Valence-Bond Theory of the Hydrogen Molecule
The first successful explanation of chemical bonding using quantum mechanics was provided by the simple computation of Heitler and London on the hydrogen molecule in 1927, only one year after the Schrödinger equation was proposed. This gave the first rational explanation of the chemical concept of the covalent electron-pair bond, proposed by G. N. Lewis in 1916 and Irving Langmuir in 1919. The hydrogen molecule has a binding energy of (not including the zero-point vibrational energy). This represents the minimum of the Born–Oppenheimer potential curve for at an internuclear separation of (0.7414Å). This is shown as a thick black curve representing the ground electronic state. As , the molecule dissociates into two hydrogen atoms in their ground states, with antiparallel spins. Also shown, in red, is the antibonding potential curve for two hydrogen atoms with parallel spins.[more]
The Hamiltonian for the molecule, within the Born–Oppenheimer approximation, consists of the kinetic energies of the two electrons added to six Coulombic potential-energy contributions. In atomic units ():
where is the distance between electron 1 and proton , and so fourth, is the instantaneous distance between the two electrons, and is the internuclear separation. Since the hydrogen molecule is formed from a combination of hydrogen atoms and , one might consider as a primitive first approximation a product of hydrogen-atom functions centered on protons and , respectively: . This gives a binding energy at a nuclear separation of , indicating that the hydrogen atoms can indeed form a stable molecule. However, the calculated energy is over an order of magnitude too small to account for the strongly-bound hydrogen molecule.
Heisenberg had suggested that a wavefunction should be symmetrical (or antisymmetrical) with respect to interchange of its electron labels, to take account of the indistinguishability of identical particles. Heitler and London  took this into account by adding a term to the wavefunction in which the electron labels are reversed. The approximate wavefunction with appropriate exchange symmetry can be written
The function is normalized, with introduction of the overlap integral . The plus and minus signs apply to the bonding and repulsive states, respectively. The computed result (augmented by an integral evaluated by Sugiura ) gave a much more realistic binding energy value of 3.156 eV, with .
A further improvement was implemented by Wang , using scaled functions of the form , with treated as a variational parameter chosen so as to minimize the molecular energy (rather than keeping , as in the hydrogen atom). The variational energy can be expressed , with and . The minimum is obtained at with , giving a binding energy of . With selection of the setter "Wang", the graphic shows solid blue curves for varying in the vicinity of the minimum in the potential curve. The energy values for a given are not significant for any larger range of . Also shown, as a dashed blue curve, is the original Heitler–London computation, with throughout.
The last enhancement we consider explicitly is the inclusion of ionic-covalent resonance. Linus Pauling recognized that the "true" structure of a molecule such as was actually a "resonance hybrid", an admixture of some ionic character and into the purely covalent structure . Weinbaum  suggested a variational function of the form , to be optimized with respect to the two parameters and . You can generate the curve shown in green by varying these parameters. The optimum values are , , yielding a binding energy of 4.024 eV at . This value implies that inclusion of about 6% ionic structure optimizes the computation based on this function.
Further small improvements, which we do not enumerate, were obtained by consideration of "polarization" of each atomic function in the direction of the opposite atom by addition of some character.
The plots shown can be magnified, to focus in on the region around the minimum.
The definitive computation on the hydrogen molecule, and in a sense an affirming verification of the validity of the many-particle Schrödinger equation, was the work of James and Coolidge . They used a 13-term linear variational method based on prolate spheroidal (also known as confocal elliptical) coordinates to obtain the values , , within the known experimental values at the time. (This is a possible subject of a future Demonstration.) More recently, with the advent of high-powered computational capability, even more accurate results have been obtained, including a 100-parameter extension of Coolidge and James and corrections for the Born-Oppenheimer approximation and relativistic contributions.[less]
Snapshot 1: primitive Heitler–London function
Snapshot 2: optimized Wang function:
Snapshot 3: optimized Weinbaum function: ,
 S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004 pp. 139–141.
 L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, New York: McGraw–Hill, 1935 pp. 340–353.
Mainly for historical perspective, we also cite the original papers:
 W. Heitler and F. London, "Wechselwirkung Neutraler Atome und Homöopolare Bindung Nach der Quantenmechanik," Zeitschrift für Physik, 44, 1927 pp. 455–472.
 Y. Sugiura, "Über die Eigenschaften des Wasserstoffmoleküls im Grundzustande," Zeitschrift für Physik, 45, 1927 pp. 484–492.
 S. C. Wang, "The Problem of the Normal Hydrogen Molecule in the New Quantum Mechanics," Physical Review, 31(4), 1928 pp. 579–586.
 S. Weinbaum, "The Normal State of the Hydrogen Molecule," Journal of Chemical Physics, 1(8), 1933 pp. 593–597.
 H. M. James and A. S. Coolidge, "The Ground State of the Hydrogen Molecule," Journal of Chemical Physics, 1(12), 1933 pp. 825-835.