The Hydrogen Molecule

The other MO is formed by combining the two atomic orbitals in a way that causes the electron density to be more or less canceled in the central region where the two overlap. We refer to this as destructive combination. The process is discussed more fully in the Closer Look box later in the chapter we don t need to concern ourselves with it to understand molecular orbital bond formation. The energy of the resulting MO, referred to as the antibonding molecular orbital, is higher than the energy of the atomic orbitals. 

Destructive combination leads to antibonding H2 molecular orbital 

By analogy with atomic electron configurations, the electron configurations for molecules can be written with superscripts to indicate electron occupancy. The electron configuration for H2, then, is 

A FIGURE 9.33 Energy-level diagrams and electron configurations for H2 and He2. 

The axial components of electronic orbital and spin angular momenta add, giving as the total axial component of electronic angular momentum (A + %)h. (Recall that A is the absolute value of We consider 2 to be positive when it has the same direction as A, and negative when it has the opposite direction as A.) The possible values of A + 2 are 

The value of A + 2 is written as a right subscript to the term symbol to distinguish the energy levels of the term. Thus a A term has A = 2 and 5 = 1 and gives rise to the levels A3, A2, and Ai. In a sense, A + 2 is the analog in molecules of the quantum number J in atoms. However, A + 2 is the quantum number of the z component of total electronic angular momentum and therefore can take on negative values. Thus a 11 term has the four 

The spin-orbit interaction energy in diatomic molecules can be shown to be well approximated by A A2, where A depends on A and on the intemuclear distance R but not on 2. The spacing between levels of the multiplet is thus constant. When A is positive, the level with the lowest value of A + 2 lies lowest, and the mnltiplet is regular. When A is negative, the multiplet is inverted. Note that for A 0 the spin multiplicity 2S + 1 always equals the number of multiplet components. This is not always true for atoms. 

Each energy level of a multiplet with A 0 is doubly degenerate, corresponding to the two values for Mi. Thus a A term has six different wave functions [Eqs. (13.86), (13.88), (11.57) to (11.59)] and therefore six different molecular electronic stales. Spin-orbit interaction splits the A term into three levels, each doubly degenerate. The double degeneracy of the levels is removed by the A-type doubling mentioned previously. 

For 2 terms (A = 0), the spin-orbit interaction is very small (zero in the first approximation), and the quantum numbers 2 and O, are not defined. 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is 

The terms in square brackets are to do with the nuclear motion the first two of these represent the kinetic energy of the nuclei labelled A and B (each of mass M), and the third term in the square brackets is the Coulomb repulsion between the two nuclei. The fourth and fifth terms give the kinetic energy of the two electrons. The next four negative terms give the mutual Coulomb attraction between the two nuclei A, B and the two eleetrons labelled 1, 2. The final term is the Coulomb repulsion between electrons 1 and 2, with rn the distance between them. As in Chapter 3, I have used the subscript tot to mean nuclear plus electron. 

The first step is to make use of the Bom-Oppenheimer approximation, so I separate the nuclear and the electronic terms  

For large molecules, very many terms contribute to the electronic Hamiltonian. To simplify the notation, I am going to collect together all those terms that depend explicitly on the coordinates of a single electron and write them as 

Such operators which collect together all the variable terms involving a particular electron are called one-electron operators. The l/ri2 term is a typical two-electron operator, which we often write 

For small 5, the three-electron bond energy in He/ should be not far from (better, a little less than) that of H/. 

The results of our model are seen to agree well with experiment, and were confirmed by ab initio calculations on the same systems (Magnasco, 2008). It was shown there that the single one-electron bond energy parameter (fB—aS) occurring in Equations (25-28) is just the model representation of the one-electron part of the exchange-overlap component of the interaction due to the exchange-overlap densities [a(r)h(r)— 5it (r)] on A and [h(r)a(r)—Sh (r)] on B. 

The naive extension of the model to the bonds of the second-row homonuciear diatomics (Li2, Li2, Bc2, Bc2), mostly involving overlap 

The hydrogen molecule H2 is the simplest molecule which forms an electron-pair bond. Many calculations have been made for this molecule, which is a prototype for many other chemical bonds. One of the two basic quantum-mechanical treatments of the hydrogen molecule involves constructing a molecular orbital for the bond from a linear combination of atomic orbitals (LCAO method). The other involves constructing the molecular orbital as the product of wave functions for each of the two electrons forming the bond. Both of these methods will be outlined. 

In this expression, although the electrons and the nuclei have been labeled, identical roles are assigned to the two electrons and the two nuclei. 

When the integrals in equation (1.32) are evaluated, the calculated energy is the sum of two terms 

The Heitler-London treatment can be improved in various ways. For example, we can add terms corresponding to ionic states. If both electrons are associated with nucleus A, we have the function 

Instead of constructing the molecular orbital for the electron-pair bond by taking a linear combination of atomic orbitals, we can take the product of two molecular orbitals One for electron 1 and the other for electron 2. The function for electron 1 can be the sum of two atomic orbitals 

In the hydrogen molecule two electrons exist in the field of two nuclei and attempts to solve the Schrodinger equation for this case have proved unsuccessful. An approximate method similar to that given above for the simpler case of the hydrogen molecule ion hiay, however, be solved. This was first carried out by Heitler and London3 in 1927. 

Previously we have only considered the application of the Schrodinger equation to problems involving a single electron. In the case of several electrons, the wave function depends upon the coordinates of all the 

Thus the Schrodinger equation for the case of n electrons will be  

Rat = the distance between the nuclei a and b ria = the distance between the two electrons. 

The potential energy term will consist of the various energies of interaction of the electrons with the nuclei, of the nuclei with each other and of the electrons with each other. Thus in atomic units the potential energy is given by, 

The treatment which we shall employ here is similar to that used for the hydrogen molecule ion. The two atoms will be considered as being initially a sufficient distance apart to prevent interaction. Under these conditions the atoms may be considered separately and electron i will be at nucleus a and electron a at nucleus b. The total energy of the system will be the sum of the energies of the two atoms i.e. 2-Eo The state of the system is described by the expression 

We might well be tempted to take the molecular orbitals obtained for H2 and put in two paired electrons in the lowest level to calculate the energy of This procedure would predict that if a + p is the electronic binding energy for H2 , then Za + 2p would be the binding energy for Hjj. In fact, the calculated values of a + P for both systems (29.7 e. v. for 1 2 and 26. 5 e. V. for H2) are amazingly 

The molecular orbital treatment of can be applied to organic molecules such as CH4 or CH2=CH2 in two different ways First, molecular orbitals can be setup as linear combinations of all of the atomic orbitals of the molecule, their energies can be calculated, and the appropriate number of electrons can be put,in. This is necessarily a complicated procedure and not of great interest to organic chemists because absolute numbers for CH4 and CH2=CH2 are less useful than comparisons relative to other molecules of the same general type. The second and simpler approach is to make the approximation that electrons in some, or most, of the bonds are localized . Localized electrons are assumed not to contribute importantly to the electronic character of the bonds in the rest of the molecule. 

for ethylene, we might consider each of the bonds to be localized and the electrons in each to act independently of 

We are then taking each bond as a sort of localized molecular orbital of the type involved in Hz but are considering different kinds of atomic orbitals. Generally-speaking this approximation is quite useful. The 

Butadiene is known to be a substance in which the double bonds can react simultaneously as, for example, in the Diels-Alder reaction and in 1,4 additions of halogens. In the simple molecular orbital treatment, butadiene is treated as a system with localized cr bonds and delocalized u bonds. 

The first step in the discussion of H2, the simplest many-electron diatomic molecule, is to build the molecular orbitals. Because each H atom of H2 contributes a Is orbital (as in HJ), we can form the lOg and loj bonding and antibonding orbitals from them, as we have seen already. At the equilibrium internuclear separation these orbitals will have the energies represented by the horizontal lines in Fig. 10.26. 

We can conclude that the importance of an electron pair in bonding stems from the fact that two is the maximum number of electrons that can enter a bonding molecular orbital Electrons do not want to pair they pair because, as we show in the following brief Justification, the Pauli exclusion principle implies that  

The spatial wavefunction for two electrons in a bonding molecular orbital y/ such as the bonding orbital in eqn 10.4b (with the plus sign) is y/ l)y/ 2). This two-electron wavefunction is obviously symmetric under interchange of the electron labels. To satisfy the Pauli principle, it must be multiplied by the antisymmetric spin state a(l)(3(2) - (3(l)a(2) to give the overall antisymmetric state 

Because a(l)(3(2) - P(l)a(2) corresponds to paired electron spins, we see that two electrons can occupy the same molecular orbital (in this case, the bonding orbital) only if their spins are paired. 

Fi) 10.25 The types of molecular orbital to which d orbitals can contribute. The a and n combinations can be formed with s, p, and d orbitals of the appropriate symmetry, but the 5 orbitals can be formed only by the d orbitals of the two atoms. 

The value of A + 2 is written as a right subscript to the term symbol to distinguish the energy levels of the term. Thus a A term has A = 2 and 5 = 1 and gives rise to the levels 

In this section the molecular orbital and valence bond approaches to bonding in the hydrogen molecule will be compared. In their simplest forms we shall find that valence bond theory is better than MO theory, but as the models become more sophisticated the results obtained by the two methods converge to give the exact experimental result. 

and are the Lapladan operators for the two electrons. The first term represents the kinetic energy of the two electrons, and the other terms the various electrostatic attractions and repulsions. 

As before, the expectation value for the energy of this trial wave-function can be calculated by inserting the wavefunction into equation (8.3). This gives  

The first term in this equation represents the electronic energy of two hydrogen atoms, and the second term is the electrostatic repulsion between the two nuclei. The term labelled Cl represents the coulom-bic interactions of various charge distributions with one another. The integrals can all be evaluated analytically and the potential energy curve 

When hydrogen atoms A and B are an infinite distance apart, the electronic wavefunction is given accurately by the equation  

In order to demonstrate the difficulty of including explicitly all the interaction effects in a system with more than one electron, we discuss briefly a model of the hydrogen molecule. This molecule consists of two protons and two electrons, so it is the simplest possible system for studying electron-electron interactions in a 

If we were dealing with a single electron, then this electron would see the following hamiltonian, in the presence of the two protons  

We define the expectation value of this hamiltonian in the state 5i(r) or 52(r) to be 

Notice that si(r) or S2(r) are not eigenstates of and e eo- Also notice that 

We call the very last term in this expression, the electron-electron repulsion, the interaction term. It will prove convenient within the 5i(ri), S2 t2) basis to define the so called hopping matrix elements 

The orbital electronic structures of molecules with more than one valence electron are built up by placing the valence electrons in the most stable molecular orbitals appropriate for the valence orbitals of the nuclei in the molecule. We have constructed the molecular orbitals for the system of two protons and two Ij atomic orbitals. This set of orbitals is appropriate for H2+, H2, H2, etc. The hydrogen molecule, H2, has two electrons that can be placed in the molecular orbitals given in the energy-level diagram (Fig. 2-8). Both electrons can be placed in the o level, provided they have different spin ( 2s) quantum numbers (the Pauli principle). Thus we represent the ground state of H2 

This picture of the bond in H2 involving two electrons, each in a a orbital but with opposite spins, is analogous to the Lewis electron-pair bond in H2 (Fig. 2-3). It is convenient to carry along the idea that a full bond between any two atoms involves two electrons. Thus we define as a useful theoretical quantity the number of bonds in a molecule as follows  

One electron in an antibonding MO is considered to cancel out the bonding stability imparted by one electron in a bonding MO. Using this formula we see that H2+ has half a o bond and H2 has one o- bond. 

The chemist is accustomed to think of the chemical bond from the valence-bond approach of Pauling (7)05), for this approach enables construction of simple models with which to develop a chemical intuition for a variety of complex materials. However, this approach is necessarily qualitative in character so that at best it can serve only as a useful device for the correlation and classification of materials. Therefore the theoretical context for the present discussion is the Hund (290)-Mulliken (4f 7) molecular-orbital approach. Nevertheless an important restriction to the application of this approach must be emphasized at the start viz. an apparently sharp breakdown of the collective-electron assumption for interatomic separations greater than some critical distance, R(. In order to illustrate the theoretical basis for this breakdown, several calculations will be considered, the first being those for the hydrogen molecule. 

The Schroedinger equation for the hydrogen molecule that corresponds to equation 1 for the atom is 

A general formulation for the symmetric and antisymmetric coordinate wave function is 

Although Herring (271a) has raised a fundamental theoretical difficulty by showing that the HL description is incorrect at very large distances ( 50 atomic spacings), the HL approach is here assumed valid for the values of R Re of physical interest. 

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Kolos W and Wolniewicz L 1968 Improved theoretical ground-state energy of the hydrogen molecule J. Chem. Phys. 49 404-10... 

Kolos W and Wolniewicz L 1963 Nonadiabatic theory for diatomic molecules and its application to the hydrogen molecule Rev. Mod. Phys. 35 473-83... 

Feibelman P J 1991 Orientation dependence of the hydrogen molecule s interaction with Rh(OOI) Phys. Rev. Lett. 67 461... 

In Figure 2, we show the total differential cross-section for product molecules in the vibrational ground state (no charge bansfer) of the hydrogen molecule in collision with 30-eV protons in the laboratory frame. The experimental results that are in aibitrary units have been normalized to the END... 

There are many compounds in existence which have a considerable positive enthalpy of formation. They are not made by direct union of the constituent elements in their standard states, but by some process in which the necessary energy is provided indirectly. Many known covalent hydrides (Chapter 5) are made by indirect methods (for example from other hydrides) or by supplying energy (in the form of heat or an electric discharge) to the direct reaction to dissociate the hydrogen molecules and also possibly vaporise the other element. Other known endothermic compounds include nitrogen oxide and ethyne (acetylene) all these compounds have considerable kinetic stability. 

Calculating the Energy from the Wavefunction the Hydrogen Molecule... 

Ale now substitute the hydrogen molecule wavefunction into Equation (2.73) to provide the ollowing ... 

I Liming now to the numerator in the energy expression (Equation (2.95)), this can be broken do, n into a series of one-electron and two-electron integrals, as for the hydrogen molecule, l ach of these individual integrals has the general form ... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

The hamionic oscillator of two masses is a model of a vibrating diatomic molecule. We ask the question, What would the vibrational frequency be for H2 if it were a hamionic oscillator The reduced mass of the hydrogen molecule is... 

Among the few systems that can be solved exactly are the particle in a onedimensional box, the hydrogen atom, and the hydrogen molecule ion Hj. Although of limited interest chemically, these systems are part of the foundation of the quantum mechanics we wish to apply to atomic and molecular theory. They also serve as benchmarks for the approximate methods we will use to treat larger systems. 

The hydrogen molecule ion is best set up in confocal elliptical coordinates with the two protons at the foci of the ellipse and one electron moving in their combined potential field. Solution follows in mueh the same way as it did for the hydrogen atom but with considerably more algebraic detail (Pauling and Wilson, 1935 Grivet, 2002). The solution is exact for this system (Hanna, 1981). 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Approximate Theoretical. The simplest molecular orbital problem is that of the hydrogen molecule ion (Pig KJ-3), is a preliminary example of all molecular orbital problems to come, w hich, although they may be very complicated, are elaborations on this simple example. 

Even though the problem of the hydrogen molecule H2 is mathematically more difficult than, it was the first molecular orbital calculation to appear in the literature (Heitler and London, 1927). In contrast to Hj, we no longer have an exact result to refer to, nor shall we have an exact energy for any problem to be encountered from this point on. We do, however, have many reliable results from experimental thermochemistry and spectroscopy. 

A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

Sketch the hydrogen molecule system (2 protons and 2 electrons) and verify the Hamiltonian 10.3.1. 

Carry out a series of calculations comparable to those in Computer Project 10-1 on the hydrogen molecule. Estimate the conelation energy from the GAUSSIAN calculations. 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

Asimple example is the formation of the hydrogen molecule from two hydrogen atoms. Here the original atomic energy levels are degenerate (they have equal energy), but as the two atoms approach each other, they interact to form two non degenerate molecular orbitals, the lowest of which is doubly occupied. 

The simplest example of covalent bonding is the hydrogen molecule. The proximity of the two nuclei creates a new electron orbital, shared by the two atoms, into which the two electrons go (Fig. 4.5). This sharing of electrons leads to a reduction in energy, and a stable bond, as Fig. 4.6 shows. The energy of a covalent bond is well described by the empirical equation... 

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