84.443/543 Advanced Inorganic Chemistry

The d orbitals

Unusual Aspects of Inorganic Compounds The use of d orbitals enables transition metals to form quadruple bonds. Sigma (σ) bonds can be formed using p orbitals, or the d z 2 orbitals. The use of d orbitals enables transition metals to form quadruple bonds. Sigma (σ) bonds can be formed using p orbitals, or the d z 2 orbitals.

Unusual Aspects of Inorganic Compounds Pi (π) bonds can be formed using the d xz and d yz orbitals. Pi (π) bonds can be formed using the d xz and d yz orbitals.

Unusual Aspects of Inorganic Compounds In addition, “face-to-face” overlap is possible between the d xy orbitals on each metal. This forms a delta (δ) bond. In addition, “face-to-face” overlap is possible between the d xy orbitals on each metal. This forms a delta (δ) bond.

Unusual Aspects of Inorganic Compounds The existence of δ bonds is usually determined by measuring bond lengths and magnetic moments. The existence of δ bonds is usually determined by measuring bond lengths and magnetic moments. [Re 2 Cl 8 ] 2- has a quadruple bond between the metal atoms.

Unusual Aspects of Inorganic Compounds The coordination number for transition metals can be greater than 4, with coordination numbers of 6 being quite common. In addition, 4- coordinate metal complexes need not be tetrahedral. The coordination number for transition metals can be greater than 4, with coordination numbers of 6 being quite common. In addition, 4- coordinate metal complexes need not be tetrahedral.

Unusual Aspects of Inorganic Compounds When inorganic compounds have tetrahedral geometry, it may be quite different from organic compounds. P 4 has tetrahedral geometry, but lacks a central atom. When inorganic compounds have tetrahedral geometry, it may be quite different from organic compounds. P 4 has tetrahedral geometry, but lacks a central atom.

Unusual Aspects of Inorganic Compounds Cluster compounds, in which there are metal- metal bonds can be formed. The structure of Mn 2 (CO) 10 has the two Mn atoms directly bonded to each other. Cluster compounds, in which there are metal- metal bonds can be formed. The structure of Mn 2 (CO) 10 has the two Mn atoms directly bonded to each other.

Unusual Aspects of Inorganic Compounds Cage compounds lack a direct metal-metal bond. Instead, the ligands serve to hold the complex together. Cage compounds lack a direct metal-metal bond. Instead, the ligands serve to hold the complex together.

Unusual Aspects of Inorganic Compounds Organic molecules may bond to transition metals with σ bonds or π bonds. If π bonded, some unusual “sandwich” compounds may result. Organic molecules may bond to transition metals with σ bonds or π bonds. If π bonded, some unusual “sandwich” compounds may result.

Quantum Numbers n principal quantum #; n = 1, 2, 3, etc. Determines the major part of the energy of the electron l angular momentum quantum # = 0,1,2…n-1 Describes angular dependence and contributes to the energy mlmlmlml magnetic quantum # = - l …0…+ l Describes the orientation in space. (ex. p x, p y or p z ) msmsmsms Spin quantum # = +1/2 or -1/2 Describes orientation of the electron’s magnetic moment in space

Common Orbital Designations spdf l0123 In the absence of a magnetic field, the p orbitals (or d orbitals) are degenerate, and have identical energy.

Wave Functions of Orbitals Wave functions can be factored into two angular components (based on θ and φ ), and a radial component (based on r). Wave functions can be factored into two angular components (based on θ and φ ), and a radial component (based on r).

Angular Functions The angular functions, based on l and m l, provide the probability of finding an electron at various points from the nucleus. These functions provide the shape of the orbitals and their spatial orientation. The angular functions, based on l and m l, provide the probability of finding an electron at various points from the nucleus. These functions provide the shape of the orbitals and their spatial orientation.

The d-orbitals

Radial Functions Radial functions are determined by the quantum numbers n and l, and are used to determine the radial wave probability function (4πr 2 R 2 ). Radial functions are determined by the quantum numbers n and l, and are used to determine the radial wave probability function (4πr 2 R 2 ). R is the radial function, and it describes the electron density at different distances from the nucleus. r is the distance from the nucleus.

Radial Functions Radial functions are used to determine the probablity of finding an electron in a specific subshell at a specified distance from the nucleus, summed over all angles. Radial functions are used to determine the probablity of finding an electron in a specific subshell at a specified distance from the nucleus, summed over all angles.

Radial Wavefunctions The radial wave functions for hydrogenic orbitals have some key features:

Radial Wavefunctions Key features: 1. All s orbitals have a finite amplitude at the nucleus. 2. All orbitals decay exponentially at sufficiently great distances from the nucleus.

Radial Wavefunctions Key features: 3. As n increases, the functions oscillate through zero, resulting in radial nodes.

Radial Nodes Radial nodes represent the point at which the wave function goes from a positive value to a negative value. They are significant, since the probability functions depend upon Ψ 2, and the nodes result in regions of zero probability of finding an electron. Radial nodes represent the point at which the wave function goes from a positive value to a negative value. They are significant, since the probability functions depend upon Ψ 2, and the nodes result in regions of zero probability of finding an electron.

Radial Nodes For a given orbital, For a given orbital, the number of radial nodes= n- l -1

p orbitals The radial wave functions of p orbitals show a zero amplitude at the nucleus. The result is that p orbitals are less penetrating than s orbitals.

Radial Probability Functions Radial probability functions (4πr 2 Ψ 2 or 4πr 2 R 2 ) are the product of the blue and green functions graphed for a 1s orbital.

Radial Probability Functions The orange line represents the probability of finding an electron in a 1s orbital as a function of distance from the nucleus.

Radial Probability Functions Note the zero probability at the nucleus (since r=0). The most probable distance from the nucleus is the Bohr radius, a o = 52.9 pm.

Radial Probability Functions The probability falls off rapidly as the distance from the nucleus increases. For a 1s orbital, the probability is near zero at a value of r = 5a o.

Radial Probability Functions In a 1 electron atom, the 2s and 2p orbitals are degenerate. In multi- electron atoms, the 2s orbital is lower in energy than the 2p orbital.

Radial Probability Functions On average, the electrons in the 2s orbital will be farther from the nucleus than those in the 2p orbital. Yet, electrons in the 2s orbital have a higher probability of being near the nucleus due to the inner maximum.

Radial Probability Functions The net result is that the energy of electrons in the 2s orbital are lower than that of electrons in the 2p orbitals.

The d orbitals

The f orbitals

The Aufbau Principle The loss of degeneracy in multi-electron atoms or ions results in electron configurations that cannot be predicted based solely on the values of quantum numbers. The aufbau (building up) principle provides rules for obtaining electron configurations.

The Aufbau Principle 1.The lowest values of n and l are filled first to minimize energy. 2. The Pauli Exclusion Principle requires that each electron in an atom must have a unique set of quantum numbers. 3.Hund’s Rule requires that electrons in degenerate orbitals will have the maximum multiplicity (or highest total spin).

Electron Configurations

Klechkowsky’s Rule states that filling proceeds from the lowest available value of n + l. When two combinations have the same sum of n + l, the orbital with a lower value of n is filled first.

Electron Configurations The electron configurations of Cr and Cu in the first row of the transition metals defy all rules, as do many of the lower transition elements.

Shielding The energy of an orbital is related to its ability to penetrate the area near the nucleus, and its ability to shield other electrons from the nucleus. The positive charge affecting a specific electron is called the effective nuclear charge, or Z eff.

Shielding Z eff = Z actual – S or Z eff = Z actual – σ Where S or σ is the shielding factor. Both the value of n and l (orbital type) play a significant role in determining the shielding factor.

Slater’s Rules 1. The electronic structure of atoms is written in groupings: (1s)(2s, 2p)(3s, 3p)(3d)(4s, 4p)(4d)(4f ) 2. Electrons in higher groupings do not shield those in lower groups.

Slater’s Rules- Calculation of S 3. For ns or np electrons: a) electrons in the same ns and np as the electron being considered contribute.35, except for 1s, where.30 works better. a) electrons in the same ns and np as the electron being considered contribute.35, except for 1s, where.30 works better. b) electrons in the n-1 group contribute.85 b) electrons in the n-1 group contribute.85 c) electrons in the n-2 group or lower (core electrons) contribute 1.00 c) electrons in the n-2 group or lower (core electrons) contribute 1.00

Slater’s Rules- Calculation of S 4. For nd or nf electrons: a) electrons in the same nd or nf levelas the electron being considered contribute.35 a) electrons in the same nd or nf levelas the electron being considered contribute.35 b) electrons in the groups to the left contribute 1.00 b) electrons in the groups to the left contribute 1.00

Problem: Z eff Use Slater’s rules to estimate the effective nuclear charge of Cl and Mg. Use Slater’s rules to estimate the effective nuclear charge of Cl and Mg.

Periodic Trends Z eff increases across a period. This is due to the addition of protons in the nucleus, accompanied by ineffective shielding for the added electrons. As a result, the valence electrons experience a greater nuclear charge on the right side of the periodic table.

Ionization energy Ionization energy is the energy required to remove an electron from a mole of gaseous atoms or ions. A n+ (g) + energy  A (n+1) (g) + e - Ionization energy increases going across a period, and sometimes decreases slightly going down a group.

Ionization energy

Electron Affinity Electron affinity has several definitions. Originally, it was defined as the energy released when an electron is added to a mole of gaseous atoms or ions. A(g) + e -  A - (g) + energy Under this definition, the elements in the upper right part of the periodic table (O, F) have relatively high (and positive) electron affinities.

Electron Affinity Your text still uses this basic definition, but defines electron affinity as the energy change for the reverse process. A - (g)  A(g) + e - EA = ∆U The values of electron affinity are the same, with positive values for elements that readily accept an additional electron.

Electron Affinity There are no real trends in electron affinity. The affinities of group IA metals are slightly positive, near zero for group IIA, and then increase in groups IIIA and IVA. They drop (but remain positive) for group VA, and then increase through group VIIA. The values are negative for the noble gases.

Atomic Radii The determination of atomic radii is difficult. The method used depends upon the nature of the elemental structure (metallic, diatomic, etc.). As a result, comparisons across the table are not straightforward. In general, size decreases across a period due to the increase in effective nuclear charge, and increases going down a group due to increasing values of n.

Atomic Radii

A close examination of the radii of elements in periods 5 and 6 shows values which defy the trends. Group 4 (4B) Group 5 (5B) Group 11 (1B) Zr = 145 pm Nb = 134 pm Ag = 134 pm Hf = 144 pm Ta = 135 pm Au = 134 pm

Atomic Radii There is a large decrease in atomic size between La (169pm) and Hf (144 pm). This is due to the filling of the f orbitals of the Lanthanide series. As a result, the elements Hf and beyond appear to be unusually small. The decrease in size is called the lanthanide contraction, and is simply due to the way elements are listed on the table.

Ionic Radii Determining the size of ions is problematic. Although crystal structures can be determined by X-ray diffraction, we cannot determine where one ion ends and another begins.

Ionic Radii Cations are always smaller than their neutral atom, since removal of an electron causes an increase in the effective nuclear charge.

Ionic Radii Anions are always larger than their neutral atom, since additional electrons greatly decrease the effective nuclear charge.

Ionic Radii For isoelectronic cations, the more positive the charge, the smaller the ion. For isoelectronic anions, the lower the charge, the smaller the ion.

Ionic Radii Determining ionic radii is extremely difficult. Ionic size varies with ionic charge, coordination number and crystal structure. Past approaches involved assigning a “reasonable” radius to the oxide ion. Calculations based on X-ray data and electron density maps provide results where cations are 14pm larger and anions 14pm smaller than previously found.