J. Phys. Chem. 1992, 96, 5825-5829
582s
Application of Potential Constants: Molecular Chemical Potentials of Heteronuclear Diatomic Molecules. 5 Ken Obwada Division of Chemistry, Japan Atomic Energy Research Institute, 319-1 I , Tokai-mura, Naka-gun, Ibaraki-ken, Japan (Received: February I , 1992)
The harmonic and anharmonic potential (force) constants of heteronuclear diatomic molecules, which are usually available from normal coordinate analyses, are applied to problems of determining the molecular chemical potentials of such molecules. The approach developed here is mainly based on density-functionaltheory; that is, the respective atomic energies in a molecule are expanded with the numbers of electrons and the nuclear potentials. These expansions are allowable because the ground-state energy for a system of N electrons and given nuclear potential u(r) is a functional of Nand u(r). To test the reliability of the approach, we have calculated the molecular chemical potentials of alkali metal halides and other heteronuclear diatomic molecules, and their results have been compared favorably with the data obtained from Sanderson’s principle as well as the abinitio SCF calculation. Consequently, the present approach is found to be simple and adequate for evaluating the approximate molecular chemical potentials of heteronuclear diatomics.
I. Introduction In studies of molecular force fields, one of the most significant problems is to determine the reliable general valence force constants such as bond stretching, angle deformation, bond-bond interaction, angleangle interaction, and bond-angle interaction in pdyatomic molecules. As is well-known, thii problem has been approached from various points of view by many investigators, and relatively sufficient data on it now exist. The next problem is to make the best use of these potential constants for an understanding of physical and chemical properties of molecules. For this purpose, we have recently developed the methods’ of determining empirically diatomic molecular energy components (electronic kinetic energy, total electrostatic potential energy, electron-nuclear attraction energy, electron-electron repulsion energy, and Hartree-Fock eigenvalue sum) by making use of harmonic and anharmonic potential constants spectroscopically available. The methods are based on the inhomogeneous linear second- and third-order differential equation^"^ which are derivable from the quantum-mechanical virial theorem6 that is theoretically exact to all intents and purposes. It has been shown that the methods are not only simple and powerful in evaluating the molecular energy components but a h applicable to know the charge transfer and electric dipole moment change on formation of heteronuclear diatomic molecules.’ In the present study, attention focuses on the molecular chemical potentials of heteronuclear diatomic molecules. Such a study is of great importance in chemistry because the electronic chemical potential, which has analogy with the chemical potential of ordinary macroscopic thermodynamics, measures the escaping tendency of an electronic cloud in an atom, molecule, or solid for the ground state.8 This report is organized in the following manner. In section 11,we attempt to derive some basic formulas necessary to calculate the molecular chemical potentials of alkali metal halides and other heteronuclear diatomic molecules. In section 111, we give an outline of the methods of determining empirically molecular energies by applying the potential constants spectroscopically available to general solutions of potential-energy differential equations derived from the quantum-mechanical virial theorem. The reliability of the method developed in sections I1 and I11 is evaluated in comparison with the data obtained from Sanderson’s principle and the abinitio SCF calculation. This is described in section IV. Brief discussions on the molecular chemical potentials are also given in section IV. 11. Mokculrr Chemical Potentials For the sake of simplicity, let us take two neutral atoms, A and B, and imagine these atoms to be brought together from infinite internuclear distance RAB(a). We shall further assume that atom
A is more electronegative than atom B. Then density-functional theory of electronic systems8tells us that electronic charge will flow from atom B to atom A, with the possibility of nonintegral electron numbers, producing some positive charge on atom B and negative charge on atom A. It could accomplish this by tunneling through the potential barrier between the atoms, and the flow of electrons from atom B to atom A will stabilize the system of AB. This behavior parallels the behavior of the chemical potential in classical macroscopic thermodynamics. To obtain the explicit formulas for the above descriptions, we begin by expanding the respective atomic energies (EA)and (EB) in the molecule with respect to the electron numbers (NA,N B ) and the nuclear potentials (v,,(r), vB(r)):
and
where r is the space coordinate vector and the subscripts zero indicate the standard state for which the partial derivatives are evaluated. Here, we shall take the standard state with energy ( E ) oto be the ground state of the isolated atom. These expansions are allowable because the ground-state energy for a system of N electrons and given nuclear potential u(r) is a functional of N and u(r), (E) = E[N,u].~In eqs 1 and 2, we shall carry, to a first approximation, our treatment only as far as the quadratic terms. In addition to the sum of the two expressions (EA)and (EB)of eqs 1 and 2, the total energy of the system will include electrostatic terms such as a nuclear-nuclear repulsion term, which are disregarded if we deal only with the isolated atoms. At this stage, let us introduce the following several definitions, which originally come from the density-functional theory of electronic systems,8 to eqs 1 and 2:
0022-365419212096-5825$03.00/00 1992 American Chemical Society
Ohwada
5826 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 a(Ex)/aNx= Px
(34
6(Ex)/6ux(r) = px(r)
(3b)
aZ(EX)/aNX2
= 2.11,
(3c)
K a / a N x ) [ 6 ( E x ) /bvx(r)lI = apx(r)/aNx = ([6/6ux(r)l [a(Ex)/aN~lI= & ~ / b o ~ ( r ) = fx(r) ( 3 4
b2Wx)/6ux(r"ux(d = bx(r)/6ux(r') (3e) where p x is the chemical potential? px(r) the electron density, vX the chemical hardness,I0 andfx(r) the Fukui function." Their physical and chemical meanings have already been discussed in detail elsewhere (see refs 8-1 1). Then eqs 1 and 2 become (EA)
=
(EA)o
+ P A O U A+ JPA(r)AuA(r) ANAJ~A(~)A~A(~)
(1/2)1
dr
dr
+?
A ( ~ A ) ~
+
J [6pA(r)/6uA(IJ)10AuA(r')AuA(r)
dr' dr
In the equilibrium configuration, the chemical potential of atom A becomes equal to that of atom B; Le., pA = pB = PAB. The quantity of PAB is called the molecular chemical potential that is definable for the molecular system in equilibrium. Then we have, from eqs 9 and 10, an equilibrium condition: C(A
- PB = ( P A o - kBo) + 2 ( ( 1 A ) + (%))me + ... = 0
or (VA)
+ (7B)
=
(MU,'
-PA0)/20e
(14)
Furthermore, let define a new parameter y as =Y
(vA)/(?B)
(15)
whose property will be discussed later in some detail. By using eqs 14 and 15, we obtain apparent chemical hardnesses (vA) and (Q) of atoms A and B as follows:
+ ..* (4)
and
(13)
= ?@Bo
(I)A)
- p A 0 ) / [ 2 ( 1 + ?)",I
(16)
- PA0)/[2(1 + ?)me]
(17)
and
(EB)= (EB)o+ P B O U B
+
J P B ( ~ ) A ~ B (dr ~ )+
m B S f e ( r ) ~ u B ( r )dr
(1 /2)
rtdmd2+
+
j J [apB(r)/6uB(r')lOAuB(r')AuB(r)
dr' dr + *.. (5)
Now, differentiating eqs 4 and 5 with respect to ANA and ANB, we have the chemical potentials, bAand pB,in the molecule AB: [ ~ ( E A ) / ~ N=APA IV PA' 27AAN sfA(I)AUA(I) dr + (6)
+
+
and [d(E,)/aN~]v = PB = PB'
-2
S ~ + m Sfs(r)&(r)
dr
+ ...
(7) where equality ANA - A N B = AN has been used. AN is of course the number of electrons transferred from atom B to atom A . Equations 6 and 7 just correspond to the expressions derived on the basis of the second-order atoms-in-molecules approach by Nalewajski.I2 To obtain more tractable forms of the chemical potentials, let us here introduce the following drastic approximation for the integral terms involving the Fukui functions in eqs 6 and 7:
dr
(l/an?JfAr)AuAr)
e (l/me)[
JfAr)AuAr)
drIe (8)
where ANedefines the number of electrons transferred from atom B to atom A in equilibrium. Equation 8 means that the integral term including the Fukui function evaluated in some region more or less near the equilibrium position nearly equals that in equilibrium. With this approximation, eqs 6 and 7 become PA
= PAo + 2 4 7 . + ( 1 / 2 m S f , ( r ) A u A ( r ) =
FAo
dr] +
'.'
+ 2 ( 7 A ) m + ...
(9)
and I*B
= PBo - 2
4
- (1 /2mS/,(r)AuB(r)
= Pgo - 2 ( ? B ) m
where
(qA)
dr] +
+ ..(
..' (10)
and (vB) stand for
(VA)
= ?A + ( 1 / 2 ~ v v ) J j A ( r ) ~ u A ( r )
7A
+ (1/2me)[
sfA(r)AcA(r)
(7s) = (fiBo
The apparent chemical hardnesses of various diatomic molecules will be predicted by making the best use of AN, determined spectroscopically under some reasonable assumption. Hereafter, we shall discuss the numbers of electrons (AN) transferred from atom B to atom A, which play important roles in eqs 16 and 17. Prior to this, consider the total molecular energy of the present A B system. Such an energy will include the nuclear-nuclear repulsion term in addition to the sum of the two expressions (EA)and ( E B ) of eqs 4 and 5. Therefore, the total energy of the system is given by (EAB) = (EA)+ (EB)+ Avnn = (EA)o + (EB)o+ (PA0
[
- P B o ) +~ JpA(r)~uA(r) dr
dr]e
(1
/2)(11 J
[ JfA(r)~uA(r)
dr
+
dr -
dr' dr +
[6PA(r)/6uA(r')10AuA(IJ)AUA(r)
dr' drl
[GpB(r)/6uB(IJ)lOAuB(IJ)AuB(r)
+ ...
(18)
AVm is the nuclear-nuclear repulsion energy, which is disregarded if we deal only with the isolated atoms. Equation 18 is theoretically exact to all intents and purposes to second order. However, implementation of eq 18, specially evaluation of the integrals in it, is very difficult at the present time. Therefore we intend here to take the following approximate way. Noting eqs 8, 11, and 12 and Slater's X a approximation technique13-16that takes into account the spherical symmetry of each atom in an isolated state, eq 18 may be reduced to (EAB) e ((EA)O
+ (EB)O) + (PA' - P [((TA)
B O ) ~
+
+ (?El)) - 2 / R A B l ( a n ? 2
(19)
This simplication implies that the integral terms including AVnn are equivalently replaced by the single attraction between the net charge of AN electrons on atom A and an equal and opposite charge on atom B; i.e.,
(1/2)(s
(ll )
JpB(r)~uB(r)
S f s ( r ) ~ u B ( r ) d r ] +~ ~
[ JpA(r)~uA(r) dr
+ ~B)(AN)* +
AV,,] +
+
s1
dr
s
+ JpB(r)~uB(r) dr + ~
[6PA(r)/auA(IJ)10AuA(r')AuA(r)
[6pB(r) /buB(r')lOAuB(r')AuB(r)
dr' dr)
~ n + n ]
-
dr' dr + -2(w2/RAB
(20)
and (qe) = OB 7B
- ( 1 / 2 m S f B ( r ) ~ u B ( r ) dr
- (1 /2ANe)[
jfB(r)AuB(r)
dr],
(12)
in Rydberg atomic units. Imposing the equilibrium condition of eq 14 and comparing it with the energy ( ( E A ) O + (EB)O) at infinite internuclear distance, the molecular energy change is approximately given by7
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 5827
Application of Potential Constants
~ ( E A B )(EAB)- ((EA)o+ (EB)o)
Q ~ R A B=)
= (PAo - rBO)M+ (1/2)[(pBo - pAo)/Me - 4 / R A B I ( m 2 which is the desired result. Evaluating eq 21 at the equilibrium position, we obtain where A(EABle and RA& are the molecular dissociation energy and the equilibrium internuclear distance, respectively. Now one can easily solve eq 22 as a second-order algebraic equation with respect to the number of electrons transferred at the equilibrium position since the quantities A(EAB)e,pAo,pBo,and RA& are all available experimentally or empirically. Lastly, we shall make a remark on the interatomic energy given by eq 20. This energy is originally derived from the X a method by which all the electrostatic-intetaction terms add to a single attraction between the net charge of AI? electrons on atom A and an equal and opposite charge on atom B.13-16 However, in a realistic point of view, it seems more essential to take the charge separation R i B instead of the internuclear distance RABin eq 20 because the charge center is not necessarily consistent with the center of gravity of atom due to atomic polarization and bond charge effect. It has already been shown in a previous paper’ that such a modification is very effective for evaluating the exact dipole moment change on formation of heteronuclear diatomic molecules. 111. Application of Vibrational Potential Constants In order to grasp apparent chemical hardnesses as well as the molecular chemical potentials of heteronuclear diatomic molecules, we have need to implement eqs 16 and 17. For this purpose, it is first of all necessary to compute the number of electrons tiansferred from atom B to atom A near the equilibrium position. This is possible with the aid of eqs 21 and 22 if exact knowledge of the molecular energy change A(EAB)is available in some way. In the present section, we shall deal with this problem. There are several ways for obtaining exactly molecular energies of diatomic molecules. As is well-known, the most traditional methods will be those using functions of the Morse p~tential’~ and the RKR potential.I* However, we here intend to use somewhat different methods’ previously developed on the basis of the quantum-mechanical virial theorem? It has already been shown that the methods give good predictions for molecular energies over a relatively wide range of internuclear distances. An outline of such methods will be given below. Within the Born-Oppenheimer approximation, the quantummechanical virial theorem6 for a diatomic molecule AB relates the total molecular energy (EAB,(RAB)) and its first derivative to the electronic kinetic energy ( TAB,(RAB))as a function of the internuclear distance RAB:
+ (EABn) =
(EABn)]
(28) (29)
m#n
where VAB is the electrostatic potential energy operator and Qn(RA~)is also given in another form as
A(EAB)~ = (1 /2)(pA0 - rBo)hnT,- ( ~ / R A & ) ( M C ) ~ (22)
RAB[d(EABn)/dRABI
21(nlTABlm)12/[(EABm) -
21(nlVABlm))2/[(EABm)- (EABn)]
=
(21)
m#n
(23)
( E A B= ~ )(EAB~(RAB)) = (~IHABI~)
(24)
TAB^) = (TABARAB)) = (~ITABI~)
(25)
Here, HAB and TAB are the molecular Hamiltonian and the electronic kinetic energy operator, and In) is the nth electronic state. The nuclear repulsion term can always be added to (EABn) since eq 23 is invariant to addition to the left-hand side of a term ~/RAB. If eq 23 is differentiated one time with respect to RAB, one finds that RAB[d2(EABn)/dRAB21 + 2[d(EABn) /dRAB] -d(TABn)/dRAB = - 2 ( n ( T ~ ~ ( n ’(26) ) where In’) = dln)/dRAB. The calculation of the right-hand-side term of eq 26 yields the inhomogeneous linear second-order differential equation of the quantum-mechanical virial theorem: RAB2[d2(EABn)/mAB21+ 4RAB[d(EABn) / ~ R A B I+ ~(EAB =~ -Q~(RAB) ) (27)
QLRAB) (1 /RAB){(d/dRAB) [RAB2( TABn(RAB))I) (30) Equation 27 seems uniquely appropriate and has been derived and discussed in detail first by Clinton2and subsequently by Parr and B~rkman.~ The general solution of eq 27 is given by the usual manner:
into which the equilibrium conditions of [ d ( E ~ e ) / d & )=] 0~ as well as (EAB(RA&)) = ( E A B ) e have been introduced. The integrations in the q u a r e brackets are made in the range from RAh to RAB. We see that eq 31 completely determines the total molecular energy for a molecule if one has the exact knowledge of Qn(RAB), RA&, and (EAB),. Unfortunately, any exact tractable Qn(RAB) function is not known at the present time. Therefore, it seems reasonable to seek other ways for obtaining approximate potential energy functions that give good predictions for molecular energies without knowledge of exact Q,(RAB) functions. In the preceding reports,lJ9 we have proposed an approximate method of determining the total molecular energy (E-) from eq 3 1 under the assumption of Qn(RAB) = constant in some region more or less near the equilibrium position. Such an approach effectively uses the quadratic potential constant that is usually available from the normal coordinate analysis of the observed vibrational frequency. The details of the approach are here omitted to avoid repetition (see refs 1 and 19). In the case of requiring more accurate predictions of the molecular energy under consideration, further developments of the method as described above have to be made. To do this, we must coIlstruct inhomogeneouslinear higher-order differential equations, from which several molecular constants may be determined by using higher-order potential constants spectroscopicallyavailable. In the rest of this section, we shall take up the inhomogeneous linear third-order differential equation4v5that is given by RAB3[d3(EABn)/dRAB3] + 9RAB2[d2(EABn)/dRAB21 + ~ ~ R A B [ ~ ( E A B ~+) /~~(REAABB~~ P ~) R A B(32) ) PARAB)= (-1 /RAB2)(d/dRAB)(RAB2(d/dRAB)
[RAB2TABn(RAB)l1 (33)
Equation 32 can be obtained by taking RAB times eq 27, subsequently differentiating it with respect to RAB, and finally adding the resultant to 2 times eq 27. The alternative intrinsic theoretical derivationS of eq 32 may be performed by taking the second derivative of the quantum-mechanical virial theorem: RAB[d3(EABn)/dRAB’l + 3[dZ(EABn)/dRAB21 = -d2(nlTAB(n)/dRAB2= -[(nlT,&’)l + [ ( ~ I T A B ~ (34) ~”)] where In’) = dln)/dR,B and In”) = d21n)/dRm2. The calculation of the right-hand-side terms of eq 34 leads to eq 32 and furthermore determines the exact form of the P,(RAB) function (see ref 5). Since eq 32 is the linear differential equation, one can easily solve it by the usual manner. The general solution for the total molecular energy is given by (EABn)
+ (1/2)jPn(RAB) dRAB] + (1 /RAB2)[ c2 - j R A B P n ( R A B ) dRAB] + (~/RAB’)[C~ + ( ~ / ~ ) S R A B ’ P ~ ( R~ARBA) B(35) ]
= (I/RAB)[CI
5828 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 which is exact and contains three integration constants C1,C2, and C,. Again, introducing the reasonable assumption of Pn(RA~) = constant for RABnear the equilibrium position, eq 35 is immediately integrated to the approximate potential-energy function that gives more exact predictions for the molecular energies than those by the method as described earlier (see ref 1). It goes without saying that such a potential-energy function makes the best use of the cubic potential constant as well as the quadratic potential constant.
IV. Results and Discussion The (electronic) chemical potential is the key concept in density-functional theory and has analogy with the chemical potential of ordinary macroscopic thermodynamicx8 It is constant, through all space, for the ground state of an atom, molecule, or solid, and equals the slope of the ( E ) versus N curve at constant u(r). The physical meaning of the chemical potential is that the chemical potential is the negative of the electronegativity of Pauling and Mulliken9 and therefore measures the escaping tendency of an electronic cloud.8 Thus, for an understanding of the chemical and physical properties of atoms, molecules, or solids, it is of much significance to predict their chemical potentials for the ground states. In the present study, we mainly deal with the molecular chemical potential that is definable for the molecular system in equilibrium. The basic working equations are eqs 9 and 10 as already derived in section 11. Implementing these expressions, we encounter with one difficulty, that is, how to evaluate the parameter y that is defined as the ratio of apparent chemical hardnesses (see eq 15). Unfortunately, there is no way to assess exactly such a parameter at the present time. Therefore we intend here to employ the following drastic approximation for y: bearing in mind that the chemical hardness of the isolated atom is given by the finite-difference approximations*10as (37) Here, Zx and Ex are the ionization potential and the electron affinity, and their data were taken from refs 21 and 22. On the other hand, pxo in eqs 9, 10, and 14 is the chemical potential of the isolated atom, and is given by the three-point finite-difference approximation:8*20 which is the negative of Mulliken’s electronegativity. The remaining quantities undirected in eqs 16 and 17 are the numbers of electrons transferred (AN). We are able to reasonably evaluate these quantities by solving eqs 19 or 21 with the aid of vibrational potential constants as described in section 111. In the calculation of such quantities, we used four basic parameters of equilibrium internuclear distances (Ita), quadratic (K,)and cubic potential constants (Le),and molecular dissociation energies (A(EAB),)in addition to two parameters of the ionization potentials (Zx) and the electron affinities (Ex).The values of equilibrium internuclear distances and molecular dissociation energies were taken from refs 23-25. In particular, the quadratic and cubic potential constants were calculated by normal coordinate analyses using the observed vibrational f r e q ~ e n c i e s . ~ ~ - * ~ Now, we are in a position to evaluate apparent chemical hardnesses and molecular chemical potentials of heteronuclear diatomic molecules. Therefore we shall first note the molecular chemical potentials that are computed from eqs 9 and 10 with the aid of eqs 16 and 17 at the ground (equilibrium) states. The calculated results of the molecular chemical potentials are listed in Tables I and 11. Table I includes the results for the alkali metal halide molecules, and Table I1 lists those for the heteronuclear diatomic molecules excluding the alkali metal halides. For comparison, both the tables also include the molecular chemical potentials that were calculated on the basis of Sanderson’s principle2’ of electronegativity equalization. It is Sanderson’s principle that,
Ohwada TABLE I: Molecular Chemical Potentials (eV) for Alkali Metal Halide Molecules (in Equilibrium) molecule” R,b calc p A B S /LAB‘ F pABd LiF 2.9553 -4.8848 -5.5935 -6.3810 LiCl 3.8186 -4.7924 -4.9980 -5.3747 LiBr -4.6605 4.1016 -4.7794 -4.8775 LiI 4.5201 -4.4765 -4.5058 -4.5051 NaF 3.6395 -4.7095 -5.4399 -5.8487 NaCl 4.4615 -4.6384 -4.8607 -5.0809 NaBr 4.7282 -4.5 150 -4.648 1 -4.6008 -4.3424 NaI 5.1239 -4.3821 -4.3848 KF 4.1035 -4.1383 -5.0206 -5.0100 KCI 5.0396 -4.1303 -4.4860 -4.3404 KBr 5.3305 -4.0363 -4.2899 -4.1599 -3.9032 KI 5.7596 -4.0443 -3.9628 -4.0144 -4.9266 -4.8284 RbF 4.2903 RbCl 5.2665 -4.0181 -4.4021 -4.1945 RbBr 5.5648 -3.9300 -4.2096 -4.0322 RbI 6.0035 -3.8049 -3.9686 -3.8515 CsF 4.4321 -3.7970 -4.7671 -4.7085 CSCl 5.4924 -3.8196 -4.2595 -4.0341 CsBr 5.8057 -3.7417 -4.0733 -3.8780 CSI 6.2649 -3.6304 -3.8401 -3.7086 ‘Isotopes are ’Li, 23Na, 39K, 85Rb, IJ3Cs,19F, j5C1, 79Br,and ‘*’I. Equilibrium internuclear distances in bohr units. Calculated from Sanderson’s principle. Fermi energy from the ab-initio S C F calcuiation. TABLE 11: Molecular Chemical Potentials (eV) for General Diatomic Molecules (in Euuilibrium) molecule R,“ calc p A B S pABb F pABc -4.6440 -4.1141 LiH 3.0147 -4.1347 2.3301 -5.3978 -5.5473 BH -4.0044 -6.6625 -6.7041 -5.0945 CH 2.1241 -7.3629 -7.3557 -5.4598 OH 1.8349 -8.7229 -8.6439 -6.6645 HF 1.7329 -6.5940 -6.6803 -3.6960 HS 2.5323 -7.8325 -7.7236 -4.8336 HCI 2.4075 HBr 2.6740 -7.4325 -7.3859 -3.8779 HI 3.0406 -6.9095 -6.9631 -3.7954 -6.8386 -6.8714 -5.5143 co 2.1316 -7.3991 -7.3834 -4.5287 NO 2.1751 -9.1543 -9.3028 -5.9481 FCI 2.9518 FBr 3.3184 -8.6597 -8.8960 -4.0574 FI 3.6073 -8.0173 -8.3868 -5.5024 -7.9488 -5.9204 4.0403 -7.9385 ClBr -5.3132 -5.7746 -4.7020 TIF 3.9390 TIC1 4.6956 -5.1545 -5.1598 -4.4883 -4.1204 -3.9761 -2.7767 SrO 3.6281 -4.4327 -2.3988 3.6661 -4.0868 BaO Equilibrium internuclear distances in Bohr units. Calculated from the Sanderson’s principle. CFermienergy from the ab-initio SCF calculation.
on formation of a molecule, electronegativities (chemical potentials) of constituent atoms or groups equalize (neutralize), all becoming equal to the electronegativity of the final molecule. That principle was first formulated by Sanderson2’ as a geometric mean equilization principle: xAB
=
(XAOX~~)”*
(39)
or
where xABis the molecular electronegativity and xA0and XB’ are the electronegativities of isolated atoms A and B. It has been shown by Parr and Bartolotti28that this rule is roughly correct, although it cannot explain electronegativity changes that occur on homonuclear bond formation. It is found from Table I that the calculated molecular chemical potentials for alkali metal halide molecules are in the range from -4.8848 to -3.6304 eV, and are nearly consistent with those calculated from Sanderson’s principle within the difference of 1.O eV. For general molecules, the calculated values also agree with
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 5829
Application of Potential Constants
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF
KCI KBr
K1 RbF RbCl RbBr RbI CsF CSCl CsBr CSI
3.0754 2.0004 1.6807 1.3184 3.0921 2.0255 1.7154 1.3625 3.4570 2.2938 1.9573 1.5805 3.5801 2.3680 2.0324 1.6468 3.8331 2.4957 2.1347 1.7072
1.0460 1.0157 0.9460 0.8510 1.0126 0.9902 0.9297 0.8468 0.9464 0.9315 0.8868 0.8212 0.9420 0.9302 0.8850 0.8223 0.9416 0.9093 0.8623 0.7907
2.2428 1.6695 1.5566 1.3018 2.4743 1.7815 1.6677 1.3386 2.9766 2.1785 1.8894 1.5474 3.1245 2.2707 1.9758 1.6208 3.3048 2.3765 2.0594 1.6645
1.8786 1.3467 1.0700 0.8675 1.6304 1.2342 0.9774 0.8708 1.4268 1.0528 0.9547 0.8542 1.3976 1.0275 0.9416 0.8483 1.4637 1.0284 0.9376 0.8334
"All the values are in electron volt units. (7J defined by eq 1 1 and 12 in the text. Isotopes are 7Li, 23Na,39K, 8SRb,13,Cs, I9F, 3sCl, 79Br, IZ7I. dCalculated from Fermi energy.
those from Sanderson's principle within a small difference of 0.5 eV (see Table 11). These results suggest that the approach developed in the present study is very useful in predicting simply the approximate molecular chemical potentials for alkali metal halides as well as the other general heteronuclear diatomic molecules. We should make a remark here on the reliability of the molecular chemical potentials calculated by the present method. Unfortunately, we cannot assessexactly the results described above at the present time since there are no comparative data of "true" values of molecular chemical potentials for molecules under consideration. However, there will be remarkably needed some comparison with "true" values of molecular chemical potentials. For this purpose, we have carried out abinitio SCF calculations (Gaussian of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and then calculated the Fermi energy (average of HOMO and LUMO energies). The results are listed in the last columns in Tables I and 11. As seen from Table I, the calculated molecular chemical potentials for alkali metal halides are in good agreement with the results of abinitio SCF calculations excluding some alkali metal fluorides for which there are differences of about 1 eV. On the other hand, for general molecules in Table 11, the calculated chemical potentials have somewhat lower values by 0-3 eV than those of ab-initio SCF calculations. Perhaps this is due to the crude approximation of equation (36). We next discuss the apparent chemical hardneses given by eqs 11 and 12. These contain the integral terms involving the Fukui functions and are easily evaluated from eqs 16 and 17 by using AN determined spectroscopically. The calculated results of a p parent chemical hardnesses for only alkali metal halide molecules are listed in Table 111. Here, it is very interesting to compare
these results with those derived from the Fermi energies that are obtained by abinitio SCF calculations. As seen from this table, their agreements are moderately good for all the molecules under consideration. These analyses also suggest that the present approach is very useful in evaluating molecular chemical potentials of heteronuclear diatomic molecules. Last of all, we shall stress that much further study is needed to test the present approach. For this purpose, experimental values of molecular chemical potentials are much needed for diatomic as well as polyatomic molecules. Theoretical studies extending the present method to polyatomic systems will also be valuable.
Acknowledgment. I am much indebted to the reviewer for many instructive and encouraging comments in preparing the revised manuscript. I also thank Dr.K. Yokoyama for his computational support.
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