Interaction of Electromagnetic Radiation with Matter. I. Theory of

T H E

J O U R N A L

OF

PHYSICAL CHEMISTRY Regisiered in U . S. Patent Ofice

@ Copyright, 1964, b g the American Chemical Society

VOLUME 68, NUMBER 4 APRIL 15, 1964

Interaction of Electromagnetic Radiation with Matter. I. Downloaded by UNIV OF PRINCE EDWARD ISLAND on September 8, 2015 | http://pubs.acs.org Publication Date: April 1, 1964 | doi: 10.1021/j100786a001

Theory of Optieal Rotatory Power: Topic A.

Trigonal Dihedral Compounds'" by Andrew D. Liehrlb Bet1 Telephone Laboratories, Inc., Murriiy (Received July S4, 1068)

Hill,New Jersey, and Melloiz Institide. Pittsburgh, Pennsylvania

Who shall be a light between truth andl intellect.-Dante

A complete theory of the electronic properties of trigonal dihedral compounds is outlined, with special emphasis upon their optical rotatory power. Molecular orbitals are derived) many electron configurations constructed, and various spectral parameters computed. A number of points of general interest have been uncovered and elucidated: (1) a zerothorder molecular orbital theory is capable of explaining the observed optical rotary strengths of trigonal dihedral compounds; (2) the electron-hole formalism is not directly applicable to optical rotatory or spectral strength calculations; (3) there is a direct correspondence between the rotational and spectral powers of d" and d5'" (n = 0, 1, 2, 3, 4,5 ) spin-freie systems, between d134.6~~spin-free systems, and numerous indirect relationships between these and other systems; (4) strong natural electronic optical rotatory dispersion, duchroism, and dissymmetry are possible in the infrared and microwave regions of the spectrum; (5) spin magnetic dipole transition moments alone, in conjunction with an allowed electric dipole transition moment, may give rise to large rotational strengths; (6) t o good approximation, U- and u-,n-bonded compounds are formally identical in their dihedral optical rotativity and intensitay behavior; (7) electronic transitions t o or from bonded and antibonded dihedral molecular states rotate electromagnetic radiation parallelly or oppositely dependent upon conditions; (8) for a fixed molecular conformation, the sign and magnitude of the rotational, dichroic, dissymmetric, and spectral strengths depend upon the degree of local electronic trigonal dihedral asymmetry as measured by such parameters as the covalency and hybridization constants and the angle of metal-ligand orbital mismatch ; (9) the absolute and relative geometrical configurations of ground electronic states of chemically distinct and variegated series of trigonal dihedral compounds can be theoretically fixed and correlated, and possible geometric deformations of these configura,tions in excited electronic states can sometimes be inferred, and their nature guessed, frorn optical rotatory, circular dichroic, and circular dissymmetric behavior; (10) cubical opticstl rotatory selection rules do not always hold in zeroth order for trigonal dihedral compounds. Examples have been given, a course for future research outlined, and a plan for closer theoretical-experimental cooperation entered.

666

ANDREWD. LIEHR

TABLE OF I. Introduction..

CONTENTS

, l l , l , l , , , , , l l l l l . I I . I . I . I I I

...

. . . ., , , , . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . .

673 .................................... 677 $3. T h e Optical Rotatory Power.. . . . . . . . . . . . . . . . . . . . . . , . . . , . . , , , , . , , , < , , , , , , , , , , , , , , , , , , , , , , 677 3.1 General Concepts , , , ........................................................... 877 3.2 Mathematical Consid ........................................................... 679 3.3 Matrix E l e m e n t s . , . , . . . 682 3,4 Rotational and Speotr .................................................................. 686 3.5 Dissymmetry Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . < 693 Applications, , ,, , ,,,, ,, . , ., ,, . ., ........................................ 693 BY3 54. T h e Many-Electron Systems: Wave Functions and Matrix Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The One- a n d ?Tine-Electron 4.2 The Two- and Eight-Electro 4.3 T h e Three- a n d Seven-Electron Cam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 T h e Four- And Six-Electron Case, ................................... 702 4.5 T h e Five-Electron Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 A Numerical Example: t h e Discussion ................................................................................. 713 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 . Acknowledgments,, , , , . , , , , , ,, ,,,, ,, ,,,, ,, ,,, ,,, ,,, , , , . . . . . . . . . . . . - ,. . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Appendix.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... 718 55. Evaluation of t h e Integrals. . . . . . . . . . . ...,,,,,,.. 718 5.1 Algebraic Reduction. . . . . . . . . . ................................................................ 718 5.2 Angular Momentum Integrals. . . . . . . . . . . . . . . . . . . . . 5.3 Local Electronic Tr&,mit.ion Moment Integrals, , , , , , .................................... 718 5.4 Local Overlap Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 , ,,,, , 5,s Algebraic Evaluation of the Baaio Integrals., , , , , , , , , , , , , , , , , , , , 5.6 Group Electronic Transition Moment Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 5.7 Group Overlap Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , . . , . . ,, ,,

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Downloaded by UNIV OF PRINCE EDWARD ISLAND on September 8, 2015 | http://pubs.acs.org Publication Date: April 1, 1964 | doi: 10.1021/j100786a001

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IV. V. VI. VII.

G66

.........................................

................................ ,, ,

111.

666

.............................................................................................

11. T h e o r y . . . . . . . . . . . . . . . . . 41, Nolecular Orbitals fo 1.1 T h e u-Molecu

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I. Introduction Of all the natural phenomena observed in nature, none has had so profound an effect on chemical thought as that of natural optical rotatory power. Discovered by Biot in 1812, physically justified by Fresnel in 18221824, and structurally interpreted by Pasteur in 18461860, this phenomenon has been the foundation upon which all modern structural theory has been built. The modern concept of chemistry in space iA a direct sutgrswth of its discovery and explanation. Indeed, the tetrahedral geometry of four-coordinate carbon compounds and the octahedral geometry of six-coordinate metal compounds to a large extent owe their final acceptance to the incontrovertible evidence provided for these structures by the resolution of optically active stereoisomers.2 In this paper we wish to demonstrate the role played by geometry in inducing optical activity in six-coordinate dihedral metal compounds. In later works we shall take up the problem of four-coordinate dihedral compounds and of nondihedral compounds of metals and nonmetals, and in particular of the transition metal and carbon compounds. The Journal of Phvaical Chemistry

I

I

I

I

I

11. Theory $1. Molecular Orbitals for Dihedral ComPunda : zero SPin-Orbit Forces 1.1 The u-Molecubr Orbituls. In Fig. 1 we schematically sketch the geometry of a typical u-bonded (neglecting so-called “ammonia” framework hypercon(1) (a) This paper was initially prepared and submitted for possible presentation a t the Quantum Chemistry Conference held at Oxford University, Oxford, England, April 10-14, 1961, It was not Qcheduled in the final select program and, henoe, was not officially presented until August, 1961 at the Sixth International conference on Coordination Chemistry, Wayne State Universitir, Detroit, Michigan, August 27-September 1, 1961. Its main oontents duo have been disoloaed subsequently in the open diacusniottal semions of th% Seventh International Conference on Coordination Chemistry, Stockholm, Sweden, June 25-29, 1962, and of the International Symposium on Molecular Structure and Spectroscopy, Tokyo. September 10-15, 1962. Other more specific portions were detailed a t the Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus, Ohio, June 10-14, 1963. (b) Mellon Institute, Pittsburgh, Penna. (2) I n this regard, it is interesting to note that it took nearly 20 years for Werner’s theory of octahedral metal complexes to gain acceptance, and that this acceptance was finally granted only after he prepared and characterized the optically active dihedral complex potassium trisoxalatochromium(II1). For a masterly discussion of the history, theory, and practice of optical rotary phenomena, read T. M. Lowry, “Optical Rotatory Power,” Longmans, Green, and Co., London, 1935.

INTERACTION OF ELECTROMAGNETIC

RADIATION WITH

jugative effects) complex cation, the copper(I1) trisethylenediamine cation. I'igure 2 illustrater; the idealized disposition of the ligand a-bond orbitals which we assume in our later numerical work, and Fig. 3 a typical, but not universal, level pattern. Note especially the explicit allowance made in Fig. 2 for an angle of cant, a, between the ligand orbitals and the metal orbitals, which are here imagined to be directed octahedrally along the 2,y, and z axes of the figure.


667

MAnER

t'

X .'

t ..-.

'

I

/

/'

Figure 2. T h e local ligand orbital spatial disposition for t h e idealized planar righbangled geometry of the

trisethylenediaminecopper(11) cation.

Figure I. T h e idealized planar righbangled geometry of the trisethylenediaminecopper(I1)cation.

If we designate the localized ligand a-bond orbitals by the symbol, XI, ( j = 0, 1, 2, 3, 4, 5 ) , we may approximate the trigonal dihedral one electron a-molecular orbitals, A,, as a linear combination of these localized orbitals.3

Now since our molecular system has the symmetry Da, the threefold axis being the z' axis of Fig. 2, and the three twofold axes being the y' axis and its trigonal equivalents of y" and y"' [y" equals e3(z')y' and y"' equals ea*(Z')y' equals ea(Z')y'', where e&')is a counterclockwise replacement rotation (i.e., the so-called rotation operator of the Da point group) by 120' about the z' axis defined by Table I], we can always require that

our molecular orbitals, A,, be trigonally symmetric; that is, that' c2aA, = u'A,, w = e2

4 3

, (s

= 0,

+1)

(2)

(3) It is important to keep in mind that the existence of one-electron molecular orhitals (or atomic orbitals for that matter) is a complete figment of mm*s imagination. Although pictorially and physically appealing. it ia nonetheless only B very rough and ready picture of the actual electronic paths in real molecules. Peruse. r.0.. (a) "Papers from the Conference om Molecular Quantum Mechanics. Cniverjity of Colorado. Boulder. Colorado. June 21-27. 195!1." Rev. Mod. Phus.. 32, 169 (1959): and (h) H. Conroy. J. C h m . Phua.. 40, 603 (1964). el aep. The use of misguided localized orbitals t o form one electmu molecular orbitals like those of eq. 1 is not new with the author: they have been employed M o r e hy Spitzer. Coulwn. and Moffitt. and others. For a review of these uses and of the pertinent literature. read ( c ) W. H. Flygare. Science. 140, 1179 (1963). and (d) C. A. Coulson and T. H. Goodwin. J. C h m . Snc.. 3161 (1963). (4) For a simpl-minded justification of this procedure read A. D. Liehr. "Theory and Structure of Complex Compounds." R. JemwsknTmhistowsks. Ed., I'ergsmon Press. London and New York. 19rd. pp. 95-135. and Prwr. Innro. C h m . . 5. 3115 (1963).


D3 MOLECULAR ORBITALS OF THE COMPOVNOS

PRIMDRY ATOM ORBITALS

p3 Y LIGAND U-MOLECVLAR ORBITALS

Figure 3. -4schematic molecular orbital energy level diagram for a trigonally dihedral transition metal cornpound in the #-bond approximation.

In the localized orbital repreaentation, eq. 1, for the oneelectron molecular orbitals, A*, we find that the a,, equal w*lsao,,( j = 0, 1, 2)) and d’a3$,( j = 3 , 4 , 5 ) ) or that [since by eq. 2 , u3+?equals w J , ( j = 0 , 1, 2)14 2

A,

=

aor C

wS3Xj

j=0

+

0

CL~$

C wS3Xg,

(S

=

0, kl)

3=3

(3)

where the constants ao, and a3,are jointly determined by normalization, further symmetrization, orientation, and/or energy minirniEation requirements, It will be convenient for later work to define the first member of the two-part summation in eq. 3 as As(1)and the second as

As(’)

2

= ao,

(4) ws’Xj,

~3~

(S

=

0, A I )

3=3

s

+

,

5

I n the case that requirement

(6) One such linear combination is readily obtained by the orientational requirement CZALI = or - ALI, This requirement imoliea +u0-,, a particular solution of wvhioh i s given by thet a,,

W%J 3-0

A,(2) =

when (s = 0). For the case s = * 1, however, no such simple symmetrization is possible as the functions iill are eurythniically doubly degenerate and by definition thus cannot be made simultaneous eigenfuiictioris of e2(y’) and e&’). R7e must therefore be content to work with the functions defined in eq. 4, or suitable linear combinations of these as indicated in eq 3.5 In

equals zero, the symmetrization e2iio

= *tho

(5)

where e2 is the 180’ rotation, e&’),about the y’ axis of Fig. 2 which is specified in Table I, fixes u3, as fao,, ( S = 0), in eq. 3. Hence, with the neglect of localized ligand-ligand overlap, we may write The Journal of Ph,ysical Chemistry

where = (s = +l). In terms of these functions, the functions of eq. 4 may be written as b r ( l ) = aop(’)d/3/2[As(+) A s ( - ) ] a n d h ~ (=~ a3,(2)d\/3/2[AS(+) ) - A s ( - ) ] , where a g P ( l ) and ~ ~ ( are 2 ) arbitrary real or complex constants, so that e,A,(lj = b-5(2jand &A,(2) = A - , ( l ) whenever aor(l)-equals c z - , ( z J , The- suitable redefinition of ‘Ilr(-)as A _ ] ( - ) = - A - I [ + ) and A + l I - ) = .I+ I ( + ) a k w s S ( + ) to satisfc both As(+) and A- the same G ( y ’ ) equations, t?2.1,1(+) = A , l ( + ) and ezA+,(+) = A = l ( + ) , so that both sets of douhly degenerate (E) functions may be properly oriented for combination with compatible central atom orbitals for (D3) compound formation [see Tables I1 and 1111. I t is not impropitious to remember that onlu properly, compatibly oriented symmetrized junctions may be combined to form molecular orbitals.

+

669

INTERACTION OF ELECTROMAGNETIC RADIATION WITH MATTER

the point group TIs,he(+) is of species AI, A,(-) of species Az, and A*, or (we now drop the subscript 1 in A 1 for notational c~nvenience)~ of species E, individually as indicated in Table 11.

x+(+)


Table I1 : T h e D, Point Group a n d Transformation Standards

At this point it is convenient to decompose the local-

k , cos a and lc, sin

01 appear in the definitions of the uand d i k e localized orbitals, The a-portion of the localized ligand functions, A,, may be further arranged to yield a pseudo-octahedral a-bond system. This arrangement is accomplished by choosing the sums, Z,, (S = 0, k l ) , of eq. 9 to transform as do the true octahedral u-molecular orbitals under the counterclockwise replacement rotation] &(z) about the e axis of Fig. 2 and under the inversion operation i about the coordinate origin of Fig. 2 as indicated in Tables I11 and IV. Care must be exercised in so doing in order that the resulting functions are properly oriented with respect t o the central atom functions with which they are t o combine. The result is that the &(+) sum is already the proper pseudo-octahedral molecular orbital of species AI, and equal I;,(2),with no,and that 2, is equal to to w s / . \ / 6 , [s = k l ) , that is

+

ized ligand orbitals A, into a “u-part” and a “7r-part” which are separately directed along the axes 5, y, and z of Fig, 2 as indicated in Fig. 4.6 We may then write xo = 11

QZO

+

= rul

r z o , x2

+

“=LJ

=

Xa

where [w= x, g, z; j

+

nt2 m13

=

7rZl27 1 4

+

rea,

h5

=

ur4

+

mi

=@I

+

(7)

?T~y6

0, 1, 2, 3,4, SI8

urn; = k,ns, + k , cos a npuwl,

= + k p sin a npT,,,,

(8) and where a superscript c and s have been added in eq. 8 to indicate the presence of the factors k , cos a and k , sin cy in the a- and d i k e functions] separately. [If these factors are absent, the superscripts c and s are to be discarded.] The r and factors in eq, 8 give the phases needed for maximum bonding power [view Fig. 2 and Table VI. In terms of this symbolism we may define

2

(6) For simplicity of illustration, we have assumed the ligating atom to be situated on the x, y, or n axes, I n actual practice this will seldom be the case for polydentate ligands due to steric encumbrances [I$. Fig. 16-19, for example]. However, even in the more general case we may still decompose our charge amplitudes into portions clirected along lines parallel to metal-ligand nuclear directions [or, if desired, parallel to the z, y, or z axes] and perpendicular to it. This decomposition is most simply and rigorouslu accomplished by the expansion of the ligand charge amplitude function in a Fourier series [if f tis taken to be initially axially symmetric] or a spherical harmonic series [if it is not taken to be initially axially symmetric] about the metal-ligand lor coordinate axes] directions, the polar angle being measured from this latter direction and the azimuthal angle [if necessary] being measured about this new direction. If it is assumed, as we shall do presently, that the ligand charge amplitude function is expansible in a cosine Fourier series in which only the first two terrns need be kept, we obtain

=

X ( ~ j ~ # i= ) k(o)(ri)f

& X(l)(ri) +

cos

. ..

In the two-dimensional case pictured in Fig. 4a. we see that 8j equals 185 - a1 and so hj = =

A(Vj,ij)

Xo(r,)

= hqrj)

+ cos

+ cos (8j - r y ) h ( l ) ( V i )-+ cos a

,

..

+ sin e j h ( l ) ( r jsin ) + (X

, ,

,

More general!y, in the three-dimensional case drawn in Fig. 4b, we sin B j sin a cos(9j - go). and hence find that cos 6’i = cos B j cos a for 9pn equals 0 or 90’ as in Fig. 2, we have

+

5 j=O

6

i=3

,

(9)

where the subscript TN equals x, y, z , which is rendered redundant by Fig. 2, has been dropped, and where the superscripts c and s must be inserted if the factors

X i = kO(ri) f COB e j X ( ’ J ( r i )cos

01

+

When the radial functions hO(ri) and h ( ’ ) ( r j ) are identified with atomic-like ligand functions [this does not imply that these functions are actually atomic in nature-see A. D. Liehr, to be published, For further discussion, generalizations, and ramifications of this as well as other related matters] of the types ns and np, we have the desired atomic-like basis decomposition X i = X(rj,#j) = k,nsj

-t- k p cos 01

+ k, sin a np.,

npCI

= uj

+ rj

where k8 and kn are the fractional concents [the so-named “hybridization” ratios (see A. D. Liehr, ibid.) ] of the ns and np atomic-like basis functions in the Fourier expansion.

Volume 68, Nirmber $,

April, 1964

ANDREWD. LTEHR

670

is the proper pseudo-octahedral molecular orbital of species E,.' The n-portion of the localized orbitals cannot be SO oriented since the six n-like orbitals of eq. 7 do not form a symmetrically complete eet under e&) and i [for exampIe from Tables IV and V, irSaequals - az5and %nE1equals - rye, neither of which again belongs to the original *-orbital set of eq. 71. Only when all twelve varieties [Table V ] of localized *-orbitals are available [as in the T-bonded complex anions (such as

epII*(l)= IIT(%). It is now a simple matter t o write down the one-electron molecular orbitals of a trigonal dihedral com-

pound: all that need be done is to add, with suitable multiplicative variational constants, the appropriately oriented and symmetrized localized ligand molecular orbitals to the proper central atom orbitals. If we let these multiplicative constants be 11, [ ( I ) , [ ( 2 ) , p(I), and d2)for the ligand functions ITo*(*), n and &'(z), respectively, we find that*

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the chromium(I1I) trisoxalato anion) of the next paragraph, $ 1.21 can a pseudo-octahedral combination of d i k e orbitals be obtained. By footnote 5, however, we do see that these orbitals are eurythmically complete under e&') [as they must be since this is an operation of the D3 point group] and obey the relationship

t

+ qno8(+'), = n,"'-'

al(tzo) = N,,(b){ttDa

a2

tt where N a , ( t ) , and No(,)are the molecular normalization constants which we choose to be real, and where .'. In t h e h b plone c o s 8 - c o s ( a - 8 ) - c o s 8 c o s u + s i n f f sin

(r

(a)

the central atom orbitals are as defined in Table III and ref. 9. The multiplicative constants are not all independent, and hence, eq. 11 may be further simplified. To see this fact all one need note is that the functions e, and e,* must have the Bame energy since the complex conjugate operator, +, commutes with the Hamiltonian [Le., leaves it invariant], and hence, with N 8 ( $ and ) (7) These statements follow when it is demanded that the functions Z* equal to E*(') f E*(*),(s = z t I) satisfy the selfsame orimtation relations with respect to Ca(z). en(u'), and i as do the central atom

orbitale e,G) of Table 111. Thus the orientational requirements [Tables I11 and I V J 84E+ = ET, &E* = ET and t Z t = ZA. uniquely fix aOrand aTs,and hence PA, to be as given above in ea, 10, (8) One might think off-hand that there should be nine conatants .TI E*('), l,(z), p , ( l ) , plt(z) instead of only five aa printed in eq, 11. However this thought is quickly dispelled once i t is noted that, as before orientational and symmetn requirementa demand that ez(g')e~equal OF, mnoe the Hamiltonian is invariant of e&'), Henoe,

--

Ilr

v

r

sin B cos p

we m w t have [by Table I11 and footnote 51 ezedtzo)= hrCctr ItZ.(;) k + ( l ) W P ) k * @ ) W 1 ) \ = e,(t2,) = N,ct){ t q ; ) E?,(~)II~~(~)&@)nP)I

+

+

+

+

r.b = r b c o s g = Ub,, + b b, tlu b , = r b (cos8cosa t sin8 sincpsina sing,tsin@cosysina C O S I ~ ~ )

.'.c o s 8 = c o s ~ c o s ta sin8sinacos(p-ypo)

Figure 4. 9 diagrammatic picturization of the decomposition of a misaligned ligand orbital into U- and r-parts. Part a illustrates the divisional procedure for a two-dimensional orbital cross-cut in a coordinate plane and part b illustrates that for the t k e three-dimensional orbital Egure in an arbitrary constellation.

T h e Journal of Physical Chemistry

as only one pair of wave functions each can be constructed from the prefactors N e ( t ) t q ! )and N,(,)eo(;). Therefore, E+ ( I ) must equal E- (') must equal E+ ( 2 ) , p+ must equal p- ( a ) , and p+ ( 2 ) must Eequal p - ( ' ) , and so we arrive at eq. 11 and its five tentatively independent variational parameters. Actually, of these five parameters only three, 7. E [equals *lE(:)ll, and p [equals * i p ( k ) ~ ] are symmetrically independent. The proof of this statement is outlined in the text above and in footnote 10. (9) A. D. Liehr, J . Phys. Chem., 64, 43 (1960).

~ T E R A C T I O NOB ELECTROMAGNETIC

RADIATION WITH

N E ( , real ) in eq, 11 we must have e** equals eT. by Table I11

67 1

hfATTER

Now

+ @ ) * I I ~ ~ ( ’ ) + .$:)*II~’(~)~ = N e ( e ) * { e g (+ ;) + 1

implies1’

{

e+(tn,)* = N,cl,*{tz,(;) e,(e,)*

p ( : ) * ~ ~ ‘ ( ~ p) ( ~ ) * ~ ~ ~ ( ’ )

(124 and so with and Ne(,)real, we must have ((’) equals E(2’ equals .$(I)*, p(1) equals p @ ) * , and p ( 2 ) equals p(”*. If we absorb the arbitrary phase factors contained in $:) and ~ ( iinto ) the normalization constants a(;),”anda(;),“of the functionsU,(:)and &(;), (s = +1), severally, where5 aOsTequals a3-rPequals us:* and uo: equals a3-rcequals a3,“* [the latter equality of both the 7r and u set of equalities is one of arbitrary phase choice associated with the normalization process], and call E equal t o or - I$;)/ and p equal to or we may rewrite eq. 11 in the shapelo


t;(z)*,

+

+

I&)/,

+ t[n,“”’ + n,t8(’)11 = N,(,,{e,(:) + PP*~‘” + 11

e*(tz,) = N , ( ~t2,(:) ){ e,(e,)

E & ~ ( ~ ) (12b)

A few final remarks will allow us to deduce the terminal structure of eq. 11 and 12. First, we note that in order for e+(e,) to reduce to the octahedral eg as the angle of cant, a , goes to zero [this corresponds physically to either increasing the “chain length” of the attached polydentate ligand or to cutting the chain of the attached polydentate ligand to produce a group of monodentate ligands in its stead-study Fig. 1 and 21, the sum X+c(l) E&“(’) of eq. 12 must reduce to the special form Z+‘ of eq. 10. That is, that the constante uoEu and agruof eq. 9 must be chosen to have the values uOIu= ay--ru= a3ru*== W ‘ / ~ G (S, = h l ) . Second, we II,t(z)cannot remark that although the sum duce to an octahedral tz, .rr-typeligand molecular orbital as the angle cant, a , becomes zero since the sum vanishes in this limit [Cf. eq. 8. Physically this vanishing is due to the alignment of the ligand orbitals, A,, along the localized ligand orbital-central atom orbital axis, as shown in Fig. 2. Notice, though, that because of this null value of Ilk(;) when a vanishes, we have that ek(t2,) tends to Its octahedra,l tag(;) expression as a tends to zero as needed], it can be made to assess correctly the dihedrality of the orbitals. This assertion is readily proved by the consideration of the dihedral molecular orbitals formed by the bidentate localized pairs (15), (23), and (04) of Fig. 2.

+

+

EO = bo,(15) f b1,(23)

+ bzs(04)

(13)

The eurythmic requirement

e&

O

H

‘s 3

w d S ,L~C, =

E-s,

(S

=

0, +1) (14)

+ w’(23) + ~ ” ( 0 4 )

= bo,(16)

(15)

The identification of the bidentate localized pair orbitals ( k l ) with the pseudo-n-localized orbital pair “ k r 2of eq. 8 and 9 then yields eq. 16.

+

(10) I t is readily apparent that any choice of functions

az t

(1)’

= +v&(l),

nl(z)’

=

8ii.n

&

(2)

I

z

+

(1)‘

=

ei.iiz,(I),

&(a)’

=

eFf?Z*(Z)

will give a satisfactory basis when added to tz@) and e@(;) as these will likewise behave properly under C%(u’) and G ( e ’ ) , for arbitrary and i , as needed. Furthermore, it is evident that this choice willlead to real values of the covalency parameters f(:) and p(2) as desired, since the selection of a real basis

%ee,.(t,,)

=

+

+

iVL(l)’(zet,eUbp’ Wen+*(l)’

gme+(t,,) = N,(i,”(Smtz,af

+ +

%eee+(e,)

=

N,(,)’(%eepd

;rme+(e,)

=

Nlcu)”{3meoa

p’ xrterr+6(2)1} $1)”3m~+5(1)’ + 3m11+~(*)”]

%ez+c(i)’ + P ( ~ ) ‘u i e z + c ( Z ) ’ I

p(l)“

+

3mI;+c(1)’ ~ ( ~ ) “ ~ r n z + ~ ( ~ ) ’ J

plus the requirement that these functions transform correctly under Q(y’) and & ( z ’ ) [that is, that all these e-type functions obey the rela= We, &3me = -3me (when e is a II or 2 function tions 7(% se ds superscript (1) into superscript (2) also), and e&%e = --I/~gee - v1Z/2Sme, %3me = 45/28?ee ‘/&me] shows that NS(t)‘ = N,(t)”, Na(Aj’= Ne(,,”, f(l)’ = E‘”’ = E(l)” = €(2)“, /+I)’ = p ( 2 ) ’ p ( l ) f f = P ( ~ ) ”with all the constants real [as a real set of basis functions plus a real Hamiltonian must yield real variational parameters. The normalization constants can always be picked real 1. The redefinition of the e-type functions as e?, = 2 { W e :-t i3me) then yields the one parameter imaginary e-type trigonal dihedral functions with real variational coefficients as wanted. But note that there are a continuous infinity of such functions which formally d i f e r only in the value of T or i assigned (of course, the nalues of 5 and p are also functions of T and i, individually). Thrs continuous infinity of solutions corresponds in a sense to the formal infinite redundancy of our model, as illustrated in Fig. 5 . From this figure we see there are an infinite number of Dj geometries corresponding to each bonding scheme, that is. to each looalized orbital directive assignment, dependent upon the angle 7 or i of the skewness of the molecular framework with respect to the I‘ZZ‘ plane. It is important t o note though that although a geometry, as far as the orbital bhase requirements are concerned, which schematically looks like that of the optical enantiomorph can be obtained by setting both T and i equal t o ”13 or - 2 a / 3 , this geometry is definitely not that of the enantiomorph ae the associated electronic charge distribution, pictured by the directed ligand orbitals of Fig. 2, is not that correspondent to the enantiomorph but is still that associated with the original conformation under consideration. The true value of T or i for a set of real normalization constants a(”,),” and a(;)r“ to be assigned to the mclecular framework picked here is determined by cubical (Oh) correspondence principles as outlined in the accompanying text. This attendant complication of the trigonal dihedral molecular orbital problem seems to have escaped [because of their neglect of the cant angle a of Fig. 2 or the nonoctahedral placement of the terminal ligand atoms or both] several coetaneous workers in this field [(a) L. E. Orgel, J . Chem. SOC.,3683 (1961); (b) D. W. Barnum, J . Inorg. Nucl. Chdm., 21, 221 (1961); 22, 183 (1961); (c) G. Weber, 2 . physik. Chem. (Leipzig), 218, 204, 217 (1961)l. Mark that it is not, suprising that two parameters and 7 or p and i are needed to specify the trigonal dihedral molecular orbitals as three sets of basis functions t q : ) , & ( I ) , n,@) or e,(;), &(‘I), 2,(2) always require two sets [the third is the normalization constant N e ( t ) or N e ( , ) ]to fix the final three sets of molecular orbit,als [we here use only two of these three possible sets-the two corresponding to the “metallic” bonding and antibonding molecular orbitals]. (11) Note that i Z is not equal to * E s so these functions are not centrosymmetric.

-

~

Volume 68, Number 4

April, 1964

672

ANDREWD. LIEHR

__

~~~~~~~

Table I11 : Central Atom Orbital Transformation Properties Central atom --orbital designation-

oh

81

tzoa


D8

e&')

d and p repreaentations

dr'

=

d3a'a-r~

1 X ' I l N 1 2 ' PLANE BEHIND 21

nt,

&Y.'llN

\

t2

--

Transformation propartieg62(uf)

64cZ)

- '/st*,, i" Q/IW-"%gb

tag&

+

a/8c,r1"t2g,

2 2 ' P L A N E B E H I N D 21

12

+

Comparison with the sum IIs(l) TIs(*)of eq. 9 and 12 shows that for proper dihedrality, the constants U O / and aaSwmust be taken as u-*boS and wSEo,, distribunS@J with tively. Normalization of the sum 119(1) the neglect of localized ligand-ligand overlap finally sets aOITequal to a3-,* equal to agsir*equal to u-*/v'6, (s = 0, f 1 ) . I 2 Letting n, stand for the sum J l k ( I ) 11*(2) when aosTand assrare as determined above, we may put down the end forms of eq. 11 and 12b as

T-0

t

+

t

+

/o

T="on-zn 3

3

T = - L o n2" 3

3

Figure 5. Pictorial representation of t h e geometrical indeterminateness of the trigonal dihedral architecture. The physical angle r of relative movement of the equilateral halves of the skewed prismoidal structure can be put in one-to-one correspondence with the inathematical phase angle 7 of a molecular orbital treatment of their electronic structure. However, as the geometry remains fixed in a molecular orbital phase variation, 7, b u t not in a physical variation, r , this correspondence is not faithful.

The Journal of Phyeical Chemistry

=

+ uno ~ , ( ~ , { t z+ ~(:)

=

Ne(e){eew p z i c 1

al(tzp) = N,,(l)(taua

i3

I

e&,) e A )

+

I(+)),

=

no"-' (17)

where [the synibol S[xl; x 2 ] represents the group overlap of x1 and xZ; their detailed forms are given in t h e Appendix] (12) The normalization expressions when ligand-ligand overlap is not neglected are given in the Appendix.

673

IXTEl,


1.2 The a-.lfoleeular Orbitals. When localized ligand functions are available which offer true ligand-

central atom s-bond possibilities, the above molecular orbital analysis must be amended accordingly. I:igures G and 7a picture the geometry and idealized local ligand orbital distribution, individually, for a typical a-bonded trigonal dihedral complex anion, the copper(I1) trisoxaInto anion, and Fig. 8 illustrates a possible, but not csscntial, level pattern. Again tbc 0- and s-ligand frameworks are canted by an angle e to the central atotn-ligand atom axes, which have been here taken to bc the x, y, z axes for presentational simplicity. To get the requisite trnly s-bonding linear combination of localized ligand orbitals, me proceed as in eq. 1 through 19. We again parse the available localized ligand orbitals when possible. If me denote by the Figure la. T h e diagonal local ligand orbital spatial disposition for t h e idealized planar righbangled geometry of t h e trisoxalatocopper(I1) anion.

t

localized r-like ligand orbitals perpendicular to those of eq. 1 through 8, and which lie in the shaded planes of Fig. 7a canted at an angle p = 90' - a to the z,y, and z axes of that figure, and by ir, the s-type orbital of Fig. 7a which is mutually perpendicular to both A, and A,', we may scribeB4eq. 20 (13) Since the Z* functions are centrosymmetric hut the Ilo(*l, ones are not.ll the only tmly dihedrslly asymmetric functions of eq. 17 are m ( t d . ar. and e&.). This circumstance occurs hecause we have haniahed the eontrihutions of the t d i k e liaand 0-molecular orhitals in our treatment by our assumption that the o-bonded dihedral functions e,(e,) reduce simply to the octahedral o-bonded The omission of t h e e t d i k e o-molecul8r functions e,(;)(e.) oomhinations IS the cause of the different phase factors and the dimerent derivations of the Z* and tl+ ligand molecular orhitals Id. eq. 10. 15. 16. and 191

n,

0 Figure G. T h e idenlired plnnnr right-angled geometry of the triaoxalatoeopper(I1) mion.

(14) This decomposition i s carried out exactly 8s in footnote fi. T o obtain it from footnote 6 one need only mtate the coordinates by 90" ahout either the z axis or v axis thus replacing cos 0 hy sin 8 sin v or sin 8 cos e, consecutively. Please mark that here. in sharp contrast to footnote fi, ambiguity may occur in the final choice of B local liaand orbital basis (view Fig. 7h). The use of an alternant such hasis. l A j , A j < , ~ , { , as for example that of Fig. 7b. rather than the hesis of Fig. 7a. { h J , h j ' , G j l , in the present. slaehraie formulation. leads to no mathematical structural change: The formal theory ahides inviolate whatever the circumstantial looal sDat.ial dirporition 01 ligand orbitals chosen. 01 oourse, definitional modifications do enter. as now eq. 20 and 21 must inelude an altered m a l e of cant. n. not equal to either or 900 , hyhridization coefficients for the orhitals npnWisnd w . ~ and , , an additional nsorhital contribution for the orhitals o*.q of eq. 21. (2

-

Volume 68. Number 4

April. 1964

ANDREW D. LIEHR


674

Figure 7b. The isosceles local ligand orbital spatial disposition for the idealized plansr right-angled geometry of the tris,,xalRtoa,pper(II ) anion.

+ = uyi +

x o l = u%.a xi1

x2

T,O,

+

= ufl

= ova

uri: 'n~

= u.4

r y 2 , x,L

+ ria, xsL

=

+

uz6

TN,

+

~ " 6

(20) where [w =

5 , y,

z;

j

=

urn: = *sin a np,,,

0, 1, 2,3,4, 5Il4

I

r d = *cos a np,,,

(21)

The use of these functions in conjunction with the dialectics of eq. 1-9 and the data of Tables I-V gives the desired dihedral 0-like one-electron molecular orbitals as

z:(')

=

aoT

5

c

2

% . ,:

= U3,"*

z:(2)

j=O

w%:

(22)

j=3

The r-like orbitals generated by the r1( localized ligand orbitals [we now drop the subscript w equals 5 , y, z which is made redundant by Fig. 7a and Table VI are likewise given by

n;(*)

=

1

(crp

5

2

,=0

=t j=n

-1.1

I

I

+

+

+


INTERACTION OF

675

ELECTROMAGNETIC RADIATION WTTH n4ATTER

Figure 8. A schematic iniolecular orbital energy level diagram for a trigonally dihedral transition metal compound in t h e I- andlr-bond approximation.

os,

Also for the pure r-type molecular orbitals, (s = 0, *l),created by the fi, network [Fig. 7a and Table VI, we can scrive

in Da [Le., under c?,,(x') and e&')]as do the central atom orbitals. This orjentation [watch Table 1111 demands that U O , ' " ~ ~ ' equals u3-ru3*'*i equals u3sus'*r*, (s -0, + l ) [the latter equality arises once normalizatioii requirements are prescribed]. The imposition of t h e further stipulation that these a-, 1-bonded ligands have pseudo-octahedral [i.e., that they also transform under e&) and i in Oh as do the central atom-orbitals of Table 1111 a-molecular orbitals, &', equal to 8,1s(1) determines that ao; equals equals u s / 6 6 ; and the stipulation that the s-molecular orbitals IIkc equal to rISc(') (s = + l ) ,sets, by eq. 13-16, aos"equals a3rc*to be u U 5 / 4 6 ,(s = 0, + l ) , where

+

As in the case of a-bonded dihedral molecules, it is necessary to orient thle a- and n-like molecular orbitals of the a- and s-bonded ligands so that they transform

+

Volume 68,Number 4 April, 1964

ANDREWD. LIEHR

676


N o ( 6 ~ ~ l ~ $from ~ ) I Ithe + expressions al(tz,) and ei(tzo), serially, and subsequently employ the definitions, eq. 26.

It will be convenient for our later purposes t o construct r-molecular orbitals which possess octahedral symmetry but are of a trigonal orientation. To this goal we define as previously the functions T ~JIB(*), , II, as those functions obtained either from the functions ~ j p I,I,"'*), of eq. 8, 9, and 17 by dropping the factor k , sin or from the functions T:, IIoC(*),and n[*' of eq. 21, 23, and 25 by dropping the factor cos a. We then find that the sums IIoB(+), I, (Y

+

where fik is the sum fi*(l) equals u3,' equal to 1/46

of eq. 24 with

UO,'

Three things now strike us. One is that we can factor out of the expressions II,s'f', 11,8(:), IIo'(+),and all multiplicative quantities such as k, sin a and cos a, distributively, which appear in them by virtue of the definitions, eq. 8, 9, 21, and 23, to obtain terms of the and {'~T(~)II,(~) where no(+)and are kind qu3*IIO(+) the d i k e molecular orbitals of the type indicted in eq. 26; the second is that we can rearrange the expressions XkC(i), by virtue of eq. 8, 9, 21, and 22, to obtain terms of the kind P ~ ! ~ ( ; ) & ( : ) , where Z;*(;) are the a-like molecular orbitals of the types given by eq. 9 and 2 5 ; and the third is that the multiplicative factors vrrT, p * ( i ) , and and the localized a-basit! orbitals, cj, thus introduced take on the structure pClT(:)

transform under e&'),ez(y'), e&),and i as the species Tz,, just as do their central atom octahedral counterparts and tag(:),sequentially.16 From eq. 24, 26, and 27 we see that the imposition of octahedrality on the n-molecular orbital system determines the coefficients aos' and u3*' in the sum 0, equals f fip(z), (P = +l),to be l/d6, Although functions of the sort given by eq. 26 cannot, of course, be used directly to describe the n-bond molecular orbitalfir of a dihedral compound, they can be used indirectly as will become apparent. The molecular orbitals for a U- and n-bonded dihedral complex may thus be set as

+

+

+

p8~og(+) vc~oL(+) al(tSg) = Na,~t)u'w{t2gn e&,) = N e ( ty) t 2 Q ( t+ ) p ( ; ) r p )+ @II*S(2)

G&(+)}

+

when the coiistants equal a 3 , ~ ~ ~ ~ f +are ,c* picked uniformly [this picking can always be accomplished as all differences can be absorbed into the arbitrary constants qaVc,6,p."(:),$:), and p 8 f r ( : ) ] . A particularly simple form for eq. 28 and 29 is gotten when this uniformity is chosen t o be that of the pseudo-octa~ equals ~ d//y/6, ~ ~ u * ~~ hedral basis for which u aorul*l'*elc

(15) A particularly simple way of deriving these formulas, as well as that of eq. 10, is to substitute the a j . ~ j and , kj. orbitals of Table V , L e . , the paVj and pTwj-type orbitals into the tetragonally oriented eh(e,) oIg,cg, rig, T ~ rlu, ~ , T~~ nuclear displacement formulas tabulated in A. D. Liehr and C. J. Ballhausen, Ann. Phys. (N. Y.), 3 , 304 (1959), and diagonalize these with respect to %(d) [this diagonalization (28) is easily accomplished as these displacements behave elementarily under ea(z'). As an instance, e8Ska.b.e = Skb.c,e,( k = 3, 5 ) , and The reduction of the twelve possible ( L = 1, 2), type e3(xa,xb,xc)= (-xa.-xo,xa),(Xa.b,c = &a,b.o, R l a , b . c ) , SO that ski = 3-'/Z(sk, $- w'sk, wZtSkb), and xt = 3 - ' / I ( X a - W'xb coefficients t o the six and four, respectively, listed in w2'X,), (t = 0, fl) are diagonal]. This method shows that t,he eq. 28 is accomplished by the orientational symmetry rerelated sums no,(-), nou(-),and I&(+', Hog(-) = 1 /v'5{IIo(-) quirement that (?&/)e, equals ef, as &,(-)I, no,(*)= 1 / 2 / (rI,,(*) ~ 7 iW), are of species TI, TI^, and Tzu, respectively [the other components of the T2g, TI,, and T z u v We cad obtain a considerable simplification of eq. 28 like functions are easily generated by the nuclear displacement techif we add and subtract the quantities Nal(2)utT+II~(+) and nique just outlined].

+ 4'"(?)n*"'2' + pfIp+ g x r p ] = Nece)u,r{egw + Pc(;)&c(1) + P c ( : ) ~ * ( 2 ) + pMfl*c(l)

\

+

tkL,

The Journal of Physical Chemistry

+

~

~

67'7


equals CO-'/V%~ and Then

equals 1 / 4 6 , (s

t x ~ ~ '

=

0, +l),l')

?'6(t)(eu) = %(%)r'(*

'/d

e F: io14

r(;)(e,) 76(;)(7)(t2#)

l/i { e+(e,)r'(1/2)

=-

=

3:

+

d%tzoat'(F1/2)

ie--(eu)t'(-

1/81

1

-

d%e,(tz,)6'( Y6(;)(8)(tzo)

=

Et:

4%t z o a t l ( r 1/21

-

4%eF(tzo)i-'(* e+3/4


1/2)

1/:2)

where the octahedrally oriented functions Hag(+), e+(tz,>P'('/z) T 4 t z g ) l . ' ( - '/J y(:)(tz,) = and are as defined by eq. 26, 27 and 10, 31. When d 2 the normalization constants are elected to be real, as (34) earlier, the need that eF and e** be identical implies The substitution of either eq. 11, 12, and 17 or eq. 28, that equals PI*(:)*and f ( 1 ) equals 5 ( 2 ) * , and p u , r ( : ) 32, and 33 into eq. 34 yields the desired a-bonded or a-, equals p o ~ ~ ( f ) *Thus ~ the pairs f q 3 K ( 1 ) and [ u g * ( 2 ) , [ ( L ) a-bonded trigonal dihedral one-electron spin-orbital differ internally only by a and E(2), and p u s " ( ' ) and molecular orbitals. Figure 9 shows the connection of phase factor and nol, in amplitude. The further rethe two schemes, that with and that without spin-orbit quirement that the functions al(ho),e*(tzo), and e*(e,) forces, in both their one- and nine-electron aspects. tend to the octahedrd one-electron molecular orbitals as the angle of cant, a, tends t o zero, demands that $3. The Optical Rotatory Power equals 7' and f(:) equals p ( ; ) in this limit (this truth 3.1 General Concepts. Before entering upon a demay be conditionally confirmed by a sight of the equaltailed exposition of the optical rotatory power of d.iity of the pertinent group overlap integrals imprinted hedral inorganic compounds, it is necessary to recall a in eq. A-12 of the Appendix) so that p r and .$err(:) few basic facts. Firstly, one must remember that plane vanish [cf. eq. 30 and Fig. 21, and that g(1) equals g(2) polarized light is a superposition of right and left circuequals f and p " ' " ( l ) equals equals p"'". For the larly polarized light [Fresnel, 18221 as pictured in Fig. proper dihedrality of the pair a t nonzero 10, and that the rotation of the plane of polarization of equals i',a(2) equals $' angles, cy, we must have light incident upon an optically active medium ["optical also when a is not null [cf. eq. 13-16 and footnote 101. rotation"] is due to the different indices of refraction, Hence, the ultimate florm of eq. 28 is

e+,

1

~

iqjd;)

e-(t2,)e+(e,)l

1 a1(t

2,)

The Journal of Phgsical Chemistry

+

e-(tz,) e- (t2,) e+(t2,) e-(e,)

I1


INTERACTION OF ELECTROMAGNETIC RADIATION WITH MATTER 707

Volume 68,Number 4

A p r i l . 1964


708

The Journal

ANDREWD. LIEHR

of Physical

Chemistru

IXTERACTION OF ELIBCTROMAGKETIC RADIATION WITH MATTER


,-

709

with eq. 91 and 921, as well as several other less direct rela tionships. ( b ) N o Configuration Interaction: Nonzero SpinOrbit Forces. Five Electrons. The utterances relevant to this paragraph are coincident with those of $4.2b. (c) Configuration Interaction: Zero and Nonzero Spin-Orbit Forces. Five Electrons. The blossoms from configurational interaction are synonymical with those of $4.2c. 4.6 A Numerical Example: The Trisethylenediamine- and TrisozaEatocopper(II) Cation and Anion. To affirm that the electrical asymmetries entertained in the antecedent paragraphs are of a suitable magnitude to rationalize the observed optical rotations of transition metal compounds, we shall here present a numerical examplar. Consider the u- and u-, 7-bonded complex ions, copper(11) trisethylenediamine and copper(I1) trisoxalato, respectively, and assume for the sake of simplicity that these are describable by a linear combination of Slater type metallic and ligand atomic orbitals with Slater (effective charge) exponents 7.85 (Cu: 3d),443.80 and 3.90 ( S : 2 s , 2p),454.55 or 4.4456 (0: 2s) 2p, or 2p),4ewith the metal and ligand nuclear centers separated by a distance of 2.00 Also assume that the ligand u-bond 2s- and 2p-like content, le, and le,, are l / z and 4 3 / 2 , each,47that the

Again, as in the four- and six-electron situation, the multiplicity of the spin electronic states precludes any close uniform accord of the five-electron vortical and photical etrengths with those of the one-, two-, three-, four-, six-, seven-, eight-, or nine-electron bevies; but certain relationshibps nonetheless exist in bounded domains. As a tutorial, the four- and five- spin-paired (44) (a) J. Bjerrum, C. J. Ballhausen, and C. K. Jdrgensen, Acta Chem. Scand., 8 , 1275 (1954); (b) C. J. Ballhausen, Dan. Mat. Fys. trigonally ordered cubical electron collections of the Med., 29, No. 4 (1954). Of our two examples of trigonal dihedral optically allowed single-electron hops can be put in a optically active copper(I1) complex ions, only the copper(I1) trisethylenediamine cation has actually been prepared and resolved. one-to-one correspondence with each other so that their Unfortunately, its instability in solutions suitable for optical rotatory vertigal and photica,l strengths become (for the most dispersion and circular dichroism measurements has thus far precluded any close experimental study of these properties [read (c) part) equal when the trigonal and cubical specie subG. Gordon and R. K. Birdwhistell. J . A m . Chem. Soc., 81, 3567 scripts 1 and 2 are interchanged, that is, so that the (1959) 1. Hence, the computations to be described in what follows are at this time hut an indicatrix of things yet to come. AZ(T1,), E*(Tl,) to A2(Tlg), Ei(T,,) transition of the (45) (a) J. Higuchi, J . Chem. Phys., 24,534 (1956) [NHs, ZNeffeotive= four-electron case corresponds to the AI(Tz,), E*(Tzo) 3.801; (b) A. B. F. Duncan, ibid.,27,423 (1957) [Errata,ibid., 39,240 to Al(T2,), E*(Tzg)transition of the five-electron case, (1963)l NHs, ZNefieotive= 3.901. With the latter of these two values and the metal-ligand bond distance of 2.00 A. cited above, the releand so on [three exceptions to this rule exist: one of vant copper(I1) Kna; integrals become the four-electron A$I'lJ, Ek(T1,) to E*(E,) transitions [Z,fr = 3.901 K ~ =A 2.9898 ~ X 10-'0, K3a8 = 5.4703 X 10-11 is four times as strong as the correspondent five-electron K u S = -1.4449 X 10-l0, K7aa = -1.0120 X 10-'0 Kira, = 7.0735 X lop2, K'eap = -2.6143 X 10-2 transition and one each of the four-electron A2(T,,) K32.46 = -4.5274 X lo-' E,(TI,) to A1,2(T2,1g),Eh(T2,1,)transitions is one-third (46) (a) F. 0. Ellison and H. Shull, J . Chem. Phys., 21, 1420 (1953); as strong as its correepondent five electron transition 1. 23, 2348 (1955) [HzO, Zoeifeotive= 4.551; (b) J. Sidman, ibid., 27, It thus follows that the five-electron vertigal and photi1270 (1957) [CHrO, Zoeffeotive = 4.44561. With the latter of these two values and the metal-ligand bond distance of 2.00 A. cited above, cal strengths are also related to those of the two- and the relevant copper(I1) Knaj integrals become seven-electron spin-free (and hence other) situations, [Zeff= 4.44563 K2aa =: 2.3847 X 10-'0, K ~ =A 3.3716 ~ X 10-11 just as were the four-electron strengths of $4.4. In adKaa, = -1.1112 X K7as = -9.7103 X 10-" dition, there is a direct multiplicative relationship beKZ~A,= 6.1327 X lo-', K z e ~ ,= -2.0399 X Kmaa = -3.9575X lo-' tween the two sets of 2 A [~2 T ~ , ( t ~ 0 52Ei[2T2e(t205)] )], to (47) These values are very close to those obtained by J . Higuchi,4ba 2Ei [zE,(t~g4e,) ] five-electron spin-paired rotational and A. B. F. Duncan,,sb and H. Kaplan ( J . Chem. Phys., 26, 1704 (1957). and spectral strengths and the associate sets of [Hartree-Fock nitrogen orbitals]) for ammonia for a variety of differorbital assumptions [their values fork, are 0.551, 0.439, and 0.442, 'E* [2E,(tzs6e,)1 to 'A1[2T~u(t~g5e,2) I, 2E*P T ~ ~ ( t 2 , ~] e , ~ )ent and for lep, 0.855, 0.904, and 0.896, apiece, as compared with our conseven-electron spin-paired strengths [compare eq. 82b venient "average" of 0.500 and 0.866 ]. Volume 68, Number 4

April, 1964

ANDREWD. LIEHR

710

ligand-metal u-,?r-mixture parameters, E, q , and p , are IO, - IO, and - 2/'7I,, serially,48and that the angle of ligand-metal orbital cant, a, is 5'. Under these conservative conditions the rotational, @, and spectral, s, strengths for copper(I1) trisethylenediamine (ZN,tfrotive taken to be46 3.90) in the conformation of Fig. 2 become, in the limit to zero spin-orbit forces

-

-

@L2E(E0)

+ 'E(Tzo)] =

R[2Ai(T2r)+2E(T2,)1 = -0.22B, S['Ai[Tz,] --j 2E(Tzo)]

real]

4%ICp sin a [&A& - 7) + rlKsaalNalct)Nac~) To = E tl 4% kp sin a (43&AB f K7a&VGI(t)Ne(,) E

= =

flz

sin a

f t 4%kP sin a ( ~ ' 3 &as

- K~A&'V.(~)N~(')

Ro = p ~ N a , ( t ) N e ( t )

+0.087B

5.7 X 10-'D2 (93)

p

Fo = e

12

2A~(Tz,)l= -@['E(E,)

S[2E(E,) 'Ai(Tz0) I 1.0 X 10-6D2,S['E(E,) --+2E(T2,)]= 2.7 X 10-'D2 Downloaded by UNIV OF PRINCE EDWARD ISLAND on September 8, 2015 | http://pubs.acs.org Publication Date: April 1, 1964 | doi: 10.1021/j100786a001

where [cf. 83.3, eq. 47, with &, 7, and

mo =

~ Z W ~ N ~ ~ W Nml~ (=~ m ) z,

= ~Z~BN,(~,N,(,,

I n terms of these the appropriate single-electron rotational and spectral strengths are [compare $3.4, eq. 521 @[adtz,) .+

@ M h o+ ) e d e u ) l = nmo, @ [ e d t d-+ededl = r m , ( R [ e d t z O--+ ) e d t d l = rzmz edtdl = --o~o,

and

S[al(tzo)-+e d t ~ J = ?az, s [ a d t 2 , ) +e+(e,)l = TO^, where the symbols B and D denote the fundamental units of rotational and spectral strength, the Biot s [ e + ( t d --+ e+(e,)l = TI2,s[e,(h,) -+ e d e P ) 1 = rz2 c.g.s.1 and the Debye c.g.s.].*@ If further the Their many electron state sums are as cataloged in Tables VI and VI1 [these sums were obtained from eq. 50, 51, 53, 64, 69-71, 80-82a,b, transition frequencies involved, v [ 2 A ~ ( T ~--t9 )2E(Tz,)] 85-89, 91, and 921. The specific numerical values of the individual and V[~E(E,) -t 2A1(T2,),2E(Ttp)] are taken t o be of the electric and magnetic dipole strengths for the Cn(I1) cam may be found order of lo3ern.-' and 1.5 X lo4 em.-', ~ e p a r a t e l y , ~ ~by , ~use ~ of the recorded integral magnitudes of footnotes 46 and 46. [For a: = 5O and I(/ = 171 = IO, p = - l / v % ,the normalization (48) The sign of these constants is determined by the requirement that their associated wave functions have the antibonding radial nodal pattern, As the ligand metal overlap is negative for the e&,) molecular orbital and positive for the a1(tz0) and eh(e,) rnolecular orbitals, this nodal pattern is attained only with the sign choice given above (cf. section VII, $5.7, eq. A-12). (49) We have here introduced a new unit, the Biot, in honor of the great French scientist who first discovered optical rotatory dispersion.$ We hope that it will find favor. [An alternative unit has recently been introduced by W. E. Moffitt and A. J. Moscowitz (peruse, e.g., A. J. Moscowitz, in C . Djerassi, "Optical Rotatory Dispersion," McGraw-Hill Book Co.. New York, N. Y., 1960, Chapter 12). While convenient theoretically, it is experimentally a nuisance because of its nonunit conversion factor, just as would the Debm be if it had been defined inclusive of the magnitude of the electronic charge. The name of Biot has been used, t o the author's knowledge, but twice before as a unit. Unfortunately. one of these two uses is also in the field of optical rot,atory dispersion and again in a mensural Sense (look a t W. Heller, "Technique of Organic Chemistry," Val. I, Part 111, "Physical Methods of Organic Chemistry," A. Weissberger, Ed., Interscience, New York, London, 1960, p. 2299. The other use is in the field of electromagnetism where 10 amp. is sometimes called 1 Biot). However, as this previous employment does not seem to have gained wide use, its present proposed utilization is, to a great extent, relatively free of possible confusion. Other equally good names would be* the Arago, the Cotton, the Fresnel, and the Pasteur. The initials of several of these (Arago, Cotton, and Fresnel), ax luck would havg it, coincide with those already in wide use as common units (the Angstrom, the degree Celsius, the degree Fahrenheit, the Faraday, and the Farad (the Farad is usually abbreviated as a lower case letter)). It i s thus six of one and half a dozen of the other! However, by calling Heller's specific and molecular rotations specific Biot and molecular Cotton rotations designated by [ + ] E and [%U]c, respectively, all confusion can be eliminated, and our proposed new utilization rendered free and clear]. The individual electric and magnetic dipole matrix elements utilized in eq. 93 are


constants become Nal(t)2 = N,(t)e = 0.99105, Ne(.)*= 0.86241 for Cu(II) trisethylenediamine and the ratios of r o : r t : n become 2,34: - 1.34: - 1.001. (50) For recent quantitative estimates of typical trigonal field separations, look a t (a) S. Geschwind and J. P. Remieka. J . Appl. Phys. Suppl., 33, 370 (1962). Other estimates quantitative and qualitative may be found in (b) H. S . Jarrett, J . Chem. Phys., 27, 1298 (1957) [results marred by an error in ligand field formulation]; (c) M.H, L. Pryce and W.A. Runciman, Discussions'Faraday SOC., 26, 34 (1958): (d) S. Sugano and Y. Tanabe, ibid., 26, 43 (1958); (e) S. Yamada and R. Tsuchida, Bull. Chem. SOC.Japan, 33, 98 (1960); (f) T. S. Piper and R. L. Carlin, J . Chem. Phye., 33, 1208 (1980); 35, 1809 (1961); T. 8. Piper, ibid.. 35, 1240 (1961) [Ewata, &id,, 36, 1089 (1962)]; T . 8. Piper and R. L, Carlin, ibid., 36, 3330 (1962); Inorg. Chem., 2, 260 (1963); and in ref. 25c [in these references the trigonal field cleavage is variously referred to as 3K or v, with v equals -3K. So take care t o avoid puzzlement]. Trigonal field intensities have also been estimated by (9) T. 8. Piper and R. L. Carlin, J, Chem. Phys., 33, 608 (1960). The ref. l l b and 50b and many of the references under 601 pertain to studies of the most enigmatic acetylacetonates. Although we have not explicitly considered these compounds in this paper, it might be hoped that the theory here outlined, suitably generalized to explicitly include the molecular orbitals of predominantly ligand character [view Fig. 81, would be serviceable and help t o reveal their secret. [Our theory of inclined bonds is not confined to trisbidentate coordinate situations alone, but IS universally applicable, and hence, usable in bisbidentate and monobidentate circumstances also (as an instance, bisbidentate compounds of digonal dihedral symmetry may have their rotational and spectral strengths estimated by those of the combined trigonal states al(tz,), e+(tz,) f e-(tzg), and e+(e,) f e-(e,), respectively, in the absence of configuration interaction). A survey of some of the riddles involved in such circumstances may be found in (h) R. L. Belford, M. Calvin, and G . Belford, J. Chem. Phys., 26, 1165 (1957); (i) J. Ferguson, R. L. Belford, and T. S. Piper, ibid., 37, 1569 (1962); (j) C. K. J$rgensen, Acta Chem. Scand., 16, 2406 (1962): and (k) K. DeArmond and L. S. Forster, Spectrochim. Acta, 19, 1393, 1403, 1687 (1963); P. X. Armendarez and L. S. Forster, J . Chem. Phys., 40, 273 (1964).] Further chemicals of trigonal dihedral form which are also treatable by our theory and which experimentally and theoretically nettle are discussed in pef. 100 and (1) C. K. JGrgensen. Mol. Phys., 5 , 485 (1962); and (m) E. Konig and H. L. Schlilfer, 2. physik. Chem. (Frankfurt), 34, 355 (1962); K. Madeja and E. Konig, J . Inorg. Nucl. Chem., 2 5 , 377 (1963); E. Konig, to be published.

711

INTERACTION OF ELEICTROMAGNETIC RADIATION WITH MATTER

~~~

Table VI: The Spin-Free d n ( n

=

~~

1, 2, 3, 4, 5, 6, 7, 8, 9) Trigonal Dihedral Rotational and Spectral Strengths for Zero

Spin-Orbit Forces" n

l(s = 2) and 6Cs = 5)


2(s = 3) a n d 7(s = 4)

3(a = 4) and 8(s = 3)

4(s = 5 ) and 9(s = 2)

5

The quantities PO, ro,l.z, Ro, m

. ~me . ~as defined

generally and specially in footnote 49.

the oscillator strengths, f,and the dissymmetry factors,

G, convert to G[2E(E,)+aA1(T~p)] = 0.22 X 10-'~, f[2E(E,) + 'A1(T1,)] = 0.74 X 10-8 G[2E(Eff) +*E(T%,)] = -0.87 X 10-2~, f[2E(E,)+ 2E(T2g)]= 1.9 X loF9 G[2A1(TZff)-+ 2El(T2,)] = -1.5 X 10-2~, f[2A1(T20)--+2E(Tzg)]= 2.7 X (94) where v is the frequency of the measuring light probe in cm.-'. If all other things are kept constant except the effective ligand charge, Zeffeotive, the rotatory etrengths for the copper(I1) trisoxalato anion should be lower than those for the copper(I1) trisethylenediamine cation by a factor of about 0.8, and the spectral strengths by a factor of about 0.65, for both Zo,ffeotive = 4.55 and 4.4456. Of course, all else cannot be kept equal because of true ?r-bond possibilities for the trisoxalato anion. However, the above numbers do show

that the ligand field rotations should be of the same order of magnitude and the same sign for the same ionic conformation. In actual practice the dihedral rotational strengths run from -0.5 to -8B. Hence, the above numerical sample was much too conservative. We know from other sources that the n- and ?r-molecular orbital mixing parameters Ne(e)p,N,,ct)r, are about equal to one anothers1and large52-of the order t/'/z. Therefore, a more faithful numerical estimate would be had by taking /El equal to 171 equal to ] p i equals 1//2 [this (51) R. G. Shulman arid K. Knox, Phys. Rev. Letters, 4 , 603 (1960), et. seg. (52) This conclusion comes from electron spin resonance and optical estimates of u-, rr-bond covalency. Eye (a) J. Owen, Proc. Rog. SOC. (London), A227, 183 (1955), (b) C. J. Ballhausen and A. D. Liehr, J . Mol. Spectry., 2 , 342 (1958) [Errata, ibid., 4 , 190 (1960)], and R. Rajan and T. R. Reddy, J . Chem. Phys., 39, 1140 (1963). (The former reference estimates the hexaamminenickel(I1) covalency to be -38%, and the latter estimates the copper(I1) triethylenediamine covalency to be -50%, for example, which is not far from the values assumed above).

Volume 68,Number I, April, 1964


Table VII: The Spin-Paired d" ( n = 1, 2, 3, 4, 5, 6 , 7, 8, 9) Trigonal Dihedral Rotational and Spectral Strengths for Zero

Spin-Orbit Forces"


INTERACTION O F

ELECTROMAGNETIC RADIATION WlTH

713

n/IATTER

I n this regard, notice that the d386 correspondence inchoice makes the per cent u-covalency, equals N e ( e ) 2 p 2 , subject]. ferred in ref. 55a is well substantiated by eq. 80 and 81 of $4.3 and -43%, and the per cent n-covalency, equals N , , ~ t , 2lor~ 2 4.4 and by the discussion of 84.4. (56) These, in a sense fictious (read A. D . Liehr, to be published), N,ct)2,$2, -33%], and. also by upping the- trigonality by parameters can be fixed, for instance, by a comparison of nuclear ~ . modificationstoward reality setting as to ~ 1 5 These magnetic resonance, electron spin resonance, optical polarization and intensity, and optical rotatory dispersion and dichroism measurewould then increase the rotational etrengths by a factor ments. Such measurements determine N a l ( t ) q , N , ( t ) E , N c ( $ ) p ,k,, of -1Od2, the spectral and oscillator strengths by k,, a, and the sign and magnitude of the trigonal field splitting inde-300, and would decrease the dissymmetry factors by pendently, and their mutual cross-check should be most enlightening as t o the physical actuality of the concomitant concepts involved. -d?/30. The rotational, spectral, oscillator, and disSome such comparisons have already been made in other connections. View T. S.Piper, J . Am. Chem. Soc., 83, 3908 (196l), as a sample. symmetry strengths .would then become of the order lB, (57) This suggestion although arrived a t unaided by the author is not 10-4D2,10-6, and 2 x 1 0 - 4 ~which , are of the order original with him. Others have made the same comment. See, for of what is actually observed.5s [Usually the Slater instance. ref. 25b and (a) S. F. Allason,Proc. Chem. Soc., 137 (1962); (b) J. G . Brushmiller, E. L. Amma, and B. E. Douglas, J . Am. Chem. orbital technique of' approximating the electronic elecSOC.,84, 111, 3227 (1962); B. E. Douglas, J. G. Brushmiller, tric dipole transition moment integrals,. K % A ~under, and E. L. Amma, "Proceedings of the Seventh International Conference on Cocrdination Chemistry," Stockholm, Sweden, June estimates them,62b,b 4 and thus an additidnal amplifica25-29, 1962, p. 42, paper 1E2; B. E. Douglas, R. A. Haines, and tion springs from t'h,is source too. Also, from eq. 169 J . G . Brushmiller, Inorg. Chem.. 2, 1194 (1963); J. G. Brushmiller, E. L. Amma, and B. E. Douglas, ibid., in press; (c) R. D. Gillard, Nature, through 92 it is apparent that other multiplicative fac198, 580 (1963); (dj Th. Btirer, Helv. Chim. Acta, 46, 2388 (1963). tors enter into the calculation for d" systems where n is The optical rotatory dispersion and circular dichroism method in principle allows the determination of the symmetries (and thus not equal to one or nine. Therefore, specific computaorders) of the electronic excited states, and of the singular or multiple tions must be done for each situation separately. These nature of the associated electronic jumps [in this last regard be sure t o read the classic papers (e) W. Kuhn and H. K. Gore, 2. Physik. computations will be carried through in other publicaChem., 12B,389 (1931), and ref. 31c and 55fl. tions. 3 3 ]

IV. Discussion It is apparent from the foregoing theory that the optical rotatory power of dihedrally asymmetric coinpounds can be adequately explained on the basis of an internal inherent dihedrally asymmetric local electronic charge distribution, bot'h in quality and in quantity. It is, however, also evident that this theory unfortunately contains a number of difficultly specified parameters and integrals. Despite these hitches, much useful information may still be gathered from the theory with a minimum of effort. First, the sign of the rotation for a given conformation can be determined by simple in~pection.5~Second, the order of magnitude of the rotation may be eetimated with relative ease. Third, the degree of covalency, hybridity, and dihedrality may be inferred by comparative means.56 Fourth, the atccuracy of spectral assignments may be checked with little effort.57 Fifth, the cleavage of the threefold degenerate electronic states may be gaged and as~essed.~'@Sixth, the proportionate symmetry of ex-

(58) This task may be accomplished by noting the change of sign (or in some cases, the drastic relative reduction in magnitude) of the circular dichroism or dissymmetry f a c t 0 r . ~ 6 ~ J ~ The need for such a change (or reduction) is hurriedly seen from eq. 52 through 92 which show that within a cubical triply degenerate band the two trigonal components rotate oppositely. To expound, we consider the sums l/g ai,,where i(e& equals e&c) or YB(;)(~c),r(;)(e,) and i ( e o ) ,j ( t 2 0 )

j(tz,) equals ai(tz,), e.h(tz,j or y;)(tz,), y~(;)(g)(h0),and where g equals the degeneracy of the ground electronic state [equals two for the spinless orbital set e*(e,), al(tzc), e*(tzo) and four for the spin orbital set w(z)(e&, y(;)(e0j, Y(;)(tzJ, -q;)(~)(t20jl. From eq. 52 and 54 we see that these sums both equal

- -+d3e I B k , 2

sin

01

Nc(e)2(Nal(t)2q 4- ~ , ( t ) 2 E ) ( + d % h 8

+ K7aa)

[This statement follows since the molecular spin orbitals of eq. 34 are related t o the ordinary molecular orbitals of eq. 17 and 33 by a sort of generalized unitary transformation. Thus since by eq. 50 and 51 the sum considered, as well as the analogous one over Si,, which equals 3/se2k,Z

sin2 01

(

d/SK4a,

Ars(e)2 Nal(t)2q2(

+ K,aB)2_+ +

N,(t)zP[4K7~g2 (+d\/aK4A8 - KTAs)'] 1 is an unitarily invariant diagonal trace, the orbital and spinorbital sums must coincide. I n the general notation of footnote 49, the sums over Uiij and Sij are romo T I ~ I rzm2 and r o 2 ~ 1 % T Z Z , separately, and that over the magnetic transition dipole moments 3nij is mo2 m1* mz2. These sums reduce t o

+

+

+

+

+

+

-

(53) For example, consult A. J. MacCaffery and S.F. Mason, Trans. Faraday SOC.,59, 1 (1963). (54) Similar difficulties are encountered in the semi-empirical assessment of the electrostatic ligand field parameter Dq,44 and of the o c t a hedral vibronic oscillator strengths, f [peruse A . D. Liehr and C . J. Ballhausen, Phys. Rea., 106, 1161 (1957), and ref. 22dl. (55) The usefulness of such a determination has been re-emphasized and well illustrated in the recent papers of (a) R. E. Ballard, A J. McCaffery, and S. F. Mason, Proc. Chem. SOC.,331 (1962); (bj A. J. McCaffery and S.F. Mason, ibid., 388 (1962); (c) A. J. McCaffery and S. F. Mason, M o l . Phys., 6 , 359 (1963); (d) Proc. Chem. Soc., 211 (1963). [But be sure t o iread also the classical papers of (e) W. Kuhn and K. Rein, Z . anorg. aZZgem. Chem., 216, 321 (1934), and (f) J. P. Mathieu, Bull. Soc. Chim. Fru'rance. 6 e sbrie, 3, 463, 476 (1936), on t,his

and individually, for the molecular orbital model used here, as may readily be checked], and that for equal covalency, where N a , ( t ) 2 q equals -AN,(t)2f(recall q and E have opposite signs when both are either bonding or antibonding orbitals48),these sums vanish [for the catholic symbolism of footnote 49, this limit implies that all m z ,(i = 0, 1, 2 j , r2 r3 = 0, and that ro2 r12 T * * = 3/zr02 are equal, that T I 1/2(n - rzj21. For unequal covalencies the sum of the al(tPgj and e(tzO) trigonal component rotational strengths is not fully rompensated internally and may attain a nonzero value. As the correspondent sums for all the many electron triply degenerate electronic configurations reduce t o multiples of this one-electron sum [cf. eq. 52 through 921, the contention is proved.

+ +

+

+

Volume 68, Number

4

+

A p r i l , 1964


714

cited electronic states may be e~rmised.~'Seventh, the overlap of optical absorption bands may sometimes be reso1ved.M Eighth, the possibility of obeervable-spinonly magnetic dipole natural optical rotatory power can be demonstrated.61 Ninth, the sum of the dihedral rotational strengths within a triply degenerate band need not vanish.".6z Tenth, for a fixed electronic configuration, d", the sign of the rotation for a given conformation may conceivably change as one proceeds from the first to the second to the third transition series and from ligands in the first to the second to the third, etc., FOWS of the periodic table.6a Eleventh, the feasibility of perceptibile electronic natural optical rotary power in the infrared and microwave regions of the spectrum is manife~ted.'~ Twelfth, the variation of the optical verticity with bidentate (or polydentate for that matter) chain length and bond angle can be predicted,B5and so on. Besides the above the theory here developed has several other important advantages. One, it shows that u- and u-,r-bonded compounds, to good approximation, have mathematically the same formal expressions which relate rotary and spectral power and covalency.

Figure 16. T h e factualistic acuteangled geometry of the trisethylenediaminecopper(II) cation. (Ligand nanplanarity has heen suppressed for illustrative clarity only.)

ANDREW

D. LIEHR

Two, it reveals the mathematical conjugation of the rotary and spectral strengths of the kd" and / ~ d " + (TI ~, = 0, 1,2,3,4, .5), spin-free systems. Three, it uncovers the simple formal multiplicative and additive relations among the rotational and optical capacities of nonconjugative kd" configurations, whether spin-free or spin-paired. Your, it unveils a direct correspondence of the rotational and optical potentialities of the d'.'.6.9 spin-free arrangements and a simple multiplicative connection between these and the d' spin-paired ar(59) The admisssnhle aeometdes of excited electronic states may he enumerated hy symmetry techniques of the kind discussed in fmtr notes 10 and 25 [read A. D. Liehr. J . Phya. Chmr.. 67, 389 (1963)l The assignation BE t o which geometry in actually attained mas he made on the hasis of large deviations from the sum rule of footnote 58. Whenever the sum of the circular diohroisms and dissymmetry factors of the components of B triply degenerate hand do not any where near vanish [that is. whenever the dichroism and dixsymmetry do not come near chsnaina Si&?"* somewhere i n the handl. one of the electronic states must he structurall.v deformed IJshn-Teller or otherwise) or a most unlikely inversion of hondina mid sntihondina pmperties must have occurred. 88 only in these fashions can reduetionsol one rotational strenath with respect toanother hesttniued. [The deformational reduotion is ohtained hy B Frsnek-Cnndon dimunition--see A. D. Liehr. t o he puhlishedl. By this means one would infer that one of the excited states of ehromium(lI1) and cbnlt(11I) is distorted in certain eomplerer.'~ (60) As the circular diehmism and &symmetry factor curves are much s h a m r than the analoaous ahsorption curves occwional optical resolutions miaht he expected [careful diapemion mesurements and analyses can also yield identioal results'g]. Illuatrstively. weak spin forhidden or close spin-orhit component bands miaht he a t times s e p ~ rated out. !SI) As examples. in eq. 55 snd 58 the electronic transitions 5 8 t o Y(;) are rotationally only spin allowed. and those fmm .rs@'(t..) t o -rd"(tiO) and -rd''(hd to -r($)(td owe a nuhntantial part of their rotational StresKth [that portion which is only multiplied by two rather thnn the usual four normalization eonatants) t o spin magnetic transition moments. (62) This situation will be the normal occurrence [cJ. fwtnote 581. (63) This posihility arises from eonceivahle alterations in the r i m of the eleetmnie transition moment inteKdS K.ni without a concomitant change in that of the variational parsmeter € a n d n [since these are determined hy inteards with different spatial weiahina factors] a. onme ehanaes the radial nodal patterns of the central atom or ligand atoms. (64) The electronic transitions which might he active in these regions of the spectrum are the internal l a ~ ( t 4 e, ( t d 1 . { - r P ( t a J . -rr(ttd. -rdtd, - r d ' ) ( t d , and 1 -r&J, -r4eo), -r&)l transitionr and their many electron orhitsl and spin-orhital ~ W I O K S . [In symmetries lower t h m triwnal and eleetmnio configurations ~ r e a t e rthan dl (or lern than d') the possihilities heeome more pronousred. (65) The dependence of the rotation and circular dichroism on the local orbital miamatoh parameter. a, through the functions sin a or 7 Sin a which occur in d l the many electron rotational strenath expressions. eq. 59 thmuah 92. foretellr that a* this anale K O ~ Sto zero and thmuali it. the mtatioii will likewise go to rem. and after the anale a has pa-sed through zero.the mtation will re-emerge with unohanaed sian (as the sian of arid '1 are fixed hy the ~ i e nof sin P look st fmtnote 48 and section VII, 65.7. eq. A-12). Some experimental evidence exists to suhstmtiate the fimt of these forecasts. (F.Woldhye. "l'roeeedinar of the Seventh Iiiteriiatioiinl Conference on Coordination Chemistry. Stockholm and 1.ppsnla. Sweden. June 25-29. 1062.'' p. 41. paper IEI: private commtmicstiorm. For R mort excellent genereli%edsurvey of optiral rotatois and dichroism measurements read F. Woldhye. "Technique of Iiiorannir Chemis try,'' H. R. Jonassen. Ed.. Intersdenre. New York. London. t o he puhlished. A more specialized but equalls fine survey haR also heen given by Th. Borer. H d v . Chirn. Acla. 46,242 (19W.l



rangement. Five, it discloses that electronic transitions to or from bonded and antibonded molecular orbitals rotate electromagnetic radiation both parallely and oppositely, dependent upon conditions.66 To embellish upon the above mentioned theoretical revelations, a few more comments would not be out of place. (a) In normal circumstances configuration interaction can do little to alter the rotational and spectral trends cataloged aloft.87 ( b ) I t is only by figuration interaction that spin forbidden electronic states appear in an optical rotatory dispersion or dichroism spectrum." ( c ) Although formally forbidden to appear in an optical rotatory dispersion or dichroism spectrum on the basis of cubic selection rules, AI to TZ or Az to TI electronic hops may appear without correlational type electronic configurational interaction within the trigonal field if the molecular orbital covalencies of the al(to) and e+(t,,) one4ectron molecular orbitals differ to a sufficient degree!' (d) Due to the one-electron nature of the electric and magnetic dipole operators, no matter how complex the electronic configuration, the many electron theory of optical activity always subsumes the aspect of a one-electron summation.zJ Clearly, further remarks of a similar vein are readily conjured. A t this point it is but logical to inquire as to how the

715

theory may be improved. Several suggestions immemediately advance themselves. (i) The use of improved wave functions, for example, self-consistent field trigonal molecular orbitals should yield a better numerical accord.'0 ( i i ) The employment of the more exact (for approximate wave functions) 21.z2 differential expression hi v,, rather than the simpler multiplicative expresj

sion,

Ce?,, for the electric dipole operator should give j

improved values for the rotatory and spectral powers. ( i i i ) The application of the true ground electronic state molecular geometry to the mathematical development of the theory should produce increased concurrence [cj. IGg. lG-l9].'l (iv) The relaxation of the simplifying restriction that the one-electron molecular orbitals al(tzp), e+(tzo),and e*(eo) retain their pseudo cubical character would make an enhanced concert.'* (v) The inclusion of metal-ligand magnetic transition dipole moment contributions and ligand-ligand electric and magnetic transition dipole moment contributions (66) This remark iollowr irom the linear dependence of the rotational strength. Uiii. on the Sit?" of the variational parametem € and and from its indieial wttisymmetry* [perceive Via. 3 and 8 for examples. In B later puhlieatiom. we shall diticum charge-transfer transitions more iully. le this connection it is interesting to notice that the comparison of the sir" of the optical rotation of B riven bonded to nntihonded and sntihonded to antibonded transition pair. such as are formed hy the de.) and e ( h J psin of Fig. 3. will sflord an erwrimental prmf of the indieial antisymmetry of the rotational strength.

'I.

miii.

-----_

(67) B y unusual means. COnfiKuration interaction can sometimes ehanee the 8i.n Of the optical rotation. ii for instance. i t inverts the Dositions of two nearby electronic ritates of opposite vertioity. (68) 13luridatively. the ZE and Il? electronic states of chromium(II1) and niekel(I1). each. attain -1 to '10% AT1. and "T,. character. and hence. should capture - 1 to 10% of the 'TsO and 'Tu rotational rtresrth. neriately. through spin-orhit coupling. in particular inrtsnaei. [(a) A. D. Liehr and (:, J. Ballhausen. Ann. Phw. (N.YJ, 6 , 134 (1959): (b) A. D. Liehr. J. I'hua. C h m . . 67, 1314 (1963)l. For s recent application of this eoiicept to the 'At, 'TI, state of col,alt(III) view (e) C . J. Ballhausen. Mol. Phya.. 6 , 461 (1963). (69) This fact may he e a ~ i l ymanifested hy the substitution of the relevant expressions irom eq. 69 tlrrouah 89 into eq. 72. (70) For an evaluation of the goodnes of Slnter type wave funetions i n lieand field c d d a t i o n s read (a) 11. L. Belford and M. Kamlus, J . C h n . Phys.. 31, 304 (1959). and (h) 11. 1,:. Watson and A. J. Fleeinan. Phya. Rn,..120. 1134. 1254 (1960). stid related papem. (71) For actual reornetvies three priiwipsl modifications of the theow would enter. One. the 0-molewlnr orbital% as given in ref. 91. 5.3 Local Electronic Transition Moment Integrals. The localized electronic transition moment integrals are not so quickly assessed as were those for the angular momentum, as the integrals have as their only first-order nonzero parts those of the metal-ligand cross-term type. They come in a variety of sorts. To demonstrate the techniques let us consider the integrals KOA, f(2s)4zd2t2 d r , KIA, = f(2px)A,~d8t2 dr

KZA,= ~ ( ~ P V ) Ad7, , ~K~~ZA=,, ZS(2pz)a,~dz?z dr (A-3) The decomposition of the trigonally oriented metallic orbital d,, into its tetragonal basis, 3-‘”(dZv d,, d,,), by means of the transformation matrix printed in ref. 59, eq. 3.4-4, and the use of the cubical symmetry elements then shows that

+

r

-

l

D. LIEHR

+

Thus all integrals may be evaluated with the z-component alone. Second, as ez(y’) [z,y,xI = ( - y , - ~ , -21 and ez(y’) [al,e*l = [al,eF](Fig. 2 and 7 and Tables I and 111),we also havezb30

16

fei,(tzg)* G, ~

d7

=

and so on. 5.2 Angular M o m e n t u m Integrals. The angular momentum integrals are readily evaluated if it is assumed that ligand-metal and ligand-ligand contributions can be neglected [the neglect corresponds to a first-order rotational strength assessment in terms of the variational parameters 5, 7 , and p ] . This assumption has been utilized in the present work to simplify the end expressions; however, it is by no means a necessary one as the integrals neglected may be easily evaluated by the same techniques used for the electronic transition moment integrals to be described later.73b In this approximation the resolution of the angular momentum integrals proceeds rapidly once the metal orbitals are The Journal of Physical Chemistry

As an examplar, reflection [ ~ ( z z ) ] in the X-z plane (y replaced by --y, 1 replaced by 3) of Fig. 2 and 7 shows that f (2s)AD,I~dZ1/ d7 is zero and J(2s)A,~d,,d7 = - f ( 2 ~ ) ~ , z d , , d r , and reflection [u(y - x ) ] in the y-x plane (z replaced by - 5 , 0 replaced by 5 ) of is zero Fig. 2 and 7 shows that f ( 2 ~ ) A , , ~ xddr~ ~ d r , and so and that f(2s)A,~d,yd7 = f(2s)AS~dzy forth. Similarly a counterclockwise rotation of 90’ [e,(z)] about the z axis (0, 1, 2, 3, 4, 5 , z, y, z replaced by 0, 4, 1, 2, 3, 5, Z, - 2 , y) of Fig. 2 and 7 shows that V ’ 1 7 ~ f ( 2 ~ ) ~ , zd7d [KOA,] z ~ = V‘’~~J(~S)A,ZL ~~[KOB~I. (79) H. A. Kramers, “Quantum Mechanics,” North-Holland Puhlishing Co., Amsterdam, 1958.

719


INTERACTION OF ELECTROMAGNETIC RADIATIOK WITH MATTER

--

K I ~ A=~d2 , ~ f (2S)A2,,X&z dr ~

6

K I ~ A ,== , , - d 2 f(%S)Alr3XdxY dr, 3 ~

K I ~ A=~ ,4% , f(2s)aO,,X&2- 112 d r K I S A= ~ , d$ ~ f(2S)A2,XdZz

K I Q A=~ ,-~ 4%S ( 2 s ) ~ ~ , , ~-d112zdzr K20A2,,=

d"/3S(2s)n2,,xdzed r , K Z I A ~ ,4 ~ %S(2s)Ao,,Xdz2&

K 2 2 ~= ~ , -'/3

f(28)~,,,Xdz~ d7,

K23A,,3=

"3

f

(2S)a1,,Xdxy dr

(A-5)

which are straightwa,y obtained by likewise expressing the imaginary trigonal metallic orbitals dn,' and d6+' in terms of the real Cartesian trigonal orbitals dZtzt, dgJz1,d,, - y,2j and d z t y ,[Table 1111, and these in turn rn terms of the real Cartesian tetragonal orbitals d,.,, d,,, d,,, dx2- y2, and d,, by means of the transformation matrix, eq. 3.4-4 of ref. 59. Xote that of these 20 integrals only three, KdA,, K ~ A ,and , K I ~ A are , , really distinct: the remainder are simple multiples of these and of those previously considered in eq. A-3 and 4. The apdication of the counterclockwise 90' replacement rotation e4(x) about the x axis of the last paragraph manifests that not even all of the K,A,, (n = 4,7, 14), are distinct. As a sample K4ii4 =

4%S (2~,).4,xdZzd r e , ( ~d2/, ) S 12pz)a4xdzzd r

=

In truth, such applications show that the sole independent local overlap expressions are KZIA,, K z ~ A ,and , K~zA,. 5.5 Algebraic Evaluation of the Basic Integrals. The requisite basic integrals are directly evaluated by the successive use of the standard bipolar and elliptic coordinate transformations.80 To refresh the memory Fig. 20 and ey. A-7 and 8 exemplify the procedure (a0 is the Bohr radius) for the localized ligand orbital situated on the negative y axis of Fig. 2 and i [atom A,]. 8,

sin

y A = Y . sin ~ YM

Y.~,

cos yAU = ii - B A

COS

(-

Y*, a

3

= -

~~~

(80) See, for example, A. D . Liehr, 2 . NatwfoTsch., 13a, 429 (19,%), and ref. 23b, among other places.

Volume 68, .lrumbcr

4

A p r i l . 1064

ANDREW D. LIEHR

720

sin OA sin sin

qA =

COS

=

cos ya,sin OM sin cpM = -cos y M

sin ya sin p,

sin


COS

i?A

= sin

KIA~ 0

Y A COS

p, cos OM

eM cos q.w =

sin

YM

=

sin y M sin p

cos p

(A-7)

K4Ao,a =

Figure 20. The bipolar coordinate system utilized for the algebraic evaluation of the localized ligand-metal electronic transition dipole moment and overlap integrals.

The employment of these transformations in the integrals leads to their resolution. The emergent expres: sions are as epitomized beneath.


-

(3'x

__ 1 b7i2~/2a0 3d3

IXTERACTION OF ELECTROMBGNETIC

RADIATION WITH

721

MATTER

(3[(A, - Az)(Bo - Bz) - (Az - Ao)(Bz - B4) (As - Aa)(Bi - B3) (A3 - AI)(B~- B5)1 2[A& - ArjBi 2A3Bi - 2A1B3 - 2A4Bz 2AzB4 - B4Ao AiBsl]

+


+

K 3 3 ~ ~ = , *

+

K~ZA K34As,, ~ , ~=,

K35~,,,,

=

- 4 3 K32Ao,6

K36Al,s

=

43 K32Ao,,

+

-2K32A~,~

(A-9)

where b and c equal Z3J3 and Zz, or 2,/2, individually, and where the arguments, u, of the auxililary A, and B, integrals

A,(u)

=

mJl

Ane -"' dX,

B,(u)

=

J:l pne-"" d p (A-IO)

+

are llZii(b c) and I/zii(b - c), separately. These latter integrals may be either be evaluated recursively or by recourse to tables.81 5.6 Group Electronic Transition Moment Integrals. The group electronic transition moment integrals are speedily determined as linear combinations of the localized integrals collected in eq. A-1. Their expressions are T,[e,(;);

zkC(l)] = wT1T,[e,(;); Zic(l)]

=

+

ksKi4~,

Tz[tzga; ZkC(')I =

+ k,

(ksKo~,

T,[tz,(:); Z p ]= -2wT1[tzg(;);

1Cp COS

COS 01

K ~ A ~

KZAJ

B i C ( l ) l=

- ~ Z w = T q t z o a B; p ] T,[tzuo; Z*7

= 0, T,[tz,(:); z*c]= 0

T , [ I I ~ ~ ( epa] + ) ; = T,[II~"'+);egb]*= i w sin

01

-___

4

T , [ I I * ' ( ~ )e,(;)] ;

=

k p ( 2 / 3 Kus

-T,[n, s ( ' 4 *, e,(;)]*

K~A~)

=

~~

(81) M . Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, "Table of Molecular Integrals," Marueen Co., Ltd., Tokyo, 1955.

Volume 88, Number 4

A p r i l , 1.964

ANDREW D. LIEHR

722

2

T , [ I I ~ " + )tzg(:)] ;

= f-

2

IC, sin a w kl ( K -~K ~ ~A J ~

+

~

~

)

+

~


+

T,[tzg(:); II*~'~)I = - 4% IC, sin a ( K ~ ~w , " ~ ~ Tz [tzd:); n,'(')I = 4%IC, sin a ( K ~w ' ~l ~ ~~~ , ) T,[tz,(g); nT*(l)] = - w"T,[tz,(:); IIP] = 1 k , wF1 sin a (KZA6 K3Aa) (A-11)

46

where T,[xl; XZ] is an ersatz for JxI*xxz d ~ ,and where the functions &c'1,2), no(+), H,s(132) are as defined in the text with/ a(:),"'"l equals d%6, (s = 0, fl). c(2) The integrals T,[xI; 25 I, (XI = e&), tsgar tzg(:)),are the negatives of those for & or i c ( l ) because of the inversion relation i&'(l) = i.c(2) [cf. Table IV]. The group electronic transition moments T,[tz,(:); II*s'1,2)]do not appear in the calculations as the matrix elements (e+(t2,)$ei(tzu)) vanish identically [this fact is readily proved by use of the Hermiteness of these matrix elements and their transformation properties under ez(y'), where e&') [d, y', x', e*] = [-x', y',

-x', ei] and b

-

by'

6

=

i' + b k'),

e(x'P'

+ y'j' +

2%')

or

? (%f'ax' + t

dependent upon whether the electric

dipole length or velocity is employed in the computado not appear beti0ns].~5 Those involving 2*c(132) cause of the centrosymmetric nature of &' = &,'(')

+

&'(2)

5.7 Group Overlap Integrals. The group overlap integrals are promptly enumerated as linear combinations of the localized overlap integrals gathered before. Their forms are

~

(82) See ref. 73b, topic


B, Appendix.

Interaction of Electromagnetic Radiation with Matter. I. Theory of

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