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Atomic Structure
The theories of atomic and molecular
structure depend on quantum
mechanics to describe atoms and
molecules in mathematical terms.
Quantum Mechanics
• The Bohr Atom (quantization of energy levels)
 1
E  R H  2  2 
 nl nh 
– The equation only works well for hydrogen-like atoms.
• Wave nature of the electron
– E = h = hc/, =h/mv (de Broglie wavelength)
– Not possible to describe the motion of an electron precisely.
• Heisenberg’s Uncertainty Principle
– xpx  h/4
• Electrons in an atom have to be described in regions of space with
certain probabilities.
The Schröndinger Equation
• Describes the wave properties of an electron in
terms of its position, mass, total energy, and
potential energy.
• Based on the wavefunction, , which describes
an electron wave in space (i.e. orbital).
• The equation used for finding the wavefunction
of a particle.
– Used to find the wavefunctions representing the
hydrogenic atomic orbitals.
The Schröndinger Equation (SE)
• H = E
– H is the Hamiltonian ‘operator’ which when operating on a
wavefunction returns the original wavefunction multiplied by
a constant, E.
• Carried out on a wavefunction describing an atomic orbital would
return the energy of that orbital.
– There are infinite solutions to the SE; each solution matching
an atomic orbital.
• Each solution (or ) is represented with a set of unique quantum
• Different orbitals have different  and, therefore, different energies.
 h  2
2 
Ze 2
H  2  2  2  2  
8 m  x y z  4 ( x 2  y 2  z 2 ) 12
The Schröndinger Equation (SE)
• Properties of the wavefunction, .
– Probability of finding an electron at a given point in
space is proportional to 2.
– The  must be single-valued.
– The  and its 1st derivative must be continuous.
– The  must approach zero as r approaches infinity.
– The probability of finding the electron somewhere in
space must equal 1.  A A d  1
all space
– All orbitals must be orthogonal.   A Bd  0
all space
Quantum Numbers and Atomic
• Implicit in the solutions for the resulting orbital
equations (wavefunctions) are three quantum numbers
(n, l, and ml). A fourth quantum number, ms accounts
for the magnetic moment of the electron.
• Examine Table 2-2 and discuss.
– n the primary indicator of energy of the atomic orbital.
– l determines angular momentum or shape of the orbital.
– ml determines the orientation of the angular momentum vector
in a magnetic field or the position of the orbital in space.
– ms determines the orientation of the electron magnetic
moment in a magnetic field.
• Only three a required to describe the atomic orbital.
Hydrogen Atom Wavefunctions
• These are generally
expressed in spherical polar
– (x,y,z)(r,,)
– r = distance from the nucleus
• (0)
–  = angle from the z-axis
• (0)
–  = angle from the x-axis
• (02)
Hydrogen Atom Wavefunctions
• In spherical coordinates, the three sides of a
small volume element are rd, rsind, and dr.
– r2sindddr (important for integration, Fig. 2-5).
• A thin shell between r and r+dr is 4r2dr.
– Describes the electron density as a function of
Hydrogen Atom Wavefunctions
• The wavefunction is commonly divided into the
angular function and the radial function.
– (r,,)=R(r)()()=R(r)Y(,)
– Tables 2-3 and 2-4, respectively.
• Angular function, Y(,)
– Determines how the probability changes from point to point
at a given distance.
• Produces the shapes of the orbitals and orientation in space.
• Determined by l and ml quantum numbers.
Examine Table 2-3 and Figure 2-6 and discuss.
Hydrogen Atom Wavefunction
• Radial function, R(r)
– Determined by quantum numbers, n and l
– Illustrates how the function changes with r
– The radial probability function is 4r2R2
• Describes the probability of finding the electron at a distance r (over
all angles). Examine Fig. 2-7.
• The distance that either function approaches zero increases with n
and l.
– Why do the radial functions and radial probability functions
• Appearance of complex numbers in the wavefunction.
– Properties of these type of equations allows us to produce
real functions out of complex function (example).
Hydrogen Atom Wavefunction
• A nodal surface is a surface with zero electron density.  and 2
will equal zero. The electron is not allowed on this surface. The
radial portion or the angular portion of the wavefunction must
equal zero.
– Radial nodes, R(r) = 0
• Spherical nodal surfaces where the electron density is zero at a given
value of r.
– 4r2R2 = 0 (examine radial probability functions)
• The number of radial nodes = n-l-1
– Angular nodes, Y(,) = 0
• These are planar or conical surfaces.
– Examine the appearance of the orbitals.
• The number of angular nodes = l.
Aufbau Principle (many electron)
• Electrons are placed in orbitals to give the
lowest total energy of the atom.
– Lowest values of n and l are filled first.
• Pauli exclusion principle
• Hund’s rule of maximum multiplicity
– Coulombic energy of repulsion, c, and
exchange energy, e.
• Klechkowkowsky’s n+l rule
Shielding and Other Factors
• Each electron acts as a ‘shield’ for electrons
farther out from the nucleus.
– Degree of shielding depends on n and l.
– Slater rules for determining the shielding
constant (Z*=Z-S).
• Higher n shields lower n significantly.
• Within the same n, lower l values can shield higher l
values significantly.
Shielding and Other Factors
• The electron configurations for Cr and Cu.
– Examine Figure 2-12. In this diagram, the 3d drops
faster in energy than the 4s.
• Formation of a positive ion reduces the overall
electron repulsion and lowers the energy of d
orbitals more than that of the s orbitals according
to this figures.
For an better description of why this occurs consult
the reference listed below.
L.G. Vanquickenborne, J. Chem. Educ. 1994, 71, 469
Ionization Energy and Radii
• Ionization energy – energy required to remove an
electron from a gaseous atom or ion.
– Trends with ionization energy (Figure 2-13).
– Draw a plot of Z*/r versus ionization energy.
• Covalent and ionic radii
– As nuclear charge increases, the electrons are pulled
toward the center. More electrons, however, increase
the mutual repulsion.
– Size of cations/anions in reference to the neutral atom.
– Other factors can influence size as well.