atomic orbitals approach. This requires the choice of a set of basis function. The HF scheme predicts the properties of a system near the ground-state equilibrium geometry, however, ignoring the correlation energy. The methodology involved in density functional theory includes electron correlation and hence is a desired method for obtaining theoretical charge density in molecular crystals. DFT, over the years has become a practical tool for calculating charge-density distributions since its adaptability to high-speed computers is easy. The level chosen for electron density in the DFT method is generally B3LYP and the corresponding Gaussian basis set is 6-311G++ (d, p) [36-38] which takes into account polarization and diffuse functions.

A. Geometry optimization

In this work geometry optimization of ampicillin molecule (C16H19N3O4S) was done at the ground state level using Becke-3-Lee-Yang-Parr hybrid exchange functional (B3LYP) of the density functional theory with 6-311G++(d, p) basis sets [37, 38] using Gaussian 09W software [39].

Geometry optimization usually attempts to locate minima on the potential energy surface, thereby predicting equilibrium structures of molecular systems. Optimizations can also locate transition structures. The potential energy surface (PES) specifies how the energy of a molecular system varies with small changes in its structure. A PES is a mathematical relationship linking molecular structure and the resultant energy. For a diatomic molecule, it is a two-dimensional plot with the inter-nuclear separation on the x-axis (the only way that the structure of such a molecule can vary), and the potential energy at that bond distance on the y-axis, producing a curve. For larger systems, the surface has as many dimensions as there are degrees of freedom within the molecule. Generally, a non-linear N atomic molecule, has 3N-6 degrees of freedom, or internal coordinates. This is because all N atoms can move in three dimensions (x, y, and z) giving 3N degrees of freedom. However six of those three translations in x, y, z directions and three rotations along x, y, and z axes of the molecule as a whole do not produce any change in energy. Ampicillin has 43 atoms and it has 123 degrees of freedom [40].

The optimized molecular structure together with the numbering scheme for ampicillin is shown in Fig. 1. In the optimized molecule the electrons are delocalized in the phenyl ring. The standard values for C-C and C-H bond in the aromatic ring are 1.39 and 1.09 Å, and our theoretical calculations lie in the range 1.3913-1.4018 and 1.0834-1.0858 Å which are close to standard values [41]. The optimized bond length for other C-C bonds ranges of 1.508-1.584 Å and it agrees well with the typical bond length of 1.53 Å. The C-H bond length in methyl group is 1.09 Å and for C-N, C-S and N-H bonds the bond length lies between 1.316-1.523, 1.852-1.914, and 1.018-1.085 Å which is close to the standard value of 1.48, 1.82, and 1.01 Å, respectively. The expected bond length for C-O and C=O is 1.43 and 1.24 Å but the optimized bond length for C-O bond is 1.242 Å and for C=O it ranges from 1.206 Å to 1.249 Å [42]. In the phenyl ring the endohedral angle for 20C-19C-24C is 119.57°. All other endohedral angle in the phenyl ring lies in the range from 120.0° to 120.19° which are very close to the reported value of 120.0° [43]. The title molecule has two methyl groups and the shape of the methyl group is found to be tetrahedral with the expected bond angle of 109.5°. The bond angle for the methyl group calculated using B3LYP/6-311G++(d, p) is different for different combinations of H-C-H and are 108.92°, 109.20°, 107.50°, 107.74°, 108.68°, and 107.88° for 36H-14C-37H, 36H-14C-38H, 37H-14C-38H, 39H-15C-40H, 39H-15C-41H, and 40H-15C-41H respectively. For the tetrahedral amino group the bond angle for 29H-8N-30H, 29H-8N-31H is 110.88° and 109.41°. The experimental and theoretical bond lengths for the title molecule have matched very closely and can be verified from Table I.

FIG. 1 Optimized molecular structure of ampicillin obtained from B3LYP/6-311G++(d, p) level.
Table I Comparison of theoretical and experimental bond lengths.
B. Frontier molecular orbital analysis

Molecular orbital theory incorporates the wave like characteristics of electrons in describing the bonding behaviour. And the bonding between the atoms is described as a combination of atomic orbitals. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are the most important frontier molecular orbitals. The HOMO-LUMO analysis is carried out to explain the charge transfer within the molecule. These orbitals determine the interaction of the molecule with other species. HOMO is a donor and represents the ability to donate an electron while LUMO is an electron acceptor which represents the ability to accept an electron. The energy of the HOMO is found to be -4.461 eV and that of LUMO is -2.776 eV. The HOMO-LUMO energy gap characterizes the kinetic stability and chemical reactivity of the molecule. The molecule is more polarizable when the frontier orbital gap is small and associated with low kinetic stability, high chemical reactivity and it is termed as soft molecule. The frontier orbital gap of ampicillin is found to be 1.684 eV. The eminent polar solvent water has a frontier orbital gap of 9.308 eV. Thus taking water as a reference molecule ampicillin is considered to be a more polarizable and soft molecule [43, 44]. The encapsulations of HOMO and LUMO regions in the molecule are shown in Fig. 2(a) and (b) .

FIG. 2 HOMO, LUMO, Mulliken plot of ampicillin.

According to Koopman's theorem [45] the HOMO and LUMO energies can be used to estimate the charge derived properties. The orbital energies of the frontier orbitals are related to the ionization potential and electron affinity as: I=-εHOMO and A=-εLUMO. Using Koopman's theorem for closed shell molecules the hardness and chemical potential are given by η=(I-A)/2 and μ=-(I+A)/2. Mulliken electronegativity is defined as χ=(I+A)/2. The electronegativity and hardness are used extensively to make predictions about the chemical behaviour. The former measures the power of an atom to attract electrons to itself and the latter measures the resistance of an atom to charge transfer. Parr et al. [46, 47] have proposed the electrophilicity index of a molecule in terms of its chemical potential and hardness as ω=μ2/η2. It is a measure of energy lowering due to maximal electron flow between the donor and acceptor. When two molecules react, the molecule having higher electrophilicity index will act as an electrophile whereas the molecule having lower electrophilicity index will act as a nucleophile [48, 49]. Based on the density functional theory the charge derived properties like ionization potential, electron affinity, hardness, chemical potential, electronegativity, and the electrophilicity index are found to be 4.461 eV, 2.776 eV, 0.842 eV, -3.618 eV, 3.618 eV, and 7.774 eV. Fluorine is the most reactive non metal with the electronegativity of 4.42 eV (Pauling scale of 3.98 eV) and it has the greatest attraction for the electron. The electronegativity of ampicillin is close to that of the Fluorine and thus it has strong attraction for electrons. Hardness depends on the closeness of the frontier orbitals. When the HOMO and the LUMO are close together, the absolute hardness is low and the atoms or molecules are ready to share the electrons to create the covalent bond. The title molecule having the HOMO-LUMO energy gap of 1.684 eV forms a strong bond with other polarizable molecule and it is considered to be a soft molecule. A property most desirable for any possible intermolecular interaction between a pharmaceutical compound and a bio molecule and thus ampicillin qualifies itself to be a fast interacting drug or a good efficacy molecule.

C. Mulliken population analysis

Population analysis is the study of charge distribution within molecules. Mulliken population analysis is based on the linear combination of atomic orbitals and therefore the wave function of the molecule. It partitions the total charge among the atoms in the molecule. This method quickly gives chemically intuitive charge sign on atoms and usually reasonable charge magnitudes. Figure 2(c) represents the Mulliken charge obtained from 6-311G++(d, p) basis set for the title compound and it shows that atom 9C is more acidic due to its high positive charge [50].

D. Dipole moment

Any possible excitation of the molecule in certain environment is also investigated and the energies were calculated. The excitation energies for the three transitions HOMO to LUMO, HOMO-1 to LUMO and HOMO-2 to LUMO are 0.237, 0.667, and 0.750 eV. The excitation energy for the HOMO-2 to LUMO is high, compared to the other two transitions but the oscillator strength is very low. The oscillator strength of a transition is a dimensionless quantity and it is used to estimate the relative strength of the electronic transitions within the molecular system. The HOMO-2 to LUMO transition with the oscillator strength of 0.0002 is considered to be a forbidden transition. The coefficients of the wave function for the first two excited states are 0.70655 and 0.70641 and the percentage contribution of each excited state is around 50%.

DFT also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. The three components of the dipole moment are μX=-29.79 Debye, μY=-0.73 Debye, and μZ=5.37 Debye and the total dipole moment of ampicillin is found to be 30.28 Debye. Quadrupole moments provide a second-order approximation of the total electron distribution and it provides a crude idea of its shape. If the three components XX, YY, and ZZ are equal then it indicates a spherical distribution. One of these components being significantly larger than the others would represent an elongation of the sphere along that axis. The six components of the quadrupole moments are QXX=-175.03, QYY=-145.61, QZZ=-147.65, QXY=10.75, QXZ=8.89, and QYZ=1.29 D-Å. The values indicate that the asymmetry of the molecule is larger in x-direction and the spread of the charge distribution is the same in y and z directions.

III. Experimental charge density

The experimental electron density was derived from high quality synchrotron powder diffraction data [35]. The data was collected at the X3B1 beamline of the national synchrotron light source, Brookhaven National Laboratory with an X-ray wavelength of 0.7003 Å. The data were collected from 2θ=3° to 41.6° in steps of 0.005°. It is customary to look into the charge density derived from X-ray diffraction experiment, to understand the real charge density of the molecule in the environment of other such molecules in a solid. To start with, the X-ray data should be refined to enact the desired molecular structure and many models of treating X-ray data have been proposed and one such technique is Rietveld technique [51]. Its primary purpose is to fit a structure model to the experimental data by calculating the diffraction pattern and using the differences between the calculated and experimental spectra to improve the parameters in the model. The refinement procedure is essentially the least squares method and the refinement must be approached with caution as it involves many parameters to be corrected.

Synchrotron X-ray data have profound advantages over conventional X-ray data for structure determination. It is customary to use Debye-Scherrer geometry for powder diffraction experiment using synchrotron radiation. The structure refinement by Rietveld profile fitting was performed using the package JANA 2006 [52]. Initially the profile parameters were refined followed by the structural parameters. A Simpson's rule integration of the Pseudo-Voigt function [53] was used to model the peak profile. The low-angle asymmetry due to the axial divergence [54] was used to model the asymmetry correction with two refineable parameters. Use of capillary specimen eliminates preferred orientation effects, which are highly significant in flat plate samples [53]. The background is refined using Legendre Polynomial and all atomic coordinates, Uiso parameters, harmonic and anharmonic parameters of non-hydrogen atoms were refined [55, 56]. For hydrogen atoms the Uiso parameters were fixed to be 0.035 Å2. In refining the synchrotron data for ampicillin trihydrate, the space group was set as P21 21 21 and the initial cell parameters were set as a=15.52 Å, b=18.92 Å, and c=6.67 Å. The refined cell parameters are in good agreement with the reported ones. For the final refinement cycles the water molecules were fixed in position as the refinement was not sensitive to the position of O and H atoms in water. All 2111 unique reflections were treated and the refinement proceeded smoothly to reach a minimum characterized by an excellent fit to the diffraction profile (R=1.98%, GoF=2.48, Rp=2.73%, wRp=3.66%). The refined Rietveld profiles are shown in Fig. 3.

FIG. 3 Rietveld refined experimental X-ray profile for ampicillin trihydrate.
A. Multipole analysis

Multipole model is a versatile tool which is used in the present study to understand the experimental charge density and its deviation from theoretical models. This model was proposed by Hansen and Coopens [30] with the option that it allows the refinement of population parameters at various orbital levels where the atomic density is described as a series expansion in real spherical harmonic functions through fourth order Ylm.

According to this model the charge density in a crystal is written as the superposition of a harmonically vibrating aspherical atomic density distribution convolving with the Gaussian thermal distribution as:

(1)

where tk(u) is the Gaussian thermal distribution term and ⊕ indicates a convolution and the individual atomic charge density is defined as:

(2)

where PC, PV , and Plm are population coefficients. The canonical Hatree-Fock atomic orbitals of the free atoms normalized to one electron were used for the construction of ρcore and ρvalence, but the valence function is allowed to expand/contract by the adjustment of the variable parameter κ and κ'. The third term describes the non spherical part of the valence electron distribution and Ylm is the spherical harmonic function in real form. The neutral atom wave functions are taken from Clementi tables and the Slater type radial functions Rl are used with nl=4, 4, 6, 8 for l<4 and the atomic orbitals up to Hexadecapole were considered with double zeta function ξ as per Hansen and Coopens [30]. The refined structure factors were used to find the core, valence and the pseudo atomic electron occupancies Plm. The multipole deformation function allowed for the orthorhombic site symmetry are P20, P22+, P32−, P40, P42+ and P44+ while the other terms vanish due to the symmetry existing in the molecule [57, 58]. The valence contraction expansion parameter κ is refined for individual atoms. The κ for 1S, 2O, 4O, 5O, 6N, 9C, 1OC, 11C, 13C, 14C, 16C, 18C, 22C, 23C, 24C, 44O, 47O, and 50O is greater than one which reveals that all these atoms undergo contraction while bonding whereas other non hydrogen atoms like 7N, 8N, 17C, and 19C undergo expansion during the bond formation. The population parameters for the orthorhombic symmetry of ampicillin and the details of the hydrogen bonding scheme are listed in Table II and III.

Table II Population parameters from multipole refinement.
Table III Hydrogen bonding scheme distances and angles.
B. Analysis of charge density map

The theoretical charge density map of the phenyl ring is picturised in Fig. 4(a) and it is compared with the experimental one in Fig. 4(b). The charge density map is drawn at the level from -2.36 e/Å3 to 8.70 e/Å3 with the interval 0.5 e/Å3 and it gives the picture on C-C and C-H interactions. The experimental charge density map shows the strength of covalent interaction between the atoms while the theoretical map does not reproduce the same. The 18C-19C is covalently bonded with intermediate charges in between and having bond path bent down rather than being a straight one. The strong covalent interaction between C-C bonds in the phenyl ring is evidently visible in Fig. 4(b). Also the covalent non polar bonds involved with 35H, 34H, and 43H are better pronounced than 33H and 32H. Such kind of interactions is not visualized in Fig. 4(a) and the interaction between the atoms seems to be uniform. The understanding of bonding and its charge distribution is very important and the models adopted should be able to reproduce the charge density concerned with the bonding. Deformation density maps were constructed to see the effect of the temperature on charge density distribution. The effect of the temperature can be distinguished from the convoluted and the deconvoluted forms of thermal contribution to the charge density as dynamic and static multipole deformation maps. The deformation densities in these maps are characterized by:

(3)
FIG. 4 Charge density map of the phenyl ring. (a) Theoretical, (b) experimental, (c) deformation density map, (d) residual density map, (e) SMD, and (f) DMD. Contour intervals are 0.5 e/Å3. Solid lines indicate positive contours and dashed lines negative contours.

where F(k)multipole is the Fourier transform of the multipole charge density. The deformation density map in Fig. 4(c) shows that the bonding between C-C and C-H bonds is covalent non polar.

The carboxyl group has an electronegative oxygen atom double bonded to a carbon atom and the double bond character increases the polarity of the bond. The theoretical and experimental charge distribution in the carboxyl group was mapped, which again reveals that the bonding between O-O is covalent non polar. Between O-O, the bond path is bent and the covalent charges between 3O and 4O lie in the off bond axis which is picturised in Fig. 5(a) and (b). The deformation density map shows that the oxygen atom has two lone pair of electrons and it is visualized in Fig. 5(c). These lone pair electrons make the oxygen more electronegative than carbon.

FIG. 5 Charge density map of the carboxyl group. (a) Theoretical, (b) experimental, (c) deformation density map, (d) residual density map, (e) SMD, and (f) DMD. Contour intervals are 0.5 e/Å3. Solid lines indicate positive contours and dashed lines negative contours.

In the methyl group the charge distribution between the C-H bonds are mapped and their theoretical and experimental results are picturised in Fig. 6(a) and (b). The experimental charge density of the methyl group in the plane containing 37H-14C-36H shows the presence of hydrogen atoms and the aspherically expanded carbon atom 14C. The same that was evolved by the theoretical calculation looks very close to the experimental charge density map. The analysis of 2-dimensional charge density maps gives an opinion that DFT can faithfully reproduce bonds involving H-atoms while it deviates by a definite magnitude when dealing with bonds not involving H-atoms. The hydrogen bonding scheme shown in Table III gives the validity of both theoretical and experimental models chosen for the study.

FIG. 6 Charge density map of methyl group along the plane containing 37H-14C-36H. (a) Theoretical, (b) experimental, (c) deformation density map, (d) residual density map, (e) SMD, and (f) DMD.

The accuracy of the multipole model compared with the experimental result can be gauged by looking at the difference Fourier map. Here phenyl ring, carboxyl group, and the methyl group are presented in Fig. 4(d), Fig. 5(d), and Fig. 6(d). The positive contour with a maximum of +0.06 e/Å3 and the negative contour with a minimum of -0.12 e/Å3 show that the experimental maps are highly accurate to be studied extensively for understanding interactions between the atoms and the type of bonding. The bonding features between C-C, C-H and C-O are the ones needed to be carefully examined for the complete understanding of the molecule.

The static multipole deformation density and dynamic multipole deformation density maps are usually plotted to understand the effect of temperature and its consequence in defining the asphericity of atoms. The static and dynamic deformation densities of the phenyl ring, carboxyl group and the methyl group are plotted in Fig. 4(e) and Fig. 4(f), Fig. 5(e) and Fig. 5(f), Fig. 6(e) and Fig. 6(f). The information on the deformation due to the temperature is well pronounced in static model while dynamic model shows very little deformation and thus showing the error between experimental and theoretical models to be very low.

One dimensional charge density profiles along the bonding direction between each bonds have been mapped and picturised in Fig. 7 and Fig. 8. Figure 7 primarily explains that covalent non polar bonds exist in the system while Fig. 8 narrates all covalent polar bonds. The bond path and its charge density have been compared for the theoretical and experimental structure. The charge density estimated using theory does not accommodate the exact interaction between the atoms that exist in the molecule. Also the atoms are spherically symmetric and their valence orbitals alone contribute to the bonding interaction. The experimental charge density however explains the thermal effect on the charge density along with the interatomic interactions. Experimental charge density clearly predicts the interacting charges, localized and depleted charge region along the bond path. The investigation of charge density at (3, -1) bond critical point (BCP) clearly explains the type of interaction between the atoms and quantifies them. Figure 7(a) and (b) enumerate the bonding between C-C atoms. A clear mid bond non nuclear maximum is expected between C-C bonds and is visualized in Fig. 7(a). However the theoretical charge density of carbon atoms does not spread over the entire bond length but it falls down steeply around 0.2 Å leaving only the valence electron in the mid bond region. Figure 7(a), (c) , and (e) enact this behaviour while Fig. 7(b), (d) , and (f) show the spread of charge density along the complete bond path with a single clear saddle pointing at the mid bond region. In Fig. 7(b) along the C-C bond the bond critical points position themselves exactly at the mid bond as the interatomic interactions between the C-C bonds are equal in strength. The charge density profiles of C-H and C-S bonds in Fig. 7(d) and (f) show that the strength of inter atomic interactions between the atoms are different and hence the BCP is shifted nearer to the smaller atom. The magnitude of charge density at BCP can be used as a tool to gauge the type of bonding between the atoms. Figure 8(a), (c) , and (e) enact the C-N, C-O and N-H bonds that take a shape smeared by thermal and charge rearrangement. And hence the saddle points defining the territory of the charge distribution of these atoms were extended over a long range. However the experimental charge density profiles shown in Fig. 8(b), (d) and (f) clearly define the extent of charge density between the atoms and the mid bond region. The size of the carbon atom can very well be understood from the spread of its charge density along the bond path and it varies between 0.523 and 0.969 Å (see Table IV) which is comparable to its covalent radius 0.77 Å.

FIG. 7 One dimensional map of electron density for (a) theoretical and (b) experimental C-C bond, (c) theoretical and (d) experimental C-H bond, (e) theoretical and (f) experimental C-S bond.
FIG. 8 One dimensional map of electron density for (a) theoretical and (b) experimental C-N bond, (c) theoretical and (d) experimental C-O bond, (e) theoretical and (f) experimental N-H bond.
Table IV Critical points from multipole analysis.
C. Topology of the electron density

Topology of the electron density provides a faithful mapping of the concept of atoms, molecules, structures and bonds. Hence charge distribution constructed using multipole model is analysed using AIM theory as proposed by Bader [17]. A quantitative way to analyse the topology of ρ is to consider the first derivative ▽ (ρ). According to Bader two atoms are bonded if they are connected by a line of maximum electron density called bond path, on which lies a BCP where , and the critical points are the characteristics of the bonding existing between the atoms. The critical points are derived from the eigen values of the Hessian matrix (3×3) of which the elements are:

(4)

By diagonalisation of this matrix, the off diagonal terms are set to zero and the three principal axis of curvature were obtained. These principal axes will correspond to symmetry axes, if the critical point lies on a symmetry element. The sum of the diagonal terms is called the Laplacian of ρ and is of fundamental importance. The critical points are characterized by two numbers, r and s, where r is the number of non zero eigen values of the Hessian matrix at the critical point (rank of the critical point) and s (signature) is the algebraic sum of the signs of the eigen values. Generally for molecules the critical points are of rank 3. The four kinds of non degenerate critical points of rank 3 are maxima (3, -3), minima (3, +3) and two types of saddle points (3, +1) and (3, -1) corresponding to nuclear positions, cages, rings and bonds [59].

Topological analysis of the experimental electron density was carried out using the crystallographic software JANA 2006. The critical points are searched in the charge density distribution using the Newton-Raphson method and are given in Table IV. The (3, -1) critical points characterize and quantifies the bonding. In the heteronuclear C-N, C-S, C-O, C-H and N-H bonds, the bond critical point is shifted towards more electronegative atom in accordance with a greater accumulation of charges. In the homonuclear C-C bonds, the bond critical point is centered between the atoms as indicated by d1 and d2 values.

The Laplacian of the electron density is related to the energy density by virial theorem. When the Laplacian is negative, the potential energy dominates and the charge is concentrated in the case of covalent bonds and lone pairs; when the Laplacian is positive the total energy is dominated by the kinetic energy density and the electron density is locally depleted in the case of ionic and hydrogen bonds [59]. The theoretical Laplacian obtained by the functional B3LYP and the experimental Laplacian in the plane containing 9C-1S-11C is shown in Fig. 9 (a) and (b). In the mapping of Laplacian function L(r)=▽2ρ(r) the depleted and accumulated regions of charge are clearly depicted by means of solid and dotted lines in Fig. 9(b) while the theoretical constructions shows the same by means of blue and green/red regions.

FIG. 9 The Laplacian of the charge density in the plane containing 9C-1S-11C. (a) Theoretical result obtained from DFT/ B3LYP and (b) experimental result obtained from multipole model.
IV. Molecular electrostatic potential

Molecular electrostatic potential maps were used to visualize the charge distribution of molecules and charge related properties of molecules. It also depicts the size, shape, charge density and site of chemical reactivity of the molecule. The negative and positive regions of electrostatic potential were seen in the lock and key type of ampicillin and are shown in Fig. 10. A portion of a molecule that has a negative electrostatic potential is susceptible to electrophilic attack and positive electrostatic potential is prone to nucleophilic attack.

FIG. 10 Molecular electrostatic potential showing lock and key type in ampicillin with gridlines originating from the phenyl ring.

Sanderson principle [60, 61] is obtained from the variational principle of the density functional theory. According to Sanderson's electronegativity equalization principle when two or more atoms with different electronegativity combine chemically, their electronegativities become equalized in the molecule. The equalization of electronegativity occurs through the adjustment of the polarities of the bonds that exist due to the partial charge on each atom. The loss in electron increases the electronegativity while the electron gain decreases the electronegativity and the final molecular electronegativity is expressed as the geometric mean of the electonegativities.

The validity of the electronegativity equalization principle is identical with that of the electrophilicity equalization principle. When two molecules interact, the molecule having higher electrophilicity index will act as an electrophile and lower electrophilicity index will act as a nucleophile. When an electrophile interacts with a nucleophile, the electrophilicity of the former gets reduced and that of the latter gets increased until they get equalized to a value somewhere between the two. As shown in the lock and key type the negative electrostatic potential in the carboxyl group of ampicillin is susceptible to severe electrophillic attack. When ampicillin is taken as an antibiotic, the electrophillic region of ampicillin interacts with the nucleophillic region of transpeptidase and the electrophilicity of the new molecule gets equalized by the electronic charge transfer. Thus the growth of bacterial cell wall gets inhibited by inactivating the transpeptidase.

V. Conclusion

The quantitative analyses of theoretical and experimental charge density distributions have led to the better understanding of the charge density features. The molecule is optimized at B3LYP/6-311G++(d, p) and the molecular properties were discussed and reported. The calculated HOMO and LUMO energies can be used to quantitatively estimate the ionization potential, electron affinity, hardness, chemical potential, electronegativity and electrophilicity index. The HOMO-LUMO energy gap of the molecule is low and it is considered to be a soft molecule with low kinetic stability and high chemical reactivity, which is a desirable property for strong intermolecular interaction. The excitation energies, oscillator strength and dipole moments were obtained from TD-DFT calculations. The higher multipole moments show that the spreads of charges along y and z directions are the same while asymmetry of the molecule is more in the x direction. A quantitative characterization of the covalent bonds and the hydrogen bonds have been obtained through the topological analysis of (3, -1) bond critical points and the effect of molecular electrostatic potential is analyzed.

VI. Acknowledgments

The authors are grateful to the Principal and Secretary and the Head of PG Physics, Department of The American College, Madurai, India for their continued encouragement and support during the progress of this work. This work was supported by University Grants Commission, India in the form of Major Research Project (F.No.41-848/2012(SR)).

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利用高分辨同步辐射衍射研究氨苄青霉素三水合物的电荷密度分布:密度泛函理论和多极模型之间的相关性
Anzline C.b, Israel S.a, R. Niranjana Devia, R. A. J. R. Sheebac, P. Richard Rajkumara     
a. Research and Postgraduate Department of Physics, The American college, Madurai-625002, Tamil Nadu, India;
b. Mother Teresa Women's University, Kodaikanal-624102, Tamil Nadu, India;
c. Research and Postgraduate Department of Physics, Madura College, Madurai-625001, Tamil Nadu, India
摘要: 实验研究了氨苄青霉素三水合物的电荷密度分布,并与用密度泛函理论的量子计算结果进行比较.计算了电荷导出性质,Mulliken原子电荷,偶极矩和分子静电势.另外用多极分析对实验总体参数的进行细化.用多极处理获得的结构因子构建了傅立叶图.同时讨论电荷分布的拓扑性质,分析了(3,-1)临界点的特性.
关键词: 电荷密度    多极分析    密度泛函理论    亲电指数