Electronic Structure of Small Molecules

In describing states of the small molecule (as well as the solid) the first step is to enumerate each of the electronic states in the atom that will be used in the mathematical expansion of the electron states in (he molecule. These become our basis states. We let the index a = 1, 2, 3. n run from one up to the number of states that are used. Then the molecular state may be written (with the notation discussed in Section 1-A) as 

we must find the coefficients of Eq. (1-23) for the electron state of lowest energy, by doing a variational calculation as indicated in Section 1-A. That is, we evaluate the variation 

In obtaining the second form, we allow the to be complex, (hough ordinarily for our purposes this would not be essential. We also make use of the linearity of the Hamiltonian operator to separate the various terms in (he expectation value of (he Hamiltonian. In particular, if we require that variations with respect to a particular uf be zero (as in Eq. 1-10), we obtain 

Let us use the foregoing method to describe the states in a small molecule. The hydrogen molecule, with two electrons, is a simple case and is more closely related to the systems we shall be considering than the simpler hydrogen molecular ion, H2 For the hydrogen molecule, we use two orbitals, 11) and 2), which represent Is stales on atoms 1 and 2 respectively. Eq. (1-26) then becomes 

Substituting the eigenvalues given in Eqs. (1-29) and (1-30) back into Eq. (1-28) gives coefficients m, and 1/2. For the bonding state, m, = 2 = 2 and for the antibonding state, i = — 2 = 2 The conventional depiction of these bond orbitals and antibond orbitals is illustrated in Fig. l-10,a. 

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Huggins, 1968). An example of the variation in some of these functions is shown in Fig. 6. It should be emphasized that all of these functions are based on limited data and assumptions whose validity is difficult to substantiate. Also, it is possible that two sets of functions may be required, one for near-neighbor interactions and another for interactions between remotely connected parts of the chain (Scott and Scheraga, 1966c). Therefore, recent attempts have been made to refine these functions by using them to compute the known structures of small molecules from electron diffraction data and the known crystal structures of small molecules from X-ray diffraction data. Jacob et al. (1967), 5 ... 

For instance, solvent effects can represent much more than just a small perturbation of the electronic structure of a molecule. Sometimes these effects can be strong enough that the chiroptical response can be dominated by their influence on the chiral solute. For instance, effects that should be considered are how a solvent shell perturbs the electronic structure of the solute (and therefore its chiral response [155]), or the possibility of a solute transferring its chirality to the surrounding solvent shell (influencing one particular solvent molecule [151] or the entire solvent shell [156]). Here and in the following discussions it is assumed that the solvent itself is not chiral, and therefore the only contributions to the chiroptical response of the solution are from the solute or from chirality induced by the solute in the solvent shell. 

Dorset, D.L. (1996). Electron crystallography. Acto Crystallogr. B, 52,753-69. [112] Dorset, D., McCourt, M. R, Gao, L. and Voigt-martin, I. G. (1998). Electron crystallography of small molecules criteria for data collection and strategies for structure solution. J. Appl. Crystallogr., 31, 544-53. [112]... 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

Such maps are primarily used to refine a trial structure, to find a part of the structure that may not yet have been identified or located, to identify errors in a postulated structure, or to refine the positional and displacement parameters of a model structure. A difference map is very useful for analyses of the crystal structures of small molecules. It is also very useful in studies of the structures of crystalline macromolecules, since it can be used to find the location of substrate or inhibitor molecules that have been soaked into a crystal once the macromolecular structure is known. A formula like that in Equation 9.1.5 is then used. When a structure determination is complete, it is usual to compute a difference electron-density map to check that the map is flat, and approximately zero at all points. 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

There arc fundamental dil fcrcnees between the quantum and molecular mechanics approaches. They illustrate the dilemma that cun confront the medicinal chemist. Quantum mechanics is derived from basic theoretical principles at the atomic level. The model itself is exact, but the equations used in the technique are only approximate. The molecular properties are derived from the electronic structure of the molecule. The assumption is made that the distribution of electrons within a molecule can be described by a linear. sum of functions that represent an atomic orbital. (For carbon, this would be s./>,./>,. etc.) Quantum mechanics i.s computation intensive, with the calculation time for obtaining an approximate solution increasing by approximately N time.s. where N i.s the number of such functions. Until the advent of the high-.speed supercomputers, quantum mechanics in its pure form was re.stricted to small molecules. In other words, it was not practical to conduct a quantum mechanical analysis of a drug molecule. 

A second important mechanism for fluorine spin-lattice and spin-spin relaxation is produced by the chemical shielding anisotropy (CSA) [13, 14, 21, 71]. The magnetic field experienced by a nuclear spin depends on both the electronic structure of the molecule and how easily the electrons can move in the molecular orientations. In addition, the CSA depends on how the molecule is oriented in the magnetic field. Like spin-spin and dipole-dipole interactions, the CSA of small, rapidly tumbling molecules will be an averaged value (the chemical shift). However, these tumbling motions cause fluctuations of the local magnetic field that lead to relaxation. Also slower reorientation, or an environment that restricts the molecular motion, will result in broader lines due to CSA. 

A number of theoretical investigations of the electronic structures of small carbon-containing molecules have been undertaken.4-6 The relative... 

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