Bond orbitals, linear combinations

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

A very simple procedure was proposed by Del Re 87 for representing the <7 bonds in saturated compounds. Its essential features are as follows, (a) Treat each bond as a two-electron problem, (b) Describe each electron in the bond by a molecular orbital linear combination of... 

Since there is no simple relation between bonds and eigen-functions in a many-atom molecule, the discussion will generally use a molecular orbital rather than a valence bond approach. The molecular orbitals used to rationalise the bonding are linear combinations of atomic orbitals, this arbitrarily limited basis set being easy to visualise. We shall occasioncdly speak of altering the size and shape of these orbitals this is equivalent to using an expanded basis set. 

The bonding may be described as three centre, four electrons s -bonding. A linear combination of the atoms gives the best overlap of the orbitals. 

In binary semiconductors having a sphalerite structure (the structure of zinc blende), the valence band in the tight-binding MO model is a combination of sp -hybrid orbitals, linear combinations of the s and three p orbitals on the metal atom that are directed toward the neighboring nonmetal atoms and form bonding combinations with atomic orbitals (s and p) on them. Similarly the empty conduction band is then built up of the antibonding combinations of the sp -hybrids on the metal atoms directed away from the bonded neighbor toward the interstices of the lattice. 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

In the late 1920s, it was shown that the chemical bond existing between two identical hydrogen atoms in H2 can be described mathematically by taking a linear combination of the Is orbitals [Pg.176]

Hoffman s extended Huckel theory, EHT (Hoffman, 1963), includes all bonding orbitals in the secular matrix rather than just all n bonding orbitals. This inclusion increases the complexity of the calculations so that they are not practical without a computer. The basis set is a linear combination that includes only valence orbitals... 

The molecular orbital approach to chemical bonding rests on the notion that as elec trons m atoms occupy atomic orbitals electrons m molecules occupy molecular orbitals Just as our first task m writing the electron configuration of an atom is to identify the atomic orbitals that are available to it so too must we first describe the orbitals avail able to a molecule In the molecular orbital method this is done by representing molec ular orbitals as combinations of atomic orbitals the linear combination of atomic orbitals molecular orbital (LCAO MO) method... 

Figure 7.15 In HCl (a) the single-bond molecular orbital is formed by a linear combination of lx on H and 3p on Cl, and (b) electrons in the 3py. and 3py atomic orbitals on Cl remain as lone pairs...

Hiickel s calculations on planar conjugated systems were extensively exploited, and I refer you once again to Streitwieser s classic book. Molecular Orbital Theory for Organic Chemists. What few calculations that had been done at that time on the (T framework had used the method of linear combination of bond orbitals. 

Once NBOs have been identified, they may be written as linear combinations of the NAOs, forming a localized picture of the atomic orbitals involved in the bonding. 

Another example which illustrates beautifully the mixing of a group orbitals to form delocalized molecular orbitals is benzene. First of all the six crcc bond orbitals interact to give six linear combinations which are delocalized over the entire carbon skeleton. The amplitudes of the various bond orbitals in each [Pg.23]

The relevant orbitals to consider are the molecular orbitals 5Aj and 6A, of cyclopropene (III.46) which correlate with the lowest acc orbital of cyclopropane (3A, III.56) and with one of the higher degenerate <7CC orbitals (3E, III.56). The cyclopropene orbitals are very similar to their cyclopropane counterparts. In particular the 6Aj orbital is a linear combination of all three crcc bond orbitals and extends over the entire cyclopropene molecule. Its amplitude, shown in Fig. 46, is fairly accurately represented by the... 

The point of this little diversion has been to show how a bond orbital like y/a above can, quite generally, be thought of as some linear combination of just metal-centred orbitals. 

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