J. Phys. Chem. 1985,89, 5412-5414
5412
Molecular Softness as the Average of Atomic Softnesses: Companion Principle to the Geometric Mean Principle for Electronegativity Equalization Weitao Yang,* Chengteh Lee, and Swapan K. Ghosht Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 2751 4 (Received: July 25, 1985)
It is demonstrated that, for a molecule consisting of M atoms, the relation, 1/(I- A ) = (l/M)CY[l/(I,- A , ) ] , is valid to a reasonable approximation, where I and A are the ionization potential and electron affinity of the molecule and I,, A, denote the same quantities for the ith atom. This relation is the arithmetic average principle for molecular softness where the softness, S , is defined [W. Yang and R. G. Parr, Proc. Nutl. Acad. Sci. U.S.A.,82, 6723 (1985)l as the inverse of the hardness (9) i.e. S = l / q = 1/(I- A ) . The calculated values of I - A for 33 molecules show agreement with experimental results. The conditions for the validity of this principle are shown to be-analogous to those for the geometric mean principle for electronegativity equalization in molecules.
1. Introduction In recent years, several chemical concepts have attained rigorous foundation within the framework of density functional theory.’ Thus, the electronegativity x of an atom or a molecule has been identified* with the negative of the chemical potential p of its electron cloud
(5) Although, some earlier implied that this relation is to be expressed in terms of a sum instead of an average, the present result is established by numerical tests as well as theoretical arguments based on the conditions similar to those which validate the geometric mean principle for ele~tronegativity.~
2. Arithmetic Average Principle for Molecular Softness and the chemical hardness 17 has been defined3 as
Following Parr and Pearsod and Huheey,’* we express the chemical potential K~ for the ith atom in the molecule to first order in its net electronic charge ANi as pi = pi0 + 1 7 p ~ ~i i=, 1, ..., M (6)
where I and A denote the ionization potential and electron affinity, respectively. And also a new approach to the frontier electron theory of chemical r e a ~ t i v i t y has ~ . ~ been developed. A relation of the hardness 17 with a local or regional hardness has also been suggested.6 Very recently, the concept of softness, S , has been introduced’ as the inverse of hardness, i.e. S , = 1/17 = 1/(I- A ) , and, particularly, local softness, which gives softness upon quadrature, has been shown to be connected to chemical reactivity and of special importance in catalysis. Upon the formation of a molecule, the chemical potentials of different parts of the molecule attain the same value.2 To relate this final molecular electronegativity to those of the constituent atoms, Sandersons introduced the postulate of geometric mean: to a certain accuracy, the electronegativity of a molecule is given by the geometric mean of the original atomic electronegativities, namely
where r: and 17: refer to the original neutral atoms. By equalizing these effective chemical potentials of each atom in the molecule, i.e. p i = W , i = 1, ..., M (7) and using the condition for the molecule to be neutral, namely
(3) where the product is to be taken over the M constituent atoms and ~p refers to the original chemical potential of the ith atom. Recently, Parr and Bartolotti9 have discussed the sufficient conditions for the validity of this geometric mean principle. Now, what happens to the molecular hardness or softness? Can any relation be found for these quantities in terms of the atomic ones? The present work answers this question and demonstrates that the softness of a molecule is, to a reasonable approximation, represented by the average of the softnesses of its constituent atoms, namely 1M s = -XS, Mi
(4)
or equivalently On leave from Heavy Water Division, Bhabha Atomic Research Centre, Bombay 400085, India.
0022-3654/85/2089-5412$01.50/0
M
CAN,= o
(8)
I
one finds that eq 6-8 together give the molecular chemical potential as (9)
We now assume that (i) the hardness of an atom is proportional to its chemical potential, i.e. i = 1, ..., M (10) and (ii) the same relation (eq 10) is true also for a molecule with 7; = yp?,
(1) R. G. Parr, Annu. Reu. Phys. Chem., 34, 631 (1983). (2) R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J . Chem. Phys., 68, 3801 (1978). ( 3 ) R. G. Parr and R. G. Pearson, J . Am. Chem. Soc., 105,7512 (1983). Note that the factor in the original definition of 7 has been dropped in eq 2 and elsewhere in the present work. (4) R. G. Parr and W. Yang, J . Am. Chem. SOC.,106, 4049 (1984). (5) W. Yang, R. G. Parr, and R. Pucci, J . Chem. Phys., 81,2862 (1984). (6) M. Berkowitz, S. K. Ghosh, and R. G. Parr, J . Am. Chem. SOC.,in
press. (7) W. Yang and R. G. Parr, Proc. Natl. Acad. Sci. U.S.A., 82, 6723 (1985). (8) R. T. Sanderson, “Chemical Bonds and Bond Energy”, Academic Press, New York, 1976; R. T. Sanderson, “Polar Covalence”, Academic Press, New York, 1983. (9) R. G. Parr and L. J. Bartolotti, J. Am. Chem. Soc., 104, 3801 (1982). (10) N. K. Ray, L. S. Samuels, and R. G. Parr, J. Chem. Phys., 70, 3680 (1979). (11) J. L. Reed, J . Phys. Chem., 85, 148 (1981). (12) J. E. Huheey, ‘Inorganic Chemistry”, Harper and Row, New York, 1983, p 154.
0 1985 American Chemical Societv
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
Molecular Softness
TABLE I: Atomic Data atom H Li B C N
0 F Na A1 Si P S Cl
K V Cr Fe co Ni cu Se Br Rb
Zr Nb Mo Rh Pd Ag Sn Sb Te
I
P
A' 0.754 0.620 0.278 1.268 0.000 1.462 3.399 0.546 0.442 1.385 0.746 2.077 3.615 0.501 0.526 0.667 0.164 0.662 1.157 1.226 2.021 3.364 0.486 0.427 0.894 0.747 1.138 0.558 1.303 1.250 1.050 1.97 1 3.061
13.595 5.390 8.296 11.256 14.540 13.614 17.420 5.138 5.984 8.149 10.484 10.357 13.010 4.339 6.740 6.763 7.900 7.860 7.633 7.724 9.750 1 1.840 4.176 6.840 6.880 7.100 7.460 8.330 7.574 7.342 8.639 9.010 10.454
yb
1.79 1.59 1.87 1.60 2.00 1.61 1.35 1.62 1.72 1.42 1.73 1.33 1.13 1.59 1.71 1.64 1.92 1.69 1.47 1.45 1.31 1.11 1.58 1.76 1.54 1.62 1.47 1.75 1.41 1.42 1.57 1.28 1.09
TABLE 11: Experimental and Calculated Results (eV) molecule I' Ab I - A (I IEBldd Acalcdd
ye
Diatomic Molecules 2.42 6.98 7.39 2.55 7.24 7.90 1.66 7.74 8.28 2.60 7.96 8.48 2.40 9.08 9.40 0.65 8.95 9.74 1.13 8.87 9.85 3.54 9.06 9.99 1.24 9.40 11.24 3.82 10.68 11.84 0.44 11.62 12.15 1.83 11.35 12.49 0.38 12.72 13.63 3.08 12.62 14.02
I2 IBr
9.40 9.79 9.40 10.56 11.48 9.60 10.00 12.60 10.64 14.50 12.06 13.18 13.10 15.70
s2
Br2
c12 p2
so CZ CH CN 0 2
OH NH F2
cs2 cos so2 0 3
NH2 N20
10.45 11.11 10.36 11.84 13.01 10.48 11.77 11.26. 12.32 12.65 13.61 13.59 14.03 17.42
3.06 3.21 2.08 3.36 3.62 0.75 1.92 1.27 1.08 0.81 1.46 1.10 0.40 3.40
1.18 1.17 1.40 1.21 1.31 1.75 1.59 1.12 1.58 1.17 1.86 1.51 1.89 1.34
Triatomic Molecules 10.08 1.00 9.08 8.78 11.18 0.46 10.72 9.90 12.34 1.05 11.29 10.51 12.67 1.82 10.85 12.15 12.80 0.78 12.02 13.36 12.89 1.47 11.42 13.64
10.63 11.60 12.32 13.61 13.88 14.18
1.85 1.60 1.81 1.46 0.52 0.54
1.64 1.84 1.69 1.50 1.77 1.59
Polyatomic Molecules 1.60 8.76 8.25 9.48 0.80 9.11 1.40 10.00 9.92 10.66 0.20 9.34 1.70 9.30 10.88 1.40 11.13 8.83 0.43 10.98 11.24 11.57 0.91 10.91 0.08 9.76 11.99 12.70 0.57 10.46 0.75 14.60 12.76 0.70 9.16 11.44 11.14 1.89 7.78
11.43 11.58 11.97 12.23 12.62 14.19 12.32 14.62 12.93 13.74 16.05 12.54 12.32
2.67 2.10 1.99 1.57 1.74 3.06 1.08 3.04 0.93 1.05 3.29 1.50 1.18
1.44 1.70 1.56 1.92 1.46 1.52 1.85 1.71 1.97 1.80 1.81 1.73 1.35
PBr3 PC13 POCl, CH31
'All data from ref 9. bCalculated by eq 13. I and A taken from columns 1 and 2.
the proportionality constant y the same as that for atoms. With these two assumptions, eq 9 gives the relation
5413
9.85 9.91 11.40 9.54 so3 11.00 CF31 10.23 C2H2 11.41 CF,Br 11.82 CH3 9.84 HN03 11.03 SF6 15.35 c 6 H ~ N 0 2 9.86 C6H402 9.67
'See ref 14. bSeejef 15. eCalculated by eq 5, from atomic data taken from Table I. dCalculated by using I - A from column 4 and wmol (= -(I A)/2) from the geometric average principle for chemical potential (eq 3). CCalculatedby eq 13, I and A taken from columns 1 and 2.
+
which is equivalent to eq 4. This explains the arithmetic average principle for molecular softness. The assumption in eq 10 follows from the condition that the chemical potential ,for an atom is exponential in the amount of charge ANi and the falloff parameter y is the same for all atoms, i.e. Mi
= ~ i exp[-y"il "
-14
triatomic polyatomic
(12)
which validate the geometric mean principle for electronegativity equilization? Parr and Bartolotti9 calculated y from eq 12, i.e. using the relation y = In (Z/A)and observed that y = 2.15 f 0.59, with standard deviation indicated, for 32 atoms (listed in Table 1). Our interest, however, lies in eq 10 itself and, therefore, we verify it by directly using the commonly used values of p and 11 within the finite difference approximation of eq 1 and 2. Thus using the relation 2(Z - A ) y = I + A
and the listed experimental Zand A values, we find that y = 1.55 i 0.22 for the same 3 2 atoms of Table I and y = 1.57 f 0.25 for the 33 molecules in Table 11, in satisfactory agreement with the two assumptions. For atoms, a similar constancy of y has also been reported recently by Na1ewaj~ki.l~For molecules, no (13) R. F. Nalewajski, J . Phys. Chem., 89, 2831 (1985). (14) H. M. Rosenstock, K.Draxl, B. W. Steiner, and J. T. Herron, J. Phys. Chem. Rex Dora, 6, Supplement No. 1 (1977). (15) B. K.Janousek and J. L. Brauman, 'Gas Phase Ion Chemistry", M. T. Bowers, Ed.,Academic Press, New York, 1979.
0
diatomic
a
/* '
-12
/
h
Q
-
0 . 6 '
I
v
-10
-8 $2
14
( I -A&
Figure 1. Experimental values of I - A, in electronvolts, against the calculated values for the molecules in Table 11. such relation seems to have been studied earlier.
3. Test of the Arithmetic Average Principle With the readily available values of Z and A for atoms, listed in Table I, Z - A values for 33 molecules are calculated by using eq 5 and compared with experimental results in Table I1 and Figure 1. For most of the molecules, the calculated values agree reasonably well (within around 1 eV) with the experimental values. For diatomic molecules, a systematic correlation of the linear form
J . Phys. Chem. 1985, 89, 5414-5417
5414
is observed: I - A = 0.881(1- A)=,&+ 0.383 eV, with a standard deviation of 0.348. 4. The Difficulties with Nonneutral Molecules Suppose the molecule has charge Q,then, instead of eq 8, we have M
xANi = Q I
which, combined with eq 6 and 7, gives the chemical potential for the molecule
again disagrees with y calculated from either eq 13 (see Table 11) or the relation' y = In ( Z / A ) . In view of the results discussed in section 3, the inappropriateness of eq 16 and 17 is clear and is a direct consequence of nonzero values of Q. The expressions for the chemical potential given by eq 6 and 12 are, however, incomplete since the contribution from the potential due to the nuclei is not taken into account. Both the geometric mean principle for electronegativity and the arithmetic mean principle for softness, thus, depend only on composition and does not differentiate between different structures of a molecule. More rigorous treatment by including the external potential contribution is necessary.
5. Concluding Remarks Equation 15, implied also by some of the earlier works,lOsllgives for molecular hardness 1
M I
which differs from eq 5 by a factor of 1/M. From eq 12 and 14, one obtains for molecular chemical potential
which leads' to the result that the exponential falloff parameter for the molecule is 1 / M times that of its constituent atoms, which
The arithmetic average principle for softness (eq 4) and the geometric mean principle for electronegativity (eq 3) are parallel to each other in their nature, conditions, and validity of approximations. Together, they define a prescription for predicting the molecular ionization potential and electron affinity values from the corresponding quantities of the constituent atoms, although the accuracy of this prediction is not very satisfactory as is evident from the values so calculated and included in Table 11. However, the main emphasis of the present work is to establish that the molecular softness is an average of the atomic softness rather than their sum (eq 16) as is implied by earlier works.
Acknowledgment. The authors are most grateful to Professor Robert G. Parr for helpful discussions and suggestions. This work is supported by grants from the National Science Foundation and National Institute of Health.
Fractionation by Inclusion of Isotoplc p -Xylenes and Isotopic Benzenes in 1,2-Bis(diphenylphosphinoselenoyl)ethane, Ph,P(Se)CH,CH,P(Se)Ph,, and of Isotopic Benzenes in Bis[ bis(diphenylphosphino)methanido]platinum( II),Pt(Ph,PCHPPh,), Shahrokh Sabat and Norman 0. Smith* Department of Chemistry, Fordham University, Bronx, New York 10458 (Received: January 22, 1985)
When a mixture of two isotopic liquid guests is equilibrated with a host capable of forming a clathrate phase with them the mole ratio of the guests in the clathrate phase differs from that in the coexisting liquid. Unlike the inclusion of p-(CH~)2CsH4-p-(CDI)2C6D4 mixtures by Ni(Cme~y),(NcS)~, inclusion by Ph2P(Se)CH2CH2P(Se)Ph2shows a preference for the protiated p-xylene, with a constant separation factor of 0.963. The same host, in the inclusion of C&+,D6 mixtures, shows a smaller preference for the protiated benzene, with a separation factor of 0.983. A similar but stronger effect is observed in the inclusion of C,&-C6D6 mixtures by Pt(Ph2PCHPPh2),, with a separation factor of 0.950. It is postulated that the preference for the protiated isomer is related to the tight fit of the guest in the host lattice., although vapor pressure differences also have an influence on the results. Guest activity coefficients in the clathrates are given, the protiated guests showing negative and the deuterated guests positive deviations from ideality.
Introduction
Tetrakis(4-methylpyridine)nickel(II) isothiocyanate, 1, forms 1: 1 inclusion compounds with many aromatic guests, including p-xylene and naphthalene. When a pair of such guests, with or without a cosolvent, is contacted with 1 the resulting solid is a solid solution of the inclusion compound of one guest in the inclusion compound of the other. The ratio of the amounts of the two guests in this solid, however, differs detectably from that in the coexisting liquid, even when the members of a guest pair differ only isotopically. In our earlier s t ~ d i e s l -1~was used to fractionate isotopic p-xylenes and isotopic naphthalenes. It was found that, 'Current address: Department of Chemistry-Physics, Kean College of New Jersey, Union, NJ 07083-9982.
with these systems, the proportion of the more deuterated guest in the included material is greater than that in the equilibrium liquid phase, with separation factors of about 1.10 for p-xylenes and 1.04 for naphthalenes. Similar results were obtained with liquid chromatography of p-xylene mixtures on a stationary column of pretreated 1.2 The slight preference for the more deuterated isomer was interpreted3 as being practically entirely the result of its greater vapor pressure. If, therefore, the protiated and deuterated isomers were (1) Ofodile, S. E.; Kellett, R. M.; Smith, N. 0. J . Am. Chem. SOC.1979, 101, 7725-7126. (2) Ofodile, S.E.;Smith, N. 0. Anal. Chem. 1981, 53, 904-905. (3) Ofodile, S. E.;Smith, N. 0. J. Phys. Chem. 1983, 87, 473-476
0022-3654/85/2089-5414$01.50/00 1985 American Chemical Society
