- Email: [email protected]
Comments on vibronic intensity borrowing
Volume 7, number 1
1 October 19’70
CHEMICAL PHYSICS LETTERS
COMMENTS
Deportment
ON
VIBRONIC
INTENSITY
B. SHARI? of Chemistry. Massachusetts Cam&ridge, Massachzisetts
BORROWING
*
Institute of Technology 02139, USA
and Department
cfChemistry,
B. HONIG**
Hwvard
University , Cambddge,
Massachusetts
02138,
USA
Received 3 August 197% The approximation which uses Her&erg-Teller theory with ground state normal coordinates is examined. ft is shown that this procedure is not generalIy vatid and is inconsistent with its underlying assumptions.
The Herzberg-Teller theory [l] of vibronic intensity borrowing in poIyatomic molecules is a well established method of predicting oscillator strengths for symmetry forbidden electronic transitions. In the case where more than one vibration contributes to the coupling mechanism, the theory has been extended to account for the intensity of individual vibronic lines [Z-4], a problem which was not explicitly treated by Herzberg and Teller [I]. It was first observed by Duschinsky [5] that the normal coordinates of excited electronic states till in general be rotated relative to those of the ground state (in normal coordinate space). This effect has been usually neglected in the consideration of the vibronic structure of allowed and forbidden electronic transitions. In this paper, we show that it is theoretically inconsistent to apply Herzberg-Teller theory with ground state normal coordinates and to neglect the Duschinsky effect, since the vibronic coupling mechanism will always induce rotations in normal coordinate space. The vibronic states resulting from the usual perturbation procedure are not diagonal with respect to the molecular hamiltonian. The mixing results in intensity redistribution between the Franck-Condon progressions built on the respective false origins. Of course, if the normal coordinates and thereof the vibrational functions of the excited electronic state were known, the Herzberg-Teller scheme could be extended to account for the intensity of individual vibronic components by expanding excited state normal coordinates and vibrational wave functions in terms of those of the sound state. However, this was not done in previous discussions [2-41. In this discussion we investigate whether the usual approximate scheme can be corrected in a simple straightforward way to account for the particular characteristics, which are induced by vibronic coupling when more than one single promoting mode is active. It is useful, at this point, to examine the underlying assumptions of vibronlc coupling theory. The molecular hamiltonian is written in the form N(q, 8) = I%?) + u(g, 8) + 31(Q) 9
(1)
where I’(q) and ‘iyQ) are respectiveiy the electronic and nuclear kinetic energy operators, and Ut$,Q) is the molecular potential energy operator. The Born-Oppenheimer adiabatic wave functions are @nrj = J/m(q, Q) Xmj(Q) where Qm(4, Q) is a solution to the electronic hamiltonian
* Supported by the PRF’ undel: Grant 3574-A5. ** N?H post-doctoral fellow 1958-1910. Present address: Department of Biological Sciences, New York, New York lW.%?, USA. 132
Columbia University,
Volume 7, number 1 and the vibrational
IT(Q)
CHEMICAL PHYSICS LETTERS functions
L October 1970
satisfy
+ Em(Q)] xmj = Emj Xmj *
(31
fn the Herzberg-Teller procedure, I,&&, Q) is expanded in the crude adiabatic basis set I$$&) E e&q, Qo) - the solutions of the zeroth order electronic hamilton&n T(q) + Ulq, Qo). (Qo is the equilibrium corifigufation of the ground elecfronic state,) The pertu,rbation hamiltonian is taken as the sum of the first derivatives of the molecular potential with respect to the normal coordinates
When calculating the intensity of individual vibronic lines one should know the normai coordinates of both the initial and final electronic states. Lacking this knowledge, it has been customary to use ground state normal coordinates as an approximation to those of the excited states. In the foliowing we examine the? validity of this procedure. We start with a zeroth order functions @$&. = qm(q,Q)Qm$Q) where Qmj(Q) are the vibrational functions COrFespOnding to the crude adiabatic poZkntia1 [6f
v;(Q)
= @‘$st{
Uk?,Q) - W,Q,,~Jl~Cd)
If one assumes that the normal coordinates &j(Q) may be expanded in the form
corresponding
where Q$ are the ground state normal coordinates
V$Ql
= (+&r,
Q) 1qd
+ U(q, Q)!$@&
(51 to this potential are those of the ground state,
corresponding
to the adiabatic potentiai
QD - Eg(Qo) -
We have thus explicitly excluded the case where the normal coordinates of a given symmetry are mixed as a result of an excitation from the ground state $ (4, Q) to the excited state ~~n(~, ~0)~ This assumption is implicit in all previous treatments of intensi % borrowing [2-41. The possibility of changes in the vibrational frequency of a particular normal mode or of displacements along totally symmetry coordinates are not however excluded. We choose a model system consisting of the ground electronic state g and the excited states nz and 1~. We assume that nt is a symmetry forbidden final state. For simplicity we assert that both nt and n are non degenerate states and are coupled by two normal modes a and b which by necessity are non-degenerate. The transition probability between the zeroth vibrational level of the ground electronic state and the jth vibrational level of the excited state J/&q, Q) is given by = {x~o~Q~) bfgmtQ)
Mpo-mj
i+mjtQf%
*m(& Q) = v$&)
+ [ vnm(Q)/Am]
where the mixing term V&(Q)
VmAQl
= bdiA~>~ W,
the electric theory.
dipole operator.
The elec-
rJl%?) 3
031
is
QlI@(d)
.
A mn -- Em(QO) - EJQo> is the electronic
VM: 8) =
0
,
where Mgm(Q) = ~~&, Q) 1Si(d I9d.h 8)) and ji((r) represents tronic wave function is expanded accordmg to Herzberg-Teller
(9) energy separation
between the states q:{(1)
and I,!&)~
and
c (au$‘gQ)) Q$ c
is the perturbation written as
0
hamiltonian expanded in ground state normal coordinates.
V&Q)
may be thus
133
Volume 7. number 1
k;nn(Q) = Ka
CHEMICAL
Qg + Kb
1 October 1970
PHYSICS LETTERS
Q%
(10)
9
where
In our model, M
Mgm (Q o ) = M”pm = 0 and Min + 0, thus
km (Q) “1~
a
Qiii + Kb @r”giA
(11)
?nt~
and thence
afgO+nlj
=(XgO(Q,g)X&(Qg)IKa
Qg + Kb Q&!Q,j,(Q~)9,ib(~))~~S/A~~
9
where S is the product of the Franck-Condon
factors
final
Ql(QL$@O(QB or @O(Qg) @I(QE) where
vibrational
states
of interest
are
either
for the vibrations
(12)
not involved in the coupling. the subscript
The
m has
been suppressed *. Thus, we have obtained the familiar result which predicts the appearance of FranckCondon progressions displaced from the c!ectronic origin by wa or CJ+, the respective frequencies of normal modes a and b. The expression for the intensity ratios of individual vibronic components is given by
&oO--+m 10
(13)
PgOO-m 01 where K = K(Xo(e) bronic states
1@I@I(@)).
The subscripts
*A(43 Q) =G’m(qsQ)Ql(Qg)‘jQo(Qf$) These states are those corresponding are built. tonian
They
arise
from
diagonalizing
T(q) * L’tq, Qo) + CT QE c aQ,g
9
ml0
*Bh,Q)
and m01 correspond
=Q’mki,Q)@o(Q&@l(Q%)
to the two false origins the electronic
respectively
states
A2 The resulting
r%(Qo) L
+
hamiltonian
-
.
v!(Q)+ T(Q)]/@o(~@@~(B(~))
= BK,K~/A
to the molecularhamil-
.
(15)
matrix
* We are in effect comparing the same vibrational components of the hvo Franck-Condon pmgressions. vibrational wave functions of the normal modes not directly involved in the coupling are not inc!uded
134
(14)
on which the Franck-Condon progressions J&(q) and t/.&q) with respect to the hamil-
However the vibronic states &~(q, Q) and +B(q, Q) are not diagonal with respect tonianab,Q). Consideration of eqs. (8) and (10) leads to the expression
(KaQf + KbQ[)2
to the final vi-
Thus the explicitiy.
Volume 7, number 1
(@a HAB
=, *b
CHEMICAL
PHYSICS LETTERS
1 October
1970
>
gives rise to new eigenfunctions
(16)
ande =wa- wb is the frequency difference Notice that @AI and Qirgrare still separable In order to compare
correspoading into electronic
the intensity distribution
to the zeroth order vibrational and vibrational components.
predicted
wave functions.
by eq. (16) to the standard result req. (13)]
arising from the approximation using Herzberg-Teller procedure with ground state normal coordinates we consider the case where B&J < E and obtain a perturbation series solution for c and d. In this limit c =S 1 and d x H&E. ’
Relative transitions
~~goO-&0
&00--G'_
+ d”goo-WZol
(cMgoo-wol
pgoo-3’
Inserting the perturbation
probabilities
- ~goo-mlo
to the final states @A’ and 4&v are given by
2 > ’
limit results for c and d we obtain
(18) In order to elucidate the procedure described above, it is of interest to approach the problem from a slightly different point of view. The vibronic coupling between the two crude adiabatic electronic states @o,(q) and J/o,(4) implies a concomitant change in the electronic potential. The Herzbrg-Teller adiabatic potential may be expanded in terms of the crude adiabatic potentials If,(Q) and 5 ?&Q) and the COK-
pling element
Vmn(Q).The hamiltonian matrix
leads to
Em(Q) "%~(Qo) + V%(Q) + VL(Q)/A
= Em(QO)
+ V:(Q)
+ (&xQp f “bQg12/A
,
where it is assumed that 1V!(Q) - V%(Q)/> 1V,,(Q)]. It can be seen that the vibronic interaction will always introduce
cross terms in the Herzberg-Teller adiabatic potential described in ground state normal coordinates, and concomitant changes in the force constants. The cross terms will result in interaction El51 between the zeroth order states @A and @g, and thus induce changes in the intensity pattern of absorption. Tkese ckatzges itt tke poferr&zL wZE not This is due to the fact that the Lowest zeroth order have a prmounced e$fect on the emission fia#em. vibration state # (Q@# (Q@ interacts through the cross term only with the vibrationally doubiy excited state cj~~(Q,) separated from it by two vibrational quanta wa f “b’ * @l(Q -B ) which is energetically Since in absorption the two false origins are energetically separated by wa - #b the mixing between the corresponding final states will be much more effective particularly when wa + Ob >b foa-~b[ - fn the = 1%,Kb/A 1 the emission pattern will also be affected, however this wit: require :g;; $;e11;‘r leo.1;’ the pseudo-Jon Teller case. In the limit where Wa
+Wb
B
] &&/hi
5
IUa-Wb]
9
the emission pattern can still be predicted using Herzberg-Teller theory with ground state normal coordinates i.e., using eq. (13) whereas for absorption eqs. (17) or (18) should be used. 135
Volume 7, number 1
CHEMICAL PHYSICS LETTERS
1 October 1970
From the above discussion it is obvious that a foss of mirror symmetry between emission and absorption will result from the vibronic interactions when several promoting modes are active. To numerically illustrate the magnitude of the effect we consider the reasonable values A, = 500 cm-l, oa fob r;: 2000 cm-l. Eq. (13) predicts that - zoocf~ cm-l. ha a &, e f 000 Cm-~ #b -war the two false origins will appear with about equal intensity in absorption, whereas eq. (18)‘predicts that Ia/jb r 5. However the emission pattern will be negligibly affected by the coupling and the intensity ratio predicted by eq. (13) should be correct. A future publication [?] will deal with a similar approach to -he problem of forbidden transitions in which couphng between vibronic rather than electronic states is considered directly.
We thank Professor R. Silbey for many helpful discussions and Professor M. Karplus for his critical reading of the manuscript. We are indebted to Professor A. C. Albrecht for extensive correspondence regarding this work. Finally we acknowledge Professor D. Craig’s helpful comment that the effects discussed above result in a Duschinsky effect.
REFERENCES [l] G. Hcrzberg nnd E.TeIler, 2. Physik. Chem. (Leipzig) B21 (1933) 410. (21 J. K. Xurrell and J. A Pople. Proc. Phys. Sot. (London) A69 (1956) 245. [31 A. C. -4Ibrecht. J. Chem. Phys. 33 (1960) 169. (41 A.D. Liehr. Can. J. Pbys. 35 @SST) 1123. [S] I?.Duschinsky. Acta Physicochim, U.S.S.R. 7 (1937) 551. f&j H. C. Longuet-Higgins, Advan. Spectry. 2 (196!.) 429. [71 B. Shnrf. B. Honig and J. Jortner. to be published.
136
1 October 19’70
CHEMICAL PHYSICS LETTERS
COMMENTS
Deportment
ON
VIBRONIC
INTENSITY
B. SHARI? of Chemistry. Massachusetts Cam&ridge, Massachzisetts
BORROWING
*
Institute of Technology 02139, USA
and Department
cfChemistry,
B. HONIG**
Hwvard
University , Cambddge,
Massachusetts
02138,
USA
Received 3 August 197% The approximation which uses Her&erg-Teller theory with ground state normal coordinates is examined. ft is shown that this procedure is not generalIy vatid and is inconsistent with its underlying assumptions.
The Herzberg-Teller theory [l] of vibronic intensity borrowing in poIyatomic molecules is a well established method of predicting oscillator strengths for symmetry forbidden electronic transitions. In the case where more than one vibration contributes to the coupling mechanism, the theory has been extended to account for the intensity of individual vibronic lines [Z-4], a problem which was not explicitly treated by Herzberg and Teller [I]. It was first observed by Duschinsky [5] that the normal coordinates of excited electronic states till in general be rotated relative to those of the ground state (in normal coordinate space). This effect has been usually neglected in the consideration of the vibronic structure of allowed and forbidden electronic transitions. In this paper, we show that it is theoretically inconsistent to apply Herzberg-Teller theory with ground state normal coordinates and to neglect the Duschinsky effect, since the vibronic coupling mechanism will always induce rotations in normal coordinate space. The vibronic states resulting from the usual perturbation procedure are not diagonal with respect to the molecular hamiltonian. The mixing results in intensity redistribution between the Franck-Condon progressions built on the respective false origins. Of course, if the normal coordinates and thereof the vibrational functions of the excited electronic state were known, the Herzberg-Teller scheme could be extended to account for the intensity of individual vibronic components by expanding excited state normal coordinates and vibrational wave functions in terms of those of the sound state. However, this was not done in previous discussions [2-41. In this discussion we investigate whether the usual approximate scheme can be corrected in a simple straightforward way to account for the particular characteristics, which are induced by vibronic coupling when more than one single promoting mode is active. It is useful, at this point, to examine the underlying assumptions of vibronlc coupling theory. The molecular hamiltonian is written in the form N(q, 8) = I%?) + u(g, 8) + 31(Q) 9
(1)
where I’(q) and ‘iyQ) are respectiveiy the electronic and nuclear kinetic energy operators, and Ut$,Q) is the molecular potential energy operator. The Born-Oppenheimer adiabatic wave functions are @nrj = J/m(q, Q) Xmj(Q) where Qm(4, Q) is a solution to the electronic hamiltonian
* Supported by the PRF’ undel: Grant 3574-A5. ** N?H post-doctoral fellow 1958-1910. Present address: Department of Biological Sciences, New York, New York lW.%?, USA. 132
Columbia University,
Volume 7, number 1 and the vibrational
IT(Q)
CHEMICAL PHYSICS LETTERS functions
L October 1970
satisfy
+ Em(Q)] xmj = Emj Xmj *
(31
fn the Herzberg-Teller procedure, I,&&, Q) is expanded in the crude adiabatic basis set I$$&) E e&q, Qo) - the solutions of the zeroth order electronic hamilton&n T(q) + Ulq, Qo). (Qo is the equilibrium corifigufation of the ground elecfronic state,) The pertu,rbation hamiltonian is taken as the sum of the first derivatives of the molecular potential with respect to the normal coordinates
When calculating the intensity of individual vibronic lines one should know the normai coordinates of both the initial and final electronic states. Lacking this knowledge, it has been customary to use ground state normal coordinates as an approximation to those of the excited states. In the foliowing we examine the? validity of this procedure. We start with a zeroth order functions @$&. = qm(q,Q)Qm$Q) where Qmj(Q) are the vibrational functions COrFespOnding to the crude adiabatic poZkntia1 [6f
v;(Q)
= @‘$st{
Uk?,Q) - W,Q,,~Jl~Cd)
If one assumes that the normal coordinates &j(Q) may be expanded in the form
corresponding
where Q$ are the ground state normal coordinates
V$Ql
= (+&r,
Q) 1qd
+ U(q, Q)!$@&
(51 to this potential are those of the ground state,
corresponding
to the adiabatic potentiai
QD - Eg(Qo) -
We have thus explicitly excluded the case where the normal coordinates of a given symmetry are mixed as a result of an excitation from the ground state $ (4, Q) to the excited state ~~n(~, ~0)~ This assumption is implicit in all previous treatments of intensi % borrowing [2-41. The possibility of changes in the vibrational frequency of a particular normal mode or of displacements along totally symmetry coordinates are not however excluded. We choose a model system consisting of the ground electronic state g and the excited states nz and 1~. We assume that nt is a symmetry forbidden final state. For simplicity we assert that both nt and n are non degenerate states and are coupled by two normal modes a and b which by necessity are non-degenerate. The transition probability between the zeroth vibrational level of the ground electronic state and the jth vibrational level of the excited state J/&q, Q) is given by = {x~o~Q~) bfgmtQ)
Mpo-mj
i+mjtQf%
*m(& Q) = v$&)
+ [ vnm(Q)/Am]
where the mixing term V&(Q)
VmAQl
= bdiA~>~ W,
the electric theory.
dipole operator.
The elec-
rJl%?) 3
031
is
QlI@(d)
.
A mn -- Em(QO) - EJQo> is the electronic
VM: 8) =
0
,
where Mgm(Q) = ~~&, Q) 1Si(d I9d.h 8)) and ji((r) represents tronic wave function is expanded accordmg to Herzberg-Teller
(9) energy separation
between the states q:{(1)
and I,!&)~
and
c (au$‘gQ)) Q$ c
is the perturbation written as
0
hamiltonian expanded in ground state normal coordinates.
V&Q)
may be thus
133
Volume 7. number 1
k;nn(Q) = Ka
CHEMICAL
Qg + Kb
1 October 1970
PHYSICS LETTERS
Q%
(10)
9
where
In our model, M
Mgm (Q o ) = M”pm = 0 and Min + 0, thus
km (Q) “1~
a
Qiii + Kb @r”giA
(11)
?nt~
and thence
afgO+nlj
=(XgO(Q,g)X&(Qg)IKa
Qg + Kb Q&!Q,j,(Q~)9,ib(~))~~S/A~~
9
where S is the product of the Franck-Condon
factors
final
Ql(QL$@O(QB or @O(Qg) @I(QE) where
vibrational
states
of interest
are
either
for the vibrations
(12)
not involved in the coupling. the subscript
The
m has
been suppressed *. Thus, we have obtained the familiar result which predicts the appearance of FranckCondon progressions displaced from the c!ectronic origin by wa or CJ+, the respective frequencies of normal modes a and b. The expression for the intensity ratios of individual vibronic components is given by
&oO--+m 10
(13)
PgOO-m 01 where K = K(Xo(e) bronic states
1@I@I(@)).
The subscripts
*A(43 Q) =G’m(qsQ)Ql(Qg)‘jQo(Qf$) These states are those corresponding are built. tonian
They
arise
from
diagonalizing
T(q) * L’tq, Qo) + CT QE c aQ,g
9
ml0
*Bh,Q)
and m01 correspond
=Q’mki,Q)@o(Q&@l(Q%)
to the two false origins the electronic
respectively
states
A2 The resulting
r%(Qo) L
+
hamiltonian
-
.
v!(Q)+ T(Q)]/@o(~@@~(B(~))
= BK,K~/A
to the molecularhamil-
.
(15)
matrix
* We are in effect comparing the same vibrational components of the hvo Franck-Condon pmgressions. vibrational wave functions of the normal modes not directly involved in the coupling are not inc!uded
134
(14)
on which the Franck-Condon progressions J&(q) and t/.&q) with respect to the hamil-
However the vibronic states &~(q, Q) and +B(q, Q) are not diagonal with respect tonianab,Q). Consideration of eqs. (8) and (10) leads to the expression
(KaQf + KbQ[)2
to the final vi-
Thus the explicitiy.
Volume 7, number 1
(@a HAB
=, *b
CHEMICAL
PHYSICS LETTERS
1 October
1970
>
gives rise to new eigenfunctions
(16)
ande =wa- wb is the frequency difference Notice that @AI and Qirgrare still separable In order to compare
correspoading into electronic
the intensity distribution
to the zeroth order vibrational and vibrational components.
predicted
wave functions.
by eq. (16) to the standard result req. (13)]
arising from the approximation using Herzberg-Teller procedure with ground state normal coordinates we consider the case where B&J < E and obtain a perturbation series solution for c and d. In this limit c =S 1 and d x H&E. ’
Relative transitions
~~goO-&0
&00--G'_
+ d”goo-WZol
(cMgoo-wol
pgoo-3’
Inserting the perturbation
probabilities
- ~goo-mlo
to the final states @A’ and 4&v are given by
2 > ’
limit results for c and d we obtain
(18) In order to elucidate the procedure described above, it is of interest to approach the problem from a slightly different point of view. The vibronic coupling between the two crude adiabatic electronic states @o,(q) and J/o,(4) implies a concomitant change in the electronic potential. The Herzbrg-Teller adiabatic potential may be expanded in terms of the crude adiabatic potentials If,(Q) and 5 ?&Q) and the COK-
pling element
Vmn(Q).The hamiltonian matrix
leads to
Em(Q) "%~(Qo) + V%(Q) + VL(Q)/A
= Em(QO)
+ V:(Q)
+ (&xQp f “bQg12/A
,
where it is assumed that 1V!(Q) - V%(Q)/
cross terms in the Herzberg-Teller adiabatic potential described in ground state normal coordinates, and concomitant changes in the force constants. The cross terms will result in interaction El51 between the zeroth order states @A and @g, and thus induce changes in the intensity pattern of absorption. Tkese ckatzges itt tke poferr&zL wZE not This is due to the fact that the Lowest zeroth order have a prmounced e$fect on the emission fia#em. vibration state # (Q@# (Q@ interacts through the cross term only with the vibrationally doubiy excited state cj~~(Q,) separated from it by two vibrational quanta wa f “b’ * @l(Q -B ) which is energetically Since in absorption the two false origins are energetically separated by wa - #b the mixing between the corresponding final states will be much more effective particularly when wa + Ob >b foa-~b[ - fn the = 1%,Kb/A 1 the emission pattern will also be affected, however this wit: require :g;; $;e11;‘r leo.1;’ the pseudo-Jon Teller case. In the limit where Wa
+Wb
B
] &&/hi
5
IUa-Wb]
9
the emission pattern can still be predicted using Herzberg-Teller theory with ground state normal coordinates i.e., using eq. (13) whereas for absorption eqs. (17) or (18) should be used. 135
Volume 7, number 1
CHEMICAL PHYSICS LETTERS
1 October 1970
From the above discussion it is obvious that a foss of mirror symmetry between emission and absorption will result from the vibronic interactions when several promoting modes are active. To numerically illustrate the magnitude of the effect we consider the reasonable values A, = 500 cm-l, oa fob r;: 2000 cm-l. Eq. (13) predicts that - zoocf~ cm-l. ha a &, e f 000 Cm-~ #b -war the two false origins will appear with about equal intensity in absorption, whereas eq. (18)‘predicts that Ia/jb r 5. However the emission pattern will be negligibly affected by the coupling and the intensity ratio predicted by eq. (13) should be correct. A future publication [?] will deal with a similar approach to -he problem of forbidden transitions in which couphng between vibronic rather than electronic states is considered directly.
We thank Professor R. Silbey for many helpful discussions and Professor M. Karplus for his critical reading of the manuscript. We are indebted to Professor A. C. Albrecht for extensive correspondence regarding this work. Finally we acknowledge Professor D. Craig’s helpful comment that the effects discussed above result in a Duschinsky effect.
REFERENCES [l] G. Hcrzberg nnd E.TeIler, 2. Physik. Chem. (Leipzig) B21 (1933) 410. (21 J. K. Xurrell and J. A Pople. Proc. Phys. Sot. (London) A69 (1956) 245. [31 A. C. -4Ibrecht. J. Chem. Phys. 33 (1960) 169. (41 A.D. Liehr. Can. J. Pbys. 35 @SST) 1123. [S] I?.Duschinsky. Acta Physicochim, U.S.S.R. 7 (1937) 551. f&j H. C. Longuet-Higgins, Advan. Spectry. 2 (196!.) 429. [71 B. Shnrf. B. Honig and J. Jortner. to be published.
136