Ruedenberg's approach to bonding in H₂⁺ is applied to the two-electron covalent bond in H₂. A simplified analysis yields the same conclusion as Ruedenberg's more rigorous treatment; kinetic energy plays a crucial role in chemical bond formation.

The covalent chemical bond is a difficult concept that is frequently oversimplified as a purely electrostatic phenomenon in textbooks at all levels of the undergraduate chemistry curriculum. Therefore the purpose of this paper is to provide an elementary quantum mechanical analysis of the covalent bond appropriate for an undergraduate course in physical or advanced inorganic chemistry. It is important to emphasize that there is no acceptable classical, electrostatic explanation for the covalent bond, just as there is no classical explanation for atomic stability or atomic structure. Quantum mechanical principles are required to understand atomic and molecular stability and structure.

The Covalent Bond

Forty years ago Ruedenberg and his collaborators undertook a detailed study of the covalent bond in H₂⁺ (1-3). This incisive theoretical analysis revealed that chemical bond formation was not simply an electrostatic phenomenon as commonly thought, but that electron kinetic energy also played an essential role. Ruedenberg's contribution to our current understanding of the physical nature of the chemical bond has been discussed in a number of publications in the pedagogical literature in chemistry and physics (4-9). In addition there are excellent critiques and summaries elsewhere that are accessible to the interested non-specialist (10-12). It should be noted that Slater also recognized the importance of electron kinetic energy in chemical bond formation in a benchmark paper published in the inaugural volume of the Journal of Chemical Physics (13). He returned to the subject later (14), but never pursued it at the depth that Ruedenberg and his colleagues did.

Ruedenberg chose to study H₂⁺ because, as the simplest molecule, he could easily extract detailed information about all the contributions to the total energy from its one-electron wave function. In this study the simplest electron pair bond, H₂, will be examined. The analysis is carried out at a much more elementary level, but the same message emerges - electron kinetic energy plays a crucial role in chemical bond formation.

Theoretical analysis shows that H-H bond formation, 2H ® H₂, is an exoergic process that obeys the virial theorem: DE = DV/2 = - DT. Uncritical use of the virial theorem, therefore, may lead one to believe that stable bond formation is solely an electrostatic potential energy effect, and that a consideration of kinetic energy is neither relevant nor necessary. However, H-H bond formation can be thought of as a very simple chemical reaction, and we know that it is never justified to assume that the balanced chemical equation is also the mechanism for the reaction. For example, even a simple first-order isomerization reaction (R ® P) requires the formation of an activated form of the reactant (R ® R* ® P).

Similarly, to study the covalent bond it is instructive to postulate a "mechanism" for bond formation; a sequence of conceptual steps that are equivalent to the overall process - 2H ® H₂. Unlike a traditional chemical mechanism it can't be tested empirically and, therefore, its value or validity rests on the clarity and cogency of its basic premises, and the light it throws bond formation.

There are actually several plausible mechanisms, but our attention will be restricted to one that might appear especially cogent to undergraduate audiences. For example, when we describe the bonding in methane to undergraduates we generally invoke a mechanism that uses the concepts of atomic promotion [2s²2p² ® 2s¹2p³], hybridization [2s¹2p³ ® (sp³ hybrids)⁴] , and bond formation through the overlap of atomic orbitals. The H₂ bond formation mechanism described in this paper will consist of two steps: atomic promotion and overlap of atomic orbitals (2H ® 2H* ® H₂). This simple mechanism has previously been used to analyze the bonding in H₂⁺(12).

To carry out a quantum mechanical analysis of bond formation in H₂ it is necessary to decide at what level of theory to work. In this analysis scaled hydrogenic 1s orbitals will be used for the atomic orbitals. At the molecular level, both molecular orbital (MO) and valence bond (VB) wave functions will be considered. Labeling the nuclei a and b, and the electrons 1 and 2, we have for the atomic orbitals are,

1s_a = (a³/p)^1/2exp (-a r_a)     and     1s_b = (a³/p)^1/2exp(-a r_b)

Using this basis set the MO and VB wave functions are (neglecting spin),

Y_MO = N_MO [1s_a(1) + 1s_b(1)][1s_a(2) + 1s_b(2)]

Y_VB = N_VB [1s_a(1)1s_b(2) + 1s_a(2)1s_b(1)]

In these equations a is a variational parameter, and N_MO and N_VB are the appropriate normalization constants.

The Hydrogen Atom

Using a scaled hydrogenic wave function in a variational calculation for the hydrogen atom yields the following expression for the energy in atomic units,

E _H = a²/2 - a

where the first term is the electron kinetic energy, and the second term is the electron-nucleus potential energy. Minimization of E_H with respect to a yields a = 1, and the following expression for the energy,

E_H = < T > + < V > = 0.5 - 1.0 = -0.5

The Hydrogen Molecule

The results for the variational calculations on the hydrogen molecule using the MO and VB wave functions are presented in Table 1, providing the optimum value of a for each wave function and the total energy as found in the literature (15). The kinetic and potential energy contributions are obtained using the virial theorem. The experimental value for the ground state energy is -1.1746 E_h. (1 E_h = 27.2114 eV = 2.6255 MJ/mol)

For both wave functions the bond formation mechanism is the same. In the first step, atomic promotion, the hydrogen atom orbitals prepare for bonding by contracting from a = 1 to the optimum a value of the final molecular wave function. This step is atomic and endoergic, increasing the kinetic energy more than it decreases the potential energy. The potential energy decreases because the electrons move closer on average to their respective nuclei. The kinetic energy increases because of the greater confinement of the electrons in the contracted orbitals - kinetic energy is inversely proportional to the square of the average distance of the electron from the nucleus.

The second step consists of the formation of a molecular wave function by overlap of the promoted atomic orbitals. The constructive interference that accompanies atomic orbital overlap brings about charge delocalization and charge redistribution. Charge delocalization distributes the electron density over the molecule as a whole (each electron now belongs to both nuclei) and brings about a decrease in kinetic energy. Charge redistribution moves some electron density from the neighborhood of the nucleus into the internuclear region and involves an increase in potential energy. The second step is exoergic because the kinetic energy decreases more than the potential energy increases - charge delocalization funds the redistribution of charge from the area around the nuclei into the bond region. The results for both wave functions are summarized in the accompanying tables.

Y_MO = N_MO [1s_a(1) + 1s_b(1)][1s_a(2) + 1s_b(2)]

	Atomic State		Promoted Atomic State		Molecular State
	2H(a = 1)	D	2H(a = 1.197)a	D	H2(a = 1.197)
T/Eh	+1.00	+0.4328	+1.4328	-0.3053	+1.1275
V/Eh	-2.00	-0.3940	-2.3940	+0.1390	-2.2550
E/Eh	-1.00	+0.0388	-0.9612	-0.1663	-1.1275

^aThe entries in this column are calculated using E _H = a ²/2 - a.

In summary this simple two-step mechanism for the two-electron bond in H₂ clearly shows that covalent bond formation is driven by a decrease in kinetic energy brought about by the charge delocalization that accompanies the overlap of atomic orbitals. This is also the basic conclusion of Ruedenberg's more detailed and sophisticated analysis for H₂⁺.

Literature Cited

1. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326-352.
2. Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 54, 1495-1511. Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 55, 5804-5818.
3. Ruedenberg, K. In Localization and Delocalization in Quantum Chemistry; Chalvet, O. et al., Eds.; Reidel: Dordrecht, The Netherlands, 1975; Vol. I, pp 223-245.
4. Baird, N. C. J. Chem. Educ. 1986, 63, 660-664.
5. Harcourt, R. D. Am. J. Phys. 1988, 56, 660-661.
6. Nordholm, S. J. Chem. Educ. 1988, 65, 581-584.
7. Bacskay, G. G.; Reimers, J. R.; Nordholm, S. J. Chem. Educ. 1997, 74, 1494-1502.
8. Rioux, F. Chem. Educator 1997, 2(6), 1-14.
9. Harcourt, R. D.; Solomon, H.; Beckworth. J. Am. J. Phys. 1982, 50, 557-559.
10. Kutzelnigg, W. Angew. Chem. Int. Ed. Eng. 1973, 12, 546-562.
11. Melrose, M. P.; CHauhan, M.; Kahn, F. Theor. Chim. Acta 1994, 88, 311-324.
12. Gordon, M. S.; Jensen, J. H. Theor. Chem. Acc. 2000, 103, 248-251.
13. Slater, J. C. J. Chem. Phys. 1933, 1, 687-691.
14. Slater, J. C. Quantum Theory of Matter, Krieger Publishing: Untington, NY, 1977; pp 405-408.
15. Linnett, J. W. Wave Mechanics and Valency, Methuen & Co. Ltd.: London 1960; pp 97-100.

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