All the ways to have a bond
Roald Hoffmann
“Will you reflect for a moment on some of the things
that I have been saying? I described a bond, a normal simple
chemical bond; and I gave many details of its character (and
could have given many more). Sometimes it seems to me
that a bond between two atoms has become so real, so
tangible, so friendly that I can almost see it. And then I
awake with a little shock; for a chemical bond is not a real
thing: it does not exist: no one has ever seen it, no one ever
can. It is a figment of our own imagination.”
Charles Coulson, 1951
“I believe the chemical bond is not so simple as some
people seem to think.”
Robert Mulliken
History
1661
1803
1846
1852-66
1856
1861
1861
1860s
1874
Robert Boyle: the idea of an element
John Dalton: the atom
Auguste Laurent: the idea of a molecule
Edward Frankland: saturation capacity, valence
Archibald Couper: lines in chemical formulas
Alexander Crum Brown: graphical formulas
Alexander M. Butlerov: chemical structures
Friedrich A. Kekulé, structural formulas
Jacobus H. van’t Hoff, Joseph A. LeBel: tetrahedral
carbon atom
1916
G.N. Lewis, Walther Kossel: bond as shared electron
pair
~1912 on
chemical crystallography; bond metrics
1929
Walter Heitler, Fritz London: H2 calculation
1930s
Linus C. Pauling, Robert Mulliken: VB and MO
theories
___________________________________________________________
C.A. Russell, “The History of Valency,” 1971
M.J. Nye, “From Chemical Philosophy to Theoretical Chemistry,” 1993
A. J. Rocke, “Image and Reality: Kekulé, Kopp, and the Scientific Imagination”
2010
19th century type formulas
(C6H5)(CH3)
(C6H5)(OH)
[C6H5][Br]
C6H5-O-H
↓
Structural formulas
↓
Lewis structures: a line ▬ is an electron pair 
↓
Linus Pauling’s valence bond
19th century type formulas
Representation – symbolic or iconic
(C6H5)(CH3)
(C6H5)(OH) (C6H5)(Br)
C6H5-O-H
↓
Structural formulas
↓
Lewis structures: a line ▬ is an electron pair 
↓
Linus Pauling’s valence bond
iconic
symbolic
19th century type formulas
Representation – symbolic or iconic
(C6H5)(CH3)
(C6H5)(OH) (C6H5)(Br)
C6H5-O-H
↓
Structural formulas
reification
↓
Lewis structures: a line ▬ is an electron pair 
↓
Linus Pauling’s valence bond
19th century type formulas
Representation – symbolic or iconic
(C6H5)(CH3)
(C6H5)(OH) (C6H5)(Br)
C6H5-O-H
↓
Structural formulas
reification
↓
Lewis structures: a line ▬ is an electron pair 
cooption
↓
Linus Pauling’s valence bond
a reasonably persistent connection between atoms
ways of probing bonding experimentally
distances (from X-ray, neutron, electron diffraction,
from microwave spectroscopy)
bonding electron densities from diffraction experiments
dissociation energies
force constants, vibrational frequencies
magnetism
magnetic resonance -- shifts and coupling constants
ionization potentials
spectroscopic criteria
scanning tunneling microscopy, atomic force
microscopy
Jay Siegel
Cambridge Structural Database
I
I
I2(s), 110 K
2.66
d
I
I
d1
I
d2
I
Counts
Counts
I
–
d [Å]
Cambridge Structural Database
2003
d1,2 [Å]
P.H. Svensson and L. Kloo
2003
I
d2 [Å]
d1
I
d2
I
–
d1 [Å]
Cambridge Structural Database, 2003
[Bürgi and Dunitz; Bent, 1968]
d
I
I
d1
I
d2
I
Counts
Counts
I
–
d [Å]
Cambridge Structural Database
2003
d1,2 [Å]
P.H. Svensson and L. Kloo
2003
Bond Dissociation Energies
e.g. photoionization mass spectrometry
Blanksby and Ellison, Accounts of Chemical Research
36, 255 (2003)
Bond Dissociation Energies
H3C-CH3
88 kcal/mol
H2C=CH2
152
Bond Dissociation Energies
H3C-CH3
88 kcal/mol
H2C=CH2
152
H3Si-SiH3
74
H2Si=SiH2
63
Bond Dissociation Energies
H3C-CH3
88 kcal/mol
H2C=CH2
152
H3Si-SiH3
74
H2Si=SiH2
63
silicon is weird, of course
Bond dissociation energies
usually correlate with bond distances…but
vibrations, force fields
vibrations, force fields
Andrews, Davidson, Duce, J. Organometal. Chem., 97,
95 (1975)
Finseth, Sourisseau, Miller, J. Phys. Chem., 80, 1248
(1976)
“How Significant are the Force Constants? Two different valence
force fields are now available for
tricarbonyl(trimethylenemethane)iron, that of ADD and ours. Both
reproduce the starting frequencies well, but the individual force
constants differ considerably as shown in Table VI. Is either set right?
What is their significance? This is worth a few comments.
We believe that most force constants which have been
derived from large symmetry blocks should be regarded with
skepticism. There are several sources of uncertainty. First, a few of
the assignments are probably wrong. Second, assumptions are
invariably made about some of the interaction constants. Third, it is
known that the refinement procedure can converge on different sets
of answers for the symmetry force constants depending upon the
initial values that are assumed. The different sets will reproduce the
frequencies equally well, and sometimes there are two or more sets
which seem equally reasonable. It is then almost impossible to tell
which is "right". One is seldom aware of the existence of these other
sets...”
Finseth, Sourisseau, and Miller, 1976
magnetic criteria
b
a
1A
3B
1A
1B
3B
lowest if splitting is small, 1A if splitting large
high spin/low spin
ferromagnetic/antiferromagnetic
spectroscopic measures
a
b
less
red shift
interaction
more
blue shift
STM
Atomic force microscopy
ways of probing bonding experimentally
distances (from X-ray, neutron, electron diffraction,
from microwave spectroscopy)
bonding electron densities from diffraction experiments
dissociation energies
force constants, vibrational frequencies
magnetism
magnetic resonance -- shifts and coupling constants
ionization potentials
spectroscopic criteria
scanning tunneling microscopy, atomic force
microscopy
Raphael
ways of analyzing bonding theoretically
Molecular orbitals, bond orders, overlap populations
Bader analysis, QTAIM, bond critical points
Electron localization function (ELF, ELI)
Natural bond analysis
Energy partitioning
Molecular configurations as sketched by Niels Bohr; [from an unpublished 1912 manuscript ,
intended as an appendix to his 1913 papers].
Svidzinsky A A et al. PNAS 2005;102:11985-11988
©2005 by National Academy of Sciences
Mulliken population analysis
Ψ = c1υ1 + c2υ2
∫ Ψ*Ψdτ = 1 = c12∫υ12 + c22∫υ22 + 2c1c2∫ υ1υ2
= c12
+ c22
+ 2c1c2S12
Overlap population = 2c1c2S12
negative for
(+)(-)(+) or (-)(+)(+)
positive for
(+)(+)(+) or (-)(-)(+)
large for large c1, c2 and large S12
Mulliken, Wiberg, Meyer, Löwdin,….bond indices
Element Lines: Bonding in the Ternary Gold Polyphosphides, Au2MP2 with M = Pb,
Tl, or Hg, X.-D. Wen, T. J. Cahill, and R. Hoffmann, J. Am. Chem. Soc., 131, 21992207 (2009).
Eschen and Jeitschko
M = Hg 3.20Å
= Tl 3.19
= Pb 3.16
P-P 2.15-2.18
Atoms in Molecules (AIM, QTAIM)
A topological view of molecular structures
•
R. F. W. Bader and P. M. Beddall, "A Virial Field
Relationship for Molecular Charge Distributions and
a Spatial Partitioning of Molecular Properties",
Journal of Chemical Physics, 1972, 56, 3320-3329.
•
R. F. W. Bader, Atoms in Molecules - A Quantum
Theory, Oxford University Press, Oxford, 1990.
•
R. F. W. Bader, A Bond Path - A Universal Indicator
of Bonded Interactions, Journal of Physical
Chemistry A, 1998, 102, 7314 - 7323.
Electron density ρ(r) in (a) σh and (b) σv planes of BF3.
All the trajectories traced out by the gradient vectors ρ in the
neighborhood of a nucleus terminate at that nucleus. There
results a surface defining atomic basins – “atoms in molecules”.
Bond path = line of maximum density linking the nuclei
of two atoms
Bond critical point = the point along bond path where
density is minimum; it is a (3,-1) critical point
P. Coppens
The Laplacian L of the electron density at the BCP along
the bond path
gauges the balance between the local kinetic energy (G) and
potential energy (V) densities at the BCP.
When L is substantially negative, one typically has a
normal covalent bond. Positive L is usually associated with
closed shell or Pauli repulsion.
Another measure: local energy density H = G + V
Electron Localization Function
ELF(r)=
1
2
1  [F(rr )]
F(r)= Curvature of pair probability function of
electrons of same spin at r relative to
free e– gas
0 ≤ ELF ≤ 1.0
High F(r), low ELF; Low F(r), High ELF
A.D. Becke and K.E. Edgecombe, 1990
ELF applied
F
S
F
F
S
F
J.K. Burdett and T.A. McKormick, 1998
F
Electron Localizability Indicator ELI
Electron pair density → integral of same-spin
electron density over a region → “q-restricted
pair population” → ELI
Roald’s personal opinion:
AIM and ELF are analytical and
descriptive. They are not predictive
nor productive.
Natural Bond Orbital Analysis
• Reed, Curtis, Weinhold, Chem. Rev., 1988, 88, 899-926
• www.chem.wisc.edu/~nbo5
• Weinhold, Natural bond orbital methods, in „Encyclopedia of
Computational Chemistry“ (1998), Vol. 3, pp. 1792-1811
a sequence of transformations: input basis → NAO → NHO → NBO → NLMO
   (1,2,..., N )
wave function 
density operator 
ˆ  N  (1,2,..., N )  * (1' ,2,..., N ) d 2 ...d N
solve eigenproblem
Γi υi = γi υi
natural orbitals
occupancy-weighted symmetric orthogonalization
natural atomic orbitals (NAO)
basis transformation
density matrix  in NAO basis
 natural population analysis
Search for an optimal natural Lewis structure
natural bond orbitals (NBO) &
natural hybrid orbitals (NHO)
natural localized molecular orbitals (NLMO)
Energy Decomposition Analysis
Morokuma
Rauk and Ziegler
ADF
Krapp, Frenking
Krapp, Frenking
Krapp, Frenking
Krapp, Frenking
ways of analyzing bonding theoretically
Molecular orbitals, bond orders, overlap populations
Bader analysis, QTAIM, bond critical points
Electron localization function (ELF, ELI)
Natural bond analysis
Energy partitioning
1.23-1.54
1.19-1.48
a reasonably persistent connection between atoms
A personal view
I think that any “rigorous” definition of a
chemical bond is bound to be impoverishing, leaving
one with the comfortable feeling, “yes (no), I have (do
not have) a bond,” but little else. And yet the concept
of a chemical bond, so essential to chemistry, and
with a venerable history, has life, generating
controversy and incredible interest. Even if we can’t
reduce it to physics.
My advice is: Push the concept to its limits. Be
aware of the different experimental and theoretical
measures out there. Accept that (at the limits) a bond
will be a bond by some criteria, maybe not others,
respect chemical tradition, relax, and instead of
wringing your hands about how terrible it is that this
concept cannot be unambiguously defined, have fun
with the fuzzy richness of the idea.
covalent vs ionic bonding, a continuum
the abiding utility of Lewis structures for the normal: by far
the greatest part of organic chemistry
… and the abnormal: what to do for octet-expanded
compounds of P, S, halogens, Xe?
octet-expanded or hypervalent compounds, electrondeficient compounds, some organometallics, clusters.
Extended structures.
Per H. Svensson, Lars Kloo,
"Synthesis, Structure, and
Bonding in Polyiodide and Metal
Iodide-Iodine Systems",
Chem. Rev., (2003), 103(5), pp
1649-1684.
19th century type formulas
Representation – symbolic or iconic
(C6H5)(CH3)
(C6H5)(OH) (C6H5)(Br)
reification
C6H5-O-H
How tools and concepts change….
↓
not revolution but gradual, punctuated
Structural formulas
change,
with appropriation, cooption,
subversion, parallel and ambiguous
terminologies. Hardly ever malicious,
↓
very human, and mostly
productive.
Lewis structures: a line ▬ is an electron pair 
↓
Not Kuhn, not quite Feyerabend.
Maybe
Galison.
Linus Pauling’s valence bond