Ionization potential and hardness

Robles, J., Bartolotti, L. J. (1984). Electronegativities, electron affinities, ionization potentials, and hardnesses of the elements within spin polarized density functional theory. J.Am. Chem. Soc. 106, 3723-3727. 

The second derivative of the energy with respect to the number of electrons is the hardness r) (the inverse quantity is called the softness), which again may be approximated in term of the ionization potential and electron affinity. 

In this equation, r) the absolute hardness, is one-half the difference between /, the ionization potential, and A, the electron affinity. The softness, a, is the reciprocal of T]. Values of t) for some molecules and ions are given in Table 8.4. Note that the proton, which is involved in all Brdnsted acid-base reactions, is the hardest acid listed, with t — c (it has no ionization potential). The above equation cannot be applied to anions, because electron affinities cannot be measured for them. Instead, the assumption is made that t) for an anion X is the same as that for the radical Other methods are also needed to apply the treatment to polyatomic... 

The value j a is called the absolute hardness and can be obtained as half of the difference between the ionization potential and the electron affinity [129],... 

The alkali metals share many common features, yet differences in size, atomic number, ionization potential, and solvation energy leads to each element maintaining individual chemical characteristics. Among K, Na, and Li compounds, potassium compounds are more ionic and more nucleophilic. Potassium ions form loose or solvent-separated ion pairs with counteranions in polar solvents. Large potassium cations tend to stabilize delocalized (soft) anions in transition states. In contrast, lithium compounds are more covalent, more soluble in nonpolar solvents, usually existing as aggregates (tetramers and hexamers) in the form of tight ion pairs. Small lithium cations stabilize localized (hard) counteranions (see Lithium and lithium compounds). Sodium chemistry is intermediate between that of potassium and lithium (see Sodium and sodium alloys). 

The energy of the HOMO (EHomo) is directly related to the ionization potential and characterizes the susceptibility of the molecule to attack by electrophiles. On the other hand, EHOMO is directly related to the electron affinity and characterizes the susceptibility of the molecule toward attack by nucleophiles. Both the E, IOMO and LUMO energies are important in radical reactions. The concept of hard and soft nucleophiles and electrophiles has... 

Another important postulate, put forward by Pearson [13], is the principle of maximum hardness, according to which a system tends to attain the maximum rigidity. This principle was based also on experimental observations. According to (10), t] increases with increasing of ionization potential and with decreasing of electron affinity. Thus, the system tends neither to render its own electrons, no to get foreign ones, i.e. to remain stable. 

R. G. Pearson has reported a scale of absolute hardness for acids and bases, defining hardness to be one-half the difference between the ionization potential and the electron affinity of an atom or species.21 He has obtained good agreement in many cases between the calculated values for hardness and the experimental behavior of the acid or base. 

In this definition, 7 is the ionization potential and A is the electron affinity. It was noted that a hard molecule will have a large value of ti. For a reaction A + B 5 A B, the following expression can be used to decide which is the acid, A or B 0... 

Mendez and Garcia-Garibay analyzed carbene/alkene charge transfer by density functional theory and ab initio methods. [81] Electrophilicity and nucleo-philicity were assessed by calculating AN, which represented charge transfer between the carbene and alkene cf., Eq. 8. Here, p is the chemical potential taken as -(I+A)/2, where I represents ionization potential, and A represents electron affinity, rj is the hardness of the alkene or carbene, defined as (I-A)/2. 

The HSAB principle has been criticized because of the difficulty in defining the terms hard and soft quantitatively. More recently, Pearson (1983) has attempted to quantify the term hardness by defining hardness operationally as half the difference between ionization potential and electron affinity. 

The absolute hardness of a chemical species has been shown to be a good measure of aromaticity (88TL4843>. As a first approximation, it can be assumed to be equal to half the difference between the ionization potential and its electron affinity <83JA7512>. Alternatively, if the Hartree-Fock or the Hiickel model is used, the absolute hardness can also be represented as half the HOMO-LUMO energy gap <86PNA(83)8440>. 

Other theoretical tests of aromaticity have been studied. The graph theory has predicted thiepine to be antiaromatic <76JA2750>. A new measure of aromaticity has been studied on the basis of absolute hardness >]) known to be an index of stability and reactivity of a molecule. The operation formula is the equation, tj = (/-/l)/2 where / is the ionization potential and A the electron affinity. Alternatively, the equation is rj = (Fhomo- lumo)/2. Absolute hardness has also predicted thiepine (1) and 2-benzothiepine (5) to be antiaromatic and 3-benzothiepine (6) to be nonaromatic. However, the prediction by this approach fails for 1-benzothiepine (2) <89JA737l>. 

The hardness of a species thus can be determined from its ionization potential and electron affinity, just as can the electronegativity [Eq. (2)]. 

Chemical Potential, Hardness, Ionization Potential, and Electron Affinity. 33... 

The purpose of this work is to start from the basic equations of density functional theory to describe the changes in the energy associated with the transition from one ground-state to another, in terms of different sets of variables. In this process one will find the natural definitions of the hardness and softness kernels, the local hardness, the local softness, the global hardness and the global softness [23]. Then, we will proceed to establish their relation with ionization potentials and electron affinities, in order to confirm their behavior as a measure of chemical hardness or softness [14, 24]. Finally, this theoretical framework will be used to analyze the maximum hardness and the HSAB principles. 

This result indicates that the hardness decreases essentially as R (or as N " ") for large cluster size. In the limit of a metallic cluster of infinite size (still with a surface) the hardness becomes zero. It is well known that for a metallic surface the ionization potential and the electron affinity become equal. In this limit one simply speaks of the work function of the metal. 

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