Shapes of Atomic Orbitals

Shapes of atomic orbitals play central roles in governing the types of directional bonds an atom can form. 

Click Coached Problems for a self-study module on shapes of atomic orbitals. 

In addition to size, an atomic orbital also has a specific shape. The solutions for the Schrodinger equation and experimental evidence show that orbitals have a variety of shapes. A second quantum number indexes the shapes of atomic orbitals. This quantum number is the azimuthal quantum number (1). 

The shapes of atomic orbitals are routinely confused with graphs of the angular factors in wave functions [60] and shown incorrectly. The graph of a py orbital, for example, gives tangent spheres lying on the y-axis. 

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

The pictures normally used to represent these angular forms are shown in Fig 4.4. Such diagrams of the shape of atomic orbitals can be interpreted in two ways ... 

In closing this section, a discussion of basis sets is in order. Almost all ab initio and DFT calculations require the use of a basis set, a set of functions in terms of which the molecular orbitals are constructed. In almost all cases, the functions chosen are atom-centered functions designed to mimic the shape of atomic orbitals. It is known that orbitals for many-electron atoms resemble the hydrogenic orbitals, and it is also observed that molecular orbitals can be expanded very efficiently in terms of atomic orbitals. One might think that a relatively small set of functions, essentially the optimized occupied atomic orbitals of the atoms making up the system, could be used... 

Two assumptions are made in this choice. Core orbitals are deemed to have negligible influence on bonding, and the shape of atomic orbitals is used to describe the molecular orbitals. More complete ab initio calculations often allow for orbital variation so the latter assumption is a possible source of error. The neglect of core orbitals is justified by their localized nature, which excludes significant participation in bond formation. A recent pseudopotential formulation by Cusachs (5), in which core orbitals were included, has shown that the form of the equations used in MO theory is unchanged although the input parameters may require some modification. Thus, most workers do not consider core orbital effects significant. 

At this point we can, again, appreciate the possibility of separating the total wave function into a radial and an angular wave function. The angular wave function does not depend on n and r, so it will be the same for every atom. This is why the shapes of atomic orbitals are always the same. Hence, symmetry operations can be applied to the orbitals of all atoms in the same way. The differences occur in the radial part of the wave function the radial contribution depends on both n and r and it determines the energy of the orbital, which is, of course, different for different atoms. 

B. RAMACHANDRAN, Examining the shapes of atomic orbitals using Mathcad. J. Chem. Educ., 72, 1082 (1995). 

The mathematical expression of i incorporates quantum numbers, which are related to the energy, size, and shape of atomic orbitals. 

I See the Saunders Interactive General Chemistry CD-ROM, Screen 7.13, Shapes of Atomic Orbitals. 

The shapes of atomic orbitals shown in Fig. 1 are important in understanding the bonding properties of atoms (see Topics C4-C6 and H2). 

Before the intricacies of bonding are described, a brief review of the shape of atomic orbitals is presented in Sec. 2.2. The concept of... 

The angular momentum quantum number ( ) has integral values from 0 to - 1 for each value of . This quantum number is related to the shape of atomic orbitals. The value of for a particular orbital is commonly assigned a letter = 0 is called s ... 

It is possible to calculate the shapes and energies of atomic and molecular orbitals by quantum theory. The shapes of atomic orbitals depend on the orbital angular momentum (the sub-shell). For each shell there is one s orbital, three p orbitals, five d orbitals, etc. The s orbitals are spherical, the p orbitals each have two lobes d or-... 

Atoms that are linked by electron-pair bonds are positioned so that orbital overlap is maximised. The orbitals used are also sensitive to bond overlap and hybridisation, so that atomic orbitals frequently mix to give hybrid orbitals with greater overlapping power. The shapes of atomic orbitals and hybrid orbitals are quite definite and point in fixed directions. This leads to the fact that covalent bonding is directional. From a geometrical point of view, the array of covalent bonds in a solid resembles a net. 

In many instances later in the text you will find that knowing the number and shapes of atomic orbitals will help you understand chemistry at the molecular level. You will therefore find it useful to memorize the shapes of the s, p, and d orbitals shown in Figures 6.19, 6.22, and 6.23. 

The VSEPR model, simple as it is, does a surprisingly good job at predicting molecular shape, despite the fact that it has no obvious relationship to the filling and shapes of atomic orbitals. For example, based on the shapes and orientations of the 2s and 2p orbitals on a carbon atom, it is not obvious why a CH4 molecule should have a tetrahedral geometry. How can we reconcile the notion that covalent bonds are formed from overlap of atomic orbitals with the molecular geometries that come from the VSEPR model ... 

If the value of is large in a unit volume of space, the probability of finding an electron in that volume is high—we say that the electron probability density is large. Conversely, if for some other volume of space is small, the probability of finding an electron there is low. This leads to the general definition of an orbital and, by extension, to the familiar shapes of atomic orbitals. 

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