Multielectron Atoms and Molecules

As discussed in Section 6.1.3, we could replace tp2Py with — p2P., and this would be an equally valid stationary state (see Equation 6.9). So there is nothing wrong with writing the negative lobe on top and the positive on the bottom. 

Different combinations of degenerate orbitals simplify different problems. For example, the normal representations px, py, pz of the / = 1 orbitals are actually mixtures of different // / values. The stationary states with a single value of mi 0 have real and imaginary parts, and are difficult to visualize. 

Schrodinger s equation is not analytically solvable for anything more complicated than a hydrogen atom. Even a helium atom, or the simplest possible molecule (H+) requires a numerical calculation by computer. However, these calculations give results which agree extremely well with experiment, so their validity is not doubted. 

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To date, there is no known analytic solution to the second-order differential Schrodinger equation for the helium atom. This does not mean that there is no solution, or that wavefunctions do not exist. It simply means that we know of no mathematical function that satisfies the differential equation. In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)... 

As for aU electrostatic interactions, the strength of vdW bond should depend on charges, which in turn depend on the number of electrons per atom or molecule hence the vdW interaction should be stronger between carbon atoms than carbon and hydrogen atoms and even stronger for interactions between atoms of multielectron... 

Fig. 4.1 Schematic potentials (bottom) and K-sheU spectra (top) of atoms and diatomic molecules. Resonances in K-shell spectra arise from electronic transitions from a 1 initial state to Rydberg or unfilled-MO final states. At the IP, corresponding to the threshold for transitions to continuum states, a step-like increase in X-ray absorption is expected. These effects lead to the characteristic spectra schematically shown in the upper part of the figure. In addiction to these one electron features other structures arising from multielectron transitions may be observed (Reprinted from StShr [2], with kind permission of Springer Science (2009))...

Now that we have some better understanding of where the H atom orbitals come from, the next topic should be the electronic structure of molecules and ways to treat problems for which we are unable to solve the Schrodinger equation exactly. Recall the difficulty of solving the Schrodinger equation for just one electron in the H atom. Then perhaps you may faint when you consider the notion of how one might treat the electronic structure of benzene with 12 atoms and 42 electrons Well, there is no known exact solution for even the He atom with only two electrons so do not faint but continue to wonder about how we are going to treat the multielectron case for molecules. There are two main methods the variation method and perturbation theory. In this chapter, we will emphasize the variation method, which is the most powerful mathematical approach, and give a few key examples. However, we will first mention the basic approach of perturbation theory but without much elaboration since it is the weaker of the two methods. 

To illustrate how this works with a multielectron atom, consider the N2 molecule. Here we must accommodate 14 electrons, 7 from each atom. As the two N atoms are brought together, each of the atomic levels split into a bonding and an antibonding state as shown in Figure 3.7. 

This method, called molecular orbital theory, starts with a simple picture of molecules, but it quickly becomes complex in its details. We will provide only an overview and then focus primarily on the application of molecular orbital theory to diatomic molecules. We will begin our overview by comparing molecular orbital theory with the conceptual model we introduced in Chapter 8 for multielectron atoms. 

The electron configuration for the molecule is obtained by placing the electrons into the MOs in a particular order. This is analogous to the conceptual model we used for multielectron atoms. The only difference here is that we are filling MOs, not AOs. Not surprisingly, rules we used for atoms, such as the Pauli exclusion principle and Hund s rule, also apply to molecules. That is, the maximum number of electrons per... 

Quantum mechanics describes molecules in terms of interactions between nuclei and electrons and molecular geometry in terms of minimum energy arrangements of nuclei. All quantum-mechanical methods ultimately trace back to Schrodinger s (time-independent) equation, which may be solved exactly for the hydrogen atom. For a multinuclear and multielectron system, the Schrodinger equation may be defined as ... 

Hybrid atomic orbitals are mathematical combinations of the normal atomic orbitals. They are as real as hydrogen-like AOs Hybrid orbitals are an effective way to describe multielectron/multiatom systems, i.e. molecules. They are extremely useful in organic chemistry since the molecules are too large to treat with MO theory and extract a physical picture. 

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