Download 3. Atomic and molecular structure

Chemistry 2810 Lecture Notes
3.
Dr. R. T. Boeré
Page 11
Atomic and molecular structure
Reference:
Kotz & Treichel, 3rd edition, 7.1 - 7.6 Shriver, Atkins, Langford, 2nd edition, 1.4 - 1.6
The single most important guiding principle for our study of the chemistry of the elements is that the internal structure of the
atoms are the key to the differences in chemical behavior among the different elements. The aspect of atomic structure that is most
important for chemistry is the arrangement and behavior of the electrons, so that what we are chiefly concerned with is often called
the electronic structure of the atom. The foundation of the modern theory of atomic structure was laid in the years immediately
following the First World War, a time of tremendous social upheaval in Europe. First came the wave-particle duality of the photon,
then the same for the electron, finally the application of this to the atom.
3.1
Atomic structure
The 20th century could be described as the era dominated
scientifically by the interaction of electromagnetic radiation with matter and thus with the elements. Electromagnetic radiation is the general form
of radiant energy of which light is just one example. It consists of a selfpropagating wave in empty space, with perpendicular electric and
magnetic field components, as shown in the diagram at the left.
3.1.1 Wave-particle duality of the photon
A major development in modern physics was the recognition that
electromagnetic radiation, long known to have wavelike properties as a
form of electromagnetic radiation, also had particle-like properties.
Einstein argued on the basis of his theory of relativity that a photon, which has zero resting mass, gain mass as its speed (the
hc
2
and the Einstein equation E = mc . Setting the two equivalent
λ
hc
and letting m be the apparent mass of the photon, we get:
= mc 2 . Reorganizing this equation gives the final form which
λ
h
h
emphasizes the wave-particle duality: λ =
= . Here p stands for momentum. Thus the wavelength of a photon is inversely
mc p
speed of light). Consider the Planck equation
E = hν =
related to the momentum p via Planck's constant
3.1.2 The Bohr model
The Bohr "planetary" model
Niels Bohr was the first to apply the new quantum
Electrons move in well-defined
orbits, whose radii could be
mechanics to the atom. His theory of electrons moving in
•e calculated and were
well-defined orbits around a nucleus containing the
shown to agree with
protons and neutrons is called the "planetary" model.
the experimental
Rydberg constant.
Central to his model were two postulates:
•ePostulate (1) In the absence of radiation absorption or
nucleus
emission, electrons stay in a stationary
state.
•e•e Postulate (2) Absorption occurs only in discrete
amounts, corresponding to a change in
energy between two stationary states of
the electron. ∆E = hν then gives the
frequency of the radiation.
With this model in mind, he was able to derive a mathematical formula expressing the energies that correspond to each of the
orbits which correspond to the allowed stationary states. Here I include the form of the equation that is most easily tested by
comparison with atomic line spectra, which are due to photons released or absorbed.
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 12
Z 2 me 4  1
1
E n − Em = hν =
2 2  2 − 2 
8ε 0 h  nn nm 
This equation was of the same form as the Rydberg equation derived some 15 years earlier from measurements of line spectra.
 1
1
E n − Em = hν = R 2 − 2 
 nn nm 
where the value of R measured from spectra (R = 2.18 × 10-18 J) was in remarkable agreement with the constant term in the Bohr
equation. This correspondence did a lot to further the acceptance of the Bohr theory. There was however a nagging suspicion in
the mind of scientists, not the least of them Bohr himself, regarding the arbitrariness of the introduction of electron quantization in
the atom.
Bohr's theory also allowed for the calculation of the radius of each orbit of the electron stationary states in the hydrogen
atoms. This is given by the equation:
ε 0h 2 2
rn =
n
Zπme 2
This simple concept turned out to be closely associated to the real orbital size, as we will see. The simple Bohr model was
expanded and patched for several years to account for more sophisticated experimental probing of atomic structure by
spectroscopy, but in the end it had to give way to a fundamentally different theory, the wave-mechanical model of the atom.
3.1.3 Wave-particle duality of the electron
An electron has a rest mass of 9.1095 × 10-28 g. Inside the atom, however,
the electrons move on the order of 10% of the speed of light (this could also
be calculated from the Bohr model). Louis de Broglie therefore suggested that
the electron in an atom could have wave character, just as the photon could
have particle character, using an equation that resembles the one derived in
section 3.1.1.
λ=
h
h
=
mv p
The de Broglie postulate that electrons moving between 10% and 100% of the
speed of light would have a measurable wavelength was experimentally
confirmed by the Americans Davisson and Germer in 1927, who diffracted
electrons through a sheet of crystalline aluminum. A re-enactment of their
experiment is presented in the figure at left, produced by Dr.Donald Potter,
Department of Metallurgy, University of Connecticut. The concentric rings
on the photographic film are related to the wavelength of the electrons and
the spacing of the Al atoms in the crystalline metal.
Electron diffraction is used today as a structure-gathering technique
especially well suited to determining gas-phase structures of volatile
molecules. The German Heinz Oberhammer is a leading practitioner of this
technique.
It is a general property of waves that they can undergo constructive and
destructive interference. If an electron with de Broglie properties is constrained by the three-dimensional force-field in an atom, we
can assume certain properties: only those wavelengths corresponding to standing waves can persist. Any other choice will lead
to destructive interference and hence in collapse of the wave.
Standing waves in two dimensions: waves that are closed (a) and (b); waves that are open (c) and (d).
The open waves will continually cross each other out of phase and will decay by interference.
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 13
3.1.4 The Schrödinger wave equation
It took another brilliant physicist, Erwin Schrödinger, to apply the idea of the wave-like property of electrons to the electronic
structure of the atom. The full details of this his theory are developed in our third year quantum mechanics course (Chemistry
3730), but I will sketch a brief outline for you here. If we start with the classical equation for wave-motion in a linear sine-wave:
Ψ = A sin
2πx
λ
The symbol Ψ represents the displacement of the wave-vector from
the x axis, and A is the maximum displacement, i.e. the amplitude. The
rest of the equation is called a wave function, i.e. a mathematical
function which describes the variation in the length of the
displacement vectors as the wave propagates through space. The
analysis of such equations is usually done through a differential
equation. For this equation, we can obtain such a differential
equation by taking the first and second differentials and then rearranging the terms, as follows:
dΨ
2πx
2πx st
=A
⋅ cos
(1 derivative)
dx
λ
λ
d 2Ψ
4π 2
=
−
⋅ Ψ (2nd derivative)
dx 2
λ2
This equation has special properties in that the Hamiltonian operator d 2/dx2 operates on the function Ψ, to give some constant
times Ψ. Such equations were already known to mathematicians and physicists and were called eigenfunctions, for the German
word for "its own". Schrödinger applied such a function to a moving electron with kinetic energy T to get the new wave equation:
d 2Ψ
8π 2 m
=
−
⋅T⋅Ψ
dx 2
h2
Since the total energy E of the electron is the sum of the kinetic energy T and potential energy, V, then T = E – V. Substituting this
gives the equation in the more familiar form of the particle in a one-dimensional box, which you will see in Chemistry 3730:
d 2Ψ
8π 2 m
=
−
⋅ (E −V ) ⋅ Ψ
dx 2
h2
Thus we have a simple one-dimensional equation of the general form HΨ = EΨ. Differential equations of this type are of a
special type where there are a set of solutions (eigenvectors) characterized by a set of eigenvalues En. The situation in a simple
atom, such as hydrogen, is more complicated. The electron is in motion around a central nucleus in three-dimensional space. This
is often called the central force field situation, and when a wave equation is written for this situation, it is called the Schrödinger
wave equation for the atom. As written below, it applies to the hydrogen atom if the appropriate form of V is supplied. This form
of the equation is known as the time-independent Schrödinger equation. It is a special case of the more general time-dependent
Schrödinger equation.
∂ 2Ψ ∂ 2Ψ ∂ 2Ψ
8π 2m
+ 2 + 2 = − 2 ( E − V )Ψ
∂x 2
∂y
∂z
b
Solutions to this equation in the form of a set of equations (so-called analytic solutions)
have only been obtained for one-electron systems. The solutions take the form of a set of
wavefunctions with associated energies. Thus the Schrödinger equation predicts
quantization of atomic properties as a natural consequence of the wave-nature of an
electron moving at high velocity in the central force-field of a nucleus within an atom.
You were wondering…
What is meant by one-electron
systems? Is there more than one
example, hydrogen?
3.1.5 Solutions to the Schrödinger wave equation
The Schrödinger equation can be solved analytically for one-electron atoms.
However, it is necessary to cast the equation in the form of spherical polar coordinates,
rather than leaving it in the more familiar Cartesian coordinates used above. This coordinate system is an alternative way to
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 14
describe the location of any point in space, and is ideally suited for spherical systems. The point is labeled not by (x,y,z), but by
(ρ,θ,φ), where these three terms are best defined in a diagram such as the one shown below:
z
Spherical polar coordinates
θ
ρ is the radius
θ is the colatitude
φ
φ is the azimuth
ρ
In this coordinate system, the
x
y
equation describing a spherical
surface is simply f( ρ,θ,φ ) = ρ
i.e. if ρ = 3, a sphere of radius 3.
Separation of variables in spherical polar coordinates and resulting solutions
We now convert the Schrödinger equation to spherical polar coordinates. Then we substitute for V the appropriate Coulomb
potential for the one-electron atoms, where Z is the number of protons in the nucleus: -Ze2/4pe0ρ. This is easy to do using
spherical coordinates, but hard in Cartesian (which is why we ignored it thus far! After the conversion we obtain an immediate
bonus, because we can now separate the differential equation into three sub-functions, each in terms of one of the defining
variables ρ, θ and φ. We call the ρ function R, the θ function Θ, and the φ function Φ (i.e. using capital Greek letters).
1 d  2 dR  8π 2 m 
Ze 2  2
⋅ r
 + 2 E +
 ρ = u (R sub-function)
R dρ  dρ 
h 
4πε0 ρ 
1 d 
dΘ  v 2 Θ
⋅  sin θ ⋅
−
+ uΘ = 0 (Θ sub-function)
sin θ dθ 
dθ  sin 2 θ
1 d 2Φ
⋅ 2 = −v 2 (Φ sub-function)
Φ dφ
In these equations appear the constants u and v. These intrinsic mathematical elements turn out to be related to the quantum
numbers first obtained empirically by spectroscopy as follows:
u = l(l + 1) and v = ml
The principle quantum numb er n is found to label the energies of the solutions En. For historical reasons, chemists have stuck
with the original spectroscopic quantum numbers rather than u and v. The solutions to the above equations are a set of
wavefunctions (eigenvectors), each with its corresponding energy (the eigenvalues). For hydrogen in an unperturbed state, the
energy of the orbitals is determined only by n. The energy of the set of solutions can be plotted in the energy level diagram shown
on the next page.
Remember from General Chemistry the governing "rules" for these quantum numbers which label the orbitals:
principal quantum number
n = 1,2,3,….., ∞
angular momentum quantum number
l = 0, 1, 2, ….., (n–1)
magnetic quantum number
ml = 0, ±1, ±2, …., (±l)
electron spin quantum number
ms = +½ or –½
Also note the following consequences of the quantum numbers:
• normally we use the s, p, d, f labeling system rather than the l quantum number, but you need to know both systems.
• shells are given by the value of n (electrons in same shells tend to have similar energy and similar average distance from the
nucleus)
• subshells are given by the value of l (electrons in the same subshell have exactly the same energy, which we call degenerate),
and chemists usually use letter designations for the l value, as shown in the table below
• orbitals given by (n - l - ml), and may contain up to two electrons with ms = +½ and –½ electron spin is usually denoted as spin
up ↑ and spin down ↓
l
orbital label
0
s
Orbital names assigned to values of l
1
2
3
p
d
f
4
g
5
h
Chemistry 2810 Lecture Notes
0
E
N
E
R
G
Y
Dr. R. T. Boeré
Page 15
n=∞
etc
n=5 l=0
l=1
l=2
l=3
n=4 l=0
l=1
l=2
l=3
n=3 l=0
l=1
l=2
n=2 l=0
l=1
The energy levels of the orbitals of the hydrogen atom
n=1
Let us now briefly look at some of these solutions in algebraic form. I have chosen a few representative examples which are
shown in the table below. Each solution is a product such that Ψ = R × Θ × Φ. For simplicity, we divide these into only two
categories, R which we call the radial part of the wavefunction and Θ × Φ which we call the angular part of the wavefunction.
Some of the wavefunctions (all the s orbitals) do not have a Θ × Φ part (i.e. the term is equal to 1). To help you recognize the ones
that do have angular components, I have listed the Cartesian functions that correspond to the angular components in the last
column of the table. In our analysis of orbitals, we start out by considering the two components separately, and then later we will
combine information from both into a composite "picture" of what an orbital represents.
Some complete Hydrogenic wafefunctions (products of Radial and Angular components)
Quantum
numbers
n
l
Spherical-polar solutions to the Schrödinger equation
Cartesian angular functions
ml
3/2
Ψ100
1  1
=
 
π  a0 
Ψ210
1  1
=
 
4 2π  a 0 
3/ 2
0
1
1  1
Ψ211 =
 
4 2π  a 0 
3/2
1
2
-1
1
0
0
2
1
2
3
Ψ32−1
⋅ e − ρ /a0
1 2 1
=
 
81 π  a 0 
None
⋅
1 − ρ /2a0
⋅e
⋅ ρ ⋅ cosθ
a0
ρ ⋅ cosθ = z
⋅
1 − ρ /2 a0
⋅e
⋅ ρ ⋅ sin θ cosφ
a0
ρ ⋅ sinθ ⋅ cos φ = x
3/2
2
1
⋅   ⋅ e −ρ /3a0 ⋅ ρ 2 ⋅ sinθ cosθ ⋅ sin φ
 a0 
ρ 2 ⋅ sinθ ⋅ cosθ ⋅ sin φ = yz
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 16
The radial part of the hydrogenic wavefunctions
By definition a radial wavefunction only has a single dimension, that of the distance from the nucleus, which is the vector ρ
that R is a function of. We can graph these functions to give us a plot of distance vs. amplitude. It is instructive to consider such
plots, and the graphs below are calculated for several of the orbitals of a single hydrogen atom.
The graphs show the following: at left the single s wavefunction of n = 1. In the middle, the two for n = 2, 2s and 2p. Can you see
which is which? The l quantum number will tell you! The right-hand column are the 3s, 3p and 3d wavefunctions. Note that only
the s wavefunctions have a finite amplitude of the orbital at the nucleus, in fact this is where the intensity of these waves is the
highest. The reason that p, d and in fact all other possible orbitals are different is that these higher orbitals all posses nodes at the
origin. The nodes at the origin are easier to see when considering plots of the angular parts of the wavefunction, but it is worth
noting the following relationship:
Nodes in orbitals
# of radial (spherical) nodes is (n – l – 1)
# of angular (planar) nodes is l
Total # of nodes = # spherical + # planar = (n – 1)
The graphs show that for any given l value, the number of radial nodes goes up by one for each jump in n. The radial nodes
occur at the points where the functions cross the ρ axis (called r in these graphs) other than at the origin. Thus 1s has zero radial
nodes, 2s has one radial node, etc. Similarly, 2p has zero radial nodes, 3p has one radial node, etc. For any value of n the total
number of nodes in the wavefunction is the same, so we can immediately see that a p wavefunction will always have a single
angular node, and a d orbital will always have two angular nodes, etc.
Note also that these graphs are scaled in such a way that the integrated area "under" each curve is the same, so that the
reason that 1s is so much higher in amplitude is precisely that it decays away faster, while the very low 3d function extends further
out. We now consider the disconcerting fact that despite all the effort we have gone to, we do not really know what we have on
our hands. Wavefunctions have no physical significance as such. In order to interpret them, we have to do some further
calculations. When introducing his theory, Schrödinger noted that the square of a
wavefunction Ψ 2 should be related to the probability of finding the electron in a
certain region of space. This led to the development of what are known as radial
probability density distribution plots. Firs we square the wavefunctions (which
eliminates the negative amplitudes), then we integrate Ψ 2 over a very small but
finite volume element to obtain the probability of finding the electron at a given
distance from the nucleus. This effectively provides us with electron density, the
closest thing that wave mechanics can give us to the concept of an electron
particle in an atom. It is instructive to consider what happens when we plot Ψ 2 as
well as the volume element curve 4πρ 2 on the same graph, as is done here at the
left. We see that one decays as the square of an exponential, while the other grows
as a square function. The product of these oposing tendencies will result in a
curve with a distinct maximum. The reason that the product curve starts at zero is
that at the origin there are zero volume elements, so the high amplitude still gives
zero electron density. To our great surprise, the position of the maximum, a 0, turns
out to be exactly the same as the radius for the "orbit" calculate by Bohr's theory, and is called the Bohr radius.
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 17
In the next figure we show these new kinds of functions, 4πρ 2Ψ 2 (actually 4πρ 2R2) graphed for exactly the same orbitals as in
the previous figure. The highest probability of finding the electron is at the nucleus. However, there are very few volume elements
there. On the contrary, at a 0 there are many volume elements along the surface of a sphere. Thus we have two concepts:
1) the highest electron density is near the nucleus
2) the highest chance of finding an electron at a certain distance from the nucleus peaks at a 0.
There is now a dramatic change, because the much greater number of volume elements at greater radius causes the electron
density to maximize much further away from the nucleus than the wavefunction amplitude. The "size" of the orbital thus increases
significantly as the principal quantum number is increased, at least insofar as the position of the largest outer "hump" in these
graphs shows. This also agrees qualitatively with the Bohr model, where higher n meant orbits with larger radii. However, the
planetary model did not include the small, inner humps that wave mechanics shows. Consider the n = 3 set of orbitals. As far as
the outer humps go, the 3s is larger than the 3p which is larger than the 3d. But we find that electrons in the 3d are the least
strongly bound (higher energy) of the set. How can this be? The answer lies with the inner humps. The small percentage of time
that the s and p electrons spend in those regions is compensated for by the extremely high forces they experience from the nucleus
during that time. This results in the normal order of orbital energies which is s < p < d < f for any given value of n. Only wave
mechanics provides a satisfactory explanation for this fact, which has been verified from the atomic line spectra of the heavier
elements. In fact, the d orbitals are especially noticeable as having very little electron density near the nucleus, with the 3d
function starting to rise beyond the inner node of the 3s function, and close to the inner node of the 3p function. This is the origin
of the unusually diffuse nature of the d orbitals, and their much higher energy than the s and p orbitals of the same quantum level.
It should be remembered that these graphs all refer to a single H atom, which has a single electron. Thus the graphs show the
behavior (in terms of electron density) of this electron in the various possible states of H, including the ground state 1s and the
excited states. An electron in a 3p excited state spends most of its time about ten times further out from the nucleus than in the
grounds state, and indeed in terms of the energy level diagram we saw earlier, such an electron is very close to being ionized.
When we apply these ideas to the other elements of the periodic table, however, it would be wrong to assume that a nitrogen
atom's valence 3p electron is that far out in the ground state (which would mean that an N atom is 10 times larger than an H atom,
when in fact their radii are thought to be 0.37 and 0.75 Å for H and N, respectively). Since the volume of the atom is primarily
determined by the space occupied by the outermost electrons, this has the result that the atoms do indeed get "bigger" (although
all stretch to infinity!) as higher quantum levels are filled. Hence atoms with more electrons in higher lying shells are larger.
However, this effect is greatly counteracted by the increasing nuclear charge, which has the effect of shrinking the whole set of
orbitals. Thus the graphs will be valid when comparing orbitals on the same atom, but not between two different atoms. This is
another way of expressing the fact that the units of ρ are in values of a 0, and the Bohr radii for the heavier elements are
significantly smaller than for hydrogen. The result of the trade-of between occupation of higher quantum levels and the shrinkage
of the scale for increasing numbers of protons (Z) is that atoms sizes grow slowly down any group in the periodic table. We will
say a bit more about this later in terms of the effective nuclear charges, as well as considering how the size of the atoms change
across the periods of the Periodic Table. The conclusion reached in General Chemistry, that n gives the size of the orbital, is still
basically true, but the finer distinctions are extremely important to the chemical properties of the elements.
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 18
Angular part of the Hydrogenic wavefunctions
The shapes obtained for the orbitals come from the angular wavefunctions. By definition in spherical polar coordinates, if θ
and φ are not specified (meaning that they can have any value you please), a function in just ρ always describes a sphere! Thus all
the s wavefunctions are spherical in shape, and differ only in the number of radial (spherical) nodes that they posses. There is little
more to be said about these simple functions than can be learned form the R plots. But for the remaining types of orbitals, the
shapes come from graphing the θφ components. Take a 2p z orbital as an example:
When we plotΘΦ
210
, the plot looks like:
since these vary as cosθ
When we plot( ΘΦ
210
2
) , the plot looks like:
2
since these vary as cos θ
This is closest to the typical "p-orbitals" seen in
most first year and organic texts
These shapes by themselves are only graphs of the trigonometric functions. They have no real meaning. Thus at this stage we recombine the radial plots with the angular plots, and we do this not for Ψ, but for 4πρ 2Ψ 2 by multiplying 4πρ 2R2 × (ΘΦ)2. The
results are three dimensional electron density maps, which can only be properly described by contour maps of the electron
density. Such graphs are now shown for several important types of orbitals.
Composite representations of the Hydrogenic wavefunctions
Contour diagrams are like topographical maps which show elevation against distance. In the following diagrams, the contours
represent electron density distribution as a function of distance from the nucleus. Contour lines close together show steep R
curves; lines far apart are due to shallow R curves. We show a 2p and a 3p function, to highlight the presence of the extra node in
the latter. The heights of the peaks are
indicated, and this is just the same
information as can be obtained from the
one-dimensional R graphs we saw earlier.
Despite the much more complex structure of
the 3p wavefunction, the outside surfaces
of the two p functions is essentially the
same, and the presence of these cores can
be almost completely ignored in chemistry.
Thus, for example, a 2p and a 3p orbital
form bonds in very much the same manner.
However, it should be noted that side-on π
overlap as occurs in the formation of
multiple bonds is significantly more
effective for a 2p-2p interaction than for a
3p-3p interaction, and this will have (a)
(b)
important consequences for the energies of
Contour map of a carbon 2p z orbital
Contour map of a chlorine 3p z orbital
single and double bonds of the elements of
groups 13 to 17. On the whole, though, you can more or less ignore the differences between the same kind of wavefunction of
different principal quantum number. This is emphasized by the solid-body sketches introduced below, which cover over the inner
structure of the orbitals. One thing that should be remembered is that the squaring of the wavefunctions has removed the phases
from these plots. But the wavefunctions to have phase, and this is shown by shading the solid-body sketches darker and lighter
colors. Alternatively (+) and (–) signs can be added to the diagrams, but note that such signs do not indicate electrical charge, but
instead relative orbital phase!
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page 19
The structure of the d orbitals
is more interesting. Two examples
are shown, a 3d z2 and a 3d x2–y 2
orbital. The other kind, 3d xy etc., are
just the same as the second
example, except rotated by 45°.
Notice that these shapes look
much squatter than the pure angular
wavefunctions graphed in the
previous diagrams above. Although
(c)
(d)
there is no "right" way to represent
Contour map of a titanium 3d z2 orbital
Contour map of a titanium 3d x2–y 2 orbital
an orbital, these contour diagrams
are perhaps the most useful when we go on to consider bonding between atoms, since they represent the location of electron
density in the atom.
A related representation is to use solid-body sketches, which enclose 90% of the electron density in an orbital in a
symmetrical fashion. These shapes are now shown for (a) a 1s and (b) a 2s, as well as for 2p and 3d orbitals. These are the orbital
pictures we will use throughout inorganic chemistry, and it would do well to learn them now. You need to learn the basic shape,
nodal pattern, and phases of these s, p, and d orbitals Note the important fact that by trigonometry, the sum of 3 p orbitals, as well
as the sum of 5 d orbitals, is exactly spherical. Hence an atom like Ne or Ar is NOT lumpy!