Jan., 1960
ELECTROSTATIC AND ELECTROMAGNETIC FORCE IN LIGAND FIELDTHEORY
the sulfur atoin by two methylene groups. I n thiophenol, however, the sulfhydro group is linked directly to the phenyl group which exerts a very strong -I effect. The decrease of 4.0 kcal. for the reaction on going from thiophenol to P-mercaptopropionic acid is in harmony with this large difference in the relative electron withdrawing tendencies of the phenyl group and the carboxyethyl group. Another argument which might be adduced for the choice by the malonic acid of the electrons on the sulfur atom rather than those on the oxygen atom is the fact that monocarboxylic acids are stronger acids, and therefore weaker bases, than thiols. The electrophilic agent mill tend to
43
combine with the stronger nucleophilic agent-in this case the sulfur atom. H-bonding and therefore association cannot take place between molecules of thiophenol, whereas in the monocarboxylic acids H-bonding leads t o dimerization. This increase in complexity of the molecule, leading to increased steric hindrance, is revealed by the decrease in AS* of nearly 9 e.u. on going from thiophenol t 3 P-mercaptopropionic acid. Further investigation of this reaction is contemplated. Acknowledgments.-This research was supported in part by the National Science Foundation, Washington, D. C.
RECIPROC.ATION OF ELECTROSTATIC AND ELECTROMAGNETIC FORCES I N LIGAND FIELD THEORY'p2 BY ANDREWD. LIEHR Bell Telephone Laboratories, Incorporated, Murray Hill,New Jersey Received J u n e 9, 1969
If ionic spin-orbital forces are weak, electronic motions in inorganic complexes are governed primarily by the electrostatic coercions of the surrounding ligands, whilst the presence of feeble coulombic directives, but robust spin-orbit correlations, dictates electronic trajectories which are only slightly modified over those characteristic of the free ion. Electronic itineraries of both types have been exhaustively discussed, in the past thirty years, within the framework of the BetheKramers-Van Vleck theory of crystalline fields. However, the rather more esoteric situation in which the spin-orbital and addend field potentials are of comparable magnitude does not seem to have received as thorough a consideration. I n this re ort an account will be given of the optical and magnetic properties expected of d", ( n = 1,9), molecular systems, in severafgeometries, which exhibit equi-energetic spin-orbital and augend field interactions, and will be applied, &There observational data is extant, to the transition metal complexes of the second and third group. An experimental prospectus s outlined in the hope that such will stimulate future research into this somewhat neglected domain.
Introduction current, as well as past, applications of the crystalThe renascent exploitation of the Bethe-Kra- line field theory have dealt with either of two mers-Van Vleck theory of ligand (crystalline) limiting behaviors: (1) the electronic motions are fields in spectro- and magneto-chemical investiga- governed predominantly by the electrostatic (couelectrotions of inorganic systems has led, of late, to an lombic) forces of the ligand field-the enhanced appreciation of the puissance of sym- magnetic subjection of spin and orbital magnetic metry considerations in the resolution of complex moments (spin-orbit coupling) acts solely as a electronic problem^.^ Indeed, the explicatory suc- minor irritant to these movements; (2) the eleccess of this theory may be traced directly to its tronic spatial excursions are preponderantly guided maximal utilization of molecular r e g ~ l a r i t y . ~It by spin and orbital magnetic coercions-the has not, however, been widely appreciated that coulombic constraints serve only as a small directive such geometrical regularity also imposes quite inducement on their regional jaunts. For interstringent correl.ations upon the orbital and spin pretive investigations of the spectral and magnetic segments of the electronic charge density distribu- properties of the iron group (the 3dn elements), tion.6J This {circumstance has prevailed since approximation (1) is normally sufficient (note through reference 5); while for the lanthanide (1) The Editors after careful consideration have decided t o accept (4fn) and actinide ( 5 f n ) groupings, description ( 2 ) this manuscript in itti present form. The scientific work is not quesis usually completely adequate (but see ref. 7 and tioned seriously b y the referees. T h e style of writing differs from t h a t customarily employeil in scientific articles. T o the extent t h a t this 8). style makes difficult the understanding of much of this article the Yet for similar researches on the palladium author disagrees. On the other hand i t seems best to publish this (4d") and platinum (5d") group metals, these article as desired by the author.-Ed. (2) Presented a t the Symposium on Molecular Structure and viewpoints are woefully I n these Spectroscopy, Columbus, Ohio, June 15-19, 1959. (3) Recent reviews of this theory have been given by the following authors: (a) W. E. Moffitt and C. J. Ballhausen, Ann. Reu.*Phys. Chem., 7 , 107 (1956) (b) J. 9. Griffith a n d L. E. Orgel, Quart. Rev., 11, 381 (1957); (0) M. H. L. Pryce, Nuouo Cimenlo, S u p p l . 3 [lo], 6, 817 (1957); (d) H. Hartmann, Z . Eleklrochem., 61, 908 (1957): (e) W. A. Runciman, Repta. Pros. Phys., 21, 30 (1958). (4) J. H. P a n Vleck, J . Chem.. Phys., 3, 803, 807 (1935). ( 5 ) A. D. Liehr and C. J. Ballhausen, Ann. P h y s . [ N . Y . ] ,6 , 134 (1959). (6) C. J. Ballhausen and A . D. Liehr, Mol. P h y s . , 2, 123 (1959).
(7) G. L. Goodman, ibzd., in press. (8) G. L. Goodman and M . Fred, J . Chem. Phys., SO, 849 (1959). (9) The secular determinants given in ref. 5, which are common t o all the kdn, (k = 3, 4, 5), configurations illustrate this point quite clearly, as the syatems kd", (k = 4, 5), are characterized by the stipulation A = Dq (n.b., ref. 10 and 11 in this regard). (10) W. E. hloffitt, G. L. Goodman, hI. Fred and B. Weinstock Mol. P h w . , 2, 109 (1959). (11) G. L. Goodman, J. Chem. Phys., in press: Doctoral dissertation, Harvard, 1959.
ANDREWD. LIEHR
44
latter systems the electrostatic and electromagnetic forces present continuously reciprocate in their prescriptive action upon the itinerant electrons: the electrostatic coercions are dominant in the vicinity of the ligands, and the electromagnetic compulsions paramount in the neighborhood of the metal ion nucleus. The resultant circuit traversed by the negative particles is thus compromissary in nature. The specific compromise struck rests, of course, upon the relative importance of these two rival propensities. I n this article we shall derive “exact” energy expressions12 for a single kd, (k = 3, 4,5), electron or p o ~ i t r o n ’driven, ~ alternately, by equally robust electrostatic and electromagnetic forces. We shall also formulate a crude, but instructive, approximational program which should be useful in the delineation of configurations kd“, when n is larger than unity. Some indicative applications will be discussed and especially close attention will be focused upon the characterization of the recently recorded hexafluorides of tungsten and rnolybden~m.’~ Theory Equations of Motion.-The possible dynamical trajectories which an electron may pursue are selected, according to Schrodinger, from amongst the well-behaved solutions of the differential (eigenvalue) equation
T‘ol. 64
+
. - L +
1.l6 The numerical coefficients which relate the (s,Z,ms,m~)basis to that of {s,Z,j,m] are as given in Table I3of Condon and Shortley.16 Hence, as j is here permitted but two values, 5/2 and 3/2, we may write the free ion functions as
s
Now Bethe” has shown that a reduction in eurythmy from that characteristic of the sphere requires additional specifications of the wave functions q. It is relatively easy to visualize how this happens. For globular symmetry, an angular rotation a about the z-axis multiplies the state vector by a phase factor eima. As the magnitude of CY may be chosen arbitrarily, the halfinteger m serves to uniquely differentiate transformational properties, and hence to isolate the functions themselves,l8 for each given energy state. However, upon descent to cubic eurythmy, only those values of a which are multiples of ?r/2 may be selected, and it is thus impossible to segregate X q = E* (1) states whose m values differ by an adjunctive where multiple of four. For example, the functions 15/2,5/2> and j5/2, - 3/2> do not portray dis++ tinct states in fields of cubic (or lower) symt ( p ) 1 . s f u(r) VC.F. ( r , 81 9) (2) metry.19 I n crystalline fields, VC.F., of eurythmy lower The initial quaternary of terms in the Hamiltonian, X represents the orthodox hydrogenic Hamiltonian than spherical, an electronic state is specified by operator, with relativity c~rrection’~;the remain- (1) the value of m modulo the order of the axis of ing expression, VC.F., particularizes the non- maximum rotational symmetry2” and (2) the sphericity of the attendant potential field of the number of orthogonal states to which it is symencompassing ligands. At present there exist metrically congruent (the degeneracy factor). two analytical modes which may be employed to Hence if the potential VC.F. is of cubic constitution, the wave functions defined in equation 1 determine the permissible electronic treks-the weak and strong field formalisms. It is the former must be recatagorized into three new subclasses itemized by the quantum numbers ( y , s, I, j , m ) , that will be pondered first. Weak Field Portrait-Wave Functions.-In the where y , is the symmetry specification, and takes event the ligand (crystalline) field tic.^. vanishes, on the species cognomina 76, y7 and y8.17*20,21 the solutions of equation 1 are expressible in the The analogous generic symbols, yk, for quadrate form l5 (Y = Q) and trigonal (Y = T) potentials are (yS,y?) and (,-yT,yx,y;f), respectively.17~2”J1 Ij,m> = ~,ml> ‘s,m,> (3) The application of suitable coordinate transm l . me where lZ,mz> and Is,ms> are the usual orbital (16) The choice of phase which corresponds to the canonical - + + + (Condon and S h ~ r t l e y phases) ’~~ and spin wave ordering or angular momenta. j = s + 1, is discussed on pages 123 functions, respectively, and Ij,m> is the requisite and 270,ref. 13c.
+
--r
state function with total angular momentum j
=
(12) B y “exact” we b u t wish to imply t h a t no simplifications other than the inherent basal suppositions of ligand field theory have been invoked.
(13) It may be shown t h a t the collective motions of nine kd, (k = 3,4,5),electrons is dynamically equivalent to t h a t of a single positron. An extended discussion of this theorem may be found in (a) J. H. Van Vleck, Phus. Rev., 41, 208 (1932); (b) G. J. Kynch. Trans. Paraday Soc., 83, 1402 (1937); (0) E. U. Condon a n d G . H. Shortley, “The Theory of Atomic Spectra,” Cambridge Univ. Press, London a n d New York, 1953,Chapter X I . (14) (a) G.B. Hargreaves and R. D. Peacock, J . Chem. S o c . , 4212 (1957): (b) 3776 (1958). (15) Ref. 13c, Chapter V.
(17) H. A. Bethe, Ann. Physik, (51 3, 133 (1929). (18) As such symmetry operations leave the Hamiltonian invariant, an eigenstate is individualized by both its energy and transfigurational behavior. (19) This follows as the interaction matrix element
= (-1)i+m-1
ij, -m>, ( j = 0, ‘/z, 1,
. . .).
( 2 5 ) By definition, the tetragonal group is t h a t particular subgroup of the cubic group which is generated by the operators e r ( z ) , U ( Z - z ) , and i (inversion). Hence, the above constituted cubic functional vectors (footnotes 23 and 24) are also quadrate basis vectors. ( 2 6 ) We shall use a capital gamma to designate over-all symmetry specification and a lower case gamma t o denote a single electron symmetry state. (27) The atomic wave functions lj. m> h a r e been azimuthally quantized about the (1, 1, 1) direction (thed-axis), rather than around the (0,0, 1) position (the z-axis), as before (ref. 2 2 ) . Hence, the cubic and trigonal functions tabulated above are implicitly oriented as succeeds (u’ E u(z’ - 2 ’ ) )
e377(;)=
e = ~ ~ * 47(g); i/3
~ ~= e (~ d ~3p3( g ) ; e3y8(9= - y 8 ( ; ) ;
u’77(;) = f ~ q ; ) ; %(;)
= %(;)
u’Ts(;) = f %(;);
=
u ‘ q ; ) = F iyq;);
%(E)
yT(;) = %(:)
(28) The procedure utilized t o derive these functional forms differs from t h a t outlined in ref. 24 merely in the replacement of &(z) by e ~ ( z ’and ) ~ ( z z ) by u ( z ’ z ’ ) . By ref. 27 the operator u’ induces the same change in our new trigonally oriented functions ij, n> as u did in the old tetragonally oriented functions ‘ j , m>. The (z’, y’, 8 ’ ) coordinate system is related t o the (z, y, z ) plan b y means of the orthonormal transformation, A , where A is defined b y the equation
-
( 2 2 ) Our coordinate transfigurations are defined a s replacement, not transference, operations. E.g., the counterclockwise rotation about the z-axis b y r r / 2 replaces z b y -v. I/ b y z, and L b y E . With this definition of coordinate changes, the spin functions transform as given by H. Goldstein, “Classical hlechanics,” Addison-Wesley, Cambridge, 1951, page 11F. The orbital function alterations are readily determined from their Cartesian representations. I n reading Hellaege’s paperzoa please note the alternate choice of Cartesian coordinate transformations and Eulerian angle designations. (23) We have oriented the cubic and quadrate functions by their behavior under a s / 2 rotation (C4) about the fourfold z-axis, and under a reflection ( u ) in the z-z-plane. We have elected t o classify our symmetry cliques in 1,he ensuing manner
e4y,(;)
= -eisi./4
e4Y8(;) = - - e i . d 4
e4rqg)=
~~(6);u?’;(;) Yp (6); uYs(;)
+ e i d 4 Yp(;);
uYs(;)
=
YQ~(;)
= *Ys(L);
-e(,.)= Ye(;)
= ZtYp(2);
YQB(;) =
Yi(6)
-
e,
(e,), e&.
Alterations in the s t a t e vectors under e4 are most easily computed in the Cartesian (z’. y’, 2’) representation of the basis functions l j , m>. I n this picture the effect of e4 is ascertained from the relations
e’.[::]
= i’..i[::]; 434 =
[“ y] 0
-1
(by ref. 2 2 )
0
T o fix the needful Eulerian angles, we simply identify the matrix BBaA with t h e Eulerian matrix of Goldstein (page 109).** T h e gyrated spin functions may then be singularized b y the substitution of the angular values into the spinor array of Goldstein (page 116)22
~q;)
One must carefully note t h a t the inclusion of spin function variations (see footnote 2 2 ) implies t h a t the identity transformation is a rotation by 4a. Therefore, the operators e4 a n d u obey the relations uz = = -1. As a matter of definition, we have taken our spin functions to be even under inversion (i) in the origin. Hence, we h a r e t h a t the z ) is niatheniatically equivalent t o a rotation by reflection u ( z ,((?I) about the y-axis. (24) The explicit form of the yaexpression may be determined by about the the application of a counterclockwise 2 n / 3 rot.ation (1, 1, 1) direction, to a linear combination of functions, with arbitrary The restnccoefficients, which have the required demeanor under tion t h a t only thosefunctions whose deportment introduces a multi-
e:
-
(29) The comments in ref. 25 apply equally xell here once the word trigonal is substituted for tetragonal and quadrate, a n d the operators and r(z - z) are supplanted b y ea(z’) and u(z’ - 2’). Mark B ) ] . These well the appearance of the phase factors exp [Sr3i/4(7r have been inserted into the Fao, d basis vector definitions to retain the identity of the cubic Hamiltonian matrix in both its quadrate a n d trigonal manifestations. The angle p which equates the off-diagonal matrix element of X(lV in these two equivalent cubic representations 1/3, sin 6 = is, as may be surmised, the tetrahedral angle (cos p = 2/34).
e+)
+
-
ANDREW D. LIEHR
46
< r f > . 3 3 To date no universaly accepted notation has been adopted for the axial fragment of the ligand potential. We shall espouse the Moffitt and B a l l h a u ~ e f ilabels ~ ~ ~ ~for ~ tetragonal addend
X
+
X
rd2Ds/2) 2Dq - 8- DS - 2Dt 5
'DE/,
- 4Dy - E
=
I + 2D3/a 2Dq - E
0; X I
1
(pi)
1
(Fa)
'Da/2 -246Dq -3X-E 2 I
=
0
r~(~Da/2)
- E -246
Dy
v'6 [Ds + IODt] = 0 +5
- $ X - 75- Ds - E
+ 2Dq + 52 D u - 13-9 D r - E
- 2 4 6 Dy
- 2 4 6 [g 1 DU + 2 D T ]
=
(12)
- 2 3X - 75 DU - E
(r:) we coalesce the parametric products -2/7 B, and -88,/21 Bd into the single indivisible quadrate emblems Ds and Dt, respectively, and designate the analogous indissoluble trigonal The non-vanishing aggregates by Du and matrix elements essential for the delineation of the augend potential energy may thus be written ( a ) Quadrate Orientation , L e . , of the radial Qverage (Rd(ri))/144;. (34) C. J. Ballhausen and W. E. hfoffitt, J . I n o r g . & Suclear Chem., 3. 178 (1956). (35) In terms of the more pliant centric forms Rz(ri) a n d Ri(ri), u-here Rn(ri)a n d E i ( r l ) are proxies for 4 / 5 d G B 2 ~ a! n d 16/34GB,r: in equation 7 a n d ref. 31, the parameters Do,(o = s, t , u, 7 ) . assume the guise
Jan. , 1960
ELECTROSTATIC AND ELECTROMAGNETIC FORCE I N LIGANDFIELD THEORY
47
Strong Field Picture-Wave Functions.-Whenever the spin-orbit energy is subservient to the crystalline energy VC.F., the Hamiltonian matrix may be diagonalized, to good approximation, by suitable linear combinations of the free ion orbital functions, ll,m+.38 As demonstrated by Bethe,” such linear combinations of atomic functions no longer are degenerate in energy if the Hamiltonian X, describes a cubic conformation, but are segregated by the supplementary quantum ciphers tZg and The sums of state vectors, , Z p > , which adorn themselves with the cubic appellations tlg and eg, may be discerned by their transformative demeanor. We shall here catalogue these series for future reference (a) Cubic Functions a i t h Tetragonal Orientation40 t2gap(ti2e)
T2,:
= dX\
12, - 2 > )
If,
12, & I >
=
dZ2
m,>
.>
: 1-,
d x ~ - y 2 { ( ~ r= ~s) 1 -{ 2, +2> 2, - 2 > )
+
%2
I1-21, m s )
11
2 ma>
(13)
( b ) Cubic Functitrns nith Trigonal O r i e n t a t i ~ n ~ ~ 2T2,&: t2gap’(7?ls) =
tZg(B)f’i ni,)
=
2Eg:eg(;)p’(ms)=
dz’2 O>
:1.->
1 4:drL - 4;
171
:>
d& I
f
4; + 4; drrL
1
ds’, \ {’(m.) (14)
The relevant wave functions which convert decently under coordinate permutations are uncovered by means parallel to those already dis-
Dv =
-
= - - 1 , 144%
(Y =
t,
7)
(36) The spin-orbit energy of the weak field picture is trivial t o H
evaluate. Simple vsctor model arguments relate 1. s t o the num1 ) -. 37/41. The radial mean, , of the spinber 1 / 2 ( j ( j orbit potential € ( r i ) has been typified a s X (it is called Sd in ref. 10 and 11). (37) I n the absence of magnetic flux, the operator, m-hich reverses the direction of all angular momenta vectors,zO,b+commutes Hence, wave functions which permute with the Hamiltonian, are representable by identical matrices. under the application of As Xr,(g) equals il’s(d,), equation 6 predicts the coincidence of the T14 a n d Ts secular determinants. (38) The Hamiltonian, in this case, may be regarded as spin independent. (39) T o avoid the creation of added turmoil. we shall employ Mulliken’s (Phus. Rev.. 43, 279 (1932)) notation for ordinary quantum states and reserve the Bethe symbolism for double group quantum configurations. Bethe calls the hg and eg coteries ys and ya, each. (40) The tetragonal, or fourfold, axis is taken t o be the z-axis, as before (ref. 22-25), Spin functions have been displayed both as f(m,) and ‘1/2, m,>, as aesthetic principles demanded. (41) The prime mark indicates quantization of angular momenta along the (1, 1, 1) direction, i.e., along the 8’-axis (ref. 27-29).
+
x,
x.
x
(42) The phase factor i ensures the reality of tlw quadrate secular determinants. (43) Pursuant to our previous convention,22 the counterclockwise rotation b y 2~/3(C?a)about the ( 1 , 1 , 1) direction (the z’-axie) 1-eplacen z‘ by x’ cos 2x,’3 - g ’ s i n 2rr/3, y’ by x‘ sin 2 ~ / 3 y t cos Z x / S and z’ b y z. The occurrence of the phase angle 8 is as rationalized i n ref. 29.
+
ANDREWD. LIEHR
48
Strong Field Picture-Energy Values.-In accordance with our prior specialization of the ligand field, the parameters Dq and X suffice to detail cubic domains while two added measures Dfi, ( p = s, u ) , and Dv, (v = t, T ) , are demanded for realms of tetragonal or trigonal disposition. Equations 8 and 9, when subjoined to 13-16, allow the necessary crystalline matrix elements to be expressed in terms of these quantities, and, hence, permit the analysis of the eigenvalue problem. I n the subsequent lines are collected those strong field determinental forms from which the needed energy distribution is i ~ s u a n t ~ ~ 'Tz~
T P ~ X
- 4Dq - E
= 0,
- _2-
(ri)
(21)
If the transformation matrices exhibited overhead are depicted as T(l?j), the relationship which conjoins the weak and strong field Hamiltonian arrays, XWand Xs, is well illustrated by the exemplary formula
2E,
Xa = f'(r1)XwT(rj)*
(17)
I
I
r8('Eg)
$ (3Ds - 5Dt)
A - 4 D q + s 7D t - E
-A - 4Ds 2
(22)
Applications Optical Spectra.-The sparsity of spectral data for the second and third transition group, dn, (n = 1, 9), complexes precludes the numerical usage of the derived secular equations. 45,46 Hom-
d!Xl=o 6Dp - El
404-E (r8)
I
Vol. 64
0
=o
+ D. + 23 Dt - E
(r?)
6Dq
+~
Ds Dt
(18)
- E/
(c) Trigonal field^^'.^^
'Tz.~)
r8(
-5
4; $
r8(2EK)
- 4Dq + D a + 3 DT - E 2
X
(r:, r:) ri( T2g) X - 4Dq
-
14 DT 9
-
6Dq
ezS/2[3Du - ~ D T ] '
4- 57 DT - E
rs(2Tzg)
@ [9Du + 2 0 0 ~ 1 9
-E
1 - - 4Dq - -?[9Da + 3.1071 - E
2
9
dE
I
[3Da - ~ D T ]
I
4;+ 5 4;
I
I
=
[3Da - ~ D T ] ,
X
6Dq
Weak Field-Strong Field Conjunctive Relations,-As precisely the self-same ten orthonormal basis functions have been utilized in the formulation of the weak and strong field "exact" energy expressions, l 2 these expressions can differ only superficially. The confluence of the two descriptive processes is promptly established when the separate basis vectors are expanded, the one in terms of the other. The conjugative algebra is thus found to be as displayed
r8('Eg)
+ 73 DT - E
0
(19)
I
I I
equation 733. Please keep in mind, though, that Condonand Sliortley introduce the factor minus one into all orbital functions, 11, m i > , whenever mi is both o d d and p o s i t i c e .
(45) The detailed experimental characterization of the newly made neutral (octahedral) hexafluorides of rhenium, osmium, iridium a n d platinum has stimulated theoretical investigations somewhat similar t o our own. The numerical accord of theory and experiment for these compounds is excellent.lo~ll (46) These equations are, of course, equally applicable t o first transition group, d". ( n = 1, 91, molecular clusters. Indeed, niany authors h a r e developed theoretical expressions, of some resemblance to several given here, for usage in the delineation of such complexes: (a) 0. M. Jordshl, Phgs. K e i , . , 46, 87 (1938); (b) A. Siegert, Phgsico, (a) Quadrate O r i e n t a t i 0 n 2 ~ * 2 ~ ~ ~ ~ 3, 85 (1036); ( c ) .J. H. Van Yleck, J . Chem. Phgs., 7, 61 (1939); (d) n. Polder, Physica, 9, 709 (1912); (e) A. ilbragarn and hI. H. L. Pryce. Proc. Roy. Soc. ( L o n d o n ) , 2 0 6 8 , IR4 (1951); ( f ) F. E. Ilse and H. Hartniann, Z. p h y s i k . Cliem.. 197, 239 (1951); (g) C. J. Ballliausen, D a n . Blat. F u s . N r d . , 29, N o . 4 (1964); (ti) L. E. Orgel, J . Chem. Phys., 23, 1004 (195.5); ( i ) Q. Felsenfeld, Proc. Roy. Soc. ( L o n d o n ) , 236A, 506 (1956); ( j ) R. L. Belford, M . Calvin and 0. Belford. J . Chem. P h y a . , 26, 1185 (1957). Watchful inspection of these alternant Eorniulations will, however. reveal significant dif(44) The spin-orbit menibers of these matrix arrays can be oompiled, without stress or strain, by use of Condon and S h o r t l e y ' ~ ' ~ ferenres. ~
Jan., 1960
ELxcmosumc AND ELECTROMAGNETIC FORCE IN LIGAXD&’IELD THEORY
ever, it is to be anticipated that such will not always be so, and, hence, several phrases in the way of an experimental prospectus should not be inappropriate. The avenues of greatest promise lie in divers directions: (a) optical studies of the recently prepared hexafluorides of iLIo(V) and W (V) ; l4 (b) spectroscopic particularization of the oxy salts of Alo(V) and W(V)47; (c) spectral characterization of solid solutions of Ag(II), Au(II), Zr(III), Hf(III), Nb(IV), and Ta(IT7)488.49;(d) careful low temperature observat’ion of light absorption in tet.ra- and hexa-coordinat,ed Cu(I1) systems.jO For future reference, the solutions of the cubic secular equations will be presented. If one defines 6 X cot 7 is equal to l/zX the angle 7 such that - d 10Dq15’the eigenvalues and functions can be written in the succinct form (lower root: ir > q > O ; upperroot: 2 n > q > n )
+
E@;) = X - 4Dq E(l’8) =
- 21 X - 4Dq -’
dB X cot 7-2 2
$i(r7)= r 7 w a g ) $(rs) = sin rs(ZT2,)- COS
rs(ZEg)
(23)
(24)
The choice of equi-valued amplitudes for X and Dq requires q i;o be 166’52’6.8” (lower root) or 346’52’6.8“ (upper root). The exemplary selectionb2 of 3000 cm.-’ for Dq predicts absorption bands a t 4923 am.-’ and 32346 cm.-’. Without advance knowledge of the pertinent tetragonal or trigonal field strengths, little can be said concerning spectral features expected of quadrate and trigonal environa. At this juncture it seems apropos to mention an approximational scheme which is applicable to the many eleclxon problem. It is simply this. One assigns each electron to either one of the four 78,two y7,or four yt orbitals. The state of lowest energy is obtained by a sequential assignment. For example, thie ground state of the mono-negative rhenium hexafluoride ion would be described as y:, which procreates the terms rj, ( j = 1, 3, 5). The first excited state would then correspond to yky!, which begets the levels rj, ( j = 3 , 4 , 5 ) ; the second to y; which produces the state rl; the third to qAr*i,which sires the components rj, ( j = 1: 2,.3 (once); and 4, ;(twice)), e t c b 3 A synthesis, in this manner, of the multielectron terms from the (47) C. K. Jdrgensen, Acta Chem. Scand., 11, 73 (1957). (48) F. J. hforin (private communication) will attempt to prepare and individualize solid solutions of this type. (49) With respect t o Au(II), t h e use of these secular equations might resolve questions as t o its existence in mixtures of Au(1) and Au(II1) in solution (R. L. Rich and H. Taube, THIS JOURNAL,68, 6 (1964)). as Au(I) should show no dn transitions and .Au(III) should exhibit spectral features in accord with the dB secular equations of ref. 5. (50) Fused salt spsctral observations should also furnish helpful data. (51) This means 01 simplification of the resultant eigenvalue and eigenfunction expresshns is due to Moffitt, Goodman. Fred and Weinstock.10
(52) T h e Dq option of 3000 cm.-’ is Deculiar to neutral rhenium hexafluoride. 1 1 (53) The “exact” t v m values for this ion may be prorured by the substitution of the appropriate numerical magnitudes for the sundry parameters which appear in the energy determinants of ref. 5 .
49
single particle spin-orbit functions, with the concomitant neglect of electron coulombic repulsions, is the direct analog of the composition of term values, from the j-j coupling view, in atomic spect r o s ~ o p y . As ~ ~in~the ~ ~appositive atomic situation, a first approximation to the true energetics is then determined by the perturbative addition of the electron-electron electrostatic repulsions. A calculation along these lines has recently been utilized in the elucidation of the neutral hexafluorides of the platinum Magnetic Properties.-Remarks as to the handiness of germane magnetic facts strongly parallel those made in antecedent paragraphs concerned with spectroscopic conduct. Generally speaking, magnetic demeanor is more greatly influenced by environmental vagaries than optical deportment, as the former is dependent upon the detailed visitatorial schedule of the transient electrons. Several sentences might, however, be profitably devoted to cont,emplation of the magnetism of the newly recorded mono-negative hexafluorides of tungsten and molybdenum. l 4 It is apparent from equation 24 that values of Dq and X in the proximity of 3000 cm.-’ more than suffice to ensure the validity of Van Vleck’s “high frequency” magnetic susceptibility separation.S6 Consequently, the paramagnetic s ~ s c e p t i b i l i t y ~ ~ may be written as
where Is> symbolizes the Fas wave function, pz the z-component of the magnetic dipole moment operator, L, 2X,, and a the temperature independent paramagnetism. If p2 denotes the square of the magnetic dipole moment, the egective magnetic dipole moment is defined byss
+
(54) Ref. 13c. Chapter X. (553 In this framework, we should assign the weak absorption a t -32,000 cm. -1, which is characteristic of the dinegative platinum group hexafluorides 111. A. Hepworth, P. 1,. Robinson and G. J. Westland, J . Chem. Soc.. 011 (1958); A . G. Turner, Jr., and A. F. Clifford, Nature, 182, 1309 (195811 as ya(or 7 7 ) y~*. T h e high intensity hand a t -50,000 cm.-l is most probably due t o charge transfer. Readers interested in the basal properties of further palladium and platinum group complexes are urged to peruse the following papers: &I. A. Hepworth, P. L. Robinson and G. J. Westland, J . Chem. Soc., 4269 (1954); R. D. Peacock, ibid., 1291 (1956); 467 (1957); C. K. Jdrgensen, Acta Ckem. Scand., 9, 710 (1955); 11, 151 (1957); 166 (1957); H. Hartmann and C. Buschbeck, 2. p h y s i k . 11, 120 (1957); H. Hartmann and H. J. Chem. (Frankfurt) [N.F.], Schmidt, i b i d , , 234 (1957). The review article by C. X . J9rgensen (Reports to t h e X’th Solvay Council, Bruxelles, May, 1956) contains an exhaustive tabulation of pat spectral data. (56) J. H. Van Vleck, “The Theory of Electric and Blagnetic Susceptibilities,”Oxford Univ. Press, London and New York, N. Y., 1932, Chapter VII. (57) The srrsceptibilities of certain diamagnetic palladium group metals have been surveyed b y J. D. Dunits and L. E. Orgel, J . Chem. Soc., 2594 (1953); J. S. Griffith and L. E. Orgel, Trans. Faraday Soc., 53, 601 (1957); C. J. Ballhausen and R. W. Asmussen, Acta Chem. Scand., 11. 479 (1957). (58) I n equation 28 terms between the two different rs states have been dropped.
-
AXDREWD. LIEHR
50
Vol. 64
truly cubic site. Contrariwise, the small effective moment and complex susceptibility deportment of the tungsten salts indicates but minor deviaAnd direct computation with equations 24 and 27 tions from (octahedral) regularity. The magnereveals thats96o tism should therefore be explicable on the basis of 1 equation 20, plus a suitable thermal element, p z + - (1 + COE 0)(19 - 13 cos q - 4 4 sin q ) (29) 4 either linear (equations 26 and 28) or transcendental The omission of contributions from the I'8(2Eg) (KotanP) .'j2,'j3 fraction of the ground charge density distribution Conclusion 2P2/hv1.61 With the previously yields an a To close this study of conflictive electronic enumerated values of q and hvl, we find that peff propensities, a few comments anticipative of equals ~ ( 0 . 2 0 0.08 X 10-2T)i/~. trends will be put forth. First, it is exIf sufficiently potent tetragonal fields are future pected that solids built of octa- and dodecaextant, the orbital momentum will be partially (cubically) coordinated palladium and platinum quenched and the spin moment will be propor- group ions will be found. These systems will be tionately released from its stagnant orbital union. described by Dq values which are -l/z, This situation finds mathematical confirmation severally, times that of their octahedraland counterfrom equation 18's disclosure that a mixture of the parts (recal14Q that for tet,rahedral coordination y?(1)(zT2g)and Y $ ( ~ ) ( ~ habitats T ~ ~ ) will subsume multiplicand is - 4 / 9 ) . Katurally the spina position of minimum energy. The resultant this orbit coupling paramet.er, X, and the ligand field magnetic moment p of such an abode, sin ,$yq'" constant, Dq, will concurrently assume appropriate cos (y?@), is expressible as 4 3 Ip.1 or, equivalently, positive or negative values accordingly as t,here 43 sin 2( - 4 2 sin 2E. The specific magnitude are one or nine d electrons present.13 of ( is determinable only if the structure of the It is proper to point out that questions of inquadrate potential is known. A simple estimate herent configurational instability have been supof 15" predicts a magnetic moment of 1.11; the pressed. Consideration of such complications is larger value of 1.58is obtained if 5 reaches 45'. beyond the compass of this work: it should be Without resolved spectral lineaments, it is mentioned that Goodman64is at present investipremature to individualize theoretically the mono- gating problems of this sort. But note that in negative hexafluoride ions of molybdenum and complexes with more than one electron, but, less tungsten. Several anticipatory remarks might than nine, orbital degeneracy does nol perforce perhaps be helpful to future workers, although the imply instability. ,4n excellent example of this antiferromagnetic proclivity of the crystalline nature is already visible in the first t'ransition materials gives a tenuous air to all pronounce- series-tetrahedral divalent nickel complexes, by ments. virtue of their moderate spin-orbit, correlations, The Curie-Weiss behavior and the large attend- do not exist in a degenerate ground st,ate.j Hence, ant permanent moment of the molybdenum these compounds may exhibit, a t most, pseudo salts most probably implies a tetragonal (or trig- Jahn-Teller forces.'j5 onal) stabilization of an orbital quenched state, The predicted admixture of the ant,ibonding as a sizeable Curie constant is incompatible with a eg orbitals to the u-non-bonding t 2 g orbitals may be
-
+
+
(59) This particular structure of 1121s due t o Xioffitt. Goodman, Fred and Welnstock.10 (60) The irresponding expression in the weak field formalism may be reckoned fiom the identity of < s ' p z l s > 2 with
where the angle w is deduced from the relation #(I's) = sin wFs(2Ds/,) cos wI's(2Ds,',). The choice of -3000 cm.-' for Dq and X fixes w a t -33'. All essential matrix elements may be extracted from Condon and S h ~ r t l e y . 'Chapter ~~ 111, pages 48-49 and 63-67. (The symbol g,, j = 5/2, a/z), represents the appropriate atomic g-factors.) (61) If this same charge density function is also eliminated in the calculation of the ground state magnetic self-interaction terms, the magnetic secular determinant becomes (s = a , b )
+
r7s I
1-
d!2BH, X
I
* BH, - El
T h e plus sign holds when s equals a. T h e solution of this determinental equation b y means of second-order perturbation theory, and the subsequent use of equation 4, ref. 56, produces Kotani's ( J . Phys. Soc. ( J a p a n ) , 4, 293 (1949)) susceptibility expression. Upon expedient disregard of all exponential factors, the constant a will emerge from the Kotani susceptibility.
(62) Other discussions, not previously cited, which take cognizance of the interplay of the crystalline and spin-orbital forces may be found in papers b y B. Bleaney (Proc. Phys, Sac., 638,407 (1950)); H. Kamimura (Proc. Phys. SOC.( J a p a n ) . 11, 1171 (1956)); P.Tanabe and H. Kamimura (ibid., IS, 394 (1958)): J. S. Griffith ( T r a n s . Faradav Soc., 54, 1109 (1958)): and B. S. Figgis, ( S u t u r e . 182, 1568 (1958)). Of course, the fundamental contributions of A . Abragam and 35. H. L. Pryce ( P T O CRoy. . Sac. (London), 2 0 6 8 , 135 (1951), et sqq.) must not be overlooked. (63) Examinations of paramagnetic resonance phenomena in palladium and platinum group metals are not numerous. Apt references are: (a) S. Ramasesham and G. Suryan, Phys. Reu., 84, 593 11951): (b) K. D. Bowers, PTOC.P h y s . S o r . , 6 6 8 , Of36 (1923): ( e ) J. IT. Griffiths, J. Owen and I. 31. Ward, Proc. R o y . Soc. (London), 219A, 526 (1953); (d) K. W. 11. Stevens, ibid., 542 (1953); (e) J. H. E. Griffiths and J. Owen, Proc. Roy. Sac. (London), 2 2 6 8 , 96 (1954); ( f ) H. M. Gijsman, H. J. Gerritsen and J. Van Den Handel, Phusica, 20, 15 (1954). Excellent surreys of both the resonant and static magnetism of these metals are given in the books by D. J. E. Ingram ("Spectroscopy a t Radio and Microware Frequencies," Butterworths, London, 1955) and P. W. Selwood ("3lagnetochemistry." Interscience Publishers, Inc., New York and London, Second Edition, 1955). (64) G. L. Goodman (private communicarion). A preliminary peek into this matter has been reported by TI-. E. 3Ioffitt and W. R. Thorson, Phys. Reu., 108, 1251 (1957). (65) -1.D. Liehr, A n n . P h y s . [-Y.Y . ] , 1, 221 (1957). .1 siiiiilar conclusion may be drawn for tetrahedral cupric complexes, all of which possess Jahn-Teller resistant I'iground states (€1.-1.Jaiin, Proc. Roy. Soc. (London), 1648, 117 (1838)); see eqn. 10 o r 17 (Note that, such complexes should exhibit a n infrared transition (rr + rs) a t -1350 cm.-' (Dq = - A = 1000 c m . - l ) ) .
THEREACTION OF FERRIC IONWITH ACETOIN
Jan., 1960
51
experimentally verified by means of nuclear magnetic resonance techniques. The procedure to be followed is .that of Shulman and Jaccarinoe6 who show that the magnitude of the paramagnetic shift in the fluorine (or any other ligand with nonzero nuclear spin, for that matter) resonant frequency is directly related to the presence of unpaired electrons in antibonding trajectories. Lastly, some thought must be given as to the validity of the crystalline field approach to complexes of the highly electropositive metallic ions situated in the fifth and sixth r o w of the periodic table. A moments reflection will reveal that no approximation is actually entailed in the introduction of the parameter, Dq, for this parameter serves only to assess the symmetry induced separation of t,he bg and eg spatial orbitals.67 It thus symbolizes a known sum of interaction integrals. All approximations derive from semi-empirical specifications of this sum.68 Entirely analogous arguments may be advanced for the tetragonal and trigonal measures Dp, ( p = s, u) and Dv, (v =
procure the spin-orbit matrix. This circumstance springs from the presence of a cross integral between the trg and eg classes.70 Theoretically two spin-orbit constants, X(tTg) and X ( t f g , eg), should be employed, but simplicity dictates outlaw such finicking requirements. It is therefore perceived that loss in generality of the “exact” addend field approach12 issues from the suppression of a second spin-orbital parameter71; for the clarification of some physical phenomena, this may not be a trivial defect.
t,
Acknowledgments.-The author is extremely grateful to Gordon L. Goodman for making available to him, in advance of publication, preprints of his discourses on similar topics. This thoughtfulness on Dr. Goodman’s part has saved the author a considerable amount of futile effort and rescued his readers from dull redundancy. The present article has also benefited from several illuminating discussions with Dr. Goodman. It is also a great pleasure to acknowledge fruitful conversations with Frank J. Morin. Robert G. Shulman and James C. Phillips. Through Dr. Phillips’ kind consideration the author was privileged to read a prepublication draft of his important manuscript.
.).69
A restriction must, however, be introduced to (66) R. G. Shulman and 1’. Jaccarino, P h y s . Reu., 108, 1219 (1957). R . G. Shulman (private communication) plans to look a t the fluorine resonance in molybdenum and tungsten hexafluorides. (67) If one measures all energies from the mean of the t z s and eg orbital energies, 1/’5[8E(trg) f 2 E ( e g ) ] , and calls the energy difference, E ( e , ) - E ( h g ) ,lODq, one readily infers both the strong and weak field cubic matrix elements. T o see this one need note t h a t when the null value, :3E(hg) @ ( e g ) , is combined with the energy separation lODq, one obtains the magnitudes of the apposite strong field matrix elements, (t2glVlt~g) and (e,lVieg), as -4Dq and 6Dq, each. Similarly, the equality of 1 / d 2 [ e g b i ilns,l and 12, f 2 ) , etc., permits the deduction of the needed weak field matrix components. (68) I n an irnporta?t paper, soon t o be published, J. C. Phillips [ J . P h y s . Chem. Sols., in press] has shown, without resort t o a full scale molecular orbital treatment (vide, e . @ . ,Y. Tanabe and S. Sugano, J . P h y s . SOC.( J a p a n ) , 11, 864 (1956)), t h a t the simple point charge model” has good theoretical justification. This article thus eliminates the extremely disconcerting result of TV. H. Kleiner (J. Chem. Phus., 20, 1784 (1952)). (69) These arguments involve the specification of separations. internal to the eg and f g g states, in terms of linear combinations of t h e axial measures.
+
NOTE ADDED IN PROOF.-Recent experiments of F. J. Morin on oxides of the second and third group transition elements indicate that these are metallic in nature. Therefore, Morin suspects there will also be Pauli type (metallic) paramagnetic contributions to the observed susceptibilities (see C. Kittel, “Introduction to Solid State Physics,” John Wiley and Sons, Inc., New York, N. Y., Second Edition, pp. 259-262 (1956)). [ A general discussion of metallic behavior in the first transition series oxides is given in F. J. Morin Bell S p .Tech. J., 37,1047 (1958), and F. J. Morin, Chapter 14, Semiconductors,” A. C. S. Monograph No. 140, Editor N. B. Hannay, Reinhold Publishing Corporation, Xew York, N. Y., 1959.1
(70) An equivalent complication occurring in atomic spectroscopy is discussed in Condon and S h ~ r t l e y , ’p.~ ~376-377. (71) Naturally, there are always present inaccuracies due t o the complete disregard of configurational interaction of t;e: states with terms of disparate origin.
THE REACTION OF FERRIC ION WITH ACETOIN (3-HYDROXY-2-BUTAXONE) IS AQUEOUS SOLUTION BY J. K. THOMAS, G. TRUDEL AND S. BYWATER National Research Council, Applied Chemistry Division, Ottawa, Canada Receiued June 16, 1969
The reaction between ferric ion and acetoin (3-hydroxy-2-butanone) has been studied in aqueous perchloric acid solution. It was found that two moles of ferric ion is required to oxidize 1 mole of acetoin. A t low ferric ion concentrations the reaction is first order in acetoin and ferric ion. The variation in rate with pH is interpreted in terms of differing reactivities of the various hydrolysis products of ferric ion.
Introduction The standard method for the analysis of acetoin consists of oxidizing it with ferric chloride. I n this reaction biacetyl is formed quantitatively and can be estimated as the nickel glyoxime complex. The present investigation was undertaken in order (1) L. C E Kniphoist and C. I. Kruisheer, Z . Untersuch. Lebensm., 73, 1 (1937).
to elucidate the mechanism of this ferric ion reduction. Preliminary results were obtained using ferric sulfate solutions in dilute sulfuric acid, but all the quantitative experiments were carried out using ferric perchlorate-perchloric acid mixtures a t low pH’s since the preliminary experiments showed that the reaction rate is strongly dependent on the ferric ion species existing in the solution. The
