Dependence of spin-orbit interaction constants on electron configuration of atoms and ions

Transcription

1 Dependene of spinorbit interation onstants on eletron onfiguration of atoms and ions a P. PELIKÁN, b M. UŠKA, a P. ČERNAY^nd^. TURI NAY "Department of Physial hemistry, Slovak Tehnial University, Bratislava ь Department of Inorgani hemistry, Slovak Tehnial University, Bratislava Institute of Experimental Pharmaology, Slovak Aademy of Sienes, Bratislava Reeived Otober Aepted for publiation Marh New method for evaluation of spinorbit interation onstants was used for the systemati determination of p and, onstants from the atomi spetral data of Moore for the atoms and ions from Be to Zn. The obtained results have shown the signifiant dependene of spinorbit interation onstants on the eletron onfiguration of atoms (ions). This dependene was approximated by polynomial funtions ois,p,d atomi orbital eletron populations. The results obtained may be used in semiempirial all valene (mainly ZDO) methods of uantum hemistry. Был применен новый метод рачета контант пинорбитального взаимодейтвия р и, по пектральным данным атомов Мура для атомов и ионов от Be до Zn. Полученные результаты показывают на отчетливую завиимоть контант пинорбитального взаимодейтвия от электронной труктуры атомов (ионов). Полученные результаты можно ипользовать в полуэмпиричеких методах квантовой химии, беря во внимание ве валентные электроны (ообенно ПДП). Detailed studies of eletron struture of moleules suggest that inlusion of spinorbit interation (SOI) plays the important role for orret theoretial interpretation of some experimental results (magneti phenomena, eletroni spetra, photoeletron spetrosopy, et.). Previous works in this field dealt preferentially with tehniues permitting inlusion of SOI in the MO alulations [ ]. As the greater part of results of those alulations is approximately linear hem. zvesti () ()

2 P. PELIKAN, M. LIŠKA, P. ČERNAY, L. TURÍ NAY funtion of the used SOI onstants, the neessity of orret parametrization of these onstants beomes topial. The previous proedures of semiempirial parametrization have dealt with those ases of eletron onfigurations obtained only from multiplet struture of the basi term. The alulation in this ase is redued to enumeration of the wellknown expression [ ] where A is the Lande's parameter and is the spin uantum number. Dunn [] has evaluated by means of en () an extensive set of t; d onstants for atoms and ions of the first two transition series with eletron onfigurations d a, d a ~ l s\ d a ~ s. This author has used an extrapolation for the ases in whih the experimental data were not known. He has supposed that h, d for the given atomi number Z depends only on the number of d eletrons and its hange within the period is for a given eletron onfiguration proportional to (Za) ; a being a sreening onstant []. But one needs to inlude also general eletron onfigurations d a s ß p Y in whih our both the SOI onstants % d and p, if one wishes to omprehend the dependene of SOI onstants on the eletron onfiguration of the atom. Method and results An approximative generalization of the SOI operator for the ase of noneuiyalent eletrons in openshell levels an be formulated [] in the form v~= š (". /. * = у ('" т) # ' s < () where a is the fine splitting onstant and V(r) is the averaged oulomb potential of the nuleus and the other eletrons. We understand under the term "noneuivalent eletrons" the eletrons with different uantum numbers n or/. The en () defines simultaneously also the effetive SOI onstants for the ase of noneuivalent eletrons. In eletron onfigurations d a s ß p Y these onstants may be abbreviated as p and l; d and I may be written as a funtion [] V«(&,&) = & /* + '* () We an, assuming validity of the LS sheme and knowing wave funtions of the given atomi term, enumerate the atual spinorbit splitting as a funtion of the effetive SOI onstants by numerial optimization of the riterion of maximal agreement between the alulated and the experimental spinorbit splitting onstants of the given term. The SOI onstants obtained in this way depend on the hem. zvesti () ()

3 SPINORBIT INTERATION ONSTANTS eletron onfiguration of the atom (ion). It is advantageous to approximate these funtions of disrete arguments by ontinuous funtions dependent on the eletron onfiguration (eletron populations of individual atomi orbitals). These funtions an be used as onfiguration dependent parameters [ ] for a omputation of some physial properties of ompounds ontaining atoms from Be to Zn. The proposed method of semiempirial evaluation of the effetive SOI onstants has several advantages []. A transition from onfigurations with euivalent eletrons to onfigurations with noneuivalent eletrons in open shells does not ause prinipal ompliations if a suffiiently general algorithm for generation of wave funtions of atomi terms is used. The values, optimizing the hosen riterion of agreement between alulated and experimental splittings of the given atomi terms are taken as the best estimates of the SOI onstants. Sum of suares of deviations of measured and alulated values is one of the most used riteria. The spetral energies of / levels are in our ase the experimental values []. Beause these values have normal error distribution with onstant dispersion, we an use the riterion of the minimum of the sum of suares of deviations F(&,,) = Í {ЕЙ (/,& ё )ЕЗ, (/ &,&)} = гшп () where N is the number of experimentally available levels J, of the given term, ELS P (Ji, p,?<*) and E^(Ji, p,,) is spetral, resp. alulated energy of the ith / level of the given spetral term. Solving this problem, we an redue en () on an undetermined system of linear euations with respet to p and d, if we remember that the operator V so (^p,,) is additive with respet to values of p and d. A detailed desription of the used method for evaluating values of ES (Ji,,, d ) and for solving en () is in []. This problem is for onfigurations s m p n or s m d n redued on one linear euation for one unknown, žf p or d, respetively. For the onfigurations with halffilled p levels of halffilled d levels it is possible to determine only the d /^p ratio. But one needs one approximation more for the alulation of the individual SOI onstants, p and d, for onfigurations d a s ß p Y. The approximation, that all values p are eual and all values d are eual in all terms of a given eletron onfiguration, seems to be warranted. The obtained SOI onstants will be used for study of eletron struture of moleules and this purpose leads to use of values of SOI onstants averaged aross all terms of the given eletron onfiguration. Using the mentioned presumption, one needs for determination of the onstants p and % d only two terms as we get a system of two euations for two unknowns. If the given eletron onfiguration ontains more than two terms, one an solve the underdetermined system of linear euations using the leastsuares proedure []. The SOI onstants were obtained using the desribed method for spetral aessible eletron onfigurations of atoms from Be to Zn. The SOI onstants i;^" for atoms and ions with eletron onfigurations K()s m p n (Table ), hem. zvesti () ()

4 P. PELIKÁN, M. LIŠKA, P. ČERNAY, L. TURÍ NAY K()L(S)s m p n (Table ), K()L()M()s m p n (Table ) and SOI onstants ŠT* and ŠT" for eletron onfigurations K()L()s p d a s ß p Y (Table ) were alulated as a weighted mean value from all the available spetral terms [] in a given eletron onfiguration. One an onsider these values to be funtions of disrete eletron populations of individual atomi orbitals. The eletron onfigurations of atoms in moleules have in the framework of LAO MO method usually noninteger values ďs x p y, where v, x, у are nonnegative real numbers, obtained by population analysis. So, if one wishes to use the onfigurationally dependent values of SOI onstants in LAO MO alulation of eletron struture of moleules, the mentioned disrete funtions must be approximated by funtions of ontinuous arguments v, x, y. The dependene of the effetive SOI onstants of atoms with eletron onfigurations K()s m p n, K()L()s m p\ and K()L()M()s m p n was approximated by polynomial funtions of the type r"(*,y)=ezs«*v () /= / = where N s, resp. N p are maximal degrees of polynomial funtions in variables x, resp. у. The dependene of the effetive SOI onstants of atoms of the first transition series on the eletron onfiguration was approximated by polynomial funtion,=o /= where / is p or d, is the atomi harge defined by expression =Zrvxy () The values of the oeffiients Д, were obtained by the leastsuares method and the optimal degrees of individual variables N s, N p, and N d in the polynoms were obtained by maximization of orrelation oeffiient. The values of oeffiients B if as well as relevant statisti harateristis for individual atoms are listed in Tables,,,, and. In Tables,,, and all spetral values ^? pet, values alulated aording to ens () and () *РР ГОХ an(j their deviations are shown. It is not possible in some ases to propose the approximative expressions beause of insuffiient number of spetral data. The presented results show that the proposed regression funtions suffiiently desribe the dependene of SOI onstants on the eletron onfiguration of atoms (ions). Some greater deviations an be eluidated by the fat that the omplete sets of spetral values for all terms of some eletron onfiguration only one were taken for obtaining the onstants of SOI (e.g. the terms with soalled degeneray of name were exluded). hem. zvestí () ()

5 SPINORBIT INTERATION ONSTANTS Table Numerial таluee of SOI onetante for atoms Table (ontinued) Atom Be«В N F Ne n WH Ln eietroi i onrigurí K()s n p n [me V] ^spet *approx o.oo Atomm Ne n espet.... Table.approx Numerial values of SOI onstants for atoms and ions with eletron onfigurations K()L()s m p n [mev] Atom m Na* Mg* AI Si P S l n i tapprox O.OOO.... o.ooo o.oo... O.OO han, zvesti () ()

6 P. PELIKAN, M. LISKA, P. ERNAY, L. TURÍ NAY Table (ontinued) Table (ontinued) Ato* m n spet tарргох S P Atom m» t арргох I AT Br Кг* O.OOO O.OOO O.OO.... Number of experimental points is insuffiier for re gression analysis Table Numerial values of oeffiients, B i j for K()e m p n [mev] Table Numerial values of SOI onstants for atoms and ions with eletron onfigurations K()L()M(l)s m p n [mev] Ato» t spét t арргох ^Р.... Standard deviation orrelation oeffiient.. a e As Se i O.OOO. O.OOO.. O.OO. O.OOO. O.OOO O.OOO O.OOO hem. zvesti () ()

7 SPINORBIT INTERATION ONSTANTS Table (ontinued) Table (ontinued) Standard orrelation deviation oeffiient Standard orrelation deviation oeffiient Ne Table Numerial values of oeffiients В. for atoms and ions with eletron onfigurations I K()L()s m p n о [mev] Standard orrelation deviation oeffiient Ato» a e As Se Br Numerial values of oeffiients В. for atoms and ions with eletron onfigurations i K()L()M()s m p n j B u [mev] Standard deviation..... orrelation oeffiient..... hem. zvesti () ()

8 P. PELIKAN, M. LISKA, P. ERNAY, L. TURÍ NAY T'.ble. Numerial valuer) of SOI onstants p and d [mev] Atom ß Л (j TL f (J V r r ч, Ü. a L i i Ü.. d. ln. n... *. :. f approx P " OS. í>.íi.. J *. ;:.\....Ó d li. i *approx s d H...« О Oiem. zvestí. () ()

9 SPINORBIT INTERATION ONSTANTS Atom ß r I Mn Fe. o r а Table р &.. (ontinued) *арргох S P _ П.Ю _ ^d rapprox s d hem. zvesti () ()

10 P. PELIKÁN, M. LIŠKA, P. ČERNAY, L. TURÍ NAY Table (ontinued) Atom ß o Ni. и* Zn* r О a I approx S P d tapprox о o.o.... Kur.ber of experimental joints is ir.suf.'iient ľor regression t» no Г/r; is. han, zmti () ()

11 SPINORBIT INTERATION ONSTANTS Table» Numerial values of oeffiients В for Č onstants [mev] Table (ontinued) S Ti V r Mn Fe o Ni Ate S t г Table i í t T t. j..^ о Standard orrelation deviation oeffiient Numerial values of oeffiients B i for d onstants [mev] B ij.... Standard orrelation deviation oeffiient.. Ti V r Mn Fe o B ij * Standard orrelation deviation oeffiient hm. zvesti () ()

12 P. PELIKÁN, M. LIŠKA, P. ČERNAY, L. TURÍ NAY Table y (ontinu'rä) Table (ontinu^d) Standard orrelation, ^ + лт B Standard orrelation deviation oeffiient л J ij i j deviation oeffiient Ni г..... б.о.?.'.it. Ni и....' One must be very areful using the proposed regression funtions for extrapolation of values of the SOI onstants outer of the area defined by used spetral data beause these funtions were not tested outer of this area. The obtained values of onstants of spinorbit interation are the effetive values beause they represent the spinorbit interation and the other types of interations inluded in the Hamiltonian. Use of these parameters in methods with onfigurationally dependent parameters ensures a onsisteny between the parametrization of the SF alulation [ ] and parametrization of the postproessors. Referenes. Turner, D. W., Baker,, Baker, A. D., and Brundle,. R., MoleularPhotoeletron Spetrosopy. Wiley Intersiene, New York,.. Dunn, Т. M., Trans. Faraday So., ().. Shiff, L. I., Quantum Mehanis. MrawHill, New York,.. ondon, E. U. and Shortley,. H., Theory of Atomi Spetra. University Press, ambridge,.. Blume, M. and Watson, R. E., Pro. Roy. So. A, ().. Blume, M. and Watson, R. E., Pro. Roy. So. A, ().. Slater, J., Quantum Theory of Atomi Struture, Vol.. MrawHill, New York,.. riffith, J. S., The Theory of Transition Metal Ions. University Press, ambridge,.. Pelikán, P., Liška, M., and Černay, P., /. Mol. Strut., in press.. Turi Nagy, L., Pelikán, P., and Liška, M., hem. Zvesti, ().. Pelikán, P., Boča, R., Liška, M., and Turi Nagy, L., hem. Zvesti, ().. Pelikán, P., Liška, M., Boča, R., and Turi Nagy. L., hem. Zvesti, ().. Turi Nagy, L., Pelikán, P., and Liška, M., hem. Zvesti, ().. Moore,. E., Atomi Energy Levels as Derived from the Analysis of the Optial Spetra, Vol.. National Bureau of Standards, irular No.. Washington, D..,. Translated by P. Pelikán От. zvesti () ()