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 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 – xpx 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 numbers. • Different orbitals have different and, therefore, different energies. h 2 2 2 Ze 2 H 2 2 2 2 8 m x y z 4 ( x 2 y 2 z 2 ) 12 o 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 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.