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Generalized selection rules: An augmentation procedure for the determination of symmetry invariants
considered_
lowed” in the .OSR~ stage_ The ~overkl procedure satisfies a sort of correspondence principle in that the OSR are effectively isomorphic with the~classi-Cal selection rulek of. the corresponding classical observable, if one exists_ Both stages of the GSR reiy on a basic concept of a symmet.$ inuariark oZ a group G, which u-e may define as a quantity or function F’(G) which is symmetric to every operation of that group. Ali observables cf a system with symmetry G mu& by -definition, be such invariants, or otherwise. a method would exist for showing that the system is of lower:symmetry_ In the context of the GSR approach, the OSR stage requirks the determ&ration of invariant open&r products; and the SSR . stage that of invariant inkgrands_ :The practical determination. of the invariant function Fe(G) corresponding to a general function F under a covering’symmetry G is therefore central to analytic implementation of selection rule procedures, and is by no means-the trivial problem that it may superficialiy appear to be The prac+ 1 ‘.- cal procedure for the determination of. -e’(G) in the initial GSR paper is based -on transformation tables for- unit 7polarand :axiai.vectors. .but it+ .‘ .. implementation becomes rehxively tedious :alg+ i. braically- for- increasingly cornpI+ fm+ions ore _ increasing group size.7A s&iIar argument -applies _- to vector couplin&rreducible @&or me;h&s @J,
1,Introduction Selection rules for spectroscopic and interaction phenomena generally increase.in compkxity with the order of $&rbation theory and/or m@ipole expansion of the perturbation operators in terms of which the theoretical description is developed_ Incorporation of symmetry arguments for disparate phenomena (for. example, induced circular dicbroi.ml [1,2]. higber-oxkr spec troscopic processes such as scattering !3] and multi-photon absorption [4,5], and classical tensorial properties in the solid state [6D has been effected in a number of different ways_ In a previous paper [7J, a proccdurc rcfcrrcd to as the generalized selection rule (GSR) scheme was presented, in which an attempt was made to establish a rigorous, systematic procedure for the analytic incorporation of symmetry in simplifying general matrix ekment product sums in terms of which quantum~observables are usually express&_ The GSR procedure has two distinct stages: _- (i) the operator selections rules (OSR), and _. -. (ii) the state selection rules (SSR)_ The OSR stage det ermineswhether- the phenoinenon is potentially observable for that symmetry; the SSR stage is. the .“quantum? refmement and selects ~which states -may c&$.riiute ckcretdy to the observable provided the. phenomenon is “al-~
:
:
Science
0301-&4/85j$03.30 @ Elsevkr Publishers B-V_-. (North-HollandPhysicsPublis’ningDivisidn) ’ - -I ’
.
‘~
~-- :-
-: .~..
~
;. -. -_
as weU as to methods baaed on the symmetry of spherical harmonics [9]_ In this paper, the determination of F’(G) proceeds directly through a group augmentation procedure, which expioits the fact that F’(G’) for a group G’ may be expressed in the symbolic form
F’(G’)
A symmetry invariant F’(G) for the group G with operations {Rl}, c= l,___, n satisfies P’(G)_F’(G)=
PO(G) =
Formalism
In this section, we derive the theorems which are fundamental to both the GSR and the augmentation procedure_ Some of the results appear intuitively obvious: however, none of them are trivial, and u-e have therefore chosen to present them in a rigorous manner. derived, where possible, directly from the group muitipkation properties of the symmetry operations_ Their relevance and interpretation will become apparent in later SeCtiO-lS
0)
$ -2 R,_
(2)
i-1
P’(G) is the totally symmetric projection operator of G. and will be referred to in this work simply as the projector_ The symmetry operations satisfy R&E G and R,R,= R,R, if and only if R,= R,. where R,, R, E G_ It follows that REX&, R, = I;_, R,. This leads to the results REP’(G)
= P’(G)
if
R, E G
(3)
and [PO(G)]
m = P’(G)
(m positive integer).
(4)
From these results flow the following theorems: (i) Any function F ma-y be written in the form F=
F’(G)
+ F’(G).
(9
where F’(G) ir an inuariant of G. and P’(G) F(G) = 0. This follows directly from the- idempotence of P’(G) [see eq. (4)] and writing F in the form F=
P’(G)F+
[I-
P’(G)]
F,
where F’(G)
t
(
where
= M’(G)_
G is a subgroup of G’, and A is an augrnenrarion operator defined soieiy in terms of those addifional operations required to generate G’ from G_ ff G is of order n. G’ is of order n’, then the number of “such additional operations is n’/n (see below)_ The beauty of the augmentation procednre is that, by starting tith the Iowest symmetry groups, the invariants for all the groups for a general function F may be .systematicalIy generated, yielding an anal-tic form for the corresponding tensorial invariant of arbirrury order for all the finite point groups_ The paper is developed in a number of stages. The first stage (section 2) dkusses the basic theorems rekvant to the definition of invariants and the augmentation procedure_ The next stage (section 3) encompasses the analytic determination of tensorial invariants of arbitrary order for the finite point groups_ In section 4, the reJevance of the results in the context of the GSR scheme is discussed. together with the relationship to rotational averages and the concept of chit-al and a&it-al properties_ FmaUy, examples of applications and the relationship to other treatments in the literature are discussed in section 5_
F.‘(b).
= P’(G)F,
F’(G)
= F-
F?(G)_
(6)
(ii) An invariant of G is _ymme~ric to all oper&ions of G_ This thsrem is proved by noting that, from eqs(I) and (3), we have R,PO(G)F’(G)
= R,F’(G)
= F’(G)
(71
for ah R, E G_ The converse follows trivially from the definition of P’(G) in eq. (2)_ (iii) The product theorem: viz P’(G){
F:(G)F,}
where F,‘(G)
= Fp(G)P’(G)F,,
is an intxzriantof G_
$9
1
The .au*;ntati-&
The theorem we-now discuss is central to ah selection rule applications involving matrix elements_ We present it here from a somewhat different viewpoint, in keeping with our emphasis on invariants_ It is important to remember that a symmetry group is fundamentally dependent on the i&r&&f any measurement (obser\iabIe) to its operations_ As mentioned earlier, if any observable is not invariant to an operation, then this operation cannot belong to the symmetry group of the system by definition, for we could then have a distinguishing “label” for the orientations before and after the operationConsider some function f(q) of a system of symmetry G, where q represents the internal coordinates of the system. If the symmetry operation R
G’={
where Fe(G) is an invariant of G, must lead to the same invariant for RCf(q); viz M{
Rcf(q)}
01)
= F’(G)-
Otherwise we would have a method of labelling the orientations, which would contravene the definition of R, as a symmetry operation_ The symmetric integrand -theorem may fhen be stated s&p& as If(q)
dq=j{J”(G)f(q)}
dq,
(12)
wehere the integral is ocer ~the whate configurarion space of the system in question. The result follows from no*&@ that. ff (q) dq is a scalar independent of q. and hence an in-ant;. use of eq. (l1) and a trivial summation over R, and division by n yields :_ the final forniof eq: (I2)_.‘~
$_& f
CR, )
R,, R;R,,.-..
(13)
form a group, then G’ is &led the augmented group_ G is then a subgroup of G’_ 77refoilowing rheorem, which we refer to & the augmentation theorens can then be slated P”(G’)
= A { R; ) P’(G)
(14)
=P’(G)d{(Rj)-‘}.
(15)
where d{R;}=
x( 1
The proof sion_ That inverse of pressed in
(W
= F”@)v
exp,~~~~-.&
work may be ‘.deveIoped in the foilowing~:wa$ Consider the group-‘G (the ort$rczi group) with &.operations ( R; }. Suppose we define a set-of oper: ations-(R3), i=.l,___~nllnoneofwhichareinG, as the nugnzenror set If the n x (m i-1)operations
R;R; M{ f (d)
.ti;&ti
of eq_ (14) follows -&vially by expanof eq_ (15) requires rcco~fmition that the an operation R:Rr in G’ can be ethe general form Rz-R,; then
= (R;_R~)-’
= R,(
w
E+ZR:)-
= (R,
j-‘( R;)-’
R;.)-’
from which eq_ (15) follows_ 2.4. GSR theorems The final set of theorems are vital to the definition of the GSR, and involve the defiiition of invariants involving wavefunctions and operators We define ( 11-l)) as a set of orthonormal wavefunctions forming a basis for the jth ,irrcducibie representation (IR) of G. such that X labels the degeneracy of the state__The symmetry operations R, may be represented in the jth IR by the _ unitary matrix with elements c;;il such that
csjxx
x and -I-
= spp
(17)
(18)
~of the resuhs in order to dete;-mine the invarianti of generaI tensorial products for the fiite point groups We do this inthree stages_:‘The fir& stage develops a wnvenient notation- for. a general tensoriai function or’operator product_ The second stage diksses the augmentation procedure as applied to the systematic generation of PO for the fiite point groups_ Fiially, the tensoriaI. invariants for the finite point groups are determined.
Theorem Ic EA 1r_(>(ri 1 its im invariant of G: The proof foIIotis from noting that
..
x
RI_ Deueiopment of notation
using eqr (17) and (18)_ Theorem 111 If 8 is an operator, and 8’(G) corresponding incariant operator, then
(1%
x
x
the
The proof exploits the SIT [eq_ (1211:
$
The problem of interest involves the determination of
where each P, is taken to have the form of a general tensorial function (or operator)
=~
P, = TRr+__,._ = Cala,
=~
IO)
and
I f)
are any tn-o states
of the system,
~
Id>-
(20)
The proof involves a simpie reordering of- each product; viz ~COlelrx)
(21)
P’(G1I-k. P
of theorem II above.
3- Determination of general tensoriaI invariants Having dispensed with the underlying formalism, we now turn to the practicaI appkation
cm
-- - a-
and each ai is either (i) one of the wmponents of a polar vector denoted (XJE), or (ii) one of the components of an axial vwtcr denoted (XYZ)_ (The term “function’* will be used for wnvenienw but the resuhs aIso appIy to the operators_) Polar and axial vectors are separated in th& way as they have different symmetry properties_ A polar vector has the symmetry properties of a directed arrow or vector, whereas an axiaI vector has the symmetry properties of a vector cross product u X u, such as a current loop, or an &rguIarmomentum_ In the pmcticaI application to the GSR. ye shah see that it is cruciaI to retain the a-ordering, i-e. the P, do not commute. Furthermore within each P,. there is an ordered set of vector~components a,_ To determine a general form for eq_ (21). it is mathematicahy wnvenient to be able to wnsider each to*& product n,P,. with the symbolic in the alternative form form EIlo,a,---a~, - . x__-_xx__:xy___yY___Yz___zz__~z,
-
where the wmponents are collected accord&to the simihuity of .-their transformation broperties under the various symmetry operations_ This may
.’
_-^
-..‘.
:
_
;
-__
-_
( .f.
_
...-
.~
--_
P-E Sdrippor,A.
Rdger
be effected :without contravening -the strict~ nonof the-k’, _or:ihe tonsorial -ordering by simply labelling the position of each of the com&xtents.with index Ii [I@ a(/& where. the r&te:of.Ii denotes the,exact positi&wbich.the component ~(1,) occupied in the original product. n,P,_s -. ~-.‘ The gqeral function may then be w&ten in the following form (numerical coefficients neglected here-..:~ .:.. ~. ..
mmmutativity
x
i-I@.+ 1z(ii) II? 1 i-c-i-
z(ri).
a=a’_
$=$‘-a’,
,f=c’-_,
::__: .:-
,.
..
r_
_-.<‘:I
:-.y
__y;.;--_. .-
:$:..::-,-<~z_i;;._ i .‘*I.<
. ..
.__
__y’
y=y’-p,
1=5’-c’_
(24)
.: ~l__. ~.,...__1
-provided it. is ~R&I~@,&$ ,that :t& c&&m&t -.&P_{ position i -. in’ .-the actual- &at~ded~~ product;: -_a$:. :- written ;S the formof eq. (23), has the lab+ I+&; : p.roduct may then finally _be _writtcn in .the_slm_-... -.y_- -, __:-.-_-.I-____ plified.notation T _ _~ 1. f,x”xsy~y6rc._&
.-
-._
.
:-
-_--: cs
This is the starting point for further ~symri&y simplifications. Once the effect of the projector on this product is determined, the labels_li are suo$ssiveiy- reintiodu&$ for each t&m (as a sum of products is usually~ obtained), and- the fmalT I-& ordering to include the & and tensor labels .cf:. fected to give the fii result.
.
The basis of the augmentation method lies in the fact that the selection rules for a group will always contain at least those of its subgroups. as will be seen below. Thus. by first determining the invariants for low symmetry gro-tips, it is possible to use the. augmentation theorem and determine
pr0caJut-e for ftite point groups
o,(O.OJ)
u,
Ta
-.
S.,(O.O.1)
=’ (u.6.c) dcnotcs the axis of rotation. or the normal to a plane b! 4-~(feos-‘[~~-~~-cos720)])_
-._
-_ __: . 5;; 33.
: :.
3.2_ The augmentation procedure
It follows that Q is the number of components of type x. fl that of type X, and so on_ It is now possible to omit for convenience the labels ii,
Table I Xugmcwation
-::..
_~
_.
where
6=g’--y’.
I-
/ .GywuExdsekaim~rufes
I
C 2r+l D 2*+1 T
-
F
.~ ‘.
34
..
the invariadts for higher .symmetry~groupsWhichmay be generatedfrdm the lower symmetry groups_:. The reIe%antn&It has been introduced in.section 31 [eqs (X4) and (l5)]_ In ap$ications, we note that once Fe(G) is determined for a -grdupG, ihe. corresponding invariant for G’ is sim&
F*(G’)=A{
. .
P_ES~~~..;L~~r/Ccnuoli=edKlkiionlulr;
R:}F’(Gj..
(26)
~.
where A{ Rg j is the augkentaiion opera& for the augmentor set defiig G’ from G. As A{ RF} .is usually a yastIy sinipler- operator than the full projktor, we are -effectively doing only the additional work chara&ristic of -the further~restrictions imposed by the additional symmetry of G’_ In building up the projectors for the tinite point 3-2 AppIicaricn
co
groups, the choi*-ok start&groups &d-the partitular operatigns defining the.au&nentaeon operators &e by na mea& uniqug and may be chosen for computational co&&en& or to suit-the pr&!ek in some other w+y__In this &0*-&e choose.to develop the totality of &oups.according to the following scheme- Starting with the group C,,:the chiral groups 0,; T, 0, I are successivelygenerated using proper rotations~The achiral groups are then generated from these usiig a singIe improper rotation in e%h case. The pr+ure is Summan‘sed in table l,mfor &g&ntation based on eq. (14). In some cases, there is a practicaI advantage in using the alternativeexpression of eq. (15), but the.rmaI results are, of course, independent of which exprcssion is used.
the finire point groups
Consider the product f= x”xBply”_‘-& _
(27)
-
is a symmetry operation of the group G with the projector PO(G), then by definition
If R,
~+f=
( R+_~)~( R,x)~
__( R,z)!
(28)
where R+_x= aewxx + a_,,p -F a_,,-‘_ etc_
(29)
Determination of RJ will then lead to a sum of terms. each of which will be an ordered operator product prefixed by a product of transformation coefficients (a) which are independent of the implicit Iabcls (li). WeconsiderflntthegroupsC,(n=l,~-_.)andS,(n=4i,i=l.~.__),whichturnouttoleadtothe generaI expressionsgiven below- The apparent complexity of the expressions is due to their validity for all n- For these groups, denoted by the general _symboiU,, substitution of eqs_ (27)-(29) into eq_ (1) yieMs PO&J_)/=;
;+
xc+J.s)“(Xc+
Ys)~(,T-~)‘(Yc-~)6L’2~.
(30)
where c = cos kl3, s = sin kt?. 13= 2%/n and T= 1,
Un=Cn;
(31)
T= (_l)k‘@=+=),
U, = S,_
(32)
To simpIZy this expression further, we define P(a, (a - i)x’s and iy’s and A=a+-_B+r-f-8,
i)x”-’ y i as the sum of all. permutations of the ..-
_ .
- (i3j
I=i+jt-Z-f-m_
<*I -.
:. - For the g&e&l
&s&, H(A, Ij ‘may be determined i&&and&d. methods,. &d the ~expansion .&&i&y .cakulated._The~increasing algebraic complexity with the order of thi axk (n) &unavoidabl< ‘but f& the”’ -. common point groups these oid&s are generally low. We turn now to the results for aU thk w grou& The general result fo; C, of eq_ (36) above redU&:to drastically simpler expressions for C, and C,, which kre quoted explicitly below,-_as well as the results for the other groups exce$t I- (l%e general expression for the point group I is messy and not w&ted since I. is fly used. Table 1 contains all the n ecesSary infonr.ation for I.) . . ~.
D,:
P’(D,)f=P’(C,)f,
y+6+c+c=2p.
=o, T:
P”(T)f
= +(x‘=XBy’Y8zz’Zr a+j3=22p .ci+p+2p =o,
l39).
otherwise;
otherwise;
+-y”YBz~Z*xgXr
+ z~ZsxrX~~‘Ys),
and
y+6=2q
and
~+I=2r
and
y+6+2q
and
a+g+2r,
or
--
W
T,:
PO& If
= 0. D
s,_(2ni I)*
Eq_ (36) for U==S,, expression for PO(&)
s,:
-f-r,,
a+B=2p
and
y+S=2q
and
c+c=2r
a+j3#2p otknvise;
and
y+6#24
and
ei5#22r,
0,.
I,:
P’(G’)f
=P’(G)f, = 0.
n=4i, i=l,2.___ and T=(-l)kL-(B*‘~” is explicitly given below_
or
a+y+c=2p,.
(45)
Otherwise_
gives an
CXpfSSiOll
for PO(S,,)_
The
~o(~,)f=f{(_~~XSy~~6+(-1)=+~*‘~~~Bx~x~j~~Zi). (46)
a+jT+y-!-6=2p, = 0,
D znri-
P’(D,,,)f
othewise; =P’(S,,).
yiS+~+-5=2p,
= 0.
othemise_
(47)
It is stressed that in the above expressions, any conditions are additional to those derived for the group from which it is augmented_ Also. the quantities 2p. etc de>ote an even number in a general way, and if 2p appears in two different sets of conditions, this does not imply any sort of equality. 3_4_ Infinite point groups Some mention should be made of the infinite point groups_ i-e those with cylindrical or sphericat symmetry_ The procedure for cyiindricai groups is the same as that for finite groups, ((l/n}& cos’k6) being replaced by ((1/2z)/~= co&t d6}_ As with C -fn point groups all terms uith i# 2p vanish and for I = 2p [IO] 1
zSo/
l=cos!ede=
I! 2’[(1/2)!]‘-
(W
Spherical groups are equivalent to rotationally ave-raged systems- These have been aiscussed at length by for example Andrews and Thirunamachandran [ll] and shipper !12]; some further general discussion of rotational averages follows in section 4 4_ The genrralired s&&ion
rule approach
General expressious for the invariants for the finite point groups have been derived in section 3_ In this section, we discuss the generalized selection rule approach in which the invariants play a
central role_ in addition, the GSR scheme is compared to other selection rule schemes, highlighting its isomorphism with classical selection r&s and relationship to rotational ave.-ages_ Consider a collection of independent systems labelkdk B.-_-and fields f. g___ _ With the useof perturbation theory and muhipole expansion techniques, each term arising in the definition of the obsercabies involving the coupling of some (or all) of these systems reduces to the general form F=
CF,F,.__F,F,.__
= CnF,,
(49)
where FA is a function oiy of matrix elements centered on A, Fs on B, and so on, and F,, Fg, _ _ _ are factors pertainin g directly to the reIevant fieIds_ C contains ail other quantities such as energy and radial denominators, and s runs over the system labels (A, B,_.., f, g,___)_ We shall take each F, to be a sum of matrix element products; it isa trivial generakation to extend consideration to matrices, tensors, etc_ with such elements. The observable F may then be analytically simplified by explicitly i+zorporating the symmetry of each system_ Suppose G, is lthe point symmetry group of system S with PZ,_operations Rf_ Then, noting that observa bks m-ust be tot&y syrnrzzetric
._
-:
with respect .to any symmetry operation of _any of thesyste&itfollowsthat
--
.:
_ .: --
Each factor-F= may therefore by treated mdependcntly, as the rcmaIhC~g factors act simply .-as a multiRlicative constant withres+t to z&y symmetry arguments based on G,: AR arguments thatfolIow apply to a single F,; which may be written in -the general quantum form
._c-
$pplied to the operator. product ang the %-i3jJket-~products indeRendently: This hypotheti&l p*:: dune may be given -a rigourous. foundation_ inrtbefollowing way_ : _ ; -.:-.- ;__ ,- -~_-_ -- -The fit stage is‘ the 0perator~:selection ruie ‘(OSRj stage, whichfollows_from the result thateq. (51) may be rigourously rewritten in the form .I
F,.=x
c (uljP;.l~~)(~~lP~;:_. r’ 7.x ,___
,i;*!u;),
where
(51) The Iuf) are the state functions of system S, which are bases for the IRS of G, such that U: transforms as the yth IR of G,_ The summation over y, A,...is over all degeneracies. The operators P,’ may be cartesian components of tensorial operators. vectors etc In the latter case, the relevant products between matrix elements must be retained in the operator product- The cyclic appearance of the states results directly from perturbation theoryIt is the manner in which symmetry is incorporated in simplifying such matrix element products that defies a selection ruIe scheme- The usual procedure for small order products involves the direct application of the SIT to each matrix element in turn; however, each successr‘ve application involves state and operator specification, as well as rapid increase in the attount of work as the order of the product in~m-eases_(The justification of the procedure has its origin in the product theorem of eq_ (S), each matrix element being replaced by its corresponding invariant, so that this procedure is rigorous and must tead to the correct result.) A fundamental chemical objection that may be levied against this proced~however, is the total obscuration of the relationships to the cotresponding _ da&al invariants that results from this procedure ,. :. The GSR approach uses the product theorem in a somewhat more abstrac: manner in order to rigourously resolve the :selection rules into a “classical” .and a “quantu~m” component. peach matrix dement is effectively split into bra, Iret.and operator factors, and the product rule is then
P”(G,)~nP~=~n~:-. r =
r- =
.153)
This result is referred to as the OSR theorem. The. proof of this result follows directly from theorems I, II, and III [eqs. (17)-(20)] discussed earlier_ The effect of a symmetry operation on a product is defined through the relation
In the expression for the operator product, it is crucial to retain the ordering of the operators, each being implicitly labelled by the relevant states in the matrix element_ It is readily seen that the OSR results Iead to invariant operator products that are isomorphic to the classical invariants, so that they reduce rigour-. ously to the classic+ s&&on rules. The second stage of state selection involves the application of the SIT to only those matrix eIement products that survive the OSR procedure (which is usually the most restrictive)_-. Rotational acerages_ It is apparent that increasing the symmetry can only serve to further ~tiplify the invariants_ The_ limit of the augmentation procedure may be taken to be the spherical groups. For -the purposes of ,this. work it is important _td distinguish between the “full” rotation group ,of_ Aoroper rotations, which we shah denote by _E , and the full rotation group .of proper and improper rotations (Z,). Note that Xh may be-generated from Z in the same manner as 0, from 0; viz 1using the single augmentation operation. i_
The relevant of this division follows from noting that we may consider the projectors of-the various grou+ containing only proper rotations as effecting a “rotational” average over the orientations generated by the operations of the group_ For the fiite groups, the average is therefore over a discrete number of orientations_ For the spherical group X, the projector is equivalent to a full three-dimensionaI rotational average (in which aII orientations are assumed equally probable, i.e. denoting such an average by ( )- we have that
PJ(Z)F=
P(Z)_
(5%
It is significant that, as a physicaI average can only be effected by proper rotations, the corresponding group (incIuding Z) are chirili groups (i-e- contain no improper rotations)_ A!l finite chirai groups are subgroups of 2, and thus no further simplification can foEow by incorporation the symmetry of mokcuks of such symmetries for expressions which are already rotationally averaged_ However, for achiraI mokcuks, the addilonuf intrinsic symmetry (i-e_ the improper axes) is not incorporated in a rotational average, and can lead to further symmetry restrictions_ In fact, for achiraI systems, the rotationai averaging and intrinsic achiraIity may be effected in a single stage by the application of PO(Z,)_ We have gone into some detail regarding this point in order fo emphasize that chiraIity cannot be destroyed by rotational averaging, a fact that stili causes some conftionThis has been discussed at Iength elsewhere [12]_ In fact. decomposition of F”(2) according to eq_ (5) reIative to 2,. i_eL
F0(2)=F0
(56)
yields FO(X,) as the pan-discriminating component (achirality function) and F’(2,) as the discriminating component (chirality function) corresponding to the rotatio&Iy averaged quantity F’(Z)_ We can in fact defime the achiral and chid projectors respectively as
PG=f(E-i)PO(Z),
(58)
where these project out the rekant (rotationally averaged) achirality or chirality function corresponding to the origin% function on which it operates (k F’(2) and -F((z,) k+zctiveIy)_ The final feature that follows from this d+cussion is that-the effect of increasing *e symmetry of the jystem is to bring the form of the invariant progrt%siveIy cIo_ to that of the rotationaUy averagd form_ The lower the rank of the tensoriai elements, the lower the symmetry required_ to reduce the invariant to the same form as that of (2) or (Z,)_ We shaII comment on the physical consequences of this in the applications.
5_ Appkatio~ The results of the preceding two sections are sufficientIy expkit to not warrant a description of how they should be applied_ We concentrate therefore on a dkcussion of the scope of applications, and a comparison with other symmetry selection methods. >_i_ CIarrcai property tensors The expressions for the invariants can be applied directly to the determination of invariant cIassical property tensors, such as permanent moments and ckxical mokcuku response tensors ;,e-- ~e~iIities, hyperpohuizabilitie etc-); dt part of each component, and the symmetry relationships between various components are thereby determined- Ahmann PI has approached this problem via spherical harmonics; Birss [6] on the other hand has determined relationships between the cartesian tensor components from examing the effect of generator sets of operations, and has tab&ted these relationships for tensors of up to fourth rank for the crystalk+ graphic point groups_ Both of -these approaches increase in complexity with increasing tensor size and group size, in contrast to the GSR approach5.t
Two-photon processes
The GSR approach does not give different re-m s&s for two-photon processes [3-5.131: it does,
however, provide anaherrkve perspective on the -previous work McCIain’s work on two-photon absorption involves an intuitive use of the OSR theorem in order to effect the rotati&aI averagingTheorem III [eq_ (2011 is then.effectiveIy assumed in order to derive his fmaI results as the SSR stage applied to twophoton absorption Similar comments appIy to conventional Raman scattering selection rule approaches, which ako implicitly assume eq_ (20)
where
and.
5_3_ Circ2fik.rdichroirm The- most effective apphcations of the GSR approach ak in chiraI~spectroscopic processes. the operator products being generahy of mixed ekctric and magnetic parentage We consider two types of appIi&tions: the first to natural (mtriusic) circular dichroism, and the -second to induced circular diChroism
X3. I. Intrinsic circular dichroism The rotationally averaged circular dichroism (CD) strength has the form Im(Olpln)
(5%
-
where p is the ekctric dipole operator [transforming as (x. y. r)] and m the magnetic dipole operator (transforming as (X, Y, Z)]_ 10) and I a> are the initial and fmat stages, of the system, respectively_ This expression is an invariant of the group Z_ Any single improper rotation wih augment Z to 2, (i-e_ c= refkction; ir inversion, S, = general improper rotation) as alI the improper rotations wiII then be generated_ Hence noting that P”(Z,)=+(EtijPof2)
(60)
and that (E+i)p-m=O
(61)
it follows that there is no intrinsic CD achiraI molecuk
for any
53.2. Induced circular dichroism CDcanbeinducedinanachiraIsystembya chiraI perturbation- The simplest models descri-bing the induced CD arise in firstorder perturba-. tion-theory, and can be expressed in the form R,=Rf,,kj,,:
]_
_.
_.
(69
The ach.iraIsystem, A, is represented by the ground state IO>, and excited states I a> and lb). such that the iuduced CD appears at the transition energy co of the 10) +. I a> transition of A which is assumed to be magnetic dipole allow&IO’), fu’) are the ground and &cited states of some and u’is perturbing system C_ Summa tionoverb impircit in eqs. (63) and (64). pA, mA arc the ekctric and magnetic dipole operators on A, so that zL!kb=
(65)
and so on_ The RS, contribution is referred to as the srnric coupfing [14] contribution, (bO’I V Id’) reducing simply to the coupling of a transition muhipole on A with a static field component due to system C The Rd, contribution may be referred to as the dynamic coupling [15] contribution, as it invoivti the coupling of transition muhipoks on A and C It is significant that both terms vanish on rota; tional averaging sq&sting that both terms be-come progressively smaher~with iucreasing symmetryofA_(Thishasbeenputfokardasan~ment for the importance of second-order pe?urbation terms, which survive such averaging [NJ].) We may consider the fmt-order terms as being_highIy symmetry&zpendent Schelhuan [l] has discussed the symmetry &kction rules for static &upIing terms By using the SIT on the matrix eIemen_ts m~“~~~. he deduces that, V must behave like a- pseudoscalar under the point symmetry.of A_ By expansion of V in terms of a Fourier &es about the unperturbed ~ymme-. ~.
try* Schelhnan analyses the A-dependent part of V in terms of spherical harmonics_ For the speciaI situation where the chiral perturber is represented by point charges, the syxnmetry behaviour of the A-dcpendcnt part of V corresponds exactly with the disposition of the charges, and so defies a sector rule_ This will be discussed further in a forthcoming publication ou CD mechanisms~ Schelhnan’s procedure becomes progressivelymore difficult for higher symmetry groups, to the extent that general expansion for 0, and T, arc not quoted_ In the GSR approach, the molecular operator product is anaIysed directly, yielding the muhipoIar components on A that can couple to the f&d (and thereby the symmetry pro_pertiesof the tieId) for non-zero CD strength_The augmentation procedure and the explicit use of symmetry4efined cartesian axes encounters no difficuh.ieswith the higher symmetry groups such as 0, and T+ with anaIytic expressions obtained for any order from cqs (3~(4t3)_ For exampIe for 0, the generaI form of the symmetry invariant A-dependent partof Yis V.-.= c
(_c)(X@,.%=c - X”J.9
(66)
where a. b. c m odd and a#bfc_ (x.y,z commute within y,_) The simplest form. for which a = 5, b = 3. c = 1, is the one given by SchelIman. The generality of the GSR approach is highlighted by its ready application to the dynamic coupling terms. for which SchcIIman’sapproach is inappropriate_Mason [2] has discussed the nature of Y in the dynamic coupling terms for metal complexes, by essentially using the SIT on the single Y matrix element The GSR scheme allows a determination of the exact form of the invariant operator product and hence the explicit form of the coupling term V_ For example. for 0, = C
(f
- zbyc)
)[mA,x’(ybz’
a.6.c
fmAryu(zbxXC-xbzIC) + mAZzP(xbyyC -ybzc)],
for a=2p;
b=2qti;
c=2rtl;
._
.:
” We have attemR&d to show that the Girt a&preach arzd the formalism presented in this paper provides a consistent framework for the definition of sekction rules for a variety of phenomena In addition, the invariants which are central to the approach have been analytically determined for the fmite point groups for arbitrary cartesian products in terms of which observables are generaIly expressed. The relationship to rotational averaging allows for a definition of obscrvablcs which arc chiral or a&i&, and in addition, those that are highIy symmetry sensitive (i-e_ those which vanish on averaging), and those that arc less so (i-c_ have non-vanishing rotationally averaged invariants). Highly symmetry sensitive terms appear particularly in first-order CD models, suggesting that higher-order perturbation terms couId easily be domiaaut over first-order terms for high symmetry systems. A detailed analysis of CD models and the role of symmetry selection rules will be diskszd in a forthcoming publication_
t _+‘z”
-X&_v=zc+ X’iezb - xcYbz$
PO(O,)[m,V,]
6_Gmdusions
q#r_
(67)
RckIV?u~ [II JA khdfma. J. Ghan Phys. 44 (1966) 55. PI SF. hfuoa. MoIaxIar optical xfivity and the cbiral dkrimkations (-bridge Univ_ plcss. London, X982)_ [3] LX Ovandcr. Opt Spuay_ 9 (1960) 302_ [4j P-R MOand W-M &fcCbin. J. Cbam Phys 53 (1970) 29. [S] T_R Bada and A_ GoId, Pbys. Rev_ 171(1968) 997_ [6] RR Bit-s. Symmetry and magnetism (North-Ho&and. Anrsterdam. 1964). m P-E Scbippcr. J_ Am C_Xmn_ Sot 100 (1978) 3658_ [8) G-F_ Kostcr. Psopcnies of the thirty-wo poinl groups (Ml-T Press. Cambridge. 1963)s 191SL Ahmann, Cambridge Phil. Sot (Ma*) Proc 53 (1957) 343. [IO] Handbook of physics and chanis~~. 60th Ed_ (CRC Press. Ckvdand 1979) P_ A-66_ &am&an. J. Chcms Phys 1111DL Andram and T_ l-him 67 (1977) 5026m 1121P-E Schippa. Ghan Phyx 44 (1979) 26L 1131WM. McChin. J. Chan Phys 55 (XVI) 23’89s 1141FS. Richardson. Clmn R&s (1979) 17. 1151EG_- H6hn and 0-E Wci~ Jr, J. Cban Phyx 48 (1968) 1127_ [16] P_E Scbipper. J_ Am Cban Sot 100 (1975) 1433s
lowed” in the .OSR~ stage_ The ~overkl procedure satisfies a sort of correspondence principle in that the OSR are effectively isomorphic with the~classi-Cal selection rulek of. the corresponding classical observable, if one exists_ Both stages of the GSR reiy on a basic concept of a symmet.$ inuariark oZ a group G, which u-e may define as a quantity or function F’(G) which is symmetric to every operation of that group. Ali observables cf a system with symmetry G mu& by -definition, be such invariants, or otherwise. a method would exist for showing that the system is of lower:symmetry_ In the context of the GSR approach, the OSR stage requirks the determ&ration of invariant open&r products; and the SSR . stage that of invariant inkgrands_ :The practical determination. of the invariant function Fe(G) corresponding to a general function F under a covering’symmetry G is therefore central to analytic implementation of selection rule procedures, and is by no means-the trivial problem that it may superficialiy appear to be The prac+ 1 ‘.- cal procedure for the determination of. -e’(G) in the initial GSR paper is based -on transformation tables for- unit 7polarand :axiai.vectors. .but it+ .‘ .. implementation becomes rehxively tedious :alg+ i. braically- for- increasingly cornpI+ fm+ions ore _ increasing group size.7A s&iIar argument -applies _- to vector couplin&rreducible @&or me;h&s @J,
1,Introduction Selection rules for spectroscopic and interaction phenomena generally increase.in compkxity with the order of $&rbation theory and/or m@ipole expansion of the perturbation operators in terms of which the theoretical description is developed_ Incorporation of symmetry arguments for disparate phenomena (for. example, induced circular dicbroi.ml [1,2]. higber-oxkr spec troscopic processes such as scattering !3] and multi-photon absorption [4,5], and classical tensorial properties in the solid state [6D has been effected in a number of different ways_ In a previous paper [7J, a proccdurc rcfcrrcd to as the generalized selection rule (GSR) scheme was presented, in which an attempt was made to establish a rigorous, systematic procedure for the analytic incorporation of symmetry in simplifying general matrix ekment product sums in terms of which quantum~observables are usually express&_ The GSR procedure has two distinct stages: _- (i) the operator selections rules (OSR), and _. -. (ii) the state selection rules (SSR)_ The OSR stage det ermineswhether- the phenoinenon is potentially observable for that symmetry; the SSR stage is. the .“quantum? refmement and selects ~which states -may c&$.riiute ckcretdy to the observable provided the. phenomenon is “al-~
:
:
Science
0301-&4/85j$03.30 @ Elsevkr Publishers B-V_-. (North-HollandPhysicsPublis’ningDivisidn) ’ - -I ’
.
‘~
~-- :-
-: .~..
~
;. -. -_
as weU as to methods baaed on the symmetry of spherical harmonics [9]_ In this paper, the determination of F’(G) proceeds directly through a group augmentation procedure, which expioits the fact that F’(G’) for a group G’ may be expressed in the symbolic form
F’(G’)
A symmetry invariant F’(G) for the group G with operations {Rl}, c= l,___, n satisfies P’(G)_F’(G)=
PO(G) =
Formalism
In this section, we derive the theorems which are fundamental to both the GSR and the augmentation procedure_ Some of the results appear intuitively obvious: however, none of them are trivial, and u-e have therefore chosen to present them in a rigorous manner. derived, where possible, directly from the group muitipkation properties of the symmetry operations_ Their relevance and interpretation will become apparent in later SeCtiO-lS
0)
$ -2 R,_
(2)
i-1
P’(G) is the totally symmetric projection operator of G. and will be referred to in this work simply as the projector_ The symmetry operations satisfy R&E G and R,R,= R,R, if and only if R,= R,. where R,, R, E G_ It follows that REX&, R, = I;_, R,. This leads to the results REP’(G)
= P’(G)
if
R, E G
(3)
and [PO(G)]
m = P’(G)
(m positive integer).
(4)
From these results flow the following theorems: (i) Any function F ma-y be written in the form F=
F’(G)
+ F’(G).
(9
where F’(G) ir an inuariant of G. and P’(G) F(G) = 0. This follows directly from the- idempotence of P’(G) [see eq. (4)] and writing F in the form F=
P’(G)F+
[I-
P’(G)]
F,
where F’(G)
t
(
where
= M’(G)_
G is a subgroup of G’, and A is an augrnenrarion operator defined soieiy in terms of those addifional operations required to generate G’ from G_ ff G is of order n. G’ is of order n’, then the number of “such additional operations is n’/n (see below)_ The beauty of the augmentation procednre is that, by starting tith the Iowest symmetry groups, the invariants for all the groups for a general function F may be .systematicalIy generated, yielding an anal-tic form for the corresponding tensorial invariant of arbirrury order for all the finite point groups_ The paper is developed in a number of stages. The first stage (section 2) dkusses the basic theorems rekvant to the definition of invariants and the augmentation procedure_ The next stage (section 3) encompasses the analytic determination of tensorial invariants of arbitrary order for the finite point groups_ In section 4, the reJevance of the results in the context of the GSR scheme is discussed. together with the relationship to rotational averages and the concept of chit-al and a&it-al properties_ FmaUy, examples of applications and the relationship to other treatments in the literature are discussed in section 5_
F.‘(b).
= P’(G)F,
F’(G)
= F-
F?(G)_
(6)
(ii) An invariant of G is _ymme~ric to all oper&ions of G_ This thsrem is proved by noting that, from eqs(I) and (3), we have R,PO(G)F’(G)
= R,F’(G)
= F’(G)
(71
for ah R, E G_ The converse follows trivially from the definition of P’(G) in eq. (2)_ (iii) The product theorem: viz P’(G){
F:(G)F,}
where F,‘(G)
= Fp(G)P’(G)F,,
is an intxzriantof G_
$9
1
The .au*;ntati-&
The theorem we-now discuss is central to ah selection rule applications involving matrix elements_ We present it here from a somewhat different viewpoint, in keeping with our emphasis on invariants_ It is important to remember that a symmetry group is fundamentally dependent on the i&r&&f any measurement (obser\iabIe) to its operations_ As mentioned earlier, if any observable is not invariant to an operation, then this operation cannot belong to the symmetry group of the system by definition, for we could then have a distinguishing “label” for the orientations before and after the operationConsider some function f(q) of a system of symmetry G, where q represents the internal coordinates of the system. If the symmetry operation R
G’={
where Fe(G) is an invariant of G, must lead to the same invariant for RCf(q); viz M{
Rcf(q)}
01)
= F’(G)-
Otherwise we would have a method of labelling the orientations, which would contravene the definition of R, as a symmetry operation_ The symmetric integrand -theorem may fhen be stated s&p& as If(q)
dq=j{J”(G)f(q)}
dq,
(12)
wehere the integral is ocer ~the whate configurarion space of the system in question. The result follows from no*&@ that. ff (q) dq is a scalar independent of q. and hence an in-ant;. use of eq. (l1) and a trivial summation over R, and division by n yields :_ the final forniof eq: (I2)_.‘~
$_& f
CR, )
R,, R;R,,.-..
(13)
form a group, then G’ is &led the augmented group_ G is then a subgroup of G’_ 77refoilowing rheorem, which we refer to & the augmentation theorens can then be slated P”(G’)
= A { R; ) P’(G)
(14)
=P’(G)d{(Rj)-‘}.
(15)
where d{R;}=
x( 1
The proof sion_ That inverse of pressed in
(W
= F”@)v
exp,~~~~-.&
work may be ‘.deveIoped in the foilowing~:wa$ Consider the group-‘G (the ort$rczi group) with &.operations ( R; }. Suppose we define a set-of oper: ations-(R3), i=.l,___~nllnoneofwhichareinG, as the nugnzenror set If the n x (m i-1)operations
R;R; M{ f (d)
.ti;&ti
of eq_ (14) follows -&vially by expanof eq_ (15) requires rcco~fmition that the an operation R:Rr in G’ can be ethe general form Rz-R,; then
= (R;_R~)-’
= R,(
w
E+ZR:)-
= (R,
j-‘( R;)-’
R;.)-’
from which eq_ (15) follows_ 2.4. GSR theorems The final set of theorems are vital to the definition of the GSR, and involve the defiiition of invariants involving wavefunctions and operators We define ( 11-l)) as a set of orthonormal wavefunctions forming a basis for the jth ,irrcducibie representation (IR) of G. such that X labels the degeneracy of the state__The symmetry operations R, may be represented in the jth IR by the _ unitary matrix with elements c;;il such that
csjxx
x and -I-
= spp
(17)
(18)
~of the resuhs in order to dete;-mine the invarianti of generaI tensorial products for the fiite point groups We do this inthree stages_:‘The fir& stage develops a wnvenient notation- for. a general tensoriai function or’operator product_ The second stage diksses the augmentation procedure as applied to the systematic generation of PO for the fiite point groups_ Fiially, the tensoriaI. invariants for the finite point groups are determined.
Theorem Ic EA 1r_(>(ri 1 its im invariant of G: The proof foIIotis from noting that
..
x
RI_ Deueiopment of notation
using eqr (17) and (18)_ Theorem 111 If 8 is an operator, and 8’(G) corresponding incariant operator, then
(1%
x
x
the
The proof exploits the SIT [eq_ (1211:
$
The problem of interest involves the determination of
where each P, is taken to have the form of a general tensorial function (or operator)
=~
P, = TRr+__,._ = Cala,
=~
IO)
and
I f)
are any tn-o states
of the system,
~
Id>-
(20)
The proof involves a simpie reordering of- each product; viz ~COlelrx)
(21)
P’(G1I-k. P
of theorem II above.
3- Determination of general tensoriaI invariants Having dispensed with the underlying formalism, we now turn to the practicaI appkation
cm
-- - a-
and each ai is either (i) one of the wmponents of a polar vector denoted (XJE), or (ii) one of the components of an axial vwtcr denoted (XYZ)_ (The term “function’* will be used for wnvenienw but the resuhs aIso appIy to the operators_) Polar and axial vectors are separated in th& way as they have different symmetry properties_ A polar vector has the symmetry properties of a directed arrow or vector, whereas an axiaI vector has the symmetry properties of a vector cross product u X u, such as a current loop, or an &rguIarmomentum_ In the pmcticaI application to the GSR. ye shah see that it is cruciaI to retain the a-ordering, i-e. the P, do not commute. Furthermore within each P,. there is an ordered set of vector~components a,_ To determine a general form for eq_ (21). it is mathematicahy wnvenient to be able to wnsider each to*& product n,P,. with the symbolic in the alternative form form EIlo,a,---a~, - . x__-_xx__:xy___yY___Yz___zz__~z,
-
where the wmponents are collected accord&to the simihuity of .-their transformation broperties under the various symmetry operations_ This may
.’
_-^
-..‘.
:
_
;
-__
-_
( .f.
_
...-
.~
--_
P-E Sdrippor,A.
Rdger
be effected :without contravening -the strict~ nonof the-k’, _or:ihe tonsorial -ordering by simply labelling the position of each of the com&xtents.with index Ii [I@ a(/& where. the r&te:of.Ii denotes the,exact positi&wbich.the component ~(1,) occupied in the original product. n,P,_s -. ~-.‘ The gqeral function may then be w&ten in the following form (numerical coefficients neglected here-..:~ .:.. ~. ..
mmmutativity
x
i-I@.+ 1z(ii) II? 1 i-c-i-
z(ri).
a=a’_
$=$‘-a’,
,f=c’-_,
::__: .:-
,.
..
r_
_-.<‘:I
:-.y
__y;.;--_. .-
:$:..::-,-<~z_i;;._ i .‘*I.<
. ..
.__
__y’
y=y’-p,
1=5’-c’_
(24)
.: ~l__. ~.,...__1
-provided it. is ~R&I~@,&$ ,that :t& c&&m&t -.&P_{ position i -. in’ .-the actual- &at~ded~~ product;: -_a$:. :- written ;S the formof eq. (23), has the lab+ I+&; : p.roduct may then finally _be _writtcn in .the_slm_-... -.y_- -, __:-.-_-.I-____ plified.notation T _ _~ 1. f,x”xsy~y6rc._&
.-
-._
.
:-
-_--: cs
This is the starting point for further ~symri&y simplifications. Once the effect of the projector on this product is determined, the labels_li are suo$ssiveiy- reintiodu&$ for each t&m (as a sum of products is usually~ obtained), and- the fmalT I-& ordering to include the & and tensor labels .cf:. fected to give the fii result.
.
The basis of the augmentation method lies in the fact that the selection rules for a group will always contain at least those of its subgroups. as will be seen below. Thus. by first determining the invariants for low symmetry gro-tips, it is possible to use the. augmentation theorem and determine
pr0caJut-e for ftite point groups
o,(O.OJ)
u,
Ta
-.
S.,(O.O.1)
=’ (u.6.c) dcnotcs the axis of rotation. or the normal to a plane b! 4-~(feos-‘[~~-~~-cos720)])_
-._
-_ __: . 5;; 33.
: :.
3.2_ The augmentation procedure
It follows that Q is the number of components of type x. fl that of type X, and so on_ It is now possible to omit for convenience the labels ii,
Table I Xugmcwation
-::..
_~
_.
where
6=g’--y’.
I-
/ .GywuExdsekaim~rufes
I
C 2r+l D 2*+1 T
-
F
.~ ‘.
34
..
the invariadts for higher .symmetry~groupsWhichmay be generatedfrdm the lower symmetry groups_:. The reIe%antn&It has been introduced in.section 31 [eqs (X4) and (l5)]_ In ap$ications, we note that once Fe(G) is determined for a -grdupG, ihe. corresponding invariant for G’ is sim&
F*(G’)=A{
. .
P_ES~~~..;L~~r/Ccnuoli=edKlkiionlulr;
R:}F’(Gj..
(26)
~.
where A{ Rg j is the augkentaiion opera& for the augmentor set defiig G’ from G. As A{ RF} .is usually a yastIy sinipler- operator than the full projktor, we are -effectively doing only the additional work chara&ristic of -the further~restrictions imposed by the additional symmetry of G’_ In building up the projectors for the tinite point 3-2 AppIicaricn
co
groups, the choi*-ok start&groups &d-the partitular operatigns defining the.au&nentaeon operators &e by na mea& uniqug and may be chosen for computational co&&en& or to suit-the pr&!ek in some other w+y__In this &0*-&e choose.to develop the totality of &oups.according to the following scheme- Starting with the group C,,:the chiral groups 0,; T, 0, I are successivelygenerated using proper rotations~The achiral groups are then generated from these usiig a singIe improper rotation in e%h case. The pr+ure is Summan‘sed in table l,mfor &g&ntation based on eq. (14). In some cases, there is a practicaI advantage in using the alternativeexpression of eq. (15), but the.rmaI results are, of course, independent of which exprcssion is used.
the finire point groups
Consider the product f= x”xBply”_‘-& _
(27)
-
is a symmetry operation of the group G with the projector PO(G), then by definition
If R,
~+f=
( R+_~)~( R,x)~
__( R,z)!
(28)
where R+_x= aewxx + a_,,p -F a_,,-‘_ etc_
(29)
Determination of RJ will then lead to a sum of terms. each of which will be an ordered operator product prefixed by a product of transformation coefficients (a) which are independent of the implicit Iabcls (li). WeconsiderflntthegroupsC,(n=l,~-_.)andS,(n=4i,i=l.~.__),whichturnouttoleadtothe generaI expressionsgiven below- The apparent complexity of the expressions is due to their validity for all n- For these groups, denoted by the general _symboiU,, substitution of eqs_ (27)-(29) into eq_ (1) yieMs PO&J_)/=;
;+
xc+J.s)“(Xc+
Ys)~(,T-~)‘(Yc-~)6L’2~.
(30)
where c = cos kl3, s = sin kt?. 13= 2%/n and T= 1,
Un=Cn;
(31)
T= (_l)k‘@=+=),
U, = S,_
(32)
To simpIZy this expression further, we define P(a, (a - i)x’s and iy’s and A=a+-_B+r-f-8,
i)x”-’ y i as the sum of all. permutations of the ..-
_ .
- (i3j
I=i+jt-Z-f-m_
<*I -.
:. - For the g&e&l
&s&, H(A, Ij ‘may be determined i&&and&d. methods,. &d the ~expansion .&&i&y .cakulated._The~increasing algebraic complexity with the order of thi axk (n) &unavoidabl< ‘but f& the”’ -. common point groups these oid&s are generally low. We turn now to the results for aU thk w grou& The general result fo; C, of eq_ (36) above redU&:to drastically simpler expressions for C, and C,, which kre quoted explicitly below,-_as well as the results for the other groups exce$t I- (l%e general expression for the point group I is messy and not w&ted since I. is fly used. Table 1 contains all the n ecesSary infonr.ation for I.) . . ~.
D,:
P’(D,)f=P’(C,)f,
y+6+c+c=2p.
=o, T:
P”(T)f
= +(x‘=XBy’Y8zz’Zr a+j3=22p .ci+p+2p =o,
l39).
otherwise;
otherwise;
+-y”YBz~Z*xgXr
+ z~ZsxrX~~‘Ys),
and
y+6=2q
and
~+I=2r
and
y+6+2q
and
a+g+2r,
or
--
W
T,:
PO& If
= 0. D
s,_(2ni I)*
Eq_ (36) for U==S,, expression for PO(&)
s,:
-f-r,,
a+B=2p
and
y+S=2q
and
c+c=2r
a+j3#2p otknvise;
and
y+6#24
and
ei5#22r,
0,.
I,:
P’(G’)f
=P’(G)f, = 0.
n=4i, i=l,2.___ and T=(-l)kL-(B*‘~” is explicitly given below_
or
a+y+c=2p,.
(45)
Otherwise_
gives an
CXpfSSiOll
for PO(S,,)_
The
~o(~,)f=f{(_~~XSy~~6+(-1)=+~*‘~~~Bx~x~j~~Zi). (46)
a+jT+y-!-6=2p, = 0,
D znri-
P’(D,,,)f
othewise; =P’(S,,).
yiS+~+-5=2p,
= 0.
othemise_
(47)
It is stressed that in the above expressions, any conditions are additional to those derived for the group from which it is augmented_ Also. the quantities 2p. etc de>ote an even number in a general way, and if 2p appears in two different sets of conditions, this does not imply any sort of equality. 3_4_ Infinite point groups Some mention should be made of the infinite point groups_ i-e those with cylindrical or sphericat symmetry_ The procedure for cyiindricai groups is the same as that for finite groups, ((l/n}& cos’k6) being replaced by ((1/2z)/~= co&t d6}_ As with C -fn point groups all terms uith i# 2p vanish and for I = 2p [IO] 1
zSo/
l=cos!ede=
I! 2’[(1/2)!]‘-
(W
Spherical groups are equivalent to rotationally ave-raged systems- These have been aiscussed at length by for example Andrews and Thirunamachandran [ll] and shipper !12]; some further general discussion of rotational averages follows in section 4 4_ The genrralired s&&ion
rule approach
General expressious for the invariants for the finite point groups have been derived in section 3_ In this section, we discuss the generalized selection rule approach in which the invariants play a
central role_ in addition, the GSR scheme is compared to other selection rule schemes, highlighting its isomorphism with classical selection r&s and relationship to rotational ave.-ages_ Consider a collection of independent systems labelkdk B.-_-and fields f. g___ _ With the useof perturbation theory and muhipole expansion techniques, each term arising in the definition of the obsercabies involving the coupling of some (or all) of these systems reduces to the general form F=
CF,F,.__F,F,.__
= CnF,,
(49)
where FA is a function oiy of matrix elements centered on A, Fs on B, and so on, and F,, Fg, _ _ _ are factors pertainin g directly to the reIevant fieIds_ C contains ail other quantities such as energy and radial denominators, and s runs over the system labels (A, B,_.., f, g,___)_ We shall take each F, to be a sum of matrix element products; it isa trivial generakation to extend consideration to matrices, tensors, etc_ with such elements. The observable F may then be analytically simplified by explicitly i+zorporating the symmetry of each system_ Suppose G, is lthe point symmetry group of system S with PZ,_operations Rf_ Then, noting that observa bks m-ust be tot&y syrnrzzetric
._
-:
with respect .to any symmetry operation of _any of thesyste&itfollowsthat
--
.:
_ .: --
Each factor-F= may therefore by treated mdependcntly, as the rcmaIhC~g factors act simply .-as a multiRlicative constant withres+t to z&y symmetry arguments based on G,: AR arguments thatfolIow apply to a single F,; which may be written in -the general quantum form
._c-
$pplied to the operator. product ang the %-i3jJket-~products indeRendently: This hypotheti&l p*:: dune may be given -a rigourous. foundation_ inrtbefollowing way_ : _ ; -.:-.- ;__ ,- -~_-_ -- -The fit stage is‘ the 0perator~:selection ruie ‘(OSRj stage, whichfollows_from the result thateq. (51) may be rigourously rewritten in the form .I
F,.=x
c (uljP;.l~~)(~~lP~;:_. r’ 7.x ,___
,i;*!u;),
where
(51) The Iuf) are the state functions of system S, which are bases for the IRS of G, such that U: transforms as the yth IR of G,_ The summation over y, A,...is over all degeneracies. The operators P,’ may be cartesian components of tensorial operators. vectors etc In the latter case, the relevant products between matrix elements must be retained in the operator product- The cyclic appearance of the states results directly from perturbation theoryIt is the manner in which symmetry is incorporated in simplifying such matrix element products that defies a selection ruIe scheme- The usual procedure for small order products involves the direct application of the SIT to each matrix element in turn; however, each successr‘ve application involves state and operator specification, as well as rapid increase in the attount of work as the order of the product in~m-eases_(The justification of the procedure has its origin in the product theorem of eq_ (S), each matrix element being replaced by its corresponding invariant, so that this procedure is rigorous and must tead to the correct result.) A fundamental chemical objection that may be levied against this proced~however, is the total obscuration of the relationships to the cotresponding _ da&al invariants that results from this procedure ,. :. The GSR approach uses the product theorem in a somewhat more abstrac: manner in order to rigourously resolve the :selection rules into a “classical” .and a “quantu~m” component. peach matrix dement is effectively split into bra, Iret.and operator factors, and the product rule is then
P”(G,)~nP~=~n~:-. r =
r- =
.153)
This result is referred to as the OSR theorem. The. proof of this result follows directly from theorems I, II, and III [eqs. (17)-(20)] discussed earlier_ The effect of a symmetry operation on a product is defined through the relation
In the expression for the operator product, it is crucial to retain the ordering of the operators, each being implicitly labelled by the relevant states in the matrix element_ It is readily seen that the OSR results Iead to invariant operator products that are isomorphic to the classical invariants, so that they reduce rigour-. ously to the classic+ s&&on rules. The second stage of state selection involves the application of the SIT to only those matrix eIement products that survive the OSR procedure (which is usually the most restrictive)_-. Rotational acerages_ It is apparent that increasing the symmetry can only serve to further ~tiplify the invariants_ The_ limit of the augmentation procedure may be taken to be the spherical groups. For -the purposes of ,this. work it is important _td distinguish between the “full” rotation group ,of_ Aoroper rotations, which we shah denote by _E , and the full rotation group .of proper and improper rotations (Z,). Note that Xh may be-generated from Z in the same manner as 0, from 0; viz 1using the single augmentation operation. i_
The relevant of this division follows from noting that we may consider the projectors of-the various grou+ containing only proper rotations as effecting a “rotational” average over the orientations generated by the operations of the group_ For the fiite groups, the average is therefore over a discrete number of orientations_ For the spherical group X, the projector is equivalent to a full three-dimensionaI rotational average (in which aII orientations are assumed equally probable, i.e. denoting such an average by ( )- we have that
PJ(Z)F=
P(Z)_
(5%
It is significant that, as a physicaI average can only be effected by proper rotations, the corresponding group (incIuding Z) are chirili groups (i-e- contain no improper rotations)_ A!l finite chirai groups are subgroups of 2, and thus no further simplification can foEow by incorporation the symmetry of mokcuks of such symmetries for expressions which are already rotationally averaged_ However, for achiraI mokcuks, the addilonuf intrinsic symmetry (i-e_ the improper axes) is not incorporated in a rotational average, and can lead to further symmetry restrictions_ In fact, for achiraI systems, the rotationai averaging and intrinsic achiraIity may be effected in a single stage by the application of PO(Z,)_ We have gone into some detail regarding this point in order fo emphasize that chiraIity cannot be destroyed by rotational averaging, a fact that stili causes some conftionThis has been discussed at Iength elsewhere [12]_ In fact. decomposition of F”(2) according to eq_ (5) reIative to 2,. i_eL
F0(2)=F0
(56)
yields FO(X,) as the pan-discriminating component (achirality function) and F’(2,) as the discriminating component (chirality function) corresponding to the rotatio&Iy averaged quantity F’(Z)_ We can in fact defime the achiral and chid projectors respectively as
PG=f(E-i)PO(Z),
(58)
where these project out the rekant (rotationally averaged) achirality or chirality function corresponding to the origin% function on which it operates (k F’(2) and -F((z,) k+zctiveIy)_ The final feature that follows from this d+cussion is that-the effect of increasing *e symmetry of the jystem is to bring the form of the invariant progrt%siveIy cIo_ to that of the rotationaUy averagd form_ The lower the rank of the tensoriai elements, the lower the symmetry required_ to reduce the invariant to the same form as that of (2) or (Z,)_ We shaII comment on the physical consequences of this in the applications.
5_ Appkatio~ The results of the preceding two sections are sufficientIy expkit to not warrant a description of how they should be applied_ We concentrate therefore on a dkcussion of the scope of applications, and a comparison with other symmetry selection methods. >_i_ CIarrcai property tensors The expressions for the invariants can be applied directly to the determination of invariant cIassical property tensors, such as permanent moments and ckxical mokcuku response tensors ;,e-- ~e~iIities, hyperpohuizabilitie etc-); dt part of each component, and the symmetry relationships between various components are thereby determined- Ahmann PI has approached this problem via spherical harmonics; Birss [6] on the other hand has determined relationships between the cartesian tensor components from examing the effect of generator sets of operations, and has tab&ted these relationships for tensors of up to fourth rank for the crystalk+ graphic point groups_ Both of -these approaches increase in complexity with increasing tensor size and group size, in contrast to the GSR approach5.t
Two-photon processes
The GSR approach does not give different re-m s&s for two-photon processes [3-5.131: it does,
however, provide anaherrkve perspective on the -previous work McCIain’s work on two-photon absorption involves an intuitive use of the OSR theorem in order to effect the rotati&aI averagingTheorem III [eq_ (2011 is then.effectiveIy assumed in order to derive his fmaI results as the SSR stage applied to twophoton absorption Similar comments appIy to conventional Raman scattering selection rule approaches, which ako implicitly assume eq_ (20)
where
and.
5_3_ Circ2fik.rdichroirm The- most effective apphcations of the GSR approach ak in chiraI~spectroscopic processes. the operator products being generahy of mixed ekctric and magnetic parentage We consider two types of appIi&tions: the first to natural (mtriusic) circular dichroism, and the -second to induced circular diChroism
X3. I. Intrinsic circular dichroism The rotationally averaged circular dichroism (CD) strength has the form Im(Olpln)
(5%
-
where p is the ekctric dipole operator [transforming as (x. y. r)] and m the magnetic dipole operator (transforming as (X, Y, Z)]_ 10) and I a> are the initial and fmat stages, of the system, respectively_ This expression is an invariant of the group Z_ Any single improper rotation wih augment Z to 2, (i-e_ c= refkction; ir inversion, S, = general improper rotation) as alI the improper rotations wiII then be generated_ Hence noting that P”(Z,)=+(EtijPof2)
(60)
and that (E+i)p-m=O
(61)
it follows that there is no intrinsic CD achiraI molecuk
for any
53.2. Induced circular dichroism CDcanbeinducedinanachiraIsystembya chiraI perturbation- The simplest models descri-bing the induced CD arise in firstorder perturba-. tion-theory, and can be expressed in the form R,=Rf,,kj,,:
]_
_.
_.
(69
The ach.iraIsystem, A, is represented by the ground state IO>, and excited states I a> and lb). such that the iuduced CD appears at the transition energy co of the 10) +. I a> transition of A which is assumed to be magnetic dipole allow&IO’), fu’) are the ground and &cited states of some and u’is perturbing system C_ Summa tionoverb impircit in eqs. (63) and (64). pA, mA arc the ekctric and magnetic dipole operators on A, so that zL!kb=
(65)
and so on_ The RS, contribution is referred to as the srnric coupfing [14] contribution, (bO’I V Id’) reducing simply to the coupling of a transition muhipole on A with a static field component due to system C The Rd, contribution may be referred to as the dynamic coupling [15] contribution, as it invoivti the coupling of transition muhipoks on A and C It is significant that both terms vanish on rota; tional averaging sq&sting that both terms be-come progressively smaher~with iucreasing symmetryofA_(Thishasbeenputfokardasan~ment for the importance of second-order pe?urbation terms, which survive such averaging [NJ].) We may consider the fmt-order terms as being_highIy symmetry&zpendent Schelhuan [l] has discussed the symmetry &kction rules for static &upIing terms By using the SIT on the matrix eIemen_ts m~“~~~. he deduces that, V must behave like a- pseudoscalar under the point symmetry.of A_ By expansion of V in terms of a Fourier &es about the unperturbed ~ymme-. ~.
try* Schelhnan analyses the A-dependent part of V in terms of spherical harmonics_ For the speciaI situation where the chiral perturber is represented by point charges, the syxnmetry behaviour of the A-dcpendcnt part of V corresponds exactly with the disposition of the charges, and so defies a sector rule_ This will be discussed further in a forthcoming publication ou CD mechanisms~ Schelhnan’s procedure becomes progressivelymore difficult for higher symmetry groups, to the extent that general expansion for 0, and T, arc not quoted_ In the GSR approach, the molecular operator product is anaIysed directly, yielding the muhipoIar components on A that can couple to the f&d (and thereby the symmetry pro_pertiesof the tieId) for non-zero CD strength_The augmentation procedure and the explicit use of symmetry4efined cartesian axes encounters no difficuh.ieswith the higher symmetry groups such as 0, and T+ with anaIytic expressions obtained for any order from cqs (3~(4t3)_ For exampIe for 0, the generaI form of the symmetry invariant A-dependent partof Yis V.-.= c
(_c)(X@,.%=c - X”J.9
(66)
where a. b. c m odd and a#bfc_ (x.y,z commute within y,_) The simplest form. for which a = 5, b = 3. c = 1, is the one given by SchelIman. The generality of the GSR approach is highlighted by its ready application to the dynamic coupling terms. for which SchcIIman’sapproach is inappropriate_Mason [2] has discussed the nature of Y in the dynamic coupling terms for metal complexes, by essentially using the SIT on the single Y matrix element The GSR scheme allows a determination of the exact form of the invariant operator product and hence the explicit form of the coupling term V_ For example. for 0, = C
(f
- zbyc)
)[mA,x’(ybz’
a.6.c
fmAryu(zbxXC-xbzIC) + mAZzP(xbyyC -ybzc)],
for a=2p;
b=2qti;
c=2rtl;
._
.:
” We have attemR&d to show that the Girt a&preach arzd the formalism presented in this paper provides a consistent framework for the definition of sekction rules for a variety of phenomena In addition, the invariants which are central to the approach have been analytically determined for the fmite point groups for arbitrary cartesian products in terms of which observables are generaIly expressed. The relationship to rotational averaging allows for a definition of obscrvablcs which arc chiral or a&i&, and in addition, those that are highIy symmetry sensitive (i-e_ those which vanish on averaging), and those that arc less so (i-c_ have non-vanishing rotationally averaged invariants). Highly symmetry sensitive terms appear particularly in first-order CD models, suggesting that higher-order perturbation terms couId easily be domiaaut over first-order terms for high symmetry systems. A detailed analysis of CD models and the role of symmetry selection rules will be diskszd in a forthcoming publication_
t _+‘z”
-X&_v=zc+ X’iezb - xcYbz$
PO(O,)[m,V,]
6_Gmdusions
q#r_
(67)
RckIV?u~ [II JA khdfma. J. Ghan Phys. 44 (1966) 55. PI SF. hfuoa. MoIaxIar optical xfivity and the cbiral dkrimkations (-bridge Univ_ plcss. London, X982)_ [3] LX Ovandcr. Opt Spuay_ 9 (1960) 302_ [4j P-R MOand W-M &fcCbin. J. Cbam Phys 53 (1970) 29. [S] T_R Bada and A_ GoId, Pbys. Rev_ 171(1968) 997_ [6] RR Bit-s. Symmetry and magnetism (North-Ho&and. Anrsterdam. 1964). m P-E Scbippcr. J_ Am C_Xmn_ Sot 100 (1978) 3658_ [8) G-F_ Kostcr. Psopcnies of the thirty-wo poinl groups (Ml-T Press. Cambridge. 1963)s 191SL Ahmann, Cambridge Phil. Sot (Ma*) Proc 53 (1957) 343. [IO] Handbook of physics and chanis~~. 60th Ed_ (CRC Press. Ckvdand 1979) P_ A-66_ &am&an. J. Chcms Phys 1111DL Andram and T_ l-him 67 (1977) 5026m 1121P-E Schippa. Ghan Phyx 44 (1979) 26L 1131WM. McChin. J. Chan Phys 55 (XVI) 23’89s 1141FS. Richardson. Clmn R&s (1979) 17. 1151EG_- H6hn and 0-E Wci~ Jr, J. Cban Phyx 48 (1968) 1127_ [16] P_E Scbipper. J_ Am Cban Sot 100 (1975) 1433s