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Molecular States and Molecular Orbitals.

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in Figure 4r‘51. The various designs are more than just geometrical isomers of membrane accomodation, each implying a
certain mode of operation, and each adapting differently to
environmental water conditions. Generally speaking, closed
systems (Fig. 4a and 4b) requiring an elaborate feed water
pretreatment are preferably used under conditions which favor
a continuous quantity operation, while open systems (Fig.
4c and 4d) are less sensitive towards difficult feed water conditions and more versatile in small scale operation.
5. Conclusion
Reverse osmosis represents a highly attractive separation
principle, whose somewhat uncertain credibility as a technical
unit operation is due, not to a lack of basic development,
but to overoptimism in projecting the intrinsically possible
onto the harsh reality of actual feed waters. Future develop
ments will have to focus on overall reliability rather than
selectively on miracle membranes.
Received: March 8, 1977 [A 181 IE]
German version: Angew. Chem. 89,624 (1977)
[l] R . A. Robinson, R . H. Stokes: Electrolyte Solutions. 2nd edit. Butterworths, London 1970.
[2] 0. Leuenspiel, N. de Neuers, Science 183, 157 (1974).
[3] K. K Boddeker, H . Strathmann, Chem. unserer Zeit 8, 105 (1974).
[4] J . 0. Kessler, C . D. Moody, Desalination 18, 297 (1976).
[5] S. Loeb, J. Membr. Sci. I , 49 (1976).
[6] U . Merten: Desalination by Reverse Osmosis. MIT Press, Cambridge,
Mass. 1966.
[7] R . Schlogl, Jahrb. Max-Planck-Ges. 1969, 135.
[8] H. K. Lonsdale, U . Merten, R . L. Riley, J. Appl. Polym. Sci. 9, 1341
( 1965).
[9] M . A . Frommer, J . S . Murday, R . M . Messalem, Eur. Polym. J. 9,
367 (1973).
[lo] H. Srrathmann, U.u. Mylius, Proc. 5th Int. Symp. Fresh Water from
the Sea 4, 189 (1976).
[ t t ] R . L. Riley, H . K . Lonsdale, C. R . Lyons, U . Merten, J. Appl. Polym.
Sci. 11, 2143 (1967).
[12] S. Loeb, S . Sourirajan, Adv. Chem. Ser. 38, 117 (1962).
[13] R . L. Riley et al., Prac. tst Desalination Congr. of the American Cont.
1976, Paper 11-1.
[14] L. Dresner, J . S . Johnson, Jr. in K . S. Spiegler, A. D. K . Laird: Principles
of Desalination. 2nd Edit. Academic Press, New York, in preparation.
[15] Fig. 4 a : UOP/Ajax; Fig. 4b: DuPont; Fig. 4c: PCI/Harweli; Fig. 4 d :
GKSS.
Molecular States and Molecular Orbitals[’’ 2]
By Hans Bock1*]
Dedicated t o Professor Erich Huckel on his 80th birthday and on the occasion of the
450th Anniversary of the foundation of Marburg University.
Molecules change their properties on acquisition or loss of energy. The state of a molecule
can be characterized by the difference between its energy and that of the preceding initial
state or that of the subsequent final state, as well as by the respective charge distribution.
Molecules in their various energy states each display a particular structure and characteristic
properties, and it is these entities which the chemist uses as synthetic building blocks.-The
best model hitherto available for describing molecular states is the molecular orbital representation. Molecular orbitals are especially suitable for comparing mutually corresponding molecular
states of similar compounds, and thus provide a manifold and stimulating overview of large
areas of chemistry.-An attempt will now be made to point out important characteristic
parameters of molecular states and to explain the rules governing their description by molecular
orbitals, while circumventing mathematical barriers. A principal objective is the comparison
of states of “chemically related” molecules with regard to the connection, the arrangement,
and the potential of their atoms. The simple models employed for this purpose, whose inherent
limitations and potential expansions are discussed, have meanwhile become valuable tools
in the hand of the chemist.
Between 1789 and 1793, Georg Christoph Lichtenberg made
the following entry in his Sudelhejl N o . I: “Wir sehen in der
Natur nicht Worter, sondern immer nur Anfangsbuchstaben von
Wortern, und wenn wir alsdann lesen wolIen, so finden wir,
dap die neuen sogenannten Worter wiederum blop Anfangsbuchstaben von anderen sind”. ?*I
[*] Prof. Dr. H. Bock
Chemische Institute der Universitat, Anorganische Chemie I1
Theodor-Stern-Kai 7, D-6000 Frankfurt am Main 70 (Germany)
[**I “In Nature we see not words but only the first letters of words, and
when we wish to read we find that the new so-called words are once again
mere first letters of other words”.
Angew. Chem. Int. Ed. Engl. 16. 613-637 ( 1 9 7 7 )
1. To Measure or to Calculate?
The tetra-atomic molecule thioformaldehyde H2C=SC3I,
detected in interstellar space, can be prepared pure by thermal
elimination of HCl from methanesulfenyl chloride and subsequent deposition of NH4CI with ammonia (Fig. 1:A) and
subjected to spectroscopic analysis (Fig. 1 :B). Computer simulation of an almost superposable photoelectron spectrumr4]
(Fig. 1 :C) takes about 2 CPU hours (IBM 360/91).
Procedures such as PNO/CI (Pseudonatural Orbitallconfiguration Interaction) and CEPA (Coupled Electron Pair
Approach) developed by MeyerC51, or the GF method (many613
Photoelectron Spectrometer
0
I -
I I
n
Problem
I ,
n
(Measure]
ment or
Calculation
x
Mdecxllar
I
I
8
8
9
,
10 11 12 13 14 15
16
17 18
10
16
18
12
14
I E CeV 1-
,
19
,
20
Zl
20
Fig. I.For gas-phase preparation (A) of thioformaldehyde [3], methanesulfenyl
chloride is passed through a quartz spiral heated to 860 K. Hydrogen chloride
can be removed from the resulting gas mixture by deposition as ammonium
chloride on reaction with ammonia introduced from a storage system via
a fine-control valve. The reaction conditions such as furnace temperature
or ratio of the gaseous reactants H3CSC1 to NH3 are best adjusted with
the aid of a photoelectron spectrometer, i. e. on the basis of the appearance
and disappearance of the known ionization bands of HCI and NH3. The
PE spectrum (B) also provides a check on the purity of H2-S.
Additional
calculation [3, 41 of the ionization band pattern (C), a “fingerprint” of the
molecule, also permits unequivocal identification of the reaction product
H,C=S.
body Green
utilized for reproduction of the PE
spectrum (Fig. 1:C) permit calculation of some state data
of gaseous, i. e. isolated, molecules up to the size of, say, para-diwith a surprising degree of accuracy,
fluorobenzene C6H4F2f61
e.g. ionization energies to within deviations of less than
0.3 eVl7].
Thus it is possible today to reproduce individual properties,
especially those of small molecules, to within the limits of
experimental errorr7*‘1 by calculation; or, generalizing,to satisfactorily approximate solutions of the Schrodinger equation
by numerical methods[’I. This achievement has various consequences”’] which will come to affect the interrelationships
between experimental practice and the application of theoretical tools. Some of these aspects[”] are presented in the simplified flowsheet of Figure 2[2’+d1.
The provocative question “To measure or to calculate?”
is probably best answered by pointing out how successful
and stimulating a combination of the two can be. Thus correlation of experimental and calculated data, and especially the
comparison of measured data for given states of chemically
related molecules, will continue to promote the chemist’s
614
Fig. 2. Experimental investigation ofa chemical problem (+) leads to individual experimental data which are generally compared with other results,
i.e. verified by comparison with an (empirical) model. This cycle can be
advantageously coupled with a loop (---a)in which the observations, after
correlation with the behavior of electrons in potential fields, can be described
in appropriate approximations, thus providing important information about
the basic properties and scope of models. In particular, acting as their own
computer, molecules “print out” complete solutions of the Schriidinger equation HJI = E $ (which cannot be solved exactly for many-particle systems)
in the form of their measured data. Even simple approximations such as
those proposed by E . Huckel in his epoch-making [ti] work on x-systems
[12] permit comparison of numerous individual data and derivation of farreaching consequences and predictions from the resulting models.-The recent
breakthrough achieved in numerically approximating solutions of the
SchrMinger equation with newly developed calculating procedures [9] also
enhances the importance of other correlations between experimental and
calculated quantities and of comparisons between states of chemically related
molecules. This could represent the starting point for a more intense development of a metatheory [lo] providing a unified description of regularities,
permitting manifold predictions, and also yielding models suitable for application to reaction courses as more and more detailed knowledge becomes
available.
understanding of the substances with which he is dealing.
In addition, however, greater effort could be invested in the
development of a metatheory which facilitates an overall view
of the material and energetic multiplicity by describing regularities and setting up rules, provides a unified set of symbols
and a common language, reveals tendencies, and supports
predictions to a greater extent than was formerly possible[10].
More attention should also be directed to theoretical calculation of quantities not readily accessible to experimental determination, e.g. on energy or reaction surface^['^^. This would
provide a means of incorporating the reactivity of chemical
compounds in improved models as increasingly detailed knowledge becomes available concerning the course of reactions.
The following remarks about molecular states and their
description by molecular orbitals, together with the examples
largely selected according to their didactic value, are intended
to arouse the preparative chemist’s interest in exploiting the
tools put at his disposal by theory.
2. Properties of Molecules in Their Various States
If energy is supplied to or withdrawn from a molecule
it goes into another state. If, in addition, electrons are acquired
or released during this transition then molecular ions, which
may also be radicals, are formed as new particles which can
themselves exist in numerous states of various energies. The
Anqew. Chem. Int. Ed. Enql. 16.613-637 ( 1 9 7 7 )
energy differencescan be recorded by a variety of measuring
techniques (cf., e.g., Fig. 3) and show that the total energy
of the individual molecular states increases on going from
those of a stable radical anion Mae via those of the neutral
molecule M to those of the radical cation Me' (Fig. 3).
Molecular States
I
n0.33 eV
Molecular State Data
IECeVl
-
H
Etcta
Fig. 3. A neutral molecule M in its ground state r(M) can take up certain
amounts of energy; thus on absorption of UV radiation E=hv it is transferred
into one of the numeroui electronically excited states xf(M). Larger amounts
of energy may lead to loss of an electron, for example to form the radical
cation M'": the first ionization energy I E , generates its ground state r(Mem);
the higher ionizationenergies IE*(, likewise observable in the photoelectron
spectrum (PES) produce its electronically excited states x;(Mam).In contrast,
on acceptance of electrons to form a stable radical anion M'e, i.e. given
a positive electron affinity EA, the total energy becomes more negative;
the ionization reaction r(M"+r(M)fee
requires energy. In solution, radical anions can be stabilized by ion-pair formation with the counterion and
by solvation; their energy difference relative to M can be measured polarographically (POL)as the half-wave reduction potential Ey,$. Information concerning the spin-, and hence charge-density in the radical ions Mem and Mae
can be gained from high-resolution electron spin resonance (ESR) spectra.
For the nomenclature of molecular states, cf. ref. [14].
Fig. 4. In the photoelectron spectrum of HBr [15a, 161 a total of 4 ionizations
are observed-corresponding to the presence of 8 valence electrons. Of the
three lowest energy ionizations shown, the bands at 11.68 and 12.OieV
can be assigned to the Br electron pairs, and that at 15.60eV to the BrH
bond: in the nomenclature of spectroscopy [14] the two (spin/orbit) coupled
radical cation states are designated as ' I I s i > and 2n~i2,
and the simple
one as 'Z+. They differ, among other things, in their stretching frequencies
V&,, which are hardly changed ('n) and almost halved ('Z'), respectively,
relative to that of the uncharged molecule VHHr. This result can be interpreted,
together with the differing band shapes, in terms of the potential energy
curves as follows: if the equilibrium distance Ro remains unchanged during
the vertical transition to the radical cation state, i. e. if an (almost) nonbonding
electron (nb) is ionized, then V ~ B B ~ ~ Vand
, , ~ owing to the most frequent
0-0 transition a needle-shaped principal band will result. If R o increases,
i. e . if a bonding electron (b) is removed, then V& becomes smaller relative
to VAH and the most frequent vertical transition now appears near the center
of the band. If the resulting radical cation AB@ is unstable, e . g . towards
dissociation into A'+B@, then the vibrational fine structure vanishes-the
curve of the dissociative state and that of the binding state cross in the
potential energy diagram-as in the present case for HBr'", which decomposes
into H'+Br@ at 15.85eV [16].
i. e. of equal energy, but owing to different kinds of orientation
Radical cation states of molecules are especially suitable
for an introductory discourse: they can be observed by the
straightforward and direct technique of photoelectron
spectroscopy[zC - e , Is], are characterized by a typical electron
hole, and as shown below can usually be interpreted on the
basis of simple models. A wealth of information about the
properties of radical cations can be gleaned from PE spectra[''- 15b*cl;
this will be illustrated by clear-cut examples of
relatively small molecules which in some cases experience
only slight structural deviations (Figs. 4 and 6) and in others
even undergo changes in structural type (Figs. 5 and 6).
In the PE spectrum of HBr[15a*161
(Fig. 4) the first two
ionizations can be assigned to the bromine electron pairs
perpendicular to the molecular axis. Two radical cation states
2r13,z and zIIl,zresult which are, however, not "degenerate",
Angew. Chem. Int. Ed. Engl. 16, 613-637 ( 1 9 7 7 )
of spin and orbit angular r n o r n e n t ~ r n [ ~ ~ ~
are
* ~split
* by
0.33 eV (Fig. 4). The vibrational fine structure recognizable
following the second main band exhibits a frequency which
is only slightly lower than the stretching frequency in the
neutral HBr molecule: hence the electron hole arising on
ionization of a "nonbonding" electron from the bromine electron pairs hardly exerts any significant effect on the force
constantsf& -f&, and the equilibrium distance R t accordingly remains practically unchanged compared with Roll 6!
In contrast, the third PE band, assignable to ionization from
the o H B r bond, displays an approximately halved radical cation
vibrational frequency VgBr (Fig. 4)-corresponding to a
diminished force constant fZr and an increased equilibrium
distance RZ in this state. That each individual molecular
state exhibits characteristic properties and possibly a specific
615
reactivity, is apparent from the dissociation HBrQ-+H'+ Br'
occurring in the third radical cation state above 15.85eVrecognizable from the disappearance of vibrational fine structure (Fig. 4).
The experimental data for HBr" demonstrate that molecular states have different kinds of charge distributions depending
upon their energy,and hence also different properties including
different kinds of reactions. In molecules containing more
than two atoms, the overall structure, which is not necessarily
linear, can therefore change according to the state, as will
now be exemplified in more detail for certain radical cation
states of disilane (Fig. 5) and ammonia (Fig. 6).
0.3e.V
0.35eV
n r -
energy: the pertinent bands should therefore have double
intensity but only one maximum. However, the PE spectrum
of disilane (Fig. 5 ) tells a different story: the second and
third PE bands are split by 0.3 and 0.35 eV, respectivelyr1'].
This observation cannot be attributed to a conformational
change since the degeneracy of the states would be conserved
on rotation about the SiSi bond and the energy barrier for
such a rotation is a mere 0.05 eV. Rather, a structural change
must have taken place which destroys the threefold axis, e . g .
by analogy with the H3C-CH;@ cation['*], by deformation
of SiSiH angles (Fig. 5). Restructuring of the A('E,) and 8('E,)
states of the disilane cation is not a unique occurrence: degenerate radical cation states of non-linear molecules are generally
unstable towards such a "Jahn-Teller distortion". The energy
gap arising on removal of degeneracy can exceed 1 eV['Jb'Cl.
Information like this read off from PE spectra is open
to generalization: during mono-ionization from the ground
state of a molecule having n valence electrons to generate
one of the possible n/2 radical cation states (Fig. 3), both
the molecular structure and other molecular properties can
undergo drastic changes. This is impressively demonstrated
by a comparison of vibrational frequencies of the tetraatomic
molecules NH3, PH3, and PF3 and those of their radical
cations NH;', PH;', and PF;'['.
(Fig. 6).
io
5
5;
I,
I1
lo
'
il
500cm-'
992 cmf
11.5
12.5 IEkVI
470cm-I
487cm-I
b h
I,
Fig. 5. In the photoelectron spectrum of Si2H6[ 1 8 ] 7 ionizations are expected,
corresponding to 14 valence electrons. Of these ionizations 2 each should
lead, in the preferred staggered conformation of Dad symmetry, to energetically
degenerate radical cation states. Thus a total of 5 PE bands are predicted,
assignable to the following H3Si-SiHjm states given in the order of increasing
energy (concerning nomenclature, cf. 114, 191): osisi ionizationp+.Y(2A,,),
D S ~ H , ionization+A('EE,) and $(*E.), and the ionizations having large 3ssi
components+C('AZ.) and D('A,J, with the latter outside the range of measurement. It is observed, however, that the second and third PE bands
are split by 0.3 and 0.35 eV, respectively. This is because the energetic equality
of the degenerate states A('Es) and B('E.) is removed by a "Jahn-TcllL,.ltwo SiH bonds probably bend, the threelold
distortion" of H,Si-SiHj':
symmetry axis vanishes (-C3), the structure of the resulting radical cation
belongs to the Clh symmetry group, and the degenerate states split ('Eb+*Bg
+'A,; 2E._+2A~+zBu).The splitting of the second and third PE bands
cannot be attributed to a conformational rotation (Djd+D3+D3h)about
the SiSi bond since the threefold axis and thus the degeneracy would be
retained in the process; furthermore, the rotational barrier amounts to only
0.05eV 1181.
Assuming the species to have the same structure as the
neutral molecule, the disilane radical cation H3Si-SiHj'
is
expected to display two sets of two degenerate states of equal
616
'
9OOcm-1
Fig. 6. The first band of the photoelectron spectra of NH,, PHI, and PF3
1201, which is assigned to ionization from the nN or np electron pair, exhibits
a marked vibrational fine structure. Some 20 peaks are recognizable, from
which the frequency of the symmetrical deformation v' of the radical cation
can be determined to within +20cm-'. On comparison of ' V with the
ground state frequency of the neutral molecule 0, a constant (NH,, PF3)
or halved value (PH3) is found [2e, 201. Constant values result when no
molecular inversion occurs, as a consequence of flattening of the molecular
skeleton yielding the planar NW3e species or of an excessively high inversion
harrier in pyramidal PF;'. Halving of frequency [21] occurs at low inversion
barriers, e.y. in the flattened pyramid of PH','. In the potential energy
diagrams (cf. Fig. 4), these results are described in terms of a common
potentialtrough (NHie),two extensively overlapping potential troughs (pH;@),
or two completely separate potential troughs (PF;@)[2e, 201.
The first ionization of the pyramidal compounds XH3 and
XF3 occurs from the electron pairs nx[2es
'O- "I.
Comparison
of the symmetrical deformation vibrations of the neutral molecule (Fig. 6: Fz)and the radical cation (Fig. 6: of)demonstrates
just how diverse the changes in properties can be, which
arise on removal of an almost nonbonding electron: in the
species NH;' no inversion can occur and the
(slightly lower frequency) deformation vibration can be described by a single potential trough (Fig. 6). The ionization
PH3-*PH;'+ee widens the HPH angle from 93.3" to about
1
the halving of frequency observed in the PE spectrum
(Fig. 6: 07% +V2) is explicableas due to extensive interpenetration of the potential troughs corresponding to the two invertoAngew. Chem. Int. Ed. Engl. 16,613-637 (1977)
mers['lbl. In contrast, the molecular structure is largely
retained on electron ejection from PF3-tPF;@ +ee, and the
high inversion barrier describable by two distinct potential
troughs["] prevents frequency halving of the symmetrical
deformation vibration 0;".
Even in the case of constant electron occupancy in the
initial and final states a supply of energy might effect a drastic
redistribution of charge: frequently cited structural modifications''''. "1 accompanying transitions into other electronic
states include the mutual twisting of the two halves of the
molecule of ethylene and the bending of acetylene in the
lowest excited state, as well as the bending of carbon dioxide
or the stretching of water on absorption in the vacuum UV
region (Fig. 7).
From the sheer number of structural modifications of molecules known to occur on uptake of energy and-more generally-from theaccompanying modificationsof molecular p r o p
erties caused by redistribution of charge it follows that:
The chemist's synthetic building block with a particular structure and characteristic properties is always a molecule in a
state of spec$c energy and charge distribution.
Reactions are likewise not typical for a molecule but-as
already emphasized for HBr'@ (Fig.4)-for a particular molecular state. This is strikingly confirmed by the different kinds
of reaction pathways emanating from different kinds of photochemically excited states[231,e.g. from the 3(n-+~*)triplet
state and from the '(K-+R*) singlet state of adamantanet h i ~ n d(Fig.
~ ~ ]8).
lo:
,Vacuum:
s
Fig. 7. The structure of a molecule can also be modified after energy transfer through a photon if the number of
electrons remains constant (cf. Figs. 3 to 6). Thus ethylene is twisted on n+n* excitation in the UV region; at the
minimum of the potential energy curve of the excited state &'BI.) [14] the two H2C halves of the molecule are
mutuallyperpendicular(D2dsymmetry). Linear in its ground state R('X;), acetylene is doubly bent in the first electronically
excited state &'A,). Excitation with short-wave light (h= 137nm) bends carbon dioxide; in contrast, water is stretched
in the vacuum UV region. In the other states shown (A,B,e,d...) there is either no
in its 2nd excited state &'X;)
structural change [Zl, 221 or the geometry of the excited state is (still) unknown.
-
Stereospecific
(
Excimer )
Regiospecific
(-Radical)
Fig. 8. Adarnantanethione exhibits I(%+%*) and '(n+n*) electronically excited states separated by 1.69 eV; the latter
undergoes radiationless transition into the lower-energy triplet state '(n+n*) with two unpaired ( 7 7 ) electron spins
[24]. Reaction with acrylonitrile or fumaronitrile on irradiation with 254-nm light (n+x*) or 500-nm light (n-n*)
therefore lead to different reaction types and reaction products [24]: the '(n-+n*)diradical reacts regiospecifically
with acrylonitrile via the radical intermediate of most favorable energy. In the '(n-x') singlet state with paired
electron spins ( 7 3 ) addition of fumaronitrile occurs stereospecifically via an excimer complex [24].
Angew. Chem. I n t . Ed. Engl. 16.613-637 ( 1 9 7 7 )
617
H&o
R
~a$=1.76G
R\
?
H
R/y\R
R'
R'
R
+--.I
at = 5.34 G
pJ
II
H
P;!
R+R
R
R
Fig. 9. 1,4-Bis(trimethyIsilyl)benzenecan be reduced to the radical anion by potassium in dimethoxyethane. Its ESR spectrum [25] displays the quintet of 4 phenyl protons with a gap of 1.76 G
between the corresponding signals, which are further split by the 18 equivalent methyl protons
(R =CH3) (intensity ratio outer line: inner line= 1:48620).The p-xylene radical anion (R= H), on
the other hand, yields an ESR spectrum which, while revealing a more strongly split quintet of
the 4 phenyl protons, gives no evidence for additional coupling with the methyl protons: hence
the substitution sites must lie in a nodal plane in which the spin density pn is approximately equal
to zero. The charge density is consequently also low and preparative protonation by treatment
with NHICl after prior Birch reduction (Na/NH3) accordingly proceeds exclusively a positions 2
and 5. As indicated by the drop in the phenyl proton coupling constant a$ from 5.34 to 1.76 G,
the electron density in these positions is lower in the trimethylsilyl derivative which is protonated
solely in positions 1 and 4 [26] where a high spin, and hence a high charge density is now
evidenced by the clearly resolved coupling of the methyl protons a,!i=CH3.
As shown by the excitation and reaction scheme for adamantanethione, the reactions starting from the 3(n+7r*) triplet
state proceed by a radical
and involve the
radical intermediate of most favorable energy : they are therefore regiospecificbut not stereospecific. In contrast, formation
of an excimer complex is observed to follow excitation into
the '(n+n*) singlet
acrylonitrile is not longer added
regiospecifically,while fumaronitriledoes undergo stereospecific addition.
Experimental evidence for the variety of reactions for various
charge distributions is supplied by, e. g., combined ESRLz51
and preparative studies[26]on the reduction of 1,Cdialkyland of 1,4-bis(trimethylsilyI)benzenederivatives (Fig. 9).
From the ESR spectra of benzene radical anions (Fig. 9)
we can deduce coupling constants a; which are directly proportional to the 7r-spin populations p r at the relevant 7r
centersK2', 281. They furnish the following information about
the distribution of the additional electron in variously substituted reference compounds : in the trimethylsilyl derivative
the substitution sites 1 and 4 bear the greatest charge, whereas
in the p-xylene radical ion these positions lie in a nodal
plane of negligible spin density (Fig. 9). The ESR evidence
can be tested and exploited preparatively: protonation with
NH4Cl after prior Birch reduction with Na/NH3 always occurs
at the centers of greatest electron density. In full accord with
the charge distribution estimated from the ESR spectra, the
1,4-bis(trimethylsilyl)compound is hydrogenated only in positions 1 and 4 to yield the corresponding cyclohexadiene,
whereas the p-xylene radical anion is protonated solely at
positions 2 and 5 [ 2 6 ](Fig. 9).
These examples, selected from a wealth of similar ones,
document that the molecules in their individual states exhibit
specific properties, including a specific reactivity. The number
618
of building blocks available for chemical synthesesis multiplied
several-fold by this energetic diversification: the number of
lo7 compounds so far known to result from variation of
constituent elements and connections has to be increased
for molecules containing N atoms and n valence electrons
to include 3N - 6 vibrational modes, at least n/2 ionizations,
far more than ($)2 electronic excitations, as well as states
arising by other processes. In order to gain some kind of
overview of such an enormous variety, to recognize inherent
trends andregularities, and to formulate predictions the chemist
needs a model containing the parameters of importance for
molecular states (Fig. 3) in suitable form, and which is also
flexible and open to extension (Fig. 2).
The best model hitherto available for a comprehensive description of molecular states is the molecular orbital representati~n'~~].
3. Description of Molecular States by Molecular Orbitals
What demands does the chemist place on a model for
molecular states? As illustrated for H2CF2, it must contain
the number of atoms, their connection (topology) and their
arrangement (symmetry):
&.. . .
......
Connection
(Topology)
Arrangement
(Symmetry)
Structure
-
Atoms
(Potentials)
-
Electron
Distribution
Energy
(Measurement)
Angew. Chem. Int. Ed. Engl. 16,613-637 (1977)
Furthermore, potentials have to be defined for the individual
centers, representing various kinds of atoms, in such a way
that the resulting electron distribution reproduces all important details of molecular structure such as distances and conformation. The energy must be included in the model as a parameter of molecular state, and changes in characteristic molecular properties should be determinable as a consequence of
changes in energy (Fig. 3). For the chemist it is important
that such a model permits comparison of experimental data
applying to corresponding states of chemically related molecules (Fig. 2).
In simple molecular orbital models, whose utility and limitations will now be considered for various examples, the procedure is to determine the distribution or motion of an electron
in the time-averaged field of the atomic nuclei and the other
electrons of a molecule. In order to answer the question "when
and with what probability will an electron be located at which
site?", the 3N-dimensional total function resulting for N electrons is approximated by a product of three-dimensional functions having particular qualities[30- 321. The (one-electron)
molecular orbitals thus obtained are especially suitable for
discussions of one-electron properties such as the states of
radical cations M'@ arising on ionization of an electron, in
which the electronic interaction and electron distribution still
resemble those in the ground state of the neutral molecule
M. In such cases Koopmans'
is valid
according to which the vertical ionization energies IE: measured by photoelectron spectroscopy, i. e. the energy differences
between the particular states AE,,,,, for ejection of an electron
M+hv+M"+ee
(Fig. 3), can be correlated with calculated
(one-electron)orbital energies -&jCF
of "Self Consistent Field"
qualityL3'- 341. Let us therefore begin our description of molecular states by molecular orbitals with the didactically advantageous example of the four states of the triatomic radical cation
HSH'@ resulting on valence electron ionization and discussible in terms of theorem (2) (Fig. 10).
Ejection of a valence electron from the HSH molecule,
which exhibits an almost spherical charge distribution in the
ground state, leads to four states of the radical cation HSH'@
observable in the PE spectrum. In these states the distribution
of the electron hole can be described by calculating the difference electron density between the initial state and pertinent
final state (Fig. 10): evidently, zero site planes can result
whose positions relative to the molecular skeleton are reflected
by the symmetry classifi~ation['~]
of the radical cation state[l41.
If Koopmans' theorem (2) applies-i. e. if the redistribution
of charge (relaxation) and changes in the mutually dependent
motion of electrons (correlation) accompanying the ionization
process are negligiblysmall" 5b, 'Lthen the electronic arrangements (configurations) in the individual states can be described by a single set of molecular orbitals +. Such a set
can be approximately constructed by linear combination of
bond or atomic orbitals (Fig. 10). There result symmetryadapted molecular orbitals JI which are themselves not experimental observables; however, their squares $' can be correlated with experimental electron densities. The electron-hole
....%
zero site planes in the radical cation states apparently correJ
10.; 11 12 1 3 i 14 15 )', 11 18
Lev1 22 23
1
__
_
spond to those of the squared
functions J12 and _
thus to the
W
LJ
nodal planes resulting from changes of sign in the molecular
orbitals
(Fig. 10). The utility of such molecular orbitals
or of their squares + 2 , sketched by hand or plotted by
computer, will now be demonstrated by PE spectroscopic
comparison of the iso-valence electronic compounds HSH
and HOH (Fig. 11).
HSH and HOH differ primarily in the higher effective nuclear
potential of the oxygen and in the larger HOH angle.
Fig. 10. In the gas phase the HSH molecule has a CZ"structure; the electron
Under the assumption, corroborated in the following, that
density distribution in the ground state 8('A1) is totally symmetric and
almost spherical. Ionization of one of the 8 valence electrons leads to the
no essential structural change will occur on ionization to
ground state 8('B,) or one of the three excited states [14, 191 A(*A,),
the radical cation ground state 8(2B1)[321,the potential differ~('Bz),or ~ ( ' A I )of the radical cation HSH", which exhibit 0 or 1 zero
site planes with respect to the resulting electron hole. All four states of
ence can be accounted for by shifting one of the two PE
HSHem are observed in the PE spectrum between 10.47 and 22.2 eV in
spectra along the ionization energy scale (Fig. 11). The angle
the He11 region 1351. The ionization energies can be satisfactorily correlated
opening effects which then become clearly recognizable were
[36]. The
with SCF orbital energies according to Koopmans ( I E ; = -EJ"")
relevant orbital diagrams can be qualitatively depicted by symmetry-adapted
correctly predicted by Wulsh in 1953[371-ten years before
linear combinations ( L C ) of Bond Orbitals JlLcs0or of Atomic Orbitals
thediscovery of PE spectroscopy: AZE1,2decreases and A I E 2 , 3
JILcA0.The resulting molecular orbitals JI frequently change their sign ;)01.(
increases. This situation is a consequence of the optimum
the resulting nodal planes of C z b symmetry types (b,), (al), and (b2) are
retained as zero site planes Iele) in the squared functions [ 2 ] . These can
electron distribution in the individual radical cation states,
be compared with the electron-hole zero site planes of the radical cation
which depends upon the given potential and can be qualitastates such as would result from the difference electron densities of the HSH
tively interpreted with the aid of the relevant molecular orbiground state and the individual HSH" radical cation states.
+
Angew. Chem. Inl. Ed. Engl. 16, 613-637 ( 1 9 7 7 )
+
619
-(&)
Fig. 1 !. On comparison of the first three radical cation states R(2B,), A('A1), and b('B2) of the
iso-valence electronic triatomic molecules HSH and HOH particular attention has to be focused
on the change of the bond angle Acp and the differing effective nuclear charge A& of sulfur and
oxygen. Assuming the first ionization, assigned to the nonbonding electron pair, to constitute a
suitable internal standard, A& can be compensated by a relative shift of the PE spectra [35] by
12.62 - 10.47 =2.!5 eV. The opening of the angle A 9 can be discussed on the basis of the molecular
orbitals (bl), (a,), and (b2) as follows 1371: diminution of the bonding interaction and augmentation
of the antibonding interaction in (al) raises (destabilizes) this orbital, while diminution of the antibonding interaction in (b,) lowers (stabilizes) that orbital. There results an experimental Walsh
diagram [37] which permits comparison of the radical cation states of the chemically related
molecules HSH and HOH.
Fig. 12. The radical cation HSH" expectedly shows structural changes (cf. Figs. 4 to 6) in some
of its states. In the ground state %('el)the angle cp remains almost constant relative to HSH.
For the 1st electronically excited state &'Al) an opening to 127" is found spectroscopically [3S]
and for the 2nd excited state a closing to 56" is calculated [36]. The distribution of the unpaired
electron, corresponding to that of the electron hole, follows from the spin population contour
diagrams in the z y and xy planes calculated for optimized geometry [36]; negative spin densities
are merely shown as dashes. There is satisfactory overall agreement between experimental and
calculated data l E = -&SCF and cp (ab initio open shell with STO 3G basis set [36]). Qualitative
molecular orbital considerations predict the following angle changes in the HSH'm states: roughly
constant according to ( bl ) because of the node in the HSH plane; widened according to (al) by
reduced antibonding interaction and analogously narrowed according to (b2).
t a l ~ ' ~if~the
] : bonding interactions characterized by like signs
in a molecular orbital increase then the orbital is stabilized,
and if the antibonding interactions characterized by a change
in sign increase then destabilization occurs (Fig. 11). Conversely, theapplicability of such perturbation arguments to the
comparison of states of chemically related compounds via
620
molecular orbitals according to (2) shows that we are dealing
with comparable states having comparable properties; it therefore supports the assignment of the PE spectra and can be
extended to other AH2 compounds which may also have
a different number of electrond3''. In sum, correlation diagrams according to W a l ~ h [ which
~ ~ ] , reproduce changes in
Angew Chem. Int. Ed. Engl. 16, 613-637 ( 1 9 7 7 )
orbital energies accompanying changes in structural parameters, furnish a far-reaching overview of numerous molecular
states.
The perturbation arguments also permit prediction of structural modifications in the individual radical cation states of
a particular molecule, if these relax towards the potential
minima after vertical excitation (cf. Fig. 4): if ionization generates a radical cation state whose representative molecular
orbital-now occupied by only one electron-is dominated
by strong bonding or antibonding interactions then structural
changes mitigating these interactions are expected. The optimum charge distribution in the individual states can be discussed in these terms as will also be demonstrated for HSH*@
as example (Fig. 12).
The calculated optimum charge distributions in the individual HSH'@
rationalizable in terms of qualitative
molecular orbital perturbation arguments, are in accord with
spectroscopic structural
ionization of the nonbonding sulfur electron pair leaves the HSH angle unchanged;
in the subsequent first excited state it is opened to 127".
In this context, it should be mentioned as a further example
of changes occurring in energetically higher states (cf. Figs.
4 to 7) that both spectroscopic
and calculation^^^^^
show the water radical cation HOH'@ to be linear in the
first electronically excited state (cf. Fig. 12: A(2Al)).
The above discussion of radical cation states of HSH and
HOH (Figs. 10, 11, and 12) demonstrates that ub initio SCF
spin populations and the "qualitative" molecular orbitals in
qualitative agreement therewith, satisfactorily describe the
electron distribution and structure, respectively, and thereby
facilitate extensive comparison of these chemically related
molecules. A further example will illustrate the feasibility of
such a state correlation also for characteristic molecular
subunits such as x bonds and the way in which the symmetryadapted qualitative MO models can be energetically parametrized with the experimentally determined energy differences
between the states. The x-ionization energies of norbornene,
norbornadiene, isopropylidenenorbornadiene, and isopropylidenenorbornane determined by PE spectroscopy represent
(Fig. 13).
the starting point of our
Three x ionizations are expected and observed in the PE
spectrum of isopr~pylidenenorbornadiene[~
(Fig. 13 : M.@).
cos
8-
9-
IE
[eVI
-EJ
CeVl
8I
019
9 - (a')
8.97
-4
.._
Fig. 13. Comparison of radical cation states M" of chemically related molecules is a valuable aid in the
assignment of PE spectra such as that of isopropylidenenorbornadiene: the first ionization of norbornene (C,
symmetry) and of isopropylidenenorbornane (C2" symmetry) leads to the x-radical cation ground state
%(2A')and R(2B2), respectively. Norbornadiene has t w o
states, %(2B2)and .%'A,). Of the three x
states of isopropylidenenorbornadiene the highest one fi(2A1) has the same energy as the A('A1) state of
norbornadiene; the two lower ones %(2B2)and A('B2) are of the same symmetry type and are split to
lower and higher ionization energies relative to the %('EB,) states of norbornadiene and isopropylidenenorbornane of equal symmetry.-This situation can be transparently illustrated with x molecular orbitals
M O 138, 391: in norbornadiene the two x orbitals (al)and (b2) split about an average value ?;rp' which
roughly corresponds to the x level of norbornane (b). Of the two norbornadiene n orbitals only (b2) has the
correct symmetry to mix with the a(bz) orbital or isopropylidenenorbornane, which leads to the two x(b2)
molecular orbitals split about the average valuc (nu) z;;". The third n molecular orbital ( a l ) cannot interact
with any other orbital of same symmetry and therefore its orbital energy expectedly resembles that of the
n(al) orbital of norbornadiene. The interaction parameters PI;"' and pp can be determined from the PE
spectroscopic Ionization energies (see text).
Angew. Chem. I n t . Ed. Engl. 16, 613-637 (1977)
621
The radical cation ground state exhibits a relatively low energy,
and the two excited states lie close together. Comparison
with then" states of other bicyclo[2.2.1] hydrocarbons reveals
how the varying energy differences can be interpreted: doubling the number of n electrons on going from norbornene
to norbornadiene leads to states of type 'A1 and 2B2 and
introduction of another exocyclic double bond of the same
symmetry type to splitting of the two 'Bz states about their
average value. This state correlation can be translated into
through-space interaction[391within the framework of a qualitative MO model (Fig. 13: MO) and described by construction
of linear combinations of appropriate symmetry. Only n orbitals of same symmetry mix with each other-in full accord
with the PE spectroscopic finding that the same ionization
energy must be expended to generate the 2A1 states of norbornadiene and isopropylidenenorbornadiene (Fig. 13: Ma@).
The qualitative MO model whose molecular orbitals reflect
of the radical cation states
the symmetry clas~ification~'~]
(Fig. 13) can be energetically parametrized as follows: The
Coulomb integrals
which represent the energy of electrons in a given potential, can be determined in the present
case by taking the average of the n-ionization energies of
norbornadiene or of the two 'B2 states of norbornadiene
and isopropylidenenorbornane (Fig. 13: ay and a';;").In many
calculation procedures-e. g. the HMO approxhationoverlap integrals S are not explicitlyaccounted for but included
by definition in the interaction parameter 8[401.The latter
can be calculated from the determinant (3), which contains
the connection of individual centers or of subunits in the
system and permits calculation of the pertinent eigenvalues
~ 5 1 by
~ ~ insertion
1
of the relevant ionization energies. For
instance, the interaction of the two R units in norbornadiene
with a",o=(8.69+9.55)/2=9.12eV and with e = I E 1 =8.69eV
(Fig. 13) is given by
Mutually corresponding states of chemically related molecules
can advantageously be compared on the basis of molecular orbitals.
4. Comparison of Molecular States Based on Molecular
Orbitals
Practically every chemist attempts to organize his set of
individual compounds by some definition of chemical relatedness, e. g. by principles of homology or by substituent
effects. Starting from (l), the following classifications are
recommended for molecular state comparison :
Buildrng Blocks
0000
---------"---oooo+....
.-...
Heteroconjugate
lsoconjugaie
+
.
I
\I
+ A Potentials
A Topology
lsoelectronic
L
I
-Extraelectronic
.7
(Z-1)H A Z
"United Atom" 1st Order
1st Order
2nd Order
Depending upon the relative change, the distinction between
iso- and hetero-conjugate, and between iso- and extra-electronic systems is of course a flexible one. The extent of perturbation often also determines the scope of a model, and, within
this scope, its quality, i. e. the deviations in correlations between
observed and calculated quantities.
From thousands of published state comparisons some exemplifying (4) will now be presented. They are chosen to illustrate
how entire classes of compounds can be surveyed by application of a network of general rules (cf. Fig. 2).
4.1. Topology-Based Comparison of States
(3)
When calculating the analogous parameters pipfor isopropylidenenorbornadiene it must be borne in mind that three 7c
subunits are interacting[401.The resulting parameters a, and
(jncommand more general interest because they can be compared with values for other 71
and can be inserted
for analogous compounds.
in qualitative MO
The satisfactorydescription,of radical cation states by molecular orbitals (Figs. 9 to 13) confirms that the requirements
specified above (1) concerning the reproduction of atom
connection and arrangement, as well as electron distribution
in the existing potentials, are fulfilled. Even the qualitative
linear combinations of bond orbitals which are readily compatible with chemical intuition can be symmetry-adapted["] and
parametrized in a transparent way with experimental state
data. The predictions of such simple MO models can be supported by the results of calculations of varying degrees of
sophistication right up to the numerical reproduction of state
data (Fig. 1).As repeatedly emphasized, there is one particular
application of particular interest to the chemist:
622
In the approximate solution of the Schrodinger equation
applying the variation principle, a secular determinant of type
(3) is passed[401which incorporates the topology of the molecule (4). Thus in the HMO approximation[z81for unbranched
polyene chains (Fig. 14) the principal diagonal contains the
energy terms (a- E ) and the adjacent diagonals contain the
n interactions Pxassumed to be equal between the connected
centers
~
.
~
.0- . The .solutions
of the corresponding eigenvalue polynomials can be derived
in closed formtzs1(Fig. 14).
From the HMO eigenvalue schemes of linear n systems
(Fig. 14) one infers for their state comparison: neutral compounds with odd-numbered polyene chains in which one electron occupies the non-bonding molecular orbital of eigenvalue
&FM0=a,
are radicals with a doublet ground state. In compounds with even-numbered n systems, the energy difference
AE= hv, between the ground state and the first (n+n*) excited
state can be compared, in a crude approximation, with the
distance between the inner molecular orbitals
and
the color of polyenes thereby satisfactorily predicted. Moreover, all unbranched polyene chains belong to the alternant
n systems[28,42-441,whose molecular orbitals show pairwise
correspondence (Fig. 14: E ~ ~ ~ = E1 ) ~and
? ~always
+
display
a.
Angew. Chem. Int. Ed. Engl. 16,613-637 ( 1 9 7 7 )
Fig. 14. The H M O model [2S) for linear n-electron systems with w= n centers, assumes in a simplifying
way that all n interactions pn are equal. A closed formula for the eigenvalues gYM0 can be derived
via the determinant [40] 11 I[ containing the connections of the individual centers. The eigenvalue
schemes in units of the parameter (Pn) permit a comparative discussion of numerous properties of
linear polyenes. Thus all odd-numbered compounds are radicals owing to the occupation of the nonbonding molecular orbital &JHM0=aby only one electron (t).For even-numbered n systems the distance
between the inner orbitals decreases with increasing number of centers and a linear regression is
obtained for the wave number Cx [cm-'1 of the longest wavelength absorption, i.e. Cx=14800+
25400 A€JHMo[28], from which the color of long-chain polyenes can be predicted. From the alternancy,
i.e. from the pairwise arrangement of the molecular orbitals $1 and
relative to Q, it follows,
among other things [28], that the charges q,, at all centers p amount to q,=l and, therefore, the
n-dipole moment pn must be identical to zero, pn = O .
,.
IN+2
'0
Breslow
O/-
1
Pettit Thiele
1
1
"Kehule" v.0oering
2
Katz
2
3
Vogel Boekelheide Sondheimer
Fig. 15. For cyclic n-electron systems [28] the interactions Pn occurring on ring closure 1-p have to
be included in the H M O determinant [40] I[ (I in addition to the linear systems (Fig. 14). The general
furnishes energy-level schemes in units of p, which display
formula resulting for the eigenvalue
the following features: regardless of the number of centers J = n , an eigenvalue &FMo=a+2pis always
obtained for J = O ; pairs of degenerate eigenvalues EI"~"=E:!'? result from the cosine term for values
O < J < n - 1;and finally E : M ~=u - 2 P follows for J = n - 1 in all even-numbered, and hence alternant,
primeters. On introducing the electrons one recognizes that closed shells with spin-paired electrons (7-1)
are obtained only if their number satisfies the Hiickel rule 4 N +2. The corresponding compounds may
be charged, exhibit characteristic properties, and have been systematically synthesized in large numbers
&YM0
1451.
a n charge qFMo= 1 at all centers p; hence a n dipole moment
pn=O is predicted for the compounds in the ground state['*].
The classical example for ground state comparisons with
the aid of MO models is found in cyclic 7c systems known
as perimetersL2'.42 -4J1. Their topologically determined secular
additionally contain the 7c interactions pi,,
Angew. Chem. Int.
Ed. Engl. 16, 613-637 (1977)
and p,, owing to cyclization of the chain by connecting centers
1 and p (Figs. 14 and 15). The resulting polynomials again
yield general solutions &YMo(Fig. 15).
As follows from the HMO eigenvalue schemes (Fig. IS),
stable nonradical n systems with closed valence shells are
obtainable only if 4 N + 2 electrons ( N = 0 , 1 , 2 , 3 ...) are sup623
plied. This conclusion, reached by E. H i i ~ k e l [ ~led
~ ] to
, one
of the greatest successes of MO theory: not only was the
tropylium ion C7H7, for example, predicted in 1931[461,long
before it was ever isolated (1954)as a salt, but a large number
of compounds such as [lo]-, [14]-, and [18]annulenes (Fig.
15) have been systematically synthesized since 1950as a conseq ~ e n c e o f t h e ( 4 N + 2 ) r u l e [ ~.~N.umerous
~ ~ - ~ ~ other
1
observations on the ground states of molecules having n perimeters
find a ready e ~ p l a n a t i o n [ ~ ~ these
, ~ ~ - include
~ ~ I : the JahnTeller distortion D4h-’D2h of the cyclobutadiene diradical,
and the oxidizability of its tetraphenyl derivative to the dication, or the flattening of the boat conformation of cyclooctatetraene on reduction to the dianion. On the other hand, caution
should be exercised when dealing with all more or less arbitrary
and artificialsupplementary definitionsof “aromatic”, “nonaromatic”, or even “antiaromatic” character of compounds[28,471.
It should also be pointed out that even-numbered perimeters
are also alternant systems128*42-44].
The fascinating topological criterion for alternancy can be formulated such that labeling
of the individual centers of a system with “*’’ or “0”leads
only to “*-o” connections. Among the numerous conse42-441, the pairwise occurquences of such a
rence of the molecular orbitals $J and $ . - J + l for linear
7-c-electronsystems (Fig. 14) or $”- for even-numbered cyclic
n-electron systems (Fig. 15)is especially striking. In this context
one might also emphasize that the calculated virtual unoccupied molecular orbitals can also be correlated with experimen-
tal data, as demonstrated here for ESR, PE, and ET spectra
(Fig. 16, 17, and 18).
The ESR coupling constants a: and a$ of radical ion ground
states (Fig. 3) can usually be satisfactorily correlated with
calculated spin populations p:.”, i.e. with the squares of
the coefficients cJ’, from the relevant linear combinations
$J
cJp+2401, which represent the electron densities, according to the relation derived by M c C ~ n n e l l [ ~ ~ ]
=F
Only in the case of alternant systems, owing to the pairing
or $,,-J, are the radical cation
properties of \IrJ and
and radical anion ground states of one and the same molecule
expected, and found experimentally, to display (almost) the
same spin populations and thus (almost) the same coupling
constants (Fig. 16)[271.
On correlation of 5 ionization energies with bond orbital
models[19],all the orbitals of the latter are doubly occupied.
For radical cation states of alternant cs systems, owing to
the pairwise correspondence of the eigenvalues EJ and E , - ~ + 1
or E . - ~ , it is expected and observed PE spectroscopically
that the 5 ionization energies of, say, methylp~lysilanes[~~~
are equally distributed to both sides of a reference state (Fig.
17).
The PE ionization pattern of decamethyltetrasilane corresponds to the fully occupied eigenvalue scheme of the chain
with n = 3 (Fig. 14), and that of dodecamethylcyclohexasilane
Fig. 16. In alternant x systems, i.e. those whose centers can be alternately starred, the pairing properties of their molecular orbitals $IJ and JI.-J+I
(Fig. 14: linear systems) or $.-, (Fig. 15: even-numbered cyclic systems) imply that the coefficients cJp of the linear combinations $,=t;cJp& [40]
differ in sign but not in magnitude [28, 42-44]. The squares of the coefficients c$,, which can be correlated ( 5 ) with the ESR coupling constants
are identical. The ESR spectra of the radial cation and the radical anion are therefore expected to be equal for the same alternant compound, but
to differ considerably for nonalternant compounds, e.g. those with odd-numbered rings (*-*). This is indeed the case, as demonstrated by the ESR
spectra of the radical ions of perylene and acenaphtho[l,2-a]acenaphthylene [27].
OF,$,
624
Angew. Chem. Int. Ed. Engl. 16, 613-637 (1977)
Fig. 17. Comparison of the lowest PE ionization energies of methylpolysilanes
.. .
R(SiR2).R and (SiRI). yields as center of gravity ctsisi the 1st ionization
energy of hexamethyldisilane. The splitting patterns correspond to the HMO
eigenvalue schemes of isoconiugate
(4) chains (Fig. 14) and rings
15),
.~
- (Fig.
with fully occupied orbitals; insofar as they are not already included in
the parameter asisi,the Sic and CH orbital components can, to a good
approximation, be neglected. Correlation with the eigenvalue coeficien xJ””
defined according to E=U+X,!’~OD
leads to a regression line with only a
slight standard deviation; its slope affords the interaction parameter t3sislisis2
=0.5eV. The alternancy of the SiSi bond skeleton is already included
in the HMO model: if Koopmans’ theorem (2) is extended to A&yMo-AIE,,
the object/mirror image arrangement of the ionization energies about
the disilane center of gravity fully confirms the above premise.
to that of the perimeter with n = 6 (Fig. 15). Comparison
of the radical cation states of methylpolysilanes on the basis
of an LCBO-MO model is at the same time a good example
of parametrization of the Coulomb integral rx and resonance
integral p“”] with the aid of ionization energies (Fig. 17).
The topological concept of alternancy can also be illustrated
for certain isoconjugate n systems by inclusion of the radical
anion
The energy differences from the ground state
of the neutral molecule up to the resonance states generated
by transient take-up of an electron are accessible by the
refined[49]experimental technique of electron transmission
spectroscopy (ETS)[”] (Fig. 18).
If electron transmission spectroscopy is regarded as complementary to photoelectron spectroscopy, and if the electron
affinities taken from the ET spectra[50]are compared with
unoccupied orbitals in an extension of (2) and contrasted
with the ionization energies[’51,there results the pairwise correspondence of orbital energies relative to a common center
of gravity expected (Fig. 18) for alternant linear and cyclic
x systems (Figs. 14 and 15).
Summarizing:Comparisons of states on the basis of topologically designed MO models emphasize the connection of
the centers in the molecule, are of considerable utility owing
to their extensively generalizable predictions-consider E.
Huckel’s (4N + 2) r ~ l e [ ~ ~ ~ - aalso
n d lead to further insights,
as shown for the examde of “ a l t e r n a n c ~ ” [ ~. Th
~ . ese
~~-~~~
insights apply strictly only to isoconjugate systems (4) such
as unsaturated hydrocarbons[28,42-461and, approximately,
to 0 systems with characteristic subunits such as peralkylated
polysilanesI”, 481, boranest5‘1, or interhalogen com’’I. However, the basic principles can also be transferred to hetero systems, e.g. after modification by the potential
perturbations mentioned in Section 4.2.
4.2. Potential-Based Comparison of States
Although they may contain different kinds of building
blocks, hetero-conjugate molecules ( 4 ) can nevertheless display the same kind of connection, comparable geometry (I),
and possibly the same number of (valence)electrons. As already
discussed for HSH and HOH (Figs. 10 to 12), however, the
varying potentials of the central atoms lead to a different
electron distribution, and thus to deviating molecular properties. If the changes remain in the range of validity of the
eeMagnetic Field
Retarding Electrodes
-1
c
Electron Transmission Spectroscopy
I
1
I
-- 8
-- 9
10
- -10
11
- -11
- -12
Fig. 18. On recording an electron transmission spectrum, a monochromatic electron beam of variable
energy is scattered by the gaseous compound 149, 501. The resonance (current) fluctuations d l
measurable in the unscattered residual transmission beam indicate the intermediate formation of
short-lived radical anions. If the electron affinities E A [eV] of the resonance states thus determined
are plotted together with the PE ionization energies I E [eV] on an orbital energy scale E, [eV]
one obtains the object/mirror image arrangement of the energy differences between the states
expected from the alternancy of the x systems (Figs. 14 and 15).
Angew: Chem. l n t . Ed. Engl. 16,613-637 ( 1 9 7 7 )
625
MO model used, then the perturbations128.39, 421 can be profitably used for comparison of molecular states.
Potential-based models, which compare the states of chemically related molecules with (largely) similar structures on
the basis of changes in potential, may open up interesting
vistas. This is demonstrated, for instance, by the photoelectron
spectra of the isoelectronic species AH, of a group (constant
n), and also within a period of the system of elements (Fig.
19), with one proton after the other with nuclear charge Z = 1
being withdrawn from the nucleus Z . An approximation of
this kind is designated as the “united atom” model (4) and
is also known as Grimm’s “hydride shift” ruler541.In the
place of the ambiguously definable and far too imprecise
a suitable parameter for comquantity “electronegati~ity”[~~],
paring the effective potential is the average ionization energy
of all n valence electrons of the central atom IE:Z&
= [ZE1(A)+ IEl(A@)+IEl(A@’).. . IE1(A+“-’)]nr561which
also serves for estimating effective nuclear charges
according to S/ater[28~31]
(Fig. 19).
The determining influence of thecore potential of the central
atom A in the compounds AH. is clearly revealed by a comparison of the atomic and molecular ionization energies IE:Z&ce/
IEl(AH,) (Fig. 19). Thus the sum of all PE ionization energies
for argon and water are almost identical. Two regions are
easily recognized: the energetically higher states with s holes
and the band of the p-type radical cation states, which are
to be assigned to the ionizations of electron pairs nA or to
the symmetry-adapted bond combinations oAH (Figs. 4 and
10). It appears, for example, that the AH2 compounds display
energetic centers of gravity which rise slower than the nsA
ionization energies (Fig. 191 corresponding to the expectation
that s electrons, being on average closer to the nucleus, are
more susceptible to changes in potential at center A.
In topology-determined systems, potential changes by
substituents within the scope of the model can profitably be
described as perturbations. One distinguishes between firstand second-order perturbations-as discussed here for the
two highest occupied, degenerate benzene orbitals
n(e1g)r28,
42, 431 (Fig. 20).
+
IFAtom
Yalenre[eV172 2
58.4
46 1
35 4
258
722
461
425
8Ea=C:X
6ctX
Perturbation
2nd Order
1st Order
BEn--
%t
an-ax
Fig. 20. Substitution i? and on the benzene ring lowers the molecular symmetry, e . g . Dsh+C2,, thereby removing the degeneracy of the elg orbital.
1st order perturbation occurs in the HMO approximation [28] via a change
in potential - 6ux (X= donor) or hax (X = acceptor) and leads to a raising
-& or lowering 6&,,of the perturbed molecular orbital, whose composition
is regarded approximately as unchanged. On substitution in a nodal plane,
i.e. coefficient at center cJx=OL5’J, 6en remains equal to zero (cf., however,
Fig. 23). 1st order perturbations are frequently isoelectronic (4), i. e. an atom
X or a group of atoms X-(Y) at the center concerned is replaced by
another atom or another group of atoms of different nuclear charge Zx.
2nd order perturbation occurs in the HMO approximation [28] on mixing
of orbitals of the same symmetry class, i.e. construction of the linear combination +=CJJIJ_+CKJIX for the complete system. The change in orbital energies
+ _ S E is
~ directly proportional to the square of the perturbing interaction
BjK and inversely proportional to the separation of the orbitals aR-aX.
2nd order perturbations are usually extra-electronic (4), i.e. the system is
extended by one or more centers.
371
Fig. 19. Comparison of PE spectroscopic ionization energies IE; [53] for 3rd row AH. species,
starting from the 3s and from 3 p states of the argon atom shows: the 3s electrons in the
proximity of the nucleus become more readily ionizable with decreasing effective nuclear
charge. represented here by the average ionization energy of the valence electrons of atom A
=I
[%I. The p electrons, which are on average farther away from the nucleus,
expectedly prove less sussceptible to the influence of the potential of atom A ; their levels split
to the radical cation states typical of the compounds which are assignable to ionizations from
nAelectron pairs or symmetry-adapted combinations of GAH bonds (Figs. 4 and lo).-In the
hydrogen compounds AH2 of group ti the energy of the radical cation states AH;’
decreases on going from 0 to Te, the rate of decrease being all the greater the higher the
ionization energy Ig, and thus the influence of the effective nuclear charge (I-).
626
Angew. Chem. Int. Ed. Engl. 16,613-637 ( I 977)
PE spectroscopy furnishes-uia Koopmans’ theorem (2)didactically excellent examples of 1st and 2nd order perturbationLZd):
these include phosphabenzene[’”. 581 and 1,4-bis(methylthio)ben~ene[~~~
591 and their homologous derivatives (Figs.
21 and 22).
H3CO
H
cos
?
I
,
“
0
GI I
“
N
CP
H
ds Sb
81
Fig. 21. In the PE spectra of benzene and its hetero derivatives C5H5X
with X = N , P, As, Sb, Bi [%I, the three I[ ionizations are observed below
12eV and, in addition, radical cation states which are assigned to the electron
pair nx(’Al) and to the benzene skeleton o(’B2) [58]. In the benzene x:
orbitals the coefficients [57] at the substitution centers are 1/3 (r,),0 (naS),
and 1/6 ( K ~ )On
. correlation of the 1st order perturbations due to the potential
changes Gax-approximated
by the valence ionization energies I E x
(4S3,2+3P0) of the atoms X-with the PE ionization energies of the C5H5X
molecules then, according to (2) A I E - A E J and to 6cJ=c$6ax (Fig. 20),
regression lines result as expected with slopes of about 1/3, 0, and 1/6.
This molecular state comparison at the same time confirms the PE band
assignment for the r ionizations.
The radical cation state comparison of benzene and its
aza, phospha, arsa, stiba, and bismuta derivatives[581demonstrates the elegance, quality, and utility of potential-based
1st order perturbation models as follows (Fig. 23): of the
three model parameters two are accessible by comparison
with experimental data, the difference in eigenvalues Lkn
according to Koopmans’ theorem (2)from differencesin molecular x ionization energies, and the perturbations of the potential 6ax from the 1st ionization energies of the atoms X[561.
The varying slopes of the resulting regression lines expectedly
correspond to the various orbital contributionst571c’,~at the
substitution center X (Fig. 21). The quality of the regression
and one of the possible applications should
(6)is very
be to predict the 1st ionization energy of the hitherto unknown
species silabenzene:
Angew. Chem. Int. Ed. Engl. 16,613-637 ( 1 9 7 7 )
Fig. 22. The PE spectra of benzene and its pbis(methy1thio)-and p-dimethoxy
derivatives contain ionizations to x:.@ states as the lowest 2,4,and 3 ionizations.
In the M O model, e.g. for the dithio r system, nz(a.)=as and xas(bZ,)=.ecn
represent internal standards which are of unique symmetry type and do
not mix with any other orbitals in the same energy region. Mixing of symmetry~.
equivalent orbitals n; +n,(b,,) leads to mirror image splitting * 6 ~ . Parameters & = -v(IE2-fE,)(IE3 - I E , ) show that 2pc/2p0 interaction effects
a larger 2nd order perturbation than 2p&ps interaction.
IE1 =5.24+0 362 I E A t a ~ 7 1
IEsi=815eVt56’
4
(6)
IEl -8.2 eV
External extraelectronic (4)substitution on the benzene ring,
e . g . with XCHj groups whose electron pairs nx extend the
n system, can usually be satisfactorily described by 2nd order
perturbation models. Para-disubstituted benzene derivat i v e ~ [constitute
~~]
an appropriate example (Fig. 22): mixing
of the symmetry-equivalent orbitals can be parametrized with
PE ionization energies, the orbitals of unique symmetry class
serving as internal standards as and a,. It warrants mention
that numerical agreement of the 3rd ionization energy of
the dithio compound (ax)and the 1st benzene %-ionization
energy, i.e. the constancy of the n,, orbitals confirms the
applicability of a 2nd order perturbation model. Parametrization by analogy with (3) yields pxo>j3s, and thereby provide
the information that the n interaction 2pc/2p0 causes a larger
perturbation than 2pJ3ps.
Considering the many known examples it is pointed out
that perturbation models, once established, also reflect other
molecular properties suitable for modeling : thus the measured
ESR spin populations of the corresponding radical cations
627
U-CH3
3.44
G
S-CH3
H 1.79 G
H 2.93G
H,C-0
5.44 G
H3C-i
must, according to the McConnell relation (5), at least qualitatively reproduce the orbital mixing ratio n,_+n$ (Fig. 22).
Accordingly, a large ring contribution IT$ is expected and
observed for the dimethoxy derivative, but a large electron
pair contribution n, for the bis(methy1thio) derivative[59!
1st and 2nd Order perturbation models of the kind discussed
in this section are commonly encountered in the literature, e.g.
for unsaturated hydrocarbons[28*3 8 , 4 1 , 42, 6ol (cf. also Fig. 13),
their halo[61]or pseudohalo derivatives[6z],and also for saturated hydrocarbons and their derivatives[17.631. The parameters
ax and PXY determinable from experimental data-especially
ionization energies-can also be used in other models and
are therefore of predictive value (6) with regard to model-determined molecular properties of chemically related compounds.
4.3. Computer-Assisted Comparison of States
The question “to calculate or to measure?” has already
been answered in Section 1 by pointing out just how successful
and stimulating a combination of the two turns out to be.
Computer assistance is usually also valuable for setting up
models for comparison of states, a whole arsenal of various
presently being available-ranging from the
optimum parametrization of MO models from a set of experimental data[62*631
via EHM0[651
(Extended Hiickel MO) as the
simplest o calculation procedure[641and numerous so-called
semi-empirical procedures[641such as CNDO (Complete Neglect of Differential Overlap) and INDOr349w (Intermediate
Neglect of Differential Overlap) or MIND0[671 (Minimum
Neglect of Differential Overlap) to ab initio calculation^^^^^ 681.
In the HMO approximation[282421,
for example, the complete
secular
containing all the centers and all the
interactions is solved instead of the incomplete perturbation
determinant of type (3). Experience shows computational aids
to be indispensable for molecules of low symmetry and special
cases, e. g. for extreme p e r t ~ r b a t i o n s(Fig.
~ ~ ~ 23).
l
In the PE spectrum of the phenyl-substituted phosphorus
ylide (Fig. 23) the characteristic benzene band at 9.25 eV (shown
as dashed line) that would have to be assigned to the n,,
ionization(Figs.20to 22)isconspicuouslyabsent.The suspected
gross perturbation of the benzene IT system by the neighboring
“ylidic” carbon electron pair n: is clearly revealed by CNDO
calculations (Fig. 23) : the computer-simulated C-C connection of the two molecular moieties without allowing for TC
interaction raises the two benzene n(elg) orbitals to different
degrees by o charge transfer. Additional splitting leads to
the final sequence of molecular orbitals. As may be gathered
from the PE spectrum (Fig. 23), the n., molecular cation
state is stabilized by 0.9 eV relative to benzene by the neighboring carbanion center-this is one of the strongest substituent
effects
The model derived by CNDO simulation
of an experimentallyinaccessible intermediate can be substantiated by state comparisons with other phosphorus ylides;
the charges q ~ ~ also
~ ’ calculated
O
for the benzylidene carbon
atoms by the CNDO program correlate satisfactorily with
I3C-NMR
(8)
+I?
-001
613C (Rel.to Benzene)
CNOO
%(a)
Fig. 23. The PE spectrum of benzylidenetrimethylphosphoranesurprisingly no longer shows a band typical for
benzene derivatives at 9.25 eV (---) [69]. This finding becomes understandable in the light of CNDO
calculations: if the partial systems benzene and methylenetrimethylphosphorane are connected together
without permitting n interaction (Ax) then a considerable raising of the n, orbital, and also of the xas orbital,
is calculated. x interaction mixes the initial orbitals nF+n,, whereas x,, remains unchanged. The PE
ionization energies are assigned thus: I E l (nz-n,), I E 2 (nss),and IEa (n,+nF). The benzene radical cation
state represented by the orbital K,, is therefore stabilized to the extent of 9.25-8.32=0.93 eV by the ylide
substituent.
628
Angew. Chem. Int. Ed. Engl. 16,613-637 ( 1 9 7 7 )
As demonstrated by the perturbation model (Fig. 23), the
phenyl ring acts as an acceptor and assists in distributing
the neighboring carbanion charge over the ylide molecule.
Generally speaking, state comparisons via computerassisted perturbation models were, and still are, frequently
utilized for detecting dominating effects and possibly defining
them as model parameters-for example, the concepts of
through space and through bond interactions[393701 (cf. also
Fig. 13[381) or of hyperconjugation[2',7'1. As a rule, the computer programs used also calculate further data for comparison
such as in (8). Among these data, particular interest attaches
to the total energies E,, and the charges q,, on the individual
centers p, which, in the HMO approximation for example,
are made up of the contributions of the individual orbitals
J f J each occupied by bJ electrons:
(9)
These data are obtained independently by semiempirical proc e d u r e ~ [ ~and
~ can
~ ~be
~ profitably
, ~ ~ ] employed in computerassisted perturbation models. Ylides have already been discussed as examples for charge distribution q:ND0[691 (cf. (8)
and Fig. 23); the use of EHMO total energies will be illustrated
by a description of the cage structure of S4N4[721
(Fig. 24).
The starting point for a discussion of the S4N4 cage struct ~ r e [is~provided
~]
by the molecular orbitals of the [8]perimeter (Fig. 15). Perturbation by S and N potentials merely
removes the degeneracy of the nonbonding orbital pair, owing
to the position ofthe nodal planes. Each nitrogen atom contributes one K electron, and each sulfur atom two rc electrons;
filling with a total of twelve K electrons affords a reactive
triplet state with two unpaired electrons. A stable singlet
state could be achieved if the highest orbital could be lowered,
by additional transannular interactions, beneath the next-highest pair (Fig. 24). Among several possible deformations,
EHMO calculations show the DZdstructure with S-S bonds
to have the lowest total electron energy[7z! This calculated
result agrees with the S4N4 structure determined by X-ray
methods. The computer-assisted perturbation model is also
valid for the ground state of other cage compounds such
as P4S7, in which the P-P connection leads to a greater
interatomic distance, and permits understanding of further
molecular properties: thus addition of the Lewis acid SbC15
generates a perturbation sufficient to stabilize opening to the
singlet ground stateC7'](Fig. 24). A planar structure was predicted by the MO
for the dication S4N:@ in 1970-in
full accord with the recently published X-ray structure analy~is['~!
Quite intentionally, the examples cited so far have not
aimed at closest possible numerical reproduction of experimental data by calculated values (Fig. I), but were instead selected
to provide an insight into the dynamic behavior of molecules
under the influence of perturbations for comparisons of states
of chemically related compounds. For the sake of completeness, however, it should be mentioned that the comparison
of experimental data and model quantities has not only often
permitted parametrization of the latter (cf., e.g., Fig. I 3 and
(3), Figs. 17, 21, or 22), but frequently leads to significant
correlations for calculated model quantities, permitting
=I
I
W
Fig. 24. The MO description of the three-dimensional cage structure of S4N4 [72] starts from the isoconjugate [dlperimeter (Fig. 15).
Lowering of the symmetry D8h+D4hmerely removes the degeneracy of the orbitals E + ~ = S L by different 1st order perturbation
(Fig. 20). Occupation by 1 N and 2s electrons shows that a planar S4N4 molecule would have to be a diradical. Between the
numerous possible structural distortions, which would either interchange the uppermost orbitals or cancel the degeneracy of the
as a
second from top pair of orbitals, no adequate qualitative distinction seems possible. Calculation of an energy surface
function of distances such as dNN shows that a Dld cage should be the most stable S4N4 structure. The model can also be applied
to other cage molecules such as P4S7 or to cage opening by perturbation in the SbCls adduct. X-Ray structure analysis recently
proved [73] the cation S&Nf' to be planar-as would be expected from occupation of the eigenvalue scheme by two electrons less.
Angew. Chem. Int. Ed. Engl. 16.613-637 (1977)
629
predictions within their confidence limits. Points to watch will
become clear in simultaneous correlations of energy - &FNDO/
IE according to Koopmans (2) and of dipole moments pcalc./
pexp.
covering 15sulfur compounds offour different types, some
exhibiting extreme substituents (F) and some with extreme
ring strain (<CSC ca. 50”)r741.
I
9
11
13
15
-d CNOOiZ
X
Y
F,CL.CH,
17 19
[eVI-
5,
0
l
1
2
3
p*“-
F,cH,,[=,oR
O,NR,S
panying loss of transparency often leads to the limits of easily
tractable MO models. Beyond these limits, the description
of molecular states on a molecular orbital basis has made
enormous progress within the discipline of theoretical chemistry-even reaching as far as the reproduction of spectra of
small molecules (Fig. 1). In what cases should the average
4
5
(10)
F, CI. C H ~ , ~
0, NH
The regression line of the Koopmans correlation (10: A)
neither passes through the origin nor exhibits a slope of 1,
and its standard deviation is relatively large. The known gasphase dipole moments are correlated directly with the calculated values: the regression line (10: B) of slope 1 passes
through the origin. Taken together, the two correlations show
the calculation procedure used to be suitable, within the limitations mentioned~28],for interpretation and for prediction of
model properties of such sulfur compounds. However, the
reader is earnestly warned against all fortuitous numerical
coincidences based on single “hit and run” calculations without
extensive testing of confidence limits (lo), or in which “homespun”parametersdefinedmore or less arbitrarily, like “aromaticity”, are backed up with some kind of calculated figures.
To summarize:
By virtue of their inherent concepts (topology, symmetry),
of their input parameters (potentials, number of electrons), and
of theirflexibility (perturbation), molecular orbital models provide a manifold and stimulating overview of types and classes
of compounds.
consumer of quantum chemistry become cautious and consult
a theory producer?
All the MO models presented are based on the implicit
assumption that the distribution or motion of an electron
can be described within the time-averaged field of the atomic
nuclei and the other electrons of a molecule (1).Such one-electron MO models describe the properties of molecules in a
particular state, in each case on the basis of a single electronic
configuration (11):for instance, all radical cation states according to Koopmans by means of a single set of molecular orbitals
(Fig. 25).
In conclusion, the limited validity and some possible extensions of one-electron MO models will be illustrated for important aspects: change in electron energy with or without changes
in the number of electrons, structural changes or reactions,
and the description of large molecules of low symmetry.
5.1. Changes in Electron Energy Due to Changes in the Number
of Electrons
One-electron MO models for molecules M with closed electron shells in the ground state (1 1:M) generally also permit
5. Limits of the Molecular Orbital Descriptionof Molecular States
Experimental molecular state data and calculated molecular
orbital parameters can be correlated in various ways: energy
differences such as ionization or excitation energies can be
compared with eigenvalues (cf. (2), (lo), or Figs. 10, 13, 17,
18) or their differences (cf. Figs. 14, 20, 21, 22). Changes in
charge such as demonstrated by ESR spin populations or
resulting from substituent perturbations can be correlated
with squared eigenfunctions (cf. (5), (7), or Figs. 12, 16, 20,
21).Furthermore, “summed-up”parameters such as total energies or total charges (9)provide a basis for discussing numerous
other molecular properties such as structures (Fig. 24), dipole
moments (lo), or NMR signal shifts (8). However, the accom630
I
M
M‘O
M .a
M’ (Degenerate)
Singlet
Triplet
description of the properties of radical ions (11:M.’ and
Mqe)with a modified number of electrons (Fig. 25). Numerous
correlationsofvariousexperimentaldatawithcalculated
parameters of the respective singly occupied molecular orbital
Angew. Chem. Int. Ed. Engl. 16,613-637 (1977)
Molecular States
(Some) Measurement Methods
Molecular Orbitals
(One-Electron Approximation)
Fig. 25. Energy differences between, and charge distributions in the various states of a molecule can be determined by a variety of experimental methods,
of which some are mentioned (cf. Fig. 3). Interpretation of experimental data with one-electron MO models is based on the fundamental assumption
of Koopmans’ theorem IE;= -&ScF (2) or the McConnell relation a’;:?= lQ16, (5), i.e. that the electron distribution can be established with a single
electron configuration. Ionization energies or half-wave reduction potentials can then be compared with the eigenvalues c J , ESR coupling constants
with the squares of the orbital coefficients cJ, at the centers p of the singly occupied molecular orbital (rJ of type 0,I[, K*, c* or R . In some (a few)
cases the UV excitation energies AE=hv can be expressed as the eigenvalue differences AcJ between the initial and final orbitals (cf. Fig. 14). The
limitations of describirig states by one-electron MO models are clearly apparent from the crossovers (+) of the individual correlations.
c0nfirm[27,28,42-441
that the electron distribution can usually
be described satisfactorily by a single electron configuration
(11: Ma@and M*&’),
The applicability of one-electron approximations requires
not only a comparable molecular structure but also that
changes in the charge distribution (relaxation) and in mutually
dependent electron motions (correlation) may together be neglected“ 5bJ. Thus the molecular structure remains essentially
s duration of vertical ionization
constant during the ca.
(cf. Fig. 4: HBr potential curves), and the deviations from
Koopmans’ theorem appearing in a particular method of calculation will be all the smaller the less the charge distribution
and electron correlation change during the ionization proc ~ s s [ Far
~ ~ ]better
.
agreement with experimental data is yielded
by the many-electron approximative solutions of the
Schrodinger equation mentioned in the introduction (cf. Fig.
I), in which the total energies of the individual states are
calculated separately and subtracted from one another according to (2): calculated values BE,,,, and vertical ionization
energies IE; generally differ by less than +0.3eV13-71.
Nevertheless, application of these highly sophisticated
methods to larger molecules still remains in the precinct of
the theoretician.
5.2. Changes in the Electron Energy not Involving Changes
in the Number of Electrons
As documented by many of our examples (Figs. 10, 13,
18, 21, or 23), ionizations can usually be described, starting
Angew. Chem. Int. Ed. Engl. 16, 613-637 (1977)
from a single electron configuration (11:M*@),by one of the
so-called canonical
which are adapted to the symmetry of the molecule and satisfy Koopmans’ theorem (2).
The electron hole generated can thus be characterized according to the orbital type and the electronic interaction regarded
approximately as unchanged. For excitations in which the
number of electrons remains constant, i. e. none of the electrons
is ejected, a description of the molecular state based on just
a single electron configuration (11: M) is generally inadequate
(Fig. 26).
As in the case of the blue compound bis(trimethylsily1)diimine[761(Fig. 26), the ionization energies are usually reproduced satisfactorily by MO calculations via Koopmans’
theorem (2), while reproduction of the excitation energies
usually requires consideration of the modified electronic interaction-in the present case electron repulsion is approximately
accounted for by Coulomb terms J as well as spin pairing
effects by exchange terms K[77J.
Generally, one-electron MO approximations, i. e. those
which treat electrons as being independent of one another,
are unsuitable for describing electronically excited states x{(M)
of neutral molecules; simple regressions between UV maxima
and eigenvalue differences, as shown in Figure 14, therefore
represent exceptionsvalidonlyforcertain excitLd states of particular series of compounds[28*441.
Thus it is impossible, with
a single electronic configuration, to establish either the energy
differences between singlet and triplet states (11: TL and TT)
or the different excitation energies to such states of like symmetry (11: hv+hv’), for which degeneracy would be predicted
631
h
Fig. 26. Bis(trimethylsily1)diimine (A) is a blue (!) azo compound [76], whose long-wave n-trr' band appears at the
edge of the IR spectral range, which exhibits a first ionization energy of only 7.1 eV, and whose unusually short
NN bond length (B) amounts to only 117 pm (azobenzene 125 pm, N=N 111 pm). The MO description in terms
of the charges Q.,.m (C) suggests that silicon (Q$yDo=+ 0 . 5 3 ) acts as a strong electron donor towards nitrogen
(QSNDO= -0.2). The CNDO calculations also satisfactorily reproduce the first ionization energy I E ; =7.1 eV (D);
the eigenvalue of the highest occupied orbital n$ amounts to &tW=7.6 eV. The n-n* excitation energy
E:+;'"'= 1.5 eV, on the other hand, cannot be related to the gap A&=9.6 eV to the lowest unoccupied K* orbital.
Only after approximate consideration of the modified electronic interaction [I 71, as expressed in the large Coulomb
term J,,*, can the n+ a* excitation energy be roughly reproduced (E) with AEg!;p= l.OeV.
without inclusion of configurational mixing[31*32, 44, 7 8 * 79!
Numerous other experimental results such as negative spin
densities (Fig. 12) can only be explained after inclusion of
electronic interaction.
5.3. Changes in the Electron Energy Accompanying Changes in
Molecular Structure
Molecular orbital models are usually based on a given,
unchanged molecular geometry-frequently an illegitimate
a s s ~ m p t i o n [ ~Rather,
~ 1 . changes in the structure of a molecule
are known to occur on transition to another molecular state
(Figs. 5, 6, 7, 12 or 24) and also within series of chemically
closely related compounds (cf. e.g. Fig. 11). In many cases,
Gillespie's electron pair repulsion modelf8'' proves very useful
in discussing possible changes; thus, along the series
H
H\kH
I
97.8"
99.8"
0
(12)
I
102.5"
widening of the FPF angle and shortening of the FP bond
can be explained by increasing electron withdrawal F-+P,
which is also observed in the PE spectra["! Precise calculation of the geometry of states requires considerable computational effort because the differences between large energy contributions AE,,,,, have to be calculated. The bending of the
S-C-N
bond in S(CN)2 is illustrative[811:
632
\N
p =3.02
[Dl
E&,
CeVl
-15827.7223
pSCf LO1
3.16
-15827.7146
3.29
Obviously, the S(CN)2structure as determined by microwavespectroscopy, and also thedipole moment, aresatisfactorily reproduced by ab initio calculation with an extensive
the slight SCN bending can be attributed to the
calculated electron density between the two carbon atoms.
However, the minor differences in total energy on bending
(13 : AE:;:, = 0.0077 eV) show how much caution should be
exercised when considering numerical data, e. g. from semiempirical procedures.
In polyatomic molecules, structure determinations via the
minimum of the total energy require calculation of a multidimensional energy surface; attempts are made to avoid prohibitive costs, e. g. by fixing some coordinates, using fast computer
programs, and optimization of coordinates (Fig. 27).
The total energy surface for rotation of SH about the CS
axis in thiphen01[~~1,
comprising 595 individual CNDO calculations (Fig. 27), and also considering deflections of C-SH,
requires some 2 hours of computing time on a medium-size
computer. It may be seen that no other conformation is preferred apart from the minima in the total energy for planar
geometry. In spite of reservations towards individual numerical
results from semiempirical calculations, such a comparison
of total energies should, in general, correctly reproduce the
energetically favored conformers.
Angew. Chem. Int.
Ed. Engl. 16, 613-637 (1977)
Hb*
u
H
H
H
H
CNDD
-E total
Lev1
Fig, 27. The total energy surface for SH rotation in thiophenol [SZ], is
made up of 1 7 x 35=595 individual CNDO calculations. In addition to
(AT) have also
twisting about the CS axis (Am). deflections of C-S-H
been considered, especially on passing of the ortho hydrogen atoms of the
benzene ring. I t is seen that no minima of the CNDO total energy occur
except for planar arrangements (m=O" and 180").
5.4. A Special Case: Molecular Reactions
As sketched out in the previous Section, changes in the
geometry of molecules can often be satisfactorily approximated
by point-for-point calculation and comparison of the total
energy. Intensive efforts are currently being invested in the
calculation of such energy surfaces also for molecular reactions
in many p l a ~ e s [ ' ~ * ~
One
~ , ~of~ the
I . many prerequisites of
such calculations is that either experimental information about
the detailed reaction course is available, e.g. from molecular
beam
or plausible assumptions are possible, as
for the presumably intramolecular isomerizations of compounds S2X21741
(Fig. 28).
Accepting the various assumptions and simplifications given
the following conclusions concerning the isomerizations
XSSXeX,SS can be drawn from the calculated energy surf a c e ~ ' (Fig.
~ ~ ] 28), in complete accord with experimental evidence: both F2SZisomers must be isolable; F2SS is more stable
than FSSF; only the HSSH and ClSSCl isomer, respectively,
should be capable of existence in the H2S2 and C12S2 systems.
What general statements can be put forward concerning
reactions
between
molecules
of
medium
size
KkLIMm+ R,S,T,-+ ? For instance, how do the successful collisions occur in the gas phase? What is the structure of the
molecular collision complex at the saddle point of maximum
energy? Are there any intermediates? How is the course of
reaction affected by the environment, e . g . in a solvent or
at a catalyst? Numerous questions require experimental elucid a t i ~ n [ ~ ~ ] - a p afrom
r t further development of suitable calculating procedures[83,841 (Fig. 2).
In this context, some attempts at mapping reaction coordinates warrant mention : thus comparison of X-ray structure
analyses furnishes information about mutually coupled
changes in molecular structures, such as angular deviation
as a function of
the "hyperspace" Hz + He has
already been ~ a l c u l a t e d ~ab
~ ~ i~n;i t i ~ [ ~831' , and MINDO/NDDO procedures[67,831compete with each other['6, 871,
Angew. Chem. Int. Ed. Engl. 16, 613-637 ( 1 9 7 7 )
Fig. 28. The two isomers FSSF and F2SS can be prepared separately: F2SS
is the thermodynamically more stable of the two. This situation-like PE
ionization energies (10: A) or dipole moments (10: B)-is reproduced by
CNDO calculations. On this basis, the isomerization energy surface
FSSFeF2SS(A)canbe approximated assuming the occurrence of an intramolecular 1,2 fluorine shift: the FSS fragment is considered fixed and the second
fluorine atom migrates along the SS axis x through 12 sectional planes
z each with 144 screening points. The geometry is completely optimized
only in the region of the minima; configuration interaction is neglected.
The surface made up of 1728 CNDO total energies clearly exhibits the
energetically preferred potential valley corresponding to isomerization. Close
to the saddle point a shallow minimum is calculated for an almost planar
The CNDO total energy for
three-membered ring arrangement F-S..
1 ".F.
s..
the more stable isomer FzSS is only slightly more negative than that for
FSSF; a relatively high barrier is found between the two. Corresponding
energy surfaces for the H I S 2 system, and likewise for CI& too [74], predict
the XSSX isomer with Cz symmetry to be the thermodynamically and kinetically stable species.
e. g. in calculation of hydrocarbon reactions; a qualitative
illustrative model for reactions from singlet and triplet excited
proves to be useful[z3];and, last but not least, the
orbital symmetry rules of Woodward and Hoffmann['ol, which
are based on configuration diagrams["], have found wide
application in organic chemistry['11.
5.5. Large Molecules of Low Symmetry
A further limitation of MO models is that characteristic
orbitals are generally no longer obtained for molecules with
more than 15 to 20 atoms and structures of low symmetry.
In particular, proximate
orbitals of the same
symmetry class usually undergo extensivemixing. For instance,
the 28-atom molecule of bis(trimethylsilyl)diimine (Fig. 2)
is calculated to have the following orbital as the seventh-fromhighest of 30 occupied molecular orbitals:
Although $24 appears to represent the XNN bond1241,its contributions at the nitrogen centers X&
amount to only about
633
50
Integrated quantities such as the total atomic charges
(9) are often much more suitable for describing the states
of such molecules. Thus the calculated high electron densities
in the NN bond of azo compounds such as bis(trimethylsily1)diimine (Fig. 26), which are due to Si-N polarization, can
be correlated with the low first ionization energies[76].
This example also marks a boundary beyond which a direct
qualitative discussion of molecular states offers several advantages. For instance, comparably low first ionization energies
are predicted for the 66- and 72-atom ethylene and benzene
derivatives in the gas phase (Fig. 29) as a consequence of
their identical substitution pattern of four tetrahedrally
arranged (H3C)3SiCH2groups, and of the excellent stabilization of the roughly spherical radical cations. Similarly ESR
spectra with extensive coupling are predicted as a result of
complete delocalization of the electron hole over the molecule
on oxidation in s~lution~~~~-regardless
of whether the central
K system is ethylene or benzene, comparable 1st ionization
energies and radical cation spin populations are observed
for the similarly substituted molecules [(H&)3SiCH2J2-n[CH2Si(CH3)3]2(Fig. 29). These compounds are already very
large for a conventional description of their states in terms
of molecular orbitals; nevertheless, in this case too-as in
the discussions of Sections 5.1 and 5.4-there is a possibility
of extending the available methods, for example by applying
calculating techniques developed for the solid state.
6. Conclusion and Outlook
The selection of examples presented in this report should
corroborate the following statement:
Chemical building blocks with defined structure and properties
are molecules in particular states which can be characterized
b y their energy and their charge distribution. The best models
yet available for their description are molecular orbitals, which
are especially suitable for comparison of corresponding molecular states of chemically related compounds. By virtue of their
inherent topology and symmetry, the potentials and numbers
/------
IE1 = 7.10 eV
a29si=11.7 G
I
II
a!
=0.77 G
I
I
I
Fig. 29. In the tetrakis(trimethylsily1methyl)-substitutedethylene and benzene derivatives CL8H44Si4and
C22H4hSi4the central systems are surrounded in similar manner by four (H3C)3SiCH2groups. The energy
ditTerences between the ground states of the neutral molecule and the radical cation are in close
agreement at 7.10 and 7.15 eV [92]. Their ESR spectra (1850 and 1988 lines, respectively) are Characterized
by the nonet of the methylene protons, of which five groups of signals are shown starting from the midpoint of the spectrum. The coupling constants for the 36 methyl protons asH3, for the 2gSi doublet
(12pSi=1/2,abundance 4.70%) and the phenyl protons a$ can be confirmed by spectral simulation [92].
Obviously, a;"' and a29si decrease on switching from the ethylene derivative to the large benzene
compound, providing further evidence for complete delocalization of the unpaired electron over the entire
molecule.
634
Angew. Chem. Int. Ed. Engl. 16,613-637 (1977)
of electrons to be inserted, and theirflexibility towards perturbations, they provide a manifold and stimulating overview covering
wide areas of chemistry.
How will molecular states be interpreted ten years from
now? Ten years ago expeditions had just started into the
terra incognita of r
s molecular skeletons, with photoelectron
spectra and all-electron calculations as navigational aids. Time
and again, MO models have done great service in the interpretation of new experimental data-some 20 years ago in the
case of the newly developed ESR technique and likewise just
recently in the novel chemical application of electron transmission spectroscopy. The stimulation of preparative work by
MO models is manifested by all those studies which are based
on the stability of cyclic compounds with (4N+ 2) electrons
or on orbital symmetry in concerted reactions. The snowballlike increase in application of MO models during the past
15 years is a classical example of the influence of large computers in the natural sciences,and even the very next generation
of computers can provide fresh impetus for advances and
innovations in calculating procedures.
Last but not least, MO models offer-in the face of the
increasing complexity and incredibly fast expansion of chemical sciences-an opportunity of gaining, and retaining, a
general understanding and overall view of chemistry.
This is an appropriate place to express my gratitude to Professor Edgar Heilbronner, who introduced me to the “HMO model
and its application”. The work of the Frankfurt group cited
above would not have been possible without the untiring efforts
of dedicated co-workers, especially Dr. Hartmut Alt (deceased),
Dr. Hans Seidl, Dr. Bahman Solouki, Dr. Klaus Wittel, Dr.
Gerhard Wagner,Dr. Herbert Stafast, Dr. Walter Ensslin, Georg
Brahler, Wolfgang Kaim, Shamsher Mohmand, Karl-Alexander
Ostoja-Starzewski, and Abbas Tabatabai. Generous support of
our work by the Deutsche Forschungsgemeinschaft, the State
of Hesse, Hoechst AG, the Fonds der Chemischen Industrie,
the Max-Buchner-Stiftung, and the Stgtung Volkswagenwerk
is gratefully acknowledged. Scientific contacts with numerous
colleagues have always had a stimulating effect; for suggestions
and critical discussion comments I wish to thank especially
Prof. E. Heilbronner, Dr. K . Wittel, Prof: G. Binsch, Prof:
W Kutzelnigg, Dr. P. Rosmus, and Dr. A. Semkow.
Received: June 29, 1977 [A 180 IE]
German version: Angew. Chem 89,631 (1977)
Based on a lecture delivered at the invitation of the Gesellschaft Deutscher Chemiker on the occasion of the Chemiedozententagung, Marhurg,
March 22, 1977.
6th Essay on “Molecular Properties and Models”.-The following
articles are to be considered as the Ist to 5th essays, respectively:
0
a) “Color and Constitution of the Azo Compounds”, Angew. Chem.
77, 469 (1965); Angew. Chem. Int. Ed. Engl. 4 , 457 (1965); b) “Die
Storung von a-Systemen als Molekiilsonde fur Substituenteneffekte”,
Jahrb. Akad. Wiss. Gottingen 1969, 13: c) “Photoelectron Spectra of
Nonmetal Compounds and Their Interpretation by MO Models” (with
B. G. Ramsey), Angew. Chem. 85, 773 (1973); Angew. Chem. Int. Ed.
Engl. 12, 734; d) “Photoelectron Spectroscopy: An Experimental
Approach to Teaching Molecular Orbitals” (with P. MollSre), J. Chem.
Educ. 51,506 (1974);e) “Photoelectron Spectra and Bonding in Phosphorus Compounds”, Pure Appl. Chem. 44, 343 (1975).
B. Solouki, P. Rosmus, H . Bock, J . Am. Chem. SOC.98, 6054 (1976)
and references cited therein; cf. also H. Bock, S. Mohmand, B. Solouki,
E. Block, L. K . Revelle, J . Chem. SOC.Chem. Commun. 1977, 287.
W u. Niessen, L . S. Cederbaum, W Domke, H. Diercksen, J . Chem. Phys.
66, 4893 (1977); concerning the G F procedure, cf. L . S. Cederbaum,
Theor. Chim. Acta 31, 239 (1973); J. Phys. B 8 , 290 (1975).
W Meyer, Int. J. Quantum Chem. 55, 341 (1971); J. Chem. Phys.
58, 1017 (1973); cf. also R. Ahlrichs, W Kutzelnigg, ibid. 62, 1225 (1975);
M . Jungen, R . Ahlrichs, Theor. Chim. Acta 17, 339 (1970).
W v. Niessen, G. H . F. Diercksen, L. S. Cederbaum, Chem. Phys. Lett.
45, 295 (1977).
Further [3] examples are: H,C=C=S [ H . Bock, 8. Solouki, G. Bert,
P. Rosmus, J. Am. Chem. SOC.99, 1663 (1977); P. Rosmus, B. Solouki,
H . Bock, Mol. Phys., in press] as well as HaCCl and HpSiCl [P.
Rosmus, A. Semkow, H . Bock, unpublished].
Cf., e. g., Nachr. Chem. Tech. 23, 24 (1975).
For a survey, see H. F . Schaefer I l l , “Molecular Electronic Structure
Theory: 1972-1975”, Annu. Rev. Phys. Chem. 1976,264.
Discussion at the CNRS Symposium “Theoretische Chemie” at Obersteigen, October 1976: E. Heilbronner, personal communication.
F . Hund, Angew. Chem. 89, 89 (1977); Angew. Chem. Int. Ed. Engl.
16, 87 (1977).
E. Hiickel, Z. Phys. 60, 423 (1930); 70, 204 (1931); 72, 310 (1931).
Cf. J . B. Coffins,J. D. Dill, E. D. Jemmis, Y. Apeloig, P. v. R. Schleyer,
J . J . Dannenberg,
R.Seeger, J. A. Popfe,J. Am.Chem.Soc.98,5419(1976);
Angew. Chem. 88, 602 (1976); Angew. Chem. Int. Ed. Engl. 15. 519
(1976).
Molecular states are generally designated r for the ground state and
xi for electronically excited states generated by excitation i-j; the
charge is also given (in parentheses), e.g. as M‘e, M, or M”.--IUPAC
rules have been published [Pure Appl. Chem. 21,29 (1970)l for specific
cases. The individual states of a particle M or Mae are denoted therein
by R (ground state) and successively by A, 8, c... (excited states),
and also by their multiplicity m = s + l , i.e. the number s of spinuncoupled electrons (singlet: s=O, m = l ; doublet: s=l, m=2, etc.)
as well as their symmetry type within the symmetry group of the
molecular structure [19].
From the wealth of review literature about photoelectron spectroscopy
the following have been selected (with contents): a) D. W Turner, C .
Baker, A. D. Baker, C. R. Brundle: Molecular Photoelectron Spectroscopy. Wiley-Interscience, London 1970 (reproductions of numerous
spectra, literature covered up to 1969); b) C. R. Brundfe, M . B. Robin
in F. C . Nochod, J . J . Zuckermanr Determination of Organic Structures
by Physical Methods. Vol. 111. Academic Press, New York 1971, pp.
Iff. (introduction with the aid of selected examples); c) J . H. D. Eland:
Photoelectron Spectroscopy. Butterworths, London 1974 (principles,
information from PE spectra, general interpretation, literature covered
up to 1972); d) J . P. Maier, Annu. Rep. Prog. Chem. B 71, 75 (1974)
(survey of recent developments). Cf. also refs. [2c, 2 e ] ,
H . J . Lempka, 7: R. Passmore, W. C. Price, Proc. Roy. SOC.London
A 3 0 4 , 53 (1968).
Cf. F. Brogli, E. Heilbronner, Helv. Chim. Acta 54, 1423 (1971); K.
Wittef,H.Bock, R. Manne, Tetrahedron 30, 651 (1974); references cited
therein.
H . Bock, W Ensslin, F. F e h t , R. Freund, J . Am. Chem. SOC.98, 668
(1976); references cited therein. The Jahn-Teller distortion . D 3 ad C 2 h
Z
b
A
g
l
A,
1
2
1
1
E,
2
1
-1
1
1
-1
@r
7
2.1
@u : :
Angew. Chem. Int. Ed. Engl. 16.613-637 (1977)
~
l
-1
l
1
0
2
1
-1
0
-1
-1
-2
3x1
I
l
1
-1
-1
-1
l
-1
0
l 24
0
12
-1
0
1
I
0
12
12
2.1
3.3
‘
2
0
1
0
1
1
63 5
of the iso(va1ence)electronic ethane radical cation is calculated by A.
Richartz, R. 3 . Buenker, P. 3. Bruna, S . Peyerimhoff,Mol. Phys., in press.
Cf. also the survey by C . Sandorfy: Chemical Spectroscopy and Photochemistry in the Vacuum Ultraviolet. Reidel, Dordrecht 1974.
[19] For designation of symmetry of states the reader is referred to the
introduction in ref. [21 c] and to F. A. Cotton: Chemical Applications
of Group Theory, Wiley-Interscience, New York 1966. A simple procedure for determining the symmetry of all radical cation states of a
molecule is illustrated for the case of disilane (see Scheme at foot of p.
635):
@ Definition of the coordinate system and bond classification; change
in sign occurs for units such as K bonds or electron pairs extending
over more than one center. Performance of all symmetry operations
@ belonging to the symmetry group @ of the molecule in order to
determine the number @ of “bond basis functions” transforming into
or “ - ” themselves. The product of the resulting characters @
of the“reducib1e”representationr@and thecharacters oftheindividual
“irreducible representations” @ of the group concerned @ are added
together @ and divided by the number of symmetry operations @
performed. One obtains the symmetry types @ of all radical cation states
that can be generated by (mono)ionizatiou of the valence electrons of
molecule 0.
[20] A. W Potts, W C. Price, Proc. Roy. SOC.London A326, 165 (1972);
L. J . Aarons, M . F. Guest, M . B. Half, I . H . Hillier, J. Chem. SOC.
Faraday Trans. I1 69, 643 (1973), discuss the results of J . P. Maier,
D. W Turner, ibid. 68, 11 (1972). Cf. also R . D. Baechler, K. Mislow,
J. Am. Chem. SOC.93, 773 (1971).
[21] Thestandard work on electronically and vibrationally excited molecular
states is G. Herzberg: Molecular Spectra and Molecular Structure,
comprising the three volumes: a) Vol. I: Spectra of Diatomic Molecules.
Van Nostrand, New York 1950; b) Vol. 11: Infrared and Raman Spectra
of Polyatomic Molecules. Van Nostrand, New York 1955; c) Vol. 111:
Electronic Spectra of Polyatomic Molecules. Van Nostrand, New York
1966 (the tables on pp. 583ff. provide a comprehensive survey of the
excited states of numerous molecules, with symmetry characterization).
[22] Cf. M . B. Robin: Higher Excited States of Polyatomic Molecules, Vol.
I and 11. Academic Press, New York 1974, 1975 (encyclopedia with
over 1000 references).
[23] Cf. G. Quinkert, Angew. Chem. 87, 851 (1975); Angew. Chem. Int.
Ed. Engl. 14, 790 (1975).
[24] A. H . Lawrence, C . C . Liao, P . de Mayo, C: Ramamurthy, J. Am. Chem.
SOC. 98, 2219, 3573 (1976); A. Couture, K . Ho, M . Hoshino, P . de
Mayo, R. Suan, W R . Ware, ibid. 98, 6218 (1976); P . de Mayo, Acc.
Chem. Res. 9, 52 (1976).
[25] F. Gerson, 3. Heinzer, H . Bock, H . Alt, H. Seidl, Helv. Chim. Acta
51, 707 (1968).The p-xylene radical anion spectrum was first published
by J . R . Bolton, A. Carrington, Mol. Phys. 4, 497 (1961).
[26] H. Alt, E. R. Franke, H . Bock, Angew. Chem. 81, 538 (1969); Angew.
Chem. Int. Ed. Engl. 8, 525 (1969).
[27] Cf. F. Gerson: Hochauflosende ESR-Spektroskopie. Verlag Chemie,
Weinheim 1967.
[28] Cf. E. Heilbronner, H . Brock: Das HMO-Modell und seine Anwendung,
Vols I and 11. Verlag Chemie, Weinheim 1968 and 1970; The HMO
Model and its Application, Vols. I and 11. Verlag Chemie, Weinheim,
and Wiley, Chichester, 1976.
[29] Review: W C . Price,S. S . Chissick, 7:Rauensdale: Wave Mechanics-the
first fifty years. Butterworths, London 1973.
[30] An important condition [31, 321 is Pauli’s principle of antisymmetry:
The overall (wave) function of a many-electron system must be antisymmetric on simultaneous exchange of site and spin coordinates ?, of
two electrons Jr(GI,X2r.:3...)= -Jr(X,,.?,,?,...).
Two electrons of a system therefore cannot exhibit identical coordinates x1= x z = x since
$(x,x,x~)=O would then hold.
[31] Among the many introductions to the principles of quantum mechanics,
one of the most recent ones is: W Kutzelnigg: Einfiihrung in die Theoretische Chemie, Vol. I. Verlag Chemie, Weinheim 1975.
[32] A comprehensible qualification of spectroscopic states and their discussion on the the basis of electron configurations is found in K . Wittel,
S. McGlynn, “The Orbital Concept in Molecular Spectroscopy”, Chem.
Rev. 1977, in press.
[33] ?: Koopmans, Physica I , 104 (1933). The scope is discussed, e.g., in
[I5 b, c] and by E. Heilbroiiner in R . Daudel, B. Pullman: The World
of Quantum Chemistry. Reidel, Dordrecht 1974, pp, 2llff.
[34] Cf. 3. A. Pople, D. L. Beueridge: Approximate Molecular Orbital Theory.
McGraw-Hill, New York 1970.
1351 A. W Potts, W. C . Price, Proc. Roy. SOC.London A326, 181 (1972).
1361 The SCF orbital energies are calculated by B. Roos, P . Siegbuhn, Thcor.
Chim,Acta21,368(1971),andthepotential surfacesandspin populations
of the individual radical cation states by H.Sakai, S. Yamabe, 7: Yamabe,
K . Fukui, H . Kato, Chem. Phys. Lett. 25, 541 (1974).
[37] A. D. Walsh, J. Chem. SOC. 1953, 2260; cf. also R . Buenker, S. D.
Peyerirnhofi Chem. Rev. 74, 127 (1974); references cited therein.
[38] E. Heilbronner, H.D. Martin, Helv. Chim. Acta 55,1490(1972);references
cited therein.
[39] R . Hojfmann, Acc. Chem. Res. 4, 1 (1971); references cited therein.
[40] Approximate solution of the Schrodinger equation HJr=&$ is discussed
at length, e.g., in ref. [31]. For the simplest case with most extensive
simplifications-a homonuclear two-center system in the HMO approximation-the procedure [28] is as given below: in order to solve the
differential equation with the two unknown eigenfunctions $ and eigenvalue E , an approximate linear combination $9.z = c I + 1 +_c2+ 2 is
assumed which is made up of atomic orbitals and initially unknown
coefficientsc for each of the two centers. The best possible fit of Jro
to Jr within the framework of the approximation procedure, i.e. determination of the coefficients cI and cz, is performed with the expanded
and rearranged equation:
+
“+”
636
the individual integrals being defined as parameters in the following
way:
i=j
ai (Coulomb)
1 (Normalization)
i+j
Bi, (Resonance)
Sij (Overlap)
Determination of the energy minimum as a function of the coefficients
6 ~ / 6 c = O leads, via a system of so-called secular equations { } and
the corresponding secular determinant jl 11, initially to the eigenvalues
€1.2:
Insertion of the eigenvalues
into the secular equations finally afTords
the coefficients
and hence the molecular orbitals
Cf. F. Brogli, E. Heilbronner, 3. Ipaktschi, Helv. Chim. Acta 55, 2447
(1972); R . Gleiter, E. Heilbronner, L. A . Paquette, G. L. Thompson,
R . E. Wingard jr., Tetrahedron 29, 565 (1973).
a) M . J . S. Dewar, R. C . Dougherty. The PMO Theory of Organic
Chemistry. Plenum, New York 1975; b) M . 3 . S.Dewar: The Molecular
Orbital Theory of Organic Chemistry. McGraw-Hill, New York 1969.
A. Streitwieser j r . ’ Molecular Orbital Theory for Organic Chemists.
Wiley, New York 1961 (complete bibliography up to 1960).
L. Salem: The Molecular Orbital Theory of Conjugated Systems. Benjamin, New York 1966 (literature up to 1965).
Cf. “Aromaticity”, Spec. Publ. 21, Chem. SOC.London 1967.
E. Hiickel, 2. Phys. 70, 204 (1931); 76, 628 (1932); 2. Elektrochem.
Angew. Phys. Chem. 42, 657 (1936).
Cf. G. Binsch, Naturwissenschaften 60, 369 (1973); M . J . Goldstein,
R. Hofmann, J. Am. Chem. SOC.93,6193 (1971).
H. Bock, W Ensslin, Angew. Chem. 83, 435 (1971); Angew. Chem.
Int. Ed. Engl. J0, 404 (1971); cf. also ref. 1181.
P. D. Burrow, K . D. Jordan, Chem. Phys. Lett. 36, 594 (1976); K .
D. Jordan, 3 . A. Michejda, P . D . Burrow, ibid. 42, 227 (1976); J. Am.
Chem. SOC.98, 1295, 7189 (1976).
L. Sanche, G. J . Schulz, J. Chem. Phys. 58, 479 (1973); I . Nenner,
G . 3 . Schulz, ibid. 62, 1747 (1975).
W N . Lipscomb: Boron Hydrides. Benjamin, New York 1963.
E. E. Hauinaa. E. H . Wiebenaa. Recl. Trav. Chim. Pavs-Bas 78. 724
(1959); E. H. Wiebenga, E. E. Havinga, K . H. Boswijk, Adv. Inorg.
Chem. Radiochem. 3, 133 (1961).
A. W Potts, W C . Price, Proc. Roy. SOC.London A326, 165 (1972);
cf. also refs. [16, 351.
CJ, e.g., Holleman-Wiberg: Lehrbuch der Anorganischen Chemie, 8190 Editions de Gruyter, Berlin 1976, pp. 734, 1137.
The electronegativity is defined by Pauling and Mulliken as an energy,
and by other authors as a force or a dimensionless ratio. The entire
range of chemistry is embraced by AEN=3.54 units (Fr 0.86 and Ne
4.40 [54]) and the effect of the central atom in various substituent
groups (C in CHs, CN, or CF,) is not accounted for. Cf. 3 . Hinze,
Fortschr. Chem. Forsch. 9, 448 (1968).
Cf. “Ionization Potentials and Ionization Limits Derived from the Analyses of Optical Spectra”,NS RDS-NBS 34, Nat. Bur. Stand. Washington
1970.
Orbital diagrams as shown in Fig. 20 are advantageously 1281 drawn
such that the radii of the circles round the sites are numerically equal
.
areas of the circles F = I I Care
~ ~then directly
to the coefficients C J ~The
proportional to the squares of the coefficients c3,.
-
[53]
[54]
[55]
[56]
[57]
I
I
_
Angew. Chem. Int. Ed. Engl. 16, 613-637 (1977)
[58] C . Batich, E. Heilbronner, !l Hornung, A. J . Ashe I l l , D. ‘I: Clark,
U . ‘I: Cobley, D. Kilcast, I . Scanlan, J. Am. Chem. SOC.95, 928 (1973);
for bismutabenzene: J. Bastide, E. Heilbronner, J . P. Maier, A. J . Ashe
111, Tetrahedron Lett. 1976, 411. Cf. also A. J . Ashe 111, F. Burger,
M. E €/-Sheik, E. Heilbronner, J . P. Maier, J.-F. Muller, Helv. Chim.
Acta 59, 1944 (1976); and references cited therein.
[59] H. Bock, G. Wagner,J. Kroner, Chem. Ber. 105,3850 (1972); Tetrahedron
Lett. 1971,3713; references cited [especially W E. Forbes, P. D. Sullivan,
Can. J . Chem. 46, 317 (1968)l.
[60] Cf. F. Brogli, E. Heilbroriner, J . Wirz, E. Kloster-Jensen, R. G. Bergmann,
K . P. C . Vollhardt, A. J . Ashe Ill, Helv. Chim. Acta 58, 2620 (1975);
M. Beez, G. Bieri, H. Bock, E. Heilbronner, ibid. 56, 1028 (1973); W.
Ensslin, H. Bock, G. Becker, J . Am. Chem. SOC.96, 2757 (1974); H.
Bock, G. Wagner, K . Wittel, J . Sauer, D. Seebach, Chem. Ber. 107,
1869 (1974).
[61] Cf. E. Heilbronner, !l Hornung, J. P. Muier, E. Kloster-Jensen, J . Am.
Chem. SOC.96, 4252 (1974); K. Wittel, H. Bock, Chem. Ber. 107, 317
(1974); C. R. Brundle, M . B. Robin, N . A. Kuebler, H. Basch, J . Am.
Chem. SOC.94, 1451, 1466 (1972).
[62] H. Stafast, H. Bock, Tetrahedron 32, 855 (1976).
[63] Cf. J. N. Murrell, M. Schmidt, J . Chem. SOC.Faraday Trans. I1 1972,
1709; H . Stafast, H. Bock, Z. Naturforsch. 8 2 8 , 746 (1973); R. N .
Dixon, J . N . Murrell, B. Narayan, Mol. Phys. 20, 61 1 (1971); references
cited.
[64] Cf. the survey of original publications in 0. Sinanoglu, K . B. Wiberg:
Sigma Molecular Orbital Theory. Yale University Press, New Haven
1970, or the review in [42b].
[65] R. Hofmann, J. Chem. Phys. 39, 1397 (1963).
[66] J . A. Pople, D. P. Santry, G. A. Segal, J . Chem. Phys. 43, 129. 136
( I 965); 44, 3289 (1966).
[67] M . J . S. Dewar, G. Klopman, J. Am. Chem. Soc. 89, 3089 (1967).
[68] W .I.
Hehre, Acc. Chem. Res. 9. 399 (1976).
1691 K . A. Ustoja-Starzewski, H . Bock, J . Am. Chem. SOC.98, 8456 (1976).
[70] As representative of numerous articles [39] one may cite: R. Hoffmann,
A. Imamura, W J . Hehre, J . Am. Chem. SOC.90, 1499 (1968).
[71] Literature surveys are found in [2c] and in C. G. Pitt, J . Organornet.
Chem. 61, 49 (1973); cf. also W Ensslin, H. Bock, G. Briihler, J. Am.
Chem. SOC.96, 2757 (1974); R. Hoffmaim, L. Radorn, J . A. Pople, P.
u. R. Schleyer, W J . Hehre, L . Salem, ibid. 94, 6221 (1972).
[72] R. Gleiter, J. Chem. SOC.A1970, 3174.
[73] R. J . Gillespie, D. R. Shin, J . D. Tyrer, J . Chem. SOC.Chem. Commun.
1977, 253.
[74] B. Solouki, H. Bock, Inorg. Cbem. 16, 665 (1977).
[75] Cf. W Kutzelnigg, Angew. Chem. 85, 564 (1973); Angew. Chem. Int.
Ed. Engl. 12, 559 (1973).
[76] H. Bock, K . Wittel, M . Veith, N . Wiberg, J . Am. Chem. SOC. 98, 109
(1976); references cited therein.
[77] E. Haselbach, A. Schmelzer, Helv. Chim. Acta 54, 1575 (1971).
[78] M. Klessinger, “Mehrelektronen-Modelle in der organischen Chemie”,
Fortschr. Chem. Forsch. 9, 354 (1968); references cited therein.
[79] J. N . Murrell: The Theory of the Electronic Spectra of Organic Molecules. Methuen, London 1963.
[SO] R. J . Gillespie, Angew. Chem. 79, 885 (1967); Angew. Chem. Int. Ed.
Engl. 6, 819 (1967); Molecular Geometry. Van Nostrand Reinhold,
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C 0 M MU N I C AT1 0N S
1-Aryl-4-methoxyarsacyclohexadiene-r2-Ar ylarsabenzene Rearrangement-An Organoelement Analog
of the Cyclohexadienol-r Benzene Rearrangement
By Gottfried Markl and Rainer Liebl“]
The cyclohexadienol-+benzene rearrangement“] has been
demonstrated on numerous occasiond2? In the present communication we report the first analogous organoelement rearrangement.
The pronounced tendency of phospha- and arsacyclohexadienes to rearrange into phospha- and arsabenzenes is well
We have now been able to show that the 4-substireadily
tuted 1-aryl-4-methoxyarsacyclohexadienes
[*] Prof. Dr. G. Matkl, R. Liebl
Chemisches Institut der Universitat
Universitatsstrasse 31, D-8400 Regensburg 1 (Germany)
Angew. Chem. Int. Ed. Engl. 16 (1977) No. 9
637
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