Powell Equivalent d-Orbitals

Once a student becomes accustomed to realizing that what counts is the probability density of orbitals as positive numbers, it should be possible to make simple freehand drawings of at least Is, 2s, 2p c, 2py, 2pj., 3s, 3px, 3py, 3p, 3d, 3dx, 3dy, 3d ,, and 3dxi yi orbitals. That is what a student could do after skipping over the tedious polynomial derivations After doing those sketches, students may wonder why there are three equivalent shapes for the p-orbitals but the d-orbitals require a different shape for the orbital. This is due to the fact that there are actually six ways to draw the four-leaf-clover shape of the d-orbitals in Cartesian representation but only (21 -f 1) = 5 orbitals for the 1 = 2 shell in the (0, ()) representation. However, one can form a d orbital by combining two four-leaf clovers as d i = N(dxz + renormalizing to find N. Thus the usual 

We need to start using matrix algebra so we introduce the idea of a matrix as an array of numbers in a table surrounded by [ ] or in some cases by ( ) and the dimension of the matrix will be 

Final Combined Forms of the H Atom Wave Functions 

Applications to Chemistry, McGraw-Hill Book Inc., New York, 1935, Chaps in V. lie copyright is now owned by Dover Press, New YotIc, and is published as an unabridged reprint. 

FIGURE 14.7 Stick models of the staggered arrangement of five-equivalent d-orbital bond directions. The application to the bonding in ferrocene seems obvious (From Powell, R.E., J. Chem. Educ., 45, 45, 1968. With permission.) 

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