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Friedrich Hund and Chemistry.

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REVIEWS
Friedrich Hund and Chemistry
Werner Kutzelnigg*
Friedrich Hund, one of the few physicists still alive of the generation that pioneered quantum mechanics, celebrates
his 100th birthday this year. This is an
occasion to recall how much chemistry
owes to him. The important contributions of Hund to theoretical chemistry
will not be presented in a primarily
historical context --the cooperation of
Hund and Mulliken during the development of M O theory will, however, be
duly discussed -but the stress will be on
the impact of the concepts introduced by
Hund on theoretical chemistry today.
Nearly all methods of quantum chemi-
cal calculations presently in use have a
first step based on Hund's MO theory.
Hund recognized that the use of delocalized molecular orbitals does not necessarily conflict with a description of a
molecule in terms of localized bonds.
Moreover, he formulated conditions for
localizability that are still pertinent today. The classification of the spectra of
diatomic molecules by Hund in 1925
was virtually definitive. Recently Hund's
rules, which, among other things, make
predictions about the energetic ordering
of states with different spin rnultiplicities within the same electronic configu-
1. Introduction
Few physicists of the generation that pioneered quantum mechanics have played as important a role for chemistry as
Friedrich Hund. who celebrates his hundredth birthday this
year. His name is omnipresent in "Hund's
and
"Hund's coupling c a ~ e s ' ' . [ ~ .The fact that Hund,I6] together
with Mulliken,"] laid the foundations of M O theory is also
widely known. Hund's localization criteria"] are probably only
familiar to the readers of my textbooksLg1(in particular Vol. 2)
and students who have had assignments on this topic.
This article reviews the above-mentioned contributions not
from the angle of science history, but in terms of the role played
by Hund's concepts in current theoretical chemistry.
2. Biographical Details
Friedrich Hund was born on February 4, 1896, in Karlsruhe.
He studied in Gottingen and Marburg to become a high school
teacher of mathematics, physics, and geography. Fortunately he
changed his mind and decided to pursue an academic career. He
[ * J Prof Dr. Werner Kulzelnigg
Lehrsruhl f u r Theoretische Chemie
Ruhr-Universitil Bochuin
D-43780 Bochuin (Germany)
Fax. l n t . code +734-7094109
e-mail: \*erner.huILe~nigg!~f
rz.ruhr-uni-bochum.de
ration. have aroused renewed interest.
The concept of atomic states of natural
and unnatural parity plays a key role in
this context. Hund's first two rules for
atomic states have been found to be special cases of more general rules.
Molecules can violate Hund's first rule
through spin polarization and through
the preference of different equilibrium
geometries of the singlet and triplet
states.
Keywords: history of chemistry . Hund,
Friedrich . Hund's rules . MO theory
obtained his doctorate in 1922 in Gottingen and while working
as Max Born's assistent completed his habilitation in 1925. Already two years later he became associate professor at Rostock
and a further two years later full professor in Leipzig, where
Heisenberg and Debye were among his colleagues. He spent the
winter of 1926/27 with Niels Bohr in Copenhagen as a visiting
scientist, and in 1929 was a visiting professor at Harvard University. Further stations of his career included Jena in 1946. and
Frankfurt in 1951, before, in 1956 he returned to Gottingen to
take the chair once held by his mentor, Max Born. In 1965
Friedrich Hund became Professor Emeritus. He lectured until
beyond his 90th birthday. primarily on the history of physics
from the viewpoint of a contemporary witness. At the age of 95
he gave his farewell lecture at Gottingen on "What I Have Not
Found".
Those of Hund's publications that are relevant to chemistry
stem from the years 1925 to 1933. The particularly important
papers on MO theory were written around 1927. For the breakthrough of MO theory, the cooperation with Robert s.Mulliken
was of crucial importance. This aspect will be outlined in the
next chapter.
Hund fulfilled the criteria that would have enabled him to
serve as the "seed crystal" in Germany for a theoretical chemistry based on quantum mechanics. When I asked him many
years later why he did not take this opportunity. he answered
with his well-known humility that his knowledge of chemistry
had been insufficient to take on this task. O n another occasion
W. Kutzelnigg
REVIEWS
he said-and here his humility is somewhat less apparent-that
he dealt with a topic only until all fundamental questions had
been answered. He has repeatedly commented on his role in the
early days of quantum chemistry.[’O.”l Hund speaks of the
early history of quantum mechanical theories of chemical bonding, when the primary goal was a qualitative understanding, as
“the happy days when the cream had not yet been scooped off
the top”.[’’] In answer to my question, Hund might have also
correctly pointed out that the German chemistry community in
the 30s, and even later, was hardly ready to accept a physicsbased theory. Eric11 Hiickel, another pioneer of theoretical
chemistry in Germany, had painful experiences in this field. His
theory of n-electron systems[I3. 14], which is based on Hund’s
M O theory, was scarcely appreciated by chemists in Germany,
even though his brother Walter-an
established chemistworked hard to bridge the gap between theory and the chemistry
of the day. Only in the 1960s, after Hiickel’s M O theory had
become popular in
Francc,“ 61 and the United
did it become accepted in Germany. Robert Mulliken,
who saw the foundation of theoretical chemistry as one of his
most important tasks,“’] found much more favorable conditions in the United States, as the chemistry community there was
less bound by tradition and, hence, more flexible. From hindsight, Hund’s departure from chemistry in the early 1930s might
not have been a mistake in the long run, especially as there
remained enough for him to d o in physics. But this is not the
topic of this article and is being appraised elsewhere.“ 2l Suffice
it to say that Hund’s experience with the quantum theory of
molecules could be applied to solids, for an understanding of
which chemical knowledge was not so important.
Now a few remarks on my personal contacts with Friedrich
Hund: I first met him in Paris in 1962 where he was giving a
lecture at the Cite universitaire. By chance, I met him again the
following weekend in Chartres, where we had the opportunity
for a longer conversation. When I came to Gottingen at the end
of 1964, I listened with great enthusiasm to a lecture of Hund on
the history of quantum theory. Hund was a member of my
habilitation committee, and of course the most prominent one.
In 1976 I was asked by Jurgen Hinze to give a lecture at a
symposium honoring Hund and Mulliken’s 80th birthdays
(Fig. 1) in Bielefeld. I titled it “Hund’s Localization Criteria and
Their Relevance to Chemistry”. I had apparently coined that
term and Hund did not appear to be too pleased. The gist of his
Fig. 1. Professors Mulliken (left) and Hund (right) on July 8. 1976. in Gottingen,
Germany (Photograph by courtesy of the h’c.ucn W~wf~i/i.sc/zcn
newspapel-.)
complaint was that he minded less being mentioned in the name
index of a scientific publication than in the subject index. His
place in the subject indexes of physics, however, was already
well established by then.
I remember one detail from a conversation with Friedrich
Hund particularly well. He explained to me that in order to
understand a physical problem, he first examined it in the
framework of classical physics. Then he would apply the correspondence principle. I responded that I had grown up with
quantum mechanics, that classical mechanics and the correspondence principle were rather foreign to me, and that I rather
regarded classical theory as a limiting case of quantum theory.
Hund replied that this was a good reason for the mandatory
retirement age for professors and that he would reach his in
two years.
Hund is a particularly friendly and also a tolerant man. Nevertheless, he is very critical and cannot be easily fooled. His
“stupid questions” after lectures at the “Physikalisches Kolloquien” in Gottingen were much feared, because they could be
quite penetrating.
3. Hund and Mulliken
The cooperation between Hund and Mulliken, who were of
the same age, proved particularly fruitful for the theory of
chemistry. Actually, this was no conventional cooperation, for
Werner Kutzelnigg, born on September f0, 1933 in Vienna, studied chemistry in Bonn
( Vordiplom in 1954) m d Freiburg (Diplom 1958). He obtained liis doctorate with R. Mecke
in 1960, working on I R spectroscopy. He was a postdoctora1,fellow in Paris with B. Pullmann
and G. Berthiev /1960-1963), andin Uppsala with P. 0.L6wdin (1963164). He completedhis
habilitation in Gottingen in 1967. From 1970- 1973 he was H2/H3prOfessor at the TU Krrrlsruhe. Since 1973 he has been H4/C4 profl?ssor at the Ruhr-Universitut Boclium. His i-eseurcli
interests are in theoretical clzeniistrj-, in particular the development qf new metlzorls, with the
emphasis on electron correlation in molecules, relativistic quantum chemistrj,, magnetic properties of molecules, theory qf chemical bonding, intermolecular,forces,and formnl and matliematical aspects of theories.
574
An,qeii~Chern. In!. 6 1 . O?,q/1996. 35,
573 5x6
~
Friedrich Hund and Chemistry
as Far as f know there tire no joint publications by Hund and
Mulliken. As an aside, neither Hund nor Mulliken had coauthors for most of their fundamental papers. Their cooperation
consisted mainly of a mutual stimulation, in sometimes similar
and sometimes complementary ideas. The two scientists met
several times and, in a way, always at the right time. Mulliken
has written nicely about these encounters.[191Reading MuIliken’s memoirs one is surprised at the ease with which he, in
European terms merely a young “Privatdozent”. approached
the leading figures in his field. and not just his peers in terms of
age and experience like F. Hund. Perhaps this unconventional
behavior was tolerated because he was an American, whereas it
would not have been accepted from a European. Recently Hund
also commented in some detail on his cooperation with Mulliken in an introduction to Mulliken’s autobiography Lzfi o f u
Scic~nti,s~.[’o’
Mulliken had made several trips to Europe during
each of which he visited a number of countries such as England,
France, Denmark, Germany, Switzerland, the Netherlands,
Austria, and Italy. The itinerary of his first European trip in
1925 included Gottingen, where he met Hund for the first time.
Hund had just finished his paper “Zur Deutung einiger
Erscheinungen in den M~lekel-Spektren”,[~I
in which he introduced the coupling cases later named after him, and was working on his book on atomic spectra.[31Two years later Mulliken
was in Gottingen again, and the cooperation with Hund on
molecular spectra continued, also during a hiking trip to the
Black Forest. Mulliken’s publication “Assignment of Quantum
Numbers for Electrons in Molecules”, Part I and II.[’] appeared
in 1928. He stated that the main ideas and methods were those
chat had been previously successfully applied by H ~ n d . How‘~~
ever, he went beyond Hund’s approach. He sent Hund a copy of
the manuscript. upon which Hund answered that it overlapped
closely with a paper that he had just finished. He did not want
to duplicate anything and therefore left out all those details
from his own manuscript[211that he considered Mulliken had
dealt with in his work. In the summer of 1929 Hund, Heisenberg. and Dirac were guests in Chicago, where Mulliken had
recently been appointed full professor. In 1930 Mulliken traveled to Europe again and spent a large part of his time in Leipzig
as Hund’s guest. Erich Hiickel was also in Leipzig during this
period. In 1930 Hiickel’s publication “Die Quantentheorie der
Doppelbindungen” appeared[131and in 1931 his habilitation
paper “Elektronenkonfiguration des Benzols und verwandter
Verbindungen”.[ 14] Hiickel expressly built upon Hund’s work.
Mulliken was again in Leipzig in 1932/33, after which he and
Hund did not meet again until 1953. when Hund was Professor
at Frankfurt University.
Attempts to disentangle Hund’s and Mulliken’s contributions
to the development of molecular orbital (MO) theory might
attribute to Hund a primary interest in the fundamental physical
concepts, whereas Mulliken was concerned with the actual application to problems of molecular spectra and to the theory of
chemical bonding. Hund much more than Mulliken was involved in comparing alternative approaches. One should also
not forget the contributions of other scientists, in particular
HerzbergI”’ and L e n n a r d - J ~ n e s , ‘to
~ ~the
] development of M O
theory. At about the same time as MO theory was developed,
another approach to the theory of molecules was proposed primarily by F. London.[z41It came to be known as the theory of
REVIEWS
spin valence, and is based on the idea of constructing the
wave function of a molecule from those of the separate atoms.
This is a generalization of the successful Heitler-London approach for the H 2 molecule (which will be discussed briefly in
Chapter 4) .[251 A related approach came from Heitler and
Rumer,[261and an important paper by Wigner and Witmer
should also be mentioned in this context.[271Another method
based on the Heitler-London approach for H, is the valence
bond (or HLSP method) of Slater1281and Pauling,[2Y1
in which
Heitler -London type pair functions are assigned to the individual bonds.
While the theory of spin valence never became very popular,
the valence bond (VB) theory turned out to be a serious competitor of M O theory. It was so successful, because it could be
viewed as the visualization of the concept of mesomerism, which
was then very popular in chemistry. Moreover. Mulliken’s comprehensive papers[’8. 311 were relatively difficult to read, in contrast to those of his opponent Linus Pauling.[’”- 301 who quickly
swayed chemists toward his theory. This is not the place to
review critically the role of Pauling in the development of theoretical chemistry. Pauling certainly had a profound knowledge
of quantum mechanics, but he was not averse to easy
success, and did not always mind dubious simplifications;
in critical situations, he relied more on his chemical intuition than on rigorous theory. It is in part due to Pauling’s
influence that the breakthrough of the MO theory took such a
long time.
While today we use the terms molecular orbital theory o r MO
theory, coined by Mulliken. Hund referred less spectacularly to
a “Zuordnungsschema” (correlation scheme) .16. ’’I A key concept in his theory is what is now generally called a correlation
diagram. Hund was the first to realize that molecules can be
constructed from molecular orbitals, even if he did not yet use
that term. He correctly classified the molecular one-electron
states according to symmetry and correctly gave occupation
schemes for molecular configurations. Instead of looking for a
way to calculate the molecular orbitals explicitly, Hund essentially proposed interpolating them from the states of the united
atom and those of the isolated atoms by means of a correlation
diagram. From these diagrams (Fig. 2) one can see which MOs
are
and which antibonding (”loosening”). Coulsaid of a similar correlation
that it deserved a place next to the periodic system of the elements in
every chemical institute.
The proposal to represent molecular orbitals as linear combinations of atomic orbitals (LCAO) goes back to LennardSuch an approach had previously been used by
P a ~ l i n g for
[ ~ ~the
~ H i ion. At this point. it might be noteworthy that the first quantitative MO calculations for diatomic
molecules were performed at the end of the 1950s at the
Mulliken Institute in Chicago by Clemens Roothan and
In 1966 Mulliken received the Nobel Prize in Chemistry for
his work on MO theory. He repeatedly stressed that he would
have liked to share this honor with his friend Hund. Erich
Hiickel would have also been a worthy corecipient.
Mulliken[”] mentioned that he never could understand why
Hund (like, by the way, Sommerfeld) used the term “die
Molekel”, while all other German-speaking colleagues referred
575
W. Kutzelnipg
REVIEWS
atoms. One then realizes that a(F2)b(Fl)is an equivalent description and that, for reasons of symmetry, the wave function must
be either a ( + ) or a (-) linear combination of these two primitive product functions. The term Fl stands for the spatial coordinates of the first electron, 1’; for those of the second electron.
Hence there are, starting from the 1s functions of the isolated
hydrogen atoms, two approximate wave functions for the molecule (Eq. (a)), where N , is a normalization factor, which is
Fig. 2. Correlation diagram for the one-electron states in a homonuclear diatomic
molecule (from Ref. [6d]).
to “das Molekiil”. A purist ought to argue that the correct
German equivalent of the Latin “molecula” or French “la molecule” should be “die Molekel” in analogy to the Latin word
“particula” (in French “la particule” and in German “die
Partikel”), and in contrast to “calculus” (“le calcul” and
“der Kalkiil”). How “molecula” became “das Molekiil” is hard
to comprehend. Anyway, terminology has become established
and “die Molekel” sounds artificial. An analogous situation arises in English: why is the word “molecule” and not
“molecle” ?
4. The Theory of Molecular Orbitals (MO Theory)
Even though Hund did not formulate M O theory-at least in
his first papers-in terms of the LCAO approximation, this
approach should be discussed here in this article, because it
allows a clear comparison with the valence-bond method. Both
the VB and the M O theory of molecules start with the theory of
atoms. In the VB approximation, one attempts to construct the
wave function of a molecule directly from those of the individual
atoms. In M O theory, on the other hand, one regards the molecule as a unit and attempts to describe it like an atom, with the
difference that several nuclei are present rather than one. In the
older German literature the term “Methode der Molekiilzustande” (method of molecular states) was popular for a while.
The direct translation of Mulliken’s term molecular orbital
(Molekiilorbital) is now standard, even in German. It is useful
to recapitulate briefly the difference between the VB and MO
descriptions using the H, molecule as example.
In view of the present special occasion I repeat something here
that a reader of Angewandte Chemie is expected to be familiar
A look at
with. More details can be found in my
overviews from the 1930s can also be recommended, in particular the book by H e l l m a r ~ n and
’ ~ ~ the
~ article by VdnVleck and
Sherman.[361
Let a and b be wave functions of the two isolated hydrogen
atoms. In VB theory (which in this case is identical to the approach by Heitler and London),’”’ one first formulates the
product a(Fl)b(Fz)-which would be exact for noninteracting
516
different for the two functions. If these two wave functions are
used to calculate the energy expectation value as a function of
the internuclear distance R, as was first done by Heitler and
London,f251one obtains two potential curves. One of them is
bonding and therefore describes the ground state of the H,
molecule. while the other one is repulsive for all internuclear
distances and does not represent a bonding state. Qualitatively,
the bonding potential curve (ENJof the ground state of the H,
molecule is rather similar to the exact potential curve (E,,,
Fig. 3).
-0.8
‘
I
I
1
t
ElHartree
Ria,----,
Fig. 3. Potential curves of the H, ground state according to the Heitler--London
approximation, the LCAO-MO approximation. and the C1 approximation, as well
as the exact curve.
As mentioned above, in M O theory one regards the H, molecule as a unit. It is then described by a configuration of molecular orbitals, just like the He atom is approximated by a configuration of atomic orbitals. Corresponding to the ground state
configuration of the He atom of 1s2, the ground state configuration of H, is lo,”.
Hence, the lowest-lying molecular orbital lo,
is doubly occupied. This molecular orbital is represented by a
linear combination of the orbitals a and b of the isolated hydrogen atoms [Eq. (b)]. An antibonding MO lo, can be constructed analogously [Eq. (c)]. The factors N, and N,, serve to normal-
A n g w . Ckem. Int. Ed. Engl. 1996. 35. 573-586
Fricdricli 1Hund and Chemistry
~
ize each M(I to 1. Hence, the MO wave function of the H,
ground state IS given by equation (d).
A calculation of the energy expectation value as a function of
the distance R between the two H atoms yields again a potential
curve ( ELcAOMO) with a minimum in the vicinity of the actual
equilibrium separation of the H, molecule. That is, the M O
approach also describes the bonding (Fig. 3).
A comparison of the VB and M O functions for the H, ground
state shows that the VB function is significantly better (Fig. 3).
Both functions exhibit a minimum in the vicinity of the actual
equilibrium separation in the H, molecule, but only the VB
wave function behaves correctly for large distances, that is.
the energy converges to the sum of the energies of two H atoms.
The M O energy shows a physically unreasonable behavior at
large internuclear separations; it is much too large. This defect
of M O theory can be corrected, as the following discussion
shows.
As is the case for atoms, molecular states may only be described by a single configuration (such as lo:) if this configuration is clearly separated in energy from other possible configurations. For large internuclear distances, however, the
configurations 10: and lo,’ have nearly the same energy (and
the same overall symmetry, ‘2: ; the configuration 1 o,l ouhas
a different overall symmetry, ‘Z;, and does not mix in). In this
case, one must not use a single configuration to describe the
molecule. but rather a linear combination of both configurations [Eq. (e)].
This is called configuration interaction (CI). The potential
curve that belongs to the MO-CI wave function (e) (Eel in Fig. 3)
gives the correct behavior for R --+ CCI and is overall a slightly
better approximation for the exact potential curve than the VB
approach. A proponent of the VB approximation would not
give up at this stage, but would point out that a further improvement of the VB approach is also feasible. This consists of adding
ionic structures. such as a(Fl)a(T2) and b(F1)b(F2),
where, in a
sense, both electrons are located on the same atom. Indeed it can
be shown that the MO approach with full configuration interaction and the VB approximation including all ionic structures are
exactly equivalent . K3 ’1
Once one proceeds beyond the H, molecule to more complex
molecules, the M O approach does have some advantages over
the VB description. Specifically, these are:
a) While the simple VB description fails for an ion-pair
molecule, such as LiH, the ionic bond results naturally from
M O theory. The valence M O of LiH is localized primarily
on the H atom, that is, the M O description reflects the idea
that a Li’ and a H- is present. between which there exists a
Coulombic attraction. The VB theory, however, requires the
inclusion of ionic structures even at the lowest level of approximation.
b) M O theory automatically yields the tetravalent nature of
the carbon atom, for example in CH,, while the VB approximation requires additional assumptions : in particular, a “valence
A n f i m Cherii
Iii/
td Engl 1996, 35. 573-586
state” is to be formed initially, and the atomic orbitals must
hybridize before the chemical bond can be dzscribed.[28-”1
Neither the concept of a valence state nor that of hybridization
is required within the M O approximation.
c) Delocalized bonds, such as in benzene or even in H:,
follow automatically from the M O approach- -the molecular
orbitals are delocalized over the entire molecule---while in VB
theory, one has to postulate the resonance of several valence
structures. Hii~kel.[’~,
14] in particular, recognized the potential of M O theory in this area and contributed greatly to its
popularization (even if this occurred only relatively late in Germany).
d) The classification of the spectra of molecules, as well
as the description of optical transitions, is much more straightforward with the M O theory than with the VB approach. The
description of spectral transitions as excitations from an
occupied to an unoccupied MO is often a good approximation. This is one of the reasons why Hund and Mulliken, who
initially studied the analysis of molecular spectra. preferred MO
theory.
e) The similarity in physical properties of isosteric compounds like N, and CO is readily understood on the basis of
M O theory. whereas this is rather more difficult in the VB approach.[221
f ) M O theory can be much more easily and better handled
than VB theory when it comes to practical calculations. Certainly, to Mulliken this argument was as important as statement d),
because he was interested in real calculations of molecular structures.
g) Woodward and H ~ f f m a n n [ ~ ’intentionally
I
based their
theory of the orbital-symmetry control of electrocyclic reactions
on MO theory. They employed correlation diagrams similar to
those used by Hund, with the difference that the correlation was
not between separated and united atoms, but between starting
materials and products of a chemical reaction.
From Hund’s point of view, a weak point of the Heitler-London approximation, which forms the basis of the method of spin
valence[241as well as of the VB method, was that it was originally formulated as a perturbation theory. The noninteracting
atoms represented the unperturbed system. The interaction of
the atoms is a rather unusual perturbation, as it also entails a
change in symmetry. The extension of perturbation theory to
the next higher order[391caused unexpected difficulties, which
were understood only later.[40-431 It is now known that
such a perturbation approach does not converge except for
H
: and H2,[40344,451
even if the first order exists and is well
defined.
One has to accept a certain hierarchy of approximation steps
for both MO and VB theory. In the early days, it only appeared
feasible to formulate theories with minimal basis sets.
The considerations about the H, molecule given above are
based on such a minimal basis set, which is constructed from the
I s orbitals of both the two H atoms. For the CH, molecule a
minimal set for the C atom would include the orbitals Is, 2s, 2p,,
2p,, and 2p,. Minimal basis sets never played an important role
in a b initio MO calculations. The results obtained in this way
are too inaccurate, and the extra effort with an enlarged basis
set, for instance of the double-zeta kind (with about twice as
many functions than the minimal set), is not all that much
577
A!!
REVIEWS
bigger. In qualitative approaches o r in a semiempirical version
of the theory (where the integrals are not solved but used as
adjustable parameters), one often employs minimal basis sets,
for example in the C N D O (complete neglect of differential overlap) approximation1461or in EHT (extended Hiickel theory) .I4’]
While the semiempirical variant of M O theory is more or less
consistent, there were fundamental difficulties with semiempirical VB theory, which were related to the nonorthogonality of
VB structures. It was often incorrectly assumed that these structures were, in fact, orthogonal. One then attempted to compensate for this error by using physically unrealistic values for the
exchange integrals (these exchange integrals in VB theory are
not to be confused with the true two-electron exchange integrals
that will play a role in the discussions in Chapter 6)
In an a b
initio version of VB theory, it was necessary to explicitly account
for the overlap integrals between valence structures, which lead
to significant practical difficulties. This was one of the reasons
why a b initio VB calculations have only played a marginal role.
It was also unclear for a long time how a VB approximation
using a non-minimal basis set should be formulated.
The M O calculations usually performed today employ basis
sets that have very little in common with the atomic orbitals of
the original LCAO-MO approximations. They can, however, be
systematically enlarged to approach a complete basis set, and all
occurring integrals can be readily evaluated.1481This approach
led to the S C F approximation o r molecular Hartree- Fock
theory, which began its conquest of this field in the 1960s. If
one is interested not only in geometries close to the equilibrium
structure, but also in the behavior at large distances from it-in
which case M O theory becomes invalid in the single-configuration approximation-the SCF approximation has to be improved by including configuration interaction (CI) .[491 In practical terms this is often difficult, but does not present any
fundamental problems these days.
There have been repeated attempts to revive the VB approximation. Recently, the modern valence-bond theory has been
propagated, in particular by Gerratt.1501In this approach, one
retains the formal structure of the VB functions; however, the
atomic orbitals are not those of isolated atoms, but are strongly
deformed. They are determined by the requirement that the
total energy becomes minimized (similar to the MOs in a b initio
S C F theory). For quite some time now, Goddard[”] has preferred the generalized valence bond (GVB) method. The term
“generalized” is somewhat misleading, because the GVB structures are chosen to be orthonormal, which is a loss of generality
relative to the original VB structures.
Nowadays, the choice between MO and VB has become obsolete. For precise calculations one uses the MO-CI method,
which includes the VB approximation as a special case, or CIlike methods of the coupled-cluster (CC) type.1521If very high
accuracy is desired, neither MO-CI nor C C procedures are sufficient, and one has to employ wave functions that depend explicitly on the interelectronic coordinates.[53- 561
It should be realized that a b initio M O theory, including its
improvements such as CI and CC, have a serious rival in density
function theory.r5’I This theory has been very popular in solidstate physics for some time, but is now also conquering chemistry. In essence, however, this alternative approach is also
based on the M O concept. Only the electron interaction is
578
Kutzelnigg
___
evaluated differently from the usual ab initio methods. At least,
one can interpret density function theory in this manner. Formally, the exchange interaction of MO theory (including selfexchange) is replaced by an exchange-correlation function. This
has the effect of partially compensating for the errors of M O
theory and of approximating correlation effects.
From today’s vantage point, one can say that the MO approximation, proposed by Hund and Mulliken in 1927, has
become fully accepted and the method of choice, although for a
long time the VB approximation of Heitler, London, Slater, and
Pauling was the more popular. Nowadays, M O theory is the
basis for the qualitative understanding of the chemical bond and
the properties of molecules, as well as the first step in sophisticated ab initio calculations.
5. Hund’s Localization Criteria
The molecular orbitals of M O theory are generally delocalized over the entire molecule. The crystal orbitals in a solid
behave similarly. In many cases, this delocalization is unavoidable and physically significant, for example in the case of metallic bonds. In other cases, however, delocalization conflicts with
physical or chemical intuition. There are molecules and even
crystals where the indications are that the bonds are localized.
Hund was one of the first to recognize that there is no inherent
contradiction between a localized and a delocalized description.
The important issue is that some situations can be described
with localized or delocalized functions, while others only correspond to the delocalized description. Diamond would be an
example of the first case, sodium metal an example of the second.
In 1931 HundC8lsucceeded in formulating criteria that indicate when an equivalent localized formulation can be valid in
addition to the always applicable delocalized description. The
equivalence of the two approaches-if the localization criteria
are satisfied-has a simple explanation, which was pointed out
much later, in particular by Lennard-Jones et al.[581A wave
function in the form of a single Slater determinant is invariant
with respect to a unitary transformation of the occupied orbitals. This is best explained with the example of the BeH,
molec~Ie.[~]
This linear, symmetrical molecule has in its ground state the
M O configuration lo:20:lo~, where the lo, M O is essentially
identical to the I s A 0 (K shell AO) of Be, and 20, and lo, are
symmetry adapted and delocalized over the entire molecule. In
the LCAO approximation (that is, using a minimal basis set),
20, and lo, can be approximated by equations (f) and (g).
Here, / I , and h, represent the Is AOs of the two H atoms, while
s stands for the 2s orbital a n d p for the 2p, orbital of Be ( z is the
molecular axis). The symbols cl -c4 are coefficients.
Contour diagrams of the 20, and 1 ouMOs resulting from a
MO-SCF calculation are depicted in Figure 4a. If one assumes
that s and p are equally capable of forming bonds to the H
Aiigru.
~ h m l Inr.
.
Ed. Enxl. 1 9 6 . 35, 573-586
Friedrich
Hun? 2nd Chemistry
- .-
REWIEUUS
atoms -which implies that L , z c 3 and ~ , % ~ , - - a n dtakes into
consideration that the total wave function does not change
when 70, and 1ouare replaced by linear combinations, for example (p, and (0, according to Equations (h) and (I), one can
cp -
1
-
r
(20, +lo,) % C,1/2 h ,
C
+2
+p)
fi
(J
I-p2
cp - ~ ,1 ( 2 0 , - l o , , ) ~ C i ) / z / 1 2 + ~ ( s - ~ )
1/2
,-12
now recognize that cpI and cp2 are two approximately localized
two-center molecular orbitals. Only the 1s A 0 of one H atom
and one sp hybrid orbital of Be participates in each MO. This
means that in this case a localized description is possible. Figure 4b shows the contour diagrams of the localized MOs 'pl and
-0.04
0.04
Let us now remove an electron from BeH, a id form the ion
B e H l ! Now Be only provides one valence elect 'on and Hund's
localization criteria are no longer satisfied. Inde$:d, now the M O
20, is doubly occupied, while lo, is only singl) occupied. One
can still construct the localized MOs according to Equations (h)
and (i), but there is no configuration of localized MOs that
would be equivalent to 20,210,. Hence, BeH: can only be described as delocalized.
Further examples can be found in my textbook.[g1There one
can also read how the localization criteria must be modified
when semipolar or multiple bonds are present. The possibility
that three critical numbers cannot be uniquely assigned is also
discussed there.
Methane (CH,) and diamond are examples of systems that
can be described by localized two-center bonds: the three numbers r, n, t are equal to 4 for each C atom. The situation is
different for benzene or graphite. If one looks at the o-system,
all three numbers are equal to 3. In the case of the n-system,
however, two neighbors are present in benzene and three in
graphite, while per carbon atom only one electron and one pA 0 are available. Hence, a description using localized n-bonds
is not possible. An extreme case in terms of non-fulfillment of
the Hund's localization criteria is illustrated by a cubic-closestpacked lattice of atoms with one valence electron per atom. The
number of neighbors is 12, the number of available valence
orbitals is 4 (s, px, p,. pJ, and the number of available valence
electrons per atom is 1. Of course, it is a characteristic of the
metallic bond that it cannot be described by localized two-center
bonding.
6 . Hund's Rules
Fig. 4. Contour diagrams of the MOs of the BeHLmolecule: a) canonical (delocalized). b) equivalent (localized) (from ref. [ 9 ] ) .The axes represent the distance from
the Be core in unitc of u o .
'p,. The localization criteria that Hund formulated can be illustrated for this example. For each atom, three numbers have to
be identical: 1 ) the number r of bound neighbors, 2) the number
n of available valence electrons, and 3) the number t of valence
atomic orbitals participating in the bonding. In BeH, these three
numbers are 2 for the Be atom and 1 for each H atom.
The requirement that r , n, and t must be equal is a necessary
condition for localization, which can be seen when one arbitrarily changes one of them. Let us assume, for example, that s and
p do not participate equally in the bond, because. for example.
p is too high in energy or overlaps poorly with h , and 12,. The
conditions e l z ('3 and c2z c4 would then no longer be valid, and
the equivalent MOs (p, and ' p 2 , which could still be constructed
according to Equations (h) and (i), would no longer represent
two-center orbitals, but rather a four-electron, three-center
bond. This is approximately the case in MgH,, where the 3p
orbital is much less involved in bonding than the 2p orbital in
BeH2. In this case, M g only provides one valence AO. Of
course. BeH, as well as MgH, fall between the limiting cases of
systems with two-center bonds and those with a four-electron,
three-center bond. BeH,, however, is closer to the first case,
whereas MgHz is closer to the second one.
Friedrich Hund's name appears in the literature probably
most frequently in connection with Hund's rules. This is somewhat surprising as these rules originate from one of Hund's
earliest scientific publications : "Zur Deutung verwickelter
Spektren, insbesondere der Elemente Scandium bis Nickel"[']
("On the interpretation of complex spectra, particularly of the
elements scandium to nickel") from 1925. The quantum mechanical tools to interpret these spectra were then available in
rudimentary form only. One was aware, for example, that the
angular momentum in the central field is a constant of motion,
and the Pauli principle was known. Quantum mechanics enabled the possible terms for a given electronic configuration to
be stated, but it was necessary to resort to empirical analyses of
spectra to order them in terms of energy. Hund made the remarkable observations that (as quoted from ref. [l]) ( I ) the
energetically lowest lying term of an electron configuration always has the highest possible spin multiplicity (2) among the
terms with the highest spin multiplicity, the one with the largest
angular momentum quantum number L appears to be lowest in
energy.
Hund later formulated these rules in somewhat more precise
but did not give a quantum mechanical explanation.
He did not state under which conditions these rules are valid,
even though one can conclude from the context that they apply
to energetically low-lying configurations with at most one open
shell with />O. that is, for example to ls2p or ls22s22p4.For
519
W Kutzelnigg
REVIEWS
such configurations, Hund's rules hold surprisingly well. Those
who have criticized Hund's rules have often intentionally assumed that they are valid almost unconditionally in order to
find many exceptions.
A few years after the publication of Hund's rules, Slater presented his theory of the spectra of complex atoms.[591If one
restricts this theory to two-electron configurations with only a
single angular momentum L, one immediately notes that, in
agreement with Hund's rules, the triplet state is lower in energy
than the singlet state. Slater, however, also found configurations, albeit having more than one open shell with 1>0, that
apparently contradicted Hund's rules, while his theory was in
accord with experimental results. Slater therefore declared
Hund's rules obsolete. Hund himself no longer mentioned them
in his 1933 review article for the Handbuch der Physik.[601Despite their popularity, Hund's rules are mentioned almost by the
way in textbooks on the electronic structures and spectra of
atorns,L6' - 6 3 1 and caution is advised when applying them.
Hund also formulated a third rule,[2.'I which is less commonly associated with his name. This rule concerns the spin-orbit
interaction and predicts whether a normal or an inverted multiplet is present. Even though the explanation of this rule requires
relativistic quantum mechanics, which did not yet exist in 1925,
Hund quite correctly traced this third rule to a magnetic interaction between angular momentum and electron spin.
It is remarkable that Hund's first two rules have enjoyed
continuous popularity until now, and that one usually turns to
them first when the ordering of energy levels in a configuration
needs to be predicted. Apparently the prevailing opinion is that
Hund's rules are based on some fundamental physical principle,
even if one does not fully understand it. Particularly for the
theory of molecules Hund's first rule has become invaluable,
though Hund did not initially consider this aspect (the second
rule does not apply to molecules).
In order to understand Hund's rules. it is useful to look at the
special case mentioned above where only the first rule is relevant, because a configuration then has, for example, only a
single value of L (like in a 1s2s or 1s2p configuration) o r because
L is not defined (as in molecules). We shall limit the discussion
to a two-electron system (or a system with two electrons outside
a closed shell). For the configuration 'pI'p2, a singlet and a
triplet wave function can be constructed; the spin-independent
parts are given by (j) and (k) and the corresponding energies by
(I) and (m) where h, and h, are single-electron contributions.
'E = h,
'E
=
h,
+ h, + (11122) + (12121)
+ h, + (11122)
-
(12121)
(1)
(m)
(11122) is a Coulomb integral, and (12121) is an exchange integral. It can now be shown that such exchange integrals are
always
This directly implies that 3 E is lower
than E.
580
In this proof, it was assumed that the two states can be described exactly by (j) and (k) and that the orbitals q , and q, are
the same for both states. Both assumptions are only approximately justified.[66.671It can be
that 3 E < 1 E remains valid if one allows different orbitals for the two states.
Even though in such an improved description the difference
between (I) and (m) is no longer equal to twice the exchange
integral (12121). this integral remains the critical quantity that
determines the energetic ordering. If one deviates from the
single-configuration description, ' E < E is no longer rigorously
valid, that is. electron correlation effects (not accounted for in
the single-configuration model) can lead to violations of Hund's
first rule. These kinds of violations are rather rare, however. The
violations of Hund's first rule due to the spin polarization[681
are more interesting. These arise for certain molecules, but
rarely for atoms. and will be discussed in more detail at the end
of this chapter.
At this point, it is tempting to see an analogy between the
singlet and triplet functions (j, k) of an atom and the VB wave
functions (a) of an H, molecule. A fundamental difference, however, lies in the fact that the two functions 'p, and 'pz in Equations (j) and (k) are orthogonal, so that the energy difference
between E and 3 E according to Equations (I) and (m) is given
by a genuine two-electron exchange integral. Such integrals are
positive, so ' E must be lower in energy than ' E . The atomic
orbitals a and h of the VB function for H, are not orthogonal,
but overlap strongly, and this overlap is at the very origin of the
bonding. Furthermore, the exchange integrals of the VB approximation still contain single-particle contributions. All this
leads to the a bonding singlet state lower in energy than the
antibonding triplet state. This has nothing to d o with Hund's
rule. One of the conditions for its validity is the orthogonality of
the involved atomic o r molecular orbitals.
Things become more complicated when several values of L
are possible for a configuration, that is, when both of Hund's
rules must be considered.
The simplest example is the configuration p2, for which the
energetic ordering of the possible terms is 'P < 'D < 'S. The fact
that 'P is lowest in energy is apparently in agreement with
Hund's first rule, but is not immediately obvious in light of the
considerations discussed above. One could come to the conclusion that the relationship I D < 'S is a manifestation of Hund's
second rule. The energetic ordering of terms for a d2 configuration. 3F< 'D < 'P < 'G < 'S, indicates that this is indeed premature. As required by Hund's first rule, a triplet term, 3F,is lowest
in energy, and the second rule applies as well inasmuch as
'F < 'P. The ordering indicates, however, that the second rule
does not apply to terms other than those with the highest multiplicity (in this case, not for the singlet terms, as 'D lies lower
than '(3). It is rather strange that Hund's second rule should
apply to the triplet, but not the singlet terms. Things get even
more intriguing when we consider states with two open shells,
like pp' (for example 2p3p) or pd configurations. For pp', there
are singlet and triplet terms for P, D, and S. The lowest-lying
term in this case is 'P, a clear violation of both Hund's first and
second rules.
This was known to some extent in the late 1920s. Russell and
MeggersL6'] formulated the alternating rule especially for twoelectron states: Among the two-electron terms from two open
'
'
A n g m . Cliern. Int. Ed. Engl. 1996. 35, 573-586
Friedrich Hund and Chemistry
REVIEWS
shells uith minimum and maximum total angular momentum
quantum numbers defined according to (n) and (o), something
similar to Hund's first rule holds for L,,, and Lmi,,as well as
L,,, - 2. L,;,, - 4, . . . Lmi, 2, that is, the triplet is lower in
energy than the singlet, while for L,,, -1, L,,, - 3 etc., the
order is inverted, and the singlet lies below the triplet. Hence, for
pp' configurations 3 D < ' D , ' P < 3 P , and 3 S < IS. This rule does
not, however. predict why 'Pis the lowest of all states.
There have been attempts to make the alternating rule plausible, but a convincing explanation has only been given recently.[hsl The key lies in the distinction between states of natural
and unnatural parity.
The parity of' an atomic state is defined as its behavior upon
the coordinate transformation F- -f It is readily apparent
that orbitals with I = 0, 2. 4 (that is, s,d,g.. .) have even parity,
while those with I = 1.3. 5 (that is, p,f,h. . .)have odd parity. For
a multielectron atom, the parities of the orbitals must be multiplied. Hence the configuration p z has even parity, the configuration p" odd parity. d 2 and d 3 are both even. d p is odd, etc.
While for one-electron states, every angular momentum / h a s
exactly one associated parity. both parities (even and odd) are
possible for each L for multielectron states. It is useful to label
states like S,. P,,. D,. etc. as having natural parity, while S,, P,.
D,, etc. are labeled ;IS states of unnatural
For oneelectron states. only natural parity is possible. that is, s,. p,, d,,
etc. Among the states of configuration dZ (which are all even),
IS,, ID,, and 'G, have natural parity, whereas 'P,. 3F, have
unnatural parity. In this case. all triplet states show unnatural
parity. For the configuration dd', however, IS,. 3S,, 'D,. 3D,,
IG,, and 'GX have natural parity. and 'P,. 'P,. IF,, 'F, have
unnatural parity. The alternating rule can now be formulated as
follows: Among the terms of a two-electron configuration with
the same L. the triplet state is lower in energy for states of
natural parity. while the singlet state is lower in energy for states
of unnatural parity.
This statement has been rigorously proven. under the assumption that the states could be described by a single configuration.'h51The proof contains several complicated and nontrivial steps and will not be given here. However, a fundamental
physical concept that forms the basis for the alternating rule will
be presented here.
A discussion of the analytical behavior of the solutions of a
two-electron Schrodinger equation reveals that in the limit
r l z - 0 ( r l z is the interelectronic distance), there are exactly
three possibilities:[691
I ) $ ( / - , , ) - r y 2 = const: relative s wave
2) $ ( r , , ) - r i 2 : relative p wave
3) $ ( r I 2 ) I-;,
: relative d wave
Case 1 is realized for a singlet state of natural parity, case 2 for
a triplet state (either natural or unnatural parity), and case 3 for
a singlet state of unnatural parity. If one plots the probability
density that the distance between theelectrons is close to r l Z= 0
(Fig. 5 ) . one recognizes the typical correlation cusp for case 1.
the well-known Fermi hole that keeps the electrons apart, for
+
-
-1
0
r12
1
Fig. 5. Schematic representation o f the probability density Q for linding t w o elecfor a singlet state of natural parit) (Cop). a triplet stilte
trons at ii distance
(center). and a singlet state of unnatural parity (bottom).
case 2.and- -and this is the surprising feature- - an even broader
symmetry hole for case 3.
In a singlet state of unnatural parity, the electrons avoid each
other even more strongly than in a triplet state. which results in
a reduction of the electron-electron repulsion (assuming that
both states have the same orbitals). This makes i t at least plausible why the energy of such a state is lower than that of the
corresponding triplet state.
Hund's 100th birthday stimulated me to continue earlier joint
work with J. D. Morgan and think in more detail about Hund's
rules.[65.691 concentrating mainly on their physical foundation,
range of validity. and extension to situations where they d o not
apply in their original form.1701
This collaboration was the source of some surprising results,
which confirmed that, up to now, no rigorous explanation of
Hund's second rule has been attempted. Details can be found in
a publication on this
The pertinent results are the
following:
1 ) The ordering of the mean values of the energy for the
singlet and triplet terms of a two-electron configuration for the
same angular momentum quantum number 1, (as functions of
L ) is governed by two effects:
a) the electrostatic interaction of the non-totally symmetric
parts of the orbital density, which are dominated by quadrupole-like charge densities. This interaction is determined by the
coefficient n,(L) of Slater's theory.r59]This n,(L) is a fourthorder polynomial in L. The minimum of u,(L). tor sufficiently
large I, and / 2 . is found when (p) is valid.[701
b) the short-range electrostatic interaction, which is markedly
smaller for states of unnatural parity than for those of natural
parity, as only for the former does the probability that
Fl/rl = F2/r2 vanish.[701
2) The alternating rule determines how the singlet -triplet
mean values are split into a singlet and a triplet terin.lh5."1
The energetic ordering of 'P < ID < 'S for p2 configurations
and 'F < D < 3P < 'G < 'S for d 2 configurations follows directly from these new rules. One finds, however. that, for example for an h' configuration, Hund's second rule no longer
applies (the ground state is not 'M with L = 9. but 'K with L
=7). It also follows that the ground state for a 2p3d configura58 1
W. Kutzelnigg
REVIEWS
tion is 'P. Our analysis suggests that it is somewhat accidental
that Hund's second rule apparently applies well to triplet states
of two-electron atoms, but not to the singlet states.
Interestingly, Hund's rules in their original form are valid for
the ground states of nearly all atoms for which Z is small enough
to allow the states to be described by L and S. A remarkable
exception is Ce, which has a ground state configuration
. . .4f5s25p65d,the ground state being 'G,, in agreement with the
alternating rule.'65'
Before concluding this section. a few words are in order about
Hund's rules for molecules. Its importance for chemistry can,
for example, be illustrated by the observation that a review
article by Berson on metaquinoid c o m p o ~ n d s ~ ' contains
~1
a
separate chapter about Hund's rules. Apart from linear molecules, such as 0, (for which the terms are 3C,- < 'Ag < ?ZP'),
only Hund's first rule plays a role for molecules. It applies surprisingly well, but there is a remarkable class of exceptions,
which was analyzed in particular by Kollmar and Staemmler.L721
Cyclobutadiene is the classic example for a violation of
Hund's first rule as a consequence of spin polarization. In this
molecule with a planar, square structure, the bonding n-MO cpl
is fully, that is, doubly. occupied, while the degenerate n-MOs
cp2 and cp3 are half-occupied, with two of four possible electrons.
These MOs are represented in Figure 6. In terms of Hund's rule,
b) Spins parallel
91
Fk
(w, + Awt&
(Ti-
hmAP
Fig. 7. Explanatlon of the spln polarization in cyclobutadiene according to ref. [72]
F1
9 2
F3
F4
Fig. 6. Schematic representation of the i[ MOs of cyclobutadiene according to ref.
[72]without (a) and with (b) d-function participation.
the exchange integral between the two degenerate n-MOs cpz
and cp3 is responsible for the singlet-triplet splitting. As these
two MOs overlap only minimally (this is a special characteristic
of the fourfold symmetry), the exchange interaction is also extraordinarily small. The spin polarization of the doubly occupied bonding MO by the singly occupied, degenerate MOs, on
the other hand, is relatively big. This spin polarization can be
understood on referring to Figure 7.
In a first approximation, the MO cpl is doubly occupied, that
is, cplx and cpIp are each singly occupied. If the electron in MO
'p2 has. for example, cc-spin, then cplz and cp,B experience a
different field and become spin polarized. This is taken into
account by replacing (cp,cc)(y,fl) with (cp;a)(cp;o), that is, a
single MO cpl is replaced by two different MOs cp', and cpz. The
singly occupied y,-MO leads similarly to a spin polarization.
The total spin polarization depends upon whether cp2 and cp3 are
coupled to a singlet or a triplet. In the former case (Fig. 7a),
a spin polarization is possible for which cp', and cp; are linear
582
combinations of the n-MOs cpl and q,. This is particularly
effective because v), is low in energy. If 'p, and 'p3 are coupled
to a triplet, there is only a spin polarization by mixing in an MO
of the same symmetry as cpi formed from the 3p, AO. This leads
only to a small energetic stabilization. Overall, this spin polarization, and the ensuing lowering of the energy, is larger for the
singlet state than for the triplet state. This effect overcompensates for the exchange effect, so that the singlet stateends up lower
in energy than the triplet state.
Actually, the situation for the cyclobutadiene molecule is even
more complicated, because in addition to the above-mentioned
effect, which counteracts Hund's rule, there is a second effect in
the same direction. It is, namely, possible to lower the energy of
the singlet state by distorting the C, frame from a square into a
rectangular structure. This distortion, termed a pseudo-JahnTeller effect, does not effect the triplet state. If one compares the
lowest-lying singlet and triplet states in their respective equilibrium geometries, one finds that the singlet state is distinctly
lower in energy than the triplet state, and the pseudo-Jahn-Teller stabilization becomes more important than the spin polarization (which is the deciding factor if one compares identical geometries).
The fact that for two states of the same configuration sometimes the triplet and sometimes the singlet state is lower in energy
is generally not only connected to Hund's rule, but also related
to the mechanisms that dictate the equilibrium geometry of the
two states.
The classic example for the singlet - triplet competition is the
CH, species. The potential energy diagrams of the lowest triplet
and singlet states as a function of the HCH angle are shown in
Figure 8a. From the Walsh diagram in Figure 8b it can be seen
that the highest doubly occupied MO, In, is doubly degenerate
Arigm'. Chrm
Inf. Ed. EngI. 1996, 35. 573 -586
REVIEWS
Friedrich Hund and Chemistry
spectra today, as Herzberg's book makes plaiiThese coupling cases are less interesting for a regular chemist than for a
molecular spectroscopist. They are only relevant for those diatomic molecules that have a nonzero angular momentum
about the molecular axis or a nonzero spin, like. for example, 0,
and NO.
One can characterize the state of a molecule in which the
nuclei and electrons move by a few angular-momentum-like
q ~ a n t i t i e s , " ~such
J as
J
= total angular momentum
N (or R ) = angular momentum of the nuclear motion (perpendicular to the nuclear axis)
L
= electronic angular momentum
A
= projection of L onto the molecular axis
S
= electron spin
c
= projection of the spin onto the molecular axis
J,
= L + S = total electronic angular momentum
c2
= A + C = projection of J, onto the molecular axis
In a strict sense only the total angular momentum J is a
constant of motion-and, hence, suitable for the classification
of states.
If in the Hamiltonian the interaction terms between different
partial angular momenta are small, other quantities can also be
regarded as almost good quantum numbers and can be used
to characterize states. For light atoms, for example, the spinorbit interaction is small and one can characterize atomic states
to a good approximation by J , L, and S, where L and S primarily determine the energy level. while J only leads to a small
splitting. This is referred to a Russell-Saunders (or L s ) coupling. For molecules there are correspondingly more possibilities.
Hund's coupling case a) assumes that the interaction between
nuclear and electronic motion is weak, but also that the motion
of the electrons is referred to the nuclear axis, even in a rotating
molecule. Here the projection Q of the electronic angular momentum (including spin) onto the nuclear axis is well-defined. 52
and the nuclear rotation N couple to give the total angular
momentum J . Figure9 shows that a rotational progression is
built up on each fine structure level.
In case b) the coupling of the spin to the molecular axis is
negligibly small. This case occurs frequently when L = 0. Here
1
Hartree
I
1
90
I
I
130
131"-
170
I
90
180
9: HAH I
O
-
Fig. X a ) Potenti;il cnergy curves of the lowest singlet and triplet stares of CH, as
I'unctions of the HCH angle 9 (from ref. 191). h) Walsh diagram for AH, molecules
For a bent pcometry. ? a , . Ib,. and 3a, are each doubly occupied in lowest singlet
state CH,
in the limiting case of linear geometry. According to Hund's
rule, for this geometry the triplet state lies lower, provided this
degenerate MO is occupied. For HCH angles close to 90", the
difference in energy between the two components 1b, and 3a, of
the original lrr,-MOs is so large, that, following the Aufbau
principle. the lower-lying 3a, MO prefers to be doubly occupied,
and the ground state is a singlet. For angles between these extremes the Aufbau principle competes with Hund's rules. This
leads to different equilibrium geometries for the singlet and the
triplet state. The actual difference in energy between the two
minima is rather small. There is no longer any doubt that the
3CH, is approximately 10 kcalmol-' more stable than
J
~ c H ~ 731
,I~.
I hope that this presentation of Hund's rules has shown that
they still are of great importance today, although they are only
rules and not theorems. In each instance where they do not
apply, a physically remarkable effect is involved. The study of
actual or apparent violations of Hund's rules allows a better
understanding of the physics of atoms and molecules.
J
8-
8%-
For Hund's coupling cases we can be brief. These cases deal
with the classification of band spectra of diatomic molecules,
which Hund, in his fundamental papersJ4. '1 essentially accomplished. His formulation is still the basis for the analysis of these
Slh-
%
6-
4?431h-
6U-
1 %-
-43
-9
6-
7-
-6
6
%
-
?I&-
S%2%-
-7
J
-%1
J
4%-
7. Hund's Coupling Cases
J
8%-
*Is%
-I
Fig. 9 Term scheme for the Hund coupling case a ) for a diatomic molecule (according to ref. [74]) for a 'P (a) and a 'D state (b).
583
REVIEWS
W. Kutzelnigg
L and N couple to give a total angular momentum K without
spin contribution. Interaction with spin then leads to a fine
structure splitting of the rotational levels. Hence, one finds rotational progressions with fine structure splitting of each rotational level (Fig. 10).
6
1%
6
4
%
6
4
2
4
f
3
‘h
k
3
t
2
k
2
2
1
4
Fig. 1 0 Term scheme for thc Hund coupling case h) (accot-ding to ref. [74]) for a ’S
( a ) and a ” S s k t e (h).
The coupling cases c), d), and e) are less important. In case c)
the interaction between L and S is stronger than the interaction
with the molecular axis. This occurs particularly for molecules
with at least one very heavy atom. In case d) the interaction
between L and the molecular axis is weak, but the interaction
with the rotational axis is strong. Case e) resembles d), but the
interaction between L and S is strong and must be considered
first.
In practice. transitions between the different coupling cases
become important. These have been extensively discussed by
Her~berg.”~]
8. Qualitative or Quantitative Theory?
Upon reading Hund’s original papers,[’-‘. 8 . 2 1 1 one always
realizes that he strives for simplicity and clarity. This is equally
true for his textbooks[751and other papers.”‘] Instead of long
mathematical derivations, one finds readily visualizable arguments. In the early days a nomenclature different from the
present was used, but this does not make the reading difficult
(for example, atomic quantum numbers were half-integral). For
Hund, apparently “understanding” enables one to present a
simple image of reality that is also convincing to a nonexpert. In
view of this attitude of Hund it is questionable whether modern
numerical a b initio quantum chemistry, which is based on the
MO theory founded by Hund, can claim him as one of its fathers. Mulliken’s position in this respect leaves no doubt. He not
only encouraged numerical calculations, but also performed
them himself after his retirement, at IBM in San Jose. He is
certainly a pioneer of a b initio theory.
584
Did Hund then prefer qualitative to quantitative theory? In
this context, a critical assessment of the progress of modern
numeric quantum chemistry is in order. The solution of the
Schrodinger equation for an atom or molecule with many electrons to any desired degree of accuracy seemed hopeless in the
pioneering era of quantum mechanics; today it is feasible.
DiracL7’I once stated that all of chemistry is in the Schrodinger
equation, but that its solution was so difficult that it would
never be achieved. With hindsight one can say that Dirac could
not anticipate the advances in computer technology. Nevertheless, he did suggest that techniques should be developed to approximate solutions of the Schrodinger equation “without too
much computation”, which one can interpret as the suggestion
to adjust the level of approximation to the available computer
power.
As an alternative to the previous definition of understanding,
one could also argue that understanding means that one can
make predictions. While the simple qualitative theories available in the early days of quantum chemistry succeeded in explaining experimental observations, predictions based on them
were uncertain at best, and even risky. Nowadays, however, in
many areas reliable predictions are possible. Present-day computer chemistry is popularly caricaturized as a black box theory.
Basically the black box predicts the same results that would be
obtained from an experiment, if this could be performed, but
gives no information about what happens between input and
output. A long time ago, E. P. W i g r ~ e r ~ argued
’~]
that such a
black box if it were available, which it was not at the timewould hardly contribute to our understanding. This opinion,
together with a manifest, and often irrational, disdain for numerical theory was very popular in researchers of Wigner’s generation (there are no indications of this in F. Hund’s work).
I believe this is based on a misunderstanding.
On one hand, such a black box can give useful information
that is not accessible by experiment. Theory, on which, after all,
the black box is founded, can look at situations that are inaccessible to experiment. Theory can focus on a single molecule,
while experiment usually has to deal with an ensemble of molecules under certain external conditions of pressure, temperature,
etc. Theory can easily prepare a single quantum state of a molecule. It can study extremely short-lived molecules, even transition states of chemical reactions. It can artificially suppress certain mechanisms (such as hyperconjugation or d-orbital
participation), and much more. All this means that the black
box furnishes a lot of information that allows a better interpretation o r classification of experimentally observed phenomena.
There is also a further aspect. The approaches used in modern
quantum chemical calculations are rarely of a brute force type
lacking transparency between input and output. To some extent
Monte Carlo-type methods belong to this group. Most methods, however, are based on initial assumptions, which are successively improved. For instance, one usually starts with M O
theory and then improves it by means of configuration interaction. It is often meaningful to notice that a certain observation
can already be explained in the framework of M O theory, and
that nothing crucial changes upon inclusion of configuration
interaction. On the other hand, there are phenomena that can
only be accounted for beyond the MO approximation, namely,
those that depend strongly on the electron correlation. Likewise,
A n p i , . Chcin. 1171.E d Engl. 1996, 35, 513 586
~
Friedrich Hund and Chemistry
some experimental results can be understood in terms of a nonrelatit istic theory, while others require a relativistic approach.
Similarly. one can distinguish between effects that can be rationalized within the framework of the Born -0ppenheimer approximation while others cannot be explained within this approximation. I t is thus generally possible to define models, that is.
simplified theoretical approaches, and to specify phenomena
that are understandable on the basis of a particular model.
Without doubt some phenomena have simple explanations,
while other empirical observations are the result of the interplay
of several different, often opposing contributions. For those, it
may be extremely difficult to give qualitative explanations.
Many physical effects can finally be fully explained in different.
and a t first glance contradictory, ways that are nevertheless
equivalent. W. H. E. Schwarz has given some convincing examples.[”l One has, of course, to face the argument that this
approach for- explaining chemical observations requires a certain hierarchy of approximations. which makes it somewhat
arbitrary. BLIL
one can also consider other types of hierarchies or
look for explanations that are independent of the chosen hierarchy. for example, in terms of observables only. Whether one
should dispense with nonobservables altogether is a matter of
discussion .
Even though the mere laconic statement that the explanation
of a chemical finding lies in the Schrodinger equation certainly
does not impart any understanding, modern quantum chemistry
does contribute to the explanation by providing not only the
black box. but also models, illustrations, and concepts, whose
importance should not be underestimated.
At this point ;I few comments on reductionism are in order. In
the present context, reductionism means, in simplified form,
that chemistry can be derived from physics. This concept is
plausible according to modern knowledge, but it is not entirely
undisputed. The question of reductionism was not relevant for
Hund. because for him molecules were objects of physics and
therefore subject to physical explanations. If one wants to enter
a realm of chemistry that is not just the theory of the physical
properties of inolecules, it makes sense to ask whether there are
intrinsically chemical qualities that d o not arise from quantum
mechanics. Primas. among others, has dealt with this question.f8’’ He has pointed out. that reductionism is already relevant within physics in the context of the derivation of thermodynamics from statistical mechanics, Even though this derivation
is correct and useful. thermodynamics can also be defined axiomatically without any reference to the statistical derivation.
Thermodynamics contains qualities that are not a direct consequence of its atomistic, statistical mechanics derivation. The
thermodynamic functions are, for example, classical, that is,
commuting variables. that have no place in the framework of
quantum theory. Whether there can be an axiomatic basis for
chemistry. as for thermodynamics, and what the classical variables corresponding to the thermodynamic functions would
look like. is yet unknown.
A famous example of a classical concept that has no place in
quantum theory -at least in that of stationary states--is the
chirality of molecules. There is an apparent paradox, namely
that stationary states must be adapted to the symmetry of the
Hamiltonian operator. so chiral molecules should not exist at
all. Hund’”] has resolved this apparent paradox by pointing out
REVIEWS
that stationary states are only realized if the tiine scale of the
measurement is large relative to the time for the tunneling between the nonstationary states of two optical isomers. which
may even be larger than the age of the universe. Then stationary
states are not realized.
It is questionable whether it is possible to specify more precisely some of the simplified concepts that are rather popular in
qualitative theoretical chemistry, such as resonance or electronegativity, in order to use them, like the thermodynamic
functions. without having to rely on their derivation from quantum mechanics. I regard it unlikely that drastically simplified
theories, which often contain semiempirical corrections, can
contribute more to the understanding of chemistry than exact
quantum chemical calculations, since the predictive power of
such simplified theories is small. However, the last word may
not yet have been said in this context.
From Friedrich Hund we can learn to separate the essential
from the marginal aspects and to search for simple explanations
even for very complex phenomena, without, of course, forcing
theory beyond its limits.
M y thanks are due to D. Herschbacli for suggesting that I write
this article, to W: Liittke for many vuluahle suggestions, to K
Staernnder as well CIS C. van Wiillen,for reviewing the manuscript,
and to E. Heilbronner,for constructive comments. I am grateful to
D. D. Morgan for helping with the English version.
Received. October 6. 1995 [A136IE]
German version: A n ~ r i rChem.
.
1996. 629
Translated by D r W C. Wilisch. Ansbdch (Germany)
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