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Associate-induced circular dichroism (AICD): A theory of the circular dichroism of associated or bonded chromophores in the dipole approximation
Chemical Physics 23 (1977) 159-166 0 North-HoUand Publishing Company
ASSOCIATE-INDUCED OF ASSOClATED
CIRCULAR
DICHROISM
OR BONDED CHROMOPHORES
(AICD):
A THEORY
OF THE CIRCULAR
DICHROISM
IN THE DIPOLE APPROXIMATION
E. SCHIPPER Departmeat of 77leoretical Chemisty, The University of Sydrrty. N.S. W. 2006. Australia Pieter
Received 1 December 1975 Revised manuscript received 22 February
1977
A theory is presented for the circular dichroism of an schira1 chromophore which is associated or bonded to some other chromophores (which may be chiral or nchiral). It is argued that second-order perturbation terms in the dipole npprosimation are at least of the same order as tiust-order perturbation terms involving hi&er multipoles. The latter are a common feature of current models of the naturcll circular dichroism of magnetic dipole allowed transitions of achiral chromophores in chin1 molecules. These second-order terms are derived and the symmetry rules for tic mechanisms discussed_ The theory is qualitatively applied to some simple cyclic peptides.
1. Introduction
The current theoretical models of the circular dichroism (CD) of achiral chromophores in chiral environments fall into two main categories: the static coupling (SC) [l-3] and dynamic coupling (DC) [4,5] models. Both have been extensively researched, and some
access
to this wide
field may
‘ae gained
through
reviews 16-91 and papers [5]. The development of such simple models has largely been stimulated by the success of empirical sector rules, and the desire to put them on some sort of theoretical basis. Both models may be established via the Rosenfeld equation [S] , which defmes the CD strength of a transition G --FE (where G designates the ground state of the system and E the excited state at which transition energy the CD appears) as a number
R,,
of recent
= Im((GlplE)
- WmlG)).
(1)
p, m are the electric and magnetic dipole operators for
the system as a whole, and may be thought of as the interacting directly with the radiation field. We initially consider the system as comprising the achiral chromophore A (with free chromophore states t) coupled via the electrostatic interaction operator V
moments
to the perturbing chromophore B (with free chromophore states u). If G corresponds to the system state where both A and B are in their ground states t = 0, u = 0, and E to the system state where only A is excited to the state t = s, the system states may be expressed in terms of the free chromophore states by using perturbation theory. Defining the product states }nc~=lt~l2l~,tnfl=~tl~rcl
(2)
the system states become, to first-order in perturbation theory:
y&s,
q
iE)= IsC!>+~ t.u (Es-Et-E,,)‘“)’
w&s=0
(3)
forG,and
The CD strength for the perturbed 0 + s transition follows simply from the substitution of G and E, noting that the zeroth-order term not containing Yvanishes (as A is achiral), and retaining only the terms that are first-order
in V. Two
types
of contributions
remain:
permanent moments on B, so that transition moments are borrowed from A itself due to the mixiig of A states by the static &Id of 6. These are the static coupling terms; (i) those involving only
P.E. Sch@per/Associate-inducedcirculardichroism (AICD)
160
(ii) those in which transition moments are borrowed from B. the perturbation acting to link the A and B states. These are the dynamic coupling terms. For reasons that wiIl be returned,to presently, these models are quite’adequate for electric dipole allowed transitions. The problems that are discussed in this paper are manifest only for magnetic dipole allowed transitions. We therefore restrict our considerations to thei latter case, for which the correction to the CD strength in first-order has the form Ras= Ln6spos-&,
(5)
where In so= (sOlm100~=m~,
(6)
and $JL
YAB(Ot,OO)Cc; v (rs,oo)pY + AB c 5 5 - %
VA&OS, $0 - ‘El, p.
1
0,)
E2 - e2 s I.i
(7)
Real wave functions have been assumed, and rnz = (~Irn~lO),~~~ = ~nIpBIO), and so on where mA, pg etc. are respectively the point dipole moment operators on the subscripted chromophores. The energies are now transition energies relative to the respective ground states. The first bracketed contribution describes the static coupling, the second the dynamic coupling. It should be noted that the two models are thus cornpIementary in describing the total CD strength in first order. Higher multipole participation may arise in two ways: firstly through the expansion of V as a multipole series [71 V=v&-~dq+vqq+.‘.,
(8)
where B is assumed uncharged, Vdd is Fe dipole-dipole interaction term, Vaq the dipoleiquadrupole term and so on; and secondly, by adding to the Rosenfeld equation terms directly describing the interaction of the radiation field&rough higher system moments [lo].. We shall restrict our attention to the former, although the general argument to be developed will also apply to the latter corrections. We are now in a position to examine criticaUy the first-order perturbation modeIs. Two types of correc-
tions that are being made to the vanishing zeroth-order CD strength should be distinguished: (i) the perturbation correction of the order of the perturbation theory employed; (ii) the multipole correction. This arises because contributions from successively higher multipoles generally converge rapidly, so that vdd is significantly greater than Vds, and so on. Thus, if Vdd is the zero&order correction to V itself, then V is the first-order correction to V. In a qualitative sen2, then, the Vdd terms constitute a first-order correction to the CD strength (first-order in perturbation theory, zero&order in correction to Y), whereas the Yds terms provide a secondorder correction to the CD strength by an analogous argument. For most achiral point groups the rddd corrections, because of symmetry arguments, vanish, so that the second-order corrections due to Vd constitute the first non-vanishing contributions to the 8D strength. Under such circumstances. one might criticaIIy argue that second-order corrections could also arise from secondorder perturbation theory, which would lead to terms quadratic in Vdd which, because of different symmetry requirements resulting from the participation of new intermediate states do not necessarily vanish. Such terms could conceivably compIete with the Vdq terms. For electric dipole allowed transitions, this problem does not arise simpIy because the first-order corrections due to Vdd do not vanish. That such contributions in the dipole-dipole approximation do not vanish turns out, in fact, to be the case, and it is the purpose of this paper to investigate such terms in detail. Such terms will be seen to involve mixed static/dynamic coupling terms, and thus cannot arise in pure SC or DC models (even if independently taken to higher order of perturbation theory [2,4]) whose complementarity is restricted to first-order perturbation theory. The model developed in this paper is referred to as AICD (associate-induced circular ctichroism) to stress the independent chromophore concept, as well as to illustrate its close affinity with DICD (dissociateor dispersion-induced CD), to which it reduces on averaging. Thus if B associates with A leading to a well defined relative AB orientation, the CD induced in A is referred to as AICD; if, however, the two are randomly orientated relative to each other, the CD induced in A is referred to as DICD. Recent DICD studies suggest that such a distinction is vita& important in interpreting CD data [13] _
P.E. ScllipperlAssociate-induced
2. The AICD model
system ground state. The states G and E are then developed to second order in the perturbation V:
The AICD model differs from the model of the introduction in two ways: firstly, it is necessary to go to second-order perturbation theory, and secondly, the possibility of terms linking a number of different perturbers arises. We consider therefore an associated SYStern of the achiraI chromophore A interacting with a number of chtomophores B,C, _.., with fixed relative orientations and negligible electron exchange- The systern as a whole will be referred to as a molecule. The states G and E now refer to the states of the multichromophoric molecule, where each chromophore is coupled to the remaining chromophores via the dipole -dipole interaction operator V. This has a radial dependence of rS3, r being used in the general sense as the distance between any two chtomophores. The presence of the interaction V will lead to the development of the CD strength as a power series in V; i.e. in rw3_ Terms up to an2 including r-e will be considered. Noting that an r term involves a maXimum of two dipole-dipoIe couplings (i.e. Vappears twice), the following types of terms may be distinguished:
@ @-%I3
M & V
one chromophore term (~‘0, two chromophore terra (F3) three chromophore term (F6) two chromophore term (r*>.
lG)=lO)+P,+T)+P;TiT),
(10)
IE) = IS) + &-I T) + gST I z?.
(111
The P’s are summation
operators defined as (12)
Degeneracies have been neglected here, but are readily included [ 121. Any states giving zero denominators are excluded from the summations_ lIre interaction operator has the form
v=fCv 43
(14)
@’
where V& = [Pa -Pa - 3(Pu *~@)(P~ -~&l/f;.
It follows that the treatment may be restricted in this approximation to a consideration of only three chromophotes (A,B,C), and that the result for the whole molecule may be determined by summing over all passible triplets in the system. 2.1. AICD theory The introduction of a three-chromophore system necessitates some change in notation. The free states of A will be denoted by t with a ground state t = 0, and the respective states of B,C will be differentiated by single and double primes. The unperturbed system states may then be written as simple products of the form
I z-3= I C,lt’)l t”>,
161
circtcb dichroisn (AKD)
69
withenergy eT= et 4 et’ +et” elatlve to the unperturbed
(1%
The indices cr,j3 run over the chromophores A&C and rti is the vectot connecting the origins of cuJ3with the cap denoting the unit vector. jr&is the electric dipole operator for chromophore a. ‘The CD of the molecule corresponding to the perturbed 0 + s transition of A is developed in temrs of the free chromophore states by substitution for G, E into the Rosenfeld equation, and expanding the molecular dipole operators as
where pa, m, are the intrinsic electric and magnetic dipole operators for chtomophore LY.(That part ofm giving rise to the socahed coupled-oscillator terms is neglected here as we are primarily interested in magnetic dipole allowed transitions.) The faNowing contributions arise: R=R,
+Rj +R;
fR&.
(17)
.
P.E. .%hipperfAssociate-induced
162
The subscript refers to the negative power of r involved in the particular term. Unprimed, primed and double primed refer to one, two and three.chromophore terms respectively. Thus R, gives the CD of unperturbed A (which must vanish by definition), and the higher terms the AICD of the s-transition. We shall refer to the ro3 terms (R;) as the first order AlCDterms, and the Rk. Ri (which fall off as r-6) as second order. Oniy two chromophore terms can arise in first order AICD. These are precisely the terms given in the introduction [eq. (5)-(7)] for the AB pair. Analogous contributions exist for each of the perturbing chromophores. There are two major simplifications that may be introduced for all second order terms. The first of these is the restriction to contributions involving only two excited states on each of the chromophores. These will be denoted ass and t for A, and u’, v’, u”, u” on B and C respectively. For A, this is equivalent to the stipulation that A has one predominant electric dipole allowed transition 0 + t through which the AICD in the s-transition is induced. r is referred to as the intermediate state. The applicability of this assumption is supported both by the close levels approximation discussed in the next paragraph, and the symmetry restrictions on the states s and t as discussed in a later section. The second important simplification may be referred to as the close levels approximation, and arises from a consideration of the relative magnitudes of the energy denominators.
AerS = or - Ed,
Writing
circular diclrroism (AICD)
are given, the corresponding
set of ABC terms foll%ing readily on inspection. There will be terms in all powers , but the leading terms are given here. For of (A# convenience, the t(‘s in the energy denominator are left unprimed, as the chromophore to which it refers is readily determined on inspection of the interaction matrix elements. (i) Terns of order (AE)-~ : R;I”(ind,disp)
[
AEusA’Ers
1
SO LLoBfl.“*,
(20a)
Ry (ind,disp)
~*,(OwJ) =lIIl
y&,00)
AElls nets
Ie
-m”Ao. (20b)
(ii) Terms of order (a~)-’ E-’ : Rp(ind,disp)
y&,00)
=Im
II
~*,m,Ou)
-df, 16;;”
(21a)
‘5s 5x
R;I” (disp) =h
Ry
AE,,~= E,/<- eS
~,,(OL~w,,(w-o
= Im
ViB (OLuv)v*B 0%a AE&~
-es)
uo.&J
1 ‘B
* ’
(21b)
(ind, disp)
(18)
and so on, and noting that for many cases, the separation between the ground and first excited states is generally significantly greater than the separation between the excited states themselves, it follows that, in a general sense, (Ae)* < EAC< E’;
(1%
i.e. terms with a (AE))’ denominator should be numerically more important than those with an EAE denominator. This may be referred to as the close levels approximation. Thus the second order AICD strength may be further expanded as a series in (Ae)-' ,and in some cases, it should be possible terms.
to consider
only the (a~)-2
The following contributions arise for magnetic dipole allowed transitions in second order. Only the ABC terms
The two moments involved in the scalar product (to the right of each term) may be referred to as the JieZd response moments, and are essentially the free chromophore transition moments that couple directly to the radiation field. For most of the contributions, the elec-
P.E. Sclt$perfAssocirrte-induced
tric and magnetic response moments are on different chromophores, aad it is this feature which indirectly allows for the relaxation of the symmetry rules responsible for the vanishing contributions of the dipole-
dipole interaction term to first-order perturbation models. The other moments are involved in the interaction matrix elements coupling the states of A,B and C. All of these terms contain a dynamic coupling component [the argument (disp) referring to a dispersiontype interaction involving the coupling of two transition moments on A and B] , and all except one contains a static coupling component denoted by (ind) which refers to an induction-type interaction involving the coupling of a permanent dipoIe on the perturber with a transition moment on A. The pure dynamic coupling term is the only one which remains on averaging over all orientations of the chromophores, and is the leading DICD (dissociate- or dispersion-induced CD) term [l I] Some other general features may be noted. Previous theories employing an independent chromophore model have restricted consideration to essentially two chromophore terms. As the radiation wavelengths employed in CD work are generally considerably larger than the chromophore separations, the radiation field sees the entire molecular system effectively as a point so that, to ,d terms, two and three chromophore terms are of equal importance. Furthermore, the majority of terms depends on both the static field at A (due to a permanent dipole on a perturber), and the polarizability of one of the perturbing oscillators. In three chromophore terms, these may originate from two quite separate perturbers. Thus for a series of compounds for which the static field component is constant, these terms will qualitatively mimic first-order dynamic coupling terms, and vice versa. This could be a rationale for the success of the polarizability model for metal complexes [4], where a dipole-hexadecapole correction to Yin first-order perturbation theory is used to explain the CD of the d-d transitions of the metal ion. Finally, the important role of the intermediate state t which should be strongly electric dipole allowed from the ground state cannot be overstressed, and will be further discussed in the symmetry arguments that follow. The importance of this state for the DICD of magnetic dipole allowed transitions has been referred to elsewhere [12,13] and is supported by experimental results for dissociated systems.
163
circular dichroism (AICB)
3. Symmetry considerations The expressions for AICD may be written in the general form
F= FAFBc__, where FA contains all the matrix elements centred on A, and FB,__ ail other quantities pertaining to the other chromophor% and all energy and radial denominators. If A belongs to the point group G, group theory restricts the possible combinations of state and operator symmetries yielding finite values of FA. These symmetry
conditions may be referred to as .he generalized selection rule for the process described by FA_ Consider, for example, the matrix element product describing the static coupling mechanism in first-order perturbation theory: toIp*lt)
m
(slm*[OL-
The selection rule is derived in two stages, so that it defines: (i) the combinations of polarizations of Pprn that are allowed under G (which we shall refer to as the operator selection n&s); and (ii) the allowed symmetries of the states s and f for a given ground state symmetry (which we refer to as the state selection rules). The operator selection rules are most restrictive, and
may be developed in the following way. Consider the general matrix element product
where the basis function Ir!!) of A transforms as the Mh row of theith irreducible representation of G. The summation over a collects all matrix products differing only in the nature of the operators, and that over K,&... is over all degeneracies (which are assumed to be not split appreciably energetically). If9 is the projection for the totally symmetric representation A1 of G, and
(24) if follows from symmetry
arguments
that
164
P.E. SdtipperlAssociate-induced circular dichroim (AICD)
Table 1 Operator selection rules for Czv Allowed polarization combinations
Order of AICD tern
Term
first first
SC
DC
PAmA
second
WI DC
CIAILA~A
second
SC/DC
PA *A
type
FA
PA-~A
Qualitatively, the operator product sums derived from simply ignoring all the wavefunctions must contain A, if FA is not to vanish. In the present case, the operators are simply components of dipole moments, so that the operator products transform as tensors. The operator selection rules then reduce to the selection rules for macroscopic property tensors, which have been discussed in detail by Birss [ 14]_ As the theory $ll be applied to the C, carbonyl chromophore in the next section, the operator selection rules for C,, are summarized in table 1, together with the operator products appearing in the expressions for the various mechanisms. For a i-polarized magnetic dipoIe allowed transition (AI + pL2under C.&, only the second order AICD terms are allowed, the moment product on A for the leading terms being
(26) This product also appears in the DICD for this transition, and is referred to as the magnetic inducibility. It follows that the state selection rules will be precisely the same as the DICD selection rules, which have been discussed in detail elsewhere [ 121. For this transition, the intermediate state must be of Bl or B2 symmetry. The allowedness of the (p X II) - m is quite general for all point groups. (For example, the B-term in MCD has this general form, although the magnetic transition moment in such a case involves the state f.) This has been discussed for DICD studies [ 12,131, and thus suggests this mechanism as an important contributor to the CD whenever first-order contributions are forbidden by the selection rules in the dipole approximation. Two systems spring readily to mind: the carbonyl chromophore, which is discussed in the next section; and the d-d transitions of metal compIexes. In the lat-
ter, the intermediate state is usually a strongly allowed (electrically) charge transfer state involving some of the orbit& on the ligating atoms, with the ground and s state being pure d states. The constraints on the A moments having been defined, it remains to consider the symmetry conditions restricting the relative orientations of the moments on the various chromophores which lead to the socalled sector rules. These will only be explicitly discussed for the (A$* terms; those for the (eAe))-I contributions are similarly derived. Two geometric constraints are imposed on the moments: (i) the field response moments lead to a vanishing contribution if they are orthogonal and a maximum contribution if parallel; (ii) the two electric moments (say pa, pO) involved in a particular interaction term Vti lead to a vanishing contribution if p.., pP, r@ are mutually orthogonal. Maximum contributtons appear for parallel moments: viz. tt, + t Note that if one of the moments is orthoif follows that Vdlpdepends directly on the gonal to r scalar pro 2’uct of the two moments. In summary, for the (~%e)-~terms the oscillatory moment pp on the perturber B must have a compon;‘l’f along rnf , and also an interactive component with uA _ The permanent moment, which may be on B or arise from any other perturber C, needs to have only an interactive component along pz which is perpendicular to the plane of the other two Amoments. That these simple geometric rules may sometimes reduce to empirical sector rules is rationalized as follows. For a particular A moment, the dipolar character leads to a natural division into (+-) and (-) charge sectors divided by the plane perpendicular to the dipole. Two orthogonal moments lead to quadrants, and three orthogonal moments to o&ants. However, such a divi-
P.E. Schipper/Associnte-induced
circular dichroism
(AICD]
165
Carbonyl transitions of two peptides: (I) L-pynolidine-2-one and (II) L-pyrrolidine-2-one. sion, though useful for highly symmetrical complexes, is usually unnecessarily complex, and the geometrical ruIes derived above are sufficientIy simple to use directly. In fact, because of the form of the dipole-dipole interaction operator, the geometric rules only reduce to sector rules for cases where the interaction terms reduce to scalar products of the interacting moments.
4. Applications to the carbony
chromophore
The carbonyl chromophore is one of the most extensitvely studied CD chromophores in the organic literature. Under effective Czv symmetry, the first transition (n +T*) from the 1AI ground state to the ‘A, excited state is magnetic dipole allowed, z-polarized, and appears at about 37000 cm-’ (in formaldehyde). Ay-polarized (n + a*) transition of symmetry IB, appears at about 54000 cm-I, and is electric dipole allowed. The axes are defined in fig. 1. Several first-order mechanisms have been suggested [5,7], but we shall consider only the second-order AICD contributions, and show that qualitatively they agree-with the empirical sector ruIes for some simple peptides. Detailed quantitative calculations to compare the various mechanisms are in progress, and the following discussion should be tentatively regarded as an illustrative application of the theory. The s-transition is taken to correspond to the ‘A, transition, and the intermediate state of symmetry ‘B,. Firat order AICD vanishes in the dipole approximation,
so that second-order AICD terms will be investigated and in particular the leading (a~)>-’ contributions. The dependence on the properties of the carbonyl cbromophore may be extracted as the magnetic inducibility factor [eq. (26)] _The calculation of this factor is a problem in itself, involving relatively sophisticated MO techniques. In this study, we shall take it to be an intrinsic constant for the carbonyl chromophore, and will be comparing the AICD of the carbonyl group in some different chiraI environments. The symmetry rules for these contributions may then be summarized as follows. The permanent dipole of the inducing species must have an x-component, and the perturbing oscillator must have atcomponent, and preferably also an appreciable component along they-axis. The theory is now applied to the carbonyl transition of two simple peptides: L-pyrrolidine-2-one (I) and L2Wiketopiperazine (II) may be defined for (I): (A): the C=O group with C,, symmetry and the states discussed above; (B): the ring skeleton; this is assumed to have an inplane transition which, due to the nitrogen, has both z- andy-components in its transition dipole; (C): the substituted groups; these are taken to contribute only the permanent dipoles perpendicular to the plane. For (II), the carbonyl transitions are degenerate.
However, they may be treated separately since their parallelism precludes any coupled-oscillator contributions, and thus taking linear combinations of their localized wave functions leads to the same expression as that obtained by treating the two chromophores sepa-
P.E. SchipperiAssociare-induced
,166
rately and simply adding their AICD. The AICD of each may be treated as for (I), and will differ only in that both R and R’ will contribute to the static field at each chromophore. Considering firstly substitution at the 3 and 5 positions for (I), the following empirical rule has been postulated in the literature 1151 for the the CD of the carbony chromophore: R nn* = c
i
Qi-&_y; (ailrim
-
bi/rF)_
The Qr are the effective charges of the groups Ci; $, 7; the direction cosines with respect to the subscripted Cartesian axes of group Ci; and ai, bi positive constants referring to 3 and 5 substitution respectively. rj is the distance of group Ci from the carbonyl chromophore A, and m, n some positive constants. This is then the empirical sector rule for this system. Application of second order AICD theory leads to the following simplified expression for the carbonyl CD, where the only variables are the permanent dipole on the groups Ci and the ACi distance:
,P&other quantities have been absorbed into K, and j; is the unit vector in the x-direction. It is not difficult to see that this theoretically derived expression leads to the same behaviour of the CD as the more complicated empirical sector rule. The effects of the substituents is simply to change the static field at the carbony1 group, and the sign of the AICD is determined simply by the sign of the x-component. The above application is qualitative, and the agreement is by no means meant to imply that other models
circuiar dichroism
(AICD)
are incapable of leading to a similar qualitative result. The importance of the above mechanism will become clearer when exhaustive quantitative tests are completed.
AcknowIedgement The author wishes to thank Professor N.S. Hush and Dr. C.J. Hawkins for discussions on the manuscript.
References [lJ W. Moffltt,Chem. Phys. 25 (1956) 1189. 121 G. Hil~es and FS. Richardson, Inorg. Chem. 15 (1976) 2582. [3] R.W. Strickland and F.S. Richardson, 3. Phys. Chem. SO (1976) 164. [4J S.F. Mason and R.H. Seal, Mol. phys. 31 (1976) 755. [SJ E.G. Hohn and O.E. Weigsng Jr., J. Chem. Phys. 48 (1968) 1127. [6J J.A. ScheUman, AccountsChem. Res. l(1968) 144. [7J A.D. Buckingham and P.J. Stiles, Accounts Chem. Res. ? (1974) 258. [SJ C.J. Hawkins. Absolute configuration of metal complexes (Wiley- Interscience, New York, 1971). [P] P. Crabbe. An introduction to chiropticalmethods in chemistry, Monograph (Syntex LA., 1971). J. Chem. Phys. [ IOJ EA. Power and T. Thirunamachandran, 60 (1974) 3695. [ll) P.E. Schipper,Mol.Phys. 29 (1975) 1705. (t2J P.E. Schipper, Inorg.Chim. Acta 14 (1975) 161. (131 P.E.Schipper.J. Am.Chem.Soc.98 (197617938. [14J R.R. Birs~;Symmetry and magnetism (North-Holland, Amsterdam, 1964). [UJ DJ. Caldwell and H. Eyrfng, The theory of optical activity (Wiley - Interscience, New York, 1971).
ASSOCIATE-INDUCED OF ASSOClATED
CIRCULAR
DICHROISM
OR BONDED CHROMOPHORES
(AICD):
A THEORY
OF THE CIRCULAR
DICHROISM
IN THE DIPOLE APPROXIMATION
E. SCHIPPER Departmeat of 77leoretical Chemisty, The University of Sydrrty. N.S. W. 2006. Australia Pieter
Received 1 December 1975 Revised manuscript received 22 February
1977
A theory is presented for the circular dichroism of an schira1 chromophore which is associated or bonded to some other chromophores (which may be chiral or nchiral). It is argued that second-order perturbation terms in the dipole npprosimation are at least of the same order as tiust-order perturbation terms involving hi&er multipoles. The latter are a common feature of current models of the naturcll circular dichroism of magnetic dipole allowed transitions of achiral chromophores in chin1 molecules. These second-order terms are derived and the symmetry rules for tic mechanisms discussed_ The theory is qualitatively applied to some simple cyclic peptides.
1. Introduction
The current theoretical models of the circular dichroism (CD) of achiral chromophores in chiral environments fall into two main categories: the static coupling (SC) [l-3] and dynamic coupling (DC) [4,5] models. Both have been extensively researched, and some
access
to this wide
field may
‘ae gained
through
reviews 16-91 and papers [5]. The development of such simple models has largely been stimulated by the success of empirical sector rules, and the desire to put them on some sort of theoretical basis. Both models may be established via the Rosenfeld equation [S] , which defmes the CD strength of a transition G --FE (where G designates the ground state of the system and E the excited state at which transition energy the CD appears) as a number
R,,
of recent
= Im((GlplE)
- WmlG)).
(1)
p, m are the electric and magnetic dipole operators for
the system as a whole, and may be thought of as the interacting directly with the radiation field. We initially consider the system as comprising the achiral chromophore A (with free chromophore states t) coupled via the electrostatic interaction operator V
moments
to the perturbing chromophore B (with free chromophore states u). If G corresponds to the system state where both A and B are in their ground states t = 0, u = 0, and E to the system state where only A is excited to the state t = s, the system states may be expressed in terms of the free chromophore states by using perturbation theory. Defining the product states }nc~=lt~l2l~,tnfl=~tl~rcl
(2)
the system states become, to first-order in perturbation theory:
y&s,
q
iE)= IsC!>+~ t.u (Es-Et-E,,)‘“)’
w&s=0
(3)
forG,and
The CD strength for the perturbed 0 + s transition follows simply from the substitution of G and E, noting that the zeroth-order term not containing Yvanishes (as A is achiral), and retaining only the terms that are first-order
in V. Two
types
of contributions
remain:
permanent moments on B, so that transition moments are borrowed from A itself due to the mixiig of A states by the static &Id of 6. These are the static coupling terms; (i) those involving only
P.E. Sch@per/Associate-inducedcirculardichroism (AICD)
160
(ii) those in which transition moments are borrowed from B. the perturbation acting to link the A and B states. These are the dynamic coupling terms. For reasons that wiIl be returned,to presently, these models are quite’adequate for electric dipole allowed transitions. The problems that are discussed in this paper are manifest only for magnetic dipole allowed transitions. We therefore restrict our considerations to thei latter case, for which the correction to the CD strength in first-order has the form Ras= Ln6spos-&,
(5)
where In so= (sOlm100~=m~,
(6)
and $JL
YAB(Ot,OO)Cc; v (rs,oo)pY + AB c 5 5 - %
VA&OS, $0 - ‘El, p.
1
0,)
E2 - e2 s I.i
(7)
Real wave functions have been assumed, and rnz = (~Irn~lO),~~~ = ~nIpBIO), and so on where mA, pg etc. are respectively the point dipole moment operators on the subscripted chromophores. The energies are now transition energies relative to the respective ground states. The first bracketed contribution describes the static coupling, the second the dynamic coupling. It should be noted that the two models are thus cornpIementary in describing the total CD strength in first order. Higher multipole participation may arise in two ways: firstly through the expansion of V as a multipole series [71 V=v&-~dq+vqq+.‘.,
(8)
where B is assumed uncharged, Vdd is Fe dipole-dipole interaction term, Vaq the dipoleiquadrupole term and so on; and secondly, by adding to the Rosenfeld equation terms directly describing the interaction of the radiation field&rough higher system moments [lo].. We shall restrict our attention to the former, although the general argument to be developed will also apply to the latter corrections. We are now in a position to examine criticaUy the first-order perturbation modeIs. Two types of correc-
tions that are being made to the vanishing zeroth-order CD strength should be distinguished: (i) the perturbation correction of the order of the perturbation theory employed; (ii) the multipole correction. This arises because contributions from successively higher multipoles generally converge rapidly, so that vdd is significantly greater than Vds, and so on. Thus, if Vdd is the zero&order correction to V itself, then V is the first-order correction to V. In a qualitative sen2, then, the Vdd terms constitute a first-order correction to the CD strength (first-order in perturbation theory, zero&order in correction to Y), whereas the Yds terms provide a secondorder correction to the CD strength by an analogous argument. For most achiral point groups the rddd corrections, because of symmetry arguments, vanish, so that the second-order corrections due to Vd constitute the first non-vanishing contributions to the 8D strength. Under such circumstances. one might criticaIIy argue that second-order corrections could also arise from secondorder perturbation theory, which would lead to terms quadratic in Vdd which, because of different symmetry requirements resulting from the participation of new intermediate states do not necessarily vanish. Such terms could conceivably compIete with the Vdq terms. For electric dipole allowed transitions, this problem does not arise simpIy because the first-order corrections due to Vdd do not vanish. That such contributions in the dipole-dipole approximation do not vanish turns out, in fact, to be the case, and it is the purpose of this paper to investigate such terms in detail. Such terms will be seen to involve mixed static/dynamic coupling terms, and thus cannot arise in pure SC or DC models (even if independently taken to higher order of perturbation theory [2,4]) whose complementarity is restricted to first-order perturbation theory. The model developed in this paper is referred to as AICD (associate-induced circular ctichroism) to stress the independent chromophore concept, as well as to illustrate its close affinity with DICD (dissociateor dispersion-induced CD), to which it reduces on averaging. Thus if B associates with A leading to a well defined relative AB orientation, the CD induced in A is referred to as AICD; if, however, the two are randomly orientated relative to each other, the CD induced in A is referred to as DICD. Recent DICD studies suggest that such a distinction is vita& important in interpreting CD data [13] _
P.E. ScllipperlAssociate-induced
2. The AICD model
system ground state. The states G and E are then developed to second order in the perturbation V:
The AICD model differs from the model of the introduction in two ways: firstly, it is necessary to go to second-order perturbation theory, and secondly, the possibility of terms linking a number of different perturbers arises. We consider therefore an associated SYStern of the achiraI chromophore A interacting with a number of chtomophores B,C, _.., with fixed relative orientations and negligible electron exchange- The systern as a whole will be referred to as a molecule. The states G and E now refer to the states of the multichromophoric molecule, where each chromophore is coupled to the remaining chromophores via the dipole -dipole interaction operator V. This has a radial dependence of rS3, r being used in the general sense as the distance between any two chtomophores. The presence of the interaction V will lead to the development of the CD strength as a power series in V; i.e. in rw3_ Terms up to an2 including r-e will be considered. Noting that an r term involves a maXimum of two dipole-dipoIe couplings (i.e. Vappears twice), the following types of terms may be distinguished:
@ @-%I3
M & V
one chromophore term (~‘0, two chromophore terra (F3) three chromophore term (F6) two chromophore term (r*>.
lG)=lO)+P,+T)+P;TiT),
(10)
IE) = IS) + &-I T) + gST I z?.
(111
The P’s are summation
operators defined as (12)
Degeneracies have been neglected here, but are readily included [ 121. Any states giving zero denominators are excluded from the summations_ lIre interaction operator has the form
v=fCv 43
(14)
@’
where V& = [Pa -Pa - 3(Pu *~@)(P~ -~&l/f;.
It follows that the treatment may be restricted in this approximation to a consideration of only three chromophotes (A,B,C), and that the result for the whole molecule may be determined by summing over all passible triplets in the system. 2.1. AICD theory The introduction of a three-chromophore system necessitates some change in notation. The free states of A will be denoted by t with a ground state t = 0, and the respective states of B,C will be differentiated by single and double primes. The unperturbed system states may then be written as simple products of the form
I z-3= I C,lt’)l t”>,
161
circtcb dichroisn (AKD)
69
withenergy eT= et 4 et’ +et” elatlve to the unperturbed
(1%
The indices cr,j3 run over the chromophores A&C and rti is the vectot connecting the origins of cuJ3with the cap denoting the unit vector. jr&is the electric dipole operator for chromophore a. ‘The CD of the molecule corresponding to the perturbed 0 + s transition of A is developed in temrs of the free chromophore states by substitution for G, E into the Rosenfeld equation, and expanding the molecular dipole operators as
where pa, m, are the intrinsic electric and magnetic dipole operators for chtomophore LY.(That part ofm giving rise to the socahed coupled-oscillator terms is neglected here as we are primarily interested in magnetic dipole allowed transitions.) The faNowing contributions arise: R=R,
+Rj +R;
fR&.
(17)
.
P.E. .%hipperfAssociate-induced
162
The subscript refers to the negative power of r involved in the particular term. Unprimed, primed and double primed refer to one, two and three.chromophore terms respectively. Thus R, gives the CD of unperturbed A (which must vanish by definition), and the higher terms the AICD of the s-transition. We shall refer to the ro3 terms (R;) as the first order AlCDterms, and the Rk. Ri (which fall off as r-6) as second order. Oniy two chromophore terms can arise in first order AICD. These are precisely the terms given in the introduction [eq. (5)-(7)] for the AB pair. Analogous contributions exist for each of the perturbing chromophores. There are two major simplifications that may be introduced for all second order terms. The first of these is the restriction to contributions involving only two excited states on each of the chromophores. These will be denoted ass and t for A, and u’, v’, u”, u” on B and C respectively. For A, this is equivalent to the stipulation that A has one predominant electric dipole allowed transition 0 + t through which the AICD in the s-transition is induced. r is referred to as the intermediate state. The applicability of this assumption is supported both by the close levels approximation discussed in the next paragraph, and the symmetry restrictions on the states s and t as discussed in a later section. The second important simplification may be referred to as the close levels approximation, and arises from a consideration of the relative magnitudes of the energy denominators.
AerS = or - Ed,
Writing
circular diclrroism (AICD)
are given, the corresponding
set of ABC terms foll%ing readily on inspection. There will be terms in all powers , but the leading terms are given here. For of (A# convenience, the t(‘s in the energy denominator are left unprimed, as the chromophore to which it refers is readily determined on inspection of the interaction matrix elements. (i) Terns of order (AE)-~ : R;I”(ind,disp)
[
AEusA’Ers
1
SO LLoBfl.“*,
(20a)
Ry (ind,disp)
~*,(OwJ) =lIIl
y&,00)
AElls nets
Ie
-m”Ao. (20b)
(ii) Terms of order (a~)-’ E-’ : Rp(ind,disp)
y&,00)
=Im
II
~*,m,Ou)
-df, 16;;”
(21a)
‘5s 5x
R;I” (disp) =h
Ry
AE,,~= E,/<- eS
~,,(OL~w,,(w-o
= Im
ViB (OLuv)v*B 0%a AE&~
-es)
uo.&J
1 ‘B
* ’
(21b)
(ind, disp)
(18)
and so on, and noting that for many cases, the separation between the ground and first excited states is generally significantly greater than the separation between the excited states themselves, it follows that, in a general sense, (Ae)* < EAC< E’;
(1%
i.e. terms with a (AE))’ denominator should be numerically more important than those with an EAE denominator. This may be referred to as the close levels approximation. Thus the second order AICD strength may be further expanded as a series in (Ae)-' ,and in some cases, it should be possible terms.
to consider
only the (a~)-2
The following contributions arise for magnetic dipole allowed transitions in second order. Only the ABC terms
The two moments involved in the scalar product (to the right of each term) may be referred to as the JieZd response moments, and are essentially the free chromophore transition moments that couple directly to the radiation field. For most of the contributions, the elec-
P.E. Sclt$perfAssocirrte-induced
tric and magnetic response moments are on different chromophores, aad it is this feature which indirectly allows for the relaxation of the symmetry rules responsible for the vanishing contributions of the dipole-
dipole interaction term to first-order perturbation models. The other moments are involved in the interaction matrix elements coupling the states of A,B and C. All of these terms contain a dynamic coupling component [the argument (disp) referring to a dispersiontype interaction involving the coupling of two transition moments on A and B] , and all except one contains a static coupling component denoted by (ind) which refers to an induction-type interaction involving the coupling of a permanent dipoIe on the perturber with a transition moment on A. The pure dynamic coupling term is the only one which remains on averaging over all orientations of the chromophores, and is the leading DICD (dissociate- or dispersion-induced CD) term [l I] Some other general features may be noted. Previous theories employing an independent chromophore model have restricted consideration to essentially two chromophore terms. As the radiation wavelengths employed in CD work are generally considerably larger than the chromophore separations, the radiation field sees the entire molecular system effectively as a point so that, to ,d terms, two and three chromophore terms are of equal importance. Furthermore, the majority of terms depends on both the static field at A (due to a permanent dipole on a perturber), and the polarizability of one of the perturbing oscillators. In three chromophore terms, these may originate from two quite separate perturbers. Thus for a series of compounds for which the static field component is constant, these terms will qualitatively mimic first-order dynamic coupling terms, and vice versa. This could be a rationale for the success of the polarizability model for metal complexes [4], where a dipole-hexadecapole correction to Yin first-order perturbation theory is used to explain the CD of the d-d transitions of the metal ion. Finally, the important role of the intermediate state t which should be strongly electric dipole allowed from the ground state cannot be overstressed, and will be further discussed in the symmetry arguments that follow. The importance of this state for the DICD of magnetic dipole allowed transitions has been referred to elsewhere [12,13] and is supported by experimental results for dissociated systems.
163
circular dichroism (AICB)
3. Symmetry considerations The expressions for AICD may be written in the general form
F= FAFBc__, where FA contains all the matrix elements centred on A, and FB,__ ail other quantities pertaining to the other chromophor% and all energy and radial denominators. If A belongs to the point group G, group theory restricts the possible combinations of state and operator symmetries yielding finite values of FA. These symmetry
conditions may be referred to as .he generalized selection rule for the process described by FA_ Consider, for example, the matrix element product describing the static coupling mechanism in first-order perturbation theory: toIp*lt)
m
(slm*[OL-
The selection rule is derived in two stages, so that it defines: (i) the combinations of polarizations of Pprn that are allowed under G (which we shall refer to as the operator selection n&s); and (ii) the allowed symmetries of the states s and f for a given ground state symmetry (which we refer to as the state selection rules). The operator selection rules are most restrictive, and
may be developed in the following way. Consider the general matrix element product
where the basis function Ir!!) of A transforms as the Mh row of theith irreducible representation of G. The summation over a collects all matrix products differing only in the nature of the operators, and that over K,&... is over all degeneracies (which are assumed to be not split appreciably energetically). If9 is the projection for the totally symmetric representation A1 of G, and
(24) if follows from symmetry
arguments
that
164
P.E. SdtipperlAssociate-induced circular dichroim (AICD)
Table 1 Operator selection rules for Czv Allowed polarization combinations
Order of AICD tern
Term
first first
SC
DC
PAmA
second
WI DC
CIAILA~A
second
SC/DC
PA *A
type
FA
PA-~A
Qualitatively, the operator product sums derived from simply ignoring all the wavefunctions must contain A, if FA is not to vanish. In the present case, the operators are simply components of dipole moments, so that the operator products transform as tensors. The operator selection rules then reduce to the selection rules for macroscopic property tensors, which have been discussed in detail by Birss [ 14]_ As the theory $ll be applied to the C, carbonyl chromophore in the next section, the operator selection rules for C,, are summarized in table 1, together with the operator products appearing in the expressions for the various mechanisms. For a i-polarized magnetic dipoIe allowed transition (AI + pL2under C.&, only the second order AICD terms are allowed, the moment product on A for the leading terms being
(26) This product also appears in the DICD for this transition, and is referred to as the magnetic inducibility. It follows that the state selection rules will be precisely the same as the DICD selection rules, which have been discussed in detail elsewhere [ 121. For this transition, the intermediate state must be of Bl or B2 symmetry. The allowedness of the (p X II) - m is quite general for all point groups. (For example, the B-term in MCD has this general form, although the magnetic transition moment in such a case involves the state f.) This has been discussed for DICD studies [ 12,131, and thus suggests this mechanism as an important contributor to the CD whenever first-order contributions are forbidden by the selection rules in the dipole approximation. Two systems spring readily to mind: the carbonyl chromophore, which is discussed in the next section; and the d-d transitions of metal compIexes. In the lat-
ter, the intermediate state is usually a strongly allowed (electrically) charge transfer state involving some of the orbit& on the ligating atoms, with the ground and s state being pure d states. The constraints on the A moments having been defined, it remains to consider the symmetry conditions restricting the relative orientations of the moments on the various chromophores which lead to the socalled sector rules. These will only be explicitly discussed for the (A$* terms; those for the (eAe))-I contributions are similarly derived. Two geometric constraints are imposed on the moments: (i) the field response moments lead to a vanishing contribution if they are orthogonal and a maximum contribution if parallel; (ii) the two electric moments (say pa, pO) involved in a particular interaction term Vti lead to a vanishing contribution if p.., pP, r@ are mutually orthogonal. Maximum contributtons appear for parallel moments: viz. tt, + t Note that if one of the moments is orthoif follows that Vdlpdepends directly on the gonal to r scalar pro 2’uct of the two moments. In summary, for the (~%e)-~terms the oscillatory moment pp on the perturber B must have a compon;‘l’f along rnf , and also an interactive component with uA _ The permanent moment, which may be on B or arise from any other perturber C, needs to have only an interactive component along pz which is perpendicular to the plane of the other two Amoments. That these simple geometric rules may sometimes reduce to empirical sector rules is rationalized as follows. For a particular A moment, the dipolar character leads to a natural division into (+-) and (-) charge sectors divided by the plane perpendicular to the dipole. Two orthogonal moments lead to quadrants, and three orthogonal moments to o&ants. However, such a divi-
P.E. Schipper/Associnte-induced
circular dichroism
(AICD]
165
Carbonyl transitions of two peptides: (I) L-pynolidine-2-one and (II) L-pyrrolidine-2-one. sion, though useful for highly symmetrical complexes, is usually unnecessarily complex, and the geometrical ruIes derived above are sufficientIy simple to use directly. In fact, because of the form of the dipole-dipole interaction operator, the geometric rules only reduce to sector rules for cases where the interaction terms reduce to scalar products of the interacting moments.
4. Applications to the carbony
chromophore
The carbonyl chromophore is one of the most extensitvely studied CD chromophores in the organic literature. Under effective Czv symmetry, the first transition (n +T*) from the 1AI ground state to the ‘A, excited state is magnetic dipole allowed, z-polarized, and appears at about 37000 cm-’ (in formaldehyde). Ay-polarized (n + a*) transition of symmetry IB, appears at about 54000 cm-I, and is electric dipole allowed. The axes are defined in fig. 1. Several first-order mechanisms have been suggested [5,7], but we shall consider only the second-order AICD contributions, and show that qualitatively they agree-with the empirical sector ruIes for some simple peptides. Detailed quantitative calculations to compare the various mechanisms are in progress, and the following discussion should be tentatively regarded as an illustrative application of the theory. The s-transition is taken to correspond to the ‘A, transition, and the intermediate state of symmetry ‘B,. Firat order AICD vanishes in the dipole approximation,
so that second-order AICD terms will be investigated and in particular the leading (a~)>-’ contributions. The dependence on the properties of the carbonyl cbromophore may be extracted as the magnetic inducibility factor [eq. (26)] _The calculation of this factor is a problem in itself, involving relatively sophisticated MO techniques. In this study, we shall take it to be an intrinsic constant for the carbonyl chromophore, and will be comparing the AICD of the carbonyl group in some different chiraI environments. The symmetry rules for these contributions may then be summarized as follows. The permanent dipole of the inducing species must have an x-component, and the perturbing oscillator must have atcomponent, and preferably also an appreciable component along they-axis. The theory is now applied to the carbonyl transition of two simple peptides: L-pyrrolidine-2-one (I) and L2Wiketopiperazine (II) may be defined for (I): (A): the C=O group with C,, symmetry and the states discussed above; (B): the ring skeleton; this is assumed to have an inplane transition which, due to the nitrogen, has both z- andy-components in its transition dipole; (C): the substituted groups; these are taken to contribute only the permanent dipoles perpendicular to the plane. For (II), the carbonyl transitions are degenerate.
However, they may be treated separately since their parallelism precludes any coupled-oscillator contributions, and thus taking linear combinations of their localized wave functions leads to the same expression as that obtained by treating the two chromophores sepa-
P.E. SchipperiAssociare-induced
,166
rately and simply adding their AICD. The AICD of each may be treated as for (I), and will differ only in that both R and R’ will contribute to the static field at each chromophore. Considering firstly substitution at the 3 and 5 positions for (I), the following empirical rule has been postulated in the literature 1151 for the the CD of the carbony chromophore: R nn* = c
i
Qi-&_y; (ailrim
-
bi/rF)_
The Qr are the effective charges of the groups Ci; $, 7; the direction cosines with respect to the subscripted Cartesian axes of group Ci; and ai, bi positive constants referring to 3 and 5 substitution respectively. rj is the distance of group Ci from the carbonyl chromophore A, and m, n some positive constants. This is then the empirical sector rule for this system. Application of second order AICD theory leads to the following simplified expression for the carbonyl CD, where the only variables are the permanent dipole on the groups Ci and the ACi distance:
,P&other quantities have been absorbed into K, and j; is the unit vector in the x-direction. It is not difficult to see that this theoretically derived expression leads to the same behaviour of the CD as the more complicated empirical sector rule. The effects of the substituents is simply to change the static field at the carbony1 group, and the sign of the AICD is determined simply by the sign of the x-component. The above application is qualitative, and the agreement is by no means meant to imply that other models
circuiar dichroism
(AICD)
are incapable of leading to a similar qualitative result. The importance of the above mechanism will become clearer when exhaustive quantitative tests are completed.
AcknowIedgement The author wishes to thank Professor N.S. Hush and Dr. C.J. Hawkins for discussions on the manuscript.
References [lJ W. Moffltt,Chem. Phys. 25 (1956) 1189. 121 G. Hil~es and FS. Richardson, Inorg. Chem. 15 (1976) 2582. [3] R.W. Strickland and F.S. Richardson, 3. Phys. Chem. SO (1976) 164. [4J S.F. Mason and R.H. Seal, Mol. phys. 31 (1976) 755. [SJ E.G. Hohn and O.E. Weigsng Jr., J. Chem. Phys. 48 (1968) 1127. [6J J.A. ScheUman, AccountsChem. Res. l(1968) 144. [7J A.D. Buckingham and P.J. Stiles, Accounts Chem. Res. ? (1974) 258. [SJ C.J. Hawkins. Absolute configuration of metal complexes (Wiley- Interscience, New York, 1971). [P] P. Crabbe. An introduction to chiropticalmethods in chemistry, Monograph (Syntex LA., 1971). J. Chem. Phys. [ IOJ EA. Power and T. Thirunamachandran, 60 (1974) 3695. [ll) P.E. Schipper,Mol.Phys. 29 (1975) 1705. (t2J P.E. Schipper, Inorg.Chim. Acta 14 (1975) 161. (131 P.E.Schipper.J. Am.Chem.Soc.98 (197617938. [14J R.R. Birs~;Symmetry and magnetism (North-Holland, Amsterdam, 1964). [UJ DJ. Caldwell and H. Eyrfng, The theory of optical activity (Wiley - Interscience, New York, 1971).