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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)
- 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. - Heisenbergs 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 numbers. - Different orbitals have different ? and,

therefore, different energies.

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.

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
- (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 rdr is 4?r2dr.
- 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 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).

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
- Hunds rule of maximum multiplicity
- Coulombic energy of repulsion, ?c, and exchange

energy, ?e. - Klechkowkowskys nl 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 (ZZ-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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