The Shape of Atoms

One of the intensively discussed topics was the question about the shape of atoms - a question with also high educational relevance, if we think of some students preconceptions. We can focus on two main positions One group of chemists observed that crystals always broke to parts with similar shapes to those of the original crystal. Therefore, their theory was that the smallest particles of atoms must have also the same shape as the macroscopic crystal. If we follow Domenico Guglielmini (1655-1710), the particles of sodium chloride would be cubic, those of calcspar would be triclinic. Hexagonal-prismatic and trigonal-prismatic crystals... 

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. 

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. 

Whenever the atoms under consideration in a given molecule are in the same valence states as in the reference molecule, the relaxation process is such that the potential created by the other atoms at the kth nucleus is the same as would be predicted by leaving the pertinent intemuclear distances and the shapes of atomic electron densities as they are in the reference molecule, with the electron populations changed as required by the new situation. 

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 ... 

Rost, J.M. and Pattard, T. (1997). Analytical parameterization for the shape of atomic ionization cross sections. Phys. Rev. A 55 R5-R7. 

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

According to Lucretius All nature as it is in itself consists of two things— bodies and the vacant space in which the bodies are situated and through which they move in different directions (p. 39). He addresses the question of the immense variety of material things found in nature by recognizing that there must be some way for atoms to combine and at the same time maintain their individual characters Material objects are of two kinds, atoms and compounds of atoms. The atoms themselves cannot be swamped by any force, for they are preserved indefinitely by their absolute solidity (p. 41). Lucretius does not suggest that we directly experience atoms. He makes no claims as to the shapes of atoms or any other of their characteristics, see also Atoms. 

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. 

Three-dimensional representations of the shapes of atomic orbitals... 

Besides providing a way to determine the shapes of atomic orbitals, the Schrodinger equation also provides a way to approximate the energetics of covalent bond formation. These approximations have taken two forms (1) valence bond (VB)... 

A possible explanation for the failure of LI(12,6) is the following. The LJ(12,6) is describing successfriUy atomic interactions. The shape of atoms is much better defined than the shape of amino acid side chains. Amino acids may have flexible side chains and alternative conformations, making the range of acceptable distances significantly larger. To represent alternative configurations of the same type of side chains, potentials with wide minima are required. 

Exhalations] be lifted up into a subtile or fine Gas in the most cold air. . . and do assume a condition in the shape of.. . Atomes. . . and do return unto their former Element of water. .. So the water which existed from the beginning of the Universe is the same, and not diminished, and shall be unto the end thereof. . . The auncient water always materially remaineth. ... 

Quantum Mechanics and the Atom 315 7.6 The Shapes of Atomic Orbitals 321 Key Learning Outcomes 329... 

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