J. Phys. Chem. 1988, 92, 5891-5895 the carbonyl group. Quantum effects, such as orbital interactions which are strong in propanal, seem to be relatively insignificant in IBA, since the classical contributions, such as van der Waals interactions and angle bending deformations, dominate the torsional barrier. Finally, we estimate the barrier height around the C-C bond of acetamide with FFA. The calculated value, 0.5 kcal mol-',47 is not inconsistent with the experimental value4 and reliable theoretical data.I4
Conclusion Scott and Scheraga6 reasoned that the energy dependence on internal rotation arises from two effects: exchange interactions of electrons in bonds adjacent to the bond about which internal rotation occurs and nonbonded or van der Waals interactions. One would therefore expect that the energy difference between the anti and syn conformers of IBA might be small since the C=O and C-N bonds are very similar and 0 and N atoms are also comparable. The available experimental and computational evidence discussed above clearly supports this view. IBA exhibits the following features: (i) The syn minimum is slightly higher in energy than the anti minimum, probably by less than 1 kcal mol-'; (ii) the potential energy curve has 2-fold (48) Hirota, E.; Sugisaki, R.; Nielsen, C. J.; S~rensen,G.0. J. Mol. Spectrosc. 1974, 49, 251.
5891
symmetry; and (iii) the height of the barrier separating the two stable conformations (anti and syn) is about 3 kcal mol-'. We have shown, in addition, (i) that the intrinsic torsional potential (in force field treatments) around the C(sp3)-C(sp2) bond adjacent to amide groups is negligible, (ii) that it is necessary to perform geometry optimization to obtain a reliable rotational potential energy curve, (iii) that the modified Urey-Bradely force field is effective for geometry optimization, and, finally, (iv) that a combination of ECEPP with the modified Urey-Bradley force field yields an accurate and convenient force field for amides. Even though results are presented only for IBA in this paper, extension to any other amide (or peptide) is straightforward, as long as the required partial charges and force constants are available for the desired compound. In addition, one must note that if only the domains in the neighborhood of energy minima are to be considered as is often the case in conformational analysis, inclusion of the modified Urey-Bradley force field (Le., of flexible bond length and bond angles) might not be necessary.
Acknowledgment. We gratefully acknowledge financial support from the American Chemical Society, the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the Bayer Professorship at the Department of Chemical Engineering at MIT. Registry No. IBA, 563-83-7.
Approximations for the Electric Dipole Circular Dichroism of Long Helical Polymers David A. Rabenold Department of Biochemistry and Biophysics, Iowa State University, Ames, Iowa 5001 1 (Received: October 26, 1987; In Final Form: March 14, 1988)
Approximate expressions for the electric dipole circular dichroism and absorption spectra of long helical polymers are presented. They are obtained by expanding components of the polarizability and rotary parameter in terms of certain quantities that cause the leading expansion terms to be dominant, and are discussed in the text. In the limit as the number of repeating units goes to infinity the leading terms are the results obtained from the use of periodic boundary conditions. Calculated results for the circular dichroism of an a-helix appear to be good approximations for 60 or more repeating units. We begin with expressions for the polymer polarizability and rotary parameter that are traditionally considered to apply to polymers of lengths much smaller than the wavelength of the light. But, the method is valid because the effective range of interaction present in the system is much less than the wavelength of the light.
I. Introduction Calculations of the circular dichroism (CD) and absorption spectra of helical polymers usually require eigenvectors and eigenvalues of energy matrices or inverses for each spectral frequency of matrices whose elements are complex polarizabilities and/or interaction terms.' For both cases large matrices require time-consuming computations. Some methods have been developed that yield more computationally tractable equations. One method is a partitioning scheme2-" which approximately accounts for the effect on the low-energy region of the spectrum of the (1) Tinoco, I., Jr.; Bustamante, C.; Maestre, M. F. Annu. Reu. Biophys. Bioeng. 1980, 9, 107. (2) Applequist, J.; Sundberg, K. R.; Olson, M. L.; Weiss, L. C. J . Chem. Phys. 1979, 70, 1240; 1979, 71, 2330. (3) Applequist, J. J. Chem. Phys. 1979, 71, 1983. (4) Applequist, J . J. Chem. Phys. 1979, 71, 4324; 1979, 71, 4332; 1980, 73, 3521. (5) Applequist, J. Biopolymers 1981, 20, 387. (6) Applequist, J. Biopolymers 1981, 20, 2311. (7) Applequist, J. Biopolymers 1982, 21, 779. (8) Caldwell, J. W.; Applequist, J. Biopolymers 1984, 23, 1891. (9) Rabenold, D. A . J. Chem. Phys. 1982, 77, 4265. (10) Rabenold, D. A.; Rhodes, W. J. Phys. Chem. 1986, 90, 2561. (11) Ito, H.; I'Haya, Y. J. J . Chem. Phys. 1982, 77, 6270.
interaction between transition charge densities for high and low energy transitions. This approach2-10yields, e.g., an eigenvalue problem of order equal to the number of low-energy transitions. Another method deals with very long polymers for which periodic boundary conditions can be employed.12-28 In this case, e.g., (12) Moffitt, W. J . Chem. Phys. 1956, 25, 467. (13) Tinoco, I. J. Am. Chem. SOC.1964, 86, 297. (14) Ando, T. Prog. Theor. Phys. 1968, 40, 471. (1 5) Loxsom, F. M. J . Chem. Phys. 1969,51,4899; Phys. Rev. B Solid State 1970, E l , 858; Int. J . Quantum Chem., Quantum Symp. 1969,3, 147. (16) Loxsom, F. M.; Tterlikkis, L.; Rhodes, W. Biopolymers 1971, 10, 2405. (17) Tterlikkis, L.; Loxsom, F. M.; Rhodes, W. Biopolymers 1973, 12,675. (18) Deutsche, C. W. J . Chem. Phys. 1970, 52, 3703. (19) Rhodes, W. J. Chem. Phys. 1970, 53, 3650. (20) Philpott, M. R. J. Chem. Phys. 1972, 56, 683. (21) Levin, A . I.; Tinoco, I., Jr. J. Chem. Phys. 1977, 66, 349. (22) Rhodes, W.; Redmann, S . M., Jr. Chem. Phys. 1977, 22, 215. (23) Redmann, S. M., Jr.; Rhodes, W. Biopolymers 1979, 18, 393. (24) Rabenold, D. A. J . Chem. Phys. 1981, 74, 941; 1981, 74, 5988. (25) Rabenold, D. A . J . Chem. Phys. 1984,80, 1326. (26) Rabenold, D. A,; Rhodes, W. J. Chem. Phys. 1984,80, 3866; 1984, 80, 3873. (27) Pamuk, H. 0.;Dougherty, A. M.; Johnson, W. C., Jr. Biopolymers 1985, 24, 1337. (28) Rabenold, D. A.; Rhodes, W. Biopolymers 1987, 26, 109.
0022-3654/88/2092-5891$01.50/00 1988 American Chemical Society
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The Journal of Physical Chemistry, Vol. 92, No. 21, 1988
eigenvalue problems of order equal to the number of transitions per repeating unit are obtained. However, this approach yields idealized spectra with large splittings and intensities. The periodic boundary conditions approach can be approximately applied to short helices yielding end effect corrections and reasonable splittings and intensities, but the helices must be reentrant such as perfect a-helices with multiples of 18 repeating units.29 In this paper we present another approximation method that yields computationally tractable expressions for the CD and absorption spectra of long helical polymers that have identical repeating units. For example, eigenvalue problems of order equal to the number of transitions per repeating unit are obtained; end effect corrections are included; the helices need not be reentrant; and the many transitions per repeating unit need not be located at the same center. Without using periodic boundary conditions we obtain approximate expressions for the complex polarizability and rotatory parameter for a long helical polymer with N identical m are the results obtained from repeating units, which as N the use of periodic boundary conditions. This is achieved by (1) decomposing the dipole and rotational strengths in terms of perpendicular and parallel polarized transition moments and helix structure parameters10 and ( 2 ) expanding the inverses of the above-mentioned matrices in terms of appropriate arrays that cause the leading expansion terms for the components of the polarizability and rotatory parameter to be dominant. The results are an extension of those obtained by Moffitt, Fitts, and Kirkwood30 in that, e.g., appropriate spectral shifts and splittings are obtained. The major point of this paper is the expansion technique. Its use rests on the fact that the xx* CD spectra for perfect a-helices for N m and N = 18 repeating units are qualitatively the same.29 For the infinitely long a-helix the exciton splitting of the CD radial bands and the intensity of the CD helix band are larger than that for the a-helix with N = 18. For example, the intensity of the CD helix band for N = 18 is about 40% less than that for N m. The formulation used for N is equivalent to that derived earlier26and for N = 18 the treatment is what we use here to obtain what we call exact CD spectra. By approximately employing periodic boundary conditions for N = 18 and including end effect corrections this discrepancy is reduced to about a 20% difference in intensity.29 That is, the chain length dependence of the component CD bands is not strong if end effect corrections are included. Also, from formulations that employ periodic boundary conditions it is well-known that spectral positions of perpendicular and parallel polarized polymer transitions are different because they stem from different matrices which we call inverses of normal-mode problems and which are different because of the different selection rules for the different types of transitions.12-28Our expansion technique here, which does not employ periodic boundary conditions, is essentially an expansion of the inverse of a normal-mode problem for each of the cases of perpendicular and parallel polarized transitions in terms of a norm and which mal-mode problem which is appropriate for N for finite N contains end effect corrections. This causes rapid convergence. All intrarepeating unit couplings of transition charge densities are included. The interrepeating unit couplings are handled approximately. In deriving the theory in section I1 we begin with the classical, also called the time-dependent Hartree, expressions for the electric dipole polarizability and rotatory parameter for an isotropic system " - ~expressions ~ of helical polymers with N repeating ~ n i t s . * ~ ~ ,The for the polymer polarizability and rotatory parameter that we begin with are traditionally considered to apply to polymers of length much smaller than the wavelength of the light. In section 111 we show that the method is valid because the effective range of interaction present in the system is much less than the wavelength
-
-+
-
-
-
Ra benold of the light. In section IV we present results of some model calculations for an a-helix. 11. Theory
The polymer polarizability a($ and the rotatory parameter P(D) are given by2,31-35 a(v) = (2/3hcN)C(a,a,)'/*Do,.DBo(M(~)-1),s
(la)
e ,
P(D) = (1 / 3 h c N ) C ( ~ ~ , a j S ) ' / ~ DXo Dgo*R,g(M(D)-'],p ,
(1 b)
a,
where
M(D) = B(D) + ( 2 / h ~ ) a ' / ~ [ W + 'Q ] ~ j ' +l ~( 2 / h ~ ) d / ~ -[ W Q]d/2 (IC) in which R,, = R, - R, is the vector distance between the location centers for transitions a and P and in which Do,.D, and Doa X Dm respectively are the dot and cross product of unperturbed electric dipole transition moments for transitions a and (? with Wa;and WaBrespectively are intra- and frequencies a, and interrepeating unit couplings of transition charge densities for a, B(D),and Q are block transitions a and 6. The arrays Wo, diagonal with blocks Wo, V, B(B), and Q, respectively. Each repeating unit has the same arrays Wo,8, B(u), and Q. For Lorentzian dispersions B,,(D) = vO2- i2- iDra. For a general band shape, B,,(D) is related to the absorption band shape for isolated transition a through its polarizability CY,;(V)
+
= ( 2 / h c ) ~ , I D ~ , l ~ [ g ' , ( i~g)" , ( ~ ) ]=
(2/hc)~,IDo,12~aD1(~)-' (2) where g",(p) is the absorption band shape and g',(ij) is the ' &function line shapes, Kronig-Kramers transform of ~ " J D ) . ~For which are later replaced by Gaussians, we use lim(t-0) a(v + k ) for which Baa(p) = 22 - (D ic)2. In eq 1 we have added the arbitrary matrix Q to Woand subtracted it from W. Our objective is to expand the inverse of the normal-mode problem, M(D)-',in terms of
+
E(D,Q)-I = [B(D) + ( 2 / h c ) ~ i ' / ~ + [ WQ]id/2]-1 ~ (3) which is block diagonal with blocks
+ ( 2 / h ~ ) g ' / ~ [ W+OQ]d/2]-'
E(s,Q)-' = [B(D)
(4)
There is one block for each repeating unit and all the blocks are identical. We choose Q such that the leading terms for C Y ( V ) and P(D) in the expansion are dominant. This can be achieved by first relabeling transitions. We consider indexes a and (3 to respectively refer to transitions u and T in repeating units j and k . The inverse of the normal-mode problem, {M(2)-1]oB, may then be expanded as follows:
(29) Rabenold, D. A.; Rhodes, W., to be submitted for publication. (30) Moffitt, W.; Fitts, D.; Kirkwood, J. G . Proc. Nutl. Acud. Sei. U.S.A. 1957, 43, 123. (31) DeVoe, H. J . Chem. Phys. 1964, 41, 393; 1965, 43, 3199. (32) McLachlan, A . D.; Ball, M. A . Mol. Phys. 1964, 8, 581. (33) Harris, R. A . J . Chem. Phys. 1965, 43, 959. (34) Rhodes, W.; Chase, M. Reu. Mod. Phys. 1967, 39, 348.
(35) B ( u ) in ref 2-4 is defined to be half of @ ( u ) in eq lb. But, the expression for the CD in ref 2-4 is equivalent to that used here. References 2-4 are based on the work in ref 35a while the @(r) that we employ here is derived in ref 35b via a multipole expansion of the transverse current linearly induced in the system by the light. (a) Applequist, J. J . Chem. Phys. 1973, 58, 4251. (b) Rabenold, D. A. J . Chem. Phys. 1975, 62, 376.
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 5893
Electric Dipole CD of Long Helical Polymers
w(zjk).Also W{[(Zjk)is an element of W,. Second, we express Do,.Deo and Do, X Deo.R,, in terms of perpendicular and parallel polarized components. This is done by using the ju, kr labeling scheme and helix structure parameters. From previous work we havelo
of
D0,.D,, = D&.D$) = p&'pdcos (SOZjk) + (PO, where Po,=
[
cos 8, z n 8,
P d z sin (SOZjk) + -sin 8,
0""
8,
81
%EO (7)
contains
&(~)(l)
-Zjk[P~,'Pd sin (SOZjk) - (PO, x Pd)' cos (sozjk)] For &(P)(')sUr and f ( z j k ) are either P&.Pd and - ' / z z j k sin (SOZjk) or (Po, x P,,JZ and + ' / z z j k cos (SOZjk). Similarly for flb(ij)(o)but with Z,, replacing z j k , The factor of arises because of the factor of 2 in eq 16. The second expansion term in eq 16 for all components of a@) and P(e) except &(e)(') in eq 17 is zero if
(18)
jk
(8)
DpyR,, = D@ X D$)-(Rj, - Rk,) = Hhkr
(Sozjk)]
N-'u(Zjk)J{$ = 0
and where so = OO/din which Bo is the angle of rotation about the helix axis between sequentially adjacent repeating units. For the triple product in eq l b we havelo X
and where
Do,
= R(8,)Dou
Do,
-z,~[p&'Pd sin (SoZjk) - (PO, x Pd)' cos
+ R h k i + Hjukr (9)
which gives Qft =
(19)
N-' n ( Z j k )Wrt(Zjk) jk
We use this definition of Q in E(e,Q) f o r all components of a(ij) and P(e). The first nonzero term for Ph(ij)(')derives from the second expansion term in eq 16. For aIl(ij)we obtain the first expansion term approximation
+
+
a"(e)= ( 2 / 3 h c ) C ( 8 , ~ , ) ~ / * ~ ~ ~ ( [ B(2/hc)~'/~[WO (e)
where
UT
= -(Zjk
vyo)]ijI~2]-'],, (20)
H$kr
+ ZUT)[pd-u'pd sin (SOZjk) -
pdo)zcos ( S O Z j k ) ]
(10)
where, from eq 19
Rhkr = [(bu P0u)"pd + Pdb'(b~ pd)l cos (SOZjk) + ([(bu PO.) pdIz + [PO, (b, pdO)lz) sin ( S O Z j k ) =
$UT
cos (SOZjk) + $UT sin (SOZjk)
RYukr = (b,
Po,)zP%
X
where we have used Rj, = 2,b,0 cos (soZj+ 8,)
+ Z,b,O
+ P&(b, X Pd)'
sin (soZj + 8,)
(12)
+ Zz(Zj + 2,)
@(e) = P
m + PRl(f9 + P b W
in which Wfe(md) and Wft(-md) are the first column and row eq 20 is the result obtained by of the array W,. For N employing periodic boundary conditions. The order of the normal-mode problem in eq 20 is equal to the number of transitions per repeating unit and the (1 - m / N ) factor and the finite summation in eq 21 makes a"(ij)polymer length dependent. For &(v) we obtain the first expansion term approximation
(E(ij,Q+)-'I,, + [P&+'d
jk
E(ij,Q*) = B(v)
+ (2/hc)~'/~[W + ~Vc(so)f iVs(so)]a'/2 (26)
[1E(O,Q)-IlAj, - ~ ~ E ( P , Q ) - ' ] u ~ ~ ~ 1 E ( e+ , Q...I) - '(1 } ~6)~ f€
in which the nature of the factors Surandflzjk) depends on which term Ti(p) is considered. For example, &(e) contains P,&*Pd cos (SOZjk) + (Po, x Pd)z sin (SOZjk) for which S,, and f(Zjk) are either P&-Pd and cos ( S g j k ) or (Po, x Pd)z and sin ( S G j k ) . At this point we write P i ( $ ) as where Ph(e)(O) contains
+ /3B(e)(l)
P,O)~I~E(P,Q-)-'J,,,) (22)
and where
UT
= /3~(e)(O'
X
N- 1
UT
( P U P T ) '/2s,f(zjk) x
PB(0)
+ i(P0,
(15)
Ti(e) = (2 / 3hcN)C C (Dun,) '/2suf(zjk)(M(e)-'lj,kr = (2 /3 hCN)
Q)
(14)
where the symbols H and R here indicate CD helix and radial bands. The perpendicular and parallel notation is used because the bands tend to be centered at the transition frequencies of the perpendicular and parallel polarized absorption bands. Our objective is to substitute eq 14 and 15, respectively, into eq l a and lb, employ the expansion in eq 5, and then sum over j and k . The components of a@)and @(e)contain several terms Ti(ij)because D&'i.D$j)and D&'jX D$$(Rj, - Rkr) do. Instead of dealing with each term separately we consider the general form and expansion as follows jk
(21)
UT
in eq 7 and 9 allow
+ a"(e)
-
+ W&d)l
aL(ij)= (I/3h~)C(ij,e,)'/~([P,ldP,- i(Pou X P,,JZ] X
b, = R(O,)Z,b,O, Z,, 2, - Z ,
a(e) = a l ( P )
c (1 - m/N)Wt&-md)
m>O
(11)
(13) The forms of DO,.DBoand Do, X D,.R,, us to write
N- I
(17)
in which the arrays P'/~[W + Vc(so)f N s ( s 0 ) ] ~ 1 /are 2 Hermitian. For P ~ ( V ) ( ~@;(e), ) , and Pk(e) we obtain the first expansion term approximations
PB(P)'O' = (1/6h~)~(e,~,)'/~Z,,[(Po, X Pd)Z(E(~,Qt)-'+ E(ij,Q-)-l}ur+ iP#,*P~(E(e,Q')-' - E(ij,Q-)-'],,] (27) 07
P,l(ij)
= (1 /6h~)C(e,8,)'/~[~,,(E(s,Q+)-~ + E(F,Q-)-'}urU7
~+,MP,Q+)-'- E(~7Q-)-'IuTl (28)
5894
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988
P ~ ( P =) (1 / 3 h c ) ~ ( s , , ~ , ) ~ / ~ [ (Xb ,Po,)zP%
+
bT
pfU(b7
P~o)ZI(E[a,vc(o)l-l)UT (29)
where &, and +u7 in eq 28 are defined in eq 11 and where the inverse in eq 29 is that in eq 20. The first nonzero expansion term for PA(s)(’) is the second one, which gives the approximation
PA (3)“) = -( 1 / 6hc) C C (sup,) 1’2[iP~u.P,o((+lurAf~(+),, UT
f€
Hu&-Ht7) + (Pou x where
(*Iu7
p,o)z((+lurAf€+~+l*,+ H u f A f < H € J 1 (30)
= (E(n,Q*)-’l,, and
Af€* = N-’( 2/hC)(Pfij~)’~’CZ,keXp( fisozjk) Wf&Zjk) Jk
= ( 2 / h c ) ( ~ f i j ~ ) ” ~ [ u Kf ( s iuk(so)] ~)
= AfF f iAit
(31)
where N- 1
U?&SO)
= d C ( m - m 2 / N cos (soms)[Wf{(md)- W(&-md)l m>O
(32) and N- 1
uA(so) = d
C ( m - m 2 / N sin (somd)[Wf&md)+ Wr&-md)l
m>O
(33) For calculation purposes we show below how to handle the inverses that appear in the approximate expressions for the com) b ( ~ ) For . example, if the elements of B(P) ponents of a ( ~and in E[v,VC(0)]-’ are Lorentzian dispersions all with the same half-peak width r {E[P,V~(O)]-~),~ = CXu,,A’Tn/(~n(0)2 - n2 - ivr) n
(34)
where X and a(0)2respectively are the eigenvector and eigenvalue matrices for a* (2/hc)~’/~[WO Vc(0)]~’/2.For general band shapes the inverse in eq 34 can be computed for each spectral frequency. For Lorentzian dispersions all with the same half-peak width ( [ E ( P , Q - ] - ~may ) ~ ~ be resolved as follows
+
+
(E(P,Q-]-~),, = C Y u n P , n / ( ~ n ( -s os3) 2 ior) n
n
A ~ ( P )= (32r3NA/23O3)ij2Im [ b & ( ~ ) ( ~P)h ( ~ ) ( l )
+
+ P ~ ( P +) P ~ ( P ) ]
-
(41)
where N A is Avogadro’s number. For the case of N m eq 29 for P ~ ( D )(28) , for P;(P), (30) for P ~ ( P ) ( * (27) ) , for PA(P)(O), and (22) for (u*(P) along with (20) for d(ij)are the results obtained m our by employing periodic boundary conditions. For N P ~ ( P ) ( ’ )@, ~ ( P ) ( O ) , and P i ( ? ) + P ~ ( P respectively ) correspond to Levin and Tinoco’s all, b,,,and b,.21 The results obtained here, like others for helical polymers, are applicable to wavelength regions that are very long compared to the effective range of interaction between transition charge densities. However, our results are only approximate for finite N. But, as N gets large, contributions to (Y(P) and P(P) that are of higher order than those discussed here must go to zero. If should be pointed out that some previous works by the author on infinitely long systems have neglected the arrays Vs(so)and uc(so)due to incorrectly assuming that the Coulombic coupling array w(zJk) is an even function of ~ ~ ~ , 2 4 , 2 s , 2 a Im &(VI characterizes the CD radial bands which are located at the frequencies of the parallel polarized absorption bands. Im @;(P) characterizes the CD radial bands which are located at the frequencies of the perpendicular polarized absorption bands. Im P;(P)(O) characterizes CD bands that are located at the frequencies of the perpendicular polarized absorption bands and are due to intrarepeating unit transitions having different helix axis com) the C D helix bands ponent locations. Im b ; ( ~ ) ( ’characterizes which are located at the frequencies of the perpendicular polarized absorption bands. The radial bands add together to yield a conservative CD contribution while both Im PA(P)(O) and Im (P)(]) individually yield conservative C D contributions. For helical polypeptides that have one low-energy rr* transition per repeating unit the C D is composed of two radial bands and ane helix band which has a first-derivative-like band shape. There are eight possible kinds of rr* CD spectra for helical polypeptides, although most are probably sterically forbidden. High-energy transitions, due to coupling of the low- and high-energy-transition systems, can alter the intensity of all the low-energy bands and the shape of the helix band.10,28
-
111. Validity of the Method
(35a)
We began with equations for the polarizability, (U(P), and the rotatory parameter, P(v), that are traditionally considered to apply to small molecules, yet results for infinitely long polymers have been obtained. These equations rest upon an approximation for averaging over orientations, namely
(35b)
(Do,D8o.A(S,~)lM(s)-ll,Bexp(-iq%B)) = [(I /3)Dou.D, X A(qP) + (1/6)Dom X Dpo%& X A(q,~)l{M(~)-’la,9
and similarly {E(V,Q+]-~},,= CY*,,Y,,/(P,(S~)~ - s2- i s r )
Rabenold
where, e.g., Y and P(s,)~respectively are the eigenvector and eigenvalue matrices for E(P,Q-). For general band shapes
where M(3) is the normal-mode problem and where A(q,s) and q respectively are the vector potential and wave vector of the light. The expansion
E(s,Q*)-’ = [f(s)-l A ig1-l
exp(-iqRa8) = 1 - iq-R,,
= [I f if(s)g]-’f(~)
(36)
g = (2/hC)P’/2Vs(so)3~~2
(37)
f(P)-’ = B(P) + ( 2 / h c ) ~ l / ’ [ W + ~ Vc(~o)]Pl/2
(38)
where
Expanding the inverse in eq 36 and collecting the two kinds of terms gives
+
E(v,Q*)-’ = [I F if(~)g][f(n)-’ gf(~)g]-I = X’(P)
=F
(39)
iX”(8)
The absorption and CD spectra are respectively given by2335
+
C(D) = (8r2NA/23O3)v Im [ a l ( ~ ) all(s)]
and
(40)
by itself is valid if qRa8
