In contrast, the electron density distribution in a molecule or crystal can be observed by electron diffraction and X-ray crystallography 3; and it can also, and often more readily, be o
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Understanding and Interpreting Molecular Electron Density Distributions
C F Matta and R J Gillespie*
Department of Chemistry, McMaster University, Hamilton, ON L8S 4M1, Canada; *gillespi@mcmaster.ca
Advances in the power and speed of computers have made
ab initio and density functional theory (DFT) calculations
an almost routine procedure on a PC (1) From these
calcu-lations the equilibrium geometry, the energy, and the wave
function of a molecule can be determined From the wave
function one can obtain all the properties of the molecule,
including the distribution of electronic charge, or electron
density However, the electron density is often not calculated
or discussed, perhaps because it is not widely realized that
very useful information on bonding and geometry can be
obtained from it It seems particularly important to discuss
electron densities in introductory chemistry courses because
students can grasp the concept of electron density much more
readily than the abstract mathematical concept of an orbital It
is also not as widely understood as it should be that orbitals
are not physical observables but only mathematical constructs
that cannot be determined by experiment (2) In contrast, the
electron density distribution in a molecule or crystal can be
observed by electron diffraction and X-ray crystallography
(3); and it can also, and often more readily, be obtained from
ab initio and density functional theory calculations
This article gives a simple introduction to the electron
densities of molecules and how they can be analyzed to
ob-tain information on bonding and geometry More detailed
discussions can be found in the books by Bader (4), Popelier
(5), and Gillespie and Popelier (6 ) Computational details
to reproduce the results presented in this paper are presented
in Appendix 1
The Electron Density
Quantum mechanics allows the determination of the
probability of finding an electron in an infinitesimal volume
surrounding any particular point in space (x,y,z); that is, the
probability density at this point Since we can assign a
prob-ability density to any point in space, the probprob-ability density
defines a scalar field, which is known as the probability density
distribution When the probability density distribution is
multiplied by the total number of electrons in the molecule,
N, it becomes what is known as the electron density distribution
or simply the electron density and is given the symbol ρ(x,y,z).
It represents the probability of finding any one of the N
electrons in an infinitesimal volume of space surrounding
the point (x,y,z), and therefore it yields the total number of
electrons when integrated over all space The electron density
can be conveniently thought of as a cloud or gas of negative
charge that varies in density throughout the molecule Such a
charge cloud, or an approximate representation of it, is often
used in introductory texts to represent the electron density
ψ 2 of an atomic orbital It is also often used incorrectly to
depict the orbital ψ itself In a multielectron atom or molecule
only the total electron density can be experimentally observed
or calculated, and it is this total density with which we are
concerned in this paper A more formal discussion of electrondensity is presented in Appendix 2
The electron density is key to the bonding and geometry
of a molecule because the forces holding the nuclei together
in a molecule are the attractive forces between the electrons andthe nuclei These attractive forces are opposed by the repulsionsbetween the electrons and the repulsions between the nuclei
In the equilibrium geometry of a molecule these electrostaticforces just balance The fundamentally important Hellman–
Feynman theorem (4–7) states that the force on a nucleus in a molecule is the sum of the Coulombic forces exerted by the other nuclei and by the electron density distribution ρ This means
that the energy of interaction of the electrons with the nucleican be found by a consideration of the classical electrostaticforces between the nuclei and the electronic charge cloud Thereare no mysterious quantum mechanical forces, and no otherforce, such as the gravitational force, is of any importance inholding the atoms in a molecule together The atoms are heldtogether by the electrostatic force exerted by the electronic charge
on the nuclei But it is quantum mechanics, and particularlythe Pauli principle, that determines the distribution of elec-tronic charge, as we shall see
The Representation of the Electron DensityThe electron density (ρ) varies in three dimensions (i.e.,
it is a function of the three spatial coordinates [x,y,z]), so a
full description of how ρ varies with position requires a fourthdimension A common solution to this problem is to show how
ρ varies in one or more particular planes of the molecule.Figure 1a shows a relief map of the electron density, ρ, of theSCl2 molecule in the σv (xz) plane The most striking features
of this figure are that ρ is very large in an almost sphericalregion around each nucleus while assuming relatively verysmall values, and at first sight featureless topology, betweenthese nuclear regions The high electron density in the nearlyspherical region around each nucleus arises from the tightlyheld core electrons; the relatively very small and more diffusedensity between these regions arises from the more weaklyheld bonding electrons In fact, it was necessary to truncatethe very high maxima in Figure 1a (at ρ = 2.00 au)1 to make
it possible to show the features of the electron density bution between the nuclei In particular, there is a ridge ofincreased electron density between the sulfur atom and each
distri-of the chlorine atoms The electron density has values distri-of3.123 × 103 and 2.589 × 103 au at the S and Cl nuclei, re-spectively, but a value of only 1.662×10᎑1au at the minimum
of the ridge between the peak around the sulfur nucleus andeach of the chlorine nuclei This ridge of increased electrondensity between the S atom and each of the Cl atoms, small as
it is, is the density in the bonding region that is responsible forpulling the nuclei together Along a line at the top of this ridgethe electron density is locally greater than in any direction
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away from the line This line coincides with the bond between
the S and Cl atoms as it is normally drawn and is called a
bond path (4–6, 10) The point of minimum electron density
along the bond path is called the bond critical point.
Figure 1b shows a relief map of the electron density of the
water molecule Its features are similar to those of the density
map for SCl2, but the electron density around the hydrogen
atoms is much smaller than around the oxygen atom, as we
would expect The electron density at the maximum at the
oxygen nucleus has a value of 2.947 × 102 au, whereas that
at the hydrogen nucleus is only 4.341 × 10᎑1 au,which is only
slightly greater than the value of 3.963 × 10᎑1 au at the
minimum at the bond critical point The very small electron
density surrounding the hydrogen nucleus is due to less than
one electron because the more electronegative oxygen atom
attracts electron density away from the hydrogen atom so that
it has a positive charge
Another common way to represent the electron density
distribution is as a contour map, analogous to a topographic
contour map representing the relief of a part of the earth’s
surface Figure 2a shows a contour map of the electron density
of the SCl2 molecule in the σv (xz) plane The outer contour has
a value of 0.001 au, and successive contours have values of
2×10n, 4 ×10n, 8 ×10n au; n starts at ᎑3 and increases in
steps of unity Figure 2b shows a corresponding map for the
H2O molecule Again we clearly see the large concentration of
density around each nucleus The outer contour is arbitrary
because the density of a hypothetical isolated molecule extends
to infinity However, the 0.001 au contour corresponds rather
well to the size of the molecule in the gas phase, as measured
by its van der Waals radius, and the corresponding isodensity
surface in three dimensions usually encloses more than 99%
of the electron population Thus this outer contour showsthe shape of the molecule in the chosen plane In a condensedphase the effective size of a molecule is a little smaller Wesee more clearly here that the bond paths (the lines alongthe top of the density ridges between the nuclei) coincidewith the bonds as they are normally drawn
Figure 3 shows the electron density contour maps forthe period 2 fluorides LiF, BF3, CF4, OF2, and for the iso-lated B atom In LiF each atom is almost spherical, consis-tent with the usual model of this molecule as consisting ofthe ions Li+ and F᎑ The volume of the lithium atom is muchsmaller than that of the F atom, again consistent with the ionicmodel We will see later that we can also obtain the atomiccharges from the electron density and that the charges on thetwo atoms are almost ±1, again consistent with the represen-tation of these atoms as ions Moreover, there is a very smalldistortion of the almost spherical density of each atom towardits neighbor, giving a very low ridge of density between thetwo nuclei indicating that the amount of electronic charge
in the bonding region is very small Thus the bonding in thismolecule is close to the hypothetical purely ionic model,which would describe the molecule as consisting of twospherical ions held together by the electrostatic force betweentheir opposite charges
As we proceed across period 2 the electron density ofthe core of each atom remains very nearly spherical but the
Figure 1 Relief maps of the electron density of (a) SCl 2 and (b) H 2 O
in the plane of the nuclei (density and distances from the origin of
the coordinate system in au) Isodensity contour lines are shown in
the order 0.001, 0.002, 0.004, 0.008 (four outermost contours);
0.02, 0.04, 0.08 (next three); 0.2, 0.4, 0.8 (next three) The density
is truncated at 2.00 au (innermost contour) These contours are
shown in blue, violet, magenta, and green, respectively, on the
figure in the table of contents (p 1028). HFigure 2 Contour maps of the electron density of (a) SCl2O The density increases from the outermost 0.001 au isodensity2 and (b)
contour in steps of 2 × 10 n , 4 × 10 n , and 8 × 10 n au with n ing at ᎑3 and increasing in steps of unity The lines connecting the nuclei are the bond paths, and the lines delimiting each atom are the intersection of the respective interatomic surface with the plane
start-of the drawing The same values for the contours apply to quent contour plots in this paper.
subse-a
b
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outer regions of the atom become increasingly distorted from
a spherical shape, stretching out toward the neighboring atom
to give an increased electron density at the bond critical point
(ρ b) (see Table 1) Figure 4 shows the electron density plots
for some chlorides of period 2 We see similar changes in theelectron density distribution for these molecules as we sawfor the fluorides
For a three-dimensional picture of the electron densitydistribution we can easily show a particular isodensity envelope(i.e., a three dimensional surface corresponding to a givenvalue of the electron density) The 0.001-au envelope gives apicture of the overall shape of the molecule as shown by theexamples in Figure 5 Making the outer 0.001-au envelopetransparent as in Figure 5 reveals an inner envelope, butshowing additional envelopes becomes increasingly difficult.The particular inner surface shown in Figure 5 corresponds tothe bond critical-point isodensity envelope (ρ b envelope), thesingle envelope just encompassing all the nuclei All isodensityenvelopes with ρ < ρ b will form a continuous sheath ofdensity surrounding all the nuclei in the molecule, and allisodensity envelopes with ρ > ρb will form a discontinuoussurface surrounding each nucleus separately Thus the ρ b
envelope is just about to break into separate surfaces, onesurrounding each atom, at higher values of ρ
The ρ b envelopes are also shown for some period 2fluorides and chlorides in Figure 6 These surfaces show thedistortion of the electron density from a spherical shape evenmore clearly than the contour maps in Figures 3 and 4 Forexample, in Figure 6 one can see the distinctly tetrahedralshape assumed by the part of the ρ b envelope surrounding thecarbon atom in CCl4 owing to the distortion of the electron
Figure 3 Contour maps of the electron density of LiF, CF4, a free
ground state boron atom, BF3, OF2 in the σ v (xz) plane (the plane
containing the three nuclei) and in the σ v ′ (yz) plane (the plane
bisecting ∠ FOF perpendicularly to the σ v [xz] plane) (See legend
to Fig 2 for contour values.)
N OTE : Data, taken from ref 6, were obtained using DFT/B3LYP
functional and a 6-311+G(2d,p) basis set.
2 d i r e P r o f a t a D d e t a l e R d a s h t g e L d
Figure 4 Contour maps of the electron density of LiCl, BCl3, SCl2
in the σ v (xz) plane (the plane containing the three nuclei) and in the σ v ′ (yz) plane [the plane bisecting ∠ Cl-S-Cl perpendicularly to the
σ v (xz) plane], NCl3 in the σ v plane (plane containing one N–Cl bond and bisecting ∠ Cl-N-Cl formed by the remaining two bonds), and Cl2 (See legend to Fig 2 for contour values.)
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density in the four tetrahedral directions In addition to the
distortion of the electron density toward each neighboring
atom we can see other changes Proceeding across period 2
the ligand atoms have an increasingly squashed “onion” shape,
flattened on the opposite side from the central atom These
changes can be understood in the light of the Pauli principle,
which is an important factor in determining the shape of the
electronic charge cloud The Pauli principle is discussed
below and more formally in Appendix 3
The Pauli Principle
The many-electron wave function (Ψ) of any system is
a function of the spatial coordinates of all the electrons and
of their spins The two possible values of the spin angular
momentum of an electron—spin up and spin down—are
described respectively by two spin functions denoted as α(ω)
and β(ω), where ω is a spin degree of freedom or “spin
coordinate” All electrons are identical and therefore
indis-tinguishable from one another It follows that the interchange
of the positions and the spins (spin coordinates) of any two
electrons in a system must leave the observable properties of
the system unchanged In particular, the electron density must
remain unchanged In other words, Ψ 2 must not be altered
when the space and spin coordinates of any two electronsare interchanged
This requirement places a restriction on the many-electronwave function itself Either Ψ remains unchanged or it mustonly change sign We say that Ψ must be either symmetric orantisymmetric with respect to electron interchange In fact, onlyantisymmetric wave functions represent the behavior of anensemble of electrons That the many-electron wave functionmust be antisymmetric to electron interchange (Ψ→ ᎑Ψ
on electron interchange) is a fundamental nonclassical erty of electrons They share this property with other elemen-tary particles with half-integral spin such as protons, neutrons,and positrons, which are collectively called fermions Ensembles
prop-of other particles, such as the α particle and the photon, havesymmetric many-particle wave functions (Ψ→ Ψ on particleinterchange) and are called bosons
The requirement that electrons (and fermions in general)have antisymmetric many-particle wave functions is calledthe Pauli principle, which can be stated as follows:
A many-electron wave function must be antisymmetric
to the interchange of any pair of electrons.
No theoretical proof of the Pauli principle was given originally
It was injected into electronic structure theory as an pirical working tool The theoretical foundation of spin wassubsequently discovered by Dirac Spin arises naturally in thesolution of Dirac’s equation, the relativistic version ofSchrödinger’s equation
em-A corollary of the Pauli principle is that no two electronswith the same spin can ever simultaneously be at the samepoint in space If two electrons with the same spin were at thesame point in space simultaneously, then on interchangingthese two electrons, the wave function should change sign asrequired by the Pauli principle (Ψ→ ᎑Ψ) Since in this case thetwo electrons have the same space and spin coordinates (i.e.,
Figure 5 Three-dimensional isodensity envelopes of (a) SCl2, (b)
H2O, and (c) Cl2 The outer envelope has the value of 0.001 au, the
van der Waals envelope; the inner one is the bond critical point
density envelope ( ρ b -envelope).