Atomic number nodal planes

The molecular orbitals are labelled a and ir depending on whether they are symmetrical about the internuclear axis or have a nodal plane passing through the nuclei. The m.o. s are numbered in sequence of increasing energy, independent of the numbering of the atomic orbitals. This numbering serves to avoid any confusion in cases where atomic orbitals from different levels are combined, as in... 

Fig. 4.11 Linear combination of n atomic orbitals (or, more generally, basis functions) gives n MOs. The coefficients c are weighting factors that determine the magnitude and the sign of the contribution from each basis function. The functions contributing to the MO change sign at a node (actually a nodal plane) and the energy of the MOs increases with the number of nodes...

We now categorize the basis atomic orbitals in any, roughly spherical, cluster according to the number of nodal planes they possess that contain the radius vector. Hence, s atomic orbitals are classified as cr-type cluster orbitals, and so are p and d 2 orbitals, where the local axes at each vertex are chosen with the X and y directions tangential to the surface of the sphere so that z points outwards along the radius vector. Radially directed hybrids, such as sp are also a-type under this classification. Examples will be illustrated below. 7r-type cluster orbitals have one intrinsic nodal plane containing the radius vector, for example, p , Py, and dy in the same axis system. A y and Aj 2 y2 functions contain two such nodal planes, and are known as 5 orbitals. ... 

Because the number of molecular orbitals must equal the number of atomic orbitals used, two additional tt molecular orbitals are needed. Each of these has a single nodal plane that is perpendicular to the plane of the molecule and bisects the molecule the nodes for these two molecular orbitals are perpendicular to each other ... 

Finally, note that an ns orbital has n 1 radial nodes a radial node is a spherical surface about the nucleus on which tf/ and are 0. These spherical surfaces are the analogues of the nodal planes in the wave functions for a particle in a cubic box (Figure 4.28). The more numerous the nodes in an orbital, the higher the energy of the corresponding quantum state of the atom. Just as for the particle in a box, the energies of orbitals increase as the number of nodes increases. 

El. 15 An orbital defined with the quantum number Z has / nodal planes (or angular nodes). For a 4p set of atomic orbitals 1=1, and each 4p orbital has one nodal plane. The number of radial nodes is given by n - Z - 1 and 4p orbitals have 4 1 — 1=2 radial nodes. The total number of nodes is thus 3. (Note that the total number of nodes, angular... 

Potentials in atoms are complicated as are the resulting wavefunctions of the valence electrons. However, these orbitals have some features in common with those of electrons in some simple potentials. The orbitals in a cubic flat-box potential (the potential is considered to be constant over the entire volume of the box) turn out to be sine functions if the Schrodinger equation is solved, like the eigenfunctions of a vibrating string. Each different potential imposes a different form on the wavefunctions, which are similar in shape to the sine functions of the flat-bottomed box with nodal planes that increase in number with the quantum number n. 

---