Atomic orbitals, nodes

In molecular orbitals, as in atomic orbitals, nodes are regions of zero electron density that divide an orbital into lobes with amplitudes of opposite sign. When a node coincides with a nuclear position, there are no lobes depicted on that atom. In the following diagram, we see that the bonding v molecular orbital for ethylene has no nodes perpendicular to the bond axis, whereas the antibonding tt orbital has one node perpendicular to the bond axis. 

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

The radial functions Rniir) for the Is, 2s, 2p, 3s, 3p, and 3d atomic orbitals are shown in Figure 6.4. For states with / 0, the radial functions vanish at the origin. For states with no angular momentum (/ = 0), however, the radial function Rno r) has a non-zero value at the origin. The function Rniir) has ( — /— ) nodes between 0 and oo, i.e., the function crosses the r-axis in — I — ) times, not counting the origin. 

But these rules are reversed by the presence of a node in the arrangement of atomic orbitals. Thus a system having (An + 2)n electrons and a node would be antiaromatic while that with An n electrons and a node would be aromatic and hence stable in the ground state. 

All lone pair orbitals have a node between the two atoms and, hence, have a slightly antibonding character. This destabilizing effect of the lone pair localized molecular orbitals corresponds to the nonbonded repulsions between lone pair atomic orbitals in the valence bond theory. In the MO theory all bonding and antibonding resonance effects can be described as sums of contributions from orthogonal molecular orbitals. Hence, the nonbonded repulsions appear here as intra-orbital antibonding effects in contrast to the valence-bond description. 

For a hydrogen atom, the lowest energy solution of the wave equation describes a spherical region about the nucleus, a Is atomic orbital. When the wave equation is solved to provide the next higher energy level, we also get a spherical region of high probability, but this 2s orbital is further away from the nucleus than the Is orbital. It also contains a node, or point of zero probability within the sphere... 

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

The Schrodinger equation is a differential equation, which means that solutions of it are themselves equations. The solutions, however, are not differential equations, but simple equations for which graphs can be drawn. Such graphs, which are three-dimensional pictures that show the electron density, are called orbitals or electron clouds. Most students are familiar with the shapes of the s and p atomic orbitals (Figure 1.1). Note that each p orbital has a node—a region in space where the probability of finding the electron is extremely small.2 Also note that in Figure 1.1 some lobes of the orbitals are labeled + and others -. ... 

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