1832
J. Phys. Chem. 1993,97, 1832-1834
Electronic Chemical Potentials of Polyatomic Molecules Ken Ohwada Department of Chemistry, Japan Atomic Energy Research Institute, 31 9- 11, Tokai-mura, Naka-gun, Ibaraki- ken, Japan Received: August 13, 1992
A basic formula for evaluating the chemically important electronic chemical potential of the polyatomic system is derived within the framework of density-functional theory. To confirm the utility of the expression, values of electronic chemical potentials for various kinds of polyatomic molecules are calculated with the aid of two simple modelings and compared with those calculated from the Sanderson's principle as well as the experimental values. It is shown that the basic formula derived here is closely related to the Sanderson's geometric mean principle for electronegativity equalization and is hopeful in predicting approximate electronic chemical potentials of polyatomic molecules.
1. Introduction The (electronic) chemical potential of density-functional theory is the negative of the electronegativityof Pauling and Mulliken.' The term chemical potential as it occurs in thermodynamics has long been accepted as a perspicuous description of the escaping tendency of a component from a phase. On the analogy of this, the electronic chemical potential measuresthe escaping tendency of an electron cloud. This quantity has the same equalization property as does a thermodynamic chemical potential: If free flow is allowed, electrons go from a region of high chemical potential to a region of low chemical potential, until both regions have the same chemical potential value.'-3 In other words, it is a constant, through all space, for the ground state of an atom, molecule, or solid and equals the slope of the curve for the total electronic energy versus the number of electrons at a constant nuclear potential. This is a rigorous consequence of the density functionalformulation4of the ground-state quantum mechanics. Thus, for an understanding of the chemical and physical properties of atoms, molecules, or solids, it is of much significanceto predict their chemical potentials for the ground states. In a previous report.5 we have dealt with the chemical potentials for heteronuclear diatomic molecules in equilibrium. In the present study, attention focuses mainly on the electronicchemical potentials of polyatomic molecules. In section 2, we attempt to derive, on the basis of density-functionaltheory, a basic formula for evaluating the electronic chemical potential of the polyatomic system. We next demonstrate the utility of the above expression by calculating electronic chemical potentials with the help of simple modelings for polyatomicmolecules. Thecalculated results are compared favorably with those calculated from thesanderson's principle6 as well as the experimental values. This is described in section 3. 2. Electronic Chemical Potentials of Polyatomic Molecules Following the previous paper? we can express the atomicenergy
(EX)for the xth atom in the polyatomic molecule (ABC...) to second order in its net electroniccharge (number) ANxand nuclear potential change Avx(r) as
evaluated. In this case, we shall take the standard state with energy EX)^ to be ground state of the isolated state. Also, the quantity px is the chemical potential,' px(r) the electron density, vx the chemical hardness? and fx(r) the Fukui function.8 The chemicalpotential px for atom X in the polyatomic system is given by differentiating eq 1 with respect to ANx,
[a(E,)/aNxl, = clx = P,o + 2'IXANX + Jfx(r)Avx(r)dr
which just correspondsto the basic formula derived on the basis of the second-order atoms-in-moleculesapproach by Nalewajski.9 To obtain more tractable forms of the chemical potentials, let us here define the following quantity:
Henceforth we call this term the apparent (effective) chemical hardness and shall discuss on it in some detail later. With this definition, eq 2 becomes
+ 2ANX[qx + (l/2ANx)Jfx(r)Avx(r)dr] + ... = ccXo + 2(qx)ANx + ... (4)
pX = P>
In the equilibriumconfiguration,the chemical potentials of all the atoms in the molecule become equal to each other,'-3 Le.,
0022-3654193 832S04.OolO * ,12097-1 .--. .., -
= Pe
... = pX =
(5) The quantity of MOL is the electronic chemical potential that is definable for the molecular system with fixed external potentials. Hereafter, we shall develop the methods to get formal electronic chemical potentials for polyatomic molecules. PA
PMOL
Using the condition for the molecule to be neutral, namely
CAN, =o
(x= A, B,c, ...)
(6) and imposing the equilibrium condition of eq 5, we have, from eqs 4-6,the change in the number of electrons ANx on atom X as follows:
ANx = ( 1/2) Here, r is the space coordinate vector and the subscripts zero indicate the standard state for which the partial derivatives are
+ ... (2)
[CbyO-
PX')/ ( V Y )( VX)
I/
[E/ 1
I
('~y)
(X, Y = A, B,C, ...) (7) The electronic chemical potential MOL for the polyatomicsystem (8 1993
American Chemical Society
Chemical Potentials of Polyatomic Molecules
The Journal of Physical Chemistry, Vol. 97, No. 9, I993 1833
is therefore given by replacing eq 7 into eq 4,
For theatoms A, B,C, in the homonucleardiatomicmolecules Az, B2, CZ,...respectively, we are able to define electronegativities x (in atomic units) by the formulas
...
(X = A, B, C, ...) (8) this being the expression which we were looking for. Equation 8 is similar to that derived by Yang, Lee,and Ghoshiowho have given the polyatomic electronic chemical potential to first order. If we define the apparent chemical softness ( UX) that is simply the inverse of the apparent chemical hardness 1/ (qx), eqs 7 and 8 become
XA
= S'qA/2rA, XB = hB/2rB* XC = S'qC/2rC, * * * (15)
where {is a constant ({ = 0.484,0.354,0.273 for single, double, and triple bonds, respectively, according to Ray, Samuels, and Parr)," qx (X = A, B, C, ...) is the bond charge, and rx is the covalent radius (2rx = Rxx); the factor 2 appears because in the homonuclear bond-charge model the net charge on a nucleus is (1/2)q. For the polyatomic system, we have similarly the following expressions for electronegativities
(X, Y = A, B, C, ...) (9)
The physical meaning of eq 8 and 10is that the electronicchemical potential of a polyatomic molecule is formally given by the statistical mean of the chemical potentials of constituent atoms with statistical weights of the reciprocals of their apparent chemical hardness (or with statistical weights of their apparent chemical softnesses). Here, we will mention about relation with the Sanderson's geometricmean principle for electronegativityequalization.6Since the chemical potential of an atom and molecule is the negative of the electronegativity x,' eq 8 becomes
Xc(in ABC...) = T(qc
+ Aqc)/2rc
where rx is the bond radius, Aqx the charge (transferred) increment, and (qx + Aqx)/2 the net charge on nucleus X. But, in the ABC...system, XA, XB, XC, ...must be equal to each other and q u a l to the electronegativity XMOL of the molecule:
= X A ( h ABC ...) = xB(in ABC...) = xc(in ABC ...) = ... (17) If we assume the charge conservation XMOL
where n is the number of constituent atoms in the polyatomic molecule. It should be noted that both the numerator and denominator of the last expression in eq 11 are multiplied by literal coefficients 1/n. Assuming that the arithmetic averages of both the numerator and denominator in eq 11 are replaced with the geometric means (such replacements may be allowed mathematically under certain circumstances)
CAqx =0 we have from eqs 15-1 7
(X = A, B, C,...)
(18)
where the final approximate expression has been derived under the reasonable assumption of
-
one finds that
This is none other than the Sanderson's geometric mean principle viewed from the density-functional approach. It goes without saying that the Sanderson's principle (eq 13) can also be derived from another eq 10 under similar replacements to eqs 12a and 12b:
3. Results and "%don
In the present section, we shall describe on the electronic chemical potential that is definable for the molecular system with the fixed external potential. The basic working equation is eq 8 or 10 for the polyatomic system. Implementing this expression, we encountered with one difficulty, that is, how to evaluate the apparent (effective) chemical hardness ( T X ) defined as eq 3 in the preceding section. To overcome this difficulty, we begin by modeling of the polyatomic system in some simple way as described below.
2rx Rxx (20) Since the electronegativity x of an atom and molecule is the negative of the chemical potential p of its electronic cloud (x = -w),I eq 19 can be converted into the chemical potential form:
Comparing eq 21 with eqs 8 and 10 in the previous section, it follows that l/('lx) = ( 4 = Rxx (22) This means that the reciprocal of apparent chemical hardness or the apparent chemical softness is proportional to RXX.The predictive value of eq 22 will be tested later. We next proceed to another way for predicting simply approximate electronic chemical potentials of polyatomic molecules. For this purpose, let first define a new parameter y that is expressed as the ratio of apparent chemical hardnesses: (tlX)/(tlY) = Yxy (23) With this definition, eq 8 for the polyatomic system is converted into
Ohwada
1834 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993
TABLE I: Ekctronie Chemical Potentiah (In eV Units) of Polyatomic Mokculcs molecule
calcd 1‘
calcd 2b
S-PC
Triatomic Molcculcs -6.9594 -6.9802 -1.0103 4.8521 -6.8409 -6.8972_._ -7.3852 -1.3895 -1.3192 -6.8188 -6.8578 -6.1591 -6.2313 -6.2330 -6.2292 -1.0563 -7.0810 -7.1054 -1.3018 -1.2953 -1.3392 -6.1519 -6.8411 -6.6396 -6.5900 -6.1169 -6.4106 -6.3123 -6.5631 -6.1094 -9.3603 -9.3491 -0.5607 -1.4685 -1.4581 -1.4646 -1.8428 -1.1885 -1.8881 Tetraatomic Molecules -7.1053 -1.1847 -1.0864 -8.3399 -8.7593 -8.1555 -1.0440 -7.4648 -7.18 16 -8.8556 -8.9216 -8.9456 -6.5811 -6.9162 -6.1377 -7.1545 -1.0466 -1.1104 -1.5353 -7.6618 -7.6558 -1.0222 -6.9990 -6.9941 -6.4578 -6.6994 -6.1491 -9.6512 -9.5211 -9.1791 -8.1320 -8.0468 -8.1548 -1.2990 -1.2624 -1.3143 -9.8412 -9.1480 -9.1136 -9.6231 -9.4319 -9.4121 -1.5514 -7.7705 -1.6322 -1.2021 -7.3973 -1.2562 Polyatomic Molecules -9.4883 -9.6121 -9.6561 -7.4113 -7.4419 -1.4255 -7.1555 -1.1581 -7.1355 -6.1836 -6.8257 -6.7811 -6.7355 -6.7861 -6.7125 -6.8991 -6.8083 -6.8600 -6.8297 -6.8715 -6.8087 -6,9990 -6.9941 -1.0222 -6.9513 -6,9011 -6.9186 -6.8188 -6.8578 -6.7591 -6.1083 -6.1561 -6.6134 -6.7959 -6.6941 -6.1546 -7.0191 -1.0610 -1.0935 -6.6623 -6.1039 -6.6040 -6.6623 -6.1039 -6.6040 -6.9231 -6.8600 -6.8885
exptld -6.7 -5.1 -5.4 -5.3 -5.35 -6.9 (-5.0) -6.3 -5.3 (-4.2)
-1.2 -8.3 (-6.2) -5.97 -6.3 (-5.1) -5.61 -5.6 -5.5 -4.1 -4.1
-8.0 -5.8 -5.19 -5.5 -5.35 -4.9
-4.7 -4.6 -4.5 -4.4 -4.4 -4.3 -4.2 -4.1 1 -4.1 -4.1
0 Calculated from eqs 8 and 22 in the text. Calculated from eqs 8 and 25 in the text. Calculated from the Sanderson’s principle. Data taken from ref 11.
At this stage, let us introduce the following drastic approximation for YXY: bearing in mind that the chemical hardness of the isolated atom is given by the finite-difference appr~ximation~,~ as where IX and EX are the ionization potential and the electron affinity. We are now in the position to evaluate approximate electronic chemical potentials for polyatomic molecules. In the calculations, we have used chemical potential values of isolated atoms for px0 in q 8. These quantities are given by the three-point finite difference approximationlJJ* Thevalues of 1x and EXwere taken from references (13,14). The remaining quantities necessary to the calculation are internuclear distances of homonuclear diatomic molecules. These were taken
from references (15,16). The calculated results are listed in the second and third columnsof Table I together with the experimental values. Those in the second column were calculated from q 8 to test the predictive value of q 22. Those in the third column were obtained from eqs 8 and 25. For comparison, the table also include the values that were calculated on the basis of the Sanderson’s principle6 for electronegativity equalization. As seen from this table, both our calculated results are in good agreement with those from the Sanderson’s principle. However, we become aware of the facts that some of our calculated values as well as Sanderson’svalues have somewhat lower values by &3 eV than the experimental values in the Pearson’s table” drew up by a comprehensive survey. It is important to clarify the cause of such differences on further raising the predictive value of the present approaches. Specially, more rigorous analysis of the apparent (effective) chemical hardness (qx) as defined in q 3 is necessary. For this purpose, it will be effective to carry out Kohn-Sham or Hartree-Fock SCF calculations of HOMOLUMO gap and Fermi energy (average of HOMO and LUMO energies). 4. Concludiag Remarks
Using a density-functional approach, we have derived a basic formula for evaluating the chemically important electronic chemical potential of the polyatomic system. In such a formulation, the apparent chemical hardness and the apparent chemical softness (reciprocal of the former) are defined conveniently. The result shows that the electronicchemical potential of a polyatomic molecule is formally given by the statistical mean of the chemical potentials of constituent atoms with statistical weights of the reciprocals of their apparent chemical hardnesses (or with statistical weights of their apparent chemical softnesses). To confirm the utility of the expression, we have calculated values of electronic chemical potentials for various kinds of polyatomic molecules with the aid of two simple modelings, and compared these values with those calculated from the Sanderson’sprinciple as well as the experimental values. It has been found that the basic formula derived here is closely related to the Sanderson’s geometric mean principle for electronegativity equalization and is hopeful in predictingapproximateelectronicchemical potentials of polyatomic molecules. However, many further studies are called for raising quantitative predictive value of the present approaches. At the same time, experimental I, E, and p values are much needed for polyatomic molecules.
References and Notes (1) Parr, R. G.; Donnelly, R. A.; Levy, M.;Palke, W. E. J. Chem. fhys. 1978,68, 3801. (2) Donnelly, R. A.; Parr, R. G. J . Chem. Phys. 1978,69,4431. ( 3 ) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York. 1989. (4) Hohenberg, P.; Kohn, W. fhys. Reo. E, 1964, 136, 864. (5) Ohwada, K. J . fhys. Chem. 1992. 96, 5825. (6) Sanderson, R. T. Chemical Bonds and Bond Energy, 2nd ed.; Academic Press: New York, 1916; Science, 1951, 114, 610; 1952, 116, 41; 1955, 121, 207. (7) Parr, R. G.; Pearson, R. G. J . Am. Chem. Soc. 1983, 105,1512. (8) Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1984, 106, 4049. (9) Nalewajski, R. F. J. Am. Chem. Soc. 1984,106,944; J. Phys. Chem. 1984.88, 6234; J . Chem. Phys. 1984,81.2088. (IO) Yang, W.; Lee, C.; Ghosh, S.K. J . Phys. Chem. 1985.89, 5412. ( 1 1) Ray, N. K.; Samuels, L.; Parr, R. G. J . Chem. fhys. 1979,70,3680. (12) Mulliken, R. S.J. Chem. fhys. 1934. 2, 182; 1935, 3, 513. ( 1 3) Pimentel, G. C.;Spratly. R. D. Chemical Bonding Clarified Through Quanrum Mechanics; Holden-Day: New York, 1969. (14) Hotop, H.; Linebcrger, W. C. J. fhys. Chem. Re/. Data, 1975, 4, 539. ( 15) Herzberg,G. MolecularSpectra a ~ M o l e ~ l a r S t n c c t u n -Spectra I. of Diatomic Molecules; Van Nostrand: New York, 1950. (16) Rosen. B. Spectroscopic Data Relative to Diatomic Molecules; Pergamon Press: Oxford, 1910. (17) Pearson, R. G. J . Am. Chem. Soc. 1985, 107, 6801; Inorg. Chem. 1988, 27, 734.
