Nephelauxetic Effect Revisited

ANDREI L. TCHOUGRE´ EFF,1,2 RICHARD DRONSKOWSKI2 1Poncelet Laboratory, Independent University of Moscow, Moscow Center for Continous Mathematical Education, Moscow 119991, Russia 2Institute of Inorganic Chemistry, RWTH Aachen, Aachen D-52056, Germany

Received 1 October 2008; accepted 3 November 2008 Published online 10 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.21989

ABSTRACT: We readdress the well-known nephelauxetic effect in coordination compounds of transition metal ions and analyze its possible origins presented in the literature. The initial hypothesis was to ascribe the reduction of the effective Racah parameters B and C of the electron–electron interaction in the complexes as compared to their respective free ion values (which is the essence of the nephelauxetic effect) to the expansion of d-orbitals in the complex because of their quantum mechanical mixing with the orbitals of the ligands leading to delocalization. This picture necessarily leads to a rigid positive correlation between the amount of the d-shell splitting controlled by the same delocalization and the amount of the renormalization of the interaction parameters. In fact, such a rigid relation does not exist and the so-called spectrochemical and nephelauxetic series of the ligands composed according to the aforementioned amounts do not coincide in many points. An alternative explanation based not on the delocalization, but on polarization of the ligands had been proposed at the same time as the delocalization based one. Realistic estimates had been obtained on this basis, but the scheme had never been implemented on the atomic scale, which is necessary to enable renormalization of the aforementioned interaction parameters as a “built-in” function of quantum chemical software. The required atomic resolution formulation of the polarization-based model of the nephelauxetic effect is constructed in this work, in which a relation of such an approach to the general problematics of the “next generation” of semiempirical methods of quantum chemistry is also discussed. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem 109: 2606–2621, 2009

Key words: effective Hamiltonian crystal ﬁled theory; nephelauxetic effect; next generation of semiempirical methods; transition metal complexes

Correspondence to: A. Tchougre´eff; e-mail: [email protected] Contract grant sponsor: RFBR. Contract grant number: 07-03-01128. Contract grant sponsor: DFG. Contract grant number: DR 342/20-1.

ligands according to the strengths of the crystal 1. Introduction fields induced by them (the 10Dq parameter) to the so-called spectrochemical series [3–6] having (with mong the words to be learned by heart by many omissions) the following form: A first-year inorganic chemistry students fasci- nated by the bright colors of the coordination com- Ϫ Ϫ 2Ϫ 3Ϫ Ϫ Ϫ Ϫ I Ͻ Br Ͻ S Ͻ N Ͻ F ϽOH Ͻ Cl pounds of transition elements, the term “nephelauxetic” (effect) is one of the most intriguing, but 1 Ͻ Ox2Ϫ Ͻ O2Ϫ Ͻ (1) unfortunately being rarely readdressed during fur- 2 ther years of studies or even professional carriers. 1 This beautiful Greek word introduced into theoret- Ͻ H O Ͻ SCNϪ Ͻ NH ,py Ͻ En ical inorganic chemistry by Jaurgensen [1] follow- 2 3 2 Ϫ Ϫ ing the advice of Kaj Barr—the prominent Danish Ͻ SO2 Ͻ NO Ͻ CNϪ Ͻ CO. orientalist—refers to the mental picture according 3 2 to which the cloud (␯␧␸␧´␭␩)ofd-electrons expands (␣␷␰␧␫␯) when a transition metal ion (TMI) becomes From this one can see that the crystal fields split- a central one in a coordination compound, where it tings are systematically weaker for charged ligands is surrounded by various other inorganic or organic than for the uncharged ones with the utter example ions or molecules, rather a free one. The conse- of CO inducing the strongest crystal field, but bear- quences of the coordination process for the d-shell ing neither charge nor even noticeable dipole mo- are in principle twofold: before the already men- ment. Thus, the relative strengths of the crystal tioned expansion of the d-orbitals the splitting of fields observed in the experiment cannot be ex- the d-levels degenerate in the free ions takes place, plained by the ionic model of the environment. being much more important both qualitatively and These observations clearly indicate that purely elec- energetically, namely, the splitting is ultimately re- trostatic effects may be only of minor significance in sponsible for the mentioned colors of the transition determining the strength of the effective crystal metal complexes (TMCs). Its qualitative picture is field felt by the d-shell. explained by the crystal filed theory (CFT) origi- The unsatisfactory situation with the CFT esti- nally proposed by Bethe [2] and numerously rep- mates called for the development of the ligand field resented in various other sources adjusted for the theory (LFT) [3, 4], trying to include the ligands on chemical problem setting [3–6] (originally CFT had a more realistic basis. In its simplest version, it been developed for the transition metal impurity assumes that it is enough to consider the valence ions in the crystals). The CFT bases on the very shell of the TMI, containing 3d-, 4s-, and 4p-orbitals natural picture of what happens in TMCs—all in- and to include one lone pair orbital per donor atom teresting events (low- energy excitations)—are lo- of the ligand thus giving the following picture of calized in the d-shell of a central TMI, whereas one-electronic states of the closest ligand shell other atoms or groups provide some external, orig- (CLS) of a TMI in a TMC which the octahedral local inally purely electrostatic, field responsible for the symmetry: splitting. In such a formulation of the CFT known as its ionic model, and treating the splitting of the ye a g ␺ ͑e ͒ ϭϪx ␾͑d 2͒ ϩ ͑2␹ ϩ 2␹Ϫ Ϫ ␹ Ϫ ␹Ϫ d-shell as a pure electrostatic effect, the CFT faces a gc eg z ͱ12 z z x x serious problem: the splitting parameters cannot be Ϫ ␹ Ϫ ␹ ͒ correctly estimated. Although the symmetry is per- y Ϫy fectly reproduced even by this simplistic scheme, it xe b g turns out that quantitatively the ionic model gives ␺ ͑e ͒ ϭ y ␾͑d 2͒ ϩ ͑2␹ ϩ 2␹Ϫ Ϫ ␹ Ϫ ␹Ϫ gc eg z ͱ z z x x at best 20% of the observed splitting even if unre- 12 alistically large effective charges are ascribed to the Ϫ ␹ Ϫ ␹ ͒ y Ϫy ligands. This happens due to the oversimplified ye description of the TMI’s environment (ligands). It is a g ␺ ͑e ͒ ϭϪx ␾͑d 2Ϫ 2͒ ϩ ͑␹ ϩ ␹Ϫ Ϫ ␹ Ϫ ␹Ϫ ͒ not thus surprising that the heaviest strike upon the gs eg x y 2 x x y y CFT from the (semi)quantitative side was given by xe TMC spectroscopy in 30s of the last century. Spec- b g ␺ ͑e ͒ ϭ y ␾͑d 2Ϫ 2͒ ϩ ͑␹ ϩ ␹Ϫ Ϫ ␹ Ϫ ␹Ϫ ͒ troscopic experiments allowed to range different gs eg x y 2 x x y y

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ya from the (effective) charges localized on the li- a 1g ␺ ͑ ͒ ϭ ␾͑ ͒ ϩ ͑␹ ϩ ␹ ϩ ␹ ϩ ␹Ϫ ϩ ␹Ϫ a1g xa1g 4s x y z x y gands, but is a consequence of covalent interac- ͱ6 tions—ultimately of the interplay between the de- ϩ ␹ ͒ Ϫz localization of the d-electronic states to the ligands and of the ligand electronic states to the d-shell. xa ␺b͑ ͒ ϭ ␾͑ ͒ ϩ 1g͑␹ ϩ ␹ ϩ ␹ ϩ ␹ ϩ ␹ Spectrochemical series Eq. (1) then can be thought a1g ya 4s x y z Ϫx Ϫy 1g ͱ6 to be arranged in the order of increase of not only ϩ ␹ ͒ 10Dq but also of the characteristic delocalization Ϫz 2 2 parameters xegyeg, describing the strength of the co-

yt valent metal–ligand interaction and determined by a 1u ␺ ͑t ␥͒ ϭϪx ␾͑4p␥͒ ϩ ͑␹␥ Ϫ ␹Ϫ␥͒ 1u t1u ͱ2 the same Hamiltonian matrix elements as the splitting through the relations:

xt ␺b͑ ͒ ϭ ␾͑ ͒ ϩ 1u͑␹ Ϫ ␹ ͒ t1u␥ yt1u 4p␥ ␥ Ϫ␥ . (2) ͱ2 2 2 1 1 x y ϭ ͩ Ϫ ͪ eg eg 1 ϩ ␨2 , (6) 4 1 eg The Stevens’ coefﬁcients x⌫, y⌫ subject to the normalization condition: H ␹ ␨ ϭ d eg . (7) Ϫ ␹␹ 2 2 Hdd H x⌫ ϩ y⌫ ϭ 1 (3) Getting very large ␨ yields the highest possible give the amount of the delocalization of the TMI- eg value of x2 y2 ϭ 1/4 referring to the maximal pos- centered states and have to be determined (ideally) eg eg sible delocalization of the d-orbital. from a self-consistent ﬁeld procedure applied to the These considerations dating back to 30s qualita- effective Fock operator describing the CLS of the tively explain pretty much of the spectroscopy of TMI. By this, the one-electron states of both the TMI TMCs in terms of the 10Dq splitting parameter for and of the surrounding ligand atoms are explicitly the one-electron d-levels (although the consistent included into consideration. In a symmetric envi- estimation of these and similar quantities through a ronment assumed in Eq. (2), the d-orbitals of the kind of quantum chemical procedure—from the t -symmetry neither get any admixture nor expe- 2g information on chemical composition and geome- rience any change of their energy (␲-orbitals on the try of the TMC—was not possible). ligands are not taken into account at this point). The At this point, one must say that the electronic antibonding orbitals of the e -symmetry (␺a(e )) are g g structure of TMCs is much more sophisticated and shifted upwards in energy: does not reduce to the one-electronic states as pre-

2 sented by Eq. (2). The d-shells of the TMIs within the H ␹ * ϭ ϩ d TMCs may to a large extent retain the multiplet struc- Eeg Hdd Ϫ , (4) Hdd H␹␹ ture they possess in the free state. In the free state, however, the excitation spectra of the d-shells, obvi- where Hdd, H␹␹, Hd␹ are matrix elements of the ously, cannot have anything to do with the splitting one-electron Hamiltonian of the CLS, thus giving and are controlled by the electron–electron interac- the following estimate for the splitting parameter: tions only. The characteristic parameters of that latter are known as the Slater-Condon parameters Fk, k ϭ 0, 2 H ␹ ϭ d 2, 4, or the Racah parameters A, B, and C. It is remark- 10Dq Ϫ . (5) Hdd H␹␹ able that even in the simplest version of the CFT these quantities had to be taken into account (although Within the LFT also only qualitative explana- considered as empirical parameters) to reproduce cor- tions could be obtained. It remained unclear where rect sequence of the many-electronic states of different to get the values of H␹␹ and Hd␹ on which all the total spin and spatial symmetries. Even more, in some details of composition and structure of the ligands situations it is possible exactly, although in others to a and their interaction are loaded in this model. Nev- good approximation, to express the excitation ener- ertheless, the expression Eq. (5) allows for impor- gies in the d-shells of TMCs in terms of the electron– tant conclusions different from the ionic model of electron interaction (Slater-Condon or Racah) param- the CFT. Within the LFT, the splitting comes not eters only. In the course of attempts to ﬁt the

2608 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED

(appropriate) excitation energies to the models ex- but having some contribution from the ligand pressing them linearly through the Racah parameters, states as stipulated by the LFT Eq. (2). Accepting Jaurgensen (as well as other students of that time) the SRC mechanism has two important conse- noticed that the interaction parameters for the TMIs’ quences. First is the possibility to ascribe differ- d-shells in TMCs coming from such fits are systemat- ent extent of expansion to different one-electron ically smaller than the same parameters derived from states in the d-shell. the spectra of the free ions. This phenomenon re- This option was used up to full scale in Ref. [1] ␤ ␤ ␤ ceived the aforementioned beautiful Greek name (al- where three nephelauxetic ratios ee, et, and tt ␤ ␤ ␤ though well before similar, but much less pronounced ( 33, 35, and 55 in the original notation) had been behavior had been observed in lanthanide complexes introduced to represent the effect of the expansion and had the name of the Ephraim’s effect) since it had of the orbitals in the e- and t2-manifolds of octahe- been interpreted as a consequence of certain increase dral and tetrahedral complexes on the Coulomb of the average electron–electron separation in the d- interaction matrix elements involving the orbitals shell taking place due to the expansion of the latter in from them. The numerical fits of the experimental the course of the complex formation. Two mecha- data reviewed in Ref. [1] allowed to conclude that nisms tentatively leading to the mentioned expansion in all cases the following relation holds: had been discussed in Ref. [1]. First was the central field covalency (CFC) mech- Ͼ ␤ Ͼ ␤ Ͼ ␤ anism, which is based on the idea that the sur- 1 tt et ee (9) rounding charges may contribute to the screening of the d-electrons they surround and by this may in a fair agreement with the qualitative picture of affect the rate of the radial decay of the d-states of a orbitals Eq. (2) according to which no delocaliza- TMI in a TMC. It could, however, only happen if tion (expansion) of the t-orbitals takes place (thus the spherical component of the external (ligand) ␤ ϭ giving tt 1, which must be overridden by Coulomb potential significantly depends on the dis- including ␲-overlap in the consideration) and the tance in the d-shell region. Employing the well- noticeable delocalization is expected only for the known expansion of the Coulomb potential over e-orbitals. the spherical harmonics: Second consequence is that the delocalization of the d-states due to mixing with the ligand states as 1 the only source of the orbital expansion puts the ͉R Ϫ r͉ observed renormalization of electron–electron in-

ϱ teractions in the d-shells in TMCs in a strict relation min͑r,R͒l 1 l r R ϭ ␲͸ ͸ ͩ ͪ ͩ ͪ with the crystal ﬁeld splitting of the latter. Indeed, 4 ϩ Y* Y (8) max͑r,R͒l 1 2l ϩ 1 lm r lm R as one can see, the estimates of the nephelauxetic lϭ0 mϭϪl ratios accepted in Ref. [1] are (here R and r are, respectively, the lengths of the ␤ ϭ x4 ; ␤ ϭ x2 x2 ; ␤ ϭ x4 . (10) radius vectors R and r) shows that the spherically tt t2g et t2g eg ee eg symmetric component (l ϭ 0) does not depend on the electron position r provided it is located closer Taking into account that to the coordinate origin than the ligand atom at R (R Ͼ r). Together with the assumption about the 1 compactness of the d-shell (␴R ϾϾ 1, where ␴ is the 2 Ϸ Ϫ ␨2 xeg 1 eg (11) exponent of the Slater d-orbital) this allows one to 4 exclude the CFC mechanism on the general theoretical grounds. The same reasoning explains why one may expect that the nephelauxetic series (that it is not possible to stabilize any bound state of an of the ligands with decreasing ␤’s) has to be similar electron (e.g., in a vacancy in a crystal or in a to the spectrochemical series (that of increasing “cavity” in a liquid) by external charges. 10Dq, see earlier). This, however, does not happen Second mechanism considered was the sym- and the nephelauxetic series for ligands (i.e., for metry restricted covalency (SRC), which ex- each ﬁxed TMI) can be presented as follows [1, 5, 6] plained the orbital expansion by the formation of (in order of increase of the amount of renormaliza- delocalized MOs of predominantly d-character, tion):

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TABLE I ______The estimates of the nephlauxetic ratios as extracted from experimental data given in Refs. [1, 5, 6]. ␤ ϭ ␤ ϭ ␥ ϭ B B/B0 C C/C0 C/B

2ϩ ϭ ϭ ␥ ϭ Mn , B0 860, C0 3850, 0 4.447 4Ϫ MnF6 0.819 0.948 5.186 4Ϫ MnCl6 0.743 0.893 5.382 2Ϫ MnCl4 0.612 0.920 6.736 2Ϫ MnBr4 0.623 0.917 6.585 2ϩ Mn(H2O)6 0.795 0.939 5.298 3ϩ ϭ ϭ ␥ ϭ Fe , B0 1015, C0 4800, 0 4.729 3Ϫ FeF6 0.675 0.773 5.416 Ϫ FeCl4 0.506 0.569 6.315

Racah parameters B and C are in cmϪ1

Ϫ 1 Ϫ However, the basic assumption of the rigid ratio F Ͻ H O Ͻ urea(O) Ͻ NH Ͻ EnϽ Ox2 2 th 2 3 2 F th/F 4 of 69-4/5 following from Eq. (13) and leading to Eq. (15) does not take place even in free ions, Ͻ NCSϪ Ͻ ClϪ (12) and thus F2(dd) and F4(dd) must be treated as inde- Ͻ Ϫ Ͻ Ϫ Ͻ Ϫ Ͻ Ϫ Ͻ 2Ϫ Ͻ 2Ϫ Ͻ 2Ϫ pendent parameters and so must be considered B CN Br N3 I S Se Te , and C. According to [6] the ratio C0/B0 where sub- which in many cases demonstrates not only devia- script 0 refers to the free ions is always larger than tions from the spectrochemical series Eq. (1) but four and smaller than five. Also the assumption Ϫ Ϫ Ϫ Ϫ even inversions: F vs. I ;Cl vs. Br , etc. that B and C renormalize by the same nephelauxetic It is worth noticing that the numerical values of ratio throughout the complex formation is only true the renormalization constants (nephelauxetic ra- if the renormalization takes place according to the 5 tios) ␤⌫⌫, had been derived from the experiment SRC mechanism. Using the data on simple d com- ϩ ϩ under some additional assumptions. First, it had plexes (those of Mn2 and Fe3 ) for which the been assumed that the Racah parameters B and C Racah parameters can be directly extracted from are related by a fixed coefficient about four. It is d–d excitation spectra, we establish the remarkable true for the theoretical values of these parameters deviations from the assumption of the uniform defined through the Slater-Condon parameters renormalization of the Racah parameters B and C as taken as integrals over Slater type d-orbitals: shown in Table I. According to it the B parameters are always strongly renormalized than the C pa- 5 ϫ 2093 rameters leading to the C/B ratios in the complexes Fth͑dd͒ ϭ ␴ 2 49 ϫ 76800 d to be systematically larger than five and sometimes to exceed six. Also, the complexes with very close ϫ 9 91 values of 10Dq are known to give sometimes differ- Fth͑dd͒ ϭ ␴ (13) 4 441 ϫ 9216 d ent values of renormalization constants indicating the absence of the direct relation between the (␴ is the orbital exponent for the Slater d-function). d amount of delocalization of the d-shell and the in- Then taking into account the definitions of the Ra- teraction renormalization in it. cah parameters All this calls for a new visit to this fascinating th ϭ th͑ ͒ Ϫ th͑ ͒ area of theoretical inorganic chemistry. B F2 dd 5F4 dd th ϭ th͑ ͒ C 35F4 dd (14) 2. Effective Hamiltonian of Crystal one arrives to: Field Theory

175 1 As we mentioned previously despite the correct Cth/Bth ϭ ϭ 4 Ϫ Ϸ 3.977. (15) 44 44 qualitative description of the spectrochemical series

2610 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED in the LFT, it cannot provide a quantitative method such a thing as a spectrum of approximately 5.1- or for calculation of the splittings of d-levels. The rea- 3.2-electron system. Thus, in the compounds of in- sons were mentioned earlier—the unclear source of terest it goes about some effective d-shell with the the matrix elements H␹␹ and Hd␹ to be used in Eq. fixed number electrons in it. In such a setting, the (4) and the need to take into account the details of contribution of the charge transfer states is taken the electron–electron interaction (correlation) of into account with the use of the Lo¨wdin projection electrons in the d-shells of TMIs. These deficiencies technique. It is done as follows: the total Hamilto- were the fundamental reason to develop the Effec- nian for a TMC is rewritten in the form: tive Hamiltonian Crystal Field (EHCF) theory Ref. ϭ ϩ ϩ ϩ [7], whose basic purpose was to unite the estimate H Hd Hl Hc Hr, (17) of the crystal field induced by the ligands with taking into account the correlations of electrons in where H is the Hamiltonian for d-electrons in the the d-shells. This twofold task required the usage of d field of the TMC atomic cores, H is the Hamiltonian a twofold tool: i.e., of combining the McWeeny [8] l for the l-system electrons, H and H are, respec- group function (GF) representation of the molecu- c r tively, the operators of Coulomb and resonance lar electronic structure with the Lo¨wdin partition interactions between two electron groups singled technique Ref. [9]. The GF method as applied to out in the TMC. TMCs assumes the wave functions of the ground Projecting the exact Hamiltonian Eq. (17) is per- and lower-energy excited states of the latter to have formed to the subspace spanned by the functions the form: with the fixed number nd of d-electrons. By this the original Hamiltonian Eq. (17) is replaced by the ⌿ ϭ ⌽d ٙ ⌽l n n 0, (16) effective one acting in this configuration subspace. By simple algebra the explicit form of the effective ⌽d Hamiltonian is obtained [7]: where n is some full CI function for nd electrons in the d-shell, and ⌽l is a single-determinant ground 0 eff ϭ ϩ state for the l-system. Taking this way of segment- H PH0P Hrr ing the electronic structure into groups reflects the ϭ ϩ ϩ H0 Hd Hl Hc main feature of that of the TMCs, which is the ϭ ͑ Ϫ ͒Ϫ1 presence of the strongly correlated d-shell of a spec- Hrr PHrQ EQ QH0Q QHrP (18) ified composition, i.e., containing some fixed integer number nd of (d-)electrons with low energy where P is the operator projecting to the subspace excitations localized in it and of relatively inert (i.e., of the functions with the fixed number nd of the having rather high excitation energies) ligands. To ϭ Ϫ d-shell electrons and nl N nd is that of the be able to reproduce the splitting of the states of the l-subsystem electrons; and Q ϭ 1 Ϫ P. d-shell under the influence of the l-system, the one The eigenstate energies must be obtained from electron transfers (delocalization) between the d- the relation: and l-systems have to be taken into account as in the LFT. Basically such transfers destroy the simple E ϭ ͗⌽ ͉Heff͑E ͉͒␺ ͘. (19) form of the wave function Eq. (16) by admixing the n n n n Ϯ ϯ states with nd 1(nl 1) electrons in the d- and l-systems (charge transfer states) to it. However, Eigenvalues of the effective Hamiltonian coin- experimentally we see that the number of electrons cide with exact Hamiltonian eigenvalues by con- in the d-shell is a good quantum number and is struction. Since the effective Hamiltonian depends perpetually used in chemical classification of the on energy, the last equation must be solved itera- TMCs. Of course, there are situations when classi- tively until convergence in energy is achieved. fication in terms of fixed number electrons in the However, since the charge transfer states (deter- d-shell fails, but we consciously exclude them from mining the poles of the above effective Hamilto- the current consideration. On the other hand, one nian) lay significantly higher in energy than the has to realize that TMCs conforming to this classi- d-shell excitations this dependence turns out to be fication do conform to it exactly, not approximately: weak and can be neglected, so one can set: their d–d-excitation spectra are precisely those of eff͑ ͒ Ϸ eff͑ ͒ three or five electrons in this shell since there is no H En H E0 (20)

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ϩ where E0 is the ground state energy of the Hamil- where d␮␴ (d␯␴) are operators of creation (annihila- ␴ tonian H0. Thus obtained effective Hamiltonian cor- tion) of electron with the spin projection on the responds to the second order of the Raleigh–Schro¨- ␮th d-AO; (␮␯͉␳␩) are the two-electron integrals of

dinger perturbation theory in Hr. the Coulomb interaction in the d-shell. Effective eff Variation principle applied to the effective Ham- one-electron parameters U␮␯ of the d-shell contain iltonian with the trial function of the form Eq. (16) contributions from the Coulomb and from the pro- leads to the self-consistent system of equations: jected [Eq. (5)] resonance interaction with the l- system: Heff⌽d ϭ Ed⌽d d n n n eff ϭ ␦ ϩ atom ϩ ion ϩ cov U␮␯ ␮␯Udd W␮␯ W␮␯ W␮␯ , (24) eff⌽l ϭ l ⌽l Hl 0 E0 0 where eff ϭ ϩ ͗⌽l ͉ ϩ ͉⌽l ͘ Hd Hd 0 Hc Hrr 0 , eff ϭ ϩ ͗⌽d͉ ϩ ͉⌽d͘ Hl Hl 0 Hc Hrr 0 . (21) atom ϭ ␦ ͸G W␮␯ ␮␯ͩ ␮,iPiiͪ iʦs,p eff ion ϭ ͸͑ Ϫ ͒ L In this system, the effective Hamiltonian Hd for the W␮␯ PLL ZL V␮␯. (25) d-electron subsystem depends on the wave function L ⌽l of the ligand subsystem 0, and in its turn the eff effective Hamiltonian Hl for the ligand subsystem Here Pii is one-electron density matrix element for ⌽d ϭ⌺ depends on the d-electrons’ wave functions 0. the ligand subsystem, PLL l⑀L Pll, ZL is the Lth L These equations must be solved self-consistently as atom core charge, V␮␯ is the matrix element of the well. In the EHCF method, Ref. [7], the Slater de- d-electron potential energy in the electrostatic ﬁeld ⌽l terminant 0 is constructed of MO’s of the l-system, of a unit point charge placed on the Lth ligand atom obtained from the Hartree–Fock equations in the (see later). The covalence contribution to the crystal CNDO/2 approximation for the valence electrons ﬁeld is given by of the ligands. In this case, the transition from the eff ͑MO͒ bare Hamiltonian Hl for the l-system to the corre- 1 Ϫ n n cov ϭϪ͸ ␤ ␤ ͭ i Ϫ i ͮ sponding effective (dressed) Hamiltonian reduces W␮␯ ␮i ␯i ⌬ ⌬ (26) Edi Eid to renormalization of one-electron parameters re- i lated to the TMI: ␤ where ␮i is the one-electron matrix element (resonance or hopping integral) between the ␮th d-or- 1 eff ϭ ϩ ͸G ϭ Uii Uii nd ␮i, bital and the ith ligand MO, ni ( 0, 1) is the 5 ␮ ⌬ ⌬ occupation number of the ith MO, Edi ( Eid) are excitation energies required to transfer an electron Zeff ϭ Z Ϫ n , (22) M M d from the d-shell (ith MO) to the ith MO (d-shell). The effective Hamiltonian for the d-shell Eq. (23)

where Uii is the parameter of the interaction of 4s- formally coincides with the CFT Hamiltonian. The ϭ and 4p-electrons (i 4s,4px,4py,4pz) with the TMI substantial difference is the covalence contribution ϭ ␮␮͉ Ϫ core, ZM is TMI core charge, g␮i ( ii) 1/2 Eq. (26) to the d-shell one-electron parameters Eq. (␮i͉i␮) are the parameters of intraatomic Coulomb (24), taking into account the effect of virtual charge ⌽l interactions. The 0 function thus obtained is used transfers between the metal d-shell and the ligands, further for constructing the effective Hamiltonian i.e., the delocalization of the d-states. Thereby, the for the d-shell. EHCF contains not only electrostatic but also cova- The effective Hamiltonian for the d-shell after lence terms coming from the resonance interactions averaging Eq. (21) inter-subsystem interaction op- between the d-shell and the ligands as the LFT does.

erators Hc and Hrr over the ground state of the According to calculations performed in Refs. [7, l-system takes the form: 10–13] for the TMCs of divalent cations, the covalence contribution to the splitting parameter 10Dq dominates and gives up to 90% of the total. This eff eff ϩ 1 ϩ ϩ H ϭ ͸U␮␯d␮␴d␯␴ ϩ ͸͸͑␮␯͉␳␩͒d␮␴d␳␶d␩␶d␯␴, (23) allows to state that like in the LFT in the EHCF, the d 2 ␮␯␴ ␮␯␳␩ ␴␶ splitting of the d-shell is controlled by an analog of

2612 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED the above ␨ parameter, i.e., by a ratio of the electron indicates that in the EHCF context some other rea- hopping integral to the energy of the inter-sub- sons of the renormalization of the Coulomb inter- system charge transfer state. Generally, the EHCF action between electrons in the d-must be admitted. method allowed to describe correctly the symmetry of the ground states and the optical d–d-transition Ϫ1 3.1. CONTINOUS INSULATOR MODEL OF energies with precision up to 1000 cm for about a NEPHELAUXETIC EFFECT hundred of the TMCs of the first transition row divalent cations ranging from hexafluoroanions to It is fair to say that the ligand polarization had porphyrine complexes and improved the semiem- been mentioned as a mechanism of renormalization pirical description of TMC electronic structure sig- of Coulomb interaction in the d-shells of TMCs at a nificantly. pretty early stage. If the nephelauxetic series Eq. (12) is rewritten in terms of donor atoms Ref. [5], it becomes: 3. Renormalization of Electron–Electron Interaction in the F Ͻ O Ͻ N Ͻ Cl Ͻ Br Ͻ I Ͻ Se Ͻ Te d-Shell which approximately corresponds to the order of The EHCF method as presented in the previous the atomic polarizabilities. It, however, had been section lacks an important element: no renormaliza- considered as a secondary effect in Ref. [1], al- tion of the electron–electron interaction in the d- though obvious deviations between the spectro- shell appears from the above derivation. Strict ap- chemical and nephelauxetic series were clear. The plication of the EHCF theory assumes that the free same idea had been used in Ref. [14]. According to ion values B and C of the Racah parameters are these authors, if one assumes that a TMI bearing the 0 0 ␴ used. This, however, would contradict to the exper- d-shell with the Slater orbital exponent d is placed iment and thus the experimental values of B and C inside a sphere of radius b cut in the medium with had been used throughout the calculations in Refs. dielectric constant, ␧, the Slater-Condon parameters [7, 10–13]. This clearly reduces the predictive F2(dd) and F4(dd) renormalize with the nephelaux- ␤ ␤ power of the otherwise very attractive EHCF pic- etic ratios 2 and 4 given by: ture and of the pragmatic calculation method based 1260 ␧Ϫ1 4 5 upon it. For that reason, we undertake this study in Ϫ ␤ ϭ ͩ ͪͩ ͪ 1 2 ␧ϩ ␴ which we try to obtain the estimate of the amount 299 3 2 b d of the renormalization of the electron–electron in- 20 ϫ 10! ␧Ϫ1 1 9 teractions in the d-shell within the same framework, Ϫ ␤ ϭ ͩ ͪͩ ͪ 1 4 ␧ϩ ␴ which previously allowed very precise estimates of 13 5 4 b d the splitting of the d-shell. Before doing so, we 1 Ϫ ␤ 4 5␧ϩ4 notice that these two effects: splitting and interac- 2 ϭ ͩ ͪ͑ ␴ ͒4 Ϫ ␤ ␧ϩ b d . (27) tion renormalizations appear in different orders of 1 4 5175 3 2 perturbation theory with respect to one-electron ␧ ϭ ϭ ␴ ϭ hopping matrix element responsible for charge Assuming 12, b 1.5 Å, and d 2.06, the transfers between the d- and l-systems. The splitting authors in Ref. [14] reproduced the magnitude of ␤ 2ϩ is of the second order in ␨ whereas renormalization the 4 ratio for one particular case (Co impurity 1 must be of the fourth order. This also agrees with ion in the tetrahedral site of MgAl2O4) and under- ␤ the LFT estimates Eq. (10) and explains our old estimated 2 by ca. 40%, thus yielding the correct result where only the aforementioned perturba- order of magnitude of the effect. The approxima- tional estimate of the fourth order had been tested2 tion of infinite space filled by an insulator outside and the nephelauxetic ratios turned out to be ca. the sphere reasonable in the case of a impurity TMI 0.98 Ϭ 0.99 in all considered cases. This failure in the crystals is not that much important since obvious transition to the model of a spherical layer reduces to subtracting of corresponding powers of 1 It is worth noting that the very notation of in the theory of the b*␴ (b* is the external radius of the spherical layer) nephelauxetic effect is not particularly convenient. The quantities of ␤ does not significantly affect the numerical result definite order are not the nephelauxetic ratios themselves , but Ͻ their deviations from the unity: 1 Ϫ ␤. due to high powers of the b/b* 1 ratio as entering 2Kharitonov, D. N.; Tchougreéff, A. L. unpublished. in the correcting multipliers (1 Ϫ (b/b*)5) and (1 Ϫ

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(b/b*)9) for the first two rows of Eq. (27). One can the charge transfer between the d- and l-system, but estimate to what extent the value of ␧ ϭ 12 used in does not assure the product form of Eq. (16). Prod- Ref. [14] can be reproduced by available data on ucts of different states of the d- and l-systems sat- polarizability of the ligands using the Clausius- isfying only the condition of the fixed number of Mossotti formula for ␧: the electrons in each of the singled out groups enter in the expansion of the true ground state of the 4␲ effective Hamiltonian Eq. (18). The form Eq. (16) is ␧ϭ1 ϩ ␳␣. (28) 3 a kind of self-consistent field approximation to it. To improve this description one has to perform one more projection namely to the subspace of the According to Ref. [1], they range from ca. 1 Å3 for Ϫ Ϫ products where all possible states of the l-system F to 14 Ϭ 16 Å3 for Te2 , whereas the densities to are replaced by its ground state. Formally, we pre- be used to get the above value of the dielectric viously consider the projection operator ᏼ : constant range from unrealistically high value of 2.5 3 Ϫ ions per Å for F to unrealistically low 0.16 ions l l Ϫ ᏼ ϭ I ͉⌽ ͗͘⌽ ͉, (29) per Å3 for Te2 . These estimates show that the d 0 0 polarizability-based model of renormalization of ϭ the Coulomb interaction of d-electrons can cover a and the complementary projection operator Q Ϫ ᏼ wide range of observed values of renormalization. 1 on the subspace orthogonal to it. Inserting ᏼ Remarkably enough the formulae Eq. (27) have the the projection operators and Q into general ex- multiplicative structure pressions for the effective Hamiltonian acting in the subspace Im ᏼ (spanned by the functions of the ⌽d 1 Ϫ ␤ Ϸ h͑ligands͒k͑metal) form Eq. (16)—all possible n times the ground ⌽l state 0 ), yields the expression:

prescribed by Ref. [1] with h to be identiﬁed with eff͑␻͒ ϭ ᏼ effᏼ ϩ ᏼ eff ͑␻͒ effᏼ ␧ ␴ H H H QR QH , the -dependent multiplier and k to be the d-dependent one. R͑␻͒ ϭ ͑␻Q Ϫ QHeffQ͒Ϫ1. (30)

The ﬁrst term is precisely d-electron Hamiltonian 3.2. POLARIZATION PROPAGATOR MODEL OF NEPHELAUXETIC EFFECT Eq. (23) multiplied by the l-system ground-state ͉⌽l ͗͘⌽l ͉ projection operator 0 0 . The second gives the From the quantum chemical point of view, the correction to it. Different terms in Heff behave dif- expression Eq. (27) lacks the atomic (or even orbital) ferently under this projection. Taking into account resolution necessary for quantum chemistry. To de- that ᏼ projects to the product of the eigenstates of eff eff rive the analog of the result, Eq. (27) compatible the operators Hd and Hl Eq. (21) one can see that: with quantum chemical description and particu- eff ϭ larly with the EHCF theory one has to follow the PH Q PWdlQ, (31) general theory of the GF-based semiempirical methods as described in Refs. [15–17]. where It must be noticed that the derivation of EHCF was performed in an assumption that not only the W ϭ Wˆ Ϫ ͗͗Wˆ ͘͘ Ϫ ͗͗Wˆ ͘͘ Ϯ ϯ dl dl dl d dl l (32) charge transfer excitations in the nd 1(nl 1) electrons manifold have the energy large enough to is the operator of reduced interaction of the d- and assure the validity of the perturbation theory with l-systems where: respect to small parameter ␨ but also the energies of the excitations in the l-system are large enough not W ϭ Vc ϩ Vrr to take them into account. The excitations of the dl l c ϭ c Ϫ ͗͗ c͘͘ Ϫ ͗͗ c͘͘ -system are, however, responsible for its polariza- V H H d H l tion. One has to realize that taking the wave func- rr ϭ rr Ϫ ͗͗ rr͘͘ Ϫ ͗͗ rr͘͘ tion of the TMC in the form of Eq. (16) even after V H H d H l (33) Lo¨wdin partition with use of the complementary projection operators P and Q is an approximation. where the expectation values of the interaction op- In fact, the operator P projects out the states with erators calculated for the ground state of the d- and

2614 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED l-systems are subtracted (i.e., only the ﬂuctuations The result Eq. (36) can be rewritten in terms of the of the interacting quantities are retained). With polarization correction to the two-electron matrix these notions, the correction acquires the form: elements: Ј͉ Љ ٞ͒ ϭ ͸ ͑ Ј͉ Ј͒⌸ ͑ Љ ͉ٞ Љ ٞ͒ ͑␦ (ᏼ ᏾͑␻͒ ᏼ dd d d dd ll llЈlЉlٞ l l d d . (38 Wdl Wdl . (34) llЈlЉlٞʦl

As before the idea of relative inertness of the Significant simplification can be reached in the l-system is formalized by the assumption that the basis of canonical MOs of the l-system where the excitation energies in it are large as compared with polarization propagator is diagonal in that sense ⌸ the excitation energies in the d-system, which are that only the matrix elements of the form llЈlЈl are only interesting for us. For this reason, one can nonvanishing where l and lЈ refer to the occupied guess that the dependence of the resolvent on ␻ is and vacant states, respectively, of the l-system. weak and that the values of ␻ in the interesting The correction Eq. (36) must be included if the energy range are much smaller than the resolvent fluctuation of the electronic density in the l-system poles that are all lying not lower than the first of the TMC as compared with the wave function excitation energy in the l-system. These notions obtained as solution of the system Eq. (21) are not allow to replace the resolvent R(␻) by its value at ␻ considered explicitly. This is precisely the case in ϭ 0 (by this, the electronic dynamic effects in the the EHCF theory. Incidentally, the intrashell Slater- l-system are excluded): Condon parameters Fk (dd) with k ϭ 2, 4 are precisely those which describe the interactions be- l l ͉⌽␭͗͘⌽␭͉ tween the density fluctuations in the d-shell and ᏾͑ ͒ ϭϪ͸ 0 l , (35) E␭ those which one should expect to renormalize to ␭0 reproduce the nephelauxetic effect. The Hamilto- nian matrix elements capable to couple the density l where E␭ are excitation energies in the l-system. fluctuations in the d-shell with those in the sur- Restricting for the simplicity in Eq. (32) by the rounding are those of the Coulomb interaction. In ϩ reduced Coulomb operator: the free ion, the fluctuation operators like d dЈ transform according to the SO(3) representations c ϭ ͸ ͑ Ј͉ Ј͓͒ ϩ Ј ϩ Ј Ϫ ϩ Ј͗͗ ϩ Ј͘͘ characterized by the angular momentum 2 and 4, V dd ll d d l l l l d d d ddЈʦd llЈʦl which are incidentally used for classification of the Slater–Condon parameters. Ϫ ϩ Ј͗͗ ϩ Ј͘͘ ͔ d d l l l , As it is shown in Ref. [7], the matrix elements of the form (ddЈ͉llЈ) describing the Coulomb interac- we get after averaging over the ground state of the tion between the electrons in the d- and l-systems l-system the following correction to the d-system fall into two types (i) intraatomic ones, describing Hamiltonian: the interaction between the d- and sp-shells of the TMI and (ii) those between the d-shell of the TMI ͸ ϩ Ј Љϩ ٞ ͸ ͑ Ј͉ Ј͒͑ Љ ͉ٞ Љ ٞ͒⌸l ͑ ͒ and the electrons in the ligand atoms. In the ionic , d d d d dd ll l l d d llЈlЉlٞ 0 ddЈdЉdٞʦd llЈlЉlٞʦl model of the CFT, the elements of the type (ii) are responsible for splitting of the d-levels. Accord- (36) ingly, the Coulomb interaction operator Hc is a sum: describing the weakening of the interaction be- c ϭ c ϩ c tween the fluctuations of the electron density in the H H1 H2 (39) d-shell due to interaction between the polarizations induced by these fluctuations in the l-system. The and we consider the effect of two contributions to polarization propagator of the l-system entering the Eqs. (36)–(38) separately. answer is: 3.2.1. Intraatomic Contribution to ⌸l ͑␻͒ ϭ ͸͗⌽l ͉ ϩ Ј͉⌽l ͑͘␻ Ϫ l ͒Ϫ1͗⌽l ͉ Љϩ ͉ٞ⌽l ͘ Nephelauxetic Effect . llЈlЉlٞ 0 l l ␭ E␭ ␭ l l 0 ␭ 0 In the intraatomic context, it is convenient to use (37) the one-electron wave functions of definite angular

VOL. 109, NO. 11 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2615 TCHOUGRE´ EFF AND DRONSKOWSKI momentum projection (complex spherical harmon- A product of two such elements (to be further ics) instead of cubic harmonics usual for the atomic multiplied by the by the polarization propagator states in quantum chemistry. We start from remind- matrix element) is: ing the form of a general intraatomic matrix ele- (ments of the Coulomb interaction between elec- ͑ddЈ͉ppЈ͒͑pЈp͉dЉdٞ͒. (43 ϭ Ϭ trons. In general, for the atomic states nilimi, i 1 Ϭ 4 ni ’s are the principal quantum numbers, li ’s are Taking into account that the element (pЈp͉dЈЈdЈЈЈ) the azimuthal quantum numbers, and mi’s are mag- is as well proportional to F2 (pd) with the symmetry netic quantum numbers of the states involved) the ͉ ϭ Ϫ multiplier matrix element (m1m2 m3m4) with m1 q m2; m3 ϭ ϩ q m4 and with respective nili ’s reads: ͑ Ј ͉ Љ ٞ͒ ϭ ͑ Ϫ ͉ ϩ ͒ ϭ p p d d m5 p,m5 m4 p,m4

ϩ ϩ Ϫ͑ ϩ ͒ l1 kl2 l3 kl4 ϩ Ϫ͑ ϩ ͒ 222 121 ͑ Ϫ ͒q l1 l3 m2 m4 ͩ ͪͩ ͪ ϫ ϭ 15͸͑ Ϫ 1͒p 3 m5 m4 ͩ ͪͩ ͪ ϫ 1 000 000 000 000 k ϫ ͱ͑ ϩ ͒͑ ϩ ͒͑ ϩ ͒͑ ϩ ͒ 2l1 1 2l2 1 2l3 1 2l4 1 222 ϫ ͩ Ϫ ͑ ϩ ͒ ͪ l1 kl2 p m4 pm4 ϫ ͩ Ϫ Ϫ ͪ q m2 qm2 1212 ϫ ͩ Ϫ Ϫ ͪF ͑ pd͒ (44) l kl p m5 pm5 ϫ ͩ 3 4 ͪ k͕͑ ͖͒ Ϫ ͑ ϩ ͒ R nili (40) q m4 qm4 subject to the selection rules ··· ··· ··· where the symbols ͩ ͪ are the Wigner ϭ ϩ ··· ··· ··· m5 q m3 (45) 3jm-symbols as described in many places, e.g., in ϭϪ ϩ Ref. [18]. For the interaction matrix elements in the m3 p m5 (46) d-shell it yields:

Ϫ͑ ϩ ͒ which immediately yields ͑ Ϫ ͉ ϩ ͒ ϭ ͸͑ Ϫ ͒q m2 m4 m2 q,m2 m4 q,m4 25 1 k p ϭ q (47) 2 k 2 2 2 k 2 ϫ ͩ ͪ ͩ Ϫ Ϫ ͪ 000 q m2 qm2 (transferred angular momentum projections must 2 k 2 be equal). Using once again the first of the afore- ϫ k͑ ͒ ͩ Ϫ ͑ ϩ ͒ ͪF dd . (41) mentioned conditions, we arrive to the following q m4 qm4 combination of the 3jm symbols depending on the Intraatomic matrix elements coupling the fluctu- momentum projections of the individual states in ations in the d- and p-shells are those coupling the the d- and the p-shells: fluctuations with k ϭ 2 (for two p-functions k ϭ 4is ϭ 2 k 2 not accessible, whereas k 0 are not the fluctua- qϪ͑m2ϩm4͒ ͑ Ϫ 1͒ ͩ Ϫ Ϫ ͪ tions). Since in the octahedral and tetrahedral com- q m2 qm2 plexes the p-shell of the TMI remains nonsplitted 2 k 2 (as in the free ion), the spherical harmonics can also ϫ ͩ Ϫ ͑ ϩ ͒ ͪ ϫ (48) q m4 qm4 be used for it. Thus we get:

͑ Ј͉ Ј͒ ϭ ͑ Ϫ ͉ ϩ ͒ ϭϭϪ ⅐ ϫ ͩ 121ͪͩ 121ͪ dd pp m2 q,m2 m3 q,m3 15 Ϫ ͑ ϩ ͒ Ϫ Ϫ ͑ ϩ ͒ . q m3 qm3 q m3 qm3

Ϫ͑ ϩ ͒ 222 121 ͑ Ϫ ͒q m2 m3 ͩ ͪͩ ͪ ϫ (49) 1 000 000

ϫ 222 First two multipliers and the phase factor yield ͩ Ϫ Ϫ ͪ Ϫ ͉ ϩ q m2 qm2 precisely that of the matrix element (m2 qm2 q m4m4)inthed-shell. Different values of m3 in the 1212 ϫ ͩ Ϫ ͑ ϩ ͒ ͪF ͑ pd͒. (42) second pair of multipliers number different (degen- q m3 qm3

2616 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED

erate) states in the p-shell, contributing to the renor- the ratio F2(pd)/F2(dd) to be ca. 0.25.3 Nevertheless, malization. These products must be summed up the earlier formula has at least the use that it dem- which yields a factor of 1/5 independent on q.We onstrates the chemical speciﬁcity of the interaction see that all matrix elements in the d-shell renormal- renormalization and describes the intraatomic con- ize accordingly so that the result can be represented tribution to it which clearly cannot be covered by as a renormalization of the Slater-Condon parame- the continuous insulator model Eq. (27). ter F2(dd). The correction it receives reads: In more general setting not limited by the CLS model of the ligands, rather including all their va-

2 2 2 lence orbitals like it is normally done in the EHCF 9 121 xt1uyt1u Ϫ ͩ ͪ ͓ 2͑ ͔͒2 F pd ⌬ (50) Eq. (52) must be modiﬁed to 5 000 t1u

2 2 2 6 ͓F ͑ pd͔͒ ͑c ␥c ␥͒ which after inserting the value of the 3jm symbol Ϫ ␤ ϭ ͸ ͸ i j 1 2 2 (53) ͑ ͒ ʦ ␧ Ϫ␧ 25 F dd ␥ʦ i occ j i reduces to x,y,zjʦvac

2 2 6 xt yt Ϫ ͓ 2͑ ͔͒2 1u 1u where the last sum is nothing but the orbital–or- F pd ⌬ . (51) 25 t1u bital polarizability of the ␥th (4)p-AO of the TMI. Sensitivity to the chemical specific of the surround- In these expressions, the last multipliers repre- ing appears due to the MO coefficients of the re- sent the contribution of the CLS as defined in Ref. spective p-AOs in the occupied and vacant MOs. [19] to the polarization propagator of the l-system; ⌬ t1u is the energy gap between the occupied and empty states of the t1u-symmetry spanned by the 3.2.2. Ionic Contribution to Crystal Field and

p-states of the TMI, and the xt1u and yt1u are the the Nephelauxetic Effect expansion coefficients of the corresponding MOs as Further and probably dominating contributions defined by Eq. (2). to the renormalization of the interaction parameters Already this result allows one to see that the of the d-shell comes from the second (interatomic) renormalization of the intrashell interaction is not term of the Coulomb interaction operator between uniform in that sense that the proposed treatment the d- and l-systems. As previously, the key element renormalizes only the Slater–Condon parameter of the construction is identifying the matrix ele- F2(dd), but not F4(dd). Thus in terms of the Racah ments of the interaction which are responsible for parameters, it means that the B parameter renor- the coupling between the density fluctuations in the malizes whereas the C parameter does not. That is corresponding systems. Incidentally, the EHCF what we could expect on the basis of our analysis contains the electrostatic interaction between the summarized in Table I where the renormalization electrons in the d- and the l-systems of the form: of the parameter B is systematically much more pronounced than that of C. Of course one should c ϭ ͸ not ascribe all this difference to the above Eq. (51) at H2 Vˆ mmЈ least for two reasons: first or all the renormalization mmЈ of C is an established fact, which needs to be ex- ˆ ϭ ͸͸ L ϩ ϩ Ϫ ␤ VmmЈ VmmЈ,dm␴dmЈ␴l␶ l␶ (54) plained, and second, the estimate of 1 2 coming lʦL from: L ␴,␶

2͑ ͒ 2͑ ͒ with 6 2 2 F pd F pd xt1uyt1u 2͑ ͒ ⌬ (52) 25 F dd t1u

3 2 2 ␴ ␴ In fact, the ratio F (pd)/F (dd) as a function of p/ d for the turns out to be too small. Indeed, the ratio F2(pd)/ principal quantum number 3 of the Slater d-orbitals nicely ap- ␴ ␴ 2 F2(dd) of the “experimental” values falls in the proximates as ( p/ d) with the Hilbert square norm of the Ϭ difference between the exact ratio and the quadratic estimate range 0.1 0.3 for all TMIs of the ﬁrst row. That being less than 2 ϫ 10Ϫ3 for the integration interval [0, 1] and ␴ ␴ Ϸ Ϫ calculated for the ratio p/ d 0.5 characteristic for becoming less than 2 ϫ 10 4 for a more realistic interval [0.2, 0.8] ␴ ␴ the Burns’ exponents used in the EHCF yields of the p/ d values.

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Ϫ 1͒m 2 k 2 ␦͑mmЈ͉mЉmٞ͒ ͑ L ϭ ͱ ␲ ͸ ͩ ͪ VmmЈ 5 4 ͱ ϩ 000 mϩmЉ kϭ0,2,4 2k 1 ͑ Ϫ 1͒ ϭϪ100␲͸ ͸ ͸ ϫ ͱ2k ϩ 1 ͱ2kЈ ϩ 1 2 k 2 LLЈ lʦL;lЈʦLЈk,kЈϭ2,4 ϫ ͩ ͪ ͑ ͒ mϪmЈ͑␪ ␾ ͒ Ј Ϫ Ϫ Ј Fk RL Yk L, L (55) mm m m Ј ϫ ͩ2 k 2ͪͩ2 k 2ͪ ϫ 000 000 where (R , ␪ , ␾ ) are the spherical coordinates of L L L 2 k 2 the ligand atom L (the TMI is located in the center ϫ ͩ Ј Ϫ Ϫ Јͪ m Ϫ mЈ ␪ ␾ mm m m of the coordinate frame); Yk ( L, L) are the spherical functions with the phases defined follow- 2 kЈ 2 ϫ ͩ Љ ٞ Ϫ Љ Ϫ ٞͪ ϫ ing Condon and Shortley [20]. Functions Fk (RL) are m m m m the integrals of squares of the radial parts R (r)of Ϫ Ј nl ϫ F ͑R ͒Ym m ͑␪ ,␾ ͒ the atomic d-functions: k L k L L C C C ЈC Ј mЉϪmٞ͑␪ ␾ ͒ ͸ il jl il jl ͒ ͑ FkЈ RLЈ YkЈ LЈ, LЈ . (58) ʦ ␧ Ϫ␧ R i occ j i jʦvac ͑ ͒ ϭ Ϫ͑kϩ1͒͵ k 2 ͑ ͒ 2 Fk R R r Rnl r r dr 0 Summation over the one-electronic states l ⑀ L; lЈ ϱ ⑀ LЈ centered on ligand atoms LLЈ can be performed ϩ k͵ Ϫ͑kϩ1͒ 2 ͑ ͒ 2 R r Rnl r r dr (56) and straightforwardly yields: R ͑ Ϫ 1͒mϩmЉ Ј͉ Љ ٞ͒ ϭϪ ␲͸ ͸ ͑␦ and depend on the distance R from the atom of mm m m 100 ͱ ϩ ͱ Ј ϩ L Ј Јϭ 2k 1 2k 1 metal to the atom L. If the Slater AOs are taken for LL k,k 2,4 Ј Rnl(r), the Fk(R) functions have been evaluated and ϫϫͩ2 k 2ͪͩ2 k 2ͪ ϫ are given, e.g., in Ref. [6]. 000 000 The approximation in Eq. (54) is to assume all 2 k 2 2 kЈ 2 two-center matrix elements of the form ϫ ͩ ͪͩ ͪ ϫ mmЈ Ϫ m Ϫ mЈ mЉ mٞ Ϫ mЉ Ϫ mٞ ϫ ͑ ͒ mϪmЈ͑␪ ␾ ͒ ͑ ͒ mЉϪmٞ͑␪ ␾ ͒␲ ͑mmЈ͉llЈ͒ (57) Fk RL Yk L, L FkЈ RLЈ YkЈ LЈ, LЈ LLЈ (59) to be diagonal with respect to AOs llЈ and to be equal for all AOs centered on the same atom L of giving the result in terms of atom-atomic polariz- ␲ the ligand. Despite this, the components of Eq. (54) abilities LLЈ of the l-system as coming from the with k ϭ 2, 4 describe precisely the interaction semiempirical SCF calculation of the latter assumed between the fluctuations in the d-shell and the elec- by the EHCF theory. This expression is of course tronic density in the l-system. The average of this not spherically symmetric any more. By itself it ion operator produces the ionic contribution W␮␯ Eq. does not represent any problem since in the TMC (25) to the effective crystal field as in the classical there is no such symmetry and in this respect the CFT. The fluctuations occurring on top of this av- result is correct. However, both the general practice erage are responsible for the effect interesting to us. of spectroscopy of TMCs and the EHCF theory base Generalizing the moves of Section 3.2.1., we obtain the description of electron–electron interactions in the correction to the matrix element (ddЈ͉dЉdЈЈЈ)in the d-shell on a tacit assumption of spherical sym- the form of Eq. (38) and then inserting explicit metry of these latter (usage of the B and C Racah forms of the matrix elements Eq. (55) we recast it in parameters or the Slater–Condon parameters). For the form of renormalization of the Slater-Condon this reason, we consider below a symmetric version

parameters F2 and F4. After substituting the matrix of the atomically resolved version of the polariza- elements and of the elements of the polarization tion stipulated nephelauxetic effect by singling out ⌸ propagator supermatrix of the form lllЈlЈ as stipu- the spherically symmetric part of the renormaliza- lated by the approximation Eq. (54) to Eq. (38) we tion and reducing it to that of the Slater-Condon

immediately get: parameters F2(dd) and F4(dd).

2618 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 11 NEPHELAUXETIC EFFECT REVISITED

For this end, we assume the environment of the of the Slater-Condon parameter Fk(dd) in the ex- TMI in the TMC (the l-system) to be approximately pansion of the two-electron matrix element spherically symmetric so that its excitation spec- (mmЈ͉mЉmЈЈЈ) so that the given result can be repre- trum can be classiﬁed according to irreducible rep- sented as a renormalization of the Slater-Condon resentations of the SO(3) group, i.e., as having def- parameters, namely: inite angular momentum and its z-projection. Then the polarization propagator ⌸ is diagonal in the 4␲ ␦F ͑dd͒ ϭϪ ͸͸ ⌸͑k␬͒ basis of such states: k ͑2k ϩ 1͒ LLЈ ␬ ͉k␬͗͘k␬͉ ͑ ͒ ͑ ͒ ␬͑␪ ␾ ͒ ͑ ͒ Ϫ␬͑␪ ␾ ͒ (⌸͑ ؅͒ ϭ ͸ ⌸͑k␬͒͑ ؅͒ ϭϪ͸ ⌸͑k␬͒͑ ؅͒ RL,RLЈ Fk RL Yk L, L Fk RLЈ Yk LЈ, LЈ . (62 R,R R,R ␻ R,R k k,␬ k,␬ On the other hand, the atom-atomic polarizabili- Ј͒ ؅ ͑ ͒ ͑ fk* R fk R R R ␲ ϭϪ ␬ ͩ ͪ ␬ͩ ͪ ties LLЈ of the l-system entering Eq. (59) correspond ␻ Yk* Yk Ј (60) k R R to the values of ⌸(R,RЈ) through the standard relation between the coordinate and the AO represen- which can be understood as a sum over excitations tations. Indeed, for a system described in the AO of a spherical layer having the excitation spectrum representation, the orbital–orbital mutual polariz- ␻ ⌸ of energies { k}. The ground state of the layer is ability lllЈlЈ spherically symmetric (k ϭ 0). The spatial angular dependence of the state k␬ʹ is that of the spherical CilCjlCilЈCjlЈ ␬ ⌸ ϭ ͸ ␪ ␾ lllЈlЈ ␧ Ϫ␧ harmonic Yk ( , ), which is complemented by i␧occ j i j␧vac some radial dependence fk(R) of the one-electron transition density to obtain its total spatial dependence. This yields the expression for the polariza- contributes to that in the coordinate representation tion propagator of the effective medium represent- according to: ing the l-system in the coordinate representation. ⌸͑ ؅͒ ϭ ͸ ͸ ͉␸ ͑ ͉͒2͉␸ ͑ ؅͉͒2⌸ Using the k␬-resolved polarization propagator R,R l R lЈ R lllЈlЈ. leads to the selection rules LLЈl␧L;lЈ␧LЈ ␸ ␸ Ј k ϭ kЈ where l(R) and lЈ(R ) are the explicit forms of the l-th and lЈ-th AOs giving necessary spatial density m Ϫ mЈ ϭ ␬ ϭϪ͑mٞ Ϫ mЉ͒ distributions. The approximation of Eq. (55) is equivalent to setting ˆ (one can think that one of the operators VmmЈ mmЈ ͒ ␬ ͉␸ ͑ ͉͒2 ϭ ␦͑3͒͑ ؊ ignites in the l-system an excitation with deﬁnite k , l R R RL ˆ ,(which must be quenched by the VmЉmٞ mЉmЈЈЈ one which reduce the general expression Eqs. (58) and for all l ⑀ L. This immediately results in (59) to ⌸͑ Ј͒ ϭ ͸ ␦͑3͒͑ ؊ ͒␦͑3͒͑ ؅؊ ͒␲ R,R R RL R RLЈ LLЈ. mmЈ͉mЉmٞ͒ LLЈ͑␦ ϩ Љ ͑ Ϫ 1͒m m 2 k 2 2 ϭϪ100␲͸ ͸ ͩ ͪ ϫ This move allows to single out the spherically 2k ϩ 1 000 LLЈ kϭ2,4 symmetric part of the renormalization:

ϫ ͩ 2 k 2 ͪͩ 2 k 2 ͪ ␦ ͑ ͒ ϭ mmЈ Ϫ m Ϫ mЈ mЉ mٞ Ϫ mЉ Ϫ mٞ Fk dd ␲ ͑k␬͒ 4 ␬ Ϫ␬ ϫϫ͸ ⌸ ͑R ,R Ј͒ Ϫ ͸ ͑ ͒ ͑␪ ␾ ͒ ͑ ͒ ͑␪ ␾ ͒␲ L L Fk RL Yk L, L Fk RLЈ Yk LЈ, LЈ LLЈ ␬ ͑2k ϩ 1͒ LLЈ ϫ ͑ ͒ ␬͑␪ ␾ ͒ ͑ ͒ Ϫ␬͑␪ ␾ ͒ Fk RL Yk L, L Fk RLЈ Yk LЈ, LЈ . (61) (63)

It is easy to check that the phase factor together which can be as well understood as a double-gen- with the 3jm symbols yields precisely the coefﬁcient eralized Fourier transform of the point–point polar-

VOL. 109, NO. 11 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2619 TCHOUGRE´ EFF AND DRONSKOWSKI ization propagator calculated in a finite number of sidered as a catastrophe: so-called ab initio methods points referring to the positions of the ligand atoms. require much more parameters to describe their The generalized Fourier transform can be used to respective basis sets or pseudopotentials used, none establish the relation with the continuous insulator of which individually has any physical sense. Po- model result Eq. (27). For that end, we represent in larizabilities, however, have transparent physical Eq. (62) the sum over LLЈ as an integral over R, R؅ meaning and can also be used in a line with the of an expression containing the LLЈ sum of the ␦(3) proposition of developing a “new generation” of densities centered in RL, RLЈ as a multiplier. Going semiempirical methods [22] characterized among to the spherical coordinates for the both integration other features by extending these latter by includ- variables R, RЈ and performing integration over the ing dispersion forces ultimately determined by the angular variables, we arrive to the following ex- same polarizabilities and being in a way another pression for the polarization-driven interaction face of the same effect: of interactions between the renormalization: fluctuations of electron densities in different parts of the system as omitted from the Hartree-Fock or 2 other mean field picture. An alternative might be 4␲ ϱ ␦ ͑ ͒ ϭϪ ͵ 2 ͑ ͒ ͑ ͒ developing estimates of the polarizabilities of the Fk dd ͫ dRR fk R Fk R ͬ . ͑ ϩ ͒␻ 2k 1 k b closed shell anions with no vacant orbitals on the basis of some analysis of interplay between the valence shell electrons considered explicitly and ϳ kϩ1 According to Ref. [6], Fk(R) 1/R , then as- ␣ Ϫ ␣ both the core and empty states not included in the ϳ k ␦ ϳ 2k 2 k suming fk (R) R we find that Fk(dd) b , calculation; however, this option will be postponed ␣ ϭ which agrees with Eq. (27) provided for all k k for the future. Ϫ1/2. This is precisely the decay exponent for the radial function of the wave in the spherical layer as given by a Bessel function with a half-integer index, which one can expect to be a model for the l-system. ACKNOWLEDGMENTS This article is dedicated to Professor Istvań Mayer on the occasion of his 65th birthday. While 4. Outlook and Discussion knowing that not numbers, but physical content concerns him most of all, the authors tried to make The derivation of the formulae for describing it to be interesting for him to read. nephelauxetic effect in TMCs in terms of the polarization propagator and reducing atom–atom polarizability for those forming the l-system of the TMCs naturally poses the question on the sources of these References quantities (polarizabilities). General expressions for it like used in Eq. (58) should not deceive the read- 1. (a) Jaurgensen, C. K. Prog Inorg Chem 1964, 4, 73; (b) Jaur- gensen, C. K. Adv Chem Phys 1963, 5, 33. er: in combination with semiempirical methods us- 2. Bethe, H. A. Ann Phys 1929, 3, 133. ing the valence basis only some (indefinite) part of 3. Ballhausen, C. J. Introduction to Ligand Field Theory; the total polarizability propagator can be repro- McGraw Hill: New York, 1962. duced. In the case of organic molecules where the 4. (a) Jaurgensen, C. K. Absorption Spectra and Chemical number of occupied and empty orbitals is some- Bonding in Complexes; Pergamon Press: Oxford, 1962; (b) what balanced it may be not that critical and ac- Jaurgensen, C. K. Modern Aspects of Ligand Field Theory; ceptable estimates of polarizabilities can be ob- North-Holland: Amsterdam, 1971. tained in semiempirical context (see Ref. [21]). The 5. Lever, A. B. P. Inorganic Electronic Spectroscopy; Elsevier: situation comes to a visible disaster with the stron- Amsterdam, 1984. Ϫ Ϫ gest renormalizers: late halogene anions (Br ,I )or 6. Bersuker, I. B. Electronic Structure and Properties of Coor- Ϫ Ϫ dination Compounds; Khimiya: Moscow, 1976 [in Russian]. halcogene dianions (Se2 ,Te2 ). When treated 7. Soudackov, A. V.; Tchougreéff, A. L.; Misurkin, I. A. Theor semiempirically these important ligands cannot Chim Acta 1992, 83, 389. have any polarizabilty at all—no vacant orbitals are 8. McWeeny, R. Methods of Molecular Quantum Mechanics, present. In this situation, one has to accept extend- 2nd ed.; Academic Press: London, 1992. ing the set of necessary semiempirical parameters 9. Lo¨wdin, P.-O. Perturbation Theory and Its Application in on account of polarizabilities. It should not be con- Quantum Mechanics; Wiley: New York, 1966.

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10. Soudackov, A. V.; Tchougreéff, A. L.; Misurkin, I. A. Int J 17. Tchougreéff, A. L. Hybrid Methods of Molecular Modeling; Quantum Chem 1996, 58, 161. Springer Verlag: London, 2008. 11. Soudackov, A. V.; Tchougreéff, A. L.; Misurkin, I. A. Russ J 18. Hammermesh, M. Group Theory and Its Applications to Phys Chem 1994, 68, 1256. Physical Problems; Addison Wesley, Reading: Massachu- 12. Soudackov, A. V.; Tchougreéff, A. L.; Misurkin, I. A. Russ J setts, 1964. Phys Chem 1994, 68, 1264. 19. Tchougreéff, A. L. Int J Quantum Chem 2007, 107, 2519. 13. Tokmachev, A. M.; Tchougreéff, A. L. Chem Phys Rep 1999, 20. Condon, E. U.; Shortley, G. H. Theory of Atomic Spectra: 18, 80. Cambridge University Press: Cambridge, 1951. 14. Morrison, C.; Mason, D. R.; Kikuchi, C. Phys Lett A 1967, 24, 21. Tchougreéff, A. L. DSc Thesis, Karpov Institute, Moscow, 607. Russia, 2004 [in Russian]. 15. Tchougreéff, A. L. Phys Chem Chem Phys 1999, 1, 1051. 22. Winget, P.; Selçcuki, C.; Horn, A. H. C.; Martin, B.; Clark, T. 16. Tchougreéff, A. L. J Struct Chem 2007, 48, S39 [in Russian]. Theor Chem Acc 2003, 110, 254.

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