The electron density and chemical bonding in organic compounds by X-ray diffraction

$Page 1: The electron density and chemical bonding in organic compounds by X-ray diffraction$
Journal of Molecular Structure (Theo&em), 255 (1992) 335-392 Elsevier Science Publishers B.V., Amsterdam

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The electron density and chemical bonding in organic compounds by X-ray diffraction

V.G. Tsirelson and R.P. Ozerov

Mendeleev Institute of Chemical Technology, Miusskaja Sq. 9,125190 Moscow (Russia)

(Received 2 July 1991)

Abstract

The principles of accurate X-ray structure analysis and its application to the study of the electron density of organic compounds are presented. It is shown how the details of the observed electron density can be associated with different features of chemical bonds. The electron density distribution in organic crystals is reviewed. The relationship between X-ray diffraction and quantum-chemical approaches is discussed.

INTRODUCTION

One of the central ideas in chemistry is the notion of the chemical-bond. By using this concept, the interactions of atoms in molecules or crystals can be looked at without having to consider the interactions of many indistinguish- able electrons. There are a number of different kinds of interatomic bond rang- ing in energy from 40 to 1000 kJ mol-‘. Bond types include covalent, ionic, donor-acceptor, metallic and hydrogen bonds; in addition there are also non- valent interactions and some specific types of interaction. Of course, such a classification is very approximate, but it enables us to divide the various interactions into groups having well pronounced special features. The ground state electron density distribution reveals such features in explicit form and for this reason the concept of electron density is at present the subject of much re- search world-wide.

There are two main approaches to investigating electron density: the first involves ab initio calculations of the electron systems studied, and the second is based on accurate X-ray diffractometry. The two approaches can give the same or different results for the same compound. Thus, as information on the possible influence of a crystal environment on a molecule within it is obtained by comparing the results obtained for the free molecule and those obtained for the same molecule in the crystal matrix, it is important to evaluate the reliability of the methods used. If the quantum-chemical approach is well known

0166-1280/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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and accepted [ 11, the validity of accurate X-ray diffraction results with regard to the electron density must be investigated and analysed. This is one of the aims of the present review.

Let us look at first at the physical principles which permit us to use the concept of electron density. Quantum theory considers a molecule or crystal as an equilibrium system of electrons and nuclei at a given temperature. In order to separate the movement of the electrons and the nuclei, the Born- Oppenheimer approximation [ 21 is usually used. The equilibrium position of each nucleus can then be considered within the framework of this approximation. The multi-dimensional vector, R,, which characterizes the equilibrium nuclear configuration corresponds to the potential energy minimum for the moving nuclei, @(R,,) . The movement of each nucleus in the crystal can be considered as a vibration about its equilibrium position and can be described by its displacement value. This representation is valid if the energies of the ground, E0 and excited E, electron states differ significantly. Here we consider only this situation and neglect the movement of the nuclei induced by the mixing of electron states which requires a quite different approach [3]. Ac- cepting that electrons “rigidly” follow the nuclei, the wavefunction of the whole system can then be written as the product of an electron part vn(r,R,) and a nuclear part, xn (u), where u is the multi-dimensional vector of the nuclear displacement from the equilibrium position. The mean values of the zero nuclear vibrations, [ h/BMc0~] ‘I2 are much less than the characteristic distances between the minima of the function @(R,). Thus the uncertainties that arise as a result of separating the electron and nuclei movements can be neglected. Therefore, by using the Born-Oppenheimer approximation the electron states of molecules and crystals can be studied with high accuracy by looking at the electrons moving in a field of motionless nuclei. In addition to its use for elucidating symmetry relationships and other attributes of chemistry and crystallography, the Born-Oppenheimer approximation can also be used to represent a molecule or crystal as a system of atoms and a network of chemical bonds. Quantum-chemical calculations of electron structure have mostly been performed at this level of approximation.

The next step in simplifying the solution of a many-electron system in the ground state is to move from the wavefunction Y( { rLsi}R,,), which depends on the coordinate ri and spins si of all N electrons, to the one-particle density function p ( r,Ro). By using the local electron density operator e CS( r - ri) and taking into account the indistinguishability of electrons, it is possible the p function can be introduced as [l]

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p(r,&)=e . ..e(rI.sl ,... rN,sN;RO)

x xa(r-rA ‘Y,(r,, sl, . ..rN. SN, R. )dr,, ch, . ..drNdsN

The integration over spin variables in this equation is equivalent to the summation over all spin directions, bearing in mind the negative electron charge. The electron density is the average of the inter-electron interactions of all N electrons in the ground state, and depends only on the coordinate of the point in real physical space. The “structure” of the p function is much more simple than the wavefunction and, therefore, the size of the problem of the description of the electronic structure decreases significantly.

During the transition from the wavefunction to the density function some information is lost due to averaging. Nevertheless, as shown by Hohenberg and Kohn [4], the electron density is sufficiently informative to describe completely the electronic ground state of a system.

It is important to note that the Fourier transformations of the electron density are equivalent to the X-ray coherent structure amplitudes, F(H). There- fore, it is possible to restore the electron density distribution in a crystal to the level of accuracy required. For this purpose it is necessary to separate the coherent part of the X-ray scatter pattern and the successive Fourier summation of the structure amplitudes. The unknown part of the experimental information-i.e. the phases of the structure amplitudes-can be calculated by using a suitable crystal structure model, by so-called direct methods [ 51, or by multi- wave X-ray diffraction [ 61. Thus, it is possible to consider the electron density function p (r ) as an observable quantity.

Until the 197Os, the data produced by X-ray diffraction measurements were not used exhaustively. For instance, the positions of the electron density maxima were usually taken as the main result of an X-ray experiment. These maxima were identified as the position of the nuclei averaged over time. But the experimental information available from X-ray diffraction studies is much wider than this. The X-rays are scattered by the whole electron cloud of a crystal and the experimental data can be used to obtain detailed information on the features of the valence electrons and the vibrational parameters of the atoms. A high level of accuracy of the X-ray experiment, a large amount of obtained data and non-traditional methods of data treatment are needed for this purpose.

Thus, the representation of a crystal as a “sea” of continuous electron den-

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sity with merged nuclei vibrating about their equilibrium positions was adopted as a model. This model allows many of the physical properties of a system to be calculated and, in addition, can be used in treating X-ray diffraction data. Thus this model has been developed intensively in recent years in order to facilitate the investigation of electron structures via X-ray diffraction experiments. X-ray diffractometry enables the necessary accuracy to be attained. The application of this approach to the study of the chemistry of organic crystals is discussed below, and the results of X-ray experiments and quantum- chemical calculations are compared.

We restrict our discussion to the well-known technique of computer con- trolled four-circle X-ray diffractometry, and leave aside the application to this problem of the modern techniques of y-rays and synchrotron radiation diffraction (see refs. 7 and 8).

These and other aspects of the methods used to determine electron density, including applications and results, are given in detail in the book by Tsirelson and Ozerov [ 91.

PRINCIPLES OF MEASURING ELECTRON DENSITY BY X-RAY DIFFRACTION

The ideal X-ray diffraction experimental situation is as follows. A monochromatic X-ray beam of wave vector Ix,, (o= ckO x 10” s-’ ) falls onto a small single-crystal specimen. After the scattering period of 10-l’ s the diffracted radiation is recorded at some angle by an X-ray photon detector. The scattering may be represented as the absorption of an X-ray photon from an incident beam and the emission of another photon in some direction k,.

Let us consider first the scattering at a one-electron atom [lo]. The X-ray scattering cross-section of an electron is about lo7 times larger than that of a nucleus and, therefore, only the interaction of the X-ray radiation with electrons need be taken into account.

The wavefunctions of the stationary states of the “electron + radiation” system prior to interaction is the product of the electron $8 (r)exp( -io+) and radiation ] lb, Okl, . . . ) wavefunctions. After interaction the total wavefunction may be represented as a superposition product of these two components. The coefficients of expansion c, represent the probability of scattering in particular directions. If the wave-atom interaction is weak, the coefficients can be determined by using the non-stationary perturbation theory. For a single scattering process on a one-electron atom in the ground electronic state

dt’ exp(io,Ot’)HL,(k,, ki, t’)

This is the well-known first Born approximation of perturbation theory [ 111,

where the frequency con0 = (E, - Eo) /fi characterizes the change in the energy due to radiation, and Hi, is a matrix element of an interaction hamiltonian, which in the non-relativistic approximation has the form

(2)

where A is the vector field potential and@ is the electron momentum operator. The first term in eqn. (2) is associated with the X-ray-photon absorption or

emission processes of the system. These processes lead either to ionization (internal photoeffect) or excitation (phosphorescence or fluorescence) of the atom, depending on whether the final state belongs to a discrete or continuous spectrum during absorption [ 121. Both phenomena influence the probability of photon scattering occurring in the direction k, and must be properly taken into account.

In order to describe the scattering process the vector field potential A can be written as the Fourier expansion over plane waves, each of which corresponds to an X-ray photon of some particular wave vectors [ 10,111

+ci+&,pu) exp[-ii(klr-o~t)l}e(kl,~) (3)

where co is the electric constant, e( kl, ,u) is the X-ray photon polarization vector, V, is the normalizing volume, and ci and ri + are the photon-absorption and photon-emission operators, respectively. The matrix element associated with the scattering is

(4)

where ro=e2/4x~Omc2, q= k,-k, and

Lo(q)=ICWexp{-iqr}&(r)dV (5)

is the atomic scattering amplitude. Equation (1) can be rewritten in the form

c,(q,t) = -i 2nroC2 VO(~O~l)” Jno(d [e(ko,pu)-dkl, v)l (‘3)

x exp{i[o,o - (o. -o,)]t’}dt’ 0

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The total probability of an X-ray photon scattering in the direction k, during time t is proportional to CcX(q,t). One term of this sum includes the

n quantity

f~o(P)={S[gS(r)12e-~rd~2 (7)

which describes the elastic coherent X-ray scattering on the electron density of ground state [@(r ) 1”. In this case o,=O and oo= ol. The other terms describe the noncoherent (Compton) scattering associated with electron transitions.

For a many-electron system, A’ must be replaced by C& and A by CA (ri,t)

(the summation is over all electrons). Equations (4)-(i) must also be’changed accordingly and the many-electron wavefunctions wn introduced into these formulae in place of the #i. Taking into account the one-electron character of the interaction operator H’ , which depends only on the electron coordinates, for coherent scattering

foe(q) =p(r)eeiq’dV (3)

If the X-ray scattering takes place not on an atom but on a multi-nuclear, many-electron system, the amplitude foe(q) becomes the static structure amplitude F(q,R,), which is connected with the electron density at the nuclear equilibrium.

Thus, in the Born single-scattering-process approximation, the coherent part of X-ray scattering has one-electron character and contains information on the electron density of a system in the ground state. Hence, information on electron density from the X-ray scattering may be restored. To solve this problem in practice, one should first describe the scattering as close to the real experimental situation as possible, and then take into account some factors which influence the measurement. In X-ray diffraction experiments the incident radiation is non-polarized and, strictly speaking, non-monochromatic. The photon counter has finite dimensions and registers all photons, scattered within a small solid angle dL2, corresponding to some range of wave vectors close to kl. The summation Cc: over k and over all possible polarization states

gives the probability of X-rai photon scattering (and recording) into the solid angle per second. This value can be related to the scattered-radiation intensity

where B. is the specimen-counter distance, p is the polarization factor [ 131 and IO is the incident-radiation intensity. This expression slightly underesti- mates the measured intensity because the non-elastic-scattering contribution

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is not taken into account. However, this contribution is nearly constant over the integration interval (approximately lo) and can be eliminated when processing the experimental data.

The atomic thermal motion, which is in general anharmonic, must also be taken into account. The measured intensity, 1, is the average value for all vibrational, rotational, etc., states. In order to introduce the thermal motion into the scattering description, the static electron density p ( r,Ro) can be presented as a sum of “atomic” (not necessarily spherical) fragments, Cpr (r - r,). It is

assumed that each fragment (pseudoatom) is rigid and adiabatically follows its own nucleus. In other words, each pseudoatom is assigned its own probability density function (PDF; pr(up) ) which describes its deviation from its equilibrium position. Then the average dynamic electron density over all nuclear displacements is given by

P(r) = $A r--l,--U,b,(u,)dur (10)

The structure amplitudes thus takes the form

F(q) = Cf,(s)T,(q)exp(iqr,) p

(11)

In general, fp and T,, are complex functions. The atomic scattering amplitude ffl describes the X-ray scattering on a pseudoatom ,u. The factor T,, = exp ( -AI,) is the Fourier transform of the PDF and is known as the “temperature factor”, and T “, is the Debye-Waller factor [ 141. The factor Mp is dependent both on the approximation of PDF, pr and on the order of perturbation theory used in describing the thermal motion. The rr in eqn. (11) is the equilibrium position vector of atom flu.

The superpositional crystal structure model is not equivalent to the inde- pendent-atom model (as it is often believed) because the atomic PDF are marginal relative to the multi-particle PDF, which correspond to the initial hamiltonian. The latter fully take into account correlations in the atomic thermal motions, whereas the set of PDFs pfl only accounts for the motion in average. For an anharmonic crystal the superpositional structure model applied is equivalent to substituting the real crystal potential O(u) by the superposition of the atomic effective potentials. Therefore the marginal atomic PDFs should be considered as an approximation to the “correct” PDF. Fortunately, both approaches depend in the same manner on the scattering vector q and temperature [ 151. This is why the simple superpositional structure model appears to be so effective in determining the atomic thermal and positional parameters from X-ray diffraction experiments, although some exceptions can be found WI.

Each pseudoatom vibrates in the force field produced by its neighbours. At

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the same time it is a non-separate and consistent part of the whole crystal. Therefore a pseudoatom should be considered as “an atom in the thermostate” and be described mathematically by the density matrix [ 11, 171. The vibrational anharmonicity can be taken into account in the potential energy operator a(u) in the nuclear vibration hamiltonian by the addition of cubic, quar- tic, etc. (over the displacements), terms. The anharmonic effects are generally very weak compared with the harmonic ones. Therefore, it is convenient to use stationary perturbation theory in order to calculate the scattered intensities, averaged over the thermal motions [ 181. This theory enables us to preserve the general harmonic treatment scheme with the thermal-motion decomposition over the normal coordinates [ 181. Bearing in mind eqns. (9) and ( 11) for a crystal, then [ 19,201

(12)

Here r,, - -R, + R, describes the position of the ,Ah atom in the nthe unit cell. The cross terms AM,, and Dz’ account for the atomic correlations in the thermal motions. The term AM,, is purely anharmonic in origin; it can influence the scattering intensity, both decreasing and increasing it depending on which interaction processes of the phonons it is associated with. It has been shown [ 201 that within the one-particle thermal-motion representation, exp[-AMdM,,,(q)]xl.ThematrixDPP “‘5 is positively defined and so the contribution of the term exp [D $ (q) ] leads to an over-estimation of the elastically scattered intensity [ 151. The eigenvalues of this matrix are much less than unity, and thus by expanding the exponent in the series with restriction to the first-order term it is easy to arrive at the expression

I(q)=Cf :(a)f,~(a)T:,(q)T,(q)C [l+q$(d+...l

=Z(q) + c &(a)

nn’

k=O

(13)

In relation to the thermal motion, the elastic part of the intensity IO (zero- phonon scattering) is separated from the intensities of the one, two, etc., phonon processes (I,, I2 etc.) In the diffraction by a single crystal, lo(q) has sharp maxima at the reciprocal lattice points, i.e. at the Bragg reflection positions with q = 2nH, where H is the reciprocal lattice vector. Thus

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I,(H)=N2Cf,(H)f*,,(H)T,(H)T;,(H)exp[i2~H(R,,-R,)l PP

=N21F(H) I2 (14)

where N is the unit-cell number in the crystal. For other terms in eqn. (13) the diffraction conditions can be represented by the equation q = 27rH+ C k,,

where k, are the wave vectors of the vibrational modes, and the index r refers to the one, two, etc., phonon processes. The acceptable values of the k, vectors lie inside the first Brillouin zone. Therefore, the intensities of Ik (lz > 1) are spread diffusely over the reciprocal space with only I, (q) centred on the Bragg position. Nevertheless, for organic compounds the parasitic contribution of I, (q) to the total reflection intensity can reach 40-60% [ 10, 13, 151. There- fore, this contribution should be subtracted from the total intensity.

Let us now turn to the structural data which can be derived from the X-ray experiment. The complete information on the crystal structure-atomic coordinates R,, temperature parameters MP and electron density p( r ) is contained in the structure amplitudes F(H). The latter amplitudes are included in the expression (eqn. (14) ) for the coherent intensities I,,(H). The I(H) values can be measured experimentally using X-ray diffractometry [ 211.

In order not to lose the information about F(H) during the determination of I0 (H) , some very important factors should be taken into account, including: radiation polarization and absorption, anomalous dispersion (the second- order effect in the Born perturbation theory [lo] ), thermal diffuse scattering (TDS ) , two- and many-fold scattering processes, and incident-reflected wave interference (extinction). Only if all these factors are considered can the diffraction information give an adequate description of the dynamic electron density p(r) with all the structural characteristics included.

Estimations show [22] that in the scheme described above the most inac- curate aspect is the decomposition of the infinite crystal electron density into rigidly vibrating pseudoatoms. However, the relative uncertainty in the intensities resulting from such a decomposition is only about 0.1% [ 231. Thus the most important factors in obtaining reliable structural and electronic parameters are the statistical precision and the completeness of the diffraction data, the correction of the data for secondary diffraction effects, and the modelling of the crystal structure.

ACCURATE X-RAY DIFFRACTION STRUCTURE ANALYSIS

Single-crystal automatic four-circle X-ray diffractometers enable us to measure X-ray intensities on a relative scale with a statistical accuracy of about 1% [9,13, 211. This is done by making a series of measurements, usually for all Laue symmetry-equivalent reflections. The agreement within each group

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of equivalent reflections can be used as a check of the precision of the measurement. The intensity of a few chosen reflections at regular time intervals are measured during data collection in order to obtain a uniform scale for all observed reflections. It is important to note whether a crystal suffers damage from X-rays during measurement [ 241.

Modern X-ray diffractometers are equipped with attachments for making diffraction measurements at low temperatures (50-120 K [25] and in some cases lo-20 K [ 26, 271) Low-temperature experiments reduce the negative influence of TDS on the measured reflection intensities and separate the effects of thermal vibrations and the aspherical nature of the atomic electron shell. For example, the measurement of 2,5-diaza-1,6-dioxa-6a-thiapentalene, C3H2N202S, at 11 K [27] allowed the electron density distribution in the regions of short non-bonding S - - 00 contacts to be studied in detail and the p orbital participation in the interaction of these atoms to be identified. This was not possible in an earlier study of the same compound done at 122 K [ 281 where the effects were masked by thermal movement. Low-temperature diffractometry also opens the way to the investigation of crystal structures and the properties of compounds which are liquid or gaseous at room temperature.

Crystal specimens for measurement are selected very carefully by means of analysing the diffraction peak profiles in different directions in reciprocal space. Usually the specimens are prepared as spheres: this facilitates the corrections that must be made for other diffraction effects and reduces the deviation of the intensities due to anisotropy of the scattering properties of the crystal. The measured data are then corrected for the above-mentioned effects which dis- tort the coherent elastic intensities (as mentioned above). For organic crystals, absorption and anomalous scattering are negligible. At the same time, TDS and extinction are of great importance in the analysis of electron density. When TDS is not taken into account, the positions of the maxima of the electron density do not change, but their heights do. Accordingly, the atomic coordinates are altered slightly by this effect, but the values of the atomic thermal parameters and the electron density in interatomic space are changed significantly. It should be noted that TDS decreases linearly with decreasing temperature. The correction for TDS can be calculated if the elastic constants of the crystal are known [ 9,15,20,29].

The interference effect which may have a significant influence on the intensity of low-angle reflections (i.e. reflections which correspond to delocalized valence electrons) is extinction. Extinction depends greatly on the character of the defect distribution in a crystal, and may be both isotropic and aniso- tropic. There are various methods for making corrections for extinction, based either on the block-mosaic model of a crystal [ 301, on a statistical model [ 31, 321, or on a combination of the two [33]. Some extinction parameters used in these methods can be introduced into a crystal model. Each approach has its own advantages and disadvantages, but none of the models provides a reliable

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correction in the case of complicated interference effects [ 91. When the extinction correction is incomplete due to defects in the model used, the picture of the electron density in the region of chemical bonds and near electron lone pairs may be distorted [ 9,341.

There are the well-known procedures and computer programs available for processing the experimental data [ 9,131. The precision of correction calculations is, in simple cases, about l-2% [ 221, which is of the same order as the statistical precision of the measurement itself. The accuracy of the calculated Fobs (H) values is thus about 1%.

In order to extract further structural information and to calculate the Fobs(H) phases, the values of Fobs(H) are approximated by using their theoretical an- alogues. The positional, thermal and electronic parameters are obtained from X-ray diffraction data by optimizing the crystal structure model. The criterion usually used is that of the minimum error of survey

& [F,,I,~(W -F(H) I2

It is clear, that the parameters obtained are model dependent. This dependency can have a noticeable influence on the values of the parameters and hence adequate models for f,, and Tp are needed. In accurate studies the Gram-Char- lier expansion [35] and Edgworth expansion [36] of Tj(H) or the Taylor expansion of atomic effective potentials [ 15,291 are frequently used. These ex- pansions allow the anharmonicity of the thermal atomic vibrations to be taken into account. The electron clouds of the atoms described by the fj (H) functions in the simplest model have been approximated by the superposition of non- interacting spherically averaged atoms. The atomic scattering factors obtained from accurate quantum-chemical calculations for such a model have been tabulated [ 371. When symmetry restrictions are neglected, errors in the coordinates for peripheral atoms in molecules, atoms with electron lone pairs, etc., are inevitable within the framework of such model. For example, the bond lengths obtained from X-ray diffraction data differ from those obtained from neutron-diffraction data by 0.005-0.015 A, and the values for bonds containing hydrogen differ by 0.1-0.2 A [ 381. The thermal atomic parameters are also distorted and the calculated phases of reflections contain errors.

In order to decrease the errors in the structural parameters that result from the inadequacy of superposition of the spherical atoms, the model is optimized within the high-angle approximation [ 391. In this case not all the measured X-ray reflections (for example, for MO K, radiation up to (sin0/3L),,,= 1.4 A-‘, where 28 is the scattering angle) are used, but only the reflections in the high-angle region with sine/d> 0.8-0.9 A-‘. These reflections contain information mostly about the inner-shell (core) electrons, for which the spherical model is quite good. Although the high angle approximation cannot compensate for all the defects of such a model, the mean-square deviations in the

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atomic coordinates obtained are a( MJ R 0.002 A besides hydrogen atoms and the deviations in the harmonic vibrational parameters are about 3-4%. The errors in the anharmonic vibrational parameters are significantly larger (up to 30-50% ).

The spherical atom model can be improved by including an additional scale factor K in the radial part of the valence atomic electron distribution functions [ 401: R(r) + R ( KR). For an atom only two parameters (the population of the valence-electron shell (P,) and K~) are needed to describe the partial interatomic charge transfer and the expansion or contraction of the atomic valence shells in a crystal. In this case the scattering factor for each atom is given by

f(h) =p f (h) +~“,1f”.¶,(~l~) core core (15)

where P,,, and PVd are the core- and valence-electron populations, respectively, and fO, and fvd are the corresponding scattering factors normalized to unity; h = sinkJ/A. Using this K model it is possible to estimate roughly the atomic charge values; however, the model cannot describe all the peculiarities of the electron density in the internuclei space.

In order to take into account the asphericity of the atomic electron shells in a crystal, of the atomic electron density is expanded in a series of real combi- nations of spherical harmonics ylrn t (multipoles)

where Rfm are the radial functions of the atomic electron density; CL are the population parameters of the expansion terms, and 1 and m are the orbital and magnetic quantum numbers. The parameters of the multipole model are the coefficient CL and scale factors &’ or exponential factors r!jP of the radial functions R/& Pexp ( - Q,). By calculating these parameters the electron density in a crystal unit cell can be calculated in analytical form (eqn. (16) ) .

There are several multipole models [ 41-451, all of which generally lead to the same results with regard to the details of the electron density [45,46].

The multipole models give greatly reduced errors in the positions of non- hydrogen atoms. For hydrogen atoms, neutron diffraction data may be used in determining the structural parameters [ 381. The accuracy of the determination of parameters within the multipole model decreases with increasing 1. For example, for hexamethylenetetramine [ 451, the error in the population of the octupole (1~3) is 15%, and the error in the hexadecupole (1=4) is about 25%.

An additional parameter in all crystal models is the scale factor, which allows the structural amplitudes to be calculated on an absolute scale. The accuracy of the scale factor has a strong influence on the electron density near the nuclei; its influence on the electron density in interatomic space is very much less.

It is necessary to mention that the refinement of the crystal structure model

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is not a trivial problem. The refinement is strongly complicated by the correlation between the refined parameters and the statistical errors in the diffraction data make the refinement even worse. There are special methods of obtaining correct values of parameters under such conditions [ 471.

The electron density in a crystal unit cell is usually calculated using a Fourier series summation

p(r) =$&F,b,exp( -2ntHr) (17)

where V is the volume of the unit cell. The wavelengths of the X-rays are usually about 0.5 A and the resolution of p( r ) is about 0.2 A. The series (eqn. (17) ) is always incomplete because of the limited number of reflections measured, and this results in a truncation error in p (I:).

Thus, it can be seen from the above discussion that the final electron density function p (r ) (eqn. ( 17) ) can be considered as an experimental value which has been subjected to many corrections. It consists of the large maxima of the vibrating atoms and the small details of the interatomic space. In order to reveal the characteristics of p, which are interesting with regard to chemical bonds, special methods are used.

The most well known method of studying chemical bonds is to calculate the deformation electron density

@=P- CPi (18)

where p is the experimental or theoretical electron density and Cpi is the electron density of non-interacting spherically averaged atoms, placed in the same positions and vibrating in the same way as in the molecule studied (this en- semble is called the “promolecule” or “procrystal” [41] ). The quantity Sp characterizes the redistribution of electrons that occurs when a molecule or crystal is formed from non-interacting spherically symmetrical atoms. The redistribution involves atoms being moved from infinity to their equilibrium positions. The calculation of dp is only one way to extract chemical information from the total electron density p. Other types of difference function are also possible, but these are not discussed here as the use of the spherical atom reference state removes the dependence of atom orientation in the promolecule. If X-rays are used eqn. (18) can be re-written as

@(r) =&W@G,.(H) -WU)exp(-2xiHr) (19)

where MH are regulating multipliers (see below). The positional and vibrational parameters needed to calculate F(H) can be obtained from the properly refined crystal structure model or from neutron diffraction data. The sum (eqn. ( 19) ) includes low-angle reflections which contain information about the dis-

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tribution of valence electrons. The truncation error in the Fourier series (eqn. (19) ) is much less than that in the series given by eqn. (18), but is still important. For example, if the structural amplitudes in eqn. (19) correspond to sine/A < 0.8 A-‘, then the truncation error near the nuclei can be up to 1.0 e A-” [48]. Thi s value is lower in the interatomic space. When the upper limit of the series in eqn. (19) is about 0.9 A-‘, then the truncation error in the area of chemical bonds is not more than the random Sp error. In the region of electron lone pair sites the Sp values can be underestimated by 0.1 e A-” because of the truncation effect.

It has been pointed out that Sp in eqn. (19) is the difference between two large values, both of which contain random errors. This results in an instability of the Fourier series summation. In order to improve the reliability of the Sp calculations, regulating multipliers MH are sometimes included in the sum (eqn. (19) ). These multipliers depend on the signal/noise ratios of the structural amplitudes [ 47,491. The use of the filtration procedure allows more accurate calculation of the Fourier series, preserving only the statistically significant details of the electron density.

Thus, the reliability of the Sp maps calculated from X-ray diffraction data depends on the statistical accuracy and completeness of the experimental data, on the reliability of the data correction according to the diffraction theory used, and on the adequacy of the crystal structure amplitude phases. The adequacy of the model used is very important for non-centrosymmetric organic crystals in which the uncertainty in Sp may be increased by up to 0.1 e A-” if the weak reflections are in the set of {F,,,J [ 501. The use of the multipole model significantly reduces this error. If only random errors are present in the diffraction data, then the error in the chemical-bond region of the Sp maps is about 0.04 in centrosymmetric and 0.06 e A-” in non-centrosymmetric structures. The filtration procedure decreases these errors by a factor 1.5 [ 91. Such accuracy is sufficient for the experimental study of the details of chemical bonds in crystals.

The methods used for the quantum-chemical calculation of the electron densities in molecules and crystals have been discussed extensively in literature [ 1,9,48, 51-541. In the following we describe briefly the main features of ab initio calculations which are nessessary for estimating the quality of reported theoretical electron densities. The main methods used for such calculations are the Hartree-Fock method, the configuration-interaction method [53, 55, 561 and different variants of density functional theory methods [ 9,54,57]. In latter case the local density approximation is usually used to describe the exchange and correlation potentials [54]. Solid-state-theory methods, such as the augmented plane waves method [ 581 and linearized augmented plane waves methods [ 591, are also used. The electron density is obtained self-consistently by solving the appropriate equations; in solving the equations the linear combination of atomic orbitals (LCAO ) approximation is often used in different

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forms. It is now established reliably enough [9,51,53] that minimal basis sets do not allow adequate deformation electron density maps to be obtained, even for simple molecules. Such basis sets give under-estimated Sp peak heights for bond lines and over-estimated ones for electron lone pairs. Satisfactory quality Sp maps are obtained if double-zeta basis sets with polarization functions are used for the calculations [9, 51, 531. If these functions are approximated by gaussian functions [ 601, it should be taken into account that Gaussian functions do not satisfy the Kato asymptotic quantum theory condition at r+O [ 611. Hence, in this case the electron density is not reproduced well near the centre of the atom, and this region should be excluded when gaussian functions are used.

The analysis of electron densities obtained with different theoretical methods shows [9,48,51,53] that the uncertainty in the chemical-bond region due to defects in the methods of calculation is not more than 0.1 e A-“. For small molecules calculations done with an extended basis set lead to errors of about 0.02-0.04 e Am3 in this region.

CHEMICAL-BOND REPRESENTATION VIA THE ELECTRON DENSITY

The formation of chemical bonds, which results in a stable system of electrons and nuclei, is the result of the mutual penetration and interaction of the atomic electron clouds, the transfer of electrons from atom to atom, and the specific modification of the electron density between nuclei and near nuclei [ 621. These processes are governed by the space limitations that result from the Pauli principle. The analysis of the electronic structure of molecules and crystals requires an understanding of the role of the electrostatic interaction of atoms with mutually penetrating electron clouds and of the quantum me- chanical effects and the relationship between these factors, and the constructive and destructive interference of atomic wavefunctions. As will be shown later, Sp maps can be used to elucidate this problem to some extent, because they show the integral effect of the electron density changes.

Covalent chemical bonds are usually classified on the basis of the symmetry properties of the corresponding molecular orbitals (MO). Such a classification is only strictly correct for diatomic molecules, but is used for multi-atomic molecules as well. Because of the limitations imposed by the Pauli principle, the formation of single 0 bonds and double x bonds between light atoms in organic molecules is possible. One may expect that the electron density will be concentrated in the bonding regions of c and K MOs where there is an in-phase combination of atomic orbitals (AOs) realize, and be depleted in empty antibonding MOs and near MO radial nodal surfaces. As the principal quantum numbers of the bonded atoms increase, the number of nodes in the radial parts of the o orbitals also increases. Accordingly, the electron density between

350

atoms is decreased and the electron density in regions of IL, tr bonds, etc., is increased. The electrostatic field thus produced stabilizes the system.

In some regions of fully occupied anti-bonding MOs, the electron density will also be increased.

Molecular regions of constructive interference coincide with the regions of in-phase A0 overlap, and regions of destructive interference coincide with regions of out-of-phase A0 overlap. MOs containing lone pairs of electrons are quite well localized and their electron density is higher than that of free-atom orbitals. When atoms of different types interact, polar covalent (partly ionic) bonds are formed with a displacement of the electron density towards the more electronegative atom. The extent of displacement depends on the type of atoms interacting and on the structure of the whole system.

Some general comments about the interpretation of deformation electron density maps may now be made. Excess electron density (areas of positive 6~) are shown on the maps as peaks in the regions to which electrons are shifted during chemical bond formation: these are, particularly, regions of in-phase overlap of atomic wavefunctions, regions containing electron lone pairs, and, in the case of predominantly ionic bonds, the sites toward which the electrons are “displaced” from the internuclei space. Areas of negative Sp are associated with the anti-bonding regions of MOs and with regions where the interference of AOs and electron delocalization results in weaker changes than, for example, the superposition of the “rigid” atomic clouds.

If the MO LCAO approximation is used, then a comparison of the spatial characteristics of the valence AOs with the features of the Sp map can give a qualitative interpretation of the map in terms of orbital representations. Such information is, however, only approximate because an MO is not an exact superposition of a few atomic orbitals. Nevertheless, such a comparison not gives only a visual picture of the well-known features of the chemical bonding but also really new information about the bonding.

By examining the features of the Sp in the molecular space and its position with respect to the interatomic regions, it is possible to establish the character of the chemical bonds. Thus o bonds are positive Sp seen as areas of positive Sp with maxima between and behind the bonded nuclei. Covalent a- bonds are indicated by peaks located outside the bond lines. The Sp peak maximum of an ionic bond is displaced towards the more electronegative atom, in accordance with conventional theory, etc. Thus an intuitive picture can be extracted from Sp maps. However, it is important that the details of the electron density on these maps not only characterize the diatomic interactions, but also reflect the presence and the nature of multi-atomic interactions. So, in molecules where neighbouring fragments have a great mutual influence on one another, displacement of the dp peaks away from the internuclei vectors may be explained as the presence of stresses in the chemical bonds (bent bonds). There are other

351

factors which result in deviations from the simple picture, and these are discussed below.

The direct approach to analysing electron density from a chemical-bond point of view is Bader’s quantum topological theory of molecular structure [ 63-651. Here we state briefly some points of this theory, which are pertinent to the further discussion.

The chemical structure of a molecule can be unambigously described by determining the number and type of critical points in its electron density p (r,R,). Critical points are points where the gradient of p vanishes. The second derivative of p, evaluated at the position of the critical point r, defines a real sym- metric (3 x 3) matrix known as the Hessian matrix. The principal curvatures of p at r. i.e. the eigenvalues of the Hessian & define the rank p and the signature q of the critical point (p,q). The rank of the critical point is given by the number of non-zero eigenvalues, and its signature is given by the algebraic sum of their signs. There are only four possible non-degenerate critical points inp: (3,-3), (3,-l), (3,l) and (3,3).

The critical point of type (3, - 3) is associated with a local maximum in p which can occur only at the position of a nucleus (see also refs. 66 and 67). The collection of all the integral curves or

r(s) = r,+ I

gradient paths

’ Pp [ r (t),R,,]dt, which are solutions of the differential equation 0

dr (s) /ds = Vp [r (s ) ,Ro] for some initial value r (0) = ro, and which will, in general, terminate at a given nucleus, defines a region in the space of the molecule which is the basin of that nucleus. Thus, the nucleus acts as an attractor for its basin. The union of an attractor and its basin defines an atom in the molecule. The nucleus is the only three-dimensional attractor in the basin; therefore the space of the molecule is partitioned into disjoint regions, each of which contains only one nucleus. Two adjacent basins are separated by an interatomic surface, S (r ) , through which the gradient vector field of p has zero flux

VP(r)-n(r) =O; reS(r)

where n (r ) is the unit vector normal to the surface at r.

(20)

Due to its zero-flux properties, an interatomic surface is generated by the gradient paths which terminate at the (3, - 1) critical point contained in the same surface. This is called the bond critical point (rb) . In fact, the eigenvector of the positive curvature at rb gives the initial direction of two gradient paths which terminate at two neighbouring nuclei, thus defining a bond path. Along a bond path p there is a maximum lateral displacement. The existence of a bond path provides the necessary and sufficient condition for bond formation. The network of bond paths for a molecule in a given nuclear configuration defines the molecular graph. All neighbouring nuclear configurations which

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have equivalent molecular graphs belong to the same structural region; this is associated with a single stable molecular structure.

The critical points of the type (3,l) and (3,3) also have chemical meaning. The eigenvectors associated with the two positive eigenvalues of the Hessian matrix evaluated at a (3,l) critical point generate a surface in which the (3,l) critical point is a minimum inp. This point is called the ring critical point. The eigenvector associated with the three positive eigenvalues of a (3,3) critical point, a local minimum in p, generate an infinity of gradient paths which orig- inate at the critical point and bound a closed region of space, i.e. a cage. A (3,3) critical point, called a cage critical point, is common to all the interatomic surfaces of the atoms forming the cage.

The cyclopropane molecule (C&H,) is an example of a system having the three types of critical point (Fig. 1). The (3,3) critical points give the positions of the nuclei. There is a (3, + 1) critical point at the centre of the bonded ring of carbon atoms. Three (3, - 1) bond critical points are situated between pairs of carbon atoms. The boundaries of the three carbon atoms meet at the (3, + 1) critical point. The unique axis of this point is the intersection of the boundaries of the basins of the atoms forming the ring.

H

I “\ /'\H

/\cAH I 'H

(a) (b)

Fig. 1. The structure (a), the electron density (b), the gradient vector field @‘p(r) (c) and the Laplacian of the electron density (d) in the plane of the carbon nuclei in the cyclopropane molecule C&H6 [98].

It has been found by means of quantum chemical calculations [ 68-701 that the value of p at the bond critical point rb correlates with the bond path length L. For the C-C bond this correlation is well described by the linear relationship

p(rb) =aL+b (21)

The p(rb) values can also be correlated with the Lewis bond order n through the relationship

n=ew{A[p(rb) -Bl) (22)

The coefficients a, b, A and B depend on the basis set used in the quantum- chemical calculations. Examples of the relationships described by eqns. (21) and (22) for C-C bonds are given in Figs. 2 and 3.

The asymmetry of the charge distribution in a plane perpendicular to the bond path is reflected in the ellipticity of the bond E = (A JA, ) - 1 (the negative eigenvalues pi are listed in increasing value). The quantity E is a measure of the deviation of the electron density from cylindrical symmetry and, therefore,

t ?g(e.Aw3)

I I c

1.2 1.3 1.4 1.5 R(A)

Fig. 2. The relationship between the electron density at the (3, - 1) bond critical point position r, and the internuclear distance Re (Hartree-Fock STO-3G calculations) [ 68-701.

Fig. 3. The correlation between the C-C bond order, n, and the electron density values pb at the (3, - 1) bond critical points (Hartree-Fock STO-3G calculations) [ 701.

can be correlated with the amount of K character of a bond. For simple bonds such as the C-C bond in free molecular ethane, 1 1 and AZ are obviously degenerate and E is zero. If 6 is greater than zero, the eigenvectors associated with A 1 and A, define a unique pair of axes perpendicular to the bond path. The electron density decreases more rapidly along the minor axis of curvature and decreases more slowly along the major one.

The Laplacian of the electron density, Pp< r ) is a useful characteristic of molecules and crystals. When Pp < 0, p is greater than its average value at neighbouring points; when Pp > 0, the opposite is true. This difference between the local value and the average value of p is a maximum at the extrema in Pp. Thus, the electron cloud is concentrated in regions where Pp < 0 and is more diffuse in regions where Pp> 0. It can be seen from Fig. 1 (d) that in the cyclopropane molecule the negative values of Pp are in the region which is usually associated with a covalent bond. Indeed, when a covalent bond is formed, p has a low positive curvature along the bond path and large negative curvatures in the perpendicular directions. The values of Pp are, therefore, negative in the bonding region.

The Laplacian of p is directly related to the local contributions to the electronic energy of a system [ 651 via the equation

(fi”/4m) Pp(r) = V(r) +2T(r) (23)

where V(r) is the potential-energy density of the electrons and T(r) is the electronic kinetic-energy density. The integrals of V( r ) and T( r ) over all space yield the total electronic potential energy, V, and the total kinetic energy of

the electrons, Z’, respectively. For all r, V(r) < 0 and T(r) > 0 and, therefore the lowering of the potential energy dominates the total energy in regions where electron density is concentrated, i.e. where Pp < 0.

According to Gauss’ theorem the integral of the Laplacian of p vanishes for an isolated system

J-Pp(r)d=+s@(r)n(r) =o s

due to the fact that p, Vand Vp vanish on the surface s at infinity. This property results in the virial theorem V+2T=O, which is valid for the whole system. However, the zero-flux condition (eqn. (20) ) is fulfilled for the bonding atom, and the integral of the Laplacian of p over a bonded atom volume (52) also vanishes. In this case the virial theorem takes the form

(fi2/4m)jPpdV= V(Q) +2T(SZ) =O (25)

This means that within the shells of an atom in a molecule, there are regions of local concentration of bonded and non-bonded electron density together with regions of depleted electron density. In a reaction of a nucleophile with an electrophile or of a base with an acid, a centre of concentrated electron density combines with a centre of depleted electron density within the valence shells of the atoms involved. The reaction corresponds to the combination of a region of excess potential energy with a region of excess kinetic energy to yield a linked pair of atoms for which the virial theorem is satisfied for each atom separately as well as for the combined pair.

Bader’s topological theory may be linked to Fukui’s frontier orbital model [71]. In many molecules, the regions where the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs) of the reactants are most concentrated coincide with the regions of concentrated and depleted electron density, respectively [ 721. Thus, the Laplacian of p provides a bridge between the p and orbital approaches to the understanding of chemical reactivity.

Following the results reported in refs. 70 and 73, the local electronic energy can be defined as

H,(r)=T(r)+V(r) (26)

The sign of I-& shows which part of the energy (kinetic or potential) is predom- inant at any point in the molecule. Taking eqn. (23) into account we can re- write eqn. (26) in the form

H,(r) = (l/2) [V(r)+&~p(r) 1. (27)

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It is remarkable that this expression does not contain any derivative of the wavefunction. In addition, it follows from eqn. (27) that, in order for a system to be bonded, it is not sufficient to fulfil only the condition Pp(r ) < 0 in the internuclear space; the value of the potential-energy density V( 1:) < 0 is of de- cisive significance.

The indications of the formation of a covalent bond are described in ref. 70. For a covalent bond to form it is necessary that: (1) there is a critical point of the (3, - 1) type (the necessary condition); and (2) the electronic-energy density has a negative value at the bond point He(r,,) (the sufficient condition). No analogous set of conditions for other chemical bond types have yet been formulated.

The topological approach does not require any a priori assumptions about the reference state, i.e. no promolecule model needs as in the case of the deformation electron density. This approach is, therefore, a direct and model-free way of elucidating the non-evident properties of p.

X-RAY DIFFRACTION DATA ON THE FEATURES OF THE ELECTRON DENSITY OF

ORGANIC COMPOUNDS

The simplest characteristics of the electron density are the effective atomic charges. The effective charge is usually defined as the sum of the positive nuclear charge and the charge of the electron-density basin belonging to that nucleus. There is no unique way to define such a basin and this decreases the value of this definition. Nevertheless, there are some approaches to defining a basin in a “sea” of electron density. Evidently the most correct way is to draw a surface which includes a nucleus and for which eqn. (20) is valid. In practice, however, either the generalized Wigner-Seitz cells (which depend significantly on the atomic/ionic radius relationship [ 741) or the pseudoatomic fragments (which are proportional to the weights of the atoms in the superpositional crystal model [ 751) are used. The first method defines the atoms with discrete boundaries, and the second defines them with fuzzy ones. The atomic charges can be derived by means of a K technique or by variation of the monopole-term parameters in a multipole model. It is clear that charges are model- dependent values. However, all these methods give similar descriptions of the general trends in the electron-density distribution [ 76-791.

An experimental estimation of the atomic charges can be very useful in some respects. The evaluation of the atomic charges in a number of predominantly organic molecules shows [ 791 that there is a linear relationship between the contraction-expansion parameters (K) and atomic charges (Fig. 4). This experimentally determined relationship agrees very well with the well-known Slater rules for selecting the valence electron radial function screening parameters (selected on the basis of atomic charges). Thus, the experimental data

357

2

I I r I

1.00 0.75 0.50 0.25 0 0.25 0.50 0.75 1.00 4-

Fig. 4. The relationship between the electron density contraction-expansion parameter (K) and the net charge, q, for N atoms in a number of molecules: (a, b) glycylglycine; (c) formamide; (d, e) p-nitropyridine N-oxide; (f) sulphamic acid, (h) NH,SCN; (j, k) KN3. Bars indicate the estimated standard deviations. The full line is the relationship resulting from the Slater rules for atomic orbital exponents [ 791.

(b) (C) Cd) W (fl

Fig. 5. Dicyandiamide (C2H,NI): (a) structural formula and atomic charges (from multipole model refinement); (b-f) resonance structures [80].

can be used successfully to determine the optimal parameters to use in re- stricted-basis-set quantum-chemical calculations.

Valuable explanations of X-ray diffraction results can be obtained on the basis of atomic charges. For example, dicyandiamide (C,H,N,) , a simple molecule (Fig. 5) consisting of atomic groups with distinctive electronic properties, has been investigated (at 83 K) in this way [80]. The actual bonding in the molecule is not clear a priori because the lengths of the three C-N bonds around the C4 atom are almost equal (1.339,1.341 and 1.333 A). The structure of this molecule was explained as arising from resonance between three main structures ( (b), (c) and (d) in Fig. 5). According to this view, it would be expected that there would be a negative charge on Nl and positive charges on the amino groups. The multipole model refinement indeed results in such a distribution of atomic charges (see Fig. 5 (a) ) and the large negative charge on

358

N3 was found. This suggests that the two additional resonance structures ( (e) and (g) in Fig. 5) may also be important.

As a second example, it was found that in the zwitterion y-aminobutyric acid ( C4H9N02, space group R&/a) the three tetrahedral carbon atoms are in fact neutral, whereas the trigonal carboyxlate C4 atom has a significant negative charge - 0.17 e (Table 1) [al]. This charge appears to result from the induc- tive effect of the carboxylate oxygen atoms. All the hydrogen atoms in the methylene chain are electrically neutral, except for H6 which has a significant positive charge of 0.11 e. Analysis of the structure of y-aminobutyric acid shows that only the H6 atom is within a short intramolecular distance of the two electronegative atoms (nitrogen and 01, the distances being 2.64 and 2.54 A, respectively). The corresponding sums of the van der Waals radii are 2.7 and 2.6 A. These short distances arise from the folding of the molecular backbone into a twisted conformation with torsional angle of t 72.6” about the C2-C3 bond. It may be imagined that the positive charge on H6 is induced as the molecule folds, and that the resulting hydrogen bridge N- * - -H+ - - -O- helps to stabilize the observed conformation.

A more general and flexible way of quantitatively describing the electron density is to use the pseudoatomic moment treatment

P~,~ ,... ,,= s &a,...,(n) (p(r)du $2

(23)

TABLE 1

Atomic charges in y-aminobutyric acid (from the multipole model) [ 811

Atom numbering Atom Atomic charge (e)

H7

N -0.23(3) Cl 0.05(4) c2 -0.04(4) c3 0.03 (4) c4 -0.17(3) 01 -0.15(2) 02 -0.27(2) Hl 0.18(2) H2 0.19(2) H3 0.20(2) H4 0.01(2) H5 0.03(2) H6 0.11(2) H7 -0.03(2) H8 0.04(2) H9 0.04(2)

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The operator i is a function of the coordinates of a point in real space: for a monopole moment y= 1, for a dipole moment I/=ra, etc. Sz is the integration volume, which can be defined in the same way as when determining the atomic charges (see above). Modifying this approach to X-ray diffraction, eqn. (28) can be re-written as

The explicit expressions for the different moments are reported elsewhere [82]. The multipole expansion method can also be used to obtain the pseudoatomic moments.

The atomic moments describe the aspherical features of atoms in a molecule, e.g. their valence states. They can be used in calculating many of the electrical properties of molecules (potential, field, field gradient, etc.). For free molecules such properties can be considered as observables. It is important that these properties are transferable among a series of analogous substances [82,83].

The moments derived by summing the structural amplitudes (eqn. (29) ) are dynamic ones. Therefore low-temperature experiments are preferable in order to decrease TDS. The multipole model can also be used to obtain pseu- dostatic moments.

We illustrate how pseudoatomic moments can be used in electron-density analysis by taking formamide (CH,NO) as an example [ 761. Two molecules in the unit cell are bonded by two hydrogen bonds, forming a centrosymmetr- ical dimer. The third hydrogen bond takes part in creating the chains of dimers. The pseudomoments calculated from the experimental electron density [84] are presented in Table 2. It can be seen from this table that the electron density in the C-H and N-H bonds lies closer to the more electronegative carbon and nitrogen atoms than to the hydrogen atoms. Most of the atoms in formamide are contracted in comparison to their free states, but the oxygen atoms are expanded in the plane perpendicular to the NCOH fragment. This pattern coincides with the Sp maps [84], which shows that the lone-pair electrons of the oxygen atoms are situated outside this plane.

The dipole moments of the formamide molecule calculated with different methods (Table 3) agree satisfactorily with one another and with the experimental data. Note, that the non-zero component of the dipole moment, which is perpendicular to the molecular plane, is related to the corresponding hydrogen bond in the crystal. Therefore, by the using pseudoatom moments we can obtain all the important features of a formamide molecule in the crystal. Anal- ogous analyses have been made for other compounds: acetylene [ 761, dicyandiamide [ 801, pyridinium dicyanomethylide [ 761, cytosine monohydrate [ 851,

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TABLE 2

Atomic deformation momenta for formamide referred to local coordinate frames (fuzzy boundaries, direct integration) [76]

Local coordinate frame

Atom p,, K PY M .k_ k (e (e (e (e A-*) ~~-z) c A-1, A-‘, A-‘, A-1, A-2,

0 -0.32 -0.047 0.013 0.012 0.066 0.072 -0.120 N -0.19 -0.088 0.027 0.004 0.123 0.150 0.020 H2 0.21 -0.079 -0.028 0.003 0.038 0.153 0.085 C 0.01 -0.035 -0.051 -0.075 0.114 0.137 0.183 Hl 0.16 -0.072 -0.015 0.020 0.003 0.135 0.080 H3 0.13 -o.OSs -0.004 0.028 0.083 0.101 0.098

TABLE 3

Molecular dipole moments (in D ) for formamide [ 761

Discrete boundary direct integration

Fuzzy boundary, direct integration

K Technique Multipole refinement Microwave measurements Population analysis of

quantum-chemistry result (extended basis set)

A & H P (Y

-4.02 -2.64 0.48 4.94 35.7

-3.25 -2.31 0.44 4.01 35.4

- 2.88 -3.31 0.15 4.39 39.0 -3.56 - 3.26 0.11 4.83 42.5 - 2.85 - 2.38 0.00 3.71 39.6 -3.06 - 2.70 -0.49 4.07 41.9

9-methyladenine [ 851, tetrafluoroterephthalonitrile [86] and others. A review is given in ref. 9.

The examples presented above show that the main characteristics of the electron structure of a molecule can be obtained from a limited set of numerical data. More detailed patterns, although semiquantitative due to TDS, can be derived from electron-density maps. We look now at the more typical maps obtained for organic molecules and consider their possible uses.

Let us consider first the electron density distribution in carbon-carbon bonds. Ethane, ethylene and acetylene, which can be crystallized at low temperature, are examples of compounds containing single, double and triple carbon-carbon bonds. The deformation electron density in these compounds has been investigated [87-891 and is illustrated in Fig. 6. The section of Sp(r ) by the H-C-

361

(c)

Fig. 6. Deformation electron density distribution in ordinary, double and triple covalent carbon- carbon bonds: (a) ethane; (b ) et$ylene.; (c ) acetylene (-) . Positive Sp values; ( ----- ) negative Sp values; contours are at 0.05 e Ae3 [87-891. For ethane, only the electron densities of carbon have been subtracted.

C-H plane of the ethane molecule is characterized by a well-pronounced, symmetrical, excessive Sp peak of 0.35 e A-” in height on the single carbon-carbon bond (the C-C bond length is 1.532 A). This peak has axial symmetry in the transverse section. The peak is associated with the formation in ethane of an ordinary covalent Q bond. The analogous peak for the double carbon-carbon bond (1.313 A) in ethylene (Fig. 6(b)) is more intense (0.50 e Am3). The formation of the a bond is reflected in the elongation in the Sp maximum perpendicular to the molecular plane and the C-C line. The intensity of the max-

362

imum for the triple carbon-carbon (1.182 A) in acetylene is 0.60 e A-” (Fig. 6 (c) ). There are two rt bonds which are preserved by the axial symmetry in the molecule. Thus, it can be seen that the intensity of the Sp peak increases with increasing bond order.

Topological analysis of the molecules mentioned above was done using the same X-ray experimental data [ 901. The results are presented in Fig. 7. In all three crystals, (3, - 1) critical points were observed at the centre of the carbon-carbon bonds. The Laplacian of the electron density is negative between the carbon atoms, in agreement with the covalent-bond picture. The Laplacian values at the critical points decrease with increasing bond order, whilst the total electron density increases at these points, in agreement with the results of quantum-chemical calculations. Thus, the experimental data support the theoretical conclusion with regard to the correlation between p (q,) and the bond order, in the spirit of the Coulson relationship [ 911.

The results of the topological analyses of the compounds mentioned are presented in Table 4. It can be seen from this table that the x-bond formation in ethylene results in a non-zero bond ellipticity, E for the C=C bond. It is interesting to note that the single bond in the crystalline ethane molecule also has non-zero ellipticity. The Hessian eigenvectors do not coincide with the main directions of the thermal motion ellipsoids of the carbon atoms in ethane. Therefore it can be suggested that the E > 0 values result from the interatomic interactions in the ethane crystal. The direction of the most gradual decrease

I I

Fig. 7. The Laplacian of the electron density for: (a) ethane; (b) ethylene; and (c) acetylene. In regions where lines are dashed, pp < 0; around nuclei, pp Q: 0 [ 901.

TABLE 4

Topological characteristics of bonds at the (3, - 1) critical points for ethane, ethylene and acetylene in the crystalline phase and in the free molecule [ 901

Bond % p VP*_ (A) @A-3, (e A-5)

1, 12 13 E

(e A-5) (e k-5) (e Ii-‘)

c-c crystal 1.510 1.61 - 16.13 - 12.09 - 9.25 5.21 0.31 Free molecule 1.528 1.70 - 15.92 - 11.49 -11.49 7.06 0.00

c=c Crystal 1.336 2.16 - 16.71 - 16.82 - 14.17 14.28 0.19 Free molecule 1.306 2.29 - 22.42 - 16.68 - 13.63 1.89 0.22

c=c Crystal 1.183 2.84 -31.24 - 22.38 - 22.36 13.24 0.00 Free molecule 1.190 2.74 -31.25 - 16.21 - 16.21 1.17 0.00

in the charge density deviates from the (l,O,O) line by 15.4”. This is the direction to the nearest neighbouring molecule in the ethane crystal.

The break in the axial symmetry of the Sp peak on the formally single C-C bond (1.554 A) was assigned experimentally to a-oxalic acid at 100 K [92]. This result was confirmed by the quantum-chemical calculations with an extended Gaussian basis set [ 931. Simultaneously the elongation of the Sp maxima in the directions, perpendicular to the formally single and double bonds was observed. These features were interpreted [92,93] as evidence of x-electron delocalization upon the a-oxalic acid molecule which is more pronounced in the solid.

The elongation of the n-bond dp peak is a characteristic feature of C-C bonds, regardless of the influence of their surroundings. To illustrate this we take as an example cumulene compounds. One of these, C24H36, is shown in the Fig. 8 [ 941. This molecule consists of linear hexapentene chains containing five C=C bonds with tetramethylcyclohexane rings at the ends. The Sp peaks are clearly elongated perpendicular to the direction of the chains. In accordance with the concept of n bonds, in cumulenes the directions of stretching of the adjacent bonds are mutually perpendicular (Fig. 8). The mesomeric structure influ- ences the a$ernative consequence of the C=C bond length (1.332,1.268,1.294 and 1.332 A at 103 K). Similar results were obtained for molecules containing butatrienic [ 951 and allenic [ 961 fragments. The elongation of the dp peaks in two adjacent double bonds was also found in bis (n-metaxiphenyl) carbodi- imine ( C15H14N202) [ 971.

Thus the transverse elongation of the dp peaks is clearly established for a- bonds. This is in good agreement with the theoretical explanation of x-bond

364

C

Fig. 8. Two mutually perpendicular sections of the deformation electron density along the molecular axis of C2.,HS8. The picture of Sp is averaged in accordance with mm2 symmetry [ 941.

.’ :

(a)’ (t

i.._.. . . . . ,,; i i ‘..._._ 2

n rd

I)

Fig. 9. Model (multipole) static deformation electron density in smallzyclic systems: (a) cyclopropane; (b) tetracyanocyclobutane. Contours: (a) 0.05 and (b) 0.1 e AP3 [99].

features. Furthermore, the mutual influence of these effects in adjacent bonds has been found experimentally.

In small atomic cycles, bonds are known as bent. Bent bonds are seen as a displacement of Sp peaks from the bond line outside of the geometrical perim- eters. Such effects in three- and four-member rings can be seen in Fig. 9. The peak displacement values are usually 0.1-0.4 A, and indirectly characterize the degree of bond strengthening.

Topological analysis of free molecules shows [ 981 that the critical points of the (3, - 1) type are located at the centre of the rings. The C-C bond paths are

bent (see Fig. 1 (b) and Table 5). The critical points of the (3, - 1) type lie outside the C-C straight lines. Hence, the bent chemical bond representation is experimentally proven. The lateral displacement of the electrons decreases the nuclear repulsion and, therefore, the bond energy decreases.

Bending of the bond paths can also be caused by the distortion of a molecule. In the ethylene molecule at the equilibrium configuration, the C-H bond paths coincide with the internuclear line. However, as the H-C-H angle decreases the crystal points appear to be out of this angle. The change in electron density caused by the distortion results in forces which try to move protons to their equilibrium positions.

As far as the Sp peaks of a electrons in cycles is concerned, these again try to become elongated and cross the C-C bond line. This Sp elongation is not necessarily in the plane of the ring. Such elongation has been observed, for example, in cyclopropane (C,H,) , bicyclopropane (C,H,,) and vinylcyclopro- pane (C&H,) [99], and rotane (C&H,,) [ 1001 and a tetratriene derivative (C&H,,) [ 941. The ellipticity in these molecules varies over a wide range [ 681. These data can be considered as evidence of the conjugation effect in small cyclic compounds.

The conjugation of the carbon-carbon bonds in aromatic hydrocarbons leads to the equal distribution of the electron density among the cyclic fragments. This can be seen, for example, in 9-tert-butylanthracene ( 1, Fig. 10) and C1eH,B [ 1011 (Fig. 11 (a) ) . The central ring of the anthracene framework is not planar (fold angle 166.2” ) and the geometry of the molecule is not ideal. However, the deformation density peaks for the conjugated bonds are the same within the limits of experimental accuracy. The deviations in the peak heights can also be attributed to steric interaction resulting from the large substituent at the 9-position.

A valence isomer of this molecule, 9-tert-butyL9,lO [ Dewar ] anthracene (2 )

TABLE 5

Bond paths (I,) internuclear distances (R,) and bond energies in some hydrocarbons [63]

Molecule

Cyclobutane Cyclopropane Bicyclo[ 1.1.11

pentane Bicyclo[l.l.l]

butane

L L/R, (3,-l) critical (A)

Bond energy point shift (kcal mol-‘)

1.554 1.000 0.013 83.6 1.528 1.018 0.096 82.8 1.550 1.004 0.067 83.5

1.503 (bridge) 1.024 0.117 71.2

1.526 1.017 0.092 77.4

Cyclopropene 1.524 1.021 0.106 1.300 1.018 0.092

-

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w 000

(1)

0 - N-N

O’(,>O

(6)

t2 f3 - - (8)

(7)

Q N-N-0

(9) (IO) (II)

OH

Fig. 10. Some molecules discussed in the text.

crystallizes in the same space group (P2Jc). The Dewar bond (C9-ClO, 1.623 A) results in the formation of a dihedral angle between the two four-membered rings of approximately 113 ’ and in bent bonds in the bicyclic fragments (Fig. 11 (b) ): the deformation electron density maxima are shifted to about 0.2 A outside of the C-C lines [ 1011. The weak Dewar bond is also strongly bent, in agreement with quantum-chemical calculations [ 1011. The peak maxima on the bonds involving the C9 atom are significantly displaced in the direction of the bridging group. There is also an additional interaction between the Dewar bond and the C9-Cl1 and ClO-HlO bonds.

Bent bonds have also been observed in small organic polycyclic cage molecules, e.g. the cubane and tetrahedrane derivatives. The Sp maxima appeared to be displaced from the internuclear vectors in cyclobutane and cyclopropane derivatives. In tetra-tert-butyltetrahedrane (C&H& [ 1021 the dp maxima are shifted away from the framework by 0.37 A (Fig. 12). In addition, a Sp maxi-

367

(b)

Fig. 11. (a), The deformation electron density in 9-tert-butylanthracene &HI8 (1) and (b), its valence isomer 9-tert-butyL9,lO[Dewar]-anthracene (2) [loll. Contours are at 0.1 e A-‘.

mum of 0.15 e A-” is observed in the centre of the tetrahedra and on the faces of the triangle. This indicates that there is a significant delocalization of electrons in this cage molecule.

The molecules considered have predominantly directed covalent chemical bonds which in some cases are bent. Another situation is illustrated by propellane molecules, which consist of three small rings, sharing a common bond (1.55-1.64 A), called the bridgehead bond. There are four bond paths, all to one side of a plane through the bridgehead carbon atoms. The model of hybridized orbitals cannot describe such systems and the nature of the chemical bond in propellanes is not clear. Furthermore, it is important to establish the difference between the electron density characteristics of the bridgehead interatomic region for propellanes and that for the closely related bicyclic compounds, where the bridgehead distances are shorter.

There are a few examples which can be used in discussing this problem. It has been shown experimentally that the bridging bonds of the bicy-

366

, IA IA

Fig. 12. The deformation electron density in tetra-tert-butylthedrane (C,H,,) [ 1021. Con- tours are at 0.05 e li-‘.

clo [ l.l.O]butane derivatives (3 [ 103],4 [ 1041 and 5 [ 1051) shows significant Sp maxima which are shifted from the internuclear axis. In a [3.l.l]propellane derivative (6) [ 1061 and a Dewar benzene derivative (7) [ 1071 the Sp values for the bridging bonds are almost zero. In [l.l.l.]propellane derivatives (8 and 9) [ 1081 the Sp values between the inverted carbon atom are negative. In molecules 6-9 the positive region of Sp lies outside of the bridgehead line. Some examples are given in Fig. 13. Unfortunately, it is not possible to state that these characteristics of the deformation electron density in the bridgehead region are seen in bridging bonds only. It is incorrect to compare the deformation electron densities for systems with different geometrical structures because the reference electron densities used to calculate Sp are different. The information on the electron density Laplacian in [ 1.1.1 lpropellane is not useful in this case. Analysis of the ,theoretical electron density has shown [ 1091 that in the bridgehead bond critical point of bicycle [ l.l.O]butane Pp< 0, but at the same time the analogous point in [ l.l.l]propellane Pp>0. This fact does not imply that the net forces exerted on the inverted carbon nuclei by the bridgehead electron density are antibonding. The atomic dipoles of the inverted carbon atoms in [ l.l.l]propellane are large and directed away from the neighbouring nucleus, being dominated by the diffuse region of the non-bonding electron density. However, the force exerted on the nucleus from its own basin electron density is bonding, being dominated by the density near nucleus. This force is directed at the other bridgehead nucleus. It should be noted that

369

lb)

Fig. 13. The deformation electron density in bridgehead carbon-carbon bonds: (a) a bicyclo[l.l.O]butanederivative (3); (b) aDewarbenzenederivative (70) [107]; (c)a [l.l.l]propellane derivative (8) [ 1081. Contours are at (a, b) 0.05 and (c) 0.025 e Ae3.

such a picture is not unique and is also observed, for example, in Cz, N2 and 0, molecules [ 1091.

The available data allow us to conclude that the bridgehead carbon atoms in the [ 1.1.1 lpropellanes differ, most probably in the degree but not the nature, of the polarization. But the question remains of whether the chemical bond between inverted carbon atoms is covalent. Thus the determination of the sign of local electronic energy He (eqn. (26) ) at bridgehead bond critical point is very desirable in order to clear up this situation.

370

It is interesting that the Pp value for the bridgehead bond in propellanes is dependent on the number of three-membered rings [ 1091. The Laplacian of p is negative and large for [2.2.2]propellane, close to zero for [2.l.l]propellane, and slightly positive for [ l.l.l]propellane. In the latter two molecules electron density is localized in the basin of the bridgehead atom. It is this factor which results in the different chemical behaviour of the fragments.

The deformation electron density maps for the [l.l.l]propellanes reveal diffuse non-bonded positive peaks outside the bridgehead bond. These peaks are probable sites of electrophilic attack. Such ease of protonation explains the pronounced susceptibility of [ l.l.l]propellane to acetolysis [ 1101.

Let us consider now heterobonds involving carbon atoms. The polarity of such bonds is not high and depends both on the nature of the partner atom and on the electronic effects acting in the system. If the electronegativities of the bonded atoms are significantly different, a large displacement of the Sp maximum toward the more electronegative atom can be expected. In organic compounds, however, mutual bond interactions, conjugation and other effects can make this displacement difficult to observe on deformation electron density maps, although such a tendency is often revealed. In inorganic compounds this tendency is much more pronounced [ 91.

Boron nitrilotriacetate (C6H6BN06, 10) gives a good representation of the electron density in carbon heterobonds [ 1111. There are six different types of interatomic bonds in this cage molecule, in particular two C-O bonds: C2-02, a double bond of length 1.206 A; and the C2-01 ester bond (1.334 A). The deformation electron density map for this molecule is presented in Fig. 14.

Fig. 14. Model (multipole) state deformatiop electron density of boron nitrilotriacetate ( C6H6BN06, 10) [ 1111. Contours are at 0.05 e Am3.

371

Only for the Cl-N bond is the Sp peak noticeably displaced towards the more electronegative nitrogen atom. Such a peak is located at the centre of the C2- 01 bond and is displaced towards the carbon atom in the C2-02 bond. How- ever, the positions of the other Sp peaks are as would be expected. The bonds in C,H,BNO, can be placed in order of decreasing peak asymmetry: B-N > B- 0 z C=O> C-N> C-C x C-O. Thus, judging by the Sp maps, the C-O bond should be less polar than the C-N bond, and B-O should be less polar than B- N. However, this contradicts the usual ideas about the ionicity of such bonds. Moreover, topological theory reveals asymmetry in the electron density distribution in C-N, C-O and C-F bonds. The Pp (r ) distributions in various molecules containing such bonds show [ 112,113] that displacement of the region of concentrated charge is observed.

The simple orbital scheme, which was considered in ref. 111, can aid in understanding the Sp peculiarity observed. Let us consider atoms A and B, of which atom A is the least electronegative, and p orbital8 $A and & oriented along A-B. The electron density on the bond is then

p=21~*~*+c~~12/(~~+c~+2c*~~~AB)

and the promolecule density is

(39)

pP =n&i +n& (31)

where SAB is the overlap integral and n is taken as one-quarter of the number of valence electrons of the atom. For the series C-C, N-N and O-O with bond distances of 1.52,1.44 and 1.47 A and using Slater-type orbital8 with exponents from ref. 114, the following of the deformation densities (Sp=p-p,) were obtained: 0.20,O.OO and - 012 e AV3, respectively. The calculated trend matches the observed one for the tetraazatetraoxatricyclic molecule C,H,,N,O, [ 1151, hence the model properly reproduces trends. It was also found that if cA x cg the bonding density is polarized towards the less eJectronegative atom A. For the B-N and B-O bonds, dpmax is 0.19 and 0.20 e Ae3, and the corresponding asymmetries are - 7 and - lo%, respectively. As can be seen, the reverse polarization is more pronounced for the more electron-rich and more electronegative atom B.

For CA < cn, 6p is polarized towards B, as expected. This polarization is less pronounced if B is more electron rich. For B-N and B-O with CA/Q w 0.47, the ~PIXS=lX values are 0.25 and 0.15 e Am3, and the asymmetries are 16 and 5%, respectively.

There is one more unexpected experimental feature of the electron density distribution: as the electronegativity of atom A increases, the Sp peaks in the C-A bond region become less intense. Relatively small dp peaks, in comparison with those for the C-C bonds, were found for the C-O bond in tram-2,5-d&

methyl-3-hexene-2,5-diol hemihydrate [ 1161, in l&crown-6 (C,,H,,O,) [ 1171; the smallest peaks were found for C-F in tetrafluoroterephthalodinitrile and

372

1,1,4,4tetrafluorocyclohexane [ 1181. Moreover, this trend is followed in other bonds in organic compounds.

The low or even slightly negative Sp density in N-N bonds has been found experimentally in N,W diformylhydrazine [ 1191, tetraformylhydrazine [ 1201, carbonohydrazide WI, 1,2,7,8-tetraaza-4,5,10,11-tetraoxatricyclo- [6.4.1.12.‘] tetradecane (11) [ 1151, N-methyl-N’-methoxydiazene N-oxide [ 1221, 4nitropyridine N-oxide [ 1231 and cis,cis-trialkyltriaziridine ( C2Hi306N3) [ 1241. A negative Sp value has been observed for C-O bonds in 11 and in hydrogen peroxide [ 1221. All these observations are in agreement with the results of the quantum chemical calculations [53,111,126] and may be considered reliable. It has been noted that the deformation electron density decreases in ordinary bonds of the C,H,,N,O, molecule ( 11) in the order C- N >> C-O > N-N > O-O [ 1151 and in bonds of the C6H6BN06 molecule in the order C=O z C-C > B-N z B-O > C-N x C-O [ 1111.

Similarly, the height of Sp peaks in the electron lone-pair regions depends on the type of electrophilic atoms present. In particular, when the electronegativity of an atom increases the Sp peak is lower in intensity and displaced towards the atomic centre.

It can be concluded that the features seen in deformation electron density maps depend on the type of atoms in a molecule and on the type of atomic densities subtracted from the molecular density. In the case of C-F, O-O and N-N bonds the use of the promolecule model with spherically averaged atomic densities reduces, or eliminates entirely, the positive features of Sp between the atoms. In the case of polar bonds, apparently unreasonable polarizations may be simulated by using the promolecule.

Hence, the reconstruction of the atomic electron shells in some systems during chemical bond formation shows that the removal of electrons from the internuclear region due to the space limitations imposed by the Pauli principle is greater than the increase in electron density due to constructive orbital interference and delocalization. As deformation density maps show the integral electron redistribution in relation to a promolecule, a negative region can be expected in the interatomic space. However, a covalent chemical bond is formed even in this case; this is confirmed by, for example, the negative Laplacian of the p region between the oxygen atoms in hydrogen peroxide [ 701.

In order to analyse the features of chemical bonds when the standard deformation electron density is not sufficiently informative, the formation of chemical bonds can be divided into three stages: hybridization and polarization of atoms; electron delocalization and transfer; and constructive interference [ 1271. The generalized valence bond (GVB) method can be used to evaluate the changes in electron density that occur at these separate stages, provided a good basis set is used. For this purpose, besides the GVB molecular density (pzz: ) it is also necessary to calculate the GVB hybrid molecule (pEm ) as the sum of the GVB valence-state hybrid atoms with one electron placed in each hybrid

373

orbital pointing along the bond direction and two electrons placed in each lone- pair orbital, and the GVB pair promolecule (&$‘F ) as the density of the singly occupied GVB pair orbitals plus the molecular core orbitals. The difference

PhO -pSph.abm gives the electron redistribution due to hybridization, the difference pit: -pkro gives the electron delocalization and electron transfer, and the difference pE:F -pEt gives the constructive interference.

Such calculations have been done for some simple molecules [ 1271 and show that delocalization and electron transfer lead to the accumulation of electrons predominantly in the internuclear space, the displacement of the electron density towards the more electronegative atom also being observed. The delocalization effect is always attractive. Constructive interference gives rise to increased (in relation to the reference density) electron density not only between the nuclei but also over all molecular space. However the changes in electron density due to constructive interference are smaller than those in the other stages. Hence, the Sp peculiarities have a complete theoretical explanation.

The important chemical information on the joint effect of electron delocalization, charge transfer and constructive interference, which is often lost in usual dp maps, can be obtained if a set of properly oriented atoms in hybridized valence-states is used as the promolecule [ 126,128]. The corresponding difference function reveals the changes that occur in the system due to short-term effects. As mentioned above, in tetraazatetraoxatricyclotetradecane ( 11)) the experimental deformation electron density of the O-O bonds is negative (Fig. 15(a)) [ 1151. The theoretical Sp map [126] is the same in all details as the experimental one (Fig. 15 (b) ). Thus the negative deformation electron density of the O-O bonds is not caused by experimental uncertainties. Two more difference maps were calculated in order to analyse the mechanism of the chemical bond formation in 11 [ 1261. One map presents the difference between the atoms in their hybridized valence states and the corresponding spherical atoms, and the second map shows the difference between the molecule and the set of hybridized valence-state atoms. A fragment of the first difference map is presented in Fig. 15 (c). The process of the “preparation” of spherical atoms for chemical bond formation can be clearly seen. The electron density concentrates in the lone pair regions of the oxygen atoms and near the nucleus of the carbon atom. As can be seen from Fig. 15 (d), the electron delocalization and constructive interference that occur due to the formation of covalent O-O and C-O bonds lead to the larger than expected electron density in the internuclear space. The same situation occurs in the C-H bond. We should also note the minima electron density which are located behind the atomic nuclei. Such a pattern of electron density is typical of chemical bonds which are of predominantly p character. Hence, electron delocalization and constructive interference in 11 are manifested as a concentration of electron density in the internuclei space in relation to “prepared” atoms and as a deficit

d)

Fig. 15. The deformation electron density of tetraaxatetraoxatricyclotetradecane (11) on the Ol- O2-C2 map: (a) experimental dp map [ 1151; (b) theoretical Sp map [ 1261; (c) difference between the electron densities of the atoms in their valence states and the spherical atoms [ 126); (d) difference between the theoretical electron density and the electron density of the atom in its valence state [ 1261. Contours are at 0.075 e A?.

in electron density behind the atomic nuclei. The first effect is not seen on Sp maps, although the second usually is.

The oriented hybridized valence-state atoms have been subtracted in an X- ray diffraction investigation of the C-Cl chemical bonds in C,I-&NCl, C,H,O,Cl and C,H,N,O,Cl [ 1291 and the S-F bond in F,S=C (CHB)CF3 [ 1301. In the latter case the positive Sp peak on the S-F bond was not observed even when the modified promolecule was used. The trigonal-bipyramidal arrangement of the sulphur atom in F4S=C(CH3)CF3 can be described as sp2, d,,, pz and dy2 hybridization for equatorial ts and 7c orbitals and the axial IS orbitals, respectively. After subtraction of the corresponding copherical atomic density from the total electron density, the charge-depleted region seemed to be unaffected. A possible reason for this is the loss of net charge at sulphur atom which cannot

375

be compensated for even in the oriented atoms reference state. Thus there are cases where a more sophisticated reference state is needed for the analysis of chemical bonds.

Schwarz et al. [ 1311 have proposed that the reference state be constructed of non-spherical ground-state atoms oriented in accordance with their local surroundings in the molecule. They called the corresponding difference densities “chemical deformation electron densities”. An algorithm for determining the orientation parameters for theoretical electron densities and X-ray diffraction data has been developed [ 131, 1321 and the first results obtained, particular for some organic compounds, have been presented [ 133,134]. It has been found, for example, that the chemical deformation density of 9-tert-butylanthracene ( 1) calculated using this algorithm is in qualitative agreement with the standard one, although the peak maxima on the bonds have smaller values. The optimized populations of the p- orbitals of the ring-carbon atoms in this molecule are p~‘p~&p~~. The low pn population results from the fact that the 2s’ density of carbon atoms is similar to some part of the 2p,, 2p,, 2p, density and it is not possible to differentiate between these components am- biguously using X-rays. Thus, such an approach meets some difficulties, which are well known in other methods [ 1351.

The electron density of organic compounds containing heterocycles has been widely investigated. The smallest three-membered cycles containing a heteroatom are highly strained systems with bent bonds [ 136-1381. The presence of the heteroatom breaks the symmetry of the electron distribution in the ring and imparts specific properties to the molecules.

Let us consider the electron distribution in the epoxide ring in truns-3,4- epoxy-1,2,3,4-tetrahydro-1,2-naphthalenediol (C,,H,,O,, 12). This molecule is a model for the ultimate carcinogenic metabolites of some polycyclic aromatic hydrocarbons. The epoxide ring is the functional group believed to be responsible for the reactivity of such compounds. An experimental investigation of 12 has been made at 105 K [ 1381. Model multipole deformation density maps of the epoxide ring are shown Fig. 16. The bent bonds are indicated by the location of the Cl-C2 covalent-bond peak outside the straight line between the atom centres. The C-O bond peaks are polarized towards the more electronegative oxygen atom. At the resolution used in the study, the C-O bond peaks merge into a single peak inside the ring. Careful tracing of the bond paths, however, also suggests bent C-O bonds. The same electron-distribution picture has been observed for the three-membered ring of tetracyanoethylene oxide [ 1371. The Cl-C2 bond in 12 also shows asymmetrical electron density where the density is polarized towards C2. This asymmetry is confirmed by the magnitude of the monopole populations which show that C2 is significantly more negative than Cl. The net atomic charges are +0.37(38) on Cl and 1.38(53) on C2.

These data can be used to predict the site on the epoxide ring which would

376

(a) (b)

Fig. 16. The model multipole (static) deformation electron density of tras-3,4-epoxy-1,2,3,4- tetrahydro-1,2-naphthalenediol (CI,JIH,oO,) [138]: (a) section in the plane of the epoxide ring, (b) section in a plane perpendicular to the Cl-C? bond of the epoxide ring and passing through the epoxide oxygen atom. Contours are at 0.05 e Ae3.

be chemically attacked by a nucleophile. Leaving aside steric considerations, attack by a nucleophile is expected at Cl, the most electropositive atom of the epoxide ring in 12. The reaction of epoxide metabolites of polyaromatic hydrocarbons has been shown to result in adducts in which the carcinogen is bound covalently to DNA bases. Such a reaction, which can be considered as a nucleophilic attack by the DNA base, has been shown to occur at a carbon atom equivalent to the Cl atom of 12. While the specific stereochemistry of the interaction of the ultimate carcinogen with the DNA molecule must be important, the electron distribution observed in 12 has been used to predict correctly the reaction sites for both bay region and non-bay region epoxides.

The deformation density calculated in a plane perpendicular to the Cl-C2 bond of the epoxide ring and passing through the epoxide oxygen atom is shown in Fig. 16 (b). The non-bonded lone-pair density on the oxygen atom is clearly evident and appears to be of predominantly p character. This phenomenon is supported by the MO coefficients obtained in an ab initio theoretical calculation on oxirane [ 1391 which indicate the presence of largely unhybridized s and p lone pairs on the oxygen atoms. It should be noted, that in tetracyanoethylene oxide [ 1371 the deformation density of the oxygen atom lone pairs is characterized by a single peak of approximately 0.4 e Am3, smeared in the ring plane.

A four-membered S-C-S-C ring is a characteristic feature of 1,3dithie- thane-1,1,3,3tetraoxide ((CH,SO,),). That there is ring strain in this molecule is shown by the shift from straight interatomic lines of the peaks in the Sp map [ 1401. However, the strain in this molecule is not as high as that in the

cyclobutadiene ring. The S-S across-ring distance of 2.593 A is short but shows no S-S bonding on the Sp map.

A number of studies have been done on organic compounds containing five- and six-membered heterocycles (for examples see ref. 9). 2-Pyridone ( C5H5NO) is a typical example (studied at 120 K) [ 1411. The Sp peaks of the C-C bonds (Fig. 17) have approxi_mately the same values, although their intensities are in the range 0.4-0.6 e Am3. There is no correlation between the peak heights and the bond lengths. Displacement of the Sp peaks on the C-N bonds towards the carbon atoms is observed. This displacement can be explained in the usual way (see above), but the influence of the oxygen atom, which can change the occupancy ratio of the Q and K bond components, should also be taken into account. The mutual influence of the bonds in 2-pyridone is very remarkable: the Sp peaks on all bonds are connected by excessive (positive) dp bridges. Referring to the lone-pair Sp peaks, the electron state of the oxygen atom can be assigned as sp2 hybridized. The results of theoretical calculations coincide with the experimental ones to within 2 0.1 e A-“.

Another example is the barbital molecule (C,H,,N,O,) [ 1421, where a mutual influence of the bonds is also seen. Displacement of the dp peaks in the 7~ region of the C-N bonds towards the more electronegative nitrogen atom is observed. In the molecular plane this displacement is towards the carbon

Fig. 17. The model (multipole) dynamic deformation electron density of the pyridone ring in 2- pyridone [ 1411. Contours are at 0.1 e Ae3.

378

atoms. The behaviour of the 7~ electron density can be explained by mesomeric electron shifts

Thus, the electron occupancies of the pn orbitals of the nitrogen and oxygen atoms are higher than that of the carbon atoms.

Carborans are interesting examples of polyheterocyclic organic compounds. The nature of the chemical bonding in these compounds has been studied extensively [ 1431. There are some indirect quantum-chemical indications [ 1441 that there is a concentration of charge inside the inner part of the icosahedron. The more accepted model is that of a delocalization of electron density along the “surface” of the cage carborane molecule [ 1451. In order to study the chemical bonding in such cage systems, the crystal of 9-azido-4carborane (N3-p- C,B,,H,,) has been investigated (at 163 K) [ 1461. Some of Sp maps for this crystal are presented in Fig. 18. There is negative (-0.16 e Am3) Sp density inside the carboran cage (Fig. 18(a) ). Positive Sp density was found to be spread nearly continuously over the surface of the carboran cage. The Lapla- cian of p (Fig. 18(b) ) inside the carboran cage is positive. The negative pp values are spread over the same surface layer, but are displaced slightly towards the carbon atoms. Hence the chemical bonding in carborans is associated with surface electron delocalization. Thus the theoretical predictions given in ref. 145 can be considered as confirmed, and can be used in analysing the “con- ducting” sphere model.

In icosahedron faces which are composed of B3 and B,C groups, the Sp distribution is continuous as is characteristic of highly delocalized electrons. This

(a) (b)

Fig. 18. (a) The deformation electron density of 9-azido-p-carbone in a section throu&h the centre of the two carborane groups and atoms Cl, B5, C7 and B12; contours are at 0.04 e Ae3. (b) The Laplacian of the electron density in the same plane as in (a); (-----) @CO [ 1461.

379

Fig. 19. (a) The deformation electron density-of 9-azido-p-carborane in a section through the three-membered rings; contours are at 0.04 e Ae3 (symbols denote the position of the (3,+3) critical point). (b) The Laplacian of the electron density in the B-B-B plane; (-.-.-) pp<O [146].

is confirmed by both the Sp map (Fig. 19 (a) ) and the Laplacian of p (Fig. 19(b) ). A small displacement of the Sp maxima can be seen on B,C! faces towards the more electronegative carbon atoms.

Let us turn now to the structure of organoelement compounds. The electron distribution in the C-X chemical bonds (X= P, S, Si, Cl, etc.) usually has no special features [ 130,147]. Bonds between atoms having more than half their electron shells filled resemble, in many respects, N-N, O-O and F-F bonds. For example, in the planar 2,5-dimethyl-6a-thiathiophthene molecule ( C7HsS3) [ 1481 which contains two fused heterocyclic five-membered rings, there is linear S-S-S fragment. The deformation electron density in the S-S bond is low compared with that in C-C and C-S bonds. The peaks of the a-electron density lie close to the terminal sulphur atoms. The low accumulation of Sp in the S- S bond is in agreement with the analogous situation for the O-O bond discussed above.

Non-standard cases of chemical bonding are of special interest. The two- coordinated phosphorus derivative ( Me2N),C=P-H [ 1471 provides such an example. This molecule belongs to a new class of unstable phosphoralkene compounds with low melting points, containing the =C=P atomic group. The high inclination of such compounds to produce the conjugated P=C double bond with substitutions at the phosphorus and carbon atoms can lead to elongation of the bond to approximately 1.76 A. In (Me2N)&=P-H, the P=C bond is 1.740 A long, which is longer than the usual non-conjugated bond (approximately 1.66 A). The section of Sp through the H-P=C plane is presented in Fig. 20. The Nl and N2 atoms deviate from this plane by AO.3 A. The Sp

’ ‘._ . __

Fig. 20. The deformation electron density of phosphoalkene ( (Me,N)&=P-H): (a) section in the plane of the Hl, P and Cl atoms; (b) section in the n-bond plane of the C=P bond. Contours areat0.05eAm3 [147].

maximum in the P-C bond (0.39 e AS3) lies between the atoms. Another maximum (0.18 e A-“) lies behind the phosphorus atom at a distance approximately 0.65 A. This maximum can be associated with the electron lone pair of the phosphorus atom; it is of predominantly s character. This is in agreement with the results of non-empirical quantum-chemical calculations [ 1491 and explains the fact that the valence angle at the two-coordinated phosphorus atom is always less than 120’ ( sp2 hybridization) [ 1501. This phosphorus lone pair plays an important role in the reduced activity of phosphoralkenes in ox- idation adduction compared with phosphines [ 1511.

In organoelement compounds hypervalent bonds can form when one atom of the molecule is in an unusually high formed hybridization valence state. The nature of hypervalent bonds is discussed in ref. 152. The pentacoordinated phosphorus molecules with the formula PX5 (X = H, F, Cl, CF,, CCL, etc. ) are such compounds. The central atom in these molecules have a quadratic-py- ramidal or trigonal-bipyramidal coordination. The electronic structure of phosphoranes have been studied using quantum-chemical methods specifically to establish the role of the outer 3d orbitals of the phosphorus atom in the chemical bonding [ 1531. The results of the calculations show that in simple phosphoranes such as PH,, PF5, etc., the occupancies of the 3d orbitals of the phosphorus atom are low. Hence these orbitals can be considered to be polarization functions only. Therefore, sp3d hybridization of the phosphorus atom by the promotion of an electron to the outer 3d orbitals is improbable, although this model is often used. A more rigorous explanation of the 5-covalency of phosphoranes requires the use of an MO approach. However, the canonical MO is delocalized over the whole molecule and this makes a pictorial description difficult. To avoid this difficulty, it is desirable to pass from the canonical MO to the localized MO. The use of the latter is preferable in the two-centre chemical bond description, especially in so-called electron-excessive com-

381

pounds which include phosphoranes. Another approach has been suggested [ 1541 which postulates a specific three-centres MO in a linear fragment X-P- X, provided that the phosphorus atom has trygonal-bipyramidal coordination. The interaction in this fragment was supposed to be the pa A0 of phosphorus which can form three Q MOs: one bonding (nodeless), one non-bonding and one antibonding. The first two MOs appear to be filled (uncoupled 2p, electrons of the X atoms plus the phosphorus atom lone pair), but the last remains empty. Thus there should be an increase in electron density and a displacement of the Sp peak towards ligands. In this way the indirect experimental data on the peculiarities of the chemical bonds in phosphoranes (X-ray, electron diffraction and spectroscopic data [ 1551) can be used within the frameworks of different models but cannot point to an unambiguous choice of model to be used.

The X-ray investigation (153 K) of bis (trichlormethyl)trichlorphosphorane (C,Cl,P) [ 1561, was allowed to restore electron density in one of phosphorane compounds. The Sp in a plane containing all indepen_dent atoms is presented in Fig. 21; the standard deviations are a(&) < 0.06 e AV3. All the Sp peaks are situated between the interacting atoms. The maxima on the hypervalent bonds P-Cl and P-C2 are 0.45 and 0.35 e AW3, respectively, and thus the shapes of the peaks are different. The peaks are also elongated perpendicular to the bond lines. This picture can not be used to describe completely the bonding in the C-P-C fragment by means of a two-centre a-MO model. The quantum-chemical topological calculations show [ 1571 that the Laplacians of the negative p regions on the P-C bond lines are undistinguishable from the other carbon

(b)

Fig. 21. The deformation electron density of phosphorane (C13P(CCl,),) [ 1561: (a) inihe axial plane of the molecule; (b) in the equatorial plane of the molecule. Contours are at 0.1 e Av3.

382

regions, as would be expected from the delocalized MO model. The main features of the hypervalent bond are well described by the scheme presented in ref. 154 and can be accepted as a first approximation.

The heights of the Sp peaks are 0.30 e A-” for the P-Cl1 bond and 0.22 e A-” for the C-Cl bond. There are two more peaks 0.7-0.8 A from Cl nucleus in directions nearly perpendicular to the P-Cl and C-Cl bond lines. These are probably associated with lone-pairs which have predominantly p character. There are also deep Sp minima ( -0.30 e A-“) behind the chlorine atoms in the direction of the phosphorus and chlorine atoms. These minima indicate that the pa orbitals of the chlorine atoms are involved in the corresponding bonding. The concentration of electron density in the P-C bonds and the high value of Sp in these regions support the well-known preference for the axial position in electron acceptor substitions. Quantum-chemical calculations lead to the same conclusion. Finally, the small deviation from a spherical form of the Sp distribution around the phosphorus nucleus may be due to a weak participation of the outer 3d electrons of the phosphorus atom axial bonds.

The investigation of (0-S;) -chloro- [l- (l,l-dimethyl-2trifluoroacetylhy- drazonium)methyl]dimethylsilane (C5H8N20F3SiC1) (Fig. 22) [ 1471 is another example of a study of hypervalent bonds. The silicon atom in this compound is trigonal-bipyramidal with an axial arrangement of chlorine and oxygen atoms; the Cl-Si-0 angle is 172 ‘, and the displacement of silicon atom away from the three carbon atom plane towards the oxygen atom is 0.076 A. The Si- Cl bond (2.434 A) is remarkably elongated and the length of the Si-0 bond (1.880 A) in the opposite direction is nearly equal to the covalent bond length.

(b)

Fig. 22. Deformation electron density of (0-Si) -chloro- [l- (l,l-dimethyL2kifluoroacetylhydra- zonium)methyl]dimethylsilane [ 1471: (a, b) two extreme valence forms of the molecule; (c) section through the axial plane which contains the Cl, Si, 0 and Cl atoms. Contours are at 0.07 e A-=.

The carbonyl bond C=O is elongated to 1.285 A and the length of the C=N bond (1.289 A) is virtually the same as the usual double -N=C (sp2) bond length (1.289 A). These geometrical features lead us to suppose that the electron structure of the molecule discussed can be represented as a superposition of two “extreme” valent forms (a) and (b) (see Fig. 22).

The Sp section through a plane containing the axial bonds and the Cl atom is presented in Fig. 22 (c ) . It can be seen that one of the Sp peaks for a lone pair of the oxygen atom is directed exactly towards the axial Si-0 bond. There is no usual covalent Sp peak for the Si-0 bond. The positive Sp peak on the Si- Cl bond is shifted towards the chlorine atom; in between the Si-Cl bond the Sp value is nearly zero. In this respect the Sp pictu.e is quite different from the usual one, with shorter Si-Cl bonds (2.09-2.15 A) in trigonal-bipyramidal l- (trichlorsilyl)-1,2,3,4-tetrahydro-l,l0-phenanthroline (C,,H,,N,SiCl,) with a &coordinated silicon atom [ 1581. In this molecule there are the usual Sp peaks in the Si-Cl bonds and electron lone-pair peaks near the chlorine atom. The latter are elongated perpendicular to the bond line.

The above facts indicate that the Si-Cl bond in C,H8N20F3SiC1 can be considered as not covalent. There is probably a coordination interaction Cl- + Si (b-form contribution) which stabilizes the system. This conclusion accounts for the elongated Si-Cl bond and explains the peculiarities of reactivity of such molecules, particularly the mobility of the Cl- anion from the axial position of the trigonal bipyramid [ 1471.

In the n-conjugated amide fragment N-C-O, the Sp peaks in the bonds appear to be elongated perpendicular to the fragment plane. The a-component peak is much more pronounced in the N-C bond than in the C-O bond (contribution of the (b) form). Thus, the electron density leads us to consider this molecule as a superposition of forms (a) and (b).

The data obtained have proved useful in modelling the “transition state” of a SN2-substitution reaction at the sulphur atom which takes place through the formation of the trigonal-bipyramidal structures.

Let us consider now the electron density distribution in hydrogen bonds. The energy of a hydrogen bond (lo-40 kJ mol-' ) is at least one order of magnitude lower than the covalent bond energy. The changes in electron density are in the formation of hydrogen bonds small; nevertheless, these changes can be detected by means of accurate X-ray diffraction measurements.

In the case of weak and intermediate hydrogen bonds, the main features of Sp in the hydrogen complexes can be represented to a first approximation as a superposition of the Sp of the isolated molecules [ 1591. This conclusion results from the observed agreement between the Sp pattern obtained by the superposition of the calculated Sp densities of the free molecules and experimental Sp maps for the interacting molecules in a crystal [ 1601. For this reason the electrostatic interactions can be considered to be predominantly responsible for weak to intermediate hydrogen bonds. The role of other factors can be

384

established by means of quantum-chemical methods. It has been shown for the formaldehyde-water system [ 1601 that the polarization part of Sp is almost the same as the total Sp. This means that the polarization stabilizes the weak hydrogen bond. The charge transfer and the exchange repultion compensate for one another. Therefore, for weak and intermediate hydrogen bonds the polarization of the functional molecular groups is the main factor in the hydrogen bond energy.

For strong hydrogen bonds the electron density distribution is quite different. a-Oxalic acid dihydrate is a good example of a compound containing a strong hydrogen bond [ 921. There are three kinds of hydrogen bond in this crystal (Fig. 23): a short strong bond (03.. *Hl-Ol_, 2.467 A) and two weak bonds (03-H3.**02,2.826 A; 03-H2.m.02,2.630 A; the 02,Ol atoms and Hl belong to the acid molecule, and 03, H2 and H3 belong to the water molecule). Maps of the modified valence density, which contain the spherically symmetrical electron density of the free hydrogen atoms only, can provide useful information. The maps show an electron density bridge with a saddle point of 0.25 e A-” between the hydrogen atom and the water atoms (strong hydro-

1 7

-2.52 X 2.52 ‘-2.52 X 2.52

(a) (b)

Fig. 23. The deformation electron density of the hydrogen bonds (hydrogen atoms are included) [92]: (a) strong hydrogen bond; (b, c) weak hydrogen bonds. Contours are at 0.05 e A-“.

gen bond). This indicates that there is a covalent contribution to the hydrogen bond. No such bridges are seen for the weak bonds, rather there is a break in the positive deformation density between the H3 and 03 atoms and a very low Sp between atoms H2 and 03. Thus, in the strong hydrogen bond contributions other than that arising from the electrostatic interaction are important.

Quantum-chemical analysis [ 1611 has shown that in cw-oxalic acid dihydrate the electron density is shifted from the 03 atom to the Hl atom. The charge transfer amounts to 0.14 e. This effect explains the approximate symmetry of the valence electron distribution observed for the short hydrogen bond [ 1591. An analogous effect must exist in the weak hydrogen bonds [ 1611; in this case the effect has a low value but can be detected by X-ray diffraction.

The Poisson equation relates the electron density to the electrostatic potential ($). The positive nuclear part of the potential is added to the electronic part and the total electrostatic potential inside the crystal ($) can be obtained by using either the multipole model [ 761 or a Fourier summation [ 162,163]. In the latter case, the combination of the Fourier summation and the direct space technique results in expression

f#J=~++&*0m+& (32)

where Sp is the deformation electrostatic potential calculated from 6p, $pprom is the electrostatic promolecule potential (including the nuclei), q&, is a constant which ensures that the mean value of $ in the unit cell is zero in accordance with the Ewald convention for Madelung sums [ 1621. The &,, term can be calculated using tabulated atomic wavefunctions (i.e. Clementi-Roetti functions [ 1641).

The potential (eqn. (32) ) characterizes the electric field inside the crystal. Both S$ and # may be used to study this field [9, 76, 1631, particularly in hydrogen bonds.

The investigation of alloxan (2,4,5,6, (lH,3H) -pyrimidinetetrone; C4H2N202) [ 1651 (Fig. 24 (a) ) is an example of the application of the electrostatic potential approach to organic compounds. The crystal structure of alloxan (space group P41212, z =4) consists of a herringbone packing of nearly planar molecules which lie on the two-fold axes. The intermolecular distances O=C* * * 0 in the carbonyl group are shorter than the sum of the van der Waals radii (3.1 A). The shortest of these, the C!5***06 distance, is 2.73 A at 123 K. The nature of such interactions in alloxan is not well understood. The hydrogen bonding in the alloxan crystal is a weak bifurcated interaction with long Ha * -0 distances (2.32 and 2.35 A).

The experimental electrostatic potential in alloxan has been calculated [ 1651 using static multipole model data. The short intermolecular C. - -0 distance at C5 appears to arise mainly from the deshielding of the carbonyl carbon atom nucleus which allows the weakly electronegative 06 atoms (charge -0.11 e) to approach closely either side of molecular plane. When molecules are brought

366

Fig. 24. Alloxane (C2H,N202): (a) atom numbering; (b) X-ray total electrostatic potential distributions in the section normal to the molecular plane; (c) as (b) b,ut in the section of the best least-squares Nl-Hl-02-06 plane [ 1651. Contours are at 0.05 e Ae3; (--) areas of positive potential.

together as is the case in this crystal, saddle points of nearly zero potential are formed along the C!5- - * 06 line (Fig. 24 (b) ) . However, the electrostatic potential around the C6 atom in the isolated molecule are nearly zero. Therefore the C- * -0 interactions are only weakly attractive. Analogous electropositive electrostatic potential bridges have been observed in parabanic acid (C,H,N,O, ) [ 1661 between the oxygen atom of the carbonyl group and the carbon atoms of the neighbouring molecules (distances are 2.75 and 2.94 A). The potential near the oxygen atoms in the two molecules mentioned are almost the same (about - 100 kJ mol-‘). However, in urea [167] the minimum electrostatic potential near the oxygen atom is -440 kJ mol-l. Thus the carbonyl oxygen atom in urea is more electronegative than the same atom in alloxan or parabanic acid. This conclusion supports the suggestion (resulting from an analysis of the geometry hydrogen bonds) that parabanic acid is more effective as a proton donor through its NH groups than as a proton acceptor through its carbonyl oxygen atoms. For urea the opposite is true. Consequently, the strong hydrogen bond with an N. - - 0 distance of 2.66 A is important for existence of the 1: 1 crystal complex involving parabanic acid as the donor and urea as the acceptor. The electrostatic potential in the 1: 1 complex of thiourea (CH,N,S) and parabanic acid has been studied at 298 K [ 1681. Unfortunately, some in- consistencies in the electron density of the thiourea in the complex and the

387

homomolecular crystal [ 1671 mean that the results obtained cannot be discussed here.

The depletion of the electron density at the carbonyl carbon atom from above and below the molecular plane and the observed electrostatic potential picture are in agreement with the supposition [ 1691 that in alloxans and several other crystals the C * * -0 distances can be regarded as normal, if a smaller van der Waals radius is assumed for the carbon atom. For this reason a value of 1.5 A (the currently accepted value is 1.7 A) was recommended for the carbonyl carbon atoms [ 1651.

In the isolated alloxan molecule the diffuse positive electrostatic potential region is around the hydrogen atom and the negative regions are around the oxygen atoms. The most negative value of potential ( - 100 (33) kJ mol-’ ) is near the 02 atom. In the alloxan crystal (Fig. 24 (c) ) the negative potential arising from the 06 and 02 atoms is compensated for by the positive contribution from the hydrogen atom in such a way that saddle points of weak positive potential are formed. The 06 and 02 points are not equivalent; the values of the potential at these positions are different. Hence, the attractive interactions in the bifurcated hydrogen bond are not equivalent, despite the small difference in the distances (AZ=0.03 A) and the almost symmetrical deformation of the electron density. The electrostatic potential approach reveals this non-equivalence.

The electropositive bridges of the electrostatic potential in the region of the hydrogen bonds were also found between a pair of molecules in parabanic acid. He et al. [ 1661 have proposed that such bridges may be a general property of short-range electrostatic intermolecular interactions.

The experimentally determined electron density can be combined with electron gas theory to determine the atom-atom potentials needed to calculate the intermolecular interaction energies in crystals [ 1701. Applications of this approach to the study of hydrogen bonds can be found in refs. 171 and 172.

CONCLUDING REMARKS

The theoretical approach to calculating the electron density distribution and investigations of the bonding in organic compounds based on accurate X-ray diffraction measurements have been described. This approach deals with problems which are traditional tasks of quantum chemistry and provides many results which justify and visualize the usual notions and models of chemical bonding. At the same time the approach reveals a number of new phenomena and demonstrates the variety of chemical bonds that exist in molecules and crystals.

Some topics, such as the chemical bonding in organometallic compounds and the calculation of the intermolecular interaction energy from X-ray data,

388

are outside the scope of this review. For such topics the reader is referred to refs. 173-175.

ACKNOWLEDGEMENTS

We thank Dr. V.K. Belsky for valuable recommendations, Prof. J.D. Dunitz for providing reprints of his works and Mrs. Z.V. Shcherbakova for technical assistance with the preparation of the manuscript.

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The electron density and chemical bonding in organic compounds by X-ray diffraction