Lewis’ “cubical atom” model. Molecular orbital calculations on the oneelectron molecule H 2 + and the twoelectron molecule H2
Lewis’ “cubical atom” model. Molecular orbital calculations on the oneelectron molecule H 2 + and the twoelectron molecule H2
Abstract and Keywords
This chapter reviews G. N. Lewis' seminal paper of 1916 that introduced the concept of the electronpair bond. The potential energy curves for the two lowest electronic states of the hydrogen molecule ion (H_{2} ^{+}) are described. The molecular orbital (MO) concept is introduced and a set of approximate molecular orbitals formed by linear combination of the 1s atomic orbitals of the two atoms (LCAO MOs). The potential energy curve for a neutral hydrogen molecule in its ground state calculated from a wavefunction consisting of the product of one LCAO MO for each electron is shown to be much higher than the experimental curve for all values of the internuclear distance R. The electron correlation energy is defined. The nonzero experimental dipole moment of hydrogen deuteride (HD) shows that the BornOppenheimer approximation is not completely valid.
Keywords: electronpair bond, hydrogen molecule ion, molecular orbital, LCAO MO, overlap integral, polarization function, hydrogen molecule, electron correlation energy, BornOppenheimer approximation
Introduction
The modern theory of chemical bonding begins with the article “The Atom and the Molecule” published by the American chemist G. N. Lewis in 1916 [1]. In this article, which is still well worth reading, Lewis for the first time associates a single chemical bond with “one pair of electrons held in common by the two atoms.” After a brief review of Lewis’ model we turn to a quantummechanical description of the simplest of all molecules, viz. the hydrogen molecule ion ${\text{H}}_{2}^{+}$. Since this molecule contains only one electron, the Schrödinger equation can be solved exactly once the distance between the nuclei has been fixed. We shall not write down these solutions since they require the use of a rather exotic coordinate system. Instead we shall show how approximate wavefunctions can be written as linear combinations of atomic orbitals of the two atoms. Finally we shall discuss socalled molecular orbital calculations on the simplest twoelectron atom, viz. the hydrogen molecule.
7.1 The electron octet, the “cubical atom,” and the electron pair bond
In 1916 Mendeleev's periodic table of the elements had been known for more than 40 years. The atoms had been shown to contain a minute, positively charged nucleus that contains most of the atomic mass. The relationship between the atomic number of an element and the charge on the nucleus had been established in 1913. The particle called an electron had been characterized, and was known to be a part of atoms and molecules.
The Bohr model of the H atom had been proposed some five years earlier, but Lewis was not impressed: One way “in which a body may hold another is that in which the planets are held by the sun, and this is the way that in some current theories of atomic structure the electrons are supposed to be held by the atom. Such an assumption seems inadequate to explain even the simplest chemical properties of the atom, and I imagine that it has been introduced only for the sake of maintaining the laws of electromagnetics known to be valid at large distances. The fact is, however, that in the more prominent of these theories even this questionable advantage disappears, for the common laws of electricity are not preserved. The most interesting and suggestive of these theories is the one proposed by Bohr and based on Planck's quantum theory. Planck in his elementary oscillator which maintains its motion even at the absolute zero, and Bohr in his electron which is moving in a fixed orbit, have invented systems containing electrons of which the motion produces no effect upon external charges. Now this is not only inconsistent with the accepted laws of electromagnetics, but (p.100) I may add, is logically objectionable, for a state of motion which produces no physical effect whatsoever may better be called a state of rest.”
Lewis then proceeded to build his model of “The Cubical Atom” based on chemical evidence. This evidence may be summarized as follows:

(i) The noble gases are chemically inert.

(ii) The alkali metal cations, the halide anions and the doubly charged oxide anion all have the same number of electrons as the nearest noble gas.

(iii) Out of the “tens of thousands of known compounds” only a handful contain an odd number of electrons, viz. NO, NO_{2}, ClO_{2} and “(C_{6}H_{5})_{3}C as well as other triaryl methyls.”

(iv) The only known molecule of composition CCl_{n} is carbon tetrachloride; CCl, CCl_{2}, CCl_{3} or CCl_{5} do not exist as stable molecules. Similarly the only known molecule of composition NF_{n} was nitrogen trifluoride, of composition OF_{n} oxygen difluoride, and of composition H_{n}F hydrogen fluoride, HF.

(v) Many compounds like the hydrogen halides that do not contain ions when pure, produce ions when dissolved in water.
On the basis of this evidence Lewis suggested that every atom consists of a “kernel which remains essentially unaltered during ordinary chemical changes and which possesses an excess of positive charges” corresponding to 1, 2, 3, 4, 5, 6, 7 or 8 in Groups 1, 2, 13, 14, 15, 16, 17, and18 respectively. Using today's terminology we would equate the kernel with the nucleus and the electrons in inner shells.
“The atom is composed of the kernel and the outer atom or shell, which in the case of the neutral atom, contains negative electrons in equal numbers to the excess of positive charges of the kernel.” Today we refer to the electrons in the outer shell as valence electrons. Lewis suggested the use of the atomic symbol in boldface to represent the kernel of the atom and that valence electrons should be indicated by dots. Figure 7.1 shows the electronic structures of the secondperiod elements from Li to F as suggested by Lewis.
After having constructed his model of the atom, Lewis goes on to describe the formation of chemical bonds between the cubical atoms as due to sharing of edges or square faces: all halogen atoms are represented by a cube with one electron at each of seven corners, all dihalogen molecules by two cubes sharing an edge. See Fig. 7.1. The threedimensional
The two electrons on the shared edge should be counted as valence electrons of both atoms, both of them are thus surrounded by eight valence electrons. This is also the case for the noble gases and presumably represents a particularly stable arrangement.
In general an atom in Groups 14, 15 or 16 would go on sharing edges till all corners were occupied by electrons: “The atom tends to hold an even number of electrons in the [outer] shell, and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube.” Thus O and the other elements in Group 16 would combine with two halogen atoms, N and the other Group 15 elements would share edges with three cubical halogen atoms, C and the other Group 14 elements would share edges with four halogen atoms. Lewis does not discuss any covalent compounds of the elements in Groups 1, 2 or 13, but the electron pair model for the chemical bond and his emphasis on the stability of the electron octet suggest that the elements in these groups would use all their electrons for bond formation.
In the following we shall refer to atoms in the Groups 1, 2, 13 or 14 that form one, two, three or four single bonds respectively, and to atoms in the Groups 15, 16, or 17 that form three, two or one bond respectively, as Lewisvalent. Atoms that form fewer bonds than expected from Lewis’ model will be described as subvalent, and atoms that form more bonds than expected from his model will be described as hypervalent.
The hydrogen atom was a special case, since it tends to hold two (or zero) rather than eight electrons in the valence shell.
Double bonds, as in O_{2}, could be represented by two cubic atoms sharing a face or by an electron dot formula with two electron pairs between the kernels. Triple bonds, as in N_{2}, would require a rearrangement of the cubical to a tetrahedral atom in which pairs of valence electrons occupy positions on the corners of a tetrahedron. The N atom would consist of an atomic kernel surrounded by one electron pair and three single electrons at the four corners of the tetrahedron, and the molecule by two tetrahedral atoms sharing a face or by an electron dot formula with three electron pairs between the kernels.
Lewis’model represented a great step forwards: It identified a single bond with an electron pair shared between two atoms, a double bond with two pairs and a triple bond with three electron pairs. It provided an easy rationalization of the stoichiometry of many chemical compounds of main group elements and of the connectivity of the constituent atoms. The model was wrong, however, in assigning more or less fixed positions to the electrons in the valence shell.
A more complete description of atoms and molecules required the discovery of electron spin, the development of quantum mechanics and the accumulation of information of the geometric structure of molecules made possible by new and improved methods of structure determination. With hindsight, one might modernize Lewis’model and describe the atoms as tetrahedral rather than cubical: The valence electrons would occupy positions at the corners of a tetrahedron, each site may accommodate one or two electrons. If two electrons occupy the same site they must have opposite spin. The electrons are not fixed to one position, but move about in the vicinity of the site. The part of space containing one or two such electrons is sometimes referred to as an electron domain.
(p.102) A single bond would then be described as two atoms sharing a corner, a double bond as two atoms sharing an edge, and the triple bond as two atoms sharing a face (as Lewis indeed suggested for the N_{2} molecule).
Problem 7.1 Use the cubical and tetrahedral models of the atom to suggest molecular structures for OF_{2}, NCl_{3}, CCl_{4} and tetrachloroethene, C_{2}Cl_{4}. Try to reproduce the angular structure of OF_{2}, the pyramidal structure of NCl_{3}, the tetrahedral structure of CCl_{4} and the planar structure of C_{2}Cl_{4} found by experiment.
7.2 Molecular orbitals: the hydrogen molecule ion, ${\text{H}}_{\text{2}}^{\text{+}}$
We shall now use quantum mechanics to describe chemical bonds and begin with the simplest of all molecules, ${\text{H}}_{\text{2}}^{\text{+}}$. The hydrogen molecule ion has not been found in solids or melts, but is easily formed by electric discharge through hydrogen gas. It is also one of the most common molecules in interstellar space. The properties are well known from experimental studies, the equilibrium bond distance is R _{e} = 106.0 pm and the dissociation energy D _{e} = 269 kJ mol^{−1}. Comparison with the H_{2} molecule, R _{e} = 74.1 pm and D _{e} = 455 kJ mol^{−1}, shows that the oneelectron bond in the ion is 43% longer and41% weaker than the twoelectron bond in the neutral molecule.
The importance of the hydrogen molecule ion for the theory of diatomic molecules is similar to the importance of the hydrogen atom for our understanding of atoms: both H and ${\text{H}}_{2}^{+}$ are oneelectron systems for which the Schrödinger equation can be solved exactly. The exact solution of the oneelectron species is then used as a starting point for the discussion of polyelectron species for which exact solutions of the Schrödinger are unavailable.
In our discussion of the ${\text{H}}_{2}^{+}$ ion we shall use several coordinate systems, sometimes we shall even mix different coordinate systems in the same expression. We begin by defining a righthanded Cartesian coordinate system with the zaxis running through both nuclei and the origin at the midpoint between them. See Fig. 7.2. We shall also use Cartesian coordinate systems with their origins at one of the nuclei, A or B. The z _{A} and z _{B} axes coincide with the
The polar coordinates r_{A} and r_{B} are given by
In Section 4.2 we defined the “electronic energy” of the molecule as the sum of the kinetic energy of the electron and the Coulomb energies due to attraction between the electron and each of the two nuclei and due repulsion between the nuclei:
Like the Schrödinger equation of the H atom, the Schrödinger equation of the hydrogen molecular ion has infinitely many solutions for each value of R. We shall, however, only discuss the ground state and the first excitedstate. The electronic energies of these states as functions of R are shown in Fig. 7.3.
When the two nuclei are far apart, the two states are degenerate, i.e. they have the same energy. This energy is equal to −1R = −1313 kJ mol^{−1}, which is equal to the energy of a hydrogen atom at a large distance from a proton. When the distance between the nuclei is reduced, the degeneracy is lifted. The energy associated with the electronic state Ψ _{b} decreases, reaches a minimum at R _{e} = 106.0 pm, and increases again when the internuclear distance is further reduced. See Fig. 7.3. The probability density calculated from the wavefunction Ψ _{b} is particularly high in the regions around and between the two nuclei, i.e. in those regions of space where the Coulomb energy of the electron is low. A wavefunction that describes one electron moving in the space around and between two or more nuclei is calleda molecular orbital, MO. An MO which describes an electron which moves in the space between and around two nuclei, is described as a twocenter, or 2c, MO. (p.104)
The energy associated with the second wavefunction Ψ _{a} increases monotonically as the internuclear distance is decreased. This means that if we somehow have succeeded in bringing the two nuclei close together, they will spontaneously move apart and accelerate as they do so. The Ψ _{a} state is therefore described as repulsive or antibonding. The probability density calculated from Ψ _{a} has maxima at the two nuclei, but the density between them is smaller than calculated by supposition of two half hydrogen atoms with the electrons in 1s orbitals. This means that an electron in Ψ _{a} is less likely to be found in the region of space where the Coulomb energy is low.
We begin our discussion of the wavefunctions by noting that when the nuclei are infinitely far apart, they are given by
7.3 Approximate molecular orbitals obtained by linear combinations of atomic orbitals
We have seen that when the distance between the two nuclei is infinite, the ground state of the ${\text{H}}_{2}^{+}$ ion is doubly degenerate and that the two wavefunctions are exactly equal to the sum or the difference of the two 1s atomic orbitals.
The shape of the exact wavefunction, Ψ _{b} (x, y, z), for the ground state of ${\text{H}}_{2}^{+}$ when R is approximately equal to the equilibrium bond distance, suggests that it may be approximated by the sum of the 1s atomic orbitals of the two atoms:
In order to find out how the wavefunction might be improved, we calculate the probability density for comparison with that obtained from the exact solution Ψ _{B}.
The probability density calculated for our approximate wavefunction is
The probability density of equation (7.11) thus consists of three parts. The first is a cloud with the shape of the 1s probability density around nucleus A, and the third corresponds to the 1s density around nucleus B, both are reduced by the factor ${\text{k}}_{\text{+}}^{\text{2}}$. The second component of the probability density has its maximum value for all points on the line connecting the two nuclei, where r _{A} + r _{B} = R. It is equal for all points where r _{A} + r _{B} are equal, i.e. for points on ellipsoids with the nuclei A and B in the focal points. This component is sometimes referred to as the overlap density or overlap cloud.
The variation of the densities of the three components along a line connecting the nuclei is shown in Fig. 7.4.
Problem 7.2 Sketch constant probability density contours of the three components in a plane containing the two nuclei.
We now turn our attention to the antibonding state. Examination of the exact solution Ψ _{a} suggests that it may be approximated by
Problem 7.3 Show that the antibonding MO is normalized if
The probability density calculated for an electron in Ψ _{−} is
The probability density in (7.16) consists of three parts. The first is a cloud of the shape of a1s electron cloud around nucleus A reduced by the factor k ^{2} _{−}. The third term corresponds to a 1s electron density around nucleus B reduced by the same factor. The second term in (7.16) has its maximum absolute value at the line connecting the two nuclei, and has a constant value at points on an ellipsoid with the two nuclei in the focal points. This term is, however, to be subtracted from the two others. The effect is to reduce the electron density between the two nuclei, i.e. in the region where the potential energy of the electron is particularly low.
Ψ _{+} and Ψ _{−} represent first approximations to the true wavefunctions of the ground state and the first excited state of ${\text{H}}_{2}^{+}$, Ψ B and Ψ _{A} respectively. Molecular orbitals that have been formed in this manner, are referred to as LCAO (linear combination of atomic orbitals) MOs. The AOs usedare referred to as the basis (for the calculation).
7.4 Improvement of the LCAO MO
As we have seen the MO calculations of the last paragraph yieldeda bond distance which was 25% too long and a dissociation energy which was 37% too low. How could the accuracy be improved?
Comparison of the probability densities calculated for the approximate wavefunction Ψ _{+} with those calculated from the exact wavefunction Ψ _{b} shows that the probability density in the lowenergy region near and in between the two nuclei is underestimated by some 40%. No wonder then that the calculations fail to reproduce the true strength of the bond!
The simplest way to increase the probability density near the nuclei is to modify the 1s orbitals:
When the two nuclei are far apart, the best value for ζ is obviously equal to 1.0. In the other limiting situation, i.e. when the internuclear distance approaches zero, the ${\text{H}}_{\text{2}}^{\text{+}}$ molecule is (p.109) transformed into a He^{+} ion, and then the best value for ζ = 2: the best value for ζ then, has to be determined for each value of R.
The calculations are carried out as outlined in the preceding section. Thus E is calculated from equation (7.10):
The next step in improvement of the calculations is to write the MO orbital as a linear combination of four atomic orbitals, one 1s type function and one 2p _{z} type function (since the coordinate system is defined in such a way that the two nuclei are lying on the zaxis) for each atom [3]. The ζ parameters of the 1s and 2p_{z}type AOs are adjusted separately. Such calculations reduce the discrepancy between experimental and calculated dissociation energies to 5 kJ mol^{−1}.
The best wavefunction obtained for R = R _{e} = 106 pm may be written on the form
What does this result mean? Are electrons in bonding molecular orbitals found in atomic orbitals outside the valence shell? Most theoretical chemists would answer no, the shape is that of a 2p_{z} orbital, but the size of the orbital is much smaller than the 2p_{z} orbital of an isolated H atom: the form of the orbitals after adjustment of ζ , is given by
7.5 The hydrogen molecule and the molecular orbital approximation
Since the hydrogen molecule ion ${\text{H}}_{\text{2}}^{\text{+}}$ contains only one electron, the Schrödinger equation can be solved exactly (once we have accepted the Born–Oppenheimer approximation). In the (p.110) preceding section we have shown how approximate solutions may be obtained by linear combinations of atomic orbitals on each atom. In this context, the term “atomic orbitals” should not be taken to mean “solutions of the Schrödinger equation for the atom” or the “best possible AOs for the atom.” The term only implies that they resemble the AOs for the hydrogenlike atoms, in particular that the angular functions correspond to H atom s, p,or d orbitals. The LCAOMO functions of ${\text{H}}_{2}^{+}$ can be brought into very good agreement with the exact solution, the discrepancy can in fact be reduced below any limit by using a sufficiently large number of basis functions. We now turn our attention to the simplest twoelectron molecule, viz. the hydrogen molecule.
The electronic energy of an H_{2} molecule contains two terms representing the kinetic energy of the electrons, four terms representing the energies due to Coulomb attraction between two electrons and two nuclei, one term representing the energy due to repulsion between the two electrons and a last term representing the energy due to repulsion between the two nuclei.
Problem 7.4 Place the H_{2} molecule in a Cartesian coordinate system in such a manner that the origin is at the midpoint of the H–H bond and the zaxis runs through both nuclei. Write down an expression for the potential energy of the molecule. Which terms depend on the internuclear distance R? What state corresponds to zero energy? Write down the Hamiltonian operator ℋ.
The hydrogen molecule, like the He atom, poses a real problem: the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrödinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the ${\text{H}}_{2}^{+}$ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals
As a first approximation for the ground state of H_{2} we use MOs of the form
When discussing the He atom, we stressed that a wavefunction written as the product of two atomic orbitals is inherently wrong since it fails to reflect the fact that repulsion between the electrons tends to keep them far apart. The energy calculated from such a wavefunction will therefore be higher than that of the real atom. What was true for the twoelectron atom, is equally true for a twoelectron molecule. Like atomic orbital calculations, molecular orbital calculations on molecules containing two or more electrons are inherently wrong.
The electronic energy associated with the wavefunction given by (7.18) and (7.19) may be calculated from
The electronic energy thus obtained has a minimum at R _{min} = 85 pm (as compared to an experimental bond distance of 74.1 pm) and an energy
Combination of equations (7.21), (7.22), and (7.23) yields an estimated dissociation energy of
(p.112) The estimated dissociation energy is much improved if we combine the calculated energy at the minimum of the potential curve with the sum of the energies obtained by calculations on the two atoms one at a time:
One way to improve the result is obviously to modify the 1slike basis functions to include an adjustable ζ parameter as indicated in (7.17): When the energy is calculated and ζ varied to give the lowest energy for each R, one obtains an electronic energy curve with a minimum at 73 pm (1 pm shorter than the experimental bond distance) and a dissociation energy of 335 kJ mol^{−1}.
Somewhat better agreement between calculated and experimental bond dissociation energies are obtained by expanding the basis, for instance by including 2p_{z} type functions. However, the highest and best estimate for the dissociation energy that can be obtained under the molecular orbital approximation is 351 kJ mol^{−1}, which still represents an error of 23%. This error is inherent in the form of the wavefunction (7.18). The difference between the experimental dissociation energy and the best value that can be obtained under the molecular orbital approximation is referred to as the “electron correlation energy.”
Molecular orbital calculations on molecules with three or more electrons are based on Slaterdeterminant wavefunctions. If the basis sets are sufficiently large, such Hartree–Fock or HF calculations are generally found to reproduce experimental bond distances to within 3 or 4 pm, and experimental valence angles to within 3 or 4 degrees. The dissociation energies obtained by such calculations are, however, too inaccurate to be useful.
Algorithms and programs that allow calculation of the correlation energy – or at least a part of it – have been developed. We shall refer to such calculations in Section 20.6.
7.6 The electric dipole moment of HD: failure of the Born–Oppenheimer (adiabatic) approximation
If we accept the Born–Oppenheimer approximation, the electron distribution in the hydrogen molecule or the isotopomer HD would always be symmetrical, in the sense that the center of gravity of negative charge would fall at the midpoint of the line between the nuclei. The center of positive and negative charges would thus coincide, and the electric dipole moment would be identically equal to zero. Careful investigation by rotational spectroscopy shows, however, that HD has a dipole moment of 0.0006 D [4].
This observation can only be explained as a failure of the Born–Oppenheimer approximation. Instead of assuming that the shape of the electron cloud always is perfectly adjusted to the instantaneous internuclear distance, let us go to the opposite extreme and assume that the electron cloud remains unchanged while the nuclei vibrate. See Fig. 7.6. During the vibration the center of gravity of the molecule must remain at rest. This means that if at one instant the proton has moveda certain distance from the equilibrium position in such a direction that R has increased, the deuteron has moved half as far in the opposite direction, (p.113)
What will be the situation when, a little bit later, the proton is moving towards the deuteron? When the proton has moved a certain distance from the equilibrium position, the deuteron will have moved only half as far. Again the molecule will have an instantaneous dipole moment, but the positive pole will now be at the deuteron.
The electron dipole moment that is measured by rotational spectroscopy represents an average over time as the molecule vibrates, the socalled”permanent” dipole moment. From what we have said, one would expect the instantaneous dipole moments of HD to average to zero during the vibrational cycle, but for some unknown reason, perhaps connected with anharmonicity, it does not. In any case, the observation of a “permanent” dipole moment of HD shows that the Born–Oppenheimer approximation has broken down.
References
Bibliography references:
[1] G. N. Lewis, J. Am. Chem. Soc. 38 (1916) 762.
[2] D. R. Bates, K. Ledsham, A. L. Stewart, Phil. Trans. Roy. Soc. London, 246 (1953) 215.
[3] B. N. Dickinson, J. Chem. Phys., 1 (1933) 317.
[4] M. Trefler, H. P. Gush, Phys. Rev. Lett., 20 (1968) 703. (p.114)