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Molecules and ModelsThe molecular structures of main group element compounds$

Arne Haaland

Print publication date: 2008

Print ISBN-13: 9780199235353

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199235353.001.0001

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Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule H 2 + and the two-electron molecule H2

Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule H 2 + and the two-electron molecule H2

Chapter:
(p.99) Chapter 7 Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule H 2 + and the two-electron molecule H2
Source:
Molecules and Models
Author(s):

Arne Haaland

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199235353.003.0007

Abstract and Keywords

This chapter reviews G. N. Lewis' seminal paper of 1916 that introduced the concept of the electron-pair bond. The potential energy curves for the two lowest electronic states of the hydrogen molecule ion (H2 +) 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 non-zero experimental dipole moment of hydrogen deuteride (HD) shows that the Born-Oppenheimer approximation is not completely valid.

Keywords:   electron-pair bond, hydrogen molecule ion, molecular orbital, LCAO MO, overlap integral, polarization function, hydrogen molecule, electron correlation energy, Born-Oppenheimer 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 quantum-mechanical description of the simplest of all molecules, viz. the hydrogen molecule ion 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 so-called molecular orbital calculations on the simplest two-electron 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:

  1. (i) The noble gases are chemically inert.

  2. (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.

  3. (iii) Out of the “tens of thousands of known compounds” only a handful contain an odd number of electrons, viz. NO, NO2, ClO2 and “(C6H5)3C as well as other tri-aryl methyls.”

  4. (iv) The only known molecule of composition CCln is carbon tetrachloride; CCl, CCl2, CCl3 or CCl5 do not exist as stable molecules. Similarly the only known molecule of composition NFn was nitrogen trifluoride, of composition OFn oxygen difluoride, and of composition HnF hydrogen fluoride, HF.

  5. (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 second-period 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 three-dimensional

                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.1. Above: the electronic structures of the second-period elements from Li to F according to Lewis [1]. Below: the electronic structure of the iodine molecule.

(p.101) figure of the two cubes might also be represented by a formula in which the atomic kernels were represented by the atomic symbol in boldface, all valence electrons are indicated by dots, and the two electrons on the shared edge are drawn between the symbols for the kernels.

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 Lewis-valent. 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 O2, 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 N2, 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 N2 molecule).

Problem 7.1 Use the cubical and tetrahedral models of the atom to suggest molecular structures for OF2, NCl3, CCl4 and tetrachloroethene, C2Cl4. Try to reproduce the angular structure of OF2, the pyramidal structure of NCl3, the tetrahedral structure of CCl4 and the planar structure of C2Cl4 found by experiment.

7.2 Molecular orbitals: the hydrogen molecule ion, H 2 +

We shall now use quantum mechanics to describe chemical bonds and begin with the simplest of all molecules, H 2 + . 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 H2 molecule, R e = 74.1 pm and D e = 455 kJ mol−1, shows that the one-electron bond in the ion is 43% longer and41% weaker than the two-electron 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 H 2 + are one-electron systems for which the Schrödinger equation can be solved exactly. The exact solution of the one-electron 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 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 right-handed Cartesian coordinate system with the z-axis 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

                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.2. Polar and Cartesian coordinate systems used to describe the position of an electron in the hydrogen molecule ion. The stippled lines indicate a plane containing the z-axis and the electron.

(p.103) original z-axis, the x A and x B axes are parallel to the original x axis while the y A and y B axes are parallel to the original y axis. Alternatively we might use polar coordinate systems with the origin at the bond midpoint (r, θ and φ p), or with the origin at one of the two nuclei A or B. See Fig. 7.2.

The polar coordinates rA and rB are given by

r A = x 2 + y 2 + ( z + R / 2 ) 2 and r = x 2 + y 2 + ( z R / 2 ) 2
where R is the internuclear distance. Note that the collinearity of the three z-axes and the parallelism of the three x- or y-axes imply that the three azimutal angles are equal: φ A = φ B = φ.

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:

E = 1 2 m e v 2 + e 2 4 π ε 0 ( 1 r A 1 r B + 1 R )
This classical expression for the energy may be transformed into the Hamiltonian operator of the system:
(7.1)
H = h 2 8 π 2 m e 2 + e 2 4 π ε 0 ( 1 r A 1 r B + 1 R )
According to the Born–Oppenheimer approximation the electronic energy and the wave-function of the electron can be determined for any instantaneous value of R by solving the Schrödinger equation
H Ψ ( x , y , z ) = E Ψ ( x , y , z )
Since the magnitude of the last three terms in the Hamiltonian operator depend on the positions of the nuclei, i.e. on R, so will the energy: E(R), and the wavefunction: Ψ(x, y, z; R).

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 two-center, or 2c, MO. (p.104)

                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.3. The electronic energies of the two lowest states of the H 2 + ion as a function of the internuclear distance R : ψb (b for bonding) and ψa (a for antibonding). Based on data from refrence [2].

Since Ψ b describes a molecule which has lower energy than the separated atoms (H and H+), it is described as a bonding MO, hence the subscript: Ψ b.

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

(7.2)
Ψ b ( x , y , z ; R = ) = [ Φ A ( x , y , z ) + Φ B ( x , y , z ) ] / 2 Ψ a ( x , y , z ; R = ) = [ Φ A ( x , y , z ) Φ B ( x , y , z ) ] / 2
where Φ A (x, y, z) and Φ B (x, y, z) are given by
(7.3)
Φ Α ( x , y , z ) = ( π a 0 3 ) 1 / 2 exp ( r A / a 0 ) Φ Β ( x , y , z ) = ( π a 0 3 ) 1 / 2 exp ( r B / a 0 )
Note that Φ A and Φ B are identical to the wavefunctions of a 1s electron on nucleus A or B respectively. (p.105) The probability densities of the two states are given by
(7.4)
Ψ 2 ( x , y , z ; R = ) = ( 1 2 ) ( π a 0 3 ) 1 [ exp ( 2 r A / a 0 ± 2 exp [ ( r A + r B / a 0 ) ] + exp ( 2 r B / a 0 ) ]
where the plus sign in front of the second term in the brackets pertains to Ψ B and the minus sign to Ψ A. Since the nuclei are infinitely far apart, (r A + r B) will be infinite for all values of x, y and z. The second term in the brackets vanishes, and the probability density of both states is given by
(7.5)
Ψ 2 ( x , y , z ; R = ) = ( 1 / 2 ) ( π a 0 3 ) 1 [ exp ( 2 r A / a 0 ) + exp ( 2 r B / a 0 ) ]
The electron density thus calculated is equal to half the probability density of a 1s electron on nucleus A plus half the probability density of a 1s electron on nucleus B: the electron occupies a 1s orbital at one of the nuclei, but we do not know which one!

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 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 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:

(7.6)
Ψ + ( x , y , z ) = k + [ Φ Α ( x . y , z ) + Φ Β ( x , y , z ) ]
The constant k + is fixed by requiring that the MO is normalized:
Ψ + 2 ( x , y , z ) d τ = k + 2 [ Φ Α ( x , y , z ) + Φ Β ( x , y , z z ) ] 2 d τ = k + 2 [ Φ Α 2 ( x , y , z ) d τ + 2 Φ Α ( x , y , z ) Φ Β ( x , y , z ) d τ + Φ Β 2 ( x , y , z ) d τ ] = 1
If the AOs have been normalized, the first and the third integrals are both equal to unity. The second integral, SAB = ∫ Φ A (x, y, z)Φ B (x, y, z)dτ , is referred to as the overlap integral. The overlap integral is a dimensionless number. If the AOs have been normalized, S will have a value between zero and one depending on the distance between the nuclei.
Ψ + 2 ( x , y , z ) d τ = k + 2 ( 2 + 2 S AB ) = 1
or
(7.7)
k + = 1 2 + 2 S AB
(p.106) The energy corresponding to such an approximate solution of the Schrödinger equation may then be calculated from
(7.8)
E + = Ψ + ( x , y , z ) Ψ + ( x , y , z ) d τ
where ℋ is the Hamiltonian operator given by equation (7.1). We now introduce the four integrals
(7.9)
H AA = Φ A ( x , y , z ) Φ A ( x , y , z ) d τ and H BB = Φ B ( x , y , z ) Φ B ( x , y , z ) d τ H AB = Φ A ( x , y , z ) Φ B ( x , y , z ) d τ and H BA = Φ B ( x , y , z ) Φ A ( x , y , z ) d τ
All these integrals have the dimension energy. Their magnitude depends on the internuclear distance R, but due to the two negative terms in the Hamiltonian operator representing the energy of attraction between the electron and the two nuclei, they will always be negative. Inserting equations (7.6) in (7.8) and using (7.7) and (7.9) we obtain:
E + = H AA + H AB + H BA + H BB 2 + 2 S AB
or, accepting that H AA is equal to H BB, and that H AB is equal to H BA:
(7.10)
E + = H AA + H AB 1 + S AB
The energy corresponding to the approximate wavefunction Ψ + for any particular inter-nuclear distance may be calculated from (7.10) and plotted as a function of R. The resulting potential energy curve has the same general shape as the exact curve in Fig. 7.3. The asymptotic value for large internuclear distances is E = −1313 kJ mol−1, corresponding to a hydrogen atom at a large distance from a proton. The minimum in the potential energy curve is found for Re = 132 pm; the corresponding energy is −1483 kJ mol−1. The dissociation energy is calculated as
D e = E + ( R = ) E + ( R = R e ) = 170 kJ mol 1
Comparison with the experimental values, R e = 106.0 pm and D e = 269.3 kJ mol−1shows that these calculations overestimate the bond distance by 25% and underestimate the dissociation energy by 37%. The wavefunction Ψ + is clearly not very good!

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

(7.11)
Ψ + 2 ( x , y , z ) = k + 2 [ Φ A 2 ( x , y , z ) + 2 Φ A ( x , y , z ) Φ B ( x , y , z ) + Φ B 2 ( x , y , z ) ]
Since Φ A and Φ B are 1s AOs at atoms A and B respectively, see equation (7.3):
(7.12)
Ψ + 2 ( x , y , z ) ) = C + 2 [ exp ( 2 r A / a 0 ) + 2 exp [ ( r A + r B ) / a 0 ] + exp ( 2 r B / a 0 ) ]
(p.107)
                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.4. The variation of the probability density along the z-axis in H 2 + in the ground state calculated from the approximate wavefunction Ψ +, equation (7.12): k 2 + Φ A 2 ( r A ) , k + 2 Φ B 2 ( r B ) , 2 k + 2 Φ Α ( r A ) Φ B ( r B ) , , and the total probability density Ψ + 2 .

Here the constant ( π a 0 3 ) 1 , and the squared normalization constant of the MO, k + 2 = 1 / ( 2 + 2 S AB ) .

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 k + 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

(7.13)
Ψ ( x , y , z ) = k [ Φ A ( x , y , z ) ) Φ B ( x , y , z ) ]

Problem 7.3 Show that the antibonding MO is normalized if

(7.14)
k = 1 2 2 S AB
and that the energy is given by
(7.15)
E = H AA H AB 1 S AB
(p.108) The energy E may be calculated for any instantaneous internuclear distance from equation (7.15) and plotted as a function of R. It is found to increase monotonically with decreasing R.

The probability density calculated for an electron in Ψ is

(7.16)
Ψ 2 ( x , y , z ) = C 2 [ exp ( r A / a 0 ) 2 exp [ ( r A + r B ) / a 0 ] + exp ( r B / a 0 ) ]
Here the constant C 2 is the product of the squared normalization constant of the 1s functions, ( π a 0 3 ) 1 , and k 2 = 1 / ( 2 2 S AB ) .

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 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 low-energy 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:

(7.17)
Φ A ( x , y , z ) = ( ζ 3 π a 0 3 ) 1 / 2 exp ( ζ r A a 0 ) Φ B ( x , y , z ) = ( ζ 3 π a 0 3 ) 1 / 2 exp ( ζ r B a 0 )
where ζ (“zeta”) is an adjustable constant much like the effective nuclear charge which we used in the calculations on atoms with more than one electron.

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 H 2 + 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):

E 1 = H AA + H AB 1 + S AB
and ζ varied to give the lowest possible energy for each value of R. The potential energy curve obtained in this manner has its minimum at R e = 106 pm (i.e. equal to the experimental value!) and the dissociation energy becomes 217 kJ mol−1 as compared to an experimental value of 260 kJ mol−1. (The optimal value of ζ at R = 106 pm is 1.23.)

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 z-axis) for each atom [3]. The ζ parameters of the 1s- and 2pz-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

Ψ ( x , y , z ) = C [ ( 1 s ) A + ( 1 s ) B + c ( 2 p z ) A c ( 2 p z ) B ]
with c = 0.15. (We have assumed that the z-coordinate of nucleus A is smaller than the z-coordinate of nucleus B. The negative coefficient in front of the function (2pz) B then assures that the positive lobe of the orbital is pointing towards nucleus A.)

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 2pz orbital, but the size of the orbital is much smaller than the 2pz orbital of an isolated H atom: the form of the orbitals after adjustment of ζ , is given by

( 2 p z ) = z exp ( 2.87 r / 2 a 0 )
This, in fact, is close to the form of the 2pz orbital of a Li2+ cation,
( 2 p z ) = z exp ( 3.00 r / 2 a 0 )
with an average electron-nucleus distance which is only 1/3 of that of a hydrogen atom 2p z orbital. The best way to view the matter is to assume that the shape of the 1s orbital of the isolated atom has to be modified in order to reproduce the real electron distribution in the H 2 + molecule. The effect of the admixture of the 2p z type orbitals is to increase the electron density between the two nuclei. We say that the H atom 1s orbitals have been polarized and refer to the 2p z type orbitals as polarization functions.

7.5 The hydrogen molecule and the molecular orbital approximation

Since the hydrogen molecule ion H 2 + 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 hydrogen-like atoms, in particular that the angular functions correspond to H atom s, p,or d orbitals. The LCAO-MO functions of 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 two-electron molecule, viz. the hydrogen molecule.

The electronic energy of an H2 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 H2 molecule in a Cartesian coordinate system in such a manner that the origin is at the midpoint of the H–H bond and the z-axis 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 H 2 + ion and assume that the wavefunction may be approximated by the product of two molecular orbitals

(7.18)
Ψ ( r 1 , r 2 ) = Ψ 1 ( r 1 ) Ψ 2 r 2 )
where Ψ 1 and Ψ 2 are molecular orbitals that resemble the solutions for the H 2 + ion and r 1 and r 2 represent the coordinates of the first and second electron respectively.

As a first approximation for the ground state of H2 we use MOs of the form

(7.19)
Ψ ( x , y , z ) = Φ A ( x , y , z ) + Φ B ( x , y , z ) 2 + 2 S AB
where Φ A (x, y, z) and Φ B (x, y, z) are hydrogen atom 1s functions centered on nuclei A or B respectively.

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 two-electron atom, is equally true for a two-electron 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

(7.20)
E ( R ) = Ψ ( r 1 , r 2 ) Ψ ( r 1 , r 2 ) d τ 1 d τ 2
(p.111)
                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.5. Schematic representation of the electronic energy of the H2 molecule calculated from the simple LCAO wavefunction, equations (7.18) and (7.19), and the experimentally determined counterpart.

where the integral extends over all values for the coordinates of both electrons. See Fig. 7.5.

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

(7.21)
E ( R min ) = 2885 kJ mol 1
which is 196 kJ mol−1 above the energy at the minimum of the experimental electronic energy curve. In order to calculate the dissociation energy D e we need to compare with the energy calculated when the two nuclei are infinitely far apart:
(7.22)
D e = E ( R = ) E ( R min )
The molecular orbital calculations yield
(7.23)
E ( R = ) = 1745 kJ mol 1
This result is obviously wrong: when the internuclear distance is very large, the ground state of the system must correspond to two separate H atoms, each in the 1s ground state, and the true energy is thus E(R = ∞) = −2R = −2626 kJ mol−1. Molecular orbital calculations on diatomic molecules with more than one electron yield energies which are higher than the true, experimental energies for all values of R. In fact, the error is larger when nuclei are far apart than when they are close together.

Combination of equations (7.21), (7.22), and (7.23) yields an estimated dissociation energy of

D e = E ( R = ) E ( R min ) = 1754 kJ mol 1 ( 2885 kJ mol 1 ) = 1131 kJ mol 1
which is more than twice as large as the experimental value.

(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:

D e = 2 E ( R min ) = 2626 kJ mol 1 ( 2885 kJ mol 1 ) = 259 kJ mol 1
as compared to an experimental value of 455 kJ mol−1. The calculations based on the simple LCAO wavefunction given by equations (7.18) and (719) thus overestimate the bond distance by some 15% and underestimate the dissociation energy by more than 40%.

One way to improve the result is obviously to modify the 1s-like 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 2pz 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 Slater-determinant 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)

                   Lewis’ “cubical atom” model. Molecular orbital calculations on the one-electron molecule                                                                            H                            2                            +                                                                   and the two-electron molecule H2

Fig. 7.6. Failure of the Born-Oppenheimer approximation for HD. The shaded circle on the left represents the deuterium nucleus, the shaded circle on the right the proton. The frozen electron distribution is indicated by an ellipse.

and the center of gravity of the positive charge no longer coincides with the center of the negative. At this instant the molecule will have a dipole moment with the positive end at the proton. If the shape of the electron cloud follows the nuclear motion, but with a certain time lag, there will be an instantaneous dipole moment in the same direction, but smaller.

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 so-called”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)