J . Phys. Chem. 1985,89, 2831-2837 NiO as thick as 136 nm, the conductivity changes were almost similar for both the positive and negative polar substrates. This is apparently due to disappearance of the band-bending effect, because of attenuation of the polarization field through the NiO layer, which gives evidence to the above explanation. It is evident that the variations in conductivity with increasing NiO thickness are closely associated with the growth of NiO particles, i.e., the density of vacant sites, grain b o ~ n d a r y , and '~ the extent of bridging network between nearby crystallites. In this regard, another possibility of different electrical properties depending on polarization direction might be a geometric effect, since the growth of metal particles deposited on ferroelectrics is influenced by the presence of the polarization field of the substrate.I4 When Ag was deposited on ferroelectric triglycine crystal, the size and/or density of particles were larger for the negative domain than for the positive domain.I5 It was also shown that an electric field applied induced coalescence of the threedimensional island structure at an earlier stage of the film growth, compared to the absence of the field.I6 The present S E M observation of NiO/LiNb03, however, showed that a dense NiO film grew homogeneously for both polar substrates and there is no such striking differences between opposite polar LiNb03 in contrast to the results described above. Thus, there seems to be no indication which shows that the geometric contribution is predominant to the different conductive properties caused by (13) C. M. Osburn and R. W. Vest, J . Phys. Chem. Solids, 32, 1355 (1971). (14) E.J. Weidmann and J. C. Anderson, Thin Solid Films, 7,27 (1971). (15) M. Takagi, S. Suzuki, and K. Tanaka, J . Phys. SOC.Jpn., 23, 134 ( 1967). (16) K. L. Chopra, J . Appl. Phys., 37, 2249 (1966).
2831
opposite polarization direction, but we have to await further investigation in this connection. In the previous study on C O oxidation over 32-nm NiO(673)/LiNb03 catalysts,2the reaction order with respect to CO and O2pressure was respectively 0.6 and 0.45 for the NiO/(+) catalyst but varied to 0.9 and 0.24 for the NiO/(-) catalyst. The activation energy of the reaction was 50 kJ mol-' for NiO/(+) and 79 kJ mol-' for NiO/(-). From mechanistic consideration, it is suggested that these kinetic changes are associated with the stronger adsorption of O2 on the NiO/(-) surface. Furthermore, the thermal desorption study demonstrated that there are clear differences in the desorption temperature of oxygen between NiO on the positive and negative polar LiNbO,; the peak maximum temperature of oxygen desorbed from NiO/(-) was higher by 16 K than that from NiO/(+) cata1y~t.I~ These results clearly indicate that the stronger adsorption of oxygen occurs on the NiO surface combined with the negative polar substrate, thus giving support to the above-mentioned view for the conductive behavior. In conclusion, the combination with ferroelectrics is a promising way to modify the surface characteristics of semiconductors. In this regard, the behavior of n-type semiconductive oxides such as TiOz and ZnO combined with ferroelectrics is interesting; the work is in progress.
Acknowledgment. This research was supported under a Grant-in-Aid for Scientific Research (No. 57470005) from the Ministry of Education, Science, and Culture. Registry No. NiO, 1313-99-1; LiNb03, 12031-63-9; 02,7782-44-7; CO, 630-08-0; Hz, 1333-74-0. (17) K. Sato and Y. Inoue, to be submitted for publication.
A Study of Electronegativity Equalizationt Roman F. Nalewajskit Institute f o r Theoretical Chemistry and Department of Chemistry, The University of Texas, Austin, Texas 7871 2 (Received: October 22, 1984) The equalization of the isolated system chemical potentials, gAD and gB0,to a common.value, wAB, in A- - -B is discussed within the following three alternative approaches: (1) the local approach based on the detailed, local changes in the electronic density and external potential due to the presence of another system; (2) the global approach using the global electron-transfer characteristics and local external potential data; ( 3 ) the intermediate, atoms-in-a-molecule(AIM) approach. As an illustrative application the AIM approach is applied to diatomics. The harmonic mean law for the chemical potential (electronegativity) heutralization is derived and the original Sanderson geometric mean principle is modified to include the external potential effects.
I. Introduction Elsewhere'*2we have shown that it is convenient to discuss the electronegativity equalization during the chemical bond formation in a diatomic molecule in terms of the atoms-in-a-molecule (AIM) model which explicitly takes into account both the electron transfer and external potential effects. This model qualitatively explains' the principle of the hard and soft acids and bases (HSAB)3,4and provides a convenient framework for interpreting the bond stability in terms of the thermodynamic-like Le Chfitelier and Le ChPtelier-Braun principles.2 In bection I1 of the present work we extend the previous diatomic analysis to the general molecular systems. We adopt three al'Supported in part by a research grant from the National Science Foundation, and a grant from the Institute of Low Temperatures and Structural Research, Polish Academy of Sciences, Wroclaw, Poland. *On leave from the Department of Theoretical Chemistry, Jagiellonian University, M. Karasia 3, 30-060 Krakow, Poland.
0022-3654/85/2089-2831$01.50/0
ternative approaches differing in the degree of the local description of both interacting molecular systems. The local d e ~ c r i p t i o n ~ . ~ uses the most detailed treatment of the changes at each local site of the one system due to the interaction with the other system. The global approach6-8uses the global electron-transfer parameters for each interacting system while preserving the local treatment of the external potential effects. Finally, the AIM approach, which is extended here to the polyatomic systems, offers (1) Nalewajski, R. F. J . A m . Chem. SOC.1984, 206, 944. (2) Nalewajski, R. F.; Koninski, M. J . Phys. Chem. 1984, 88, 6234. (3) Parr, R. G.; Pearson, R. G . J . Am. Chem. SOC.1983, 205, 7512. (4) Pearson, R. G. "Hard and Soft Acids and Bases"; Dowden, Hutchinson and Ross: Stroudenburg, PA, 1973. (5) Nalewajski, R. F.; Capitani, J. F. J . Chem. Phys. 1982, 77, 2514. Nalewajski, R. F. J . Chem. Phys. 1984, 82, 2088. (6) Nalewajski, R. F. J . Chem. Phys. 1983, 78, 6112. (7) Nalewajski, R.F.; Parr, R. G. J . Chem. Phys. 1982, 77, 399. (8) Parr, R. G.; Yang, W. J . Am. Chem. SOC.1984, 206, 4049.
0 1985 American Chemical Society
2832 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 an intermediate description based on the “local” atomic parameters which are “global” characteristics of each atom within a molecular system. This atomic “local” resolution is expected to be sufficient and adequate for most of the chemical applications. Section I11 includes a simple, illustrative application of the AIM model to the electronegativity equalization in a diatomic molecule. We emphasize there the importance of the external potential terms which are often neglected within the approximate electronegativity equalization rules. The new harmonic mean law in terms of the valence-state chemical potentials (before the charge transfer) is derived. We also briefly comment on the validity of the geometric mean rule9 and propose its modified version, in terms of the valence-state chemical potentials, which is shown to remedy most of the shortcomings of the original Sanderson principle using the isolated atom chemical potentials.
Nalewajski distances even more crude approximation to suiB(?),in which one considers only the electrostatic potential due to the net atomic charges on B, should be approximately valid. In what follows we use the symbols A’, A’, and A* to denote, respectively, the isolated A, A in the presence of B with frozen, unrelaxed density nA’(i), and the (equilibrium) valence state of A in A---B, with the relaxed density n;(?) before the charge transfer. From eq 1 one obtains the following expressions for the final (after the charge transfer) equilibrium chemical potential and electron density of A in A- - -B: piB(?,)=
11. Energy Expressions and Chemical Potential Equalization 11.1. Local Approach. Consider the isolated, ground-state
atomic or molecular system A, AO. According to the main theorems of the density functional theoryi0the system ground-state energy, EAo, chemical potential, pAo, and other ground-state properties of A are unique functionals of the ground-state electron density, nAO(?), and the external potential due to the system nuclei, uA0(?), e.g., EA-- = E A O [ ~ A O , Now, ~ A O assume ]. that the system A is interacting with another system B in the complex A- - -B. The latter perturbs both uA0(?), by 6viB(q, and nAO(?), by 6niB(?). The corresponding second-order change in the electronic energy of A in A- - -B, relative to the isolated system value, is
+
6EiB/6niB(7) = pAo
= PA+(?)
= const
+ I d ? ’ VAo(?,??6niB(??
niB(?) = 6EiB/6ViB(?)= nAo(?)
= =
[
nAo(?)
4-
(6)
AniB(?)
+ Jdf’@~o(7,7?6Vi~(??]+ 6niB(?)
+ 6nA*(?)] + 6niB(7) = nA*(?) + 6@(3
[nAO(i)
(7)
Obviously, p A + ( i ) does not include the charge relaxation contribution, before the charge transfer, and therefore is not a constant. In order to obtain the expression for the valence-state chemical p~tential’~-~~ = pA[nA*,l$B]
PA*
= const
one has to add to pA+(?) the polarization (fluctuation) term due to 6nA*(?) (eq 7), preserving the number of electrons, Jd7
6nA*(?)
= 6NA. = 0
(8)
+ Jdr“qAe(7,f?6nA*(??
PA*(?) = PA+(?)
+ Jd7’
= PA+(?)
= Jd?
[p~0(36t@(?)
JdP’ 6viB(P?Jd7
+
y 2 S d F I d ? ’ [oA.(?,??6n~B(?)6nftB(??
where
p A o ( i ) = pAo(7’)
(YAO(?,??
=
= pAo
[6pA(?)/6VA(??]AD
+ @ A 0 ( 7 , 7 ? G ~ B ( r ) B ~ i B(1)( r ? ] =:
constant and
=
[6nA(?)/6nA(??]AO
?A4(7??
=
pAO(?’,?)
(3)
= [6pA(fl/6nA(??]Aa =
VAo(?’,?)
(4)
[6nA(?,)/6vA(??]AO
The density response kernel @A’(i,?? represents the external potential stiffness modulus, while the stiffness modulus for variations in the density is given by the hardness kernel, VAO(?f?. If we neglect the nonlocal contributions to 6viB(?) from the kinetic exchange functional”
suiB(?)
5
(9)
@AO(7,7?
=0
hence, taking into account the arbitrary character of 6viB(i) and the symmetry property (3) one obtains Jd?
= 6(%?9 (2)
pAO(7,?9 =
@~.(?’,?’?6Ui~(?’?
The number preserving property of eq 8 implies an important general constraint on the response kernel PA.(?,?’). Namely, from eq 8,
+ nAO(?,)60iB(?,)] +
2.A.(?,7?6niB(?)6v~B(7?
VAo(?,P?JdP”
@AO(7,7’)
= Jd7
@Aa(7’,7)
=0
(10)
It follows from eq 6 that, in order to bring the constancy of piB(?) throughout the space, the density change contribution must
neutralize, to within a constant, the suiB(?) contribution to pA+(i). This can be explicitly demonstrated by using the definition of the hardness kernel or its equivalent obtained via the chain rule:
(5)
UBO(3
where UBo(?)is the classical electrostatic potential due to the B whole charge distribution (electrons and nuclei). At large A-B
Hence, by using the ground-state Euler-Lagrange equationI2 /.bA
~~~
= uA(?)
6F/6nA(?)
(12)
~
(9) Sanderson, R. T. J. Am. Chem. SOC.1952, 74,272, Science 1951, 114, 670; ‘Chemical Bonding and Bond Energy” Academic: New York, 1976. (10) For the recent reviews, see: Rajagopal, A. K. Adv. Chem. Phys. 1980, 41, 59. Kohn, W.; Vashishta, P. In “Theory of the Inhomogeneous Electron Gas“; Lundqvist, S.,March, N. H., Eds.; Plenum: New York, 1983; p 79. Parr, R. G. Annu. Rev. Phys. Chem. 1983, 34, 631. (1 1) Payne, P. W. J. Chem. Phys. 1978, 68, 1242.
VA(?,?’)
= 62F/6nA(q6nA(??
(13)
(12) Parr, R. G.;Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801. ( 1 3 ) Palke, W. E. J. Chem. Phys. 1980, 72, 2511. (14) Parr, R. G.; Bartolotti, L. J. J. Am. Chem. SOC.1982, 104, 3801.
The Journal of Physical Chemistry, Vol. 89, N o . 13, 1985 2833
Electronegativity Equalization where F is the universal Hohenberg-Kohn density functional for the sum of the electronic repulsion and kinetic energy components. From eq 6 and 13 we therefore get
+ JdP’
= p~+(j?
@(F)
= MA+(F)
[62F/6nA(F)6nA(r?]6niB(7?
+ 6[6F/6nA(F)l
= pA+(F) - 6viB(F) + const
(14)
where we have used the ground-state equilibrium criterion 12. Obviously, a similar cancellation of the direct 6viB(F) term and that induced by the density relaxation, anA., must also take place in the expression for the valence-state chemical potential of eq 9. A simple rearrangement of the energy expression 1, using eq 6 and 7, gives
+
AEiB= I d 7 (pA.6niB(7)
+ ) / 2 [ A n i B ( F ) 6 ~ i B ( 7 +) A p i B 6 n i B ( F ) ] ) = )/z(pAo + p i B ) 6 N i B + YZJdP [nA.(F) + nftB(F)]6viB(F) = p i B 6 N i B + d7 fiiB(F) 6viB(7) (15) in terms of the average chemical potential, piB,and the average density, of A in A---B. Next we consider the second-order change in the electronic energy of the whole interacting system, A-- -B, relative to the isolated systems value. Equation 15 gives &AB
= (piB- p i B ) S N
+
= MiB hEAB B
+ I d 7 [fiiB(7)6viB(F) + ii$B(F)6vf’(F)]
6NA =
pfiB(F) = &??
(17)
= ApOAB6N
=
[@Bo
(24) This equation, representing the charge-transfer equivalent of the polarization equation 23, clearly indicates that the charge transfer is being driven by both the differences in changes of the external potential, due to the other system, and by the initial difference in the chemical potentials. As a final simple case we consider the density changes involving two local subsystems on A and two on B, and including both the fluctuation flows between the local volume elements of the same system as well as the charge-transfer term corresponding to the flow between the site at 7 (on A) and the site at x’ (on B):
6niB(r“? = (6n
+
= p i B = ~ A + ( F ) JdP” qAe(?,7’?6niB(?’?
(19a)
+ I d ? ’ ’ v,e(7’,7”6nfB(r‘’?
(19b)
= PA+(^?
= p i B = pB+(F) = pB+(7’)
=
...
+ SdF”qB.(F,F’?6niB(P’?
+ I d ? ” vB~(F’,V~6n~B(7’’)= ...
+ 6nA)6(7”-F)
(19~)
In order to simplify the notation we adopt the matrix notation: PA+
[PA+(F)3rpA+(7?1,
[PAB&AB]r
PAB
Po(;:)
~ N =A r[6nA,-6nAI, ‘lAa = tlAo(r
>
r
PA*
=
6 N = T[6n,0]
[kA*rpA*]
]:I
tlAo(r’TI)
7)Ao(r’,r’
and similar matrices for the system B. The chemical potential equalization equations for this case become = PA+ + ( 6 N + SNA)qA. = PA* 6 N qA0
+
= pB+
+ (-SN + 6NB)qBo =
-6N ~ B D
6 N = (PB*- PA*)(TAO + ~ B O ) - ’
(26)
+ qBO)-’~AO PAB - PB* = (PB* - PA*)(VAO + - PA* = (PA’ - PB*)(‘lJAo
VBO)-’VBO
From the equality (19a,b),
6 V i B ( F ) - 6UiB(7? = s d 7 ” [qAO(7’,?’? - qAO(?,7’?]6niB(?’? = S d i ” KA.(7,?’;r“?6nfiB(7’?
(20)
one formally extracts a local change in the density attributed to A,in terms of the inverse of the hardness kernel KAo and changes in the external potential on A due to the presence of B
6niB(F) = JdP’ J d t ” KAO-1(ci”,7? [6dB(7’? - 6dB(7’)]
(27a) (27b)
11.2. Global Approach. We now briefly consider a global approach6-8 in which the ground-state energy of A is considered a functional of the average number of particles and the external potential: EA = E A [ N A , u A ] . The corresponding second-order change in E A , due to the presence of B, is AEiB = p A o 6 N i B + I d 7 nA.(F)6viB(F)
yz[7jA.(6NiB)z + 26NiBJd7
+
&Ao(F)SviB(F)
+
Jd7 I d ? ’ BA.(7,7?6viB(F)6viB(P?] (28) (21)
where, by definition
(25)
where the valence-state potentials p x = ~ px+ + 6 N x qxo include both the external potential and internal density redistribution give terms. Equations 25 when solved for SN and
PAB
(19d)
- 6nA6(7”-7?
6ngB(i’? = (-6n +6nB)6(i”-z) - 6nB6(r”’-?)
+ j d 7 [fiiB(F)6viB(F) + f i i B ( F ) 6 ~ P B ( F ) ]
The chemical potential equalization relations 17, which determine the equilibrium density distributions, 4’ and &’, must be satisfied for all local subsystems at different space locations within A and B: PAB
- PAo) + (6v$’(7? - 6 v i B ( F ) ] / [ v A o ( ~ ~+F )4B0(7’97?1
-+’
(18)
(23)
- PA+(?)] / [OAo(7,F) + ?B0(7’,F?I
6 N = [ME+(??
- pf’ = pAo - p g ~= A p 0 A ~ .Hence
and this implies AEAB
= pAB = const
[suiB(?? - 6 v ~ ’ ( 7 ) ] / [ ~ A 0 ( ? ’ , 7 ?+ TAO(7,?)]
This equation indicates that the size of local polarization fluctuation is driven by the difference in the modified external potentials at both locations and it is inversely proportional to the sum of local hardnesses. Similarly, if one assumes the simple charge transfer between local subsystems at ?on A and at ?’on B: 6niB(7”) = 6n6(7”-?) and 6nf”F”) = -6n6(7”-7’), one obtains from eq 19a,d
(16) where we have used the closure condition 6 N i B = -6NkB = 6 N . The equilibrium condition for the charge transfer requires the equalization of the chemical potential throughout the whole interacting system
(22)
The similar expression for 6nkB(F) immediately follows from the equality (19c,d). Now, if we assume a simple fluctuation on A (SN;’ = 0), representing an exchange of 6nT = 6 N A electrons between the small volume ( T ) elements located a t ?and 7’: 6niB(F”) = 6n[6(FF’? - 6(7’-7’’)], then from eq (19a,b) we immediately have
nAe(F)6viB(F)
s
Jdy KA~-’(r‘;x’,y’)KAo(z,~;7?= 6(+7?
JdS
where j j A = ( 8 p A / a N A ) , is the global hardness of A, = [6pA/6vA(F)]N = (anA(F)/aNi,), represents the N - v coupling
2834
Nalewajski
The Journal of Physical Chemistry, VoL 89, No. 13, 1985
function, and DA(?,F? = [6nA(q/6uA(??], = [6nA(??/6uA(q], is the corresponding density response kernel. As shown recentlys the coupling function tiA(F) provides a measure of the site reactivity, leading to the classical frontier theory of chemical reactivity.15 The final chemical potential and density distribution of A in A- - -B are given by the following expressions:
AEiB =
E AE$~(N$B,z$B)
XEA
+ S',J2$B+ f / z [ f j x ~ ( s N+$ ~ ) ~ 2ax.6N$B62$B + Pxo(6Z$B)2])(35) = 6NiBpi0+ 6ZiB(rf;. + y2[6NiBfl~o(6NiB)T + 26NiB&~e(6ZiB)T + 6zftBBAO(6ziB)T] =
(pxo6N$B
XEA
(35a) Here the vectors 6NiB,6ZiB,pAo, and l A o and the diagonal ma= ?Ao6NiB (29) trices qAO,&Ao, and BAegroup the parameters for all MA atoms of the system A; CX = ( a E x / a Z X ) , < 0 is the average nucleniB(?) = 68iB/6ufiB(?) = [ n A O ( 7 ) ar-electron attraction energy per unit nuclear charge, qx = Jd?' DAO(?,~?~@(??]+ t i ~ o ( q 6 N=i ~nA* ? i ~ ( q 6 N i ~ (apx/aNx)z, > 0 is the atomic hardness, ax = ( a p x / a Z x ) N x= (a(x/aNx)zx< 0 represents the atomic N - 2 coupling derivative, (30) and px = (a{x/aZx)Nx< 0 measures the overall sensitivity of the atomic nuclear-electron attraction to a change in the effective Again, the external potential (polarization) and the charge-transfer nuclear charge. The AIM parameters pxo, {XO, QX., axo,and contributions are clearly separated. This should facilitate the correspond to the atom X in an isolated system. The diagonal analysis of the interaction mechanism. Notice, however, that the second-order matrices qAO,aA0,and BA0may prove inadequate A+ stage, appearing in the local approach, vanishes within the to account for some effects, e.g., conjugate bonds, for which the global description with only the valence-state chemical potential inclusion of the nondiagonal matrix elements in eq 35a, apx/aNy, and density entering the expressions for the final state of A in apx/aZY,and aCX/aZ,, is expected to be necessary. In such cases A- - -B. the state of each AIM is defined by the whole set of the average From the chemical potential equalization equation N x and 2, within a given system, e. AEGB = AE$B(N$B,ZiB). MA* ?A.GN = /.LB* - ijBo6N (31) The AIM expressions for the p$'and SiB are
+
+
+
+
6N = SNiB = -6NtB = (FB* - MA*)/(?AO + ?BO)
(32)
piB
Hence pAB
=
[?Bo/(?Ao
+ ?BD)lpA*+ [?AO/(?A''
+ ?Bo)lpB*
(33)
Equations 32 and 33 are the global analogues of the local expressions 26 and 27. The final chemical potential is an average of the valence-state chemical potentials with the weights determined by the global hardness parameters of both interacting systems. From eq 28-30 and 32 one derives the following expression for the A- - -B stabilization energy
=
(PA"
=
((A'
+ (ZfB&Ae) + 6NiBqAo= + S Z i B B A o ) + 6NiBaA0=
/LA+
(A+
+ 6NiB+jAe + 6NiBaAo
(36) (37)
The changes in atomic electron populations, SNiB,can be partitioned into the intra- and intersystem contributions:
6NiB = (NA. - NAD)
+ ( N i B- NA*) = 6NAA)+ 6NLAB)
(38)
with the intra- and intersystem closure relations:
C6fiqj, = C6N& X
Y
C6"y
= 0,
X
= -C:BNpf) , Y
(39)
f
The intrasystem changes 6N are due to polarization, and the intersystem contributions 6N reflect changes due to a flow of electrons between A and B. The partitioning (38) enables us to rewrite eq 36 and 37, pi'
=
(FA+
+ 6NkA)qA.)+ 6NaAB)qA.=
pAi
+ 6N~AB)qA. (36a)
(iB= ((A+ + 6NLA)&,.) + 6NaAB)aA.= (A* + 6NAABkiA0 (37a) Within the AIM model the valence-state electron redistribution 6NLA) can be determined from the MA- 1 equations for the equalization of the chemical potential in the valence state (see eq 36a)
in terms of the valence-state chemical potentials and the average, transition densities ( n A ) and (nB), between the isolated and valence states, Therefore, this expression offers a way to estimate the stabilization energy of the A- - -Bcomplex from the properties of the isolated and perturbed reactants. This should be expected from our second-order perturbation expansion 28. 11.3. Atoms-in-a-Molecule Approach. In this section we summarize the relevant expressions from the atoms-in-a-molecule approach',* which represents a compromise, an intermediate description between the local and global approaches described previously. It preserves a major aspect of the local description by explicitly considering each atom of the whole interacting system. However, each atom, X, is now characterized by the global atomic parameters and its energy is considered a function of the average number of electrons, N x , and an effective nuclear charge Z x # Z x o modeling the external electric field the atom X is exposed to inside a molecular system. The second-order change in the energy of the system A in A- - -B is now given by the following expansion:
The intersystem flow of electrons, 6NAAB)and 6NBAB),follows from the MA + M E - 1 equations for the equalization of the chemical potential throughout the whole A- - -Bsystem (eq 36a)
(15) Fukui, K. "Theory of Orientation and Stereoselection"; SpringerVerlag: West Berlin, 1973; p 134; Science 1982, 218, 747.
AZx = C AZx(Y,Rxy)
PA*,]
= hA*,2 =
... -- KA*,MA
(40)
and the intrasystem closure relation 39, for a given 6ZiBwhich explicitly depend on the geometry of the interacting system. One similarly determines the polarization terms 6NBB)within the system B.
MAE
... = MAS, - AB = ... - pB,MB
= hi! = 145 =
= pi,? = p e
(41)
and the intersystem closure relation 39. In order to apply the AIM model one has to estimate the effective nuclear'charge AZx modeling the perturbation of atom X by the electric field due to a presence of all the remaining atoms in a molecular system ( R x y is the internuclear distance): Y#X
(42)
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2835
Electronegativity Equalization The crucial, pair interaction charges AZx(Y,RxY) still remain to be defined. In our previous paper2 a realistic, approximate relation AZx(Y,Rxy)
N
A - B
A(cid)
Nose)
'
( f ) x ~ z y ( R/( ) ( f 2)XORXY) (43)
has been derived by using a simple least-squares criterion. Here Z X ( R )= Zx.f(R) is the shielded nuclear charge of atom X, with an exponentially decreasing screening function f(R). An alternative expression for AZx(Y,Rxy) follows from the familiar Hellmann-Feynman theorem.16 Namely, since the effective atomic charges in X-Y are introduced to model the changes in the AIM energy due to the variations in the internuclear distance R, one should demand the exact AZx(Y,R) to satisfy the following derivative identity (see eq 35)
-F$y(R)= dE$Y(Nx.,Zxe;R)/dR = [dE$'( Nxo,2$y) /dZgy] [dAZx(Y ,R)/dR] d = -wx~AzX(Ym dR
+ Y2Bx.[Azx(Y,R)121
(44)
This identity provides after integration the quadratic equation for determinin the unknown AZx(Y,R) from, it is assumed, known and 6x0: function F f y ( R ) and the atomic parameters
cx:x.
flx.[A2x(Y,R)]2
+ 2[x:x.AZx(Y,R) + 2
The solution which vanishes a t R
-
R
s dR' FgY(R? = 0 m
(45)
is
."9':.
(46) The ground-state electronic energy E;' expectation value
in eq 44 is given by the
(47) e_xplicitlydepends on the internuclear distance 121 = 18, - 2 x1 Hxo is the isolated atom Hamiltonian and is the valence-state wave function (the ground state of the Hamiltonian (47)). By the Hellmann-Feynman theorem
Gy
$$YlaH$Y/aRI$$Y)
&-
Akid) A-B Bbe) Figure 1. Schematic chemical potential equalization diagrams for various hardness combinations within the A (acid)-B (base) diatomics.
where the AIM Coulombic Hamiltonian
F$Y(R) = -(
6N-W
P~~
(48)
Hence the final expression for AZx(Y,R): r
where ?y = 7- 2,O = O[7y,i?] and p i y is the one-electron density corresponding to &'. Obviously the exact expression 49 requires the knowledge of p$'(.F,R) and therefore appears not to be very useful in practical applications of the model. However, one would expect that a modeling of the major changes in the atomic density during the bond formation (contraction, polarization) could provide via eq 49 reasonable apDroximate expressions for AZx(Y,R), generating realistic diatomic potentials and adequate for both qualitative and semiquantitative considerations. (16) For example: Levine, I. N. "Quantum Chemistry"; A l l y and Bacon: Boston, 1970; pp 374-380.
111. Illustrative Application of the AIM Model to Diatomics III.1. Chemical Potential Equalization. Consider now a diatomic system A-B. The AIM chemical potential p;B
=
+ a p A z ~ ( B , f ? ) ]+ q ~ 0 6 N=i ~FA*+ q ~ 0 6 N i ~
[ ~ A O
(50)
explicitly includes the external potential (AZA) and charge flow (6NtB)terms. The former is often neglected in the approximate rules of electronegativity equalization, e.g., in the postulated Sanderson geometric mean principle9 PAB
-
(pAepBo)"2
(51)
in which the explicit dependence of pAB on R is disregarded. On the basis of the analysis already given,2the trends exhibited by the atomic parameters -pAe, -aAO,and qAOare very similar. Thus, the hard atoms (large q ) usually also exhibit a strong N-2 coupling (large -a) and high electronegativity (-p), while the soft atoms (small q) are accordingly characterized by a relatively weak N - 2 coupling and low electronegativity. The hard atoms are also expected to induce much larger AZ on their partners than do the soft atoms.'v2 This is because the atomic hardness should parallel the strength of the nuclear attraction, providing therefore an effective measure of the extent to which the nucleus of atom B remains unscreened; this in turn determines the electrostatic influence of atom B on the partner A, modeled by AZA(B,R). This qualitative statement is supported by the explicit, approximate AZA(B,R) derived previously2 (eq 43). Clearly, the difference pAo- pA. = -npAZA(B,R) depends on the identity of both the perturbed (A) and perturbing (B) atoms. In Figure 1 we have schematically illustrated the chemical potential equalization in a diatomic molecule A-B for various
2836
TABLE I: Numerical Test of the Constancy of the Ratio -wxe/qxe' density Hartreefunctional theoryd atom Li B C N 0 F Na AI Si
P S C1 K
V Cr Fe
co Ni cu Se Br Rb
Zr Nb
Mo Rh Pd Ag
Sn Sb
Te I (-po/vo)
"qx.
Nalewajski
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
=
exutlb 1.26 1.07 1.25 1.oo 1.24 1.49 1.24 1.16 1.41 1.16 1.51 1.56 1.26 1.17 1.23 1.04 1.18 1.37 1.38 1.52 1.79 1.26 1.13 1.30 1.24 1.37 1.14 1.41 1.41 1.28 1.56 1.83 i u 1.32 i 0.19 1/2~xD.
X.
FwkC
XGT
0.96 0.84 0.91 0.99 1.08 1.18 0.96 0.91 1.05 1.20 1.35 1.52 0.96 0.77 1.oo 0.96 0.96 1.oo 1.oo 1.40 1.63 0.87 0.86 1.07 1.oo 1.oo 0.87 1.oo 1.19 1.23 1.54 1.62
0.42 0.94 1.16 0.61 1.05 1.34 0.47 0.95 1.24 0.74 1.27 1.57 0.52 1.12 0.46 1.74 1.42 1.24 0.93 1.27 1.57 0.54 2.03 0.51 0.48 0.85 0.40 0.93 1.25 0.8 1 1.34 1.63
0.86 1.oo 1.19 0.78 1.19 1.43 0.94 1.10 1.36 1.oo 1.49 1.74 1.01 0.91 2.22 1.63 1.45
1.09 A 0.23
1.03 i 0.43
1.31 f 0.41
1.54 1.80 1.04 2.15 0.99 0.5 1 1.25 1.45 1.13 1.65 1.90
*Reference 3. 'Reference 2. dReference 19.
combinations of the constituent atom hardnesses; for definiteness, = SNiB A is assumed to be an acid and B a base (pAo < PBO,GN = -6NiB > 0 ) . Reference to the figure shows that the hard atoms are always shown to initially lower the chemical potentials of their partners to a much larger degree than the soft atoms. The expressions for the optimum 6N that equalizes the chemical potential is given by eq 32 and the equilibrium pABby eq 33 which can alternatively be written as PAB
- FA' =
~ A ~ C LA M* E * ) qAo
+ qBo
Equation 52 shows that the corresponding p A -~px* shifts are proportional to the hardness of atom X and the difference pA* - pBo,which drives the flow of electrons at a given internuclear distance. These trends are also qualitatively illustrated in Figure 1. 111.2, Harmonic Mean Law f o r Electronegativity Neutralization. A simple rewriting of eq 33 gives
(53) NAB = (QAOPB* + ~ ~ B + A * ) / ( ~ A O+ 460) In order to eliminate the hardness from this expression we observe that the basically parallel trends exhibited by the isolated atom -bxo and qxo valuesZ suggest
-
-px./fix. c (54) where C is an approximately universal constant. Since,3 by the finite differences pxe = -!/z(Ixo Axe)
+
4x0 = I x o - Ax.
(55)
where the ionization potential I,. is much larger than the electron affinity AX., 2C should be close to unity. The proportionality relation 54 is tested numerically in Table I, where -2px0/qxx.ratios from various sources are compared. Reference to this table shows that indeed the proportionality assumption is approximately valid since the reported CX.values fall in a sufficiently fairly narrow range around the average value. This tendency of the isolated atoms prompts us to assume that it also remains valid for the (slightly) perturbed AIM's, i.e. MA*/~A*
-
(56)
PB*/VB*
being a manifestation of an apparently universal feature of atomic systems. Within our second-order model the changes in the hardness parameters of the AIM's are ignored; therefore, eq 56 is equivalent to the assumption C(A-/VAO PB*/~BO C' (57) This relation enables us to eliminate qAoand qBofrom eq 53. The resulting harmonic mean expression for pAB N
PAB
-
~MA*PB*/(PA*
-
+ PB*)
gives the final chemical potential in terms of the valence-state AIM chemical potentials only. We therefore conclude that the equilibrium chemical potential in a molecule is approximately given by the harmonic mean of the valence-state chemical potentials of the constituent atoms. 111.3. Modified Geometric Mean Principlefor Electronegativity Equalization. Multiplying p i Band &(eq 50) and ignoring the second-order terms in 6Nx and AZx gives MAEz
= PiBp$B
-
PAobBo
+ ((hBOqAo - PAoqB')GN +
[ P A + P ~ B ( A , R+)I L B ~ & A O A Z A (11 B J(59a) )
-
kA*MB*
+ (PBoqAo - P A o ? B o ) 6 N
TABLE 11: Various Electronegativity Estimates (in eV) for the Alkali Metal-Halogen Pairs at Large Internuclear Distances'
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCI KBr KI RbF RbCl RbBr RbI 'Based on data from ref 3
5.59 4.99 4.48 4.50 5.45 4.87 4.65 4.39 5.01 4.48 4.28 4.04 4.94 4.41 4.22 3.98
4.28 4.42 4.26 4.06 4.18 4.31 4.16 3.97 3.84 3.96 3.82 3.64 3.77 3.89 3.75 3.58
(58)
4.65 4.41 4.30 4.16 4.47 4.24 4.15 4.01 3.91 3.74 3.66 3.55 3.82 3.65 3.58 3.48
4.11 4.33 4.14 3.90 4.09 4.25 4.06 3.84 3.81 3.95 3.79 3.59 3.75 3.88 3.73 3.53
4.40 4.50 4.38 4.22 4.27 4.38 4.25 4.10 3.87 3.98 3.85 3.70 3.79 3.90 3.77 3.62
(59b)
J. Phys. Chem. 1985, 89, 2837-2843 For the Sanderson principle (5 1) to be valid, within this first-order approximation to pm2, all terms but pAopg0 in eq 59a must vanish. This is never the case and, as suggested previously? the geometric mean rule should exhibit increasing errors as the hardnesses of the constituent atoms increase (see also Figure 1). One would therefore expect the electronegativity values from the Sanderson rule to be generally too small at the equilibrium internuclear distance, since it ignores the net stabilizing influence due to the external potential of the bond partner. For example, for homonuclear diatomics (AN = 0) one obtains the qualitatively incorrect prediction: pLAA= pAo. For H2 and N 2 the predicted molecular electronegativities, 7.17 and 7.27 eV, are smaller than the lower bound given by ZM/2 (from ref 17):7.71 and 7.79 eV, respectively. Equation 59b suggests how to remedy this fault of the geometric mean rule. Namely, one has to replace the isolated atom chemical potentials with the corresponding valence-state potentials:
We have neglected the 6N term of eq 59b since, by the approximate proportionality relationship 54, both its contributions should approximately cancel each other. By using the atomic data one can test the approximate electronegativity equalization rules by considering almost noninteracting (large R ) pair of atoms, A---B. As demonstrated by Perdew et a1.'* the chemical potential for such neutral species at zero temperatures is given by (17) Herzberg, G. 'Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules"; Van Nostrand-Reinhold: New York, 1950; p 459. (18) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Phys. Reu. Lett. 1982, 49, 1691. (19) Robles, J.; Bartolotti, L. J. J . Am. Chem. SOC.1984, 106, 3723.
2837
= -1/2(zmin + Amax) where Zmin is the minimum of (ZAe, ZBo) and A,, is the maximum of (AAo,ABO). Therefore, one can use the quantity bn, the exact chemical potential of a pair of atoms, at large internuclear distances from one another where the electrostatic influence can be neglected (pA. pAo, pB. p p ) , as a convenient criterion for evaluating the quality of predictions from the harmonic and geometric laws. In Table I1 we compare various electronegativity estimates for the alkali metal-halogen pairs against the pair electronegativity. Reference to the columns 1, 3, and 5 shows that the harmonic mean rule gives much closer estimates of the pair electronegativity than those predicted by the geometric mean rule. The results reported in these columns have been obtained by averaging the isolated atom (unbiased) chemical potentials. However, one would expect that the (open) AIM'S should exhibit very small net charges: alkali metal slight positive charge and the halogen slight negative charge. This in turn implies that it would be more realistic to average over the electronegativities of slightly charged species, rather than over the isolated atom data. Due to the discontinuity in the chemical potential'* we then have xM+= ZMoand xx = Ax.. The corresponding averages are shown in the columns 2 and 4 of Table 11. One immediately observes a dramatic improvement in the quality of predictions from the geometric mean rule, now approximately the same as that of predictions from the harmonic mean rule. On the basis of this demonstration one would therefore expect that both the geometric and harmonic mean principles should give adequate, semiquantitative predictions if one uses the A I M chemical potentials, including the effects of both the external potential and the qualitative charge transfer. ppair
-
-
Acknowledgment. I thank Professor Robert E. Wyatt for his generous hospitality in Austin.
Fluorocarbons as Oxygen Carriers. An NMR Study of Nonionic Fluorinated Microemuisions and of Their Oxygen Solutions Marie-Jose St4b4, Guy Serratrice, and Jean-Jacques Delpuech* Laboratoire d'Etude des Solutions Organiques et Colloidales, UA CNRS 406, Universite de Nancy I , B.P. 239, 54506 Vandoeuvre- les- Nancy Cedex, France (Received: October 23, 1984)
NMR studies have been performed to gain insight into the structure and dynamics of nonionic fluorinated microemulsions prepared at 28 OC from ternary mixtures of perfluorodecalin, pentaethylene glycol 7H,7H-perfluoroheptyl ether, and water in appropriate proportions. The I3Cspin-lattice relaxation times in the degassed microemulsion confirm the liquidlike structure of the oil droplets and the chemical shifts strongly suggest a trans conformation of the hydrophobic chains of the surfactant molecules in the interfacial film. The solubilization of oxygen decreases through the surfactant film from the perfluordecalin core acting as the main oxygen reservoir to the water interface, as shown by the analysis of the paramagnetic relaxation of IH and I9F nuclei induced by the dissolved oxygen molecules.
Introduction A novel class of nonionic microemulsions, highly fluorinated microemulsions, have been recently prepared in this laboratory,'-3 using for this purpose well-defined (monodisperse) fluorinated polyoxyethylene surfactants4 with an appropriate hydrophile-lipophile balance (HLB) value. These solutions may offer an alternative to commercial macroemulsions presently used as temporary blood substitute^^-^^ due to their long-term stability (1) Mathis, G. Thesis, University of Nancy I, France, 1982. (2) Mathis, G.; Delpuech, J.-J. French Patent No. 8022875, 1980. (3) Mathis, G.; Leempoel, P.; Ravey, J. C.; Selve, C.; Delpuech, J.-J. J . Am. Chem. SOC.1984, 106, 6162. (4) Selve, C.; Castro, B.; k m p o e l , P.; Mathis, G.; Gartiser, T.; Delpuech, J.-J. Tetrahedron 1983, 39, 1313.
0022-3654/85/2089-2837$01.50/0
and a better solubilization of fluorocarbons, and consequently of molecular Other potential applications for such systems are the preservation of organs or the culture of microorganisms in an oxygen-rich and chemically inert liquid. ( 5 ) (a) Riess, J. G.; Le Blanc, M. Angew. Chem., Inr. Ed. Engl. 1978, 17, 621. (b) Riess, J. G.; Le Blanc, M. Pure Appl. Chem. 1982, 54, 2383. (6) Naito, R.; Yokoyama, K. Green Cross Corporation Technical Information: Osaka, Japan; 1978, Ser. No. 5 ; 1981, Ser. No. 7. (7) Yokoyama, K.; Suyama, T.; Naito, R. In 'Biomedical Aspects of Fluorine Chemistry"; Filler, R., Kobayashi, Y., Eds.; Tokyo Kodansha: Tokyo, 1982; p 191. (8) Geyer, R. P.; Monroe, R. G.; Taylor, K. Fed. Proc. 1968, 27, 374. (9) Geyer, R. P. Prog. Clin. Biol. Res. 1983, 122, 157. (10) Yokoyama, K.; Naito, R.; Fukaya, C.; Watanabe, M.; Handa, S.; Suyama, T. Prog. Clin. Biol. Res. 1983, 122, 189.
0 1985 American Chemical Society
