**XI. HOW DO WE DETERMINE THE MINIMUM NUMBER
OF PARAMETERS (AND THEIR FORMS) NEEDED FOR LINEAR REGRESSION ANALYSIS OF PROPERTIES
OF PHOSPHINES AND THEIR COMPLEXES?**

We begin by assuming that the variations of a physicochemical property of a phosphine or of a complex is linearly related to a set of parameters (x1, x2, x3, …) whose number is unknown. For example, two different properties might be described by equations 1 and 2, where e and e’ are constants

prop1 = ax1 + bx2 + cx3 + …+ e .............................................(1)

prop2 = a’x1 + b’x2 + c’x3 + … + e’..................................... (2)

We will show that by plotting one property vs. the other (a "property vs. property plot") and examining a series of such plots for the same families of ligands, we can determine the minimum number of parameters (and their forms) needed to describe these variations of physicochemical properties.1

Consider a plot of one property vs the other for a given set of ligands. First we relate prop1 to prop2 by eliminating x1 between them to get:

prop1 = (a/a’) prop2 + (b-ab’/a’)x2 + (c-ac’/a’)x3 + … + (e-ae’/a’)............... (3)

For prop1 to be linearly related to prop2, equation 3 must reduce to the form:

prop1 = A prop2 + const........................................................ (4)

For equation 4 to hold, one of the following statements must be true.

a. Each property depends only on one and the same parameter.

b. Each property depends on a number of parameters, but only one varies while the others are constant

c. Sufficient correlation exists among all the parameters so as to reduce the effective number of parameters to one.

d. In a rarer case, each property depends on the same set of parameters and the ratio of coefficients of the xi are all the same (i.e., a/a’ = b/b’ = c/c’ = …).

First we consider a set of ligands consisting of the PR3 and PAr3 and the two properties, Bodner’s d-values2 (the relative 13C chemical shift of the CO ligands in LNi(CO)3) and the c-values3 (related to the A1 nCO values of LNi(CO)3). (Figure 1). The linearity of the plot indicates that the minimum number of parameters needed to describe each property is one. Furthermore, since the points for each family are distributed along the line, x1 must be a variable parameter. In the QALE model, x1 is the electronic parameter c.

**Figure 1.** Plot of Bodner's d values versus c. The linear relationship indicates that the minimum number of stereoelectronic parameters necessary to describe the stereoelectronic properties of the ligands is 1. The data for PAr3 are shown as open circles and the data for PR3 are shown as filled squares.

In the next example, we plot the Eo values of the h-Cp(CO)Fe(L)(COMe)o/+ couple versus nCO for h-Cp(CO)Fe(L)(COMe)o (Figure 2)4. We observe in this plot that the data for PAr3 and PR3 fall on two separate but parallel lines. A second parameter is needed and consideration of equation 5 allows us to establish the form of this parameter.

Eo = (a/a')nCO + (b-ab'/a')x2 + (e - ae'/a')............................................. (5)

In equation 5, the terms a/a', b-ab'/a', e - e'/a', are the same for both PR3 and PAr3. **Thus, to get 2 parallel lines x****2 must have different values for PR3 and PAr3 and be constant within each family**. In our QALE model, x2 is the aryl effect parameter, Ear5, which is 2.7 for PAr3 and 0 for PR3.

**Figure 2.** A plot of the Eo values for h-Cp(CO)Fe(L)(COMe)o/+ versus nCO of h-Cp(CO)Fe(L)(COMe)o (cyclohexane). The observation of two parallel lines indicates that at least two stereoelectronic parameters are operative, one that determines the slopes and one that determines the separation between the lines. The data for PAr3 are shown as open circles and the data for PR3 are shown as filled squares.

In the third example, the system is Ir(CO)(Cl)L2, and we plot in Figure 3A log k for the second order addition of MeI6 versus nCO6, and in Figure 3B we plot A1 nCO for LCr(CO)57 versus c, which is just another property of the phosphines. In Figure 3A we see that the data for PAr3 and PR3 lie along lines with very different slopes. In Figure 3B, again we see lines with different slopes. So far we have identified x1 as a variable parameter (c) and x2 as a parameter (Ear) that differs for PR3 and PAr3 but is constant within each family of ligands.

To account for a difference in slope, a third, continuous parameter, x3, is needed. Either x3 is variable for only one family and constant for the other, or it is variable for both. Furthermore, if x3 is variable, then since the data for each family still falls on a straight line, x3 must be linearly correlated with x1, and this correlation must be different for the two families.

In the QALE model, x3 is the steric parameter,q ,which is linearly correlated with c for PR3 but is constant at 145o for PAr3. (For some PR3 the linear relationship between c and q does not hold, e.g. P(i-Bu)3. These ligands are excluded from the graphical analysis but would be included in a full regression analysis.) Thus, the slope of the PAr3 line is determined by the coefficients of c, whereas the slope of the PR3 line is determined by the a combination of the coefficients of the correlated c and q parameters.

**Figure 3.** (A) Plot of log k for the addition of MeI to Ir(CO)(Cl)L2 versus nCO of Ir(CO)(Cl)L2 and (B) the a plot of A1 nCO of LCr(CO)5 versus c. The differences in the slopes of the lines indicates the need for a third parameter. The data for PAr3 are shown as open circles and the data for PR3 are shown as filled squares.

Finally, to show that still a fourth parameter is needed, we take as one property, log k for the dissociation of CO from LRu(CO)48,9 and let c be the other property. In Figure 4, we show this plot of log k versus c . We observe that the data for PAr3 lie along a single line but that there is break in the line for the PR3 data. Hence, we need a fourth parameter (x4) that breaks the trialkyl ligands into two groups. In the QALE model, x4 is the steric parameter, (q - qst)l.

**Figure 4.** Plot of log k for the dissociation of CO from LRu(CO)4 versus c. The break in the line for the PR3 demonstrates the need for a fourth stereoelectronic parameter. The data for PAr3 are shown as open circles and the data for PR3 are shown as filled circles.

Thus, we can see that at least four parameters are generally needed to describe the stereoelectronic properties of alkyl and aryl phosphines. This conclusion is based on simple linear algebra and is not dependent on any particular model.

** **Inclusion of phosphites into the set of alkyl and aryl phosphorus(III) ligands brings the possibility of M-P p bonding into the picture. If p bonding is indeed involved, then we can incorporate its presence into the QALE model as follows. To equations 1 and 2 we add a term, dx4,,where x4 is the new parameter associated with p bonding, and form equations 6 and 7

prop1 = ax1 + bx2 + cx3 + dx4 + e..................................... (6)

prop2 = a’x1 + b’x2 + c’x3 + d’xd + e’ ............................(7)

In the QALE model, the coefficients, a, b, c, e and a’, b’, c’, e’ must be invariant to the absence or presence of ligands that exhibit M-P p bonding. Now the task is to determine whether the parameter, x4, is indeed necessary, and again property vs. property plots will be used to address this question. In anticipation of the final results, x1 through x3 will be identified as cd, q, and EAR; and x4 will be denoted as pp. (Note that cd and c are identical for PR3 and PAr3, however, for p-acidic ligands, c contains a contribution from p effects. Removal of p effects, as discussed in ref. 4, results in the s-donor parameter, cd). On eliminating x1 (i.e., cd) between equations 6 and 7, we find

prop1 = (a/a’) prop2 + (c-ac’/a’)EAR + (d-ad’/a’)pp +(e-ae’/a’)..................... (8)

In the following example, we use as our properties nCO for h-Cp(CO)(L)Fe(COMe)o, and DHo and Eo (229 K) for the Cp(CO)(L)Fe(COMe)+/o couple 4. These three properties are virtually independent of q. In the plot of nCO versus Eo (229 K) (Figure 5A), we see a set of parallel lines. The similar slopes and distribution of points indicate that both properties are responding to a single parameter that is variable as well as common to all the ligands. In the QALE model this parameter is cd. The different intercepts requires that the remaining parameters differ from family to family by a constant value. For example, in Figure 5A, the line for P(p-XC6H4)3 lies below the line for PR3 because of the 'aryl effect' which is constant (2.7) for P(p-XC6H4)3 and zero for PR3. (pp is assumed to be zero for both P(p-XC6H4)3 and PR3) Notice that the P(OR)3 and P(O-p-XC6H4)3 lie on two lines on the **other side **of the PR3 line and the point for P(OCH2)3CEt lies above all the lines. This pattern of lines could be described by equation 8 with all pp values equal to zero, provided that the phosphites possess negative values of the 'aryl effect' parameter, Ear. If this be the case, then the lines for the phosphites and P(p-XC6H4)3 should always lie on opposite sides of the PR3 line in the absence of a steric effect.

However, we observe a different ordering of the lines, when we plot nCO versus DHo, Figure 5B. In this case **all the lines lie below** the PR3 line. Thus, the deviation of the phosphite data from the PR3 lines is not explainable in terms of equation 6 along with a negative aryl effect for the phosphites. Let us see if the deviation of the phosphites is explainable in terms of a new p parameter, pp, only (equation 9).

............................................. (9)

According to equation 9, the deviation of the lines for the phosphites (Figure 5A) from the PR3 line would be attributable to p effects. This same pattern of lines (PR3, P(OR)3, P(O-p-XC6H4)3, P(OCH2)3CEt) must then be observed for each property-versus-property plot. In Figure 5B, we see that this is not the case, the point for P(OCH2)3CEt lies on the PR3 line and the points for P(OR)3 and P(O-p-XC6H4)3 lie on a single line below the line for PR3. Thus, we conclude that at least two parameters, in addition to cd and q, are required to describe the phosphites.

**Figure 5**. (A) nCO
(-1900 cm-1) versus Eo (229 K), (B) nCO
(-1900 cm-1) versus DHo,
(C) nCO (calc)
versus Eo (229 K), and (D) nCO
(calc) versus DHo.
Values of nCO,
Eo (229 K), and DHo
are taken from Tables 1 and 2. nCO
(calc) were calculated via regression analysis using the Ear and pp
parameters presented in Table 1* via * equation 4 where prop2 is Eo (229
K) (Figure 1C) or DHo
(Figure 1D). The sets of ligands are represented in the following way PR3 (filled
squares), P(p-XC6H4)3 (open circles), P(OR)3 (filled triangles) and P(O-p-XC6H4)3
(open triangles). P(OCH2CH2Cl)3 is not included in this analysis.

Applying Occam's razor, we make the assumption that the first additional parameter for the phosphites is Ear which apparently describes a property not restricted to aryl phosphines. The second additional parameter (pp) is then attributed to a p interaction between the phosphites and the metal to which they are attached. As pointed out above, the set of parallel lines in Figures 5A and 5B requires that Ear and pp be virtually constant for each family.

Finally, we used the parameters in the Table of Parameters along with equation 8 to calculate the data necessary to simulate the property vs. property plots in Figures 5A and 5B. The results, which are displayed in Figures 5C and 5D are virtually identical with those in Figures 5A and 5B.

In sum, we have found that the minimum number of parameters needed to describe all the phosphorus(III) compounds listed in the Table of Parameters is five: the the two electronic parameters, cd and Ear, the steric parameter q, the steric thresholds parameter, qst, and the p bonding parameter, pp . These conclusions are based on simple linear algebra and are independent of any particular model.

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