Pi-electron approximation

Physical constants, table of, 467 Pi-electron approximation, 71 Pigpen, 320... 

The FE MO, HMO, and PPP methods are restricted to planar conjugated molecules (e.g., butadiene, benzene, pyridine), and make the pi-electron approximation of treating only those valence electrons that are in pi MOs (those that have eigenvalue — 1 for reflection in the molecular plane) they assume the existence of a pi-electron Hamiltonian Hm of the form... 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

Figure 2.36 A shows a typical low-loss spectrum taken from boron nitride (BN). The structure of BN is similar to that of graphite, i. e. sp -hybridized carbon. For this reason the low-loss features are quite similar and comprise a distinct plasmon peak at approximately 27 eV attributed to collective excitations of both n and a electrons, whereas the small peak at 7 eV comes from n electrons only. Besides the original spectrum the zero-loss peak and the low-loss part derived by deconvolution are also drawn. By calculating the ratio of the signal intensities hot and Iq a relative specimen thickness t/2 pi of approximately unity was found. Owing to this specimen thickness there is slight indication of a second plasmon.

In order to perform calculations on larger molecules in a reasonable amount of time, approximations are made, which may involve the neglect of certain terms, or the inclusion of experimentally determined parameters. The best known and simplest example of this level of approximation are Hiickel Molecular Orbital (HMO) calculations, which treat only pi-electrons, in conjugated hydrocarbons, with neglect of overlap (1). While obviously limited in use, HMO methods are still used in certain research applications. 

Another example is provided by the acidity of cyclopentadiene. This compound is approximately as strong an acid as water and is many orders of magnitude more acidic than other hydrocarbons because its conjugate base, with six pi electrons, is aromatic. 

Figure 1.6. Schematic representation of first-order configuration interaction for alternant hydrocarbons. Within the PPP approximation, conHgurations corresponding to electronic excitation from MO 4>i into and from MO., into are degenerate. The two highest occupied MOs (i =, k = 2) and the two lowest unoccupied MOs (f = r and k = 2 ) are shown. Depending on the magnitude of the interaction, the HOMO- LUMO transition

We thus find that the three pi electron pairs in C5H51 can be considered as housed in three localized regions toward which the pd2 hybrid GO s point. It may be mentioned that the approximately symmetrical distribution of pi electron pairs around any planar ring is a function only of the number of such pairs and is independent of the number of atoms making up the ring. 

Resonance structures are diagrammatic tools used predominately in organic chemistry to symbolize resonant bonds between atoms in molecules. The electron density of these bonds is spread over the molecule, also known as the delocalization of electrons. Resonance contributors for the same molecule all have the same chemical formula and same sigma framework, but the pi electrons will be distributed differently among the atoms. Because Lewis dot diagrams often cannot represent the tme electronic stmcture of a molecule, resonance stmctures are often employed to approximate the tme electronic stmcture. Resonance stmctures of the same molecule are connected with a double-headed arrow. While organic chemists use resonance stmctures frequently, they are used in inorganic stmctures, with nitrate as an example. 

Markovic, S. (2003) Approximating the total pi-electron energy of phenylenes in terms of spectral moments. Indian J. Chem., 42, 1304—1308. 

Infinite one dimensional conjugated pol3miers still maintain a finite bandgap due to a dimerization phenomonon known as the Pierels instability The free electron approximation (which views the pi electron region as an isotropic Drude electron gas) does not adequately explain this scenario. 

Aromatic hydrocarbons also decrease in solubility with size, although they are far more soluble (approximately two orders of magnitude on a molar basis) than aliphatic hydrocarbons having the same number of carbon atoms (Mackay and Shiu, 1977 Pearlman et al., 1984). The solubilities of benzene, toluene, and naphthalene are, respectively, 1750 mg/L (0.022 M), 550 mg/L (6 x 10" M), and 32 mg/L (2.5x10 M). Even a rather large aromatic compound such as phenanthrene (C Hio) has a solubility of nearly 1/tM. It is probable that there are significant charge-transfer interactions between water molecules and the pi electron clouds of aromatic species. 

The Hiickel method is simple and has been in use for decades (Hiickel 1931a, b). It is based on the a-n separation approximation while accounting for the pi-electrons only, i.e. the atomic orbitals involved refer to those 2pz for Carbon atoms as well to... 

Moreover, the benchmark ordering hierarchy was chosen as produced by Hiickel theory (since being an approximate approach for quantum chemical modeling of chemical bonding is let to be exposed in the Volume III of this work (Putz, 2016a), dedicated to quantum molecule and chemical reactivity) and approximation since closely related with pi-electrons delocalized at the ring level as the main source of the experimentally recorded aromaticity of organic compoimds imder study (Putz et al., 2010). 

Notice that the MOs are evenly distributed in energy with respect to the barycenter. If the orbitals were completely filled, the total energy would be the same (within the bounds of the Hiickel approximation anyway) as the energy of five filled 2p2 atomic orbitals. The four electrons in the 02" MOs would be destabilized by 4(1.62 ) = 6.48/ , while the six electrons in the and e/ MOs would be stabilized by the same amount 2(2/ )+4(0.62 ) = 6.48. The barycenter energy is indicated by the horizontal dashed line in Figure 10.45. Because there are only six pi electrons in the Cp ion, the total energy of these electrons is 6a + 6.48/ . The energy of the six pi electrons in three isolated ethylene molecules is 6a+ 6/ . 

By considering also relativistic effects, in addition to s electrons relativistic pi/2 electrons will also turn out to have finite density at the place of the nucleus. Then, for a uniformly charged spherical nucleus of radius R, single-electron approximation gives the following radial density distribution Pe r) ... 

The Hiickel method is simple and has been in use for decades (Hiickel, 1931a,b). It is based on the ct-ti separation approximation while accounting for the pi-electrons only, i.e., the atomic orbitals involved refer to those 2p for Carbon atoms as well to the 2p and 3p orbitals for the second and third period elements as (N,0, F) and (S, Cl) respectively further discussion on the d-orbitals involvement may be also undertaken, yet the method essence reside in non explicitly counting on the electronic repulsion with an effective, not-defmed, mono-electronic Hamiltonian, as the most simple semi-empirical approximation. In these conditions, for the mono-electronic Hamiltonian matrix elements two basic assumptions are advanced the forthcoming discussion follows (Putz, 201 Id) ... 

The particle in a one-dimensional box is a model system that consists of a single particle that can move parallel to the x axis. The particle moves without friction, but is confined to a finite segment of the x axis, from x = 0 to x = a. This interval is called a one-dimensional box, but could also be called a potential well. This model system could represent a particle sliding in a frictionless tube with closed ends or a bead sliding on a frictionless wire between barriers. One chemical system that is approximately represented by this model is a pi electron moving in a conjugated system of single and double bonds. We will discuss this application in a later chapter. 

The Hiickel method is a simple semi-empirical method for determining approximate LCAO molecular orbitals to represent delocalized bonding in planar molecules. It treats only pi electrons and assumes that the framework of sigma bonds has been treated separately. As an example we consider the allyl radical, CH2 = CH - CH2.. If the plane of the molecule is the xy plane, each carbon atom has an unhybridized 2pz orbital that is not involved in the sigma bonds, which are made from the Isp hybrids in the xy plane with the appropriate rotation of the coordinate system at each atom to provide maximum overlap. We construct linear combinations from the three orbitals, as in Eq. (21.6-2). 

Let us ignore the H atoms of benzene and note that a hexagon can be made into six isosceles triangles with approximately 1.4 A sides (actually 1.395), which is a good approximate value for a, and let us assume that the six pi electrons are three spin-pairs in levels n = 0, n=l, and n=—l. Then the HOMO —> LLfMO transition can be computed using the formula derived earlier. 

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Nowadays there is a tendency to use the word term to describe that which arises from an approximate treatment of an electron configuration, whereas the word state is used to describe something that is observable experimentally. For example, we can say that the s 2s 2p 3d configuration of C gives rise to a term which, when spin-orbit coupling is taken into account, splits into Pi, 2 Pj, states. Since spin-orbit coupling can be excluded only in theory but never in practice there can be no experimental observation associated with the P term. ... 

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