Electronic Terms of Polyatomic Molecules

Ab initio and density-functional methods that take account of electron correlation are discussed in Chapter 16. Chapter 17 discusses semiempirical methods and the molecular-mechanics method. 

A free database of references to molecular ab initio and density-functional calculations is the Quantum Chemistry Literature Database at qcldb2.ims.ac.jp. 

For polyatomic molecules, the operator for the square of the total electronic spin angular momentum commutes with the electronic Hamiltonian operator, 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 -l- 1. (The commutation of and H i holds provided spin-orbit interaction is omitted from the Hamiltonian. For molecules containing heavy atoms, spin-orbit interaction is considerable, and S is not a good quantum number.) 

For linear polyatomic molecules, the operator for the component of the total electronic orbital angular momentum along the molecular axis commutes with the electronic Hamiltonian, and the same term classifications are used as for diatomic molecules (Section 13.8), giving such possibilities as 2 , 2 , 11, and so on. For linear polyatomic 

For nonlinear polyatomic molecules, no orbital angular-momentum operator commutes with the electronic Hamiltonian, and the angular-momentum classification of electronic terms cannot be used. Operators that do commute with the electronic Hamiltonian are the symmetry operators Og of the molecule (Section 12.1), and the electronic states of polyatomic molecules are classified according to the behavior of the electronic wave function on application of these operators. Consider H2O as an example. 

The density-functional metkod (Section 15.20) does not attempt to calculate the molecular wave function but calculates the molecular electron probability density p and calculates the molecular electronic energy from p. 

The molecular mechanics method (Section 16.6) is not a quantum-mechanical method and does not use a molecular Hamiltonian operator or wave function. Instead, it views the molecule as a collection of atoms held together by bonds and expresses the molecular energy in terms of force constants for bond bending and stretching and other parameters. 

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Mulliken introduced the term "orbital" distinct from "orbital wave function" in 1932 in the second of fourteen papers carrying the general title, "Electronic Structures of Polyatomic Molecules and Valence." Mulliken defined atomic orbitals (AOs) and molecular orbitals (MOs) as something like the... 

Mulliken, Life, 90. On the "orbital," Mulliken wrote in 1932 "From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. The method followed here will be to describe unshared electrons always in terms of atomic orbitals but to use molecular orbitals for shared electrons." In Robert Mulliken, "Electronic Structures of Polyatomic Molecules and Valence," Physical Review 41 (1932) 4971, on 50. 

The Jahn-Teller theorem is important in considering the electronic states of polyatomic molecules. Jahn and Teller proved in 1937 that a nonlinear polyatomic molecule cannot have an equilibrium (minimum-energy) nuclear configuration that corresponds to an orbitally degenerate electronic term. Orbital degeneracy arises from molecular symmetry (Section 1.19), and the Jahn-Teller theorem can lead to a lower symmetry than... 

The first mention of the methylene molecule in the traditional scientific literature seems to be by Mulliken [5] in a 1932 Physical Review article titled Electronic Structures of Polyatomic Molecules and Valence, n. Quantum Theory of the Chemical Bond. Mulliken was interested in the nature of double bonds and in particular the double bond in ethylene, which he analyzed in terms of the constituent CH2 fragments. In the course of this analysis he proposed that the ground state of CH2 was of Ai symmetry with an angle of about 110° and that there was a low-lying Bi state. This was a remarkable illustration of Mulliken s legendary insight. It is humbling to note that quantum mechanics as we know it was only 6 years old. 

Thus far we have discussed the chemical bonding in polyatomic molecules in terms of the VB model, or more crudely in terms of Lewis structures. These two treatments are related in that they focus on chemical bonding in terms of the sharing of electron pairs by adjacent atoms. In many cases, however, a more sophisticated approach based on molecular orbital concepts is needed to accurately picture the electronic structure of polyatomic molecules—even on a quahtative level. 

The appearance of a vibration-rotation band is shown in Fig. 4.9. Because A/=0 is forbidden, we have a gap at the wave number of the band origin o0. (We are considering only diatomic molecules in 2 electronic states for electronic states in which the electronic orbital angular-momentum quantum number A is nonzero, transitions with AJ=0 are allowed, giving a Q branch in each band. An example is NO, which has a 2n electronic ground term, -branch transitions also occur for vibration-rotation bands of polyatomic molecules see Chapter 6.) Under low resolution, an infrared band of a diatomic molecule looks like Fig. 4.10. 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

So far we have discussed chemical bonding only in terms of electron pairs. However, the properties of a molecule cannot always be explained accurately by a single structure. A case in point is the O3 molecule, discussed in Section 9.8. There we overcame the dilemma by introducing the concept of resonance. In this section we will tackle the problem in another way—by applying the molecular orbital approach. As in Section 9.8, we will use the benzene molecule and the carbonate ion as examples. Note that in discussing the bonding of polyatomic molecules or ions, it is convenient to determine fust the hybridization state of the atoms present (a valence bond approach), followed by the formation of appropriate molecular orbitals. 

Two theories go hand in hand in a discussion of covalent bonding. The valence shell electron pair repulsion (VSEPR) theory helps us to understand and predict the spatial arrangement of atoms in a polyatomic molecule or ion. It does not, however, explain hoav bonding occurs, ] ist where it occurs and where unshared pairs of valence shell electrons are directed. The valence bond (VB) theory describes how the bonding takes place, in terms of overlapping atomic orbitals. In this theory, the atomic orbitals discussed in Chapter 5 are often mixed, or hybridized, to form new orbitals with different spatial orientations. Used together, these two simple ideas enable us to understand the bonding, molecular shapes, and properties of a wide variety of polyatomic molecules and ions. 

As mentioned above, vibronic-coupling model Hamiltonians constructed by low-order Taylor expansions of the diabatic PE functions in terms of normal coordinates are particularly suitable for the calculation of low-resolution spectra of polyatomic molecules. In resonance Raman spectroscopy, for example, the usually extremely fast electronic dephasing in polyatomic s tems limits the time scale for the exploration of the excited-state PE surface by the nuclear wave packet to about 10 jjj... 

The Born-Oppenheimer adiabatic approximation represents one of the cornerstones of molecular physics and chemistry. The concept of adiabatic potential-energy surfaces, defined by the Born-Oppenheimer approximation, is fundamental to our thinking about molecular spectroscopy and chemical reaction djmamics. Many chemical processes can be rationalized in terms of the dynamics of the atomic nuclei on a single Born Oppenheimer potential-energy smface. Nonadiabatic processes, that is, chemical processes which involve nuclear djmamics on at least two coupled potential-energy surfaces and thus cannot be rationalized within the Born-Oppenheimer approximation, are nevertheless ubiquitous in chemistry, most notably in photochemistry and photobiology. Typical phenomena associated with a violation of the Born-Oppenheimer approximation are the radiationless relaxation of excited electronic states, photoinduced uni-molecular decay and isomerization processes of polyatomic molecules. 

In principle, a description of the electronic structure of many-electron atoms and of polyatomic molecules requires a solution of a Schrodinger equation for stationary states quite similar to equation 3.36 [2]. Even for a simple molecule like, say, methane, however, such an equation would be enormously more complicated, because the hamiltonian operator would include kinetic energy terms for all electrons, plus coulombic terms for the electrostatic interaction of all electrons with all nuclei and of all electrons with all other electrons. The QM hamiltonian operator for the electrons in a molecule reads ... 

In practice, a very wide range of intensities is observed for electronic absorption bands of polyatomic molecules, with maximum molar absorption coefficients ranging from the lower limit of detection (0.1-0.01 M cm ) up to values of the order of 5 x 10 M cm. Without carrying out any actual calculation of Mjf, it is easy to identify typical cases in which the first or the second terms in (1.5) are expected to vanish, thus leading to a zero predicted intensity of the electronic... 

The o- and 7r-oibitals of diatomic molecules extend over the whole structure. In a similar way, orbitab could be constructed which extend over the whole of polyatomic molecules. In the case of conjugated systems it is normal to describe 7r-electrons in terms of orbitals of this type. Tliis is clearly in accord with chemical experience for it is found that the behaviour of one double bond is strongly influenced by other double bonds in conjugation, and that electronic effects are readily transmitted along the conjugation pathway. 

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