Molecular Potential Energies from Experimental Electric Field and

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Molecular Potential Energies from Experimental Electric Field and Electrostatic Potential at Nuclei Nour-Eddine Ghermani J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12305 • Publication Date (Web): 18 Feb 2019 Downloaded from http://pubs.acs.org on February 22, 2019

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The Journal of Physical Chemistry

Molecular Potential Energies from Experimental Electric Field and Electrostatic Potential at Nuclei Nour-Eddine Ghermani Institut Galien Paris Sud, UMR CNRS 8612, Université Paris-Sud/ Université Paris Saclay, Faculté de Pharmacie, 5 rue Jean-Baptiste Clément, 92296 Châtenay-Malabry, France.

ABSTRACT: In this paper, the molecular electrostatic potential energy V is first estimated from the electric field generated by an experimental electron density. Once the high resolution X-ray diffraction data are fitted against the Hansen-Coppens multipole model, the electric field E is analytically computed on every point inside and around the molecule by using our own software FIELD. The potential energy is then obtained by a numerical and robust integration of E2/2. The topological analysis of the electric field is carried out to reveal the specific contribution of atoms in a molecule. Application is made on a set of seven organic molecules of different sizes. The results are compared to those obtained from Politzer’s empirical calculations of the molecular energy using the electrostatic potential values at the atomic nuclei (EPN) within the framework of the Thomas-Fermi approximation. In this context, the molecular energy estimates found for the chosen molecules are presented and discussed.

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■ INTRODUCTION The nowadays diffraction instruments with highly sensitive detectors and stable X-ray sources permit to collect high resolution data even in a few hours. Available computational tools like MOPRO1,2 or XD3 are used worldwide to derive accurate electron density for molecules in the crystalline state. This experimental electron density is generally comparable to quantum calculations (DFT or HF) for molecules in the gas phase or in the crystal lattice (periodic theoretical calculations). The theory of “Atoms in Molecules” (AIM) is then applied to analyse the topology of the electron density in order to accurately characterize the atomic bonds.4,5 Moreover, global and local molecular features are computed through properties like atomic charges, electric moments, electrostatic potential and electric field E generated by the charge distribution. Some years ago, we have used this latter property to estimate atomic charges and electrostatic forces by the Gauss law using the flux through the atomic surface.6,7 In the continuity of this methodological approach, we propose here to use the electric field for the estimation of the potential (electrostatic) energy V of a molecule, for the electron density was determined experimentally. This scalar property is intrinsic to each molecule and gives a global characterization of the chemical system. Furthermore, in the X-ray diffraction experiment the nuclei are supposed to be at their equilibrium positions according to the Born-Oppenheimer approximation. Therefore, the virial theorem can be applied and the determination of the potential energy of the system results in an estimate of the kinetic energy T of the electrons (only available from the Compton effect) since T = -V/2.8 With the same idea, we have previously shown that the forces acting on the nuclei (Feynman force) and on electrons (Ehrenfest force) can be calculated from the flux of the Maxwell stress tensor related to the components of the electric field.7 By definition, the electrostatic potential energy V is equal to the work done to


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assemble a system of charges. For a density of charge (r) located at a point r in a volume d where an external electrostatic potential r) exists, the potential energy is dV = (r) r) d. For the whole system, the potential energy is the integral over the entire volume  of the system

V

1 2

  (r )(r )d

(1)



which, by exploiting Poisson’s equation and integration by parts, can be easily expressed in terms of the electric field generated by the system

V

1 E 2 ( r ) d 2



( 2)



From mechanistic consideration, this expression of the potential energy based on the energy density (or field energy) E2/2 has been first presented as an “ambiguity” in the 60’s by several authors because there was no available demonstration at this time.9,10 In 2002, R. F. Favreau provided the final evidence of the field energy using the Maxwell stresses during the assembly from infinity of a charge distribution. He proved a transfer of the self energy of a system into the field energy.11 For a system at equilibrium, the force and the electric field vanish at the nucleus position, but not the electrostatic potential. From the Hansen-Coppens Model,12 the mathematical expressions of the electrostatic potential (EPN) and electric field (EFN) at the nucleus have been given in detail by Coppens et al.13 The estimate of the electrostatic potential at the nucleus is important to reveal for example the correlation energy,14 or acidity of atoms in molecules and the EPN can be related to their pKa.15-16 Another aspect in the interest of the EPN determination is its strong correlation with the atomic and molecular energy. Politzer and co-workers have published several papers on this subject.17-19 The correlation between the EPN and the atomic or


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molecular energy was done within the statistical approximation of Thomas-Fermi. The proposed relation for the total energy Etot = T + V of a system is

E tot 

k

A

Z A  0, A (3)

A

ZA and 0,A are the nuclear charge and the potential at the nucleus of atom A, respectively; kA is a proportional and adjustable factor for each atom A close to the value of 3/7 suggested by the theory of Thomas-Fermi except for H atom (kH = 0.531278).19 The values obtained from the empirical equation 3 for the molecular energy of a set of several small organic molecules are in excellent agreement with those obtained by direct DFT calculations.17, 19 This theory was also recently reported in an excellent book.20 We propose in this paper to estimate, within the Thomas-Fermi approximation, the potential and total energies of larger molecules with the two methods described above and strictly based on the experimental electron density. The topology of the potential energy will be performed for each molecule at the atomic level. For the EPN calculation, the atomic contributions of the core, valence and environment will be detailed.

■ METHODS AND COMPUTATIONAL DETAILS Potential energy from the electric field integration. The Hansen-Coppens model is

used for the refinement of the experimental electron density.12 The valence population Pval and mutipole parameters Plm are estimated for each atom in the molecule as well as the contractionexpansion coefficients , ’. The core and valence densities core and val are chosen spherically symmetric and tabulated from Hartree-Fock calculations.21 The electric field components are analytically calculated by our program FIELD based on the Hansen-Coppens electron density


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model.6 The integrals of equation (2) are performed with BADERWIN numerical program.22,23 It is based on the robust steepest ascent algorithm and can partition the space containing the molecule into atomic basins corresponding to the zero-flux surfaces of the potential energy. The integration over each atomic basin is also given in the output. MOLEKEL program is used in this study for the visualization of the molecules and the different surfaces (electron density, electrostatic potential, electric field).24 Total and potential energies from the EPN. The program POTNUC6 is used to

compute the electrostatic potential (EPN) 0,A at each nucleus A located at RA as

0, A 

R

B A

ZB B

 RA





ρr'  d r' r'R A'

(4)

Z is the nuclear charge and (r) the electron density of the whole molecule. The contributions from core, val and the valence and multipole populations are also estimated as well as the potential generated by the other atoms of the molecule.6 From the values of EPN, equation (3) is used to estimate Etot and V energies. Politzer has also proposed the decomposition of Etot as

E tot 

3 (V ne  2V nn ) 7

( 5)

where Vne and Vnn are the total nuclear-electronic attraction and the total nuclear-nuclear repulsion, respectively.17 Vnn is easy to calculate from the nuclear point charges, then an estimate of the nuclear-electronic interaction can be obtained from equation (5). From this approach, the potential energy VEPN is then 2*Etot because T = -V/2. From the experimental electron density approach, a more detailed expression based on the multipole model was given in the pioneering work of J. Bentley.25


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The chosen set of molecules. In order to compare the results obtained for the potential

energies, we have chosen a set of 7 organic molecules containing C, N, O, H and S atoms. The experimental electron densities of these molecules have been published elsewhere. We chose the molecules for their sizes and their particular conformations. Figure 1 displays the molecular conformations of these 7 molecules.

■ RESULTS AND DISCUSSION Topological features of the electric field. The electric field was generated from the

experimental electron densities around each of the 7 molecules. In order to reveal the contribution of the electrons, we have projected the electron density  onto the E2 isosurface (0.1 au) shown in Figure 2 for molecule 1 chosen as example. Around each atom the E2 surface is almost spherical showing the predominance of the nuclear point charge (Z) contribution to the electric field. The red regions correspond to the highest cut-off chosen here for  (0.08 e.Å-3). These red regions surround the most negative atoms (O) and were also found in the middle of the atomic bonds (covalent parts). The values found for the carbon atoms are intermediate (0.04 e.Å3,

green regions). As can be expected, the lowest values (< 0.02 e.Å-3) of the electron density

were found in the vicinity of the hydrogen atoms. Conversely, we have projected the values of E2 on the isodensity surface (0.10 e.Å-3) in the top of Figure 3. The weakest values (blue regions) of the electric field are found in the vicinity of the oxygen atoms showing a total screening of the nuclear charge contribution by the surrounding electrons. An almost zero value of the electric field is also found (blue bands) in the middle of the atomic bonds where the mutual contributions of two attached atoms compensate each other. These weak parts of the electric field partition the molecule in singular atomic basins. The red regions in the top of Figure 3 correspond to the


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highest values of E2 (here with a cut-off of 0.85 au). These regions are found in the vicinity of C and H atoms where the depletion of the electron density is important increasing the nuclear charge contribution. Potential energy from the electric field. Equation (2) and Henkelman’s BADERWIN

program22, 23 were used to estimate the potential energies of the 7 molecules chosen here. The results are reported at the end of each column in Table 1. With respect to the size of the 7 molecules, the absolute potential energy magnitudes are in the range of 849.12 au (molecule 6) to 2958.20 au (molecule 4). BADERWIN program was initially dedicated to the numerical topology of the electron density permitting to estimate for instance the integrated AIM charges or higher electric moments.4 The atomic basins can be visualized by using programs like MOLEKEL.24 We have applied the same approach for the E2 values around each of the 7 molecules. The integrations of E2 over the atomic basins are also reported in Table 1. These basins are illustrated in the bottom of Figure 3 for the oxygen atoms in molecule 1. The atomic basins are separated by the zero-flux surface of E2 (potential energy density) gradient. This means that the inter-basin surfaces correspond to the regions where the total electrostatic forces vanish with equilibrium between attraction and repulsion. In Table 1 are also given the integration of E2 over each atomic basin. The values are consistent from one molecule to another. In average, we obtain atomic potential energy values of -3.7, -86.0, -107.1, -125.6 and -767.1 au for H, C, N, O and S, respectively. These values can be compared to those obtained for free atoms giving -0.9, -76.0, -108.2, -149.1 and -794.9 au for the same atoms.19 We can notice that except C and H atoms, the absolute potential energies are higher for isolated atoms revealing particular chemical bond features.


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Potential energy from the EPN. The electrostatic potential at the nuclei (EPN) is only

due to the spherical part of the electron density, all higher multipole contributions are zero.7, 13 The POTNUC program was used to calculate the EPN for the set of the 7 molecules.6 An example of the output of the program is given in Table 2 for the molecule 1 (Figure 1). We can notice that the main contribution to the EPN comes from the atomic core core,A except for H atoms. val,A is the valence contribution to the EPN and given per one electron, thus the EPN is calculated as 0,A = core,A + Pval..val,A from the values listed in Table 2 for each nucleus. The electrons of the atom A contributes largely to the EPN 0,A when compared to rest,A, the electrostatic potential generated by the rest of the molecule. The last column of Table 2 gives the EPN promol,A generated by isolated atoms (promolecule). The comparison of 0,A and promol,A shows that the absolute values of 0,A are higher than those of promol,A due to the covalent bonds and the charge transfer. An exception can be made to H atoms especially when their charges are importantly (for example, H16 in Table 2). With respect to the virial theorem and from equation (3), we have computed the potential energies for our set of molecules. The results are listed in Table 3. Compared to the previous V values obtained in Table 1, the relative errors do not exceed 5%. In Table 3 are also given the potential energies corresponding to promolecule i.e. the superposition of free atoms. As can be expected for non interacting atoms, the values of the potential energy for the promolecules are systematically higher. From the equation (5), we have estimated the nuclear-nuclear repulsion Vnn and the nuclear-electronic attraction Vne for the 7 molecules. Here also, Vne values are lower for the real molecules compared to those of the promolecules as shown in Table 3. The calculation of the EPN is straightforward using POTNUC program.6 It is thus possible to estimate the energy of interaction between dimers or clusters of molecules. We have carried out a


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calculation for the molecule 2 (paracetamol) which establishes two main hydrogen bonds in the crystal lattice as shown in Figure 4. The molecules are attached by one O-H…O hydrogen bond (A-B) and one N-H…O (A-C) involving the hydroxyl groups. The interaction energies (E1,2 = E1,2 - E1 - E2) between dimers are found equal to EA-B = -29.5 kcal/mol and EA-C = -53.3 kcal/mol, respectively. These values are in the same order of magnitude for the interaction energies found recently in paracetamol-oxalic acid interaction obtained from the topology of the electron density.26 In our previous study of the electrostatic properties of paracetamol, we have found that the contributions to the electrostatic interaction energy are -15.23 kcal/mol for the N-H…O (A-C) hydrogen bond and -11.15 kcal/mol for O-H…O hydrogen bond (A-B).27

■ CONCLUSIONS This work is a continuity of the use of the electric field to derive electrostatic properties of molecules in crystals.7 In the present study, we have shown that the potential energy can be straightforwardly estimated after a multipolar refinement of the experimental diffraction data. It is worthy to note that the Politzer’s approach developed here stays in the approximation of Thomas-Fermi. A recent study shows however that the results obtained for the total energies compare well with those derived from HF or DFT calculations.28 The results from the electric field and from the electrostatic potential at the nuclei (EPN) are in excellent agreement. The topology of the potential energy gives rise to a new partitioning of the molecular space with atomic basins separated by zero-flux i.e. zero force surfaces. The use of the present approach for the estimate of the total energy of interaction between molecules in the crystal lattice or for a cluster of molecules is in progress.


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■ AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. ORCID Nour-Eddine Ghermani: 0000-0001-7180-6624 Notes The authors declare no competing financial interest. This paper is dedicated to the memories of Professors R.F. Stewart and B.M. Craven of the University of Pittsburgh (USA) and Professor P. Coppens of the University of New York at Buffalo (USA), who passed away in these three last years. Their huge contribution to the field of the electron density and electrostatic potential is unforgettable. ■ ASSOCIATED CONTENT Supporting Information The Hansen-Coppens model of the electron density.

■ ACKNOWLEDGMENTS The financial supports of Université Paris-Sud, Université Paris Saclay and CNRS are gratefully acknowledged. ■ REFERENCES (1)

Guillot, B.; Viry, L.; Guillot, R.; Lecomte, C.; Jelsch, C. Refinement of proteins at

subatomic resolution with MOPRO. J. Appl. Crystallogr. 2000, 34, 214-223. (2)

Jelsch, C.; Guillot, B.; Lagoutte, A.; Lecomte, C. Advances in protein and small-

molecule charge-density refinement methods using MoPro. J. Appl. Crystallogr. 2005, 38, 38-54. (3)

XD2006; a computer program for multipole refinement, topological analysis of charge

densities and evaluation of intermolecular energies from experimental or theoretical structure


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factors.

Volkov, A.; Macchi, P.; Farrugia, L.

J.; Gatti, C.; Mallinson, P.; Richter, T.;

Koritsanszky, T., 2006. (4)

Bader, R. F. W. Atoms in Molecule; Clarendon Press, Oxford, 1994.

(5)

Bader, R. F. W.; Matta, C. F. Atoms in molecules as non-overlapping, bounded, space-

filling open quantum systems. Foundations of Chemistry, 2013, 15, 253-276. (6)

ELECTROS, STATDENS, FIELD, POTNUC: Computer programs to calculate

electrostatic properties from high resolution X-ray diffraction. Internal report UMR CNRS 7036, Université Henri Poincaré, Nancy 1, France and Institut Galien Paris Sud, UMR CNRS 8612, Université Paris-Sud, France and Université Cadi Ayyad, Morocco (1992-2017). N. E. Ghermani, N. Bouhmaida and C. Lecomte. (7)

Bouhmaida, N.; Ghermani, N. E. Advances in electric field and atomic surface derived

properties from experimental electron densities. Phys. Chem. Chem. Phys., 2008, 10, 3934-3941. (8)

Coulson, C. A.; Bell, R. P. Kinetic energy, potential energy and force in molecule

formation. Trans. Faraday Soc., 1945, 41, 141-149. (9)

Purcell, E. M. Electricity and magnetism. Vol. 2. McGraw-Hill, New York. 1965.

(10) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman lectures on physics. AddisonWesley, Reading, Mass, Volume I, II (1964); Volume III (1965). (11) Favreau. R. F. Transfer of the self energy into field energy of density E2/8π during assembly of a localized charge distribution from infinity. Can. J. Phys., 2002, 80, 1133-1144.


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(12) Hansen, N. K.; Coppens, P. Testing aspherical atom refinements on small-molecule data sets. Acta Crystallogr., 1978, A34, 909-921. (13) Volkov, A.; King, H. F.; Coppens, P.; Farrugia, L. J. On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model. Acta Crystallogr., 2006, A62, 400-408. (14) Alonso, J. A.; Cordero, N. A.; March, N. H. Electrostatic potential at the nucleus of a neutral atom related to electronic correlation energies of atomic ions. Mol. Phys., 1996, 88, 13651372. (15) Dimitrova, V.; Ilieva, S.; Galabov, B. Electrostatic Potential at Atomic Sites as a Reactivity Descriptor for Hydrogen Bonding. Complexes of Monosubstituted Acetylenes and Ammonia. J. Phys. Chem. A, 2002, 106, 11801-11805. (16) Liu, S.; Pedersen, L. G. Estimation of molecular acidity via electrostatic potential at the nucleus and valence natural atomic orbitals. J. Phys. Chem. A, 2009, 113, 3648-3655. (17) Politzer, P. Observations on the significance of the electrostatic potentials at the nuclei of atoms and molecules. Israel J. Chem., 1980, 19, 224-232. (18) Politzer, P.; Levy, M. Energy differences from electrostatic potentials at nuclei. J. Chem. Phys., 1987, 87, 5044-5046. (19) Politzer, P.; Lane, P.; Concha, M. C. Atomic and molecular energies in terms of electrostatic potentials at nuclei. Int. J. Quant. Chem., 2002, 90, 459-463. (20) Brändas E. J.; Kryachko, E. S. Quantum Chemistry: A tribute to the memory of Per-Olov Löwdin. KLUWER ACADEMIC PUBLISHERS. The Netherlands. 2003.


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(21) Clementi E.; Roetti, C. Atomic data and Nuclear data tables. Academic press, New York, USA, 1974, 14, 177. (22) Henkelman, G.; Arnaldsson, A.; Jónsson, H. A fast and robust algorithm for Bader decomposition of charge density. Comp. Mat. Science, 2006, 36, 354-360. (23) Sanville, E.; Kenny, S. D.; Smith, R.; Henkelman, G. Improved grid‐based algorithm for Bader charge allocation. J. Comp. Chem., 2007, 28, 899-908. (24) S. Portmann and H. P. Lüthi, MOLEKEL: An interactive Molecular Graphics Tool. Chimia, 2000, 54, 766-770. (25) Bentley, J. Determination of electronic energies from experimental electron densities. J. Chem. Phys., 1979, 70, 159-164. (26) Srivastava, K.; Shimpi, M. R.; Srivastava, A.; Tandon, P.; Sinhaa, K.; Velaga, S. P. Vibrational analysis and chemical activity of paracetamol–oxalic acid cocrystal based on monomer and dimer calculations: DFT and AIM approach. RSC Adv., 2016, 6, 10024–10037. (27) Bouhmaida, N.; Bonhomme, F.; Guillot, B.; Jelsch, C.; Ghermani, N. E. Charge density and electrostatic potential analyses in paracetamol. Acta crystallogr. B, 2009, 65, 363-374. (28) Nicolaï, B.; Fournier, B.; Dahaoui, S.; Gillet, J.-M.; Ghermani, N. E. Crystal and Electron Properties of Carbamazepine–Aspirin Co-crystal. Cryst. Growth & Des., 2019, XXXX, XXX, XXX-XXX. DOI: 10.1021/acs.cgd.8b01698.


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Table 1 Atomic Partitioning of the Total Potential Energy VE2 (in au, given in bold at the end of each column) of the 7 Molecules Using the Integral of E2 (equation (2))

1 O1 O2 O3 O4 C5 C6 C7 C8 C9 C10 C11 C12 H13 H14 H15 H16 H17 H18 H19 H20

-125.75 -125.16 -125.70 -127.14 -85.14 -86.19 -86.48 -84.74 -85.71 -84.04 -86.59 -86.25 -3.62 -3.69 -3.95 -4.60 -3.65 -3.60 -3.64 -3.83 -1219.46

2 O1 O2 N3 C4 C5 C6 C7 C8 C9 C10 C11 H12 H13 H14 H15 H16 H17 H18 H19 H20

-128.62 -128.89 -109.47 -87.18 -83.47 -87.06 -87.86 -89.06 -87.14 -86.24 -89.14 -3.61 -3.82 -3.82 -3.76 -3.88 -4.04 -3.81 -3.83 -3.97 -1098.66

3 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 O14 O15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 H28 H29 H30 H31 H32 H33

4

-85.64 -85.54 -86.10 -84.65 -83.78 -83.00 -83.81 -83.97 -82.91 -83.55 -83.54 -83.15 -84.95 -125.64 -121.60 -4.45 -3.55 -3.51 -3.52 -3.43 -3.59 -3.35 -3.34 -3.35 -3.72 -3.56 -3.73 -3.56 -3.53 -3.33 -3.76 -3.71 -3.26 -1406.07

S1 O2 O3 O4 O5 N6 N7 N8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 H24 H25 H26 H27 H28 H29 H30 H31 H32 H33 H34 H35 H36

-784.03 -129.90 -106.94 -127.71 -128.05 -108.44 -117.62 -111.01 -89.84 -90.61 -86.76 -89.56 -89.61 -75.32 -89.58 -89.04 -76.51 -85.60 -88.00 -87.45 -85.82 -85.53 -90.24 -3.54 -3.37 -3.40 -3.24 -3.71 -3.43 -3.81 -3.38 -3.45 -3.64 -3.49 -3.38 -3.22

5 S1 S2 O3 O4 O5 O6 O7 O8 C9 C10 C11 C12 C13 C14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 H28

-769.19 -765.09 -125.91 -125.49 -127.51 -125.85 -127.28 -124.67 -85.25 -85.86 -86.90 -89.77 -85.02 -88.05 -3.46 -3.42 -3.49 -3.50 -3.57 -3.46 -3.57 -3.59 -3.53 -3.54 -3.78 -3.76 -3.73 -3.74 -2861.94

7

6 O1 O2 N3 C4 C5 C6 C7 C8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19

-128.53 -127.70 -109.35 -86.11 -80.07 -91.13 -92.24 -92.78 -4.12 -4.12 -4.15 -3.87 -3.56 -3.43 -3.56 -3.46 -3.82 -3.57 -3.59 -849.12

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 O13 O14 N15 N16 H17 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 H28 H29 H30

-84.05 -84.95 -85.52 -85.39 -83.80 -84.12 -84.41 -84.23 -84.98 -84.24 -85.82 -88.37 -125.80 -124.00 -107.85 -106.37 -3.88 -3.79 -3.79 -3.86 -3.84 -3.74 -3.43 -3.65 -3.65 -4.17 -3.96 -3.94 -4.24 -4.23 -1538.04

-2958.20


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Table 2. Molecule 1: Decomposition of the EPN 0,A in Core core,A, Valence val ,A and Total Electronic e,A Contributions, respectively. rest,A is the Electrostatic Potential Generated by all Other Atoms on the Nucleus A. promol,A is the EPN for an Isolated Atom.  is the Contraction-Expansion Coefficient of the Electron Density and Pval is the Spherical Valence Population.12 All Values are in Atomic Unit.

O1 O2 O3 O4 C5 C6 C7 C8 C9 C10 C11 C12 H13 H14 H15 H16 H17 H18 H19 H20



Pval

0.99 0.98 0.98 0.99 1.03 0.98 1.01 1.00 1.00 0.99 0.99 0.99 1.26 1.17 1.24 1.15 1.21 1.21 1.21 1.21

6.15 6.38 6.38 6.25 3.52 4.47 3.88 4.04 4.04 4.17 4.26 4.25 0.76 0.85 0.76 0.62 0.82 0.82 0.82 0.78

core

val

e,A

rest,A

0,A

-15.30 -15.30 -15.30 -15.30 -11.34 -11.34 -11.34 -11.34 -11.34 -11.34 -11.34 -11.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.17 -1.17 -1.17 -1.17 -0.84 -0.84 -0.84 -0.84 -0.84 -0.84 -0.84 -0.84 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

-22.37 -22.60 -22.58 -22.50 -14.39 -15.00 -14.64 -14.75 -14.73 -14.82 -14.88 -14.89 -0.96 -0.99 -0.94 -0.71 -0.99 -0.99 -0.99 -0.95

-0.13 0.03 0.13 0.07 -0.48 0.07 -0.30 -0.16 -0.12 -0.06 0.08 0.07 -0.08 -0.11 -0.13 -0.16 -0.06 -0.07 -0.07 -0.06

-22.50 -22.57 -22.45 -22.43 -14.87 -14.93 -14.94 -14.91 -14.85 -14.88 -14.80 -14.82 -1.04 -1.10 -1.07 -0.87 -1.05 -1.06 -1.06 -1.01

promol,A -22.26 -22.26 -22.26 -22.26 -14.69 -14.69 -14.69 -14.69 -14.69 -14.69 -14.69 -14.69 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00


15


Page 16 of 21

Table 3. Total Potential Energy VEPN (in au, given in bold) of the 7 Molecules Using the EPN in Equations 3 and 4. “exp” and “promol” correspond to Calculations using Experimental and Promolecular Electron Densities, respectively. Nuclear-Nuclear Repulsion Vnn = (1/2)*ij(Zi*Zj/Rij) and Nuclear-Electronic Attraction Vne (equation (5)) are given in au. Relative Errors  = |VEPN - VE2|/ VE2 are Given at the Last Line.

1

2

3

4

exp

Promol

exp

Promol

exp

Promol

VEPN 2*Vnn 2*Vne

-1233.96 +1372.72 -5625.84

-1221.52

-1088.08 +1144.82 -4828.36

-1032.44

-1346.14 +1943.12 -7027.22

-1319.43



0.01

-5595.66

0.01 5

exp VEPN 2*Vnn 2*Vne

-2987.52 +2486.90 -11944.70



0.04

-4698.64

0.04 6

Promol -2974.40 -11914.08

exp -810.59 +822.18 -3535.72 0.05

-6964.88

exp -2902.35 +4228.94 -15230.00

Promol -2884.36 -15188.02

0.02 7

Promol

exp

Promol

-809.54

-1475.08 +2112.34 -7666.54

-1454.12

-3533.28

-7617.62

0.04


16

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1 2

4

3

6

5

7

Figure 1. The set of the 7 molecules used in this study: O (red), N (blue), C (green), S (yellow) and H (white).


17


Page 18 of 21

Figure 2. Projection of electron density  values on the isosurface of E2 (0.1 au) for molecule 1. The  values are from 0.00 (blue) to 0.08 eÅ-3 (red).


18

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Figure 3. Projection of E2 values on the isodensity surface of 0.1 eÅ-3 for molecule 1 (two orientations in top). E2 values are from 0.00 (blue) to 0.85 au (red). Oxygen atomic basins obtained from the topology of E2 (bottom).


19


Page 20 of 21

A

B

C

Figure 4. Hydrogen bonds (dashed lines) in a cluster of 3 molecules 2 (paracetamol). The surface in blue represents the isosurface of +0.4 au of the experimental electrostatic potential. Oxygen atoms are in red.


20

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TOC Graphic


21

Molecular Potential Energies from Experimental Electric Field and

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