Chapter 73 Chapter 73
4. The non-invariance groups of Racah 1. Interim experimental developments
The decade 1935-45 saw steady progress in the experimental spectroscopy of free rare-earth atoms and ions. Russell et al. (1937) were able to find almost all levels of the six configurations 4f5d, 4f6p, 5d z, 4f6s, 4f6d and 5d6s of Ce III, and, a little later, Harrison et al. (1941) used Zeeman data to help them identify low levels coming from 4f5d6s, 4f5d 2, 4f26s and 4f25d of Ceil. For P r l l , Rosen et al. (1941) gave all the levels belonging to 4f3(4I)6s and 4f3(4I)5d 5L, K. Levels from the analogous configurations 4f4(5I)6s and 4f4(51)5d of N d l I were reported by Albertson et al.
(1942). Somewhat earlier, Albertson (1936) had found 40 levels coming from 4f6(TF)6s and 4f6(VF)5d of Sm II, and he proposed a new scheme for the ground configurations of the neutral lanthanides, only two of which (Pr I 4f25d and Tb I 4f85d) have subsequently proved to be the first excited configurations instead.
Daring as he was to choose more ground configurations of the type 4f N, he did not go quite far enough. His proposal, although a step in the right direction, was in any case ignored by Meggers (1942) in his review of rare-earth spectroscopy. Albertson et al. (1940) found 4f7 (sS)5d6s l°D and 4fv(sS)5d6p X°F of Gd II, though some levels of their 8D and l°D deriving from 4fv(8S)5d6s and 4fT(sS)5d6p are spurious. The earlier work of Albertson (1935) on Gd I was extended by Russell (1942), who found the undecets 11p, X~D and ~ F of 4fT(SS)6s5d6p as well as some nonets. Oddly, Russell (1942) refers to the 6p electron as 7p.
In contrast to that inconsequential lapse, an array of major misinterpretations afflicted the spectra of lanthanide ions in crystals. Ellis (1936) proposed identifi- cations for the principal absorption bands of eight lanthanides, most of which are erroneous. Unaware of the severe perturbations in La II 4f 2 from neighboring configurations and the misidentification of 1D 2 (see section 2.2.3), Bethe and Spedding (1937) simply scaled the term scheme of La II 4f 2 and adjusted the spin- orbit coupling constant ~ ( = (~(r)~) to predict the levels of the conjugate con- figuration 4f 12 of Tm 3 + in Tm 2 (SO4)3" 8HzO. The correspondence that they drew with experiment is almost wholly fallacious. We now know (Dieke 1968, or H/ifner 1978) that the levels identified as 3F4, 3F3, 1D2 and 1I 6 a r e really 3F3, 3F2, 1G4 and 1D2, respectively. Only their supposition that part of the absorption band at 21500cm -~ should be ~G4 is correct. A subsequent analysis of Pr 3+ 4f 2 by Spedding (1940), based on a similar connection to La II 4f 2, is almost as bad. As in the case of Tm3- +, the calculated positions of 1D 2 and 1G 4 almost coincided, whereas, in reality, they are separated by about 7000cm -~. The actual 1D 2 level is misidentified as 1I 6. Other spectroscopists made equally mistaken analyses. Lang (1938) placed both Xl 6 and 1G4 or Pr 3 ÷ 4f 2 in a region now known to correspond to
ID2. Gobrecht (1938) gave a different analysis from that of Bethe and Spedding (1937) for Tm 3 + 4f 12, labelling the 3P 2 level 116. Gobrecht's over-sized scaling of the theoretical scheme had the effect, however, of accidentally making a correct assign- ment for 1D 2. This level appeared unnaturally low in the analysis of Condon and Shortley (1935) for La II 4f 2 owing to the misidentification discussed in section 2.2.3.
Among the few articles of that period that can be read today without a sense of unease are those that are purely experimental, such as that of Ewald (1939) on the neodymium bands. By 1945, the puzzle of Van Vleck (1937) concerning rare-earth spectra in solids remained as baffling as ever.
4.2. A precursor to Racah's analysis: Weyl's use o f characters
It is clear from the foregoing that, by the mid-1940's, no particular pressure to develop the theory for the general lanthanide configuration 4f N was supplied by experiment. Racah's motivation to press ahead, and erect a totally new theoretical structure in his article of 1949, came from the challenge of the theoretical problem.
Evidence for the need for some new approach, such as the unexpected simplifi- cations mentioned in sections 3.5 and 3.6, was accumulating. The use of Kramers's symbolic method (see section 2.2.2) by Bijl (1945) to analyze the configuration 3d94s5s of Cu I proved to be very unwieldy. It was Racah's genius to see a solution in the use of non-invariance (or non-symmetry) groups, that is, groups whose generators do not commute with the Hamiltonian.
Racah (1949) was not the first, however, to use non-invariance groups in atomic physics. Weyl (1931) had done so some twenty years earlier in one of the more arcane chapters of his book on groups and quantum mechanics. T o get an idea of Weyl's method, we use it to derive the allowed quartets of f3. The central idea is to use characters. F o r the irreducible representation ~l of SO(3), the character Zt(~b) corresponding to a rotation through an angle q~ can be found by choosing the z-axis as the axis of rotation. Every single-electron orbital eigenfunction ~91m behaves like
Ylm(O,
~D), s o w e haveif/tin ~ ei"~ ~lm (37)
under the rotation in question. The character, being the trace of the transformation matrix, is given by
~(t((~ ) = ei/~ + eifl-l)~b Ar - . . . + e-it~ (38)
F o r U(21 + 1), the unitary group corresponding to transformations among them- selves of the 21 + 1 eigenfunctions ~n, ~b._ 1 .. . . . ~ _ j , the substitution given by (37) is replaced by
~ktm ~ exp(iq~,,)qJl,,. (39)
We now have 2l + 1 angles (kin rather than just the one q~. If, for 13, we pick M s = a2, all mt values must be distinct. Thus the orbital parts of the three-electron eigenfunc- tions possess a character.
~exp(i~bm + iqS,., + i~bm,,), (40)
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 103 where the sum is constrained by I > / m > m ' > m">~ - I . F o r l = 3 there are 35 terms in the sum. When the transformations of U(7) are limited to those of its subgroup SO(3), (39) becomes (37). That is, qS,,=mqS. Making this substitution in (40) gives
~6 + E5 + 2e4 + 3~:3 q_ 4,~2 _I_ 4e + 5
+ 4 ~ - l + 4 e - 2 + 3 e 3 + 2 g , , + ~ ; - 5 + ~ - 6 , 1411 where e = e~L From (38) we find that this sum is
z6(4,) + z,(4,) + z3t,~) + z2(q~) + zolq~),
from which we deduce that the quartets of f3 are I, G, F, D and S, in agreement with the analysis of section 3.1. Weyl's actual handling of this problem is rather more sophisticated than the simple presentation given here, and he was able to obtain a general solution for finding the quartets and doublets of 13 for arbitrary I.
Racah (1949) examined the possibility that an intermediate group X might exist in the g r o u p - s u b g r o u p sequence
U ( 2 / + 1) = X = SO(3). (42)
The usefulness of a group X was never explored by Weyl. His preoccupation with arbitrary l seems to have distracted him from the enormously important special cases for which l = 2 and 3. Racah's realization that X can be chosen to be S O ( 2 / + 1) makes very little impression in the general case. F o r example, there are four 2I terms in i 3, only one of which can be separated out and assigned a different irreducible representation of SO(13) from that for the other three. However, for d 3 the only ambiguity arises for the two ZD terms, and the possibility of assigning them different irreducible representations of SO(5) solves a classic dilemma in an extra- ordinarily elegant way. What is more, all ambiguities in the d shell can be resol- ved with the choice X - SO(5), which turns out to provide an equivalent description to seniority for the states of the d shell (Racah 1949).
As we have already noticed (in section 3.6), the terms 2D, 2F, ZG and 2 H each occur twice in f3. Only the 2F pair can be separated by using SO(7). By a remarkable accident, the exceptional group G2 of Cartan (1894) exists as a subgroup of SO(7) and also contains as a subgroup the particular group SO(3) corresponding to rotations in physical space. Thus, for f electrons, the sequence
U(7) = SO(7) = 62 = SO(3) (43)
of groups and subgroups can be used to classify the states of fN. Racah (1949) found that the terms ZD, 2G and 2H of f3 c a n be separated by assigning them irreducible representations of Gz. We can thus write a state of fN as
[fN w u r S L M s M L ) , (44)
where W and U stand for irreducible representations of SO(7) and G2, respectively.
The additional classificatory symbol z is needed to separate a few pairs of like terms when 5 ~< N ~< 9. Racah's discovery of the usefulness of G2 made the analysis of the f shell tractable.
4.3. Racah's 1949 paper
The problem that Racah (1949) solved by using non-invariance groups was the calculation of the energy matrices for the Coulomb interaction between f electrons.
The idea is to replace the expansion
e 2 ~. (i/rij) = ~ F k ~ Pk(COStOij), (45)
i>j k i>j
which follows from eq. (29), by
e 2 ~ (1/rij) = eo E° + el E1 + e2 E2 + e3 E3, (46)
i>j
where the operators e r are the components T [wv] of the generalized tensors T [wUl corresponding to unique irreducible representations W and U of SO(7) and G 2, respectively. The E" are linear combinations of the F k. The advantage of using the tensors T [wU~ is that the Wigner-Eckart theorem can be applied to relate the matrix elements to C G coefficients for SO(7) and G2. The only complication is that, in the evaluation of the matrix element
( " ' W 1 U 1 . . . 1 T[pWU]l"'W 2 U 2 "" " ) , (47) the irreducible representations I4/1 and U1 sometimes occur more than once in the decompositions of their respective Kronecker products W × 14/2 and U × U2. The simple relationship for SO(3) between a matrix element and a C G coefficient, as exemplified by (30) and (31), no longer always holds: instead, a sum of C G coefficients appears. There are as many terms in the sum as the multiplicities of the Kronecker products demand. In spite of this complication, there is an enormous gain in efficiency, since the labels W 1 U 1 W U W 2 U 2 appearing in (47) may recur in various configurations fN.
In order to put these ideas to use, Racah (1949) had to generalize the mathematics of SO(3) to SO(7) and G2. Because of the great impact that this has had on many aspects of theoretical spectroscopy, it seems well worthwhile to describe his work in some detail.
4.3.1. Two-electron m a t r i x elements
Powerful though the generalized Wigner-Eckart theoi'em is, we are no further forward in the calculation of the matrix elements of the e, if the relevant C G coefficients are not known. N o tables for the C G coefficients for SO(7) or G2 were available in 1949; and even today we have only a very limited knowledge of them.
One way to proceed is to invert the problem and calculate the matrix elements of the er by other means: we can then deduce sets of C G coefficients that may be used in the future when similar representations of SO(7) and G: appear. Equation (1) of Racah (1949) specifies the basic mechanism for calculating matrix elements step by step along the f shell. In a slightly different notation it runs
(fUOlerlfN~ ' ) = [ N / ( N - 2)] ~
(~{1~/1)(1~1 le'rl~2)(02l}O').
(48)~//! 02
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 105 The states ~1 and ~2 belong to the parent configuration fN-1. The operator e'r acts on all pairs of electrons not involving the Nth electron. The ratio of the number of such pairs to all pairs is given by
'~-'C2/NC2 = ½(N - 1)(X - 2)/½N(N - 1) = (N - 2)/N,
which accounts for the square-bracketed factor in eq. (48). T h a t equation relates the matrix elements for a configuration fN to those for its predecessor fx 1, and hence provides a means of advancing along the shell.
The obvious difficulty with this approach is that the cfp (~9{t~1) and (~2 ]IV') have first to be worked out. Although we seem to have replaced one difficulty with another, the cfp are C G coefficients too: we have only to enlarge the group structure so that all the states of a particular configuration fN form basis functions for a single irreducible representation [2] of some group containing SO(7) and G 2 as subgroups.
The obvious choice is U(14), the unitary group acting in the space of all fourteen spin-orbitals. The cfp ( ~ { t ~ 1 ) c a n now be visualized as the C G coefficient
([;qpl[;,, ]p,, [f]p0,
(49)where [ f ] p f a r e the labels defining a state of an f electron. Such labels are a m o n g the most elementary that we are likely to encounter. The C G coefficients in which they appear should therefore be a m o n g the easiest to calculate.
4.3.2. Racah's lemma
The correspondence between (~b{lOl) and (49) is not quite exact because the definition of the cfp given in eq. (35) involves states that are coupled in the spin and orbital spaces. It is m o r e accurate to write
(~{Iqz,)(S M s IS, Ms, , sms)(LM t [L, MLl, fmf)
= ([,~]P[[,~I]P,, [f]Pf). (50)
This connection reveals an important property of the generalized C G coefficients:
they can be factorized. In section 3 of his paper, Racah (1949) proved this for a group G (with irreducible representations A) containing a subgroup H (with irreduc- ible representations B possessing bases defined by labels b):
(~ A fl B b ] A1 fll B, bl,A2fl2 B2b2)
= ~ ( T B b ] B , b , , B 2 b z ) ( ~ A f l B [ Alfll B1 + A2f12B2),,.. (51)
7
The symbols a and 7 are multiplicity labels that are required if A and B occur more than once in the respective decompositions of A 1 x A 2 and B 1 x B2. The symbol fli is a multiplicity label of a different kind: it is required when the reduction G ~ H yields more than one identical irreducible representation Bi in a given Ai. Like the proof of the W i g n e r - E c k a r t theorem, the derivation of (51) depends crucially on Schur's lemma. The second factor on the right-hand side of eq. (50), for want of a better expression, has become known over the years as an isoscalar factor, in analogy to the situation for SU(3) (Edmonds 1962).
For us the simple relation G ~ H can be replaced by the very much richer scheme U(14) ~ SOs(3) × U(7) ~ SOs(3) × SO(7) ~ SOs(3) x G 2
SOs(3) x SOl.(3) ~ SOs (2) x SOL(2), (52)
which separates the spin space from the orbital space. The groups SOs(2) and SOL(2) are included for completeness, since they provide the labels M s and ML.
The factorization of the isoscalar factors and hence of the cfp, by itself, brings us no nearer to finding their numerical values. It is here that the detailed properties of the representations of G 2 and SO(7) play a role. Racah, of course, had to discover many of these properties for himself; today we can refer to the tables of Wybourne (1970) for much of what we need. Because of the variety of aids now at hand, it is quite difficult to get a sense of the problems that Racah faced. No tables of 3--j or 6-j symbols were available in 1949, though their algebraic forms were known, of course.
It therefore seems worthwhile to show how some non-trivial isoscalar factors can be calculated by elementary techniques. We cannot know in detail how Racah built up his tables, but we can at least glimpse the kinds of methods available to him.
4.3.3. Sample calculation of isoscalar factors using Slater determinants
F o r our example, we pick G2 and its subgroup SOL(3). The irreducible repre- sentations U of G2 are defined by the coordinates (Ul u2) that specify the highest weight of the representation. They are the analogs of the maximum value L of ML that is used to label the irreducible representation ~L of SOLO) (see section 2.2.1).
Racah needed to know the structure of Kronecker products of irreducible repre- sentations of G2 as well as the decompositions of the representations into direct sums over the ~L. It is likely that he pieced together the required formulas from general expressions for the dimensions of irreducible representations of G2, such as those given by Weyl (1925) for an arbitrary Lie group. T o d a y we simply turn to tables E-4 and E-3 of Wybourne (1970) to find
(10) x (10) = (00) + (10) + (11) + (20), and
(00) -'~ ~ 0 , (10) ---~ ~ 3 , ( 1 1 ) ~ ~1 + ~ 5 , (20)---~ ~ 2 + ~ 4 + ~ 6 . (53) Thus the terms of f2 can be assigned G2 labels as follows:
(00)1S, (ll)3p, (20)lD, (10)3F, (20)1G, (ll)3H, (20)11. (54) There are two 2H states in f3: one belongs to (IlL the other to (21). Wybourne's table E-4 reveals that (11) × (10) does not contain (11) in its reduction: thus (f2 3Hi}f3 (ll)2H) is zero, and the symbols 7 and y' of section 3.6 can be given a precise meaning, namely, 7 -- (11) and " / = (21). We can construct if3 (21)2H) by actually carrying out the coupling implied by if2 2H, 2f, 2H); but since there is no 4H state in f3 we can ignore the spin space and simply take Ms~ = + 1 for f2 and
A T O M I C T H E O R Y A N D O P T I C A L S P E C T R O S C O P Y 107 TABLE 1
Coefficients of Slater d e t e r m i n a n t s in the expansion of states 2H of f3 for which M s = ½ and M L = 5.
D e t e r m i n a n t States of fa
1(21)2H) If z ID,f,2H) If 2 I G , f , 2 H ) if 2 ll,f,2H) {3 + 2 + 0 } (30/91) 12 - ( 5 / 2 1 ) 1/2 (240/100111/2 (50/3003)1 2 13" 1 + 1 } - (36/91) 1/2 (2/7) 1'2 - (288/1001) 12 - (20/1001 t 1 2 {2 + 1 + 2 - } (20/273)]/2 0 (105/286) 1 ~2 - (243/1001 )1 z 1 3 + 0 + 2 } (40/273) 1/2 - t 5 / 2 1 ) 1~2 (15/4004) 12 37/(600611-' 13 + - 1 * 3 - ] - ( 5 / 2 7 3 ) 1/2 (5/42) 1/2 (135/2002) I'2 - 3 2 / ( 3 0 0 3 ) 12
12 + 0 + 3 } - (10/273) 1/2 0 _ (105/572)12 (243/200211 2
m~ = - ½ for f. Thus, for M s = ½ and M L = 5, we get
]f3(21)ZH, 1~,5) = ~ (5Mt.~, 3mt155)lf2H, 1, ML];f, --½, m t )
M LI ,ml
= • a i [ m i + m i ' + m l - }, (55)
where the sum runs over the various Slater determinants for which m'~ + mi' + rn~ = 5. The non-vanishing coefficients are given in table 1.
By a similar coupling process we can work out If 2 tL, 2f, 2H) for L = 2, 4 and 6.
The results are listed in table 1. As indicated in (54), the states 1D, 1G and lI of f2
belong to (20) of G2. Thus the overlaps
( f 2 ( 2 1 ) 2 H l f 2 1L, 2f, 2 H ) ( 5 6 )
must be proportional to the isoscalar factors
((21)H [ (20)L + (10)f). (57)
The former can be easily found from table 1. They turn out to be - ( 1 3 / 1 8 ) 1/2 , (8/11) 1/2 , (5/99) x/2 for L = 2, 4, 6, respectively.
These numbers can be normalized by multiplying them by (2/3) 1/2 . In this way we find that the isoscalar factors (57) are given by
- ( 1 3 / 2 7 ) 1/2, (16/33) 1/2, (10/297) 1/2 for L = 2, 4, 6, respectively, (58) The magnitudes agree with the original calculation of Racah (1949, table IVa), but the signs are reversed. This illustrates an extremely c o m m o n problem in all branches of spectroscopy: establishing phase conventions. Every state q; is associated with an arbitrary phase, and it is impossible to be sure that one's own choices coincide with those of others unless some specification is made. Fortunately, the tables of Racah (1949) for fN do just that. T o bring our results in coincidence with those of Racah we have only to replace our proportionality constant (2/3) 1/2 by - ( 2 / 3 ) I/2
4.3.4. Reciprocity
All kinds of isoscalar factors can be calculated with sufficient ingenuity. The proportionality factor - ( 2 / 3 ) 1/2 of section 4.3.3 is itself an isoscalar factor for
U(14) = SOs(3) × U(7). Many more isoscalar factors can be calculated by generaliz- ing that particular symmetry property of an SO(3) C G coefficient that interchanges a component ~s with a resultant ~ j :
(J MsIS Ms, LML) =
( - 1) L+ML [(2J + 1)/(2S + 1)] 1/2 (S - M s [J - M j , LML). (59) Edmonds (1957, eq. 3.5.15) has shown that this equation can be derived from the recursion relations satisfied by the C G coefficients. In seeking to generalize it, Racah (1949, section 5.1) used a result of Wigner (1931) for SO(3). The C G coefficients arise when triple products of the elements ~ , ( o ~ ) of the rotation matrices are integrated over the Euler angles ~, suitably weighted. This procedure has the advantage that the three numbers of the type j that enter do so on a par. The disadvantage is that the result of the integration yields a product of a pair of C G coefficients rather than just one. In spite of this complication, Racah was able to extend the procedure and obtain analogs of eq. (59) for the isoscalar factors. For example, his eq. (49) runs
I U ' r ' L ' I u r L + r i o ) f ) =
( _ 1)L - L, [ ( 2 L + 1)D(U')/(2L' + 1)D(U)] 1/2 (U z LI U' r' L' + ( 1 0 ) f ) , (60) where D(U) is the dimension of the irreducible representation U of G2. Equations such as this are very useful for finding new isoscalar factors from those already known.
4.3.5. Calculation of the e,
A c o m m o n difficulty in appreciating Racah's use of groups lies in the transfor- mations themselves. It is not too hard to understand how the seven orbital wavefunctions ~fm (for - 3 ~< m ~<3) can be subjected to the unitary transfor- mations of U(7), or how successive constraints on these transformations lead to the subgroups SO(7), G2 and SOL(3); but to assign the group labels W and U to operators we need to know how the spherical harmonics Ykq(Oi, C~i) stand with respect to SO(7) and G2. Substitutions of the type (0~, ~b~)~ (01, ~bl) merely yield transformations belonging to SOL(3). Some generalization is required.
We can take a step in that direction by first recalling (from section 3.3) that the Coulomb interaction between two electrons involves sums over products of the type Y'q(01, q~l) Ykq (02 ,1#2 ). The ranks k match the labels of the Slater integrals F k and are thus limited to 0, 2, 4, and 6 for f electrons. If, however, we consider the trans- formations among the 28 components of Yo, 112, Y4, and Y6, we are led to U(28) and its subgroups. This generalization evidently goes too far.
The correct approach is to consider the operators that perform the transfor- mations of the particular group under study. In analogy to exp(iOkJk) of section 2.2.1 for SO j(3), we introduce operators of the type exp(i0tpV~), where the tensors V "~ are the generators of the group and the generalized angles 0tp are its parameters. Since our Lie groups work on the single-electron orbitals ~br,~, the generators are
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 109
operators that act on one electron at a time. T h a t is,
N
V " j = ~ v~ ~ (61)
i = 1
for fu. F o r U(7), all possible values of t are called for: that is, 0 ~< t ~< 6. As we proceed to the subgroups of U(7), operators are successively rejected from the set of operators. Just how this is done was not made explicit by Racah in his 1949 article, since he defined his Lie groups by specifying the algebraic forms that they left invariant. However, the paring away of the generators is remarkably simple: for SO(7) we have t = 1, 3 and 5; for G2 we retain t = 1 and 5; while for SOL(3) we are left with t = 1, corresponding to the vector V "1 that is necessarily proportional to the orbital angular m o m e n t u m L. The crucial connection can now be made. It turns out that the c o m m u t a t o r s of the generators of SO(7) (or G2) with the spherical harmonics Y2q, Y4q and Y6q, suitably renormalized, produce residues that exactly match the results of acting with these generators on the components of the states 1D, I G and 1I o f f 2 for which M L =q. We have already assigned the G2 label (20) to 1D, ~G and ~I in section 4.3.3; it follows that the same label can be attached to the collection of 27 spherical harmonics Y2q, Y4~ and Y6q. Precisely similar methods allow us to also assign to these harmonics the SO(7) label (200), which exactly encom- passes (20) of Gz.
The rest follows without too much trouble. The products of spherical harmonics arising from P2(cos~oij ), P4(cos(oij) and P6(cos~o~j) must transform under the operations of SO(7) and G2 like those parts of the Kronecker squares (200) 2 and (20) 1 that contain an SOL(3) scalar. F r o m tables D-4, E-2 and E-3 of W y b o u r n e (1970), we find that the acceptable irreducible representations are given by
W U - (000)(00), (400)(40), (220)(22). (62)
The linear combination of the Pk (COS O~j) corresponding to a particular W U entails the isoscalar factors (WU0[ (200)(20)k + (200)(20)k) as well as the renormalization factors for the Ykq. Only one complication remains, and it is a very minor one. The constant Po(cos~o~;) is also present in the expansion of eZ/r~j and this produces a second operator for which W U =- (000)(00). The particular mixtures picked by Racah (1949) for eo and el simplify the expressions for the energies of the terms of m a x i m u m multiplicity. Putting the pieces together, we can relate the parameters and operators of eqs. (45) and (46) by writing
E ° = Fo - 10F2 - 33F4 - 286F6, E 1 = ~(70F2 + 231F4 + 2002F6),
E 2 = ~(F2 - 3F4 + 7F6),
E 3 = ~(5F2 + 6F4 - 91F6) (63)
for the parameters and e, [(000)(00)] =~o,
e, [(000)(00)] = ~ (594Zo + llZ2 + 6-~4 + 36),