The hydrogen-molecule ion

The atomic unit of energy, is called the hartree (symbol E  

The ground-state energy of the hydrogen atom is - hartree if nuclear motion is neglected. The atomic unit of length is called the bohr  

To find the atomic unit of any other quantity (for example, time) one combines h, trie, and 47TS0 so as to produce a quantity having the desired dimensions. One finds (Prob. 13.14) the atomic unit of time to be h/E = 2.4188843 X 10 s and the atomic unit of electric dipole moment to be ea = 8.478353 X 10 ° C m. 

Introducing the reduced quantities into the Schrodinger equation, we find (Prob. 13.13) that the reduced H-atom Schrodinger equation is — (1/7 ) / = ffriAr 

We now begin the study of the electronic energies of molecules. We shall use the Born-Oppenheimer approximation, keeping the nuclei fixed while we solve, as best we can, the Schrodinger equation for the motion of the electrons. We shall usually be considering an isolated molecule, ignoring intermolecular interactions. Our results will be most applicable to molecules in the gas phase at low pressure. For inclusion of solvent effects, see Sections 15.17 and 17.6. 

The Hydrogen Molecule-Ion.—The structure of the hydrogen molecule-ion, H2, as of any molecule, is discussed theoretically by first considering the motion of the electron (or of all the electrons in case that there are several) in the field of the atomic nuclei considered to be fixed in a definite configuration.19 The electronic. energy of the molecule 

For the hydrogen molecule-ion our problem is to evaluate the energy as a function of the distance tab between the two nuclei A and B. For 

However, the structure assumed is too simple to represent the system satisfactorily. We have assumed tfiat the electron forms a normal hydrogen atom with nucleus A  

In this discussion another type of interaction between the hydrogen atom and ion has been neglected to wit, the deformation (polarization) of the atom in the electric field of the ion. This has been considered by Dickinson,22 who has shown that it contributes an additional 10 kcal /mole to the energy of the bond. We may accordingly say that of the total energy of the one-electron bond in (61 kcal/mole) about 80 percent (50 kcal/mole) is due to the resonance of the electron between the two nuclei, and the remainder is due to deformation. 

The simplest conceivable molecule is H2+, the hydrogen molecule ion with two nuclei and one electron. It is true it does not occur in chemical compounds but a spectrum can be observed in discharges in hydrogen gas which must be attributed to H2+. Thus the heat of dissociation and the nuclear separation are also known. 

For this hydrogen molecule ion H2+ we must substitute in the wave equation  

We start with diatomic molecules, the simplest of which is Hj, the hydrogen molecule ion, consisting of two protons and one electron. Just as the one-electron H atom serves as a st u ting point in the discussion of many-electron atoms, the one-electron H2 ion furnishes many ideas useful for discussing many-electron diatomic molecules. The electronic SchrOdinger equation for is separable, and we can get exact solutions for the eigenfunctions and eigenvalues. 

The first term is the electronic kinetic-energy operator the second and third terms are the attractions between the electron and the nuclei. In atomic units the purely elec- 

In Fig. 13.3 the coordinate origin is on the intemuclear axis, midway between the nuclei, with the z axis lying along the intemuclear axis. The H2 electronic Schrodinger equation is not separable in spherical coordinates. However, separation of variables is possible using confocal elliptic coordinates 17, and . The coordinate f is the angle of rotation of the electron about the intemuclear (z) axis, the same as in spherical coordinates. The coordinates f (xi) and 17 (eta) are defined by 

We must put the Hamiltonian (13.32) into these coordinates. We have 

To see how this works, we will consider the simplest possible molecule, H2+. This is the hydrogen molecule Ion, which consists of two nuclei of charge +1, and a single electron shared between them. 

Just as the electronic configuration of an atom is built up by stepwise population - electron by electron - of hydrogen-like atomic orbitals, that of a diatomic molecule is constructed by successively filling the molecular orbitals derived from the hydrogen molecule ion, H2 [1]. 

The first thing to note about Fig. 3.1 is that the two hydrogen nuclei define an internuclear axis, which is conventionally labelled z the x and y axes remain undefined, but the xy plane can be - with care. In principle, either the center of nuclear mass or the center of nuclear charge can be chosen to serve as the origin of our coordinate system. When both Hj and are protons, either criterion fixes the origin at i, the point midway between them. However, when is a 

Assume that the electron is momentarily situated at an arbitrary point p. Its potential energy is uniquely determined by two distances, Ta and r. Neither of these distances is altered when the electron is rotated about the axis through any angle (j), say to point p, as long as the perpendicular distance to the axis remains constant (rp/ = Tp). One particular rotation of this kind, specifically hy j = 180°, brings the electron to q, which can also be related to p by reflection in the plane (not drawn in Fig. 3.1) which includes the -sr axis 

Chapter 3. Diatomic Molecules and their Molecular Orbitals 

At infinite nuclear separation, the electron has to choose between one nucleus and the other, producing either (Ra + Hj) or (Hj + Hb). In either case, it is localized on one of the nuclei in what is effectively an isolated atom, the electron being too far from the the second nucleus for its attraction to have an appreciable effect on the energy. However, in order to define the 2 axis, the bare nucleus - say Hj - must be sufficiently close to H for its attraction to be perceptible, if only as a minute perturbation of the energy of the bound electron. Therefore, although r, the cylindrical symmetry of the molecule, i.e. its invariance to symmetry operations that interchange and rg, ensures that 

In Chapter 1, we dealt at length with molecular mechanics. MM is a classical model where atoms are treated as composite but interacting particles. In the MM model, we assume a simple mutual potential energy for the particles making up a molecular system, and then look for stationary points on the potential energy surface. Minima correspond to equilibrium structures. 

Classical descriptions of molecular phenomena can be remarkably successful, but we have to keep our eye on the intrinsic quantum nature of microscopic systems. 

The traditional place to begin a quantum-mechanical study of molecules is with the hydrogen molecule ion H2+. Apart from being a prototype molecule, it reminds us that molecules consist of nuclei and electrons. We often have to be aware of the nuclear motion in order to understand the electronic ones. The two are linked. 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities, The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rA, and the scalar distance between the electron and nucleus B is rB. I will write Rab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb. 

It is an easy step to write down the Hamiltonian operator for the problem 

The first two terms represent the kinetic energy of the nuclei A and B (each of mass M), whilst the fourth term represents the kinetic energy of the electron (of mass m). The fifth and sixth (negative) terms give the Coulomb attraction between the nuclei and the electron. The third term is the Coulomb repulsion between the nuclei. 1 have used the subscript tot to mean nuclei plus electron, and used a capital I.  

A procedure that is commonly used in such cases is to construct trial functions for molecules from the exact eigenfunctions that apply to the atoms from which the molecules are formed. The eigenfunction for the Is state of the hydrogen atom is given by equation (1.27) with Z = 1, and we may therefore write the eigenfunction for the electron bound to proton A as 

The corresponding wave function for the electron associated with nucleus B is 

In the variation method, we can use any trial function we like, and a reasonable procedure is to use the sum of the above functions. 

The fact that this sum is not normalized does not matter, since this is taken care of in the denominator of the expression in equation (1.32). 

In the procedure outlined we have used a linear combination of atomic orbitals as the trial wave function. This procedure is known as the LCAO method or as the LCAO MO method, the latter designation meaning that we have constructed a molecular orbital (MO) as a linear combination of atomic orbitals. The LCAO method is frequently employed,-but unless additional terms are added the agreement with experiment is never very close. 

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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. 

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. 

This section briefly considers the proton H+, the hydride ion H, the hydrogen molecule ion H2, the triatomic 2-electron species H3+ and the recently established cluster species +... 

The species H2 and H3+ are important as model systems for chemical bonding theory. The hydrogen molecule ion H2+ comprises 2 protons and 1 electron and is extremely unstable even in a low-pressure gas discharge system the energy of dissociation and the intemuclear distance (with the corresponding values for H2 in parentheses) are ... 

Why is the hydrogen molecule ion H2 stable, and what should its bond length be ... 

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

In Chapter 3, I showed you how to write a simple LCAO wavefunction for the electronic ground state of the hydrogen molecule-ion, H2 ... 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

In Chapter 4,1 discussed the concept of an idealized dihydrogen molecule where the electrons did not repel each other. After making the Bom-Oppenheimer approximation, we found that the electronic Schrddinger equation separated into two independent equations, one for either electron. These equations are the ones appropriate to the hydrogen molecule ion. 

In our study of the hydrogen molecule-ion in Chapter 3, we considered the electron density map shown in Figure 18.8. It is obvious by inspection that the... 

Molecular structure theory is a fast-moving subject, and a lot has happened since the First Edition was published in 1995. Chapters 3 (The Hydrogen Molecule-ion) and 4 (The Hydrogen Molecule) are pretty much as they were in the First Edition, but 1 have made changes to just about everything else in order to reflect current trends and the recent literature. I have also taken account of the many comments from friends and colleagues who read the First Edition. 

The problem has already been solved for the normal state of the hydrogen molecule-ion (ZA = ZB = 1) by the use of numerical methods. A rather complete account of these calculations of Burrau (30) will be given here, since the journal in which they were published is often not available. 

Fia. 4. The Electronic Energy of the Hydrogen Molecule-ion in the Normal State as a Function of the Distance Between the Two Nuclei (Burrau)... 

Although no new numerical information regarding the hydrogen molecule-ion can be obtained by treating the wave equation by perturbation methods, nevertheless it is of value to do this. For perturbation methods can be applied to many systems for which the wave equation can not be accurately solved, and it is desirable to have some idea of the accuracy of the treatment. This can be gained from a comparison of the results of the perturbation method of the hydrogen molecule-ion and of Bureau s accurate numerical solution. The perturbation treatment assists, more-... 

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

The above perturbation treatment of the hydrogen molecule-ion has not before been published. 

B. N. Finkelstein and G. E. Horowitz (Z. f. Physik, 48, 118 (1928)) have similarly applied the Ritz method to the hydrogen molecule-ion, obtaining the following values ... 

In Sections 42 and 43 we shall describe the accurate and reliable wave-mechanical treatments which have been given the hydrogen molecule-ion and hydrogen molecule. These treatments are necessarily rather complicated. In order to throw further light on the interactions involved in the formation of these molecules, we shall preface the accurate treatments by a discussion of various less exact treatments. The helium molecule-ion, He , will be treated in Section 44, followed in Section 45 by a general discussion of the properties of the one-electron bond, the electron-pair bond, and the three-electron bond. 

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