Dependence of Slater- Condon parameters on electron configuration. II. Slater Condon parameters of elements of the first transition series

Transcription

1 Dependence of Slater- Condon parameters on electron configuration. II. Slater Condon parameters of elements of the first transition series "P. PELIKÁN, b R. BOČA, Ь М. UŠKA, and C L. TURI NAÜY 'Department of Physical Chemistry, Slovak Technical University, Bratislava "Department of Inorganic Chemistry, Slovak Technical University, Bratislava "Institute of Eperimental Pharmacology, Slovak Academy of Sciences, Bratislava Received July There are proposed approimative relations for the dependence of values of the Slater Condon parameters F {s,s), F (s,p), F"(s,d), F (p,d), F (d,d), G'(i,p), F (d,d), F\d.d), G (s,d), G l (p,d), F (p,d), and G (p,d) on electron configuration of atoms (ions) of the first transition series. The proposed functions are useful in LCAO MO SCF methods including the dependence of Hartree Fock matri on electron configuration of atoms in molecules. В работе предложены апрокимативные оотношения для завиимотей величин параметров лейтера Кондона F (s,s), F (s,p), F"(s,d), F (p,d), F (d,d), G\s,p), F (d,d), F\d,d), G (s,d), G } {p,d), F (p,d) и G (p,d) от электронной конфигурации атомов (ионов) первого переходного периода. Предложенные регреные функции применимые в ЛКАО МО П методах включением завиимоти матрицы Хартри Фока от электронной конфигурации атомов в молекулах. The semiempirical methods with an eplicit inclusion of electron repulsion were up to this time used for calculation of electron structure only for a small number of coordination compounds. At the present state of parametnzation they allow a comparison of some physicochemical properties in series of similar molecules. In contrast to IEHT method they give good results for calculations of equilibrium geometries. In spite of a considerable effort epended in this field (see for eample review articles [ ]), the results have not realized epectation, because they are Chem. zvesti () - ()

2 SLATER CONDON PARAMETERS. II far from the level of the results of analogical methods applied to organic compounds []. A great number of semiempirical methods of this kind mutually differ mainly in using different combinations of some fundamental types of the originally proposed approimations to epress different integrals []. On the basis of the analysis of atomic spectra [, ] it was found that all the monocentric integrals needed for the construction of Hartree Fock matri are firmly dependent on the charge and electron configuration of the given atom. The first attempts to regard this fact [] have shown a considerable improvement of results, mainly for polar systems. Because the bonds in coordination compounds are predominantly strongly polar and a great part of these systems is not electroneutral, the consideration of this fact in parametrization is etraordinarily important. In the present work there is epressed the dependence of Slater Condon parameters on electron configuration of atoms and ions of the first transition series. Method and results If we wish to calculate values of all monocentric integrals of electron repulsion which arise in the basis of s, p, d atomic orbitals, we need to know values of these integrals of the radial part of atomic orbitals []: F (s,s), F (s,p), F (p,p), F (s,d), F (p,d), G\s,p), F\d,d), F (p,p), F\d,d), G\s,d), G\p,d), F (p,d), G (p,d), R\s,d,d,d), R\s,p,p,d), R (s,p,p,d). The last three integrals are not the Slater Condon parameters and we cannot determine them by analysis of atomic spectra, because they do not occur in epressions for energies of atomic terms. From available eperimental data there is noi possible to determine the Slater Condon parameters F (p,p) and F (p,p). The net paper of this series will be devoted to evaluation of parameters R'(s,d,d,d), R\s,p,p,d), R (s,p,p,d), F (p,p), and F (p,p). In determination of values of the Slater Condon parameters F k and G k (кфо) Hinze and Jaffé [], Tondello, de Michelis, Olearí, and Sipio [], and Anno and Teruya [] have been engaged. In this work we use the results obtained by Anno and Teruya [], because only these are determined systematically with a correct employment of all available spectral data. These values are listed in Tables under the symbol Eperimental. The basis for the calculation of values of Slater Condon parameters of the type F are the energetic balances of reactions of the electron echange A ( + l)+ + A iq + l)+ -+A q+ + A (q+)+ () Anno [, ] has shown that energetic effect of this electron transmission can be epressed by using the values of the Slater Condon parameters relevant to the Chem. zvestí () - ()

3 P. PELIKÁN. R. BOČA, M. LIŠKA, L. TURÍ NAGY most negative ion in eqn (). Then the parameters F for atoms and ions with electron configuration d"s ß p Y can be enumerated from the following relations E l+l)+ (d a s ß - i p Y ) = E^(d a s ß p Y ) + E i<!+)+ (d a s ß - p Y ) + AE(s i s) () E ( «+l)+ (d a s ft p Y - l ) = E q+ (ď's p p Y ) + E (q+)+ (d"s p p Y - ) + AE(p,p) () E u,+l)+ (d a s ß p Y - l ) + E l «+n+ (d a s ß - l p Y ) = E q+ (d a s ß p Y ) + + E (q+)+ (d a s p - { p Y - l ) + ÁE(s,p) () E lq+l)+ (d a - l s ß p Y ) = E q+ (d a s ß p Y ) + E iq+)+ (d"- s ß p Y ) + AE(d,d) (). where E (q + l)+ (d a s ß - i p Y ) + E (q+,)+ (d a - i s ß p Y ) = E q+ (d a s ß p Y ) + + E (q+)+ (d n - l s ß ' l p Y + ÄE(s,d) () E iq+n+ (d a s ß p Y - l ) + E (q+i)+ (d a - s ß p Y ) = E q+ (d"s ß p Y ) + + E (q+)+ (d- l s ß p Y - l ) + AE(p,d) () AE(s,s) = F (s,s) () AE(p,p) = F"(p,p)-^F (p,p) () AE(s,p) = F"(s,p)-±G\s,p) () AE(d,d) = F a (d,d)-^f (d,d)-]-f (d,d) () AE(s,d)^F ] (s,d)-y^g (s,d) () AE(p,d) = F'(p,d)-^G\p,d)-^G (p,d) () We obtain the values AE as the difference of average energies of electron configurations on the right and the left side of eqns ( ). We calculate the values F according to eqns ( ) in which we replace F k and G k (кфо) by values obtained from approimative epressions according to eqns ( ). Average energies of configurations were enumerated as. the weighted mean of energies of individual terms with the weighting factor equal to the multiplicity of the given term. Values of the energies of terms from Moore's spectral tables [] were used. The obtained values of parameters F are listed in Tables under the symbol Eperimental. Chem. zvesti () - ()

4 SLATER CONDON PARAMETERS. II Table. Eperimental and calculated yaluee of F (a,e) (ev Table (Continued) Z S P D mental Approimative UoŤ~ Z S P T) Eperimental Approimative Deviation б S Table. Eperimental and calculated values of F (s,p) fevl s P D Table. Eperimental and calculated values of F (s,d) %v] Z S P D Eperi Eperimental Approimative Deviation Approi- Devia Table. Eperimental and calculated values of F (p,d) [ev] Z S P D mental Approimative о.ззоз diem, zvestí () - ()

5 P. PELIKAN. R. BOCA, M. LISKA, L. TURÍ NAGY Table (Continued) Table (Continued) z z S ť D Eperimental Approimative Deviation Eperimental Approimative Deviation Б Table * Eperimental and calculated values of F (d,d) [ev] Table «Eperimental and calculated values of G (s,p) [ev Z S D Z S P D mental Eperimental Approimative Deviation Approimative D ut -! ó ,....., H ie * ! Table. Eperimental and calculated values of F (d,d) [ev Ь ť ú Eperimental Approi mative- Deviation Chem. zvesti () - ()

6 SLATER CONDON PARAMETERS. II Table (Continued) Eperi- Approi- Deviamental native tion? ] Ö \ о.ооо О.ОО Chem. zvestí () - () Table (Continued) Eperi- Approi- Deviamental mative tion ó? , c C -. p Table. Eperimental and calculated values of P (d,d) ev] Eperi- Approi- Devia- Z P D., mental mative tion

7 Table (Continued) Eperi- Approi- Deviamental mative tion б.h Ö. P. PELIKAN, R. BOCA, M. LISKA, L. TURÍ NAGY Table (Continued) Eperi- Approi- Deviamental mative tion C c C Chem. zvesti () - ()

8 W M W ř O W W ř J W l O W W I U W r W W r f r r y M W ^-^ -^-^OJUJUJLJUiUJ ГО ГО fv) го го го О О О О О О О О О О О О О О О О О О О О О O O О О О О ) «v O ^ O Ď N O ^ O ^ ^ ' ^ V J l V J I ^ ^ U ^ > S J V f l < ^ l «U ^ > ^ - r O W U U ^ - ' - w r O U - - ' N ^ * * p ^ p p p o p О p - * - * О О О О О О О О О О О O -» V J i r g v O W ) - - ' V Í ) W - J - ' - * \ O U» Q D f u - f \ ) v o D - J W O a ) ^ O Vfl M Vfl -* *" - U \ J l O \ - J p ) O l - J a ) W M O - í - J - U l f O U l - ' OD ^ O D O O O C D L * J O O - J - a «r O - J O D O O D U J k - f v ů W U l W O \ - ^ - * l J l C > - M - ' V O O - O O V J J V O U J ^ J - J O D - J CD Ы Ц {Л U > - Л \J M w ^ O U i y v O f O W O U I f f l - J - г - О ^ ч л о л з - ^ - ^ о г о - о о о о о о о о о о о о о о о о оо о о о о о О -* ЧЛ ГО ЧО О SO О coro r o c o v o - О vo CD - J - ^ v n o O D ^ W U v ß - v i o a u v ß - ' v O -» ч О ч л - * Ш < ^ О - ^ ^ О А CTv Ov \л го чл O v \ O C D O ^ m O - -._ ON *. o -J -J -J -J - A (^ VO *» ( ^ u - J - J ^ ^ C B C D ^ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o - o o o o o o ^ ^ O - u ui ч л г о г о ч о ^ л о ^ л - ^ о ^ ч л " O -J vi -> vo U -J ЧЛ ЧЛ ы o u ы \o o\ \Л O O O O O O O O O O O O O O O O Q O -» u u» í > - * y i u O b J r o r o r o ) U O U U - J A v O - J > J - J - J M \ J I «J l Ч Л - Г О Т \ Ч Л О Ч Л ч О и > О -» - ^ О к э ю и г о г о г о г о г о г о ю г о г о г о г о vov)voo)a)a)--o-g-- M W M M o o oo o o o o o o o ooo o o o o o ooo o oo o <DDvo\O-^JCCvOONCT\-g^JCD>" \Л o\ ^J OJ -^ > o o o o o o o o o o o o o o o o o o o o o > - ^ ГО ^ CD ч О - Ы Г О Л - ^ о U U -J -о ЧЛ vl vo VO ЧЛ cr» o -^ UJ o\ o o\ ГО a\ O * - M - u ЧЛ Ov ГО LJ u-ř*- i\) (u u - ro o o o o o o o o o o o o o o o o o o o o o o ^CD-J^vO^^-JCA C T A O - P - I - J O V O O Q - * -JvOO M U VO У N -J -ŕ--jrocororolovooro-r». Р.Й vc vo o o o o o o o o o o o o o o o o o o o o o o u ю í. > o * ^)OUItfVJ) M o\ ^ M *. -g o o o o o o o o o o o o o o o o o o o o o Ü D * U W * D ( B v ß U i y i - V O > J a \ y i \ o - J { J y y i -J -~ О ЧЭ ' OJ ON t\ ЧЛ OD o o o o o o o o o o o o o o o o o o o o o o o o o Р р р о о о о о о о о о о о о о о о Ы UJ -J - O *. o o o o o o o o o o o o o o o oo o p o o g o o o o o o o M O -J - O U I U - K J U O - W - I O ^ O U - M M чл a\ en -» o * O o o u» U Q O» * - O O O O C^OUlvDvOOf^O) M U ) u z t

9 P. PELIKAN, R. BOCA, M. LISKA, L. TURÍ NAGY Table. Eperimental and calculated valuee of F (p,d) fev] Table. Eperimental and calculated valuee of Q (p,d) fev^ Eperi- Approi- Deviamental mative tion Eperi- Approi- Deviamental mative tion. ' C '. -.C ".. -..! A : H C C C C O.OOOC C Chem. zvesti () - ()

10 SLATER CONDON PARAMETERS. II In the framework of LCAO MO approimation electron configuration of atoms in molecules acquires as a rule noninteger values d v s p y, where u,, у are nonnegative real numbers obtained by population analysis. Thus, if we want to use the configurationally dependent values of monocentric integrals, obtained by the analysis of atomic spectra in MO calculation of electron structure of molecules, we must approimate the discrete functions with the continuous ones. Then we consider individual monocentric integrals to be functions of continuous arguments which are represented by populations of the individual levels of the valence sphere of an atom, or functions of atomic charge and its atomic number. The dependence on the atomic number is included to render the maimum possible number of known eperimental data in the regression. The most suitable polynomial functions are of the following types F(Z,Q) = B * Z I Q ' ( i ) F(Z, Q) = B.Z'iQ + )' () = / = N z N d F(Z,u,Q) = >, ZV + i= =Q N N N N + B IO *Z'Q* + д*и'о* (i) i= к = / = k = l N z N d N z N s F(Z,v,,y) = JB*.ZV + B, oto ZV + i= /= = fc = l N z N N d JV + Biooizy + Ba&v'" + i = / = / = J fc = l + Д, ои ху () k=l= F(Z,u,,y) = ^*»V' + ^,B Ma> Z, k +,= / = = lc = l N z N p N d N + S B^z'iy +)' + B^V'" + = / = / = fc = l + Z,B oou k (y + iy- - () Z is the atomic number, Q is the atomic charge,, y, v are the populations of s, p, d valence orbitals, respectively. The constants В were determined by the least-squares method and the optimum degrees of polynomials N z, N, N s, N p, N d were determined by the maimization of the correlation. Chem. zvesti () - () ßQJ

11 P. PELIKÁN, R. BOČA, M. LIŠKA, L. TURÍ NAGY Table. Numerical values of В. 's for function F (s,s) [ev] Table. Numerical values of В ik for function F (p,d) TeV] i B i i к B ik T ID" J ю г......c.... ю-' ю-' Table. Numerical values of B.. 's for function F (s,p) fevl.. Table. Numerical values of B.... s for function F (d,d) ['ev] ikl -..C Table. Numerical values of В ik for function F (s,d) [ev].. к в... ik \d \о г "' id for function,?) [ev] J C Chcm. zvesti () - ()

12 SLATER CONDON PARAMETERS. II Table (Continued) Table (Continued) i J B i ДЮ.. Q *,,C \о г " Table. Numerical values of B. fcl '. к for function F (d,d) G B ikl * C C O > «O o ^ c c' < o i c" C J J o c o ^ I o o' "' [ev] '. ' C Table. Numerical values of В.... 's к for function F (d,d) [ev] о- íkl бе У. " i о ю ю- ю- ю- Ó- io- J ^ ю" diem, zvestí () - ()

13 P. PELIKÁN, R. BOČA, M. LIŠKA, L. TURÍ NAGY Table (Continued) Table. Numerical values of В... 's. for function G Up,d) [evl i lei ". " -. "'.. Table. Numerical values of В ik i k for function G (s,d) ev] B i* - G..C O. ". -. C J.!C J H W A C J (У - IQ"«* "' ^ i C c C c le C c c C lation coeff B i* ю O.O. C..... O.I.#.....H.C.... О.З..... o c b * o ^ c o" ^ b c b O Q> O J C J o C J o ^ c J o c o" o" o" c" c- J o- J - o- J C".. Chem. zvesti () - ()

14 SLATER CONDON PARAMETERS. II Table» Numerical values of В ik Table (Continued) i з о з к for function F (p,d) ev] B ik \о г ю-' J. ^. J to *. ^. ^.. G l ю-. "' IQ" ^ io- J. ^. ID" * ю-' i к ' lation coeff B ik ' ю' ю'.. "'. IG" ю-' ю- ю-' io- J ю- ю' ю' ^. ю- IQ"' Table. Numerical values of В ik for function J (p,d) fevl к B ik. -. ю' Chem. zvesti () - ()

15 P. PELIKAN, R. BOČA, M. LIŠKA, L. TURÍ NAGY Resulting functions for F (d,d) and F (d,d) have the form of eqn (), for G (s,d); G l (p,d), F (p,d), G\p,d), F (s,d), and F (p,d) the form of eqn (), for G\s,p) the form of eqn (), for F (s,s) and F (s,p) the form of eqn () and for F (d,d) the form of eqn (). Numerical values of s B m, B ik, and B ti are listed in Tables, where also the correlation s of individual regression functions are included. Functional values of approimative epressions for electron configurations, available from spectral data, are listed in Tables under the symbol Approimative. In these tables under the symbol Deviation the s of approimative values from eperimental values of relevant integrals are also listed. From the comparison of eperimental and approimative values of individual Slater Condon parameters it follows that suggested functions well agree with the dependence of spectral values on the electron configuration of an atom (ion). Therefore, we can use them for interpolation of the values relevant to noninteger electron configurations obtained by the population analysis in LCAO MO calculations. The analysis of the parameters F (p,p), F (p,p), R\s,d,d,d), R l (s,p,p,d), and R (s,p,p,d) which are necessary for enumeration of the monocentric repulsion integrals in the framework of LCAO MO approimation should be the content of the net paper of this series. References. Burton, P. G., Coord. Chem. Rev., ().. Pelikán, P., Turi Nagy, L., Liška, M and Boča, R., Chem. Listy, ().. Davies, D. R. and Webb, G. A., Coord. Chem. Rev., ().. Cársky, P., Chem. Listy, ().. Klopman, G. and O'Leary, В., Fortschr. Chem. Forsch. IS, ().. Anno, T. and Teruya, H J. Chem. Phys., ().. Anno, T. and Teruya, H., Theor. Chim. Acta, ().. Klopman, G. and Polák, R., Theor. Chim. Acta, ().. Pelikán, P. and Turi Nagy, L., Chem. Zvesti, ().. Hinze, J. and Jaffé, H. H., /. Chem. Phys., ().. Tondello, E., de Michelis, G., Oleari, L., and di Sipio, L., Coord. Chem. Rev., ().. Anno, Т., /. Chem. Phys., ().. Moore, C. E., Atomic Energy Levels as Derived from the Analysis of the Optical Spectra, Vol.. National Bureau of s, Circular No.. Washington, D.C.,. Translated by M. Kozlík Chem. zvesti () - ()