- Email: [email protected]
rij mechanism. Inhomogeneous line broadening in the Rydberg spectrum of naphthalene
Volume 8, pumber 2
CHEMICAL
EVliDENCE INHO-MOGENEOUS
FOR LINE
SPECTRUM ~e~~~en~
THE
l&j
MECHANISM.
.BROADENING
OF
15 January 1971
PHYSICS LE’TTERS
IN
THE
RYDkERG
NAPHTHALENEt
B. SHARF
Clientfsfry, Mussaciruseffs institute of Techndogy, Cambridge, lCIassacitu.setts 02139, USA 0-f
Received 24 November
1970
Theoretical discussion of the Iine &ape of vibronic resonances in the Rydberg spectrum of naphthalene is presented.‘ The spectrum provides strong evidence for the I/Tie mechanism recently proposed by the author. The results of this work are in disagreement with the war-t, of Jortner and Morris.
The line shape of &homogeneously broadened vibronie resonances in the electronic spectra of large polyatomfc molecules was recently discussed [I, Z]. The nature of the vibronic interactions fnduc$g ~~omogeneo~s line broadening w+ elucidated by Sharf and Silbey {Z&4]. Burland and Robinson [5,6] have independatiy . arrived at the same mechanism when discussing electronic relaxation processes in large polyatomic mblec&es. The convenient representation was shown to be the crude adiabatic scheme. The -procedure previously used by Bixon and Jotiner [?] and others [8,9] to evaluate the necessary matrix elements was shown to be invalid’ WI* In contradiction to the predictions of Bixon ,I and Jortner f7] it was shown that generally the lihe shape of an inhomogeneously broadened resonance is expected to be asymmetric - on the basis of energy and intensity considerations. Symmetry considerations, or better to say se&xCion rules, determine whether the fine shape 1 will be symmetric OT asymmetric. Moreover, , 41 general, the line shape was shown to be depen~ dent on light polarization Ill. Thus, .the one and the s&e resonance may appear as a symmetric peak, an aggmmetric resonance, or as a dip -when the polarization of tight is varied. . Recently, anti-resonances have been det&ed in the gas phase Rydberg spectrum’of naphtha:: lene [lo], .where the n=5 to n=13 Rydberg levels overlap a’m&iium iptetisity (fw 0.1) of.-?r*transi. ~~&~p+.&by 238
the l?Rl? +mdeir Oran& 3s?4~-A%:_ .: I
tion broadened by quasi degenerate vibronic transitions to a lower excited R-Z* state. Similar, -but less pronounced, anti-resonances have been claimed [II] to appear in the 56000 cm-l region of anthracene [lZ] and the 59000 em-l region of pyrimidine [ 131. Jortner and Morris [II] have recent@ discussed the interference effects leading to anti-resonances in the naphthalene Hydberg spectrum. We feel that this discussion is incorrect, and incomplete, and the present note ts an attempt at a proper theoretical interpretation of this interesting phenomenon. fn particular this case seems to provide strong experimental evidence for the l&j mechanism 1141. The I/Yij mechanism was recently proposed by the author to be operative in inducing inhomogeneous fine broadening of resonances below the first ionization threshold, i.e., when _ auto - itmization does not pwvaiE 1141. In the naphthalene molectlies of n2h symmetry the coltnear case prevails [I, 21, and the line sh.ape is given by the original Fano expression [1 5]
where 5, the reduced energy parameter, and.p, t?e line prpfile index, we defined by _ ER ‘1E ER-E qtpR 1 PR V ES-------= &$.
.&A-
. * . .
-r-=Yr-~* .p.%Vp
IfJ
zA
p is the mtiifc$d density bf s&e& ik,, he nuti&r of states per pnii- energy:: V is fhk [email protected] ‘_
Volume 8, number 2
CHEMICAL PHYSICS LETTERS
between the discrete zeroth order state GR which energy is denoted by ER- and each of the zeroth order manifold states ($BB}. ER and {Jim) are transition moments connecting the initial vibronic state with the final vibronic states $R and {qJj respect.ively. All (PJ) are assumed to be equal and colinear with ,5R [definition of the colinear case [l, 21). The zeroth order Rydberg states (c$& interact with the TI-TI* manifold {@d which can be cast in the form:
where (GS} denote the sparse vibronic manifold of the upper z-~i* electronic state quasi degenerate with the electronic Ryclberg states and {Q} denote the dense manifold of high excited vibrational levels of the lower excited a-z* state in which {$S] and {$R} are embedded. We will assume that the Rydberg states (+) interact with the {QJ} manifold due to its {@s) character, this assumption is justified in view of the favorable overlap factors connecting the vibrationally low excited r#+Iwith the vibrationally low excited (4-q). Thus, the interaction of $R with the (@R) manifold - which density is approximated by the density of the {I#Q}manifold - is given by
where +i and Q!+,are the electronic wave functions for the nth Rydberg and for the n-n* state respectively. fis the sum of the corresponding weighted vibrational factors. The eigenstates 9, of this system in the isolated resonance approximation have the form
whereas the ‘line width’ A (if 1q 1-=J) is given by since the Rydberg transition n= 5.to n=13 which extend over the entire electronic width Av of the broadened n-n* transition have about the same width, we assert that f$ ,pK and estimate Thus from the line bvidth equation f2p=Ad. KA M (A.Av/2s)1/2.-
15Janua~
1971
The profile index q is eqanded to give CR
1
‘=qs
ER 1 =gi,f;rKfp
’
where iI, corresponds to the electronic dipole moment andj corresponds to the weighted sum of the overlap factors between the groun_dstate lowest vibrational state and 9, Sinceff Lies between f 2 andf2, and as pf2=l/Av=pf2 we
get
FRAU ---. q - ,ii= nzc
From the experimental data we estimate q -1; ,iiR/,ii B Y (A/AV) 'I2 and thus Kq= (A.Av/;;)'/~ in
good agreement with ZcA. Introducing t.e experimental values A= 30 - 100 cm-l; AU= 3000 - 5000 cm -1 we get K= 100 - 300 cm-l. Jortner and Morris have asserted that the interaction has to be vibronic and have estimated that the relevant electronic matrix element between the spatially extended Rydberg orbital and the n* orbital is of the order of 1 cm-l. This estimate seems to be very reasonable. However, interactions of this magnitude cannot account for the recorded experimental results and will evidently lead to a line width of 0.01 cm-l and q values of -100 which are far off. The discrepancy between the present conclusions and the results of Jortner and Morris is due to the use by these authors of incorrect expression for the line profile index, viz.: q=
Pm 1 --. jiJ7qI
As pointed out previously [L4] the l/~-i - interactions are generaBy stronger than v1-I!ronic interactions. For instance, vibronic interactions between two s orbitals are usually 10 -LOO times weaker than the corresponding l/rij interactions. Moreover, the l/r - - interaction wicriL1 _ not decrease as fast as vibr 2.nit interactions on passing from J -0 interactions to in-_R interactions. And thus n-R interelectronic interactions of the order of several hundreds of cm-l are very reasonable. Therefore we are forced to conclude that the interference effects recorded in the naphthalene Rydberg spectrum are due to the 1/7ij mechanism. I am grateful to Professor ful diecussions.
R Silbey for help-
239
_ ,.Voiume
._
._
8; number 2 .
CHEMICAi
Pi-KS&
I.,
‘I.IET+ERS..
: . 15 Jam&y :-
19’71. .
:
~prsss) 1543. _
‘:- [I; Bkharf; Chim. Phys. Ltetiers 5 (1970) 4;6 fz] E’khiqf,, Chem. ‘Phys. Letters 5 [iSlO)459: . . [3f .B.‘Shati and .R.Silbey,‘ Chem.. Phys.. Letters 4 (1970) 423:f4] B.Sharf ind’R.Silbey, Chem. Phys. Letters 4 * (1970) 561: [S] D,M. Burland and G. W. Rcrbinson, J. Chem. ‘Phys.
$1 f1969) 454q.
[6] 0. ;M;Badand.and
G. XV.Robinson, Proc..Nafi. Acad._Fci. 66 ~~9~6~257. : 1’71gaBixon and J. Jorfner, J, Chem. Phya. 48 (1968)
.-
: ~ “(9f S.Ii;Lin; J;-Chem. Phys:k4jf3& 3759. .- ” , _ I [Ia] S..G;~t&s, B+.Christ and _G; C. Mori’is; -’ Austra&$ J. Che& 21 (l966)‘2163. . _. . - [Zl]J.5Srttier and G:C. M&is, J. Chem..Phys. 21 I (1969) 3689.. . [12].J. G. Angus and G. C; Morris, Y: Mol. Spectry. 21. . (196s) 310. -. ‘.
ii31 J. E. PsrMn and K.K.I~es,
f1965)40?.
J. Mol. Specfry. 15
.
CHEMICAL
EVliDENCE INHO-MOGENEOUS
FOR LINE
SPECTRUM ~e~~~en~
THE
l&j
MECHANISM.
.BROADENING
OF
15 January 1971
PHYSICS LE’TTERS
IN
THE
RYDkERG
NAPHTHALENEt
B. SHARF
Clientfsfry, Mussaciruseffs institute of Techndogy, Cambridge, lCIassacitu.setts 02139, USA 0-f
Received 24 November
1970
Theoretical discussion of the Iine &ape of vibronic resonances in the Rydberg spectrum of naphthalene is presented.‘ The spectrum provides strong evidence for the I/Tie mechanism recently proposed by the author. The results of this work are in disagreement with the war-t, of Jortner and Morris.
The line shape of &homogeneously broadened vibronie resonances in the electronic spectra of large polyatomfc molecules was recently discussed [I, Z]. The nature of the vibronic interactions fnduc$g ~~omogeneo~s line broadening w+ elucidated by Sharf and Silbey {Z&4]. Burland and Robinson [5,6] have independatiy . arrived at the same mechanism when discussing electronic relaxation processes in large polyatomic mblec&es. The convenient representation was shown to be the crude adiabatic scheme. The -procedure previously used by Bixon and Jotiner [?] and others [8,9] to evaluate the necessary matrix elements was shown to be invalid’ WI* In contradiction to the predictions of Bixon ,I and Jortner f7] it was shown that generally the lihe shape of an inhomogeneously broadened resonance is expected to be asymmetric - on the basis of energy and intensity considerations. Symmetry considerations, or better to say se&xCion rules, determine whether the fine shape 1 will be symmetric OT asymmetric. Moreover, , 41 general, the line shape was shown to be depen~ dent on light polarization Ill. Thus, .the one and the s&e resonance may appear as a symmetric peak, an aggmmetric resonance, or as a dip -when the polarization of tight is varied. . Recently, anti-resonances have been det&ed in the gas phase Rydberg spectrum’of naphtha:: lene [lo], .where the n=5 to n=13 Rydberg levels overlap a’m&iium iptetisity (fw 0.1) of.-?r*transi. ~~&~p+.&by 238
the l?Rl? +mdeir Oran& 3s?4~-A%:_ .: I
tion broadened by quasi degenerate vibronic transitions to a lower excited R-Z* state. Similar, -but less pronounced, anti-resonances have been claimed [II] to appear in the 56000 cm-l region of anthracene [lZ] and the 59000 em-l region of pyrimidine [ 131. Jortner and Morris [II] have recent@ discussed the interference effects leading to anti-resonances in the naphthalene Hydberg spectrum. We feel that this discussion is incorrect, and incomplete, and the present note ts an attempt at a proper theoretical interpretation of this interesting phenomenon. fn particular this case seems to provide strong experimental evidence for the l&j mechanism 1141. The I/Yij mechanism was recently proposed by the author to be operative in inducing inhomogeneous fine broadening of resonances below the first ionization threshold, i.e., when _ auto - itmization does not pwvaiE 1141. In the naphthalene molectlies of n2h symmetry the coltnear case prevails [I, 21, and the line sh.ape is given by the original Fano expression [1 5]
where 5, the reduced energy parameter, and.p, t?e line prpfile index, we defined by _ ER ‘1E ER-E qtpR 1 PR V ES-------= &$.
.&A-
. * . .
-r-=Yr-~* .p.%Vp
IfJ
zA
p is the mtiifc$d density bf s&e& ik,, he nuti&r of states per pnii- energy:: V is fhk [email protected] ‘_
Volume 8, number 2
CHEMICAL PHYSICS LETTERS
between the discrete zeroth order state GR which energy is denoted by ER- and each of the zeroth order manifold states ($BB}. ER and {Jim) are transition moments connecting the initial vibronic state with the final vibronic states $R and {qJj respect.ively. All (PJ) are assumed to be equal and colinear with ,5R [definition of the colinear case [l, 21). The zeroth order Rydberg states (c$& interact with the TI-TI* manifold {@d which can be cast in the form:
where (GS} denote the sparse vibronic manifold of the upper z-~i* electronic state quasi degenerate with the electronic Ryclberg states and {Q} denote the dense manifold of high excited vibrational levels of the lower excited a-z* state in which {$S] and {$R} are embedded. We will assume that the Rydberg states (+) interact with the {QJ} manifold due to its {@s) character, this assumption is justified in view of the favorable overlap factors connecting the vibrationally low excited r#+Iwith the vibrationally low excited (4-q). Thus, the interaction of $R with the (@R) manifold - which density is approximated by the density of the {I#Q}manifold - is given by
where +i and Q!+,are the electronic wave functions for the nth Rydberg and for the n-n* state respectively. fis the sum of the corresponding weighted vibrational factors. The eigenstates 9, of this system in the isolated resonance approximation have the form
whereas the ‘line width’ A (if 1q 1-=J) is given by since the Rydberg transition n= 5.to n=13 which extend over the entire electronic width Av of the broadened n-n* transition have about the same width, we assert that f$ ,pK and estimate Thus from the line bvidth equation f2p=Ad. KA M (A.Av/2s)1/2.-
15Janua~
1971
The profile index q is eqanded to give CR
1
‘=qs
ER 1 =gi,f;rKfp
’
where iI, corresponds to the electronic dipole moment andj corresponds to the weighted sum of the overlap factors between the groun_dstate lowest vibrational state and 9, Sinceff Lies between f 2 andf2, and as pf2=l/Av=pf2 we
get
FRAU ---. q - ,ii= nzc
From the experimental data we estimate q -1; ,iiR/,ii B Y (A/AV) 'I2 and thus Kq= (A.Av/;;)'/~ in
good agreement with ZcA. Introducing t.e experimental values A= 30 - 100 cm-l; AU= 3000 - 5000 cm -1 we get K= 100 - 300 cm-l. Jortner and Morris have asserted that the interaction has to be vibronic and have estimated that the relevant electronic matrix element between the spatially extended Rydberg orbital and the n* orbital is of the order of 1 cm-l. This estimate seems to be very reasonable. However, interactions of this magnitude cannot account for the recorded experimental results and will evidently lead to a line width of 0.01 cm-l and q values of -100 which are far off. The discrepancy between the present conclusions and the results of Jortner and Morris is due to the use by these authors of incorrect expression for the line profile index, viz.: q=
Pm 1 --. jiJ7qI
As pointed out previously [L4] the l/~-i - interactions are generaBy stronger than v1-I!ronic interactions. For instance, vibronic interactions between two s orbitals are usually 10 -LOO times weaker than the corresponding l/rij interactions. Moreover, the l/r - - interaction wicriL1 _ not decrease as fast as vibr 2.nit interactions on passing from J -0 interactions to in-_R interactions. And thus n-R interelectronic interactions of the order of several hundreds of cm-l are very reasonable. Therefore we are forced to conclude that the interference effects recorded in the naphthalene Rydberg spectrum are due to the 1/7ij mechanism. I am grateful to Professor ful diecussions.
R Silbey for help-
239
_ ,.Voiume
._
._
8; number 2 .
CHEMICAi
Pi-KS&
I.,
‘I.IET+ERS..
: . 15 Jam&y :-
19’71. .
:
~prsss) 1543. _
‘:- [I; Bkharf; Chim. Phys. Ltetiers 5 (1970) 4;6 fz] E’khiqf,, Chem. ‘Phys. Letters 5 [iSlO)459: . . [3f .B.‘Shati and .R.Silbey,‘ Chem.. Phys.. Letters 4 (1970) 423:f4] B.Sharf ind’R.Silbey, Chem. Phys. Letters 4 * (1970) 561: [S] D,M. Burland and G. W. Rcrbinson, J. Chem. ‘Phys.
$1 f1969) 454q.
[6] 0. ;M;Badand.and
G. XV.Robinson, Proc..Nafi. Acad._Fci. 66 ~~9~6~257. : 1’71gaBixon and J. Jorfner, J, Chem. Phya. 48 (1968)
.-
: ~ “(9f S.Ii;Lin; J;-Chem. Phys:k4jf3& 3759. .- ” , _ I [Ia] S..G;~t&s, B+.Christ and _G; C. Mori’is; -’ Austra&$ J. Che& 21 (l966)‘2163. . _. . - [Zl]J.5Srttier and G:C. M&is, J. Chem..Phys. 21 I (1969) 3689.. . [12].J. G. Angus and G. C; Morris, Y: Mol. Spectry. 21. . (196s) 310. -. ‘.
ii31 J. E. PsrMn and K.K.I~es,
f1965)40?.
J. Mol. Specfry. 15
.