Paolo Mirone

University of Modena and Reggio Emilia, Department of Chemisty



At about the middle of the past century atomic orbitals made their entry into the university textbooks of general chemistry.  Shortly after, they began to be adopted also in the teaching of chemistry in the high schools of some countries: for example, within a sample of 19 textbooks issued in Italy between 1968 and 1979, only one did not present the so called ‘orbital model’ of the atom [1].  The reason of the quick fortune of orbitals in the teaching of general chemistry is to be traced to the belief of chemistry teachers that orbitals are an indispensable tool for an up-to-date teaching of the subject, especially in order to depict the electronic structures of atoms and to explain the molecular geometries.  As a matter of fact, the grounds for this belief  were rather weak.  The term orbital was introduced by Robert Mulliken as an abbreviation of the expression “orbital unielectronic  wave-function” [2].  This reveals that the orbital concept pertains to the domain of wave mechanics: the wave-function is one of the solutions of the Schroedinger equation for the hydrogen atom; it is orbital since it describes (of course in quantum-theoretical terms) the motion of the electron about the nucleus, unielectronic inasmuch as it depends on the position of one electron.  Finally the use of the term wave-function is justified inasmuch as it behaves as a wave amplitude; this means that two wave-functions can interfere in the same way as two physical waves of an identical nature, giving rise to a new wave-function whose amplitude is given by the algebraic sum of their amplitudes.  Being functions of the coordinates of a single electron, orbitals always have an approximate character in multielectron atoms since the effects of interelectronic repulsions can be accounted for only in part.


The orbital has the character of a wave amplitude, but it is not to be confused with a physical wave since its value may be a complex number.  What has a physical significance is the square of its amplitude (more exactly, the square of its modulus), which in any case is a real positive number proportional to the probability to find an electron in the given position.  The introduction of orbitals in the teaching of general chemistry at the freshman or high school level has been a true educational anomaly, comparable to the pretension of a mathematics teacher to teach the subject of derivatives without having previously explained the mathematical concept of limit.  Indeed, the attempt at presenting a typically quantum theoretical concept such as the orbital to students devoid of any knowledge of the basic principles of quantum mechanics was bound to failure, unless the concept was somehow ‘tamed’.  Of course, the process of taming could not avoid some serious distortion of the concept.  In the majority of cases, it consisted simply in saying nothing about the quantum theoretical character of the orbital and in defining it as “the complete description of the probability of finding the electron in various points of space” (i.e.  the description of  the space distribution of the electronic density), or “a region of high electron charge density”, or similar expressions.  22 university textbooks of General Chemistry were examined; 15 of these were of North-American authors and 7 of Italian authors.  Only 8 textbooks treated the topic of orbitals satisfactorily or at least in a non-misleading way. Among the remaining 14, 10 adopted the above definition in terms of electron density.


Table 1. Textbooks examined


Brady, Holum, Fondamenti di chimica (1985)

Brown e Le May, Chimica: centralità di una scienza (1986)

Campbell, Chemical systems (1970)

Chiorboli, Fondamenti di chimica (1975)

Corradini, Chimica generale (1973)

Dickerson, Geis, Chimica, materia e universo (1980)

Dickerson, Gray, Haight, Principi di Chimica(1984)/

Gillespie, Humphreys, Baird, Robinson, Chimica (1988)

Kotz, Purcell, Chimica (1994)

Mahan, Chimica generale e inorganica (1971 )

Malatesta, Chimica generale (1963)

Masterston, Slowinski, Principi di chimica (1972)

Mortimer, Introduzione alla chimica (1971)

Nardelli, Chimica generale (1970)

Oxtoby, Nachtrieb, Freeman, Chimica (1997)

Petrucci, Harwood, Chimica generale (1995)

Pimentel, Spratley, Chimica generale (1977)

Sabatini, Chimica generale (1996)

Sacco, Fondamenti di chimica (1996)

Silvestroni, Fondamenti di chimica (1996)

Sienko, Plane, Chimica (1977)

Zumdahl, Chimica (1993)


From such definitions to the identification of orbitals with charge clouds there was a short step, with the paradoxical consequence that the formation of the covalent bond had to be explained as the effect of the overlapping of two negative charge clouds which on the contrary should repel themselves (actually the formation of a covalent bond requires that two orbitals interfere constructively so that their amplitudes sum up and the passage to the probability by squaring of the resulting amplitude creates a concentration of negative charge at the middle of the bond balancing the repulsive forces between the atomic cores).  A further source of distortion is hybridization.  Many textbooks describe it vaguely as a mixing or combination of orbitals, but several others seem to regard hybridization as a real phenomenon, as shown by the use of expressions such as “the carbon atom undergoes a sp3 hybridization” or “the atomic orbitals of carbon are transformed so as to accept the best arrangement of electrons” or “the approach of H atoms to an isolated C atom causes a rearrangement of the four s and p orbitals of carbon”.  Actually, as explained by Coulson [3]:


We must not … allow ourselves to believe that it (hybridization) represents any real ‘phenomenon’, any more than resonance between different structures such as the covalent and ionic ones of a polar bond may be called a ‘phenomenon’.


Obviously, even the misunderstanding of hybridization is a consequence of having ignored the wave-like character of orbitals.  Finally, to the best of my knowledge no textbook of general chemistry treats the issue whether orbitals are observable.  Given their immaterial nature of wave-functions, they should be readily recognized as unobservable.  However, for those who accept the image of orbitals popularized by many textbooks of general chemistry the issue of their observability remains open.  In fact, the front cover of the issue of Nature dated 2 September 1999 announced “Orbitals observed” in big letters, referring to a paper by Zuo, Kim, O’Keeffe and Spence entitled “Direct observation of d-orbital holes and Cu-Cu bonding in Cu2O” and reporting the results of a study of electron distribution in the crystal of cuprite by means of X rays and electron diffraction.[4]  The authors present their results in the form of a difference map obtained by subtracting from the observed density the electron density calculated for the hypothetical crystal formed of spherical Cu+ and O2- ions.  They remark:


The correspondence between our experimental map and the classical diagrams of dz2 orbitals sketched in textbooks is striking.  All our difference maps show strong non spherical charge distributions around the copper atoms with the characteristic shape of d orbitals”.


On the other hand the editorial article [5] goes well beyond the recognition of a ‘correspondence’.  The headline states: “The classic textbook shape of electron orbitals has now been directly observed.” Moreover in the body of the editorial one reads:


For the first time the striking shape of some of the electron orbitals is revealed experimentally. The paper by Zuo et al. is remarkable because the quality of their charge-density maps allows, for the first time, a direct experimental picture to be taken of the complex shape of the dz2 orbital.


The news of the direct experimental observation of orbitals was readily divulged by a number of magazines.  Among these, Le Scienze, that is the Italian edition of Scientific American, reproduced the colour map by Zuo et al. accompanying it with a comment, entitled “Orbitali molecolari con vista” (Molecular orbitals with view), where the terms orbital and electron cloud are used as if they were strictly synonyms.[6]  The didactic journal of the Italian Chemical Society attempted to re-establish the truth with a short article [7], recalling that:


no physically observable entity may be associated with a single orbital, the only entity susceptible of direct observation being the electron cloud formed of all the electrons of an (isolated) atom or of a  molecule or of the unit cell of a crystal.


A more thorough criticism of the Nature editorial was published in the same year by Eric Scerri. [8].  He corroborates the mathematical nature of orbitals and refutes the conflation of the terms orbital and electron density.  The following quotation neatly summarizes the main thesis of Scerri’s paper:


According accepted current theory atomic orbitals serve merely as basis sets – that is, as types of coordinate systems that can be used to expand mathematically the wave function of any particular physical system.  Just as the coordinate system of x, y, and z used to describe any particular experiment in classical physics is unobservable, so too atomic orbitals are completely unobservable even in principle.


At about the same time, a paper by Wang and Schwarz [9] showed that the unobservable character of orbitals does not preclude the observation of charge distribution differences simulating their shapes.  As they state in the conclusions of their paper:


There are many cases found in the literature, especially for transition metal compounds, where an appropriately defined difference electron density is dominated in some region of space by the difference of two atomic-orbital densities, or even by a single orbital density, if also the orbital densities are appropriately chosen.


Closing his paper, Scerri writes: “As is often the case in scientific research, the appeal of realism seems to be irresistible in some quarters and this is perhaps why the reports were not seriously challenged – or at least no objections were raised in letters to the editor of any of the magazines that carried the story.”  But he adds in a footnote: “This was not for want of trying.  My letters and those of others to these editors were all denied publication”.  In this way a chance was missed for making clearer a main concept and for dispelling a misconception that has persisted for nearly fifty years in the teaching of general chemistry.



1. P. Mirone, La Chimica nella Scuola, 1980, n. 2, p. 8.

2. R. Mulliken, Physical Review, 41 (1932), 49.

3. C. A. Coulson, Valence, Oxford University Press, 1961, p. 233.

4. J. M. Zuo, M. Kim, O’ Keeffe, J. C. H. Spence, Nature, 401 (1999), 49.

5. C. J. Humphreys, Nature, 401 (1999), 21.

6. M. Cattaneo, Le Scienze, 374, 1999, p.18.

7. P. Mirone, La Chimica nella Scuola, 22 (2000), 14.

8. E. R. Scerri, J. Chem. Educ., 77 (2000), 1492.

9.S. G. Wang, E. Schwarz, Angew. Chem. Int. Ed., 39 (2000),1757.