Atomic Structure

1. Atomic Structure The theories of atomic and molecular structure depend on quantum mechanics to describe atoms and molecules in mathematical terms.

2. 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 • xpx  h/4 • Electrons in an atom have to be described in regions of space with certain probabilities.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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 • (02)

8. 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 distance.

9. 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.

10. 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 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).

11. 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.

12. 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

13. 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.

14. 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

15. 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.