Electronic Terms of Diatomic Molecules

Examples of diatomic van der Waals molecules and their and values indude Ne2, 3.1 A, 0.0036 eV HeNe, 3.2 A, 0.0012 eV Ca2,4.28 A, 0.13 eV Mg2, 3.89 A, 0.053 eV. Observed polyatomic van der Waals molecules include (02)2, H2—N2, Ar—HCl, and (02)2-For van der Waals bonding, is significantly greater and is very substantially less than the values for chemically bound molecules. The Be2 bond length of 2.45 A is much shorter than is typical for van der Waals molecules the closeness of the 2p orbitals to 2s orbitals in Be allows snbstantial 2s-2p hybridization in Be2 and perhaps gives some amount of covalent character in addition to the dispersion attraction. For more on van der Waals molecules, see Chem. Rev., 88, 813-988 (1988) 94, 1721-2160 (1994) 100, 3861-4264 (2000). 

We now consider the terms arising from a given diatomic molecule electron configuration. 

the operator commutes with H. For a many-electron diatomic molecule, one finds that the operator for the axial component of the total electronic orbital angular momentum commutes with H. The component of electronic orbital angular momentum along the molecular axis has the possible values Mih, where Mi = 0, 1, 2, . To calculate Mi, we simply add algebraically the m s of the individual electrons. Analogous to the symbol A for a one-electron molecule, A is defined as 

For A 0, there are two possible values of Mi, namely, -l-A and —A. As in, the electronic energy depends on Mi, so there is a double degenaacy associated with the two values of Mi. Note that lowercase letters refer to individual electrons, while capital letters refer to the whole molecule. 

Just as in atoms, the individual electron spins add vectoriaUy to give a total electronic spin S, whose magnitude has the possible values [5(5 + 1) ] with 5 = 0,5,1,. The 

A filled diatomic molecule shell consists of one or two filled molecular orbitals. The Pauli principle requires that, for two electrons in the same molecular orbital, one have nis- +2 and the other have mj = -j. Hence the quantum number M, which is the algebraic sum of the individual values, must be zero for a filled-shell molecular configuration. Therefore, we must have 5 = 0 for a configuration containing only filled molecular shells. A filled tr shell has two electrons with m = 0, so M is zero. A filled IT shell has two electrons with m = + and two electrons with m = — 1, so M (which is the algebraic sum of the m s) is zero. The same situation holds for filled shells. 

Thus a closed-shell molecular configuration has both 5 and A equal to zero and gives rise to only a 2 term. An example is the ground electronic configuration of H2. (Recall that a filled-subshell atomic configuration gives only a 5 term.) In deriving molecular terms, we need consider only electrons outside filled shells. 

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Usually the electronic structure of diatomic molecules is discussed in terms of the canonical molecular orbitals. In the case of homonuclear diatomics formed from atoms of the second period, these are the symmetry orbitals 1 og, 1 ou, 2ag,... 

Most of the infrared laser lines originating from transitions between vibrational-rotational levels in different electronic states of diatomic molecules, e. g. N2, Oj, H, D2, CO, CN, etc., have been meanwhile correctly indentified Some lines and term systems have been found which had never been observed before 354a)... 

For polyatomic molecules the operator 5 for the square of the total electronic spin angular momentum commutes with the electronic Hamiltonian, and, as for diatomic molecules, the electronic terms of polyatomic molecules are classified as singlets, doublets, triplets, and so on, according to the value of 25 + 1. (The commutation of 5 and H holds provided spin-orbit interaction is omitted from the Hamiltonian for molecules containing heavy atoms, spin-orbit interaction is considerable, and 5 is not a good quantum number.)... 

FIGURE 15.9 Electronic spectra of diatomic molecules are more specifically defined in terms of A, fi, and S. A is defined in terms of the orbital angular momentum of the electrons, L. Cl is defined in terms of the total angular momentum, /. The vector difference between L and / is, of course, S. 

For linear molecules or ions the symbols are usually those derived from the term symbols for the electronic states of diatomic and other linear molecules. A capital Greek letter E, n, A, O,... is used, corresponding to k — 0,1,2,3,..., where A. is the quantum number for rotation about the molecular axis. For E species a superscript + or - is added to indicate the symmetry with respect to a plane that contains the molecular axis. 

Before we proceed to these details we must describe some aspects of the theory of the electronic and vibrational states of diatomic molecules. To this end we return to the master equation displayed at the end of chapter 3, and develop the consequences of some of the terms contained therein. This is a huge subject, described in many textbooks, and at any level of detail which one might require. In this chapter we present what we consider to be the minimum required for a satisfactory understanding of what follows in later chapters. What is satisfactory is a subjective matter for the reader, and in many cases there are aspects to be explored in much greater depth than is to be found here. Some of these aspects are presented in later chapters, but here we deal with the essential fundamentals. 

We will see in due course that there are important correlation rules between atomic term symbols and molecular electronic states, rules that are important in understanding both the formation and dissociation of diatomic molecules. Elementary accounts of the theory of atomic structure are to be found in books by Softley [3] and Richards and Scott [4], Among the more comprehensive descriptions of the quantum mechanical aspects, that by Pauling and Wilson [5] remains as good as any whilst group theoretical aspects are described by Judd [6],... 

After a general introduction, the methods used to separate nuclear and electronic motions are described. Brown and Carrington then show how the fundamental Dirac and Breit equations may be developed to provide comprehensive descriptions of the kinetic and potential energy terms which govern the behaviour of the electrons. One chapter is devoted solely to angular momentum theory and another describes the development of the so-called effective Hamiltonian used to analyse and understand the experimental spectra of diatomic molecules. The remainder of the book concentrates on experimental methods. 

In solving problems in this book, we shall not obtain wave functions by solving differential equations such as Eq. (1-5), but shall instead assume that the wave functions that interest us can be written in terms of a small number of known functions. For example, to obtain the wave function ij/ for one electron in a diatomic molecule, we can make a linear combination of wave functions j/i and l/2, where 1 and 2 designate energy eigenstates for electrons in the separate atoms that make up the molecule. Thus,... 

The coefficients from Table 2-1 and atomic term values from Table 2-2 will suffice for calculation of an extraordinarily wide range of properties of covalent and ionic solids using only a standard hand-held calculator. This is impressive testimony to the simplicity of the electronic structure and bonding in these systems. Indeed the. same parameters gave a semiquantitativc prediction of the one-electron energy levels of diatomic molecules in Table 1-1. However, that theory is intrinsically approximate and not always subject to successive correc-... 

A basic assumption made earlier in the discussion of diatomic molecules and crystalline CsCl is that electronic states can be written as linear combinations of atomic orbitals. We do not need to depart from that assumption now as we begin to describe electronic states in solids as linear combinations of bond orbitals, since bond orbitals can be written as linear combinations of atomic orbitals, and vice versa. Bond orbitals and atomic orbitals arc equivalent representations, but thinking in terms of bond and antibonding orbitals, which can be made to correspond with occupied and empty states of the covalent solid (as was shown in Fig. 2-3), is essential in making approximations. 

It is necessary to mention here simply that the application of these 1 ules is not automatic, because the theoretical treatments from which they are derived refer to ideal cases to which real molecules approximate more or less well. Indeed, the whole discussion of diatomic molecules in terms of potential energy curves and distinguishable electronic states is itself an approximation, though a good one. 

The one-dimensional potential energy curve describes precisely the electronic term of a system only in the simple case of diatomic molecules. With A H bonds, where A is part of a multiatom molecule, the diatomic approximation is valid. But in the general case the potential energy as a function of the A H bond length, F(r), should be expressed using a polynomial expansion... 

Actually, however, in the case of diatomic molecules, we have to deal with systems where, in addition to the nuclei which are practically points of large mass, a number of electrons are present, which move around the nuclei and may, under certain circumstances, possess angular momentum about the line joining the nuclei. This system may be roughly compared to a top, whose moment of inertia Az about the nuclear axis is small in comparison with the moment of inertia A about a perpendicular direction. For an invariable electron configuration, the quantum number n, and consequently the second term in the energy (16), is a constant. For the dependence of the energy on the state of rotation we have therefore... 

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