3408
J. Phys. Chem. 1980, 84, 3408-3411
(16) Freed, J. H. J. Chem. Phys.1985, 43, 1710. (17) For example, see McDowell, C. A.; Raghunathan, P.; Shlmokoshl, K. J . Chem. Phys. 1973, 58, 114. (18) Kroh, J.; Plonka, A. J. Phys. Chem. 1975, 79, 2600. (19) Pacansky, J.; Coufal, H. J. Chem. Phys. 1979, 77, 2811. (20) Bouldin, W. V.; Gordy, W. Phys. Rev. 1964, 135, A806. (21) Morton, J. R.; Preston, K. F.; Strach, S. J.; Adrian, F. J.; Jette, A. N. J. Chem. Phys. 1979, 70, 2889. (22) Kinugawa, K.; Miyazaki, T.; Hase, H. J. Phys. Chem. 1978, 82,
1697. (23) Miyazakl, K.; Kasugai, J.; Wada, M.; Kinugawa, K. Buii. Chem. Soc. Jpn. 1978, 51, 1676. (24) TOriyama, K.; Iwasaki, M.; Nunome, K. Int. Congr. Radtat. Res. 6th, 1979, Abstract p 176. (25) Adltya, S.; WIky, D. D.; Wang, H. Y.; Willard, J. E. J. Phys. Chem. 1979, 83, 599. Wang, H. Y.; Willard, J. E. IbM. 1979, 83, 258. (26) Torlyama, K.; Nunome, K.; Iwasakl, M. J. Phys. Chem. 1980, 84, 2374.
General Formulas for the Evaluation of the Valence Orbital Ionization Potentials for the K(2)L(8)3sm3p" Atoms and for the Lower Excited Configuratlons of the Flrst- and Second-Row Atoms Yoshlko Sakal and Toslnobu Anno" College of General Education, Kyushu lJnivers& Ropponmatsu, Fukuoka, 810 Japan (Received: March 26, 1980; In Final Form: July 21, 1980)
Formulas with which the 2s,2p, 3s, 3p, and 3d VOIP of any atom in any configuration of the type ls22sm2p"Xr (x = 3s, 3p or 3d; r = 0 or 1) (type 111) and the 3s, 3p, 3d, 48, and 4p VOIP of any atom in any configuration of the type K(2)L(8)3sm3pnx'(x = 3d, 4s, or 4p; r = 0 or 1)(type IV) may be calculated are obtained. The procedure adopted is a least-squares fitting of all of the empirical data available on VOIP of atoms and ions with these configurations to quadratic functions of m,n, r, and the atomic number, in much the same way as was adopted in a previous work. The resulting formulas should be useful in predicting VOIPs of these atoms and ions for which direct evaluation is impossible for lack of experimental data. The formulas should thus facilitate the parametrization of the semiempirical molecular orbital theory and should be useful in locating the average energy of a configuration for which direct experimental values are unknown for lack of the experimental spectroscopic data.
I. Introduction
perimental spectroscropic data. Since the energy difference between a spectroscopic term and the average energy In previous p a p e r ~ , l -we ~ have made an empirical of a configuration may be expressed in terms of the Slaanalysis of the 3d, 4s, or 4p VOIP (valence orbital ioniter-Condon parameters,13which may be calculated easily zation potential) of atoms4with the electron configurations if suitable forms of the atomic orbitals are assumed, this of the type K(2)L(8)3s23p63da4s@4py (type I) and the 2s means that the VOIP formulas should be useful even in or 2p VOIP of those with the ls22sm2pn(type 11) configlocating the spectroscopic terms unknown as yet experiurations and have derived general formulas for such VOIPs mentally and should facilitate the analyses of atomic of these atoms. In the present paper, the results of a spectra. similar attempt for the 3s and 3p VOIP of the K(2)L(8)Technical details of the present work will be described 3sm3pnatoms as well as for the VOIPs corresponding to in a supplementary report to be published elsewhere,14 the removal of an electron from various orbitals of the firstwhere some insight into the electronic structure of atoms and second-row atoms in their lower excited configurations and ions will also be discussed on the basis of various will be given. The formulas should be useful because the information on VOIP, all of which can be obtained from VOW is an important quantity in the semiempirical theory of molecular electronic structure, as was stressed b e f ~ r e . ~ ~our ~ VOIP formulas. It is true that the nonempirical methods for the molecular 11. Procedure electronic structure have progressed very much in recent The electron configurations of atoms and ions to be years6 and that general interests have waned in some of covered in the present work can be summarized either as the semiempirical methods such as the extended Hiicke17 ls22sm2pnxr(x = 3s, 3p or 3d; m = 0, 1, or 2; n = 0, 1, ..., and the Pariser-Parr-Pople methods: but semiempirical 6; r = 0 or 1) (type 111) or as K(2)L(8)3sm3pnxr(x = 3d, methods such as CNDO? IND0:O and MINDOll still have 4s, or 4p; m, n, and r range as above) (type IV). In the to be used especially in calculating large molecules of present work, formulas for the 2s,2p, 3s, 3p, and 3d VOIP chemical interest.12 The valence-state ionization potential of the type I11 atoms as well as those for the 3s, 3p, 3d, (VSIP) is one of the fundamental empirical data in the 4.9, and 4p VOIP of the type IV atoms are to be given. parametrization in some versions of the above-mentioned The form of the formulas to be obtained is expected to semiempirical theories,lZand the VSIP can easily be obbe of the following type tained from VOIP, as was described b e f ~ r e .Therefore, ~ the VOIP still remains an important quantity in eluciVOIP = B1 + Bzm B3n + B4m2+ B5mn + B6n2 + dating the molecular electronic structure. (B, Barn + B9n)Z' + B I J a + (Bll + Blzm + B13n + The VOIP is by definition a difference in average energy, Bl4r + B15Z?r (1) referred to the same energy reference, between a pair of configurations, so that the VOIP formulas may be used in from Slater's simple theory15 for the total energy of an locating the average energy of a configuration for which atom with the idea of a screening effect due to inner direct empirical values are unknown for lack of the exelectrons, in just the same way as was adopted previous1y.l2
+
0022-3654/a012084-3408~0 I ,001o
+
0 1980 American Chemical Society
Formulas for Valence Orbital Ionization Potentials
The Journal of Physical Chemistry, Vol. 84, No. 25, 1980 3408
In eq 1,Z'may be taken either as the atomic number or
as the charge of the core, although the physical meaning and the numerical value of the coefficientsBk depend upon this alternative choice. We are going to obtain the formulas required by fitting eq 1 to the empirical values of VOIPs of a given kind as far as the data are available, with a least-squares method. To do this, one must first evaluab the average energy of a configuration (E,,) with reference to the ground state for various configurations of various atoms and ions, since the VOIP is defined as the difference between the E,, vallues of atomic species present before and after the removal of an electron from a given orbital, and the ionization potentials connecting the ground states are readily available.16 The E,, values of ls22sm2pnatoms have been taken from Anno and T e r ~ y a ;while ~ those of the type IV atoms have been taken from an unpublished work similar to that of Anno and Teruya.17J8 Experimental values of spectroscopic energy levels have been obtained mostly from Moore's table16 in just the same way as was described p r e v i o ~ s l y . ' ~For - ~ ~ some of the atoms and ions, more recent datalv have also been incorporated. In the case of the type 111atoms with r = 1, E,, has mostly been obtained by taking the weighted mean20of the experimental term values directly. When r = 1, however, in some cases for both the type I11 and the type IV atoms, a special procedure has been found to be necessary and useful because the number of the observed term values are so small. The details of the procedure adopted, which is based on the concept of parentage of the multiplet or configuration21 and uses the data on the "parent ion", will however be described in B supplementary report." With procedures outlined above, the E,, values may be obtained for a fairly wide variety of atoms that we are interested in, and the VOIP may then be obtained from such data by incorporating the data of the ionization potentiaP connecting the ground states.
111. Determination of Bk We are now in a position to determine the Bk values of eq 1and to obtain the VOIP formulas. The empirical data of VOIP for atoms with q (ionic charge) = 0, 1, and 2 only have been used, exclept in the cases of the x VOIP of the type IV atoms. In these exceptional cases, the data for q = 3 atoms have also been included because the number of reliable pieces of data is small in these cases. For reasons similar to those described previously,3 there are various identity ]relationsamong Bk values for various kinds of VOIPs for both the type I11 and the type IV atoms, which are liisted explicitly e1~ewhere.l~These identity relations have been imposed in our least-squares fitting of VOIP, so that all kinds of VOIP under consideration for the type I11 atoms in the present work have been treated simultaneously and similarly for the type IV atoms. Moreover, there are several combinations of Bk values which cannot be separated into individual Bks. Presence of such "inseparable combinations", which are listed elsewhere,14 ii3 due partly to the Pauli exclusion principle, as was described previously,3 but is also due to the fact that r cannot take a value greater than 1for lack of data. Some Bk values must therefore be fixed to zero rather arbitrarily. For this purpose, of Bks involved in an inseparable combination, Bk or Bks bearing larger k value(s) has or have been chosen to be fixed to zero. Thus, we have eventually two least-squares problems, one for each set of the type I11 and type IV atoms, both involving 45 parameters to be determined. The number of the data to be fitted amounts to 420 altogether for the type I11 problem, while it amounts to 297 in the case of the type IV atoms.
a
hl
4
2
3
4
3410
Sakai and Anno
The Journal of Physical Chemistry, Vol. 84, No. 25, 7980
TABLE 11: Numerical Values of B,,-B,, for Various Kinds of VOIP Corresponding t o the Removal of an Electron from an Inner s or D Orbital (cm-' la 2sm 2pn 3s
2P 2s" 2p" 3p
2s" 2pn 3d
30922.5* 13945.5 16157.1
36409.2* 17965.7 19 607.9
39 159.8* 24481.0 25468.3
29
configuration 2sm 2pn 3s Bll Bl2 B, 3 Bl4 Bi 6
30479.0* 11 621.2 13945.5
o.o*
-19605.7
configuration 3s" 3p" 3d B, 1 B, 2 Bl 3
B, * B;
;
a
-5270.7* 27 764.0 22097.4
o.o*
-28998.3
2s" 2p" 3p 36602.1* 15003.6 17965.7
o.o* - 21 455.4
2sm 2p" 3d Type I11 Atom 40041.3* 22084.0 24481.0
o.o* - 24 699.7
3s 3s" 3pn4s
3sm 3p" 4p
-13440.2* 7 496.4 8811.5
" b e IV Atom -10095.6* 10 516.7 11 509.3
- 1 2 689.3
- 1 3 742.3
o.o*
o.o*
o.o*
- 21 075.1
o.o* - 22 830.7
- 25419.0
3s" 3p" 3d
3P 3s" 3pn4s
3sm 3p" 4p
-4259.8" 22097.4 17 867.1
-14 155.5* 8811.5 9 799.4
-10 977.8* 11 509.3 11 395.8
-13360.8
-14186.6
o.o* - 24 276.5
o.o*
o.o*
o.o*
See footnotes a and b to Table I. 1 dcm"
20-
15-
10-
I 3 d o
c.
B C N 0 F Ne Na Mg A l Figure 1. 39, 3p, and 3d VOIP plotted as functions of atomic number for atoms with electron configuration ls22s22p"x(x = 3s, 3p, or 3d) in their various stages of ionization q. Lines drawn correspond to those calculated with our VOIP formulas (see tables), while empirical values are indicated by points (O,O,A) (in lo4 cm-').
In least-squares fittings, each of the empirical VOIPs was given a weight proportional to its inverse square in order to minimize the sum of the squares of the relative deviations rather than of the absolute deviations. The charge of the core, Le., the atomic number minus 10, was taken as 2' of eq 1 for the type IV atoms, although the atomic number itself was taken as Z'in the case of the type 111 atoms. Tables I and 11list the numerical values of Bk derived as outlined above. An asterisk attached to a Bk value indicates that that value is obtained by fixing some of the Bks to zero as mentioned before. Table I refers to B1-BIO,while Table I1 to Bll-B15. The reason that Bk values are given in two separate tables is that all of the Bll-B15 values for the x VOIP have had to be fixed to zero because of the inseparability described before, although they can be separated to some extent for the inner s or p VOIP. Now that we have obtained the numerical values of Bk, calculations of VOIPs have been tried by putting the numerical values of Bk into eq l to see how good our Bks are
AI
SI
P
S
Ci
Ar
K
Ca
Sc
Figure 2. 3d, 4s, and 4p VOIP plotted as functlons of
atomic number (x = 3d, 48, or for atoms with electron configuration K(2)L(8)3s23pnx 4p) in their various stages of ionization q . See also caption to Flgure 1.
relative to the empirical VOIPs. In Figure 1, the calculated values of the x VOIP of the type I11 atoms are shown as curves, while the empirical values are given as points of various kinds although the comparison is limited to the m = 2 cases to avoid complexity in the figure. Figure 2 gives a similar comparison for the type IV atoms with rn = 2. It may be seen that the empirical data and our VOIP formulas are fairly consistent with each other. I t is to be noted that empirical data for some kind of VOIPs, especially for the 3d VOIP of the type IV atoms, are scattered to an appreciable extent. This is believed to be due to the fact that the empirical values of such VOIPs had to be obtained by using speculative estimates of the E,, values (section I1 and ref 14). In such cases, the values obtained from our formulas are believed to be more accurate than the empirical values because the error arising from the estimation of the E,, values is expected to be smoothed out in the course of the least-squares fitting. A measure of the deviation of the empirical data from values given by our formulas may be given by u defined as ~7= ((1/ N)C[(VOIP,,l,d - VOIP,mp)/VOIP,mp121"2 x loo(% 1, where VOIPdd and VOIP,, stand for the calculated and the empirical values of VOIH, respectively, N is the total
J. Phys. Chem. 1980, 84, 3411-3417
number of cases fior which VOIP,,, is available, and the summation extends over all such cases. The value of u thus defined is 0.46,O.59, 1.55,0.97, and 1.25% for the 2s, 2p, 3d, 39, and 3p V(31P, respectively, of the type I11 atoms and 1.13, 1.69,4.04,0.79, and 1.27% for the 3s, 3p, 3d, 49, and 4p VOIP, respectively, of the type IV atoms. In conclusion, we must mention the fact that our VOIP formulas have been found to be very useful in evaluating E, of configurations for which direct empirical values are unknown for lack: of experimental spectroscopic data, in conformity with the expectation outlined in the Introduction. Detailed examples will be given e1~ewhere.l~ It should however be admitted that very few pieces of experimental data are available for the spectroscopic term values to either the pnx of the sp"-l type configuration.22 It is expected therefore that our VOIP formulas are yet to be improved for evaluating the average energy of a configuration of riuch a type, although our formulas are based on the best set of spectroscopic data available at present.
Acknowledgment. We are grateful to Professor Inga Fischer-Hjarmars of the University of Stockholm for reading an earlier draft of the present paper and giving several useful suggestions. Calculations reported in the present paper have been done by using a FACOM M190 at the Computer Center of Kyusliu University. We are grateful to the staff of the Center for their help in using the computer.
References and Notes (1) T. Anno and Y. Sakai, Theor. Chim. Acta, 18, 208 (1970). (2) T. Anno, Theor. Chim. Acta, 18, 223 (1970).
341 1
(3)T. Anno and Y. Sakai, J. Chem. Phys., 56, 922 (1972). (4) In the presont paper, "atoms" often mean both neutral atoms and ions.
(5) T. Anno and Y. Sakai, J. Chem. Phys., 57, 4910 (1972);58, 5190 (1973). (6) See, for example, various review articles appearing in Mod. Theor. Chem., 4 (1977). (7) M. Wolfsberg and L. Helmholz, J. Chem. Phys., 20, 837 (1952);R. Hoffmann, !bid., 39, 1397 (1963). (8) R. Pariser and R. G. Parr, J. Chem. Phys., 21, 466,767 (1953);J. A. Popie, Trans. Faraday SOC.,49, 1375 (1953). (9)J. A. Pople and 0.A. Segai, J . Chem. Phys., 43, S136 (1965). IO) J. A. Popie, D. L. Beveridge, and P. A. Dobosh, J. Chem. Phys., 47, 2026 (1967). 11) N. C. Baird and M. J. S. Dewar, J. Chem. Phys., 50, 1262 (1969); M. J. S. Dewar and E. Haseibach, J. Am. Chem. SOC.,92, 590 (1970);R. C. Bingham, M. J. S. Dewar, and D. H. Lo, ibhf., 97, 1285 (1975). 12) See, for exalmple, varlous review articles appearing in Mod. Theor. Chem., 7-81 (1977). 13) J. C. Slateir, "Quantum Theory of Atomic Structure", Voi. 1, McGraw-HIii, New York, 1960,Chapter 14. 114) . . Y. Sakai and T. Anno. Mem. Fac. Sei.. Kvushu Univ.. Ser. C . 12.. 137 (1980). Reprints are available upon'request to T. Anno. (15) Reference '13,pp 366-72. (16) C. E. Moore, Natl. Bur. SM.(U.S.),Circ., 1, 467 (1949). (17) T. Anno and H. Teruva. Theor. Chlm. Acta, 21. 127 11971). (18j T. Anno and H. Teruya, J. Chem. Phys., 52, 2840 (1970).' (19) L. J. RadzieinskC Jr., K. L. Andrew, V. Kaufman, and U. Liken, J. Opt. SOC. Am., 57 336 (1967)[Si]; Y. G. Toresson, Ark. Fys., 18, 389 (1961)[Si2']; W. C. Martin, J. Opt. SOC.Am., 49, 1071 (1959) P, PI]; L. J. Radzlemski, Jr., and V. Kaufman, bhf., 59 424 (1969)\a];A. Borgstrom, Ark. Fys., 38, 243 (1968)[ca2+j (20) In taking the weighted mean, each term is given a weight (2L 1x2s I),where Land S are the quantum numbers representing the total
+
+
orbltai and h e total spin angular momentum, respectively, appropriate to the term. (21) Reference 13,pp 242-3. (22) In specifylng the electron conflguratiin, the core (K shell for the type 111 atoms and K and L shell for the type I V atoms) Is omitted for brevity; s or p stand for the inner (2sor 2p for the type 111 atoms and 3s or 31) for the type IV atoms) orbitals.
Reactive Channels of the CH302-CH302Reaction Charles S. Kan,+Jack 0. Calveit,*+ and John H. Shawt Depaartment of Chemistry and Department of Physics, The Ohio State University, Columbus, Ohio 432 10 (Received: July 22, f980; In Final Farm: August 25, 1980)
Kinetic studies of the products of the CH302-CH302reactions have been made by using long-path FT-IR spectroscopy. These allow an evaluation of the relative importance of the four suggested channels: 2CH302 '2CH30+ O2 (la); 2CH302 CH20+ CH30H+ O2 (lb); 2CH302 CH300CH3+ O2 (IC);2CH302 CH302H + (2HzOz(Id). The data from experiments at 25 "C give klb/kla = 1.32 f 0.16; klb/klc 1 7; kld
