2866
J . Phys. Chem. 1990, 94, 2866-2872
Charge-Dependent Hamiltonian for First- and Second-Row Atomic Properties John David Baker and Michael C. Zerner* Quantum Theory Project, University of Florida, Gainesville, Florida 3261 I (Received: October 5, 1989)
Charge-dependent parameters have been developed for hydrogen through argon that allow the calculation of average valence energies for a wide range of atomic charge. These parameters allow atoms in molecules to contract or expand with charge, emulating the capability of more complex multiple-!: ab initio calculations while retaining the simplicity of a minimal basis formalism. In addition to promotion energies being reproduced, Mulliken electronegativities are also successfullyreproduced, as are absolute hardness values.
I.
Introduction
Today, semiempirical electronic structure formalisms are firmly established as necessary tools for the understanding of chemical phenomena. They are used when rigorous ab initio calculations are too difficult even for the most powerful computers and are frequently used as starting points for accurate ab initio calculations of small systems. While the basic form of ab initio theories are maintained, semiempirical theories are conceptually different due to the use of parameters in the semiempirical model that have been determined by experiment. In this way, accurate calculations can be performed with minimal effort. Successful implementation of semiempirical theories is therefore often dependent upon the optimal use of the available experimental data in the interpretation and parametrization of the terms in the model. It has been shown that considerably improved agreement with experiment can be obtained if the parameters assumed in semiempirical models are allowed to be a function of the charge of the atom, instead of being considered as constants depending only on atom type.'-5 These previous schemes, however, do not attribute the reason for this as we do below nor do they present a systematic study of what the charge dependencies should be for all the parameters involved in a calculation. I n this study, a functional form of charge-dependent parameters for atoms is developed and fitted to the experimental data. I n section 11, we review the link between experiment and the parameters that are currently in use. The proposed parametrization developed in section 111 removes some of the approximations that are used in developing atomic based parameters. The assumption is still made that atomic energies can be accurately modeled with a valenceorbital-only minimum basis formalism, but the parameters that arise from this assumption no longer need be constants for a given atom. By design, the new parameters will reproduce atomic energies and in this way also reproduce the energy differences necessary for describing atomic promotion energies, ionization potentials, and electron affinities. As a consequence of ionization potentials and electron affinities being reproduced, orbital electronegativities6 and absolute hardness values' are also obtained. The ability of the new model to accurately duplicate these data is demonstrated while exposing the inability of current models to perform as well. The use of atomic based parameters in molecular theories is well developed. I t is here, of course, that the real interest lies, and we are presently examining the use of these parameters in such molecular models. Although there are no clear guidelines for the development of terms that are two-center in nature, pre( 1 ) Wheland, G . W.; Mann, D. E. J . Chem. Phys. 1949, 17, 264-268. (2) Brown, R. D.; Heffernan, M. L. Trans. Faraday SOC.1958, 54,
757-764. ( 3 ) Parks, J . M.; Parr, R. G.J . Chem. Phys. 1960, 32, 1657-1681. (4) Iffert, R.; Jug, K. Theor. Chim. Acfa 1987, 72, 373-378. ( 5 ) Kovesdi. I. THEOCHEM 1987, 152, 341-346. ( 6 ) Mulliken, R . S. J . Chem. Phys. 1934, 2. 782-793. (7) Pearson, R. G.J . Chem. Educ. 1987, 64, 561-567.
liminary work within the zero differential overlap (ZDO) formalism is very ~ n c o u r a g i n g . ~ - ' ~ 11.
Atomic Parametrization
We seek to empirically parametrize atomic integrals. Use of experimental data to fit these parameters has three p i t i v e results. The first is to give a feel for the impact the terms in the model have on calculated properties. The second is that use of experimental data in the model moderates the approximations invoked. In other words, a balance is achieved between losing accuracy due to the neglected terms and maintaining in the model the essential physics that is necessary to determine electronic structure in a meaningful fashion. A third benefit that arises is that parametrized models may perform more accurately than an exact implementation of the theory used. We limit here our discussion to atoms whose valence shell consists of only S- and P-type orbitals. We can calculate from experimental data the average valence electronic energy for an atom with a fixed number of S and P electrons. The valence energy is calculated via a weighted average of the term energies arising from a given configuration. The term energies are weighted by their degree of degeneracy determined from the term S and L quantum numbers. Total degeneracy for a term energy would be (2s 1 ) ( 2 L
+
+
term
The average configuration energy can be expressed as a sum of one-center core terms (U,and Up,) and a sum of average pair energies.16 The core terms consist of the electron kinetic energy, the attraction of the electron to the core, and the core-valence potential energy.
U;,S,,,,
by orbital symmetry (2)
The average pair energies are composed of two-electron coulomb and exchange integrals of the type (8) Pople, J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory; McGraw-Hill: New York, 1970. (9) Pople, J . A.; Santry, D. P.; Segal, G.A. J . Chem. Phys. 1965, 43, s 1 2 9 4 135. (IO) Pople, J. A.; Segal, G. A. 1. Chem. Phys. 1965, 43, S136-SI51. ( I I ) Pople, J. A.; Segal, G. A. J. Chem. Phys. 1966, 44, 3289-3296. (12) Santry, D. P.; Segal. G. A . J. Chem. Phys. 1967, 47, 158-174. (13) Pople, J. A.; Bevertdge, D. L.; Dobosh, P. A. J . Chem. Phys. 1967, 47, 2026-2033. (14) Sadlej, J. Semi-Empirical Methods of Quantum Chemistry; Wiley: NPWYnrk 19RS
( I 5 ) Slater. J . C. Quanrum Theory of Atomic Structure; McGraw-Hill: New York, 1960; Vol. I , pp 296-3 IS. (16) Karlsson, G . ; Zerner, M. Int. J . Quantum Chem. 1973, 7, 35-49.
0022-3654/90/2094-2866$02.50/0 0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 2867
Hamiltonian for First- and Second-Row Atomic Properties
(x,x~IxAx~)
= Jdrl
1
-!
d72~,*(1) x,( 1) - x A * ( ~ ) ~ ~ ( 2 ) (3) Rl2
14
One can integrate over the angular part of the two-electron coulomb and exchange integrals that are not zero by symmetry and express these as a function of Condon-Shortley integral^.'^ These integrals will become the empirical parameters. We further simplify the equations by assuming the effective exponents for valence S and P orbitals are the same for a given atom. We can now write expressions for one-center two-electron integrals.
J,, = J,, = (sslss) = (sslp,p,) = P
(4)
Ksp, = (SP,lSP,) = Y3G’
(5)
= (P,PvlP,Pv) =
(6)
Kpppl
3/Sp
I 01
2
Jprpr Jprpv
= (PpP,IPpP,) = = (PpP,lP#u) =
+ ?2Sp - 2/25p
i
2t
3
4
5
(7)
(8)
6
7
8
9
10
11
I
12
ATOMlC NLJMBER(Z)
Figure I . Row 1 experimental values17 for Slater-Condon integral P: (0)szpo: (+) S P I ; (*) SPZ; ( X ) SZP’; ( # ) S 2 P .
P,GI, and F2 are the Condon-Shortley
parameters of interest. By averaging over all term energy expressions for a fixed valence configuration and then expressing all two-electron integrals in terms of the above parameters, we obtainI6
+
+
E(Z,n,,n,) = nJUss+ npUpp f/z(n,+ np)(ns np - l ) P j/gnsnpc’ - ‘/25np(np(9) This equation is our working model for atomic parametrization. Standard values for GI and F2 derived from atomic spectroscopy have been available in the literature for many years.I7J8 Ionization potentials (IP‘s)19 and electron affinities (EA’S)%are also available for both S and P electrons for most atoms. Standard parametrizations assume that Up,, P,GI, and p are fixed values for any given atom. One can then use eq 9 to calculate expressions for Us,and Uppin terms of known IP’s or EA’s. As an example, Pople’s original parametrization for U , in the CNDO was derived on the basis of IP’s alone.”
+ np - 1)P + (10) - (n, + np - I)P + ‘/6n,G’ -k 2/25(np- 1)p ( 1 1 )
UzsZs= -IPS - (n, u2p2p=
-IP,
16 -
This has been the parametrization of choice for many programs currently in use. Another popular choice uses the average of IPS and EA’s to evaluate the U,,,’S.~~The only remaining quantity is P,which is generally calculated theoretically with Slater S-type atomic functions with exponents derived from empirical rules for screening constants such as those of Slaterz1or Clementi and Raimondi.22 111. Proposed New Atomic Parametrization It is well accepted that ab initio minimal basis set calculations in quantum chemistry at the self-consistent-field (SCF) level are not generally reliable. One problem with minimal basis set calculations is that the effective valence exponents are fixed so that the electronic density in an atom cannot contract or expand as the atom loses or gains charge. The basis functions should, however, have the flexibility to expand or contract as necessitated by the environment of an atomic center in the molecule. In ab initio theories, this implies that the exponents in the basis functions should also be optimized according to the variation principle along with the orbital coefficients. The optimization of orbital exponents is a complex nonlinear problem that is also highly dependent on (17) Hinze, J.; J a m , H. H. J . Chem. Phys. 1963, 38, 1834-1847. (18) Skinner, H. A.; Pritchard, H. 0. Truns. Furuduy SOC.1953, 49, 1254-1 262. (19) Moore, C. E. Atomic Energy Leoels; National Bureau of Standards Circular No. 467, National Bureau of Standards: Washington, DC, 1949;
Vol. I .
(20) Huheey, J. E. Inorganic Chemistry; Harper & Row: New York, 1983. (21) Slater. J. C. Phys. Reu. 1930, 36, 57-64. (22) Clementi, E.; Raimondi. D.L. J . Chem. Phys. 1963,38,2686-2689.
s s
5
14
-
12
-
10-
1 6-
4-
’2
3
4
5
6
7
8
9
1 0 1 1 1 2
ATOMIC NUMBER(ZJ
Figure 2. Row I experimental values’7 for Slater-Condon integral GI: (0)SPO; (+) S P I ; (*) SP2; ( X ) S P 3 .
the form of the basis set functions in its implementation. This problem is alleviated in ab initio theory calculations by use of multiple-{ basis sets. Instead of a single basis function with a fixed exponent for an orbital, orbitals are represented as a sum of basis functions with differing exponents. In this way, the nonlinear optimization of exponents in a minimal basis calculation is linearized. Due to the fact that ab initio calculations grow as N4, the use of multiple-{ basis sets further restricts the size of systems that can be studied. The reliability of semiempirical theories, which are most often minimal basis set formalisms, is aided by the experimentally calibrated parameters. These theories are less reliable, however, when bonds are highly polar in nature. For example, the calculation of electronic spectra of carbonyls appears to require a different set of two-center parameters than molecules in which the atoms remain primarily neutral in order to get transition energies that match e ~ p e r i m e n t . ~We ~ - ~are ~ not free to apply the variational principle to optimize effective exponents in semiempirical theories since we have not explicitly dealt with the core electrons. Core-valence potentials are built into the oneelectron parameters, but without restraining the valence exponents to remain in the valence space, optimizing these exponents will result in a collapse of the valence orbitals into the core space.2s We then propose to maintain the experimental determination of the one-center integrals but remove the restriction that these parameters are constants. Instead, they will be expressions dependent on the charge of the atom. In other words, we envision our orbitals contracting and expanding with charge and proceed (23) Del Bene, J . ; JaffE, H. H. J . Chem. Phys. 1968, 48, 1807-1813. (24) Ridley, J.; Zerner, M. Theor. Chim. Acta 1973, 32, 1 11-1 34. (25) Zerner, M. Mol. Phys. 1972, 23, 963-978.
2868
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
TABLE I: F2,Comparison of Some Calculated (Equation 12) vs E~perimentally’~ Obtained Values element Z 0 exm, eV model. eV Be 2 0 2.6564 2.625 I B 3.4809 3 0 3.3337 5.8986 4 +I 5.8856 C 4.5100 4.8376 C 4 0 6.459 6.0858 N 5 0 6 +I 8.8866 8.484 I 0 6.9029 6 0 7.0268 0 9.1718 7 +I 9.271 1 F Ne 8 +2 1 1.363 11.514 TABLE 11: C ’ , Comparison of Some Calculated (Equation 13) vs Ex~erimentallv’~ Obtained Values
Be B C C C
0 0 2 I 0 2
2 3 4 4 4 5 5 5 6 6 6 7 7 8 8
N N N 0 0
0 F F Ne Ne
1 0 2 1
0 2 1
3 2
3.9104 5.6653 9.1220 8.2712 7.4203 10.877 10.026 9.1752 12.632 11.781 10.930 14.387 13.536 16.993 16.142
3.8282 5.4015 9.2959 8.3779 6.8979 I 1.036 10.087 8.9585 12.666 1 1.500 11.816 14.196 13.850 16.813 15.788
to determine the effect this would have on the parametrized integrals. As previously, our working model will be eqs 1 and 9. We first consider the Condon-Shortley parameters in the literature.,’ The values available are for atomic cations as well as for neutral atoms. We begin by assuming that the parameters are a function of charge and atomic number. We then examine the experimental data to see if any pattern emerges that would suggest a specific functional form. At this point, the data is fit in a least-squares fashion to the model. A plot of the data for elements Be through F for both and GI are included in Figures 1 and 2, according to the number of valence electrons. Lines have been drawn through the experimental data representing isoelectronic curves. From left to right, the curves represent valence configurations S2PN for N = 0, I , 2, 3, and 4. In this way, the functional dependence of these parameters with respect to charge and atomic number is evident. The regularity of the data has allowed for the fitting of a single model for each parameter for these elements that incorporates both charge and atomic number. In all cases, the models are linear in net charge (Q) once the valence core charge (Z) is fixed.I8 For elements Be through Ne, we obtain
P (eV) = -2.1972
+ 1.99502 - 0.0249Z2 - 0.0086Z3 + (0.2294
+ 0.2047Z)Q
STD = 0.323 eV
GI (eV) = 0.4004
+ 1.75492 + 0.8509Q
STD = 0.323 eV and for Na through Ar,
P (eV) = 15.4738 - 9.94742 + 2.2126Z2 - 0.1430Z3 + ( 1 1.5939 - 4.46032 + 0.5035Z2 - 0.0134Z3)Q (14) STD = 0.143 eV
+ +
G’ (eV) = -8.9073 + 8.85722 - 1.8726Z2 0.1204Z3 + (-4.4977 + 1.97602 - 0.1706Z2 0.0007Z3)Q ( 1 5)
STD = 0.415 eV Tables I-IV compare some of the experimental values for F2 and G’ with those calculated from eqs 12-15.
Baker and Zerner TABLE 111: F’, Comparison of Some Calculated (Equation 14) vs E~perimentally’~ Obtained Values element Z Q expt, eV model, eV 3.2851 Mg 2 0 3.2132 AI 3 +I 4.1721 4.0662 AI 3 0 1.6025 1.683I Si 4 +I 2.591 3 2.8834 Si 4 0 2.2627 1.9317 P 5 +2 3.6390 3.5841 P 5 0 2.9477 3. I729 4.741 5 S 6 +3 4.7796 S 6 +2 4.75 17 4.6772 S 6 +I 4.7 145 4.6129 0 4.5379 4.5485 S 6 CI 7 +3 6.3759 6.5427 CI 7 +2 6.1 156 6.0953 CI 7 +I 5.6878 5.6478 8.0937 Ar 8 +3 8.1830 Ar 8 +2 6.75 10 6.8 193 TABLE IV: C ’ , Comparison of Some Calculated (Equation 15) vs Experimentallv” Obtained Values element Z Q expt, eV model, eV 2.2802 2 0 2.4765 Mg AI 3 +I 4.1532 3.9771 3.3591 4.0627 AI 3 0 5.5782 5.7141 Si 4 +2 4.9299 4.9909 Si 4 +I Si 4 0 4.8124 4.2677 7.2426 5 +3 6.8469 P P 5 +2 6.4385 6.0344 5 +I P 4.7748 4.8262 S 6 +3 7.6545 6.9577 S 6 +I 3.6452 4.2009 S 6 0 3.0757 2.8360 CI 7 +2 4.3299 5.0938 c1 7 +I 4.0863 3.8690 Ar 8 +3 6.0301 6.0607 Ar 8 +2 5.5273 5.2956
Of immediate interest here is the strong dependence on these integrals with charge for a given atom. For a fixed minimum basis, these values would be considered constant. Moderate deviations between calculated and experimental f and GI integrals will have a negligible effect on the total energy in eq 9 in that they contribute the smallest proportion to the total energy. Having now formulas for and GI, we turn our attention to Us,Upp,and F“. As previously discussed, these parameters have been estimated in the past as charge-independent constants determined via ionization potentials or ionization potentials and electron affinities. Karlsson and Zerner16 proposed calculating the core terms from the spectroscopic data alone for neutrals by fitting eq 9 to total valence energies determined from multiple ionization energies and valence promotion energies. In their proposal, the value for p continues to be calculated theoretically over Slater functions with exponents determined from Slater’s rules. It is not possible to uniquely determine Us,, Up*,and p from a linear least-squares fit of eq 9 due to the interdependence of the variables that are coupled. It must be further pointed out that any reasonable form chosen for P can be used and the fitting procedure applied successfully. Since we cannot allow all parameters to vary freely in the fit, we seek to use a form for p that has a simple interpretation. We perform the fit using two such F”s that have been used successfully in semiempirical calculations. We first assume the theoretical form for F” in a fashion consistent with Slater’s rules and note that is linear with respect to the exponential factor {,, in the Slater basis. FO(s,s) = FO(s,p) = P ( p , p ) = K(n)(,, = (nsnslnsns)
K( 1) = 0.6250 au = 17.0075 eV K(2) = 0.3633 au = 9.8856 eV (16) K(3) = 0.2581 au = 7.0244 eV From Slater’s rules for minimal basis set calculations { = (Z,,,,
Hamiltonian for First- and Second-Row Atomic Properties - o ) / n . From these rules, we derive for
F’, in
electronvolts,
F’(2s,2s) = 3.2128 F’(3s,3s)
(17)
where 2 = Zvalence = (Z,,,, - no. of inner electrons) and Q is the net charge for the center. Although Slater’s rules are frequently used in semiempirical calculations to evaluate F’, it is often convenient to evaluate F’ via Pariser’s observation,26 especially in the calculation of electronic spectra. Pariser’s observation is to set F’ equal to the ionization potential minus the electron affinity for neutral atoms. We can fit the values from Pariser’s formulation with a simple A BZ model for each row and use the slope term in charge from Slater’s rules above to obtain
+
F’(ls,ls) F’(2s,2s) F‘(3s,3s)
+ 11.90532 + 5.1023Q = 3.7271 + 1.57362 + 1.7300Q = 2.1443 + 1.32392 + 0.8195Q
= 0.9447
(23) Na-Ar: Slater Fo U3s3s=
+
0.7922 - 4.68 10 2 - 1.52 1 52’ (0.5146 - 0.63262 0.0089Z2)Q + (0.2017
+
= 3.0984 - 4.20002 - 1.45122’ (0.8865 - 0.31622 - 0.0O04Z2)Q
U3,3,
(18)
H-He: Slater Fo
+ (-0.4781 + 0.0103Z)Q
(20)
+ (-2.5569 + O.Ol03Z)Q
(21)
H-He: Pariser Fo
Li-Ne: Slater Fo
+ + (0.6291 + 0.0088Z)Q’ U2,2, = 1.1008 -1.74622 - 3.09802’ + (1.2714 - 0.60432 - 0.00162’)Q + (0.2392 + 0.0045Z)Q2
U2,Zs = 1.6381 - 3.86142- 3.12752’ (1.6649 - 0.96322 - 0.0065Z’)Q
(22) (26) Pariser, R. J. J. Chem. Phys. 1953, 21, 568-569. (27) Hinze, J.; Jaffe, H. H. J. Am. Chem. SOC.1962,84, 540-546.
+ 0.0255Z)Q2
+ + (0.2945 - 0.01832)Q’ (24)
Na-Ar: Pariser Fo
+
0.0497 - 4.03752 - 1 .4224Z2 (-0.2279 - 0.73162 0.0089Z2)Q + (0.2017
In this equation, we assume a scale of zero energy for the atomic core with all valence electrons separated. In this way the energy of the core is maintained as a constant. Each addition of a valence electron thus lowers the atomic energy by the negative of the corresponding ionization potential. We can additionally use electron affinity data to add electrons beyond the neutral atom. In order to obtain ,Epromenergies for anions, we extrapolate the promotion energies27of atoms with the same valence configuration in terms of atomic number since these data are not readily available for anions. The correlation coefficients for the interpolation of Eprom’s with atomic number are essentially 1.000 for a simple linear model in all cases. The extrapolated values for the lowest valence states are usually small as would be expected for orbitals on an atom of loosely held electrons. Using these experimental data and the modeled Slater-Condon integrals, we obtain for the core integras, in electronvolts,
Ulsls = 24.6886 - 38.28362
+ + 0.0088Z)Q’ U2,2, = 1.3580 - 2.82292 - 2.2784Z2 + (1.5286 - 1.42392 - 0.0016Z’)Q + (0.2392 + 0.0045Z)Q’
U3s3s=
Now that we have general expressions for the Slater-Condon integrals, we seek to generalize the core terms in a similar fashion by fitting eq 9 for not only neutral but also cationic and anionic valence configuration energies determined from multiple ionization energy data. If we replace E,, in eq 1 with the spectral transition energy for that specific term from the ground state, we obtain an equation for the calculation of the promotion energy (Eprom) from the ground state of the atom to its average valence configuration energy. We can now express the valence configuration energy of an atom in terms of experimentally known quantities.
Ulsls = 26.7674 - 40.26242
Li-Ne: Pariser Fo
UzsZs= 1.8954 - 4.93812 - 2.3079Z2 + (1.9221 - 1.78282 - 0.0065Z’)Q (0.6291
F’(ls,ls) = 5.1023 + 11.90532 + 5.1023Q
+ 3.21282 + 1.7300Q = 3.6293 + 1.52202 + 0.8195Q
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 2869
+
+ 0.0255Z)Q2
+
= 2.3559 - 3.55662 - 1.3522Z2 (0.1440 - 0.41532 - 0.0004Z’)Q + (0.2945 - 0.0183Z)Q’ (25) The number of parameters included in the fit for a row is 16 once the two-electron integral forms are specified. The number of atomic states included in the fit for row 1 is 80 and for row 2, 58. A summary of the results obtained by using these equations in calculating average configuration energies of atoms and ions is given in Table V for first-row species and Table V I for second-row species. The use of Slater Fo’s or Pariser Fo’s are equally successful in this rather remarkable fit. The standard deviation for row 1 is 0.24 eV, while that for row 2 is 0.32 eV. The data for aluminum, silicon, phosphorus, and sulfur are shown in Figures 3-6 where they are compared with results with INDO/ 1 atom fixed parameter^.^^ Three possible energies are available for an atom of a given charge depending on the promoted state of the atom. These are S2PN,SIPNf’,and SopN+’.The INDO/I parameters are based on Slater p s for neutrals and eqs 10 and 1 1 for the core terms. Several important observations can be made. In all cases, the new parametrization is able to reproduce the experimental values quite well. Since one can obtain total atomic energies by adding a constant to the valence energies, the new model is capable of reproducing total energies as well. The INDO/l average energies not only do a poorer job as might be expected but also shift the minimum of the curve to the right. Since the position of the minimum determines the charge at which atoms are most stable, this shift of minimum cannot account for the fact that most anions are more stable that neutrals. In addition, the use of fixed two-electron repulsion integrals causes the slope of the energy to increase too rapidly as the atom becomes anionic in INDO/ I . The curve for sulfur is a dramatic example of this effect. I n this curve, the slope of the experimental data is much “softer” than that of the INDO/I data. It is again pointed out that the new parametrization takes into effect the softening of the repulsion integrals as atoms become anions and reproduces the experimental data well. A second observation one can make from the figures is that of the differences in slope of the curves of the neutral species for the two parametrizations. The slope of atomic energy with respect to the number of electrons is directly related to the electronegativity of the atom. In the early 1960s, workers fit empirical quadratic formulas to atomic multiple ionization potentials and electron affinities and used these formulas to calculate empirical slopes.28 These slopes correlate quite well with Mulliken’s electronegativity ~ c a l e . ~Hinze , ~ ~and Jaffe27developed a similar model based on the use of spectroscopic energies to fit a quantum U3,3,
(28) Iczkowski, R. P.; Margrave, J . L. J. Am. Chem. SOC.1961, 83, 3547-3551. (29) Pritchard, H. 0.; Skinner, H. A . Chem. Reo. 1955, 55, 745-786.
2870
Baker and Zerner
The Journal of Physical Chemistry, Vol. 94, No. 7. I990
TABLE V: Row 1 Energies
element H
no. of S 2
0 -I
H
1
He He
2
0 0
1
1
Li Li Li
2 I 0 2 2 I 0 I 0 2 2
0
1
Be Be Be Be
Be Be B B B
0 I 1
0 1 2 0 I 2
-1 0
0 -I 0 0 0 1 1
-I 0 0
1
B B
2
2 0
1
1
I
B B B C C C C
0 I 0 2 2
2 0
1 2
1
2
C C C C C C C
1
2 I 0 2 1
N N N N
0 I 0 2 2 2 2 2 I 2 I 0
h
2
N N N N
N
no. of s
no.
of P
3 2 3 1
2 3 0 I 2 0 1
6 5 4 3 2 3 I 2 3 0
1
-I 0 0 1 1 1
2 2 2 3 3 -3 -2 -I 0 I 1
2 2 2 3
exot. eV
model. eV
-14.349 -13.595 -79.002 -54.415 -6.01 2 -5.392 -3.544 -27.534 -27.534 -24.1 71 -20.170 -18.21 1 -14.252 -71.367 -7 1.384 -65.654 -63.087 -57.339 -50.442 -37.93 1 -31.933 -147.87 -147.42 -1 38.80 - I 36.75 - I 27.98 -1 17.63 -I 12.37 -104.33 -94.61 3 -64.494 -56.493 -245.23 -258.60 -266.26 -264.70 -251 S I -239.12 -222.79 -2 1 1.09 -197.01 -175.36
-14.349 -13.595 -79.002 -54.41 5 -6.121 -5.351 -3.743 -27.207 -27.551 -24.392 -20.059 -18.236 -14.479 -71.414 -7 1.542 -65.734 -62.861 -57.549 -50.399 -38.038 -32.121 -1 47.99 -147.55 -138.69 -1 36.69 -128.05 -117.13 - I 12.02 -104.54 -94.562 -64.745 -56.651 -245.44 -258.50 -266.07 -264.82 -25 I .44 -239.06 -222.61 -21 1.12 -196.69 -175.01
element N
.I
P
.
-25
'
1
N N
0
N
0 2 2 2 I 2 I 2
1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1
2 1
2 1 0 2 1
2 I 0 2 2
Ne
i
Ne Ne Ne
1
-40
5
E!
1
2 I 0 2
0 0 1
I 2 2 2 3 3 4 4 4 0
1
2
Ne Ne Ne
-1
2 3 6 5 6 4 5 3 4 5 2 3
1
Ne Ne
-165.32 -152.51 -98.341 -88.053 -426.04 -434.61 -432.46 -41 6.02 -41 6.34 -399.7 1 -383.02 -367.08 -347.73 -329.26 -314.88 -296.93 -25 1.80 -239.88 -224.21 -662.3 7 -659.37 -638.24 -640.36 -6 19.00 -602.47 -58 1.54 -556.69 -542.23 -522.69 -456.58 -439.29 -417.78 -954.24 -932.64 -905.66 -889.95 -863.41 -823.10 -797.82 -767.94 -729.00 -705.81
1
1
1
model. eV
-165.06 - 1 52.55 -97.890 -87.898 -426.54 -434.63 -432.22 -415.53 -416.38 -399.84 -383.18 -367.21 -348.10 -329.42 -314.82 -297.14 -252.02 -239.45 -224.16 -662.33 -658.91 -638.03 -640.25 -618.73 -602.49 -581.61 -556.74 -542.35 -522.78 -456.55 -439.03 -417.77 -954.36 -932.74 -905.88 -890.14 -863.69 -823.1 1 -797.92 -767.82 -728.95 -705.79
2 2 2 3 3 3 4 4 4
2 6 5 6 4 5 3 4 5 2 3
2 2
exDt. eV
3 3 4 4 -2 0 0 1
1
I 0
Q
-1
2 3 0
2
Ne
I
6 5 4 5 3 4 2 3 4
0 2 I 0
F F F F F F F F F F F F F
-30
no.
of P
1 1
2 2 3 3 3 4 4
1
1
1
-50-
-60-
zi
4
3
-70-
F
3
-80-
4 -55
c
i
I - 4 5
-1
mechanical expression similar to the one used in this proposal but with charge-independent parameters. With their model as well as the one proposed here, electronegativities for atoms are reproduced well. As can be seen in the graphs, the slopes for the INDO/I data are much too small for neutral atoms to reproduce atomic electronegativities. If we take the derivative of eq 9 with
0
0'5
I
1'5
2
3
2'5
3'5
ATOMlC CHARGE
ATOMIC CHARGE
Figure 3. Average valence configuration energies vs charge for alumiexperimental values: results from the present model (--) for num: (0) the 3S23P" configuration (- -) for the 3S13P"+' configuration, and (-.-) for the 3S03P"+2 configuration, INDO/I results (+) for the 3S23Pn configuration, (*) for the 3S13P"+' configuration, and ( X ) for the 3S03Pn+2configuration.
.os
Figure 4. Average valence configuration energies vs charge for silicon: (0) experimental
values; results from the present model
(-)
for the
3S23P" configuration, (--) for the 3S13P"+'configuration, and (-.-) for the 3So3P"+*configuration, INDO/I results (+) for the 3S23P" configuration, (*) for the 3S13P"+' configuration, and ( X ) for the 3S03P"+2
configuration.
respect to the number of P electrons, we obtain the negative of the electronegativity of P-type electrons for atoms in an average valence state. These derivatives have a paricular significance in density functional t h e ~ r y . ~ ~Mulliken .~' electronegativities are (30) Parr, R. G.: Donnelly, R. A.; Levy, M.; Palke, W.E. J . Chem. f h y s . 1978, 68, 3801-3807.
Hamiltonian for First- and Second-Row Atomic Properties
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 2871 40
TABLE V I Row 2 Energies
no.
no.
element Na
of S
of P
0
expt, eV
model, eV
2
Na Na
1
0 0
-I 0
1 1
-I
-5.685 -5.139 -3.036 -22.681 -22.681 -19.561 -15.035 - 10.607 -53.485 -53.256 -47.277 -41.937 -28.449 -21.778 -103.48 -102.71 -94.9 37 -87.762 -.78.611 -7 1.1 32 -62.648 -45.143 -36.275 -177.52 -175.79 -169.00 -165.65 -157.04 -146.52 -137.09 -127.13 -I 16.39 -106.79 -95.626 -65.026 -5 3.98 5 -272.64 -278.74 -276.09 -264.49 -242.19 -207.79 -196.04 - 160.55 -148.84 -41 2.65 -408.98 -395.26 -383.70 -369.98 -331.41 -278.74 -264.7 1 -577.99 -562.14 -5 33.69 -519.48 -49 1.08
-5.816 -5.410 -2.553 -22.083 -22.638 -19.469 -15.1 I8 -10.596 -53.695 -53.219 -46.924 -42.005 -28.438 -21.735 -103.94 -103.09 -94.827 -88.297 -78.259 -71.208 -62.561 -45.163 -36.083 -177.23 -175.83 -168.92 -165.99 -157.66 -146.52 -1 37.40 -126.56 -1 16.23 -106.94 -95.542 -6 5.087 -53.752 -272.87 -278.43 -267.05 -264.40 -242.14 -207.92 -196.30 -160.43 -149.07 -412.29 -408.72 -394.92 -383.77 -369.54 -33 I .24 -278.66 -265.00 -578.43 -562.53 -533.83 -519.1 6 -49 1.08
Mg Mg Mg Mg
Mg
AI AI AI AI AI AI
Si Si Si Si Si Si Si Si Si P P P P P P P P P P P P P
S S S
S S S S S S CI CI CI CI CI CI CI CI Ar Ar Ar Ar Ar
0 2 2
I 1 0
2 2 2 1 1
0 2 2 2 1
2 1
0 1
0 2 2 1
2 1
2 1
0 2 1 0 I 0 2 2 2 2 2 2 1
2
0 1 0 1
2
1
2 0 1 6 5 4 3 2 I 2 0 1
2 2 2 1
2
-1 1 1
6 5 4 5 3 2 1 2 6 5 4 5 3
1
1 1
0 1 0 1 3 2 1 2 0 1 2 0 1 4 3 4 2 3 1 2 3
1
2 2 2
0 0
0
2 2 2 1
0
1
0
C
do
2 2 -1 0 1 1
2 2 2 3 3 -1
0 0 1
I 2 2 2 3 3 3 4 4 -2 -1
0 I 2 3 3 4 4 -1
0 1 1
2 3 4 4 0 1
2 2 3
f
t6
.I
-80-
-loo-120-
I::I: F:
-1801 -e-
-2
-1
0
2
1
3
4
5
ATOMC CHARGE
Figure 5. Average valence configuration energies vs charge for phosphorus: (0)experimental values; results from the present model (-) for the 3S23Pnconfiguration, (- -) for the 3S13P”+lconfiguration, and (-+-) for the 3S03P”+2configuration; INDO/I results (+) for the 3S23P” configuration, (*) for the 3S13P*l configuration, and ( X ) for the 3S03Pn+2configuration. TABLE VII: Comparison of Electronenativities (eV) element Q CHI’ CHIb B 0 3.70 4.14 B -1 -3.38 C +I 17.20 17.53 C 0 5.08 5.56 C -1 -3.63 N +I 20.55 20.96 N 0 6.76 7.38 N -I -3.71 -3.05 0 +I 24.18 24.52 0 0 8.59 9.13 0 -1 -3.75 -2.84 F +I 27.92 28.21 F 0 10.43 11.04 F -1 -3.90 Ne +I 3 1.63 32.1 1 Ne 0 12.10 AI 0 3.31 3.10 AI -I -2.28 Si +I 12.26 12.05 Si 0 4.41 4.27 Si -I -2.55 P +I 14.46 14.64 P 0 5.42 5.94 P -I -2.42 S +I 16.74 16.95 S 0 6.79 7.13 S -1 -1.82 -1.73 CI +I 19.36 19.50 CI 0 8.46 8.70 CI -I -1.09 Ar +I 22.09 22.15 Ar 0 9.92
CHIc 4.29 6.27 7.27 7.53 10.4 I
3.21 4.76 5.62 6.22 7.3 1
‘From eq 9, CHI = -(C3E/C3N,)Z.N: b C H I = 0.5(IP + EA), IP and EA corrected for use of average conhguration energies. < C H I = 0.5(IP + EA), IP and EA from ground state3’
calculated as the average of the valence-state ionization energy and electron affinity for the atom in its ground state. To compare Mulliken electronegativities in the literature and the model used here, it is necessary to correct the literature values for the use of average configuration energies used to fit eq 9. Table VI1 is provided for this comparison. Electronegativities can be calculated in closed form for anions and cations as well as neutrals for all elements in a row with a single formula. Reasonable estimates can be obtained even if ionization potential and electron affinity information is not readily available. Finally, we consider the concept of absolute orbital hardness as it relates to hard and soft acids and bases’ and density functional
theory.32 Recent studies have associated analytic second derivatives of the energy with respect to the number of electrons with an absolute hardness scale in much the same way as the absolute electronegativity is defined above. A finite difference value for absolute hardness is obtained by33
(31) Parr, R. G.; Bartolotti, L. J . J . Am. Chem. SOC. 1982, 104. 380 1-3803.
(32) Berkowitz, M.; Ghosh, S.K.;Parr, R. G. J . Am. Chem. SOC.1985, 107. 681 1-6814. (33) Parr, R. G.; Peanon, R. G. J . Am. Chem. Soc. 1983,105,7512-7516.
hardness = 0.5(
2)
= 0.5(IP - EA)
(26)
2872
The Journal of Physical Chemistry, Vol. 94, No. 7 , 1990
TABLE VIII: ComDarison of Hardness Values (eV) element Q hard." hard.b B 0 4.40 4.16 B -I 2.67 6.91 6.86 C +1 5.21 C 0 5.1 1 c -I 3.5 I N +I 7.23 1.11 I\( 0 6.07 5.82 h 4.4 I 4.61 -1 0 +I 8.68 8.60 0 0 6.98 6.12 0 -I 5.25 5.36 +I 9.54 F 9.55 F 7.96 0 1.62 F -I 6.37 Ne +I 10.54 10.49 NC 0 8.99 AI 0 2.91 2.88 AI 2.69 -1 Si +1 4.15 4.28 Si 3.70 0 3.50 SI 3.25 -1 P +I 4.82 4.50 4.22 P 0 4.21 P -1 3.62 S +1 5.31 5.35 4.64 S 0 4.48 3.91 S -1 4.38 5.19 CI +I 5.78 0 CI 5.1 1 5.03 -1 4.44 CI Ar +I 6.40 6.30 5.77 Ar 0
-140 0
hard.' 4.0 1 5.00
8 P
6 7.21 6.06
-160
-
:P
i
-180-
-200-220-
F
3u
g
-240-
-260-
U
7.01
-280-300
2.77 3.38 4.86 4.12 4.70
From eq 9, hard. = 0 . 5 ( ~ 3 ~ E / d N , 2 ) ~ hard. , ~ . = 0.5(IP - EA), IP and EA corrected for use of average contiguration energies. chard. = 0.5(1P - EA), IP and EA from ground state.j3
As for electronegativity, we can obtain analytic second derivatives for hardness. The hardness values can easily be estimated in this way for anions, which are difficult to obtain experimentally. Table VI11 compares some hardness values obtained by finite differences and by the present model. The agreement is outstanding as expected from the manner in which these values are obtained.
IV. Conclusion An atom in a molecule gains or loses charge to its neighbors. Although quantum chemical methods to measure a nonobservable like atomic charge are somewhat ambiguous, the observation that the electronic distribution about an atom expands or contracts with gain or loss of electrons is not. Minimum basis set calculations choosing basis sets from neutral atoms are handicapped in this regard. They have fixed extent.
I
