Jan. 7, 1952

To: Dr. Fernandez

From: Linus Pauling

Subject: Proposed investigation of electronic structure of phosphorus pentachloride

I think that the work that you have been carrying out on the electric dipole moments of halogen derivatives of methane is not interesting enough to justify your continuing with it. I suggest instead that you carry out some calculations on the electronic structure of molecules such as phosphorus pentachloride.

Phosphorus pentachloride has the configuration of a trigonal bipyramid. According to an unpublished investigation by Schomaker, the two axial chlorine atoms are 2.11 Å. from the phosphorus atom, and the other three chlorine atoms, in equatorial positions, are 2.04 Å. from the phosphorus atom. Rouault at first gave the values 2.25 and 2.10 Å. for these distances, and then, after the publication of Schomaker's values in the Nature of the Chemical Bond, he reported 2.07 and 2.01 Å.

I think that some theoretical work on PCl_{5} has been reported, but I do not have the references at hand. I suggest that you look through the literature, especially the Journal of Chemical Physics for the last few years, to see what you can find about orbitals for the trigonal bipyramid.

In the Nature of the Chemical Bond the statement is made that the phosphorus atom can form 5 covalent bonds, in these atoms, by the use of 1 3d orbital. I suggest that an investigation of the electronic structure of PCl_{5} be made, by methods somewhat similar to those used by Dr. Simonetta in the paper on bond orbitals and bond energy in elementary phosphorus. An outline of the proposed calculations is given in the following paragraphs.

Let us assume first that the 5 bonds are pure covalent bonds, formed with use of one 3d orbital. The problem is then to find the best sp^{3}d orbitals that can be formed. It is necessary that the orbitals be orthogonal to one another, and we define as the best orbitals the orbitals that make the bond energy for the molecule a maximum. We assume that the bond energy for a P-Cl bond is proportional to the product of the strength S of the chlorine bond orbital and the strength of the phosphorus bond orbital. The strength of the chlorine bond orbital is assumed to be constant; hence our problem is to find the set of 5 orthogonal sp^{3}d orbitals for which the sum of the strengths in the respective bond directions (toward the corners of the trigonal bipyramid) is a maximum. I suggest that you solve this problem first, in case that it has not already been solved and reported in the literature. If you find that it has been reported in the literature, please check the calculations.

It will be interesting to see whether or not the two kinds of bond orbitals for the phosphorus atom, as determined by this treatment, differ in their bond strengths, and especially whether they differ in such a way that the three equatorial bond orbitals are better than the two axial ones. Next, we must consider the partial ionic character of the phosphorus-chlorine bonds. Phosphorus has electronegativity 2.1, the same as hydrogen, and accordingly we predict about the same amount of partial ionic character for the phosphorus-chlorine bond as for the hydrogen-chlorine bond. The latter is indicated by the electric dipole moment of hydrogen chloride to be 17% If we assume that there is synchronism in the achievement of partial ionic character by the 5 bonds, each having the ionic aspect P^{+}-Cl^{-} 17% of the time, the five ionic structures of the sort Cl-Cl-P^{+}-Cl-Cl Cl^{-} make a total contribution of 85%, leaving only 15% for the completely covalent structure. It is, of course, possible that the amount of partial ionic character is somewhat greater, and that the completely covalent structure makes a still smaller contribution than 15%. In any case, it is clearly important to investigate the configurations of the phosphorus atom in which 4 covalent bonds and 1 ionic bond are formed.

I suggest that as your second job in the attack on the problem you discuss the nature of the best bond orbitals that can be formed by a phosphorus atom in this situation. There are two structures to be considered. The first is that in which one of the axial bonds is ionic, and the second is that in which one of the equatorial bonds is ionic. The first problem is to be tackled by essentially the same methods as that used by Simonetta in the discussion of the P_{4} molecule. Equations are to be formulated for three equivalent equatorial bond orbitals, projecting approximately in the directions of the three equatorial chlorine atoms, and a fourth axial bond orbital, extending in the positive direction of the z axis. I think that the discussion on page 6 of the elementary phosphorus manuscript relates to the three bond orbitals, which together have the symmetry given by a three-fold axis, the z axis. You need in addition a fourth orbital of the spd type that is orthogonal to each of these three. You are then to evaluate the strength of each of the three orbitals in the direction of its respective equatorial chlorine atom, and the strength of the fourth orbital in the direction of the positive z axis. It might be interesting for you to maximize the sum of these four bond strengths, without consideration of the energy of promotion of an electron from the s or p orbital to the d orbital. The actual problem, however, is to minimize the energy for the phosphorus atom with consideration both of the bond energy and of the promotional energy. The secular equation given on page 7 of the manuscript is not the correct one for you to use. I suggest that you set up the expression for the total energy of the molecule, including bond energy and promotional energy, on the assumption that there are three equatorial covalent bonds and one axial covalent bond, and then minimize the total energy with respect to all of the parameters.

In setting up the energy for the molecule you may make use of the atomic energy values given on page 5 of the manuscript; namely, you may assume that the energy of promotion of a 3s electron to a 3p orbital is 194 kcal/mole^{-1} and that the energy of promotion of a 3p electron to a 3d orbital is also 194 kcal mole^{-1}.

In addition you need to make some assumption about the phosphorus-chlorine bond energy. The energy of a normal phosphorus-phosphorus bond is given in the manuscript as 51.6 kcal mole^{-1} (please check this value with the published paper - perhaps a change has been made). The chlorine bond energy, in Cl_{2}, is 57.8. The average of these two is 54.7 kcal mole^{-1}, which can be taken as the normal covalent bond energy for these two atoms. The strength of a normal phosphorus bond orbital is given on page 7 of the manuscript as 2.118. Hence the energy of a phosphorus-chlorine bond can be taken as 25.9 S kcal mole^{-1}, where S is the strength of the phosphorus bond orbital in the direction of the bond; that is, in the direction of the line from the phosphorus nucleus to the chlorine nucleus.

After you have solved this problem, the next calculation to be made is the same one for the case that two covalent bonds are formed to the two axial chlorine atoms, and two covalent bonds to two of the three equatorial chlorine atoms. This calculation is to be carried out in the same way as the preceding one.

The ionic aspect of the phosphorus pentachloride molecule may now be represented approximately by five ionic functions, with equal weight; these are tow functions of the first kind, in which there is an ionic bond to one of the two axial chlorine atoms, and three functions of the second kind, in which there is an ionic bond to one of the three equatorial chlorine atoms. If the average strength of the covalent bonds to an axial chlorine atom turns out to be less than the average strength of the covalent bonds to an equatorial chlorine atom, as given by this calculation, we may consider that the difference in phosphorus-chlorine distance reported by Professor Schomaker, and also by Rouault, has been given a theoretical explanation. It might even be possible to discuss the magnitude of the difference in interatomic distance in terms of the nature of the bond orbitals.

Finally, a discussion of the significance of the ionic structures and of the completely covalent structure should be carried out. This can be done by setting up a secular equation for the six structures. The energy values to be introduced in the diagonal terms of the secular determinant are those that have been calculated by the methods described above. The non-diagonal matrix elements are to be obtained with use of the formulas published by Schomaker and Simonetta. These formulas permit a reasonable estimate to be made of the resonance integral corresponding to change from a covalent to an ionic structure. The Coulomb integral for the ionic structure should be given such a value as to correspond to about 17% partial ionic character for an isolated phosphorus-chlorine bond. The resonance integral between one ionic structure and another ionic structure would have to be estimated, with use of the expressions given by Schomaker and Simonetta. The secular equation would then be solved, and the solution would give a value for the total energy of the phosphorus pentachloride molecule (uncorrected, however, for the energy of van der Waals repulsion of adjacent chlorine atoms), and also values for the contributions of the five ionic structures and the covalent structure to the normal state of the molecule. This final calculation should permit a statement to be made as to the extent to which the phosphorus atom has exceeded the octet in forming five bonds to chlorine.

There are a number of other problems related to this one, that might be tackled later on. One of them is the problem of the configuration of the TeCl_{4} molecule. The configuration of other halogen halides, such as IF_{7}, might also be discussed.

Linus Pauling:W

cc: Prof. Schomaker