The Concept of Rediagonalization Ligand-Field Applications within the Parametrical dq Model Mlchael Brorson, Gunnar S. Jensen, and Claus E. Schaffer H. C. 6rsted institute, University of Copenhagen, Universitetsparken 5. DK-2100 Copenhagen 6 , Denmark This paper has many aspects that are more general than the present ligand-field applications would seem to indicate. However, those who would like to use thrm in other contexts will understand their potential through the illustrations given here, which are based upon the parametrical dq model.' This model may describe the effects on a partially filled d shell of any combination of the following: ligand fields, interelectronic repulsion, spin-orbit coupling, and external fields. I t is common nractice to write the lieand-field ooerator as a sum of terms, each one corresponding to a particular symmetrv-determined vart of the total field. The interelectronic repulsion operator, which is of spherical symmetry, may for a d electron system be written as a sum of two terms, while the spin-orbit coupling operator only gives rise to a single term. The magnitudes of all these individual terms depend on the particular transition-metal complex examined. Generally, the jth term of the model Hamiltonian operator is written as a product (2,3) of an operator, &+,and an associated empirical parameter P,. The total Hamlltonian operator is the sum over all such terms (eq 1).
We call the expression 1aparametrical q-electron Hamiltonian. I t is not imnortant here to worrv about the form of the operators Qp. since in this paper they are always represented bv m a t r i ~ e s . ~ ~ ~ o w eitvis e rimnortant . that the ooerators are t i e same for all complexes of the same dq conf&ration. In contrast to the operators, the values of the parameters vary from one complex to the other and are determined empirically. While the parameters have the dimension of energy, their associated operators have the dimension of a pure number. A cubic dq complex (q z 1and 9) can serve as an example of the terms which may appear in eq 1.If spin-orbit coupling is neelected. the descriotion of this svstem involves three independent terms and ihus three paraketers. The parameters are conventionallv chosen as the lieand-field narameter A, measuring the eneigy difference between the ;,(oh) and tzg(Oh) orbitals, and the Racah interelectronic repulsion parameters, B and C. This choice of parameters results, according to eq 1,in unnormalized o p e r a t o ~ s . ~ Operatorsacting on a given (function) space may be represented by matrices. For a given operator, these matrices depend on the set of orthonormal basis functions chosen to span the space in question, i.e., the operator may be represented by an infinite number of matrices. All these matrices can be transformed into each other by basis shifts, or more precisely, by means of unitary transformation^.^ Given a set of orthonormal basis functions, one can write the matrix eq 2 corresponding to the operator eq 1.
The empirical values of the parameters P, are those from which one calculates the experimentally observed differences between the higher energy levels and the ground level. The energies of all the levels are the eigenvalues of the energy matrix H.
Provided the dq functions form an orthonormal basis, spanning the total dq space, they may, as far as the eigenvalues are concerned, be chosen in any way desired. For a deeper understanding of the physical situation within the model, one may, however, he interested in using special sets of basis functions which have well-defined properties. Strong-field and weak-field functions constitute examples of such sets of functions (see later section on this topic). The nresent naoer contains an analvsis and several oractical examples to iilustrite the concept of rediagonaliktion, which loosely speaking means shift of basis. This concept is not new, but we claim that our approach contains major methodological, practical, and also pedagogical novelties. The Rediagonalization Process A matrix representation of an operator generally contains both diagonal and nondiagonal elements.'l'he eiprnuoluea of such a matrix may be found by a diagonali&tion, i.e., a unitary transformation which brings the matrix to a diagonal form. The diagonal elements of this matrix are the eigenvalues and its basis functions are the eigenfunctions of the ooerator. The eigenfunctions of the Hamiltonian operator of a system are esoeciallv"imnortant because all observable orooer. ties of thesystem's ground state and excited statedcan be derived from these functions. If. for exam~le.one wants to know the orbital angular momentum L of s'ome stace, L t L + I1 is calculated as the value of the matrix element of 1.2 for this state. This value is called an expectation value of the for the Hamiltonian operator eigenstates. Inoperator stead of calculating each such expeetation val;e separate1 one can transform any arbitrary matrix representation of to the eigenhasis of the Hamiltonian. The diagonal elements of this transformed matrix are then the expectation values of for the eigenstates of .?i Such a matrix of will only he diagonal for a free atom which also provides the natural 2S+'L bases in which to set un amatrixof e2in the first nlace. For metal complexes eigenf;nctions of the ligand-field-model Hamiltonian are usuallv not simultaneouslv eieenfunctions of e 2 , and the matrix is therefore not diagonal. The particular importance of the diagonal elements as expectation values should be noted. The above-exem~lifiedtransformations to the eigenbasis of the ~amiltonian-operatorof a system are classica within quantum mechanics. We now want to introduce a generaliza$on by considering transformations to the eig&bases of
e2
8
e2
e2
The parametrical dq model is identical to the expanded radial function model ( I ) . It is usually referred to simply as ligand-field theory. In previous papers (2.3)we used the one-electron operator of one-electron coordinates, normalized to unity over the d space. Here we are concerned with operators acting on q-electron functions, and these may be one-electron or twoelectron operators and as such may be normalized within the full dq space or a suitably chosen subspace thereof. We shall illustrate this in forthcoming papers. In the present paper we only need to consider orthogonal transformations, i.e.. the formationof reallinear combinations of the basis functions. The more general unitary transformations operate with complex linear combinations of basis functions. Volume 83 Number 5 May 1986
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hermitian operators in general. Such transformations will be called rediagonali~atiom.~ Assume that we have a space and two hermitian operators A andB actingon this space. Assume further that we have an arbitrarily chosen basis for this space giving the matrices A and B which in general both will be nondiagonal. If A is diaeonalized to nroduce the diaeonal matrix A' bv the unitar; transformaiion of eq 3, we &all say that the mitrix Brof ea 4 is the matrix B rediaeonalized with respect to the operator A.
-
A' = UAU-'
(3)
In eqs 3 and 4 U is the unitary matrix containing the eigenvectors of A in its rows. B' will, in general, not be diagonal. A special type of rediagonalization occurs when the matrix B of eq 4 is a parametrical energy matrix like that of eq 2. As it can be seen from eq 5, this is a transformation of each of the matrices associated with the individual parameters.
All the applications we shall consider in the present paper involve the rediagonalization of parametrical energy matrices with respect to certain model Hamiltonian operators. Such Hamiltonians are specified by the values of their em~ i r i c a parameters, l and their eieenfunctions are found by hiagoni~iein~ the total matrix that arises when these eter values are inserted into eq 2. These eigenfunctions may then be used to transform the parametricaienergy matrix as shown in eq 5. The matrices Q'p, produced by this rediagonalization will in general not be diagonal. However, their weighted sum is, of course, diagonal if the weights are the associated parameter values by which the Hamiltonian operator is specified. Insertion of parameter values belonging to a different Hamiltonian into the expression for H' will not, on the other hand, produce a diagonal matrix. For very similar Hamiltonians (corresponding to closely related chemical systems) the nondiagonal elements may, however, be quite small. The Hamiltonians with respect to which a rediagonalization is performed need not belong to a real chemical system. For example, using a Hamiltonian where some of the terms in the sum of eq 1have been omitted can produce matrices in conceptually interesting bases. In the following, we shall illustrate this by producing cubic strong-field and weakfield matrices from each other. Certain symmetry-defined intermediate-field matrices will also be produced by rediagonalizations. The Cubic Strong-Field and Weak-Field Schemes Twosetsof hasis functions tor twoschemesJ have recrived s ~ e c i aattention l in the treatment of octahedral com~lexes: the cubic strong-field scheme and the cubic weak-field scheme. The strone-field functions are eieenfunctions of the cubic ligand field but usually not of theinterelectronic repulsion. When the ligand field is large relative to the interelectronic repulsion,the strong-field functions are close to being eigenfunctions of the total Hamiltonian. The weakfield functions, on the other hand, are eigenfunctions of the interelectronic repulsion5 but only incidentally of the ligand field. The ~ h v s i c asituation l encountered in actual comnlexes is always intermediate between the strong-field and weak-field situations. We then talk about intermediatefield schemes." The so-called strong-field approximation has been, and is A
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Journal of Chemical Education
still. widelv used. Here one avoids a diaeonalization of the eneigy matrix by approximating the eigenvalues with the diagonal elements of the strong-field matrix. Such an approximation is never very good and is not to be recommended. The weak-field apwoximation where the diagonal elements of the weak-fieid matrix are used as an appr&imation t o the eigenvalues is even worse. The application of the strong-field and weak-field approximations to the same complex gives, in general, quite different results. ~ k v e v e rprovided , that the lull energy matrices are used, there is uvthrn th~parnmrrrrcaldq modd no physical difterence hrtwren thr strune-field and weak-field schemes 181. In fact, the strong-field a i d weak-field matrices are interrelatand one can he obtained ed by a unitary transformation (9), from the other by a rediagonalization process. We illustrate this bv the well-known (10)strone-field energy matrix for the spin-quartet terms of the d3 configuration in cubic svmmetrv. This matrix is given in ex~ression6 where the parameters A and b represent the cubic ligand field and the interelectronic repulsion, respectively. The parameter b is equal to (3/2)B, whereB is the Raeah parameter, and we shall soon see that the zero point of energy for the repulsion coincides with the 4F term of the d3 configuration and that the repulsion energy of the 4Pterm then becomes lob.
All the off-diagunal zeros in this matrix arise from symmeir\.. 'l'hr energies of the IA9 and 'T1terms are linear funrtiins of A a n i b; the energies of thd 4T1 terms do not have this simple property and must be found by diagonalization of the 2 X 2 submatrix in the upper left corner of the matrix. The spectrochemical parameter A is absent in the nondiagonal elements of 6 hecause the strong-field basis functions are eieenfunctions of the ligand-field operator Qn. This prope;tv has the consequence that each strong-field basis function can be associated with a cuhic subconfiguration. For example, the lowest strongfield 'TI term has two elec-
'We are only aware of two cases (4,5) where this word has been used before. in none of these cases was a precise definition given. Our present use is not in disagreement with the previous uses. We find the word rediagonalization very apt because it focusses attention upon the close connection which exists between a basis and the diaoonal of its matrices of different ooerators whose exuectation values sit in this o~agonal.What we do inour transformationihen is to prodxe a new representation of The diagonai elements of the operator. Therefore, we like to say that we rediagonaiize. Actually, a more general definition of a weak-fieldscheme would beone requiring the basis to bea basis for irreducible representations of the three-dimensional rotation group R, (or, with spin-orbit coupling, of SL42)). For our present purpose this distinction is not so important though. When repeated terms occur within a do configuration, however, thedistinction is necessary. Forthe twoZDtermsofthe d3 configuration, any linear combination makes up a weak-field set. One such set can be specified by symmetry labels called seniority numbers. This set is, however, not diagonal in the interelectronic The set that repulsion (expressedby the Racah parameters Band 0. is diagonai in the repulsion, on the other hand, cannot be specifiedby symmetry alone and depends on the ratio between the values of the empirical parameters B and C. The two kinds of basis functions mentioned will of course give different weak-field matrices. "Kng and Kremer (6) use the term "intermediate field in a. slightly different way, namely to specify a situation where the ligand field is smaller than the interelectronic repulsion but larger than the spin-orbit coupling. We do not here consider spin-orbit coupling effects, and the present usageof the term is in accordance with ref 7. ~
-~~
~
~
~
~~~
trans in the d orbital subset t2and one in the subset e, and so belongs to the subconfiguration t22e. The weak-field basis functions are eigenfunctions of the interelectronic repulsion operator &, and the weak-field matrix therefore does not contain matrix elements expressed in terms of b off the diagonal. This matrix can be produced by a rediagonalization of matrix 6 with respect to &, i.e., a transformation which brings b into the diagonal. The two 1X 1suhmatrices are, of course, already diagonal in b, and for these there is, therefore, no distinction between the strong-field and weak-field forms. Consequently, we are only concerned with the 2 X 2 4T1submatrix in the following rediagonalization process. First, we set A equal to zero and obtain the repulsion matrix (eq 7).
'TI he2 k2e (7)
Diagonalization of the matrix in eq 7 produces the eigenvalues (diagonal elements of the diagonalized matrix) given in the matrix Rand t h r rigenvectorsgiven ineq 9. Both of these sets of quantities refer to the weak-field situation. The components (coefficients)of the eigenvectors express the weakfield functions in terms of strong-field functions.
A comparison of the matrices 7 and 8 reveals some of the fundamental properties of unitary transformations. First we notice the well-known trace invariance. Both eqs 7 and 8 have a trace of lob. Another invariant is the sum of the squared matrix elements. The matrix 7 here gives a square = 100b2,the same as the square sum of (2b)2 2(4b)2 sum of the elements of the matrix 8. Notice that, according to eq 5, these two invariances will apply independently for each parameter matrix Qp,. Theeigenvectors of eqs 9 are now, as the second step in the rediagonalization process, used to transform the 2 X 2 submatrix of the matrix 6 into the weak-field basis. The result for the repulsion part is, of course, already given in expression 8 from which the notation used on the left-hand side of eqs 9 is also understood by noting that the lower energy gaseous term is 4F.The unitary transformation of expression 10, where U is given as an overlap matrix in eq 11, also converts the ligand-field part of the energy matrix into the weak-field scheme.
+
+
The total cubic weak-field matrix for the quartets of d3 is finally given in expression 12.
tion to their cuhic term symbols, specified hy a spherical symbol (P or F),while the strong-field functions are specified by the cuhic subconfigurations. I t is seen that in a spherical situation, where A is vanishing, the three cubic terms associated with 4F are degenerate. The energy of 4F (Ob) is lower than that of 4P (lob) in accordance with Hund's
.-.-.
ml.
If we had considered the weak-field matrix of expression 12 to be the known matrix in the first place, we would have generated the strong-field matrix by rediagonalizing the weak-field matrix with respect to @a. This would have led to the inverse unitary matrix to that of eq 11. In the matrix U.7given in eq 11, the squares of the individual elements can be given a simple symmetry interpretation either by looking a t the rows or the columns. Looking a t the rows provides an analysis of the weak-field 4T1 terms expressed in trrms of thestrong-fielddT, terms, and one reads directly, for example, that the 1' term has 20% ~ ~ e ~ c h a r a r t e r (and 80% tz2e chiracter). Looking a t the columns, one ohtains an analysis of the strong-field 4T1terms in terms of the weak-field 4T1 terms, and bne observes that the higher strong-field term 4Tl(t2e2) has 80% of 4F and 20% of 4P character, while the opposite distribution of percentages is valid for the lower term 4Tl(t22e).In other words, the following statement can be made: The higher4T1strong-field term predominantly consists of the lower spherical term (4F) and the lower 4T1 term predominantly of the higher spherical term (4P). This is not completely trivial since in inorganic textbooks the term containing (in the squares of the function) up to 80% F character is labelled 4T1(P) and that containingup to 80%P character4Tl(F). Such anotation can only he made meaningful if (P) and (F) are reduced to mean "higher" and "lower" in energy. This is, however, conceptually confusing since the causeof the higher and lower energies for the strong-field functions is the ligand field rather than the interelectronic repulsion, which, of course, is responsible for the fact that 4P has higher energy than 4F. For the d2 confieuration this discussion is different (21. Here.~,for example, thk higher energy strong-field term 3 ~ ~ t z cone) sists of 80% 3P and 20% 3F, and the notation 3T1(P) is not unreasonable provided it is explained properly. The 80120 ratio just discussed on the basis of an analysisof the eigenvectors can also he seen by looking at the energy matrices themselves, notahlv a t the diagonal elements of the strong-field and weak-field matrices. ~ k c the e matrices are diagonal in A and b, respectively, one has to focus upon the ligaid field when interpreting a given set of basis functions in terms of the strong-field scheme and upon the repulsion when interpreting them in terms of the weak-field scheme. Let us begin by analyzing the weak-field matrix of expression 12. Here one observes. as antici~ated.that 4T.(PI has ) the repulsion energy lob while'^^(^ f, T ~ ( F ) ,a n d 4 ~ 2 ( kall have the repulsion energies Oh. The fact that all three cuhic terms have the same repulsion energy is, of course, a consequence of the fact that thev are all Dure 4F terms in the weak-field scheme. The cir&nstance that their repulsion energies are Ob onlv means that the repulsion enerav ... of 4F has twen chospn as the zero point for the repulsion operator. The coefficient to A is zero fur 'l'l(P) because 4 P onlv contains that one cuhic term. heref fore, the center of gravity rule (harycenter rule) requires the "splitting" to he zero. This zero can also be obtained as 20% of (415)A plus 80% of -(115)A. Similarly, one observes that the weighted sum of the ligand-field energies of the seven 4F components is vanishing as shown in eq 13. The coefficient 315 can also be directly obtained as 80% of
(12)
As illustrated in eqs 9, the weak-field functions are, in addi-
'This matrix for the d3 confiouration is related to that of the d2 conffgurat8on through the van ~ l & krelation ( 11) The matrlx forthed2 conlfgurationis a sc-called Racah-lemma mahlx (2)
Volume 63 Number 5 May 1986
389
415 plus 20% of -115. We then analyze the strong-field scheme (the matrix of which is given in expression 6) in terms of the weak-field scheme. Remembering that the repulsion energies of 4P and 4F are lob and Ob, respectively, and observing that the repulsion energies of strong-field terms 4Tl(t2e2)and 4Tl(tZ2e)are 26 and 8b, respectively, one sees that the term 4Tl(t2e2)contains 20% 4P (and 80% "F) while the term 4Tl(t22e) contains 80% 4P (and 20% 4F). We finally note that the diagonal elements represent the tangents (or asymptotes) to the curves in Tanabe-Sugano diagrams (12). In these diagrams the energy eigenvalues ( E ) in units of the renulsion narameter are olotted versus A in units of the same repulsion parameter. For example, using the first diaeonal element of matrix 6. the expression aiven in eq 14 is tLe equation of the upper &ong-field 4T1 asimptote (see figure).
+
Elb = (4/5)(Alb) 2
(14)
In this section. we have illustrated the cubic strona-field and weak-field schemes and their interrelation by considering the soin-auartets of the d:'confiruratiun. These schemes are two bf in'finitely many. They represent extreme situations, which are easy to define by symmetry, but which do
not correspond to experimental situations. In the next section, we introduce intermediate-field schemes by discussing two such symmetry-defined schemes, again by way of a d3 spin-quartet example. Intermedlate-Field Schemes
The results of the previous section may be summarized for the two 'TI termsas follows. For Alb = 0 the uoner and lower 4T1 terms i r e 4P and 4F, respectively. The &per term can also be analyzed as containing 80% of the subconfiguration tZ2e(and 20% of t2e2)and the lower term contains what is left over of the subconfigurations. As Alb now increases from zero, the t22e-contribution in the upper 4T1 term quickly decreases from 80% and a t some point both 4T1 terms contain 50% of each of the suhconfigurations. I t is this intermediate-field situation we shall first examine here. Let us for a moment consider the general matrix 15.
+
v) and (.x - ". v) and eigenvectors i j 2 / ~ ) ( $ ~ $2; and (\iz12)(~1*~ *z), respectivelv. An eoual mixine of the basis functtons is thus obtained when the two diagonal elements are equal. We shall now apply this general property to the 4T1block of the strong-field matrix 6 to obtain the parametrical conditions under which the cubic subconfigurations aremixed in a 50150 manner. Requiring that the diagonal elements should he equal gives eq 16.
A matrix of this tvDe has the eieenvalues ( x
-
+
+
(415)A 26 = - ( 1 / 5 ) A
+ 8b
-
Alb = 6
(16)
Notice that the desired basis is specified completelv by the rotlo between the twoparametero.%andb.Thi; is used in the Tannbe-Sucano diaerams where the abscissa is 1\ divided by the repulsi& According to eq 16 a basis corresponding t o an equal mixing of the two subconfigurations is an eigenbasis when Alb = 6. It is thus possible to obtain this particular intermediatefield matrix by rediagonalizing any 4T1 matrix with respect to 6Qn + &. Choosing the strong-field matrix for this rediagonalization, we obtain the eigenvector matrix V of expression 17.
@% t,e2 + @% t,2e \/50% t2e2- @F% k2e
Tsnabe-Suganc-like diagram for the spin-quartet terms of the d3 configuration. This graph is a TanabsSugano diagram except for the fact that the ground term graph4A2 does not play the particular role of defining the zero-point for the other graphs, Instead, ail the graphs are on an equal footing and represent theeioenvaiuesof anv one of the tour matrices6. 12. 16. or 21.Thetwo terms ' A , anddT,da not interact wnthother terms and are. therefore, represented oy Straight ihnes inthe d agram The slopesand lnterceptsof tnese lhnes are g.#en as the coelfic ents to 3 and b ,n the re want d agonal elements of the malroces 6 or 12.The two4T,terms, aand b, on the other hand, interact and their graphs thereforeappear curved. In this case the coefficients to Aand b in thediagonal matrix elements define the tangents or asymptotes of the graphs. These lines areshown for ail the four schemesdiscusssd in the main text.The strong-field asymptotes obtained from 6 are labeiied *.a. and s.b. while the weak-field tanoents obtained fmm 12 are labeiied w.a. and w.b. The two intermediatefoe d scnemes are labellea 1 and 12 where it corresponds to Alb = 6 ,the mat, x 1Blano 210 X o = 5013 (the mat, x 2 1 ) h o l m tor 11that tne slopes of tnerwoAT,tangentsaresquai. wnereasfor 2 1 isthe lnlsrceplsthat areequal
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Journal of Chemical Educatlon
t2e2
t,2e (17)
Bv lookine at this eieenvector matrix. one can immediatelv re"cognize -the definiEion of our interhediate-field scheme. The matrix could of course have been constructed directly without first finding the Alb ratio and diagonalizing the strona-field matrix. However, if we had chosen t o rediaeonalize any other matrix than the strong-field matrix, V could not have been found without a knowledge of Alb. Transformation of the strong-field matrix S with V of expression 17 according to VSV-I gives the intermediatefield matrix 18.
The nmdiagunal elementsof the matrix 18vanishasexpected for 4 = 6b. The coefficientsto A in the diagonal elements are both equal to 3110. This is a consequence of the equal mixing of the subconfigurations. The number 3/10 may be obtained from the coefficients of A in the strong-field matrix
as 50%of 415 (corresponding to t2e2) plus 50%of -115 (corresponding to tz2e). The coefficients 3/10 can be further interpreted as the slopes of the tangents to the 4T1 graphs in the TanabeSugano diagram for Alb = 6. These tangents are thus parallel for Alb = 6. The iuterceots of the tanaents are eiven as the coefficients t o b in t h e diagonal, i.e, as 9 a d 1. These tangents are shown in the figure. The vertical distance between the two 4T1 graphs in the Tanahe-Sugano diagram has a minimum for a certain value of Alb. This value may be found from eq 19 where E(q4Tl)lb is the ordinate of the q4Tl graph (q = a for the lower 4T1 graph and q = b for the higher).
From eq 19 i t can he concluded that an extremum is found when the slopes of the two 4T1 graphs are equal. Inspecting the figure shows that i t is a minimum distance that is found for Alb = 6. From the eigenvalues of the matrix 15 we know that this closest distance is equal to twice the nondiagonal element of the strong-field matrix 6. For the d2configuration the whole discussion is different. Here no intermediate-field situation corresponds to a minimum distance between the two 3Tl terms. There are no octahedral d3 complexes whose values of Alb are close to 6, but the tetrahalido complexes of cobalt(I1) are much better represented by the basis functions of the matrix 18 than by the strong-field or weak-field functions.8 The intermediate-field matrix just discussed was derived and svmmetrv-defined from the strone-field scheme. There ~~~~~~~-~ - - ~ - exists a quite analogous development based upon the weakfield scheme. Here the intermediate-field situation is defined by a 50% mixing of 'P and 4F, i.e., the eigenvectors relative to the weak-field matrix W are given by V of expression 20. P F (20) ~
~
~
~~
"I
One obtains for VWV-' the matrix (21).
The matrix corresponding to the basis functions 22 is given in expression 23. q
$1
1
4%
-0.320A+ 4.994b
61
1
This matrix is, of course, diagonal for the parameter values A = 17.4 kK and b = 1.114 kK (1 kK = 1000 em-') corresponding to [Cr(H2O)6I3+.Inserting these parameter values into the matrix 21, which is numerically very similar to the matrix 23, produces the almost diagonalmatrix 24 whose eigenualues, 17.76 and 3.82, barely deviate from its diagonal elements, 17.75 and 3.83. The matrix 23 is a third example of an intermediate field matrix. 17.75 0.35 0.35 3.831 (24)
[
We repeat that the strong-field, weak-field, and three intermediate-field matrices all represent equivalent ways of describing a d3 system in octahedral symmetry. The question now arises: What is the use of having several different energy matrices when they all contain the same information? The answer to this question is the following: If we have a scheme, i.e., a particular choice of hasis functions for a matrix, and this matrixis diagonal for the empirical parameter values of a particular complex, then all the energies, as wellas the energy differences, have the linear form of expression 25. ~
SA
+ Bb
(25)
This means that the partial derivatives of the energies (or energy differences) with respect t o A and b a t the particular point in parameter space are equal to 6 and 0, respectively. This observation is of utmost importance for the analysis of the properties of the eigenstates and transitions of ligandfield systems and will be extensively used in forthcoming papers. Here we only want to mention that such partial derivatives can be made directly comparable by suitable renormalization of the individual parameter mat rice^.^ Conclusion
One observes that the diagonal elements of W (the matrix 12) are equal for A = (5013)b, and, for the same value of Alb, the nondiagonal elements of the intermediate-field matrix 21 vanish. The tangents to the4T1graphs here have different slopes but their intercepts are equal, in agreement with the fact that the mixed terms in this case both have 50% P and 50%F character (see the figure). We notice that the distance between the two curves for the abscissa Alb = 5013 is twice the nondiagonal element of the weak-field matrix 12 whereas the distance between the curves at Alb = 6 is twice that of the strong-field matrix 6. I t is noteworthv that the basis functions of the intermediate-field matrix 21 are very similar to the eigenfunctions 22 of the hexaaquachromium(111) ion. $,=\1524P+\147.5%4F $2=\147.5%4~j52.5%%
In the present paper a few examples of the utility of rediagonalizatious have been given. However, there are numerous other awwlications of both concewtual and nractical charac.. ter. T o mention just a few of them, the effects of low-symmetry distortions, spin-orbit coupling, and external fields on the individual eigenstates of a given system can be examined by means of rediagonalizations. Rediagonalizations of parametrical energy matrices with respect to Hamiltonians based on guessed values of parameters are useful when fitting parameters to experimental spectroscopic data. In a forthcoming paper, we will describe the foundations of a computer program which can generate and symmetry-adapt ligand-field matrices. The symmetry adaptation there is based on consecutive rediagonalizations. Literature Clted (11~ $ r ~ ~C.n K~ ~ei s~e u, s s~orodwsoe. . 19~8.26.110. 121 sch&ffer,C. E ~ h 1982, ~ J J ~ A2s. ~, i ~ ~ 131 Brorron, M.: Darnhus, T.: Sehiffer, C. E. Comrnenlslnorg. Chom. 1983.3.1. 141 Jbrgensen, C. K. '"EnemLevels of Complexes and Gaseous Ions'' (Thesis); Jul.
Gjellerups Forlag: Copenhagen, 1957: p 17.
(51 Sehaffer, C. E. Pror. Roy. Soe.,London
(22)
It is well known that if A for a tetrahedral complex is defined as the energy of the b(Td orbitals minus that of the e(T.)orbitals,then the energy matrices of a d7system in Td symmetry are identical to those of a d3 system in 0,symmetry.
1967.A297.96.
(61 Kbnig, E.; Kremer. S. "Liand Field Energy Diagrams": Plenum: New York. 1977: p 22. (71 Sehaffer, C. E. In "Wave Meehani-Tho First Fifty Years"; Price, W. C.; Chissick, S. S.; Revenadale, T., Eda.; Butteworths: Landon. 1973,Chap 12. (81 Hartmann, H. J . horg. Nuei. Chom. 1958.8.64. (91 Schaffer, C.E. ' ' ~ ~ t s v i l i d ~ ( l~~~~ ~~ ri ~ rof8 'e' ~d i ~ ~~ ~ held ~ in vgl.5da~ l 18". Sluedon):Quantum Chamisby Group: Uppsda University. 1958: p 64. Tanah*, Y.;Sugano, S. J. Phys. Sor.Jpn. 1954,9,753. (111 Schiffer. C.E.;~ bC.K. MU. ~ FB.~ ~ ~ dD~". d .vid. ~ sslsk.~196.5, ~ 1~3 1 . ~ (121 T S ~ SY ~. ; ~s u, ~ ~ ~J~. ,Ps~. ~ ~ . sJP". o c . 1954.s.766.
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