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)
– 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 xpx h/4
• Electrons in an atom have to be described in regions of space with certain probabilities.
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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 numbers.
• Different orbitals have different and, therefore, different energies.
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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.– All orbitals must be orthogonal.
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Quantum Numbers and Atomic Wavefunctions
• 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 coordinates.– (x,y,z)(r,,)– r = distance from the nucleus
• (0) = angle from the z-axis
• (0) = angle from the x-axis
• (02)
Hydrogen Atom Wavefunctions
• In spherical coordinates, the three sides of a small volume element are rd, rsind, and dr.– r2sindddr (important for integration, Fig. 2-5).
• A thin shell between r and r+dr is 4r2dr.– Describes the electron density as a function of
distance.
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 4r2R2
• 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 differ?
• 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.
– 4r2R2 = 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.
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