The following complete article may be found under :
C. J. Ballhausen, Quantum Mechanics
and Chemical Bonding in Inorganic Complexes.
J. Chem. Ed. 56, 215-218 (1979).
Quantum Mechanics and Chemical Bonding
in Inorganic Complexes. I.
C. J. Ballhausen
University of Copenhagen,
Copenhagen, Denmark
I. Static Concepts of Bonding;
Dynamic Concepts of Valency
The Static Concepts of Chemical Bonding
It is a fact that chemical entities exist which are made up of
atoms and which have a well-defined composition and steric
arrangement; the atoms are "bonded" together to form
molecules. "Why?" and "How?" were questions asked by chemists
ever since Dalton formulated his empirical laws. Around the
turn of the nineteenth century the chemists were well
acquainted with the rules governing the formation of molecules
and ions from their constituents, and could taik about polar
or non-polar linkages. To this came the coordinate links of
Werner.
Valency is, of course, just a general term used to describe
the power which atoms have of combining with each other to
form molecules. We know now that chemical bonding
parameters span a wide range, from several electron volts/mole
found in the so-called "covalent" bonds through
kilocalories/mole for "hydrogen bonds" and the fraction of
kilocalories/mole observed in the "Rydberg molecules." Here I shall,
however, only be concerned with molecular bonding in the
electron volt class.
Quite early after the discovery of the electron by J. J.
Thomson in 1897 many scientists suggested that molecules
might be held together by the electrostatic force resulting from
a transfer of an electron from one atom to another. A great
advance was made by G. N. Lewis (1), who in 1916 suggested
that it was possible for an electron to be shared between two
atoms, thereby giving a stability to both. A divalent atom like
oxygen needs, for instance, two more valency electrons to fill
an octet of electrons. It can gain these by sharing two electrons
belonging to another atom or atoms, thereby filling both their
octets of electrons at the same time.
Lewis did not discuss how this sharing took place, and of
course his static model of the chemical bonding would require
some unknown forces. Sixteen years later he wrote (2)
... when I first deduced the idea of the electron pair bond from an
analysis of chemical facts... it was obviously incompatible with the
then accepted laws of electromagnetics and mechanics. The qualitative
principles of molecular structure were presented, so to speak, as
the minimum demands of the chemist which must eventually be met
by the more far-reaching and quantitative work of the mathematical
physicist.
No understanding of the mechanisms and dynamics of a
chemical bond was indeed possible before the advent of
Schrödinger's wave equation in 1926. Yet it should not be
forgotten that in a qualitative way the Lewis concept of the
chemical bonds offers a remarkably successful picture of many
valence phenomena. Its impact on the understanding and
teaching of chemistry can hardly be exaggerated.
The Dynamic Concepts of Valency
The year 1926 clearly marks the outset of a new era in
theoretical physics and chemistry. We shall pause at this moment
in the history of wave mechanics and chemical bonding and
take stock of the situation. In the preface to his book "The
Electronic Theory of Valency" published in 1927 N. V.
Sidgwick writes:
It has been suggested that the development in the last few years of the
theory of wave mechanics necessitates a fundamental change in our
views of atomic structure. This theory, in the hands of de Broglie,
Heisenberg, Schrödinger, and others, has had the most remarkable
success in dealing with problems of atomic physics ... [but] the
electron as a separate entity seems to disappear from physics.... The
theory of wave mechanics, although there can be no doubt of its value
as a calculus, has not yet reached the stage at which one can say
definitely how it is to be translated into physics.... It has as yet given
no proof that the physical concepts which led Schrödinger to his
fundamental differential equation should be taken so literally as to
be incompatible with the conceptions of the nature of electrons and
nuclei to which the work of the last thirty years has led.
What is reflected here is the reluctance to making a
transition from a pure pictorial static description of a chemical
bond to a dynamic mathematical theory. Yet the writing was
on the wall. December 17, 1926, N. Bohr brought before the
Royal Danish Academy of Science a paper by Ø. Burrau (3),
"A Calculation of the H2+ Ion in its Groundstate,"
based on the Schrödinger equation.
The problem was not new. An abortive attempt to account
for the stability of the hydrogen molecule ion had been made
in 1922 by W. Pauli (4), using the Wilson-Sommerfeld
quantization rule. It is therefore suggestive that Bohr took the
agreement between Burrau's calculation and the experiments
as providing a support for the new wave mechanics rather than
drawing attention to the importance of the paper in relation
to the problem of chemical bonding between two protons.
Burrau was inspired to his work by 0. Klein and the
Norwegian astronomer S. Rosseland. His paper was published
March 19, 1927, and it must have reached Munich, Germany,
very fast. Here E. U. Condon worked as a postdoc with A.
Sommerfeld and April 18, 1927, he communicated a paper (5)
"Wave Mechanics and the Normal State of the Hydrogen
Molecule" to the National Academy of Science. Orally the
work was presented in The Physical Society of Berlin (6) May
13, 1927.
Condon took the paper by Burrau as his point of departure.
He wrote (5)
In [Burrau's] work the tacit assumption is made that because of the
large masses of the nuclei, the problem can be solved regarding the
nuclei as fixed at a distance which is one of the parameters of the
problem.... Turning now to the neutral (H2) molecule one expects,
on the Pauli principle of assigning quantum numbers, that the two
electrons will be in equivalent orbits .... The electronic energy of this
model at each distance is evidently just twice that for
H2+.... Turning
now to the electronic interaction, the analysis of Hund provides the
important result that the electronic term of the lowest state of a
molecule changes continuously from its value for a neutral atom of
equal number of electrons to its value for the dissociated atoms.
The electronic energy of He was known to be 5.818 e2/2a0.
In the limit of small R, the distance between the protons,
Condon took therefore for the electronic energy E
E = - [ (2 EB(R)) / ( 8.000) ] * 5.818
where EB(R) is the electronic energy as calculated by Burrau.
Adding the nuclear repulsion he obtained the potential curve
for H2. This showed a dissociation energy of 4.4 eV and an
equilibrium distance of 0.7 Å in good agreement with
experiments.
Condon's attack on the hydrogen molecule is seen to contain
the first example of what was later to be called the method of
molecular orbitals (7). His papers were, however, quite
overshadowed at the time by the calculations of Heitler and
London (8) on the same subject. Their paper, executed in
Zürich, where E. Schrödinger also was at the time, was
published in what was at that time the leading journal of physics,
Zeitschrift für Physik, June 30, 1927. An account of it had
been given at a physicists meeting in Freiburg, Germany, June
12, 1927.
In their paper Heitler and London introduced what was
later to become known as the Valence Bond Method. It is
noteworthy that just as Condon had approximated the
electron-electron repulsion term, so did Heitler and London. First
Sugiura (9), working in Göttingen completed later that year
the evaluation of the 1/r12 integral.
Heitler and London's paper was immediately recognized
as a milestone in the history of chemistry. Here was found the
mathematical dynamic formulation of Lewis' covalent bond,
the energy of the electron pair bond being given as a resonance
energy due to the interehange of two electrons.
Heitler and London obtained two wave functions for the
hydrogen molecule, which in a modern language we can
write
1sA (1) 1sB (2) + 1sA (2) 1sB (1)
1sA (1) 1sB (2) - 1sA (2) 1sB (1)
The first wave function is symmetric when the electron
coordinates 1 and 2 are exchanged; the second is
antisymmetric. Only the symmetric state leads to an energy
stabilization of the two hydrogen atoms. Heitler and London
thought that the repulsive antisymmetric state corresponded
to the Van der Waals' repulsion between the two hydrogen
atoms. Had they read Heisenberg's two papers (10) on the
helium atom, they would have known that the incorporation
of the spin-coordinates of the electrons would have identified
the antisymmetric solution as a spin-triplet state. However,
the idea of an excited state of the hydrogen molecule was
evidently at that time foreign to Heitler and London.
The problem of how to construct many-electron wave
functions which behaved properly was not considered an easy
task. J. C. Slater wrote (11)
During the years 1926-28, there bad been a great deal of effort to
adapt Heisenberg's simple ideas of symmetry in two-electron systems
to atoms and molecules with more than two electrons. Wigner, Hund,
Heitler, and others had been studying the general symmetry
properties of many electron systems using group theory, but their results
were so involved that it was difficult to see through them and arrive
at practical results.... It seemed to me of great importance to see how
to tie in the ideas of symmetry and anti-symmetry of wave functions,
which Heisenberg and Dirac had formulated in 1926, with the
practical study of complex spectra....
Slater (12) discovered that by introducing the electronic
spin at the very beginning of a calculation an enormous
simplification could be achieved. The use of determinental wave
functions, built from spin-orbitals, made it easy to write down
anti-symmetric electronic wave functions of the proper space
and spin symmetry. His important paper was published in
Physical Review in 1929. By that time the German
predominance in molecular science was slowly ebbing, to be taken over
by the Anglo-Saxon countries.
Thus quantum chemistry came of age 50 years ago. Even
to sketch all that took place in the next years is not possible
in a short paper. Suffice it is to say that already in 1929 Dirac
(13) could state "the underlying physical laws necessary for
the mathematical theory of a large part of physics and the
whole of ehemistry are thus eompletely known." Schrödinger's
wave equation and the Pauli exclusion principle form the
foundation on which ultimately all chemical understanding
must rest; however, an additional discipline, group theory, is
very useful.
The first person who applied group theory to physics was
probably Wigner (14). Many years later Wigner wrote (15)
When the original German version [of the book "Gruppentheorie"]
was first published, in 1931, there was a great reluctance among
physicists toward accepting group theoretical arguments and the
group theoretical point of view. It pleases the author that this
reluctance has virtually vanished.
Wigner showed in 1927 that the rotation group of the sphere
possesses irreducible representations of dimensionality
2l + 1 (l = 0, 1, 2 ...).
Those are of course just the s,p,d ... . set of
functions. If we now insert the atom in a crystal or form a
molecule, then the symmetry of the potential energy is
reduced from spherical symmetry to the symmetry of the
position that the atom occupies in the crystal or molecule. Wigner
had indicated the way to deal with such a diminishing of the
symmetry, but the actual working out of the details was left
to H. Bethe (16).
His very important paper which introduced
point group symmetry in the handling of the electronic
structures of polyatomic molecules and ions was published in
1929, the year Bethe became 23 years of age.
The development of quantum chemistry has amply
manifested the elegance and power of group theory. Looking aside
from diatomic molecules, with which we shall not be
concerned here, Van Vleck (17)
was the first to recognize the merit
of symmetry arguments in molecular calculations on
polyatomic molecules. The same year (1933) saw a paper by
Mulliken (18)
in which extensive use was made of group
theory. Interestingly enough, in Mulliken's footnote 6 we
read:
The writer is greatly indebted to Professor J. H. Van Vleck for calling
his attention to the applicability of Bethe's results to molecular
electron wave functions
However, the application of symmetry arguments was slow
in penetrating the chemical community.
By this time the game was afoot. The first quantum
mechanical treatment of a double bond in a polyatomic molecule
was given by E. Hückel (19) in 1930. The molecular orbital
method was used in 1931 by F. Hund (20,21) to account for
the water molecule, and R. S. Mulliken (22) started his
investigations into the "Electronic Structures of Polyatomic
Molecules and Valence" in 1932.
The Valence Bond Method, generalized from the Heitler-
London treatment of H2 was shown by J. C. Slater (23) and
L. Pauling (27) to lead to directional properties of the valence
bonds. How to calculate the molecular energy levels in this
approximation was likewise looked into by Slater (24) in 1931
and Pauling (25).
The pioneering years were now over. In the first issue of The
Journal of Chemical Physics, January 1933, the Editorial by
Harold C. Urey said
"The Journal of Chemical Physics... is a natural result of the recent
development of the chemical and physical sciences. At present the
boundary between the sciences of physics and chemistry has been
completely bridged .... Most important of all, the experimental and
theoretical work associated with the quantum theory has made a
profound contribution to our knowledge of chemistry and physics.
Moreover, the history of these sciences in recent years teaches the
effectiveness of applying the exact logic of mathematics to chemical
as weh as physical problems.
For an authoritative and well-written account of just how
the "Quanturn Theory of Valence" appeared and was under-
stood in 1935,I point to the big review paper written by Van
Vleck and Sherman (26). In 1935 also came the book by L.
Pauling and E. Bright Wilson, Jr: "Introduction to Quantum
Mechanics" with the significant subtitle: "With Applications
to Chemistry." Further, from the middle thirties courses in
quantum mechanics were given in the chemistry departments
of the foremost American and English universities. Before that
time a chemist could only learn that discipline in the physics
departments.
Quantum Chemistry as a separate discipline thus came of
age in the thirties. As is well known it has flowered since. I shall
in the rest of this paper concentrate my efforts on showing how
the quantum mechanical understanding which had been
reached influenced the thinking and development of the
chemistry of transition metal complexes.
Literature Clted
(I) Lewis, G. N., J. Amer. Chem. Soc., 38,762 (1916).
(2) Lewis, G. N., J. Chem. Phys., 1, 17 (1933).
(3) Burrau, 0., Kgl. Danske Vid. Selsk. Matt.-Fys. Medd. 7, [14](1927).
(4) Pauli, W., Ann. Phys. Leipzig., 68, 177 (1922).
(5) Condon, E. U., Proc. Nat. Acad. Sci., 13, 466 (1927).
(6) Condon, E. U., Verh. d. Deut. Phys. Ges., 8, 19 (1927).
(7) Condon, E. U., Int. J. Quantum Chem. Symp., 7, 7 (1973).
(8) Heitler, W., and London, F., Zeit. Physik, 44, 455 (1927).
(9) Sugiura, Y., Zeit. Physik., 45, 484 (1927).
(10) Heisenberg, W., Zeit. Physik., 38, 411(1926);
39, 499 (1926).
(11) Slater, J. C., Int. J. Quant. Chem. Symp., 1, 1 (1967).
(12) Slater, J. C., Phys. Rev., 34, 1293 (1929).
(13) Dirac, P. A. M., Proc. Roy. Soc., A123, 714 (1929).
(14) Wigner, E., Zeit. Physik., 43, 624 (1927).
(15) Wigner, E., "Group Theory", Preface,
Academic Press, New York, 1959.
(16) Bethe, H., Ann. Physik, 5, 133 (1929).
(17) Van Vleck, J. H., J. Chem. Phys., 1, 177 (1933).
(18) Mulliken, R. S., Phys. Rev., 43, 279 (1933).
(19) Hückel, E., Zeit. Physik., 60, 423 (1930).
(20) Hund, F., Zeit. Physik., 73, 1(1931).
(21) Hund, F., Zeit. Physik., 74, 429 (1932).
(22) Mulliken, R. S., Phys. Rev., 40, 55 (1932).
(23) Slater, J. C., Phys. Rev., 37, 481 (1931).
(24) Slater, J. C., Phys. Rev., 38, 1109 (1931).
(25) Pauling, L., J. Chem. Phys., 1, 280 (1933).
(26) Van Vleck, J. H., and Sherman, A., Rev. Mod. Phys., 7, 167 (1935).
(27) Pauling, L., J. Amer. Chem. Soc., 53, 1367 (1931).
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