Correlation of molecular and atomic electronic states

A correlation diagram relating the molecular orbitals to the united atom orbitals can be drawn for heteronuclear systems, as shown in figure 6.9 it is similar to that for homonuclear systems. The combination of like atomic orbitals on atoms a and b will, in each case, give rise to two complementary molecular orbitals, as shown on the right-hand side of figure 6.9. 

We have described the orbital approaches to the electron configurations of diatomic molecules, both the molecular orbital and the united atom models. We now turn to the question of what types of molecular states result from given states of the separate atoms. If Russell Saunders coupling is valid for the separate atoms, the correlation rules, due to Wigner and Witmer [16] provide a valid and complete summary of the molecular states. This information is extremely important for an understanding of both the formation and dissociation of diatomic molecules. 

The possible spin multiplicities are readily determined because the resultant spin quantum number S is given by 

Similar rules hold for the combination of two like atoms, the main difference being that the resulting molecular states must be either even or odd. Again an abbreviated list is presented in table 6.5 more comprehensive lists are available in the literature. Note that it is not necessary to specify the parities of the separated atoms, since we are dealing with combinations of like atoms. 

Similar correlation rules exist for atoms which follow jj coupling, rather than Russell-Saunders coupling, but such cases are relatively rare and will not be discussed here. 

---

F. Hund, "Zur Deutung der Molekulspektren. IV," ZP 51 (1928) 759795 R. S. Mulliken, "The Assignment of Quantum Numbers for Electrons in Molecules. II. Correlation of Molecular and Atomic Electron States," Physical Review 32 (1928) 761772 E. Hiickel, "Zur Quantentheorie der Doppelbindung,"... 

Mulliken, R. S. 1928a. The assignment of quantum numbers for electrons in molecules. 11. The correlation of molecular and atomic electron states. Physical Review 32 761-772. 

APPENDIX C. The Correlation of Molecular and Atomic Electronic States... 

There are a number of different approaches to the description of molecular electronic states. In this section we describe molecular orbital theory, which has been by far the most significant and popular approach to both the qualitative and quantitative description of molecular electronic structure. In subsequent sections we will describe the theory of the correlation of molecular states to the Russell Saunders states of the separated atoms we will also discuss what is known as the united atom approach to the description of molecular electronic states, an approach which is confined to diatomic molecules. 

There are three different schemes for building up the electronic states of diatomic molecules (a) from separated atoms, (b) from the united atom, and (c) from the molecular orbitals of the diatomic molecule itself. It is the correlation between the electronic states of the diatomic molecule as built up from the separated atoms and as determined from the molecular orbitals of the diatomic which is most valuable for any general consideration of reactions and excited states. The correlation of molecular states obtained by these two methods is not limited solely to diatomic molecules but also forms a valid approach for polyatomic molecular systems. The correlation of separated atoms with the hypothetical united atom has value for diatomics and has been applied to simple polyatomic molecules, especially those with a heavy atom or two and a number of hydrogen atoms. However, it is conceptually less appealing even for simple polyatomic molecules and completely inapplicable for complex polyatomic molecules. 

Table 7.1 summarizes the correlations between separated (R —> oo) atomic L-S-J-Q and molecular 2S+1Aq states for the 2P + 2S separated atom example. It also gives the explicit linear combinations of molecular 25+1 An states that correlate to each separated atom L-S-J-Q, combination and vice versa. These molecular —> separated atom linear combinations describe the expected fine structure branching ratios that would result without any electronic orbital rearrangements, i.e., the diabatic limit. 

For carbon atoms, however, the ab initio configuration interaction calculations of Blint and Newton on the reactions of S, D, and P electronic states with molecular hydrogen have made an important contribution (73). A schematic representation of states and pathways is given in Figure 4. The calculations indicate that carbon will not react with H2, a result also obtained by Husain from a symmetry-derived correlation diagram (74). Therefore, the reactions of only the two lowest states of atomic carbon, and P, were considered. The formation of methyne CH from the state and insertion by P recoiling carbon atoms was indicated. 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

---