The atomic orbital model

This probability wiU be a number between 0 (impossibility) and 1 (certainty). 

If we are interested in calculating the probability of finding electron number 1 in dT regardless of the position of electron number 2, we must integrate the probability expression in (2.5) over all values for the coordinates of electron 2  

Finally, we define the total electron density in ra as the sum of the probability densities of electrons 1 and 2  

Due to the term in the Hamiltonian operator that represents the electron-electron repulsion energy it is impossible to find an exact solution to the SchrOdinger equation (2.3) in closed form. 

A common way to find an approximate description of the bound states (E 0) of two-electron atoms is to assume that the wavefunction of the atom may be written as the product of two functions, each of which depends on the coordinates of one electron only  

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Starting with Bohr s version of 1913, the evolution of this model was examined in an attempt to highlight the assumptions and approximations that were made at each stage. As in the case of many other papers in this volume, there is an educational motivation for raising these questions, especially in view of the central role of the atomic orbital model at all levels of chemical education. My suspicion is that many chemical educators do not appreciate the extent to which this model is an approximation and the conditions under which it ceases to be applicable. 

Propadiene adds hydrogen chloride to yield 2-chloropropene. However, the possibility exists that initial attack of a proton might lead to the 2-propenyl cation (Section 6-6), which then would react with chloride ion to form 3-chloropropene. Using the rules for application of the resonance method (Section 6-5B) and the atomic-orbital model for 1,2-propadiene (Figure 13-4), rationalize why a 2-propenyl cation might not be formed easily by addition of a proton to 1,2-propadiene and why 2-chloropropene is the observed product. 

Extension of the basic ideas of the VB treatment described in Section 21-2 to the atomic-orbital model of benzene is straightforward. We can write VB structures that represent pairing schemes of electrons in the atomic orbitals as shown in 9 through 13 ... 

MO treatment of the 2-propenyl cation begins with the atomic-orbital model 22 ... 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The neglect of electron-electron interactions in the Extended Hiickel model has several consequences. For example, the atomic orbital binding energies are fixed and do not depend on charge density. With the more accurate NDO semi-empirical treatments, these energies are appropriately sensitive to the surrounding molecular environment. 

Carbon has six electrons around the atomic core as shown in Fig. 2. Among them two electrons are in the K-shell being the closest position from the centre of atom, and the residual four electrons in the L-shell. TTie former is the Is state and the latter are divided into two states, 2s and 2p. The chemical bonding between neighbouring carbon atoms is undertaken by the L-shell electrons. Three types of chemical bonds in carbon are single bond contributed from one 2s electron and three 2p electrons to be cited as sp bonding, double bond as sp and triple bond as sp from the hybridised atomic-orbital model. 

In the 1930s a theoretical treatment of the covalent bond was developed by, among others, Linus Pauling (1901-1994), then at the California Institute of Technology. The atomic orbital or valence bond model won him the Nobel Prize in chemistry in 1954. Eight years later, Pauling won the Nobel Peace Prize for his efforts to stop nuclear testing. 

As pointed out in Chapter 7, the atomic orbital (valence bond) model regards benzene as a resonance hybrid of the two structures... 

This paper deals with some questions in the foundations of chemistry. The atomic orbital (or electronic configuration) model is examined, with regards to both its origins and current usage. I explore the question of whether the commonly-used electronic configuration of atoms have any basis in quantum mechanics as is often claimed particularly in chemical education. 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The structure of CaB contains bonding bands typical of the boron sublattice and capable of accommodating 20 electrons per CaB formula, and separated from antibonding bands by a relatively narrow gap (from 1.5 to 4.4 eV) . The B atoms of the B(, octahedron yield only 18 electrons thus a transfer of two electrons from the metal to the boron sublattice is necessary to stabilize the crystalline framework. The semiconducting properties of M B phases (M = Ca, Sr ", Ba, Eu, Yb ) and the metallic ones of M B or M B5 phases (Y, La, Ce, Pr, Nd ", Gd , Tb , Dy and Th ) are directly explained by this model . The validity of these models may be questionable because of the existence and stability of Na,Ba, Bft solid solutions and of KB, since they prove that the CaB -type structure is still stable when the electron contribution of the inserted atom is less than two . A detailed description of physical properties of hexaborides involves not only the bonding and antibonding B bands, but also bonds originating in the atomic orbitals of the inserted metal . ... 

Figure 2. [TCNE] [Bu,N].+ spin density obtained by MaxEnt reconstruction using an atomic orbital model, and subsequent projection onto the molecular plane of [TONE]-. Positive contour steps are 50 mpE/A- and negative contours are dashed (step 10mpB/A2). A significant off-centring is present.

When the electron configurations of the elements were worked out, it became clear that the valence electrons of the period 2 elements must be accommodated in just four orbitals, the 2s and the three 2p orbitals. In the localized orbital model it is assumed that each bond can be described by a localized orbital formed by the overlap of one orbital on each of the bonded atoms. According to this model, therefore, a period 2 element can form bonds with at most four ligands so that electron configurations appeared to provide a justification for the octet rule. 

The model of the metal surface, as described above, has been used recently by several authors for the interpretation of their results of chemisorption and catalytic studies 67-72), mostly without stating explicitly that the atomic orbitals are probably hybridized and can be approximated by purely atomic orbitals during the interaction only. The hybridization... 

In the familiar standard Huckel treatment2174 of planar 7r-systerns, the basis functions are atomic orbitals (AO) 2pz /x = (plL, z, being the coordinate perpendicular to the molecular plane. In analogy to the previous model, the basis energies of the atomic orbitals and the cross terms between neighbouring pairs of AOs, [Pg.203]

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