1.1: Quantum Mechanics and Wavefunctions There is a relationship between the motions of electrons in atoms and molecules and their energies that is described by quantum mechanics. Because of wave–particle duality, scientists must deal with the probability of an electron being at a particular point in space. To do so required the development of quantum mechanics, which uses wavefunctions to describe the mathematical relationship between the motion of electrons in atoms and molecules and their energies. 1.2: Quantum Numbers Quantum mechanics uses four quantum numbers (n, l, ml, and ms) to define wavefunction. The first three quantum numbers provide information about the spatial distribution of an electron. The principal quantum number n describes the distance of the orbital from the nucleus. The angular momentum quantum number l describes the shape of the orbital, and the magnetic quantum number l specifies its orientation in space. The spin quantum number ms allows for two electrons to occupy the same orbital. 1.3: Probability Density and Nodes Orbitals with l = 0 are s orbitals and are spherically symmetrical, with the greatest probability of finding the electron occurring at the nucleus. Orbitals with values of n > 1 and l = 0 contain one or more nodes. Orbitals with l = 1 are p orbitals and contain a nodal plane that includes the nucleus, giving rise to a dumbbell shape. Orbitals with l = 2 are d orbitals and have more complex shapes with at least two nodal surfaces. l = 3 orbitals are f orbitals, which are still more complex. 1.4: Representations of Atomic Orbitals Drawings of atomic orbitals will include information given by the principal quantum number n and the angular momentum quantum number l, which specify the shell and subshell of an orbital. 1.5: Energies of Atomic Orbitals Although we have discussed the shapes of orbitals, we have said little about their comparative energies. For atoms or ions with only a single electron (such as H or He+), all orbitals in the same shell are degenerate. For atoms or ions with more than one electron, the values of quantum numbers n and l determine the energies of each orbital. 1.6: Rules Governing Ground State Electron Configurations The Aufbau Principle (also called the building-up principle or the Aufbau rule) states that, in the ground state of an atom or ion, electrons fill atomic orbitals of the lowest available energy level before occupying higher-energy levels. 1.7: How to Write a Ground State Electron Configuration Ground state electron configurations are the foundation for understanding molecular bonding, properties, and structures. From the electrons in an atom, to the differing orbitals and hybridization, the ground state electron configuration sheds light on many different atomic properties. Fundamentally, understanding electron configuration leads to an understanding of the periodic table. 1.8: Magnetic Properties The magnetic moment of a system measures the strength and the direction of its magnetism. The term itself usually refers to the magnetic dipole moment. Anything that is magnetic, like a bar magnet or a loop of electric current, has a magnetic moment. A magnetic moment is a vector quantity, with a magnitude and a direction. An electron has an electron magnetic dipole moment, generated by the electron's intrinsic spin property, making it an electric charge in motion. There are many different magne 1.9: Electron Configurations for Transition Metal Elements Writing an electron configuration for a transition metal element follows the same basic steps as for writing an electron configuration for an element in the s-block or p-block. List each subshell, and then fill each subshell with an appropriate number of electrons until all electrons in the element are accounted for. Transition elements have electrons in the d orbital, which introduces some additional nuance in the electron configurations. 1.10: The Periodic Table and Periodic Properties The Periodic Table of Elements categorizes like elements together. On the periodic table, elements that have similar properties are in the same groups (vertical). From left to right, the atomic number (z) of the elements increases from one period to the next (horizontal). 1.11: Atomic Radius The atomic radius is one-half the distance between the nuclei of two atoms (just like a radius is half the diameter of a circle). Atomic radii can be obtained from quantum mechanical calculations or can be determined experimentally. When calculated, the atomic radius is defined as the radius of the spherical volume in which the electron can be observed with 90% probability. Atomic radius decreases from left to right within a period. This is caused by the increase in the number of protons and ele 1.12: Ionization Energy Ionization energy is the energy required to remove an electron from a neutral atom in its gaseous phase. Ionization energy is always positive. The ionization energy of the elements within a period generally increases from left to right. This is due to valence shell stability. The ionization energy of the elements within a group generally decreases from top to bottom. This is due to increasing electron energy shell. The noble gases possess very high ionization energies because of their full valen 1.13: Electron Affinity Electron affinity is a quantitative measurement of the energy change that occurs when an electron is added to a neutral gaseous atom. The more negative the electron affinity value, the higher the electron affinity and the more easily an electron is added to an atom. Electron affinity can be either positive or negative. The greater the negative value, the more stable the anion is. Electron affinity increases (becomes more negative) from left to right within a period. This is caused by the decreas 1.14: Other Periodic Properties of Atoms Metals generally possess a high melting point. Most non-metals possess low melting points. The non-metal carbon possesses the highest melting point of all the elements. The semi-metal boron also possesses a high melting point. Metallic character relates to the ability to lose electrons, and nonmetallic character relates to the ability to gain electrons.Metallic characteristics decrease from left to right across a period. Metallic characteristics increase down a group. 1.15: Effective Nuclear Charge and Shielding Coulomb's Law is from classical physics; it tells us that particles with opposite electrostatic charge are attracted to each other, and the larger the charge on either particle or the closer the distance between them, the stronger the attraction. Coulomb's law is insufficient for predicting the energies of electrons in multi-electron atoms and ions. To explain these things, we need to consider how both electron shielding and penetration result in variations in effective nuclear charge (Z*) that 1.16: Slater's Rules We have previously described the concepts of electron shielding, orbital penetration and effective nuclear charge, but we did so in a qualitative manner. In this section, we explore one model for quantitatively estimating the impact of electron shielding, and then use that to calculate the effective nuclear charge experienced by an electron in an atom. The model we will use is known as Slater's Rules (J.C. Slater, Phys Rev 1930, 36, 57). 1.17: Periodic Trends and Effective Nuclear Charge There are some predictable trends in Zeff. The Zeff for electrons in a given shell and subshell generally increase as atomic number increases; this trend holds true going across the periodic table and down the periodic table. However, trends in the Zeff for valence electrons is more complicated and warrants additional consideration. 1.18: The Effects of Shielding on Periodic Properties The attraction of the nucleus to the valence electrons determines the atomic radius, ionization energy, and electron affinity. The stronger the attraction, and the stronger Zeff, the closer the electrons are pulled toward the nucleus. This in turn results in a smaller size, higher ionization energy, and higher electron affinity. The Lanthanide Contraction describes the atomic radius trend that the Lanthanide series exhibit. Another important feature of the The Lanthanide Contraction refers to th