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Interference effects in natural circular dichroism spectra
Volume 57, number4
CHEMICAL PHYSXCSLETPERS
1.5 August 1978
INTERFERENCE EFFECTS IN NATURAL CIRCULAR DICHROISM SPECFRA Aase E. HANSEN Department of Physid
Gremistry, H-C_ &rsred Ihstitu te, Universify of Copenhagen. DK-2100 Copenhagen. Denmark
Received20 March 1978
Etis shown that Fano-type interferencebetween a discrete(e-g_Rydberg) transitionand a continuous(e.g. valenceshell) transitioncan lead to naturalcircukudichroismline shapeswhicS deviatesignificantlyfrom shnpIesuperposition.The resuitsarc ilXustxtedfor a &iml moIecuIeof C, symmetry_
Ccmparisons between computed and measured signs and magnitudes of circular dichroism (CD) spectra are important both in the study of the structure of chiral rnoiecu!es, and in the assignment of spectral transitions. However such comparisons can be hampered by spectral congestion of strongly overlapping excitations_ We shall consider here the case of a discrete excitation (e-g_ of Rydberg type) embedded in an essentially continuous excitation (e-g_ the dense viironic manifold of a valence shell excitation) in a chiral moiecuie, and derive expressions for the resuhing CD line shapes which suggest that neglect of the interference between these excitations can Iead to misinterpretations of the chiroptical properties of the discrete exci’kation For ordinary absorption spectra such interference effects were frrst discussed by Fano [I] for autoionizing atomic transitions, and subsequently by a number of authors for moIecuhu and solid state systems, where the interference is usuahy assumed to be generated by viironic interactions (see e.g. refs. [2-101). For the chiroptical properties of chi.ralmolecules the effect of tibronic coupling between discrete excitations has been studied in detail for many different coupling cases (see ref. [I I] and references therein); in contrast to these approaches the present treatment utilizes specifkaliy that one of the excitations has a very high density of fmai states_ The molecubrr model envisaged here is a nondegencrate ground state IO), and a set of excited states con-
588
sisting of a non-degenerate discrete state is) embedded in a dense manifold of states lk)_ These states are essentially adiabatic molecular states, however the exact details of the choice of zeroth order states are not required in the present formal development (see e.g. refs. [2,3,8] for discussions of various choices of basis states). The molecular hamiltonian can be separated into the following form HI
=iYo +ks,
(1)
where I?’ represents the vibronic coupling’. We shah assume that IO>is an eigenfunction of HM to a suffrciently good approximation, i_e_ ~oIH~Io~=~ow~Is~=
(2)
which also serves to define the zero point of energy, and that 6s)and lk>are chosen such that “llT,~)=MH$s>=E,, Wf,lk’~
= WHolk’)
cslH3,ifk) = @pI’lk>
(34 = E&n, = v,
(=I
(34
* We shall neglectspin-orbit coupling,aad assumethat there are no staticexternalmagaetic&kls present_He aad HM are thenpurely zeal,and the couplingconstantY of eq. (3c) canbe takenas real with impunity.
CHEMICAL PHYSICS LETTERS
Volume 57. number 4
15 August 1978
where we have assumed the standard approximation [3], that the coupling WZ’lk) is so weak a function of the index k that it can be treated as a (real) constant in tie energy range of interest (namely the vicinity of E,). For the moment the spectrum of IZc)is treated as disc:ete for notational corkvenience. The perturbation treatment of the interaction between the discrete and continuous excitations can be facilitated by introducing the molecular resolvent G(p) = (E - HM + it))-l [2,4,12], where it is implied that aIf reMions involving G(p) are to be evaluated in the limit q+ O+. In addition to the assumption of constant V, eq_ (3c), we shall assume that the manifold lZc1has a constant density of states p(Ek) in the vicinity of E,. This implies that we can use the identification [ 121
all the excited molecular states, and c+, is the angular Bohr frequency for the excitation 0 + ZI.The index i runs over all the electrons in the molecule; the neglect of nuclear contriiutions in the transition moment operators is consistent with the neglect of vibronic coupling between the ground state and the excited states [ 141. Notice that IO)and lb) are the total vibronic states of the molecule, so that the intensity distribution is governed by the energy conservation expressed in the delta function [12] _ Using the relation [12]
I&
ee<=D=,---1 Im{(OI [Z ‘i X *J
5
(E - Ek + +)-I
= -inp(Ek),
(4)
and the exact relation G = 8 + 8 H’G 1123, where G’(I?j is the resolvent corresponding to Ho, then leads to the following matrix ekments
Jin G(E+) = -ilrc
Ga=Gk,
G,.=
+ r/2)-‘,
%k*
- [G(E+) Z+J IO,), (9)
which can be expanded in the truncated set IO), Is>and Ik>to yield G 1 C ss
(W
= G, V(E+ -E&-l,
@I
which holds in the limit q + Oi-, and where lb>are the eigenstates of HM , we observe that the CD line shape is governed by the function
~?~=n--~lm Gsp = GIG(E+)W = (E -Es
lb) 6(E - Eb) 01,
(=)
GssV
(E’ - Ek) + (E+ - Ek) (p - EkeI
(W
where the optical matrix elements are defmed as follows 0lE
VJO) = VbO’
COlzr~x Vi@’ = ZOb.
(11)
where the line width is given by r = r(E)
= 2&@j,
(6)
and where we have used that I’ is non-vanishing so that E* can be identified with E in eq. (Sa). As the measure of CL\intensity we shall take the ellipticity per unit length e(w) [13], which for natural (i.e. not magnetically induced) CD can be written in the following form (in cgs units) 8(o)
y = y@‘j = r/V=
2sVp(E),
(12a) (12bj
e=&Z)=(E-E,)IV
and using the explicit relations in eqs. (5), the CD line shape function then becomes
= -(2n2e2ik2/3mc2jN
X T
The matrix elements Vko and ZOkfor the excitations into the dense manifold are weak functions of k and will in fact be treated as constants (i.e. independent of k), in analogy with the approximation in eq. (3~). Introducing the scaled (dimensionless) quantities
6(Zz:ho- hc@
(7) JzW
for randomly oriented moiecules. Here IV is the number of molecules per unit volume, the index t, runs over
= -PQ
(E2 f r2/4j-l
C[Z&. V&l
f E[‘ok - V&3 f e2[$-Jk - VJ-J)] E >
(13) 569
Vohune 57, nuder
where we have used that VMt and ZOkare considered constant. Eq. (13) shows that the resulting CD line shape function contains three distinct contributions, nameiy a Iorentzian governed by the rotatory strength* of the discrete excitation, a dispersion type contriiution governed by cross-terms between the transition moments of the two excitations, and an anti-resonance governed by the rotatory strength of the continuous state. Notice that in the limit of vanishing coupling, i.e. Vgoing to zero, the lorentzian goes Into a delta function_ This is an artifact arisiig from the negIect of the coupling to the radiative continuum buIIt upon the molectdar ground state [O>.Roper incIusion of this radiative coupling wotdd restore the natural radiative Line width for the discrete excitation in this Iimit ]9] _ In the case of a general chiraI moIecuIe the syrnmetry is so low that none of the involved electric and magnetic transition moment vectors are strictly parallel or perpendicular, and in adarGon the sundry products of the transition moments can be both positive and negative. The CD Iine shapes are therefore somewhat more co_mpIicated than the interference line shapes displayed by Fano [I] for ordinary absorption (see also ref. ]6]). We shah use here the speciaf case of a chiraI moIecuIe belongIng to the C2 point group to iIlustrate some of the important features of the resulting Iine shapes. For a C2 mofectde *he transition moments are &beer pohu-ized along the C2 axis or perpendicular to this axis_ Consider fust the case where 0 + s and 0 + k are both polarized along *he C2 axis. Using the notation f =
(u -Z,,)l(u -l*&
v = (u
l
V,)l(~
l
Vm),
04)
where rf is a unit vector afong the C axis, eq_ (13) takes th& form
05) * The rotatory streqgrh of an ekixronic
transitionisgivenas
f131 (incgsurlits)
wfiere web is the resonancefrequencyfor the ekctroaic WtiOn.
590
15 August 1978
CHEMICAL PHYSICS LETTERS
4
In the case where the discrete excitation 0 * s is poIarized along u, and the continuous excitation is polarized perpendicular to u, the factor multiplying the dispersion-type contribution vanishes, and eq. (13) becomes
fP
= -(ZOk
- V&
P(E) (e2 + R)@
+ r2/4).
(16)
R being the ratio of the two rotatory strengths
(17) Eqs. (15) and (17) both take the form of the CD line shape for the continuous excitation modulated by appropriate proftie functions. These profde functions are shown in tig_ 1 for a number of values of the parameters_ For vani&ing rotatory strengths of the discrete excitation both types of polarizations show pure antiresonance, whereas the line profties approach shnple superposition when the rotatory strengths of the discrete excitation is significantIy larger (numerically) than r2/4 times the rotatory strength of the continuous excitation_ When the numerical ratios of the two rotatory strengths are cIose to jLf4, fig. I shows that the profile functions for these situations depend cruciaIl upon the relative directions of rhe transition moments. The latter observation parallels the results of Sharf [6] who finds that the relative directions of the electris dipok transition lmoments are crucial for the ordinary absorption profde, when the ratio of the oscillator strengths for the discrete and the continuous excitations is of the order of y2/4 ( in our notation). In addition to these exampiles we wish to caII attention to the particular situation, which arises when the electric transition moment of one excitation and the magnetic transition moment of the other are both almost vanishing (e-g_ Z, c=0 and Pm z 0) In that ease only one cross-term survives in eq. (14), and the CD line shape becomes a pure dispersion-type curve. Experimental techniques in CD measurements are ROWso refined (see e.g. ref. 115J), that CD data on Rydberg-type transitions are becoming increasingly avaiIabIe. The present results suggest that a corm& correlation between experimental and caIcuIated chiroptiCal properties of these excitations in many cases may require a detaiIed analysis of vibronicaI.Iy induced Literference effects.
CHEMICAL
Volume 57, number 4
PHYSICS
A 4 3 2 1
8
I ,‘.. _----\ _________ -z-r;-~___ \ -----____ \;-a
c I-8
:’
--,(l,l)
--__
-4
V’
4 3 2
-_
,,,,___;;_~._---_:-~-I
/-
6
1
c
_
-8
15 August 1978
s
(1,4)
i ,
LETTERS
8
0 -1 -2 -3 -4
Fig- 1. Shows the profde functions of eqs. (15) and (17) for 7 = 2. The curves in A and B are labelled hy (I, V) and R, respectively. __
The main part of this work was carried out during a visit at the Department of Chemistry, University of Tel-Aviv, Israel, with support from the Programme of Cultural, Educational and Scientific Cooperation between Denmark and Israel. The author is grateful to Professor J. Jortner for numerous valuable discussions, and for the hospitality of the Department.
] l] U. Fano, Phys. Rev. 124 (1961) 1866. [2] R.A. Harris, J. Chem. Phys 39 (1963) 978. [3] M. Bixon and J. Jortner, J. Chem. Phys. 48 (1968) 715. [4] A. Shiiatani and Y. Toyozawa, J. Phys. Sot. Japan 25 (1968) 33.5. [S] J. Jortuer and G.C. Morris, J. Chem_ Phys 510969) 3689.
[B] B. Scharf, Chem. Phys. Letters 5 C1970) 456. [7] M.C. Sturge, H.J. Guggenheim and M.M.L. Ryce, Phys Rev. B2 (1970) 2459. PI D. Florida, R Scheps and S.k Rice, Chem. Phys. Letters 15 (1972) 490. 191 A. Nitzan, Mol. Phys. 28 (1974) 65. I101 J_ Jortner and S. Mukamel, in: Proceedings of the First international Congress on Quantum Chemistry, eds R. Daudel and B. Pulhnan (Reidel, Dordrecht, 1974). D. Caliga and F.S. Richardson, Mol. Phys 28 (1947) !ll] 114.5. [19-l A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1963). 1131 AE. Hansen and J- Avery, Chem. Phys. Letters 13 (1972) 396. [141 CJ. Ballhausen and A.E. Hansen, Ann. Rev. Phys. Chem. 23 (1972) 15. A. Gedanken and 0. Schnepp, Chem. Phys. 12 (1976) WI 341.
CHEMICAL PHYSXCSLETPERS
1.5 August 1978
INTERFERENCE EFFECTS IN NATURAL CIRCULAR DICHROISM SPECFRA Aase E. HANSEN Department of Physid
Gremistry, H-C_ &rsred Ihstitu te, Universify of Copenhagen. DK-2100 Copenhagen. Denmark
Received20 March 1978
Etis shown that Fano-type interferencebetween a discrete(e-g_Rydberg) transitionand a continuous(e.g. valenceshell) transitioncan lead to naturalcircukudichroismline shapeswhicS deviatesignificantlyfrom shnpIesuperposition.The resuitsarc ilXustxtedfor a &iml moIecuIeof C, symmetry_
Ccmparisons between computed and measured signs and magnitudes of circular dichroism (CD) spectra are important both in the study of the structure of chiral rnoiecu!es, and in the assignment of spectral transitions. However such comparisons can be hampered by spectral congestion of strongly overlapping excitations_ We shall consider here the case of a discrete excitation (e-g_ of Rydberg type) embedded in an essentially continuous excitation (e-g_ the dense viironic manifold of a valence shell excitation) in a chiral moiecuie, and derive expressions for the resuhing CD line shapes which suggest that neglect of the interference between these excitations can Iead to misinterpretations of the chiroptical properties of the discrete exci’kation For ordinary absorption spectra such interference effects were frrst discussed by Fano [I] for autoionizing atomic transitions, and subsequently by a number of authors for moIecuhu and solid state systems, where the interference is usuahy assumed to be generated by viironic interactions (see e.g. refs. [2-101). For the chiroptical properties of chi.ralmolecules the effect of tibronic coupling between discrete excitations has been studied in detail for many different coupling cases (see ref. [I I] and references therein); in contrast to these approaches the present treatment utilizes specifkaliy that one of the excitations has a very high density of fmai states_ The molecubrr model envisaged here is a nondegencrate ground state IO), and a set of excited states con-
588
sisting of a non-degenerate discrete state is) embedded in a dense manifold of states lk)_ These states are essentially adiabatic molecular states, however the exact details of the choice of zeroth order states are not required in the present formal development (see e.g. refs. [2,3,8] for discussions of various choices of basis states). The molecular hamiltonian can be separated into the following form HI
=iYo +ks,
(1)
where I?’ represents the vibronic coupling’. We shah assume that IO>is an eigenfunction of HM to a suffrciently good approximation, i_e_ ~oIH~Io~=~ow~Is~=
(2)
which also serves to define the zero point of energy, and that 6s)and lk>are chosen such that “llT,~)=MH$s>=E,, Wf,lk’~
= WHolk’)
cslH3,ifk) = @pI’lk>
(34 = E&n, = v,
(=I
(34
* We shall neglectspin-orbit coupling,aad assumethat there are no staticexternalmagaetic&kls present_He aad HM are thenpurely zeal,and the couplingconstantY of eq. (3c) canbe takenas real with impunity.
CHEMICAL PHYSICS LETTERS
Volume 57. number 4
15 August 1978
where we have assumed the standard approximation [3], that the coupling WZ’lk) is so weak a function of the index k that it can be treated as a (real) constant in tie energy range of interest (namely the vicinity of E,). For the moment the spectrum of IZc)is treated as disc:ete for notational corkvenience. The perturbation treatment of the interaction between the discrete and continuous excitations can be facilitated by introducing the molecular resolvent G(p) = (E - HM + it))-l [2,4,12], where it is implied that aIf reMions involving G(p) are to be evaluated in the limit q+ O+. In addition to the assumption of constant V, eq_ (3c), we shall assume that the manifold lZc1has a constant density of states p(Ek) in the vicinity of E,. This implies that we can use the identification [ 121
all the excited molecular states, and c+, is the angular Bohr frequency for the excitation 0 + ZI.The index i runs over all the electrons in the molecule; the neglect of nuclear contriiutions in the transition moment operators is consistent with the neglect of vibronic coupling between the ground state and the excited states [ 141. Notice that IO)and lb) are the total vibronic states of the molecule, so that the intensity distribution is governed by the energy conservation expressed in the delta function [12] _ Using the relation [12]
I&
ee<=D=,---1 Im{(OI [Z ‘i X *J
5
(E - Ek + +)-I
= -inp(Ek),
(4)
and the exact relation G = 8 + 8 H’G 1123, where G’(I?j is the resolvent corresponding to Ho, then leads to the following matrix ekments
Jin G(E+) = -ilrc
Ga=Gk,
G,.=
+ r/2)-‘,
%k*
- [G(E+) Z+J IO,), (9)
which can be expanded in the truncated set IO), Is>and Ik>to yield G 1 C ss
(W
= G, V(E+ -E&-l,
@I
which holds in the limit q + Oi-, and where lb>are the eigenstates of HM , we observe that the CD line shape is governed by the function
~?~=n--~lm Gsp = GIG(E+)W = (E -Es
lb) 6(E - Eb) 01,
(=)
GssV
(E’ - Ek) + (E+ - Ek) (p - EkeI
(W
where the optical matrix elements are defmed as follows 0lE
VJO) = VbO’
COlzr~x Vi@’ = ZOb.
(11)
where the line width is given by r = r(E)
= 2&@j,
(6)
and where we have used that I’ is non-vanishing so that E* can be identified with E in eq. (Sa). As the measure of CL\intensity we shall take the ellipticity per unit length e(w) [13], which for natural (i.e. not magnetically induced) CD can be written in the following form (in cgs units) 8(o)
y = y@‘j = r/V=
2sVp(E),
(12a) (12bj
e=&Z)=(E-E,)IV
and using the explicit relations in eqs. (5), the CD line shape function then becomes
= -(2n2e2ik2/3mc2jN
X T
The matrix elements Vko and ZOkfor the excitations into the dense manifold are weak functions of k and will in fact be treated as constants (i.e. independent of k), in analogy with the approximation in eq. (3~). Introducing the scaled (dimensionless) quantities
6(Zz:ho- hc@
(7) JzW
for randomly oriented moiecules. Here IV is the number of molecules per unit volume, the index t, runs over
= -PQ
(E2 f r2/4j-l
C[Z&. V&l
f E[‘ok - V&3 f e2[$-Jk - VJ-J)] E >
(13) 569
Vohune 57, nuder
where we have used that VMt and ZOkare considered constant. Eq. (13) shows that the resulting CD line shape function contains three distinct contributions, nameiy a Iorentzian governed by the rotatory strength* of the discrete excitation, a dispersion type contriiution governed by cross-terms between the transition moments of the two excitations, and an anti-resonance governed by the rotatory strength of the continuous state. Notice that in the limit of vanishing coupling, i.e. Vgoing to zero, the lorentzian goes Into a delta function_ This is an artifact arisiig from the negIect of the coupling to the radiative continuum buIIt upon the molectdar ground state [O>.Roper incIusion of this radiative coupling wotdd restore the natural radiative Line width for the discrete excitation in this Iimit ]9] _ In the case of a general chiraI moIecuIe the syrnmetry is so low that none of the involved electric and magnetic transition moment vectors are strictly parallel or perpendicular, and in adarGon the sundry products of the transition moments can be both positive and negative. The CD Iine shapes are therefore somewhat more co_mpIicated than the interference line shapes displayed by Fano [I] for ordinary absorption (see also ref. ]6]). We shah use here the speciaf case of a chiraI moIecuIe belongIng to the C2 point group to iIlustrate some of the important features of the resulting Iine shapes. For a C2 mofectde *he transition moments are &beer pohu-ized along the C2 axis or perpendicular to this axis_ Consider fust the case where 0 + s and 0 + k are both polarized along *he C2 axis. Using the notation f =
(u -Z,,)l(u -l*&
v = (u
l
V,)l(~
l
Vm),
04)
where rf is a unit vector afong the C axis, eq_ (13) takes th& form
05) * The rotatory streqgrh of an ekixronic
transitionisgivenas
f131 (incgsurlits)
wfiere web is the resonancefrequencyfor the ekctroaic WtiOn.
590
15 August 1978
CHEMICAL PHYSICS LETTERS
4
In the case where the discrete excitation 0 * s is poIarized along u, and the continuous excitation is polarized perpendicular to u, the factor multiplying the dispersion-type contribution vanishes, and eq. (13) becomes
fP
= -(ZOk
- V&
P(E) (e2 + R)@
+ r2/4).
(16)
R being the ratio of the two rotatory strengths
(17) Eqs. (15) and (17) both take the form of the CD line shape for the continuous excitation modulated by appropriate proftie functions. These profde functions are shown in tig_ 1 for a number of values of the parameters_ For vani&ing rotatory strengths of the discrete excitation both types of polarizations show pure antiresonance, whereas the line profties approach shnple superposition when the rotatory strengths of the discrete excitation is significantIy larger (numerically) than r2/4 times the rotatory strength of the continuous excitation_ When the numerical ratios of the two rotatory strengths are cIose to jLf4, fig. I shows that the profile functions for these situations depend cruciaIl upon the relative directions of rhe transition moments. The latter observation parallels the results of Sharf [6] who finds that the relative directions of the electris dipok transition lmoments are crucial for the ordinary absorption profde, when the ratio of the oscillator strengths for the discrete and the continuous excitations is of the order of y2/4 ( in our notation). In addition to these exampiles we wish to caII attention to the particular situation, which arises when the electric transition moment of one excitation and the magnetic transition moment of the other are both almost vanishing (e-g_ Z, c=0 and Pm z 0) In that ease only one cross-term survives in eq. (14), and the CD line shape becomes a pure dispersion-type curve. Experimental techniques in CD measurements are ROWso refined (see e.g. ref. 115J), that CD data on Rydberg-type transitions are becoming increasingly avaiIabIe. The present results suggest that a corm& correlation between experimental and caIcuIated chiroptiCal properties of these excitations in many cases may require a detaiIed analysis of vibronicaI.Iy induced Literference effects.
CHEMICAL
Volume 57, number 4
PHYSICS
A 4 3 2 1
8
I ,‘.. _----\ _________ -z-r;-~___ \ -----____ \;-a
c I-8
:’
--,(l,l)
--__
-4
V’
4 3 2
-_
,,,,___;;_~._---_:-~-I
/-
6
1
c
_
-8
15 August 1978
s
(1,4)
i ,
LETTERS
8
0 -1 -2 -3 -4
Fig- 1. Shows the profde functions of eqs. (15) and (17) for 7 = 2. The curves in A and B are labelled hy (I, V) and R, respectively. __
The main part of this work was carried out during a visit at the Department of Chemistry, University of Tel-Aviv, Israel, with support from the Programme of Cultural, Educational and Scientific Cooperation between Denmark and Israel. The author is grateful to Professor J. Jortner for numerous valuable discussions, and for the hospitality of the Department.
] l] U. Fano, Phys. Rev. 124 (1961) 1866. [2] R.A. Harris, J. Chem. Phys 39 (1963) 978. [3] M. Bixon and J. Jortner, J. Chem. Phys. 48 (1968) 715. [4] A. Shiiatani and Y. Toyozawa, J. Phys. Sot. Japan 25 (1968) 33.5. [S] J. Jortuer and G.C. Morris, J. Chem_ Phys 510969) 3689.
[B] B. Scharf, Chem. Phys. Letters 5 C1970) 456. [7] M.C. Sturge, H.J. Guggenheim and M.M.L. Ryce, Phys Rev. B2 (1970) 2459. PI D. Florida, R Scheps and S.k Rice, Chem. Phys. Letters 15 (1972) 490. 191 A. Nitzan, Mol. Phys. 28 (1974) 65. I101 J_ Jortner and S. Mukamel, in: Proceedings of the First international Congress on Quantum Chemistry, eds R. Daudel and B. Pulhnan (Reidel, Dordrecht, 1974). D. Caliga and F.S. Richardson, Mol. Phys 28 (1947) !ll] 114.5. [19-l A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1963). 1131 AE. Hansen and J- Avery, Chem. Phys. Letters 13 (1972) 396. [141 CJ. Ballhausen and A.E. Hansen, Ann. Rev. Phys. Chem. 23 (1972) 15. A. Gedanken and 0. Schnepp, Chem. Phys. 12 (1976) WI 341.