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Jahn-Teller coupling and the photoelectron spectrum of cyclopropane
Volume 9, number 2
CHEMICALPHYSICS LETTERS
JAHN-TELLER THE
15 April 1971
COUPLtNC
A’ND
PHOTOELECTRONSPECTI?UMOF
CYCLOPRQPANE
C. G. ROWLAND Labmutoire de Citimie TMorique*, Cenh? d’Orsny, Universitk de Paris-Sud, 91-Orsay, Frawe Received 8 March 1971
Reparameterized CNDO calculations on the ground and ffrst excited states of the cycfopropaue caD”on sboowthat both states are subject to significant J&n-Teller distortions. First and second order vibroaic coupling parameters derived from the calcuIations are used to predict the shape of spectra which would be observed for electronic transitions to these degenerate states. Good agreement with the observed pbotaehctron spectrum is found. The inclusion of second order terms increases the complexity of the spectra but does not greatly alter the shape of the band cootours.
of e’ symmetry [2E The lower sheet of the JahnTeller potential surface is characterized by one point at which the ion has D2h symmetry, where the energy is a miuimum with respect to all tvtally symmetric coordinates, and two sets of three equivalent points at which the symmetry is Czv [3j, One of the sets of C2y points corresponds to the energy minima on the surface: the other set corresponds to the saddle points separating these equivalent minima A cycEopropane fvn of Czrr symmetry has seven independent coordinates. %I order to minimize the energy with respect to all of these with the expenditure of a moderate amount of computer time it ia necessary to use a semi-empirical method of ealcul~on The CNDO/2 method
In the photoelectron spectra vf highly symmetric, nonlinear molecules bands which correspond to the ionization of degenerate orbita& may show evidence of Jahn-Teller splitting. The first band in the photoeiectron spectrum of cyclopropane [1] corresponds to ionization of the highest occupied e’ (in plane) orbital and shows two maxima separated by 0.77 eV. However the second band, which may be associated with ionization of the eW(out of plane) orbit& shows little evidence vf splitting. fn this work reparameterized UNDOcalculations on the 23’ and 23” states of the cyclopropane catfon have been used to obtain parameters which govern the first and second order dependence of the electronic energy on the distortion coordinates. The vibronic energy levels and wavefunctions for the idedlized surfaces defined by these parameters have been calculated~andare used to obtain the spectra which would be observed for transitions to the Jahn-Telter
of Pople and Segti (41 gives reasonable values for equi~bri~ geometries, but heats of atomization, excitation energies and force constants are
generdly overestimated These defects have largely been corrected for in t&e reparameterized version of C!NDOfor hydrocarbons of Fischer and Kollmax [5] and this method has been used in the present work. Open shell c~cu~at~vns were peyote on the species derived from the 23’ state of the un-
distorted states,
2. SCF ~AL~LATIONS
distorted_ion- Tiie energies of the species derived from the 2E” state were obtained by the application of Koopmans’ theorem to the results of c!vsed shell calculations on Thaineutral mole-
. The 23’ and 233” states of the ~yc~o~rv~~e cati& . are anstable with respect to distortions
CUM; The.result$ of the energy minimization for the ?I& species are shown !n table 1 and for Qv species in tai$e 2.. The coordinates for the C&k--. rjipeqleg are explairied inn‘fig. 1.
* T&e Laboratoire de ChimieThQrfque is part of the -L+ratoire
. _, ,.
de Physico-Cbimie
des Rayonnements,
dependentoxit&eUnivsrsM dd Paris XI, asaocjated . witpthe C.NJ2.S. : . .
.
.’ ‘,
.. : ._
.;
.
. .
;
I
_.
. ..
.-.
-~_.
.,.., *
;
‘_,._
.‘” ‘,
:
.
_.
‘.
..:,.>‘. ._:.
.’
:
.'l@
-Y
._
;._.‘,
-. ,_
..,’ ,
,_ .‘_
_’ -’
..
-’ t,-.
>
.V+ti%
9. nuhber 2
CHEWCAL
PHYSICS LETTERS
15 April 1971
:
Table 1
_.
Geometriesandenergiesof D3h species
.
Neutral
Neutral calculated
expt. f6l
I
235
2E_
+&I
1.509
1.512
1.544
1.481
RCH @I
1.089
1.111
1.117
1.132
a t&s, Energy (au)
100.0
110.2 -_
- 21.3594
- 21.2544
--
-w
- 21.3553
- 21.2467
Energy of ion formed in vertical trknsitfon from calculated neutral
Geometries
Table 2 and energies of C&v species
Species derived from 2E1 saddle minimum point
Speciesderivedfrom 2E* saddle minimum point
1.676
1.437
1.512
1.475
1.492
1.604
1.472
1.503 1.110
1.118
1.114
1.168
1.117
1.119
1.116
113.2
112.4
1.14J
70.6
113.0
109.5
91.0
112.5
112.3
16.2
- 16.0
-
5.5
6.0
- 21.3745
- 21.2682
- 21.2631
10.5
9.5
8.7
5.5
iA1
2B2
2%
2A2
- 21.3762 relative to D3h minimum
113.9
113.6 _-
(kcalmole-l) Symmetry species
energies in b&h sets of calcuiations are in good agreement although the energy diiference between the minima and the as Iale points is more pronounced in the CNDOresutz. On bflizaiion
forma&ion of the 23” state there is a ht.rge reduction in the HCR angles. The Jabn-Teller distortion involves further change in these angles and
leadsj to mintni and saddle points differing in energy by 2.2 b.al mole-l.
3. VR3ROF-X INTERACTION
Fig. 1. Coordinates for C2v ~~clopropane.
,Sirtiffean_t
distm-tions
kre obs&ved
for botd
states, The zig’ state is stabilized most by ring deformation and by wagging of the CH2 groups. The geometries of the distorted species are in qualitative agre‘ement with those found by Baselbath in his MINDOlz calculations f?J. The &a-_ i,l?O, .,..
.“,.
:
-_ 1:
:
._
.. :
Moffitt and Thorson [g] and S.&&yet-Riggins et al. t2f have studied the quantum mechanical problem of the coupling.between the com~neats of a doubly degenerate electronic state of a non-
linear molecule induced,b$ a doubly degenerate, vibration. EIectronic transitip~~ from a-non_ . : _ . ,_ : : : I-
._-
VoluniS
9,
number
CHEMICALPHYSICS LETTERS
2
degenerate state to the Jahn-Teller distorted state are accompanied by an irregularly spaced progression in the doubly degenerate vibration. (Ifthere were no Jahn-Teller coupling the excitation of a non-totally symmetric vibration would be forbidden.) At large values of the first order coupling parameter a double humped structure appears in the spectrum, essentially because the probability distribution in the lowest vibrational level of the non-degenerate state is maximum at a non-zero value of the radial coordinate [lo, 111. The vibronic wavefunction for a doubly degenerate electronic state may be written: *(q, Q) = @e(Cl,8) h(Q) + +el(qr 8) h’(Q) * in numerical work on the Jahn-Teller and Renner
effects it is usual to neglect the dependence of the electronic functions Ge, I&~Ion the nuclear coordinates Q so that
*Iq,Q) = @$?)xJQ)
+ $~,k?)xJQ).
j=m+$h
(n, mlpe’i@(n+l,
m+l>
= (n+l, m+l Ip&jn,
* = &+x+(Q) + J/,x_(Q). Following Longuet-Higgins [12] first and second order dependence on Q is introduced into the hamiltonian. The matrix elements of H(Q) are expanded as a Taylor series in the nuclear coordinates up to terms of the second order. The terms in the totally symmetric coordinate may be eliminated by a suitable choice of origin and the vibrationai factors x, and x. are obtained by solution of the coupled equations
kpe-W + -&&#P
= I[1 x,
0.
+ $g$e-W
does not pass through its vertex. If g is non-zero the term in 3 @ introduces three equivaIent minima at cos3@ = 1 with energy -ak2/(1 -g) and saddle points at cos3@ = -1 of energy -fk2/(l+g). The vlbronic factors h and x_ may be expanded in terms of the two dimensional harmonic oscillator wavefunctions In, m). is the expectation vaIue of the enern=l,2,..., gy of the state [n, m) and the vibrational angular momentum quantum number m can take the valuesn-1, n-3, . . . . -rtcl. It is convenient also to introduce an electronic quantum number A, which is 1 for q+ and -1 for @_. Wheng = 0 the only levels which are mixed are those with the same value of and @+ and @_ can be expanded in terms of harmonic oscillator functions with m = j - $ and m = j + 2 respectively. The only non-zero matrix elements are then [9,13]
Taking complex forms of the wavefunctions
iYo - E
15 April 1971
Ho - E
x,
Ho is the unperturbed hamiltonian for the two dimensional harmonic oscillator and:
(rz, m Ipe-W In-l, m+l) = (n-l, m+l IpeW In, m> = En, m, where A n, m = ~(?z+m+l))l’2
Q, = pei@ , The associated
Q, = pe-i@; electronic
U, U’ = $A~ f (kp + &g2cos
potential is 3@),
where h is the force constant of the unperturbed vibration. A = 1 if the energy units are chosen as those of the zero point energy of the unperturbed vibration When g = 0 the electronic potential is a parabola revolved ab0ut.a line which
)
Bn, m = 6_@-m-l)}~/?
When g is different from zero, j is no longer a good quantum number and alI levels with the same value of 2jmod3 are mixed [13]. The relevant matrix elements can easily be calculated from expressions such a~: (12,m (p2e-2iQ ]nr, m’> =nmTmW(n, m Ipe+ and the only non-zero
Q, and Q_ are the complex conjugate forms of the real degenerate coordinates Qx, Q,. In polar coordinates :
m> = A,, m ;
InN,nrR> (ttrr,mm[pe+
!n’,m’!
values are
(n,mlp2e’2i+ ltw2,m+2) = (n+2,m+21p2e2i@ [qm) = 42, mh+l,
m+l ;
(n,m~p2e’2i0 ln,m+2) = (n ,m+21p2e2iG In,m> = An, mBn+l, m+i + %, m&-l,
m+l ;
(n,mlp2,-2id)(n-2, mc2) =~(n-2,mc21p2e2i~ 1n.m> =~B&mB+l
* WI.:,.
171
: -.
Volume 9. number 2.
CHEMICAL
PHYSICS
LETTERS
15 April 197l
Analogous matrix elements may be derived from
0.
the general expressiqns given by Bell [14]..
hi ~6.643
Knowing the matrix ekemdnts of HO, pe*l@ and $PWJ eq. (1) is easily converted into matrix form. The eigenvalues of this matrix are the vibronic energy levels for a particular value of 2jmod3 and the eigenvectors give the expansion coefficients of the harmonic oscillator wavefunc-
tions in bland x_.
The intensities of transitions from the lowest vibrational level of a degenerate state to the vibronie levels of the degenerate state are given, exactly if the vibration frequency is the same in the two states and to a good approximation otherwise, by the square of. the coefficient of 11,O> in the expansion of >G[9]. 11,0) is only present in the expansion for 2jmod3 = 1. Calculated spectra for transitions to the 2E’ state akshown in fig. 2. The value k2 = 6.643 was derived from the CNDO calculations assuming that the‘e’ ring deformation mode is excited on distortion and that in the undistorted ion this mode hasthe same normal frequency of 1050.6 -1 as in the neutral ground state [S]. CompaEEon of fig. 2a (g = 0.0532) and fig. 2b (g = 0) shows that the main effect of the presence of weak second order coupling is greatly to increase the number of allowed transitions in the spectrum. The band contour shows hvo maxima in both cases with a separation of 0.75 eV for g = 0.0532 and 0.55 eV for g = 0. There is, dowever, no guarantee that the normal coordinates and the vibration frequencies are the same in the ion and the neutral molecule and there is evidence of a 48 0 cm-l progression in the first band of the photoelectron spectrum [l], 0 = 480 cm-l corresponds to k2 = 14.54, but the order of the matrix which must be diagonalized in order to obtain a good representation of the spectrum rises very rapidly with k2 if g is non-zero. If g = 0 many fewer basis functions are needed for a given I22 and the matrix may be written in tridiagonal form. For k2 = 14.54, g = 0 the two maxima in the spectrum are separated by CL.37eV. If it is assumed that the 1478.1 cm-l CIi2 de-formation mode [6] is excited in transitions to the 2,” state the coupling parameters are k2 = 3.169, g = 0.2266. In the spectra calculated for .tiese parameters_ (fig. .3a) tici for g = 0 (fig. 3b) t.he,‘s&comJ maximum is reduced to a shoulder ‘.on the high energy side of the band. h-for the -‘i72
b.
-3940123456
the J&n-Teller
ted et vibrational structure for transiO> level of the neutral ground state to distorted 2El state of the positive ion
g=. -llLdLL 0.
k2 c 3.169 1133
-2
-1
0
1
2
3
c
5
b.
9=Q
-_lllL -2 -1
0
1
2
3
L
5
for transiFig. 3. CalctiI ted 6’ vibrational titrucke tfon from the tq,O> level of the neutral ground state to the Jabn-Teller distorted 2Ew &ate of thhepositive ion.
Volume
9,
number 2
CHEXEAL
23’ state the ~troduction of second order coupling does not greatly titer the shape of the band contour. If the vibration frequency were greatly reduced on ionization this shoulder would develop into a separate maximum, but its energy separation from the first maximum would be decreased. The results of the UNDO calculations
for the 23” state show a vertical-adiabatic transition energy difference of 4.8 kcaf for the D3h ~pectes.
Afairly
long
progression
in the totally
symmetric CH2 deformation mode should therefore be observed in the spectrum. In the semiclassical configuration coordinate model 1111 this would bave a width of 0.46 eV. The observed band in the photoekectron spectrum will be a fold of the a’, and e’ progressions and the width of the a’ progression may be expected to obscure any structure specifically a~ribut~le to the
presence
of J&m-Teller
coupling.
The calculations were carried out on the Univac PLO8at the Centre de Caleul of the Universite de Paris-Sud and on the IRM 360/75 at the Centre Inter-discipline Regional de Calcul Electronique of the C.N.R.S., Orsay. I would like to thank Mr. Xavier ~apuisat for the use of his reparameterized version of the CNDO/2 pro-
15 April 19n
PXYslCS LETTERS
gramme written by &gal and dfstributed by the QCPEi Professor Lionel Safem for his hospital-
ity at the Laboratoire de Cltimie ThPorique and the generous provision of compufer time and the Royal Satiety for the award of a European Exchange Programme Fellowship.
REFEZEUZNCES [I)
FLBasch,
M. B. Robin,
and D. W. Turner,
[2] G. Herzberg,
N.A.Kuebler,
f. Chem. Phys.
C, Baker St
(1969) 52.
Electronicspectraof polyatemic
Ecolecules fvan Nostrand, PrincetrJa, L966). f3I R. S. Da.@ andI. B. Leviuson, Opt. Spectry. Suppl. 3, Mol.Spectry. II (.lSSS) 1. (41 J.A.PopIe and G.A. &gal. 3. Chem.Phys. 44 (1966)
3289.
(1958)
141.
151H. Fischer and H. Kollmar. Theoret. Chim.Acta 13 (IQSS) 213. [6] S, J. Cyvin. Spectrochim.Acts 16 (rSS0) 1022. [?I E. Raeelbach. Cbam. Phys. Letters 7 (l970) 428. [8] W. Moffitt and WK. Thorson. Co% Iat. C.N.R.S. 82 19) H. 6. Lo~et-bins, U.&J&. M. K. L. Puce and %A.-ack, %oc. key. E&:A244 (lSS6) -1. ilO] M. C. M.O’Brien. Proc. Phys. Sot. (London) 86 (1965) 847, Ill] M.D. Sturge, Solid Stat-ePhys. 20 (196’1) 91. 1121&Z. Loaguet-Kigeas, Advan. spectry. 2 (Ls6.2) [X3] W.&offftt and A. D. Lie&r. Pbya. Rev. LO6 (1956) 1195. [14J S.Be!l. J. Phys. B (Atom. &Tot. Phys.) 3 (1970) 745.
173
CHEMICALPHYSICS LETTERS
JAHN-TELLER THE
15 April 1971
COUPLtNC
A’ND
PHOTOELECTRONSPECTI?UMOF
CYCLOPRQPANE
C. G. ROWLAND Labmutoire de Citimie TMorique*, Cenh? d’Orsny, Universitk de Paris-Sud, 91-Orsay, Frawe Received 8 March 1971
Reparameterized CNDO calculations on the ground and ffrst excited states of the cycfopropaue caD”on sboowthat both states are subject to significant J&n-Teller distortions. First and second order vibroaic coupling parameters derived from the calcuIations are used to predict the shape of spectra which would be observed for electronic transitions to these degenerate states. Good agreement with the observed pbotaehctron spectrum is found. The inclusion of second order terms increases the complexity of the spectra but does not greatly alter the shape of the band cootours.
of e’ symmetry [2E The lower sheet of the JahnTeller potential surface is characterized by one point at which the ion has D2h symmetry, where the energy is a miuimum with respect to all tvtally symmetric coordinates, and two sets of three equivalent points at which the symmetry is Czv [3j, One of the sets of C2y points corresponds to the energy minima on the surface: the other set corresponds to the saddle points separating these equivalent minima A cycEopropane fvn of Czrr symmetry has seven independent coordinates. %I order to minimize the energy with respect to all of these with the expenditure of a moderate amount of computer time it ia necessary to use a semi-empirical method of ealcul~on The CNDO/2 method
In the photoelectron spectra vf highly symmetric, nonlinear molecules bands which correspond to the ionization of degenerate orbita& may show evidence of Jahn-Teller splitting. The first band in the photoeiectron spectrum of cyclopropane [1] corresponds to ionization of the highest occupied e’ (in plane) orbital and shows two maxima separated by 0.77 eV. However the second band, which may be associated with ionization of the eW(out of plane) orbit& shows little evidence vf splitting. fn this work reparameterized UNDOcalculations on the 23’ and 23” states of the cyclopropane catfon have been used to obtain parameters which govern the first and second order dependence of the electronic energy on the distortion coordinates. The vibronic energy levels and wavefunctions for the idedlized surfaces defined by these parameters have been calculated~andare used to obtain the spectra which would be observed for transitions to the Jahn-Telter
of Pople and Segti (41 gives reasonable values for equi~bri~ geometries, but heats of atomization, excitation energies and force constants are
generdly overestimated These defects have largely been corrected for in t&e reparameterized version of C!NDOfor hydrocarbons of Fischer and Kollmax [5] and this method has been used in the present work. Open shell c~cu~at~vns were peyote on the species derived from the 23’ state of the un-
distorted states,
2. SCF ~AL~LATIONS
distorted_ion- Tiie energies of the species derived from the 2E” state were obtained by the application of Koopmans’ theorem to the results of c!vsed shell calculations on Thaineutral mole-
. The 23’ and 233” states of the ~yc~o~rv~~e cati& . are anstable with respect to distortions
CUM; The.result$ of the energy minimization for the ?I& species are shown !n table 1 and for Qv species in tai$e 2.. The coordinates for the C&k--. rjipeqleg are explairied inn‘fig. 1.
* T&e Laboratoire de ChimieThQrfque is part of the -L+ratoire
. _, ,.
de Physico-Cbimie
des Rayonnements,
dependentoxit&eUnivsrsM dd Paris XI, asaocjated . witpthe C.NJ2.S. : . .
.
.’ ‘,
.. : ._
.;
.
. .
;
I
_.
. ..
.-.
-~_.
.,.., *
;
‘_,._
.‘” ‘,
:
.
_.
‘.
..:,.>‘. ._:.
.’
:
.'l@
-Y
._
;._.‘,
-. ,_
..,’ ,
,_ .‘_
_’ -’
..
-’ t,-.
>
.V+ti%
9. nuhber 2
CHEWCAL
PHYSICS LETTERS
15 April 1971
:
Table 1
_.
Geometriesandenergiesof D3h species
.
Neutral
Neutral calculated
expt. f6l
I
235
2E_
+&I
1.509
1.512
1.544
1.481
RCH @I
1.089
1.111
1.117
1.132
a t&s, Energy (au)
100.0
110.2 -_
- 21.3594
- 21.2544
--
-w
- 21.3553
- 21.2467
Energy of ion formed in vertical trknsitfon from calculated neutral
Geometries
Table 2 and energies of C&v species
Species derived from 2E1 saddle minimum point
Speciesderivedfrom 2E* saddle minimum point
1.676
1.437
1.512
1.475
1.492
1.604
1.472
1.503 1.110
1.118
1.114
1.168
1.117
1.119
1.116
113.2
112.4
1.14J
70.6
113.0
109.5
91.0
112.5
112.3
16.2
- 16.0
-
5.5
6.0
- 21.3745
- 21.2682
- 21.2631
10.5
9.5
8.7
5.5
iA1
2B2
2%
2A2
- 21.3762 relative to D3h minimum
113.9
113.6 _-
(kcalmole-l) Symmetry species
energies in b&h sets of calcuiations are in good agreement although the energy diiference between the minima and the as Iale points is more pronounced in the CNDOresutz. On bflizaiion
forma&ion of the 23” state there is a ht.rge reduction in the HCR angles. The Jabn-Teller distortion involves further change in these angles and
leadsj to mintni and saddle points differing in energy by 2.2 b.al mole-l.
3. VR3ROF-X INTERACTION
Fig. 1. Coordinates for C2v ~~clopropane.
,Sirtiffean_t
distm-tions
kre obs&ved
for botd
states, The zig’ state is stabilized most by ring deformation and by wagging of the CH2 groups. The geometries of the distorted species are in qualitative agre‘ement with those found by Baselbath in his MINDOlz calculations f?J. The &a-_ i,l?O, .,..
.“,.
:
-_ 1:
:
._
.. :
Moffitt and Thorson [g] and S.&&yet-Riggins et al. t2f have studied the quantum mechanical problem of the coupling.between the com~neats of a doubly degenerate electronic state of a non-
linear molecule induced,b$ a doubly degenerate, vibration. EIectronic transitip~~ from a-non_ . : _ . ,_ : : : I-
._-
VoluniS
9,
number
CHEMICALPHYSICS LETTERS
2
degenerate state to the Jahn-Teller distorted state are accompanied by an irregularly spaced progression in the doubly degenerate vibration. (Ifthere were no Jahn-Teller coupling the excitation of a non-totally symmetric vibration would be forbidden.) At large values of the first order coupling parameter a double humped structure appears in the spectrum, essentially because the probability distribution in the lowest vibrational level of the non-degenerate state is maximum at a non-zero value of the radial coordinate [lo, 111. The vibronic wavefunction for a doubly degenerate electronic state may be written: *(q, Q) = @e(Cl,8) h(Q) + +el(qr 8) h’(Q) * in numerical work on the Jahn-Teller and Renner
effects it is usual to neglect the dependence of the electronic functions Ge, I&~Ion the nuclear coordinates Q so that
*Iq,Q) = @$?)xJQ)
+ $~,k?)xJQ).
j=m+$h
(n, mlpe’i@(n+l,
m+l>
= (n+l, m+l Ip&jn,
* = &+x+(Q) + J/,x_(Q). Following Longuet-Higgins [12] first and second order dependence on Q is introduced into the hamiltonian. The matrix elements of H(Q) are expanded as a Taylor series in the nuclear coordinates up to terms of the second order. The terms in the totally symmetric coordinate may be eliminated by a suitable choice of origin and the vibrationai factors x, and x. are obtained by solution of the coupled equations
kpe-W + -&&#P
= I[1 x,
0.
+ $g$e-W
does not pass through its vertex. If g is non-zero the term in 3 @ introduces three equivaIent minima at cos3@ = 1 with energy -ak2/(1 -g) and saddle points at cos3@ = -1 of energy -fk2/(l+g). The vlbronic factors h and x_ may be expanded in terms of the two dimensional harmonic oscillator wavefunctions In, m). is the expectation vaIue of the enern=l,2,..., gy of the state [n, m) and the vibrational angular momentum quantum number m can take the valuesn-1, n-3, . . . . -rtcl. It is convenient also to introduce an electronic quantum number A, which is 1 for q+ and -1 for @_. Wheng = 0 the only levels which are mixed are those with the same value of and @+ and @_ can be expanded in terms of harmonic oscillator functions with m = j - $ and m = j + 2 respectively. The only non-zero matrix elements are then [9,13]
Taking complex forms of the wavefunctions
iYo - E
15 April 1971
Ho - E
x,
Ho is the unperturbed hamiltonian for the two dimensional harmonic oscillator and:
(rz, m Ipe-W In-l, m+l) = (n-l, m+l IpeW In, m> = En, m, where A n, m = ~(?z+m+l))l’2
Q, = pei@ , The associated
Q, = pe-i@; electronic
U, U’ = $A~ f (kp + &g2cos
potential is 3@),
where h is the force constant of the unperturbed vibration. A = 1 if the energy units are chosen as those of the zero point energy of the unperturbed vibration When g = 0 the electronic potential is a parabola revolved ab0ut.a line which
)
Bn, m = 6_@-m-l)}~/?
When g is different from zero, j is no longer a good quantum number and alI levels with the same value of 2jmod3 are mixed [13]. The relevant matrix elements can easily be calculated from expressions such a~: (12,m (p2e-2iQ ]nr, m’> =nmTmW(n, m Ipe+ and the only non-zero
Q, and Q_ are the complex conjugate forms of the real degenerate coordinates Qx, Q,. In polar coordinates :
m> = A,, m ;
InN,nrR> (ttrr,mm[pe+
!n’,m’!
values are
(n,mlp2e’2i+ ltw2,m+2) = (n+2,m+21p2e2i@ [qm) = 42, mh+l,
m+l ;
(n,m~p2e’2i0 ln,m+2) = (n ,m+21p2e2iG In,m> = An, mBn+l, m+i + %, m&-l,
m+l ;
(n,mlp2,-2id)(n-2, mc2) =~(n-2,mc21p2e2i~ 1n.m> =~B&mB+l
* WI.:,.
171
: -.
Volume 9. number 2.
CHEMICAL
PHYSICS
LETTERS
15 April 197l
Analogous matrix elements may be derived from
0.
the general expressiqns given by Bell [14]..
hi ~6.643
Knowing the matrix ekemdnts of HO, pe*l@ and $PWJ eq. (1) is easily converted into matrix form. The eigenvalues of this matrix are the vibronic energy levels for a particular value of 2jmod3 and the eigenvectors give the expansion coefficients of the harmonic oscillator wavefunc-
tions in bland x_.
The intensities of transitions from the lowest vibrational level of a degenerate state to the vibronie levels of the degenerate state are given, exactly if the vibration frequency is the same in the two states and to a good approximation otherwise, by the square of. the coefficient of 11,O> in the expansion of >G[9]. 11,0) is only present in the expansion for 2jmod3 = 1. Calculated spectra for transitions to the 2E’ state akshown in fig. 2. The value k2 = 6.643 was derived from the CNDO calculations assuming that the‘e’ ring deformation mode is excited on distortion and that in the undistorted ion this mode hasthe same normal frequency of 1050.6 -1 as in the neutral ground state [S]. CompaEEon of fig. 2a (g = 0.0532) and fig. 2b (g = 0) shows that the main effect of the presence of weak second order coupling is greatly to increase the number of allowed transitions in the spectrum. The band contour shows hvo maxima in both cases with a separation of 0.75 eV for g = 0.0532 and 0.55 eV for g = 0. There is, dowever, no guarantee that the normal coordinates and the vibration frequencies are the same in the ion and the neutral molecule and there is evidence of a 48 0 cm-l progression in the first band of the photoelectron spectrum [l], 0 = 480 cm-l corresponds to k2 = 14.54, but the order of the matrix which must be diagonalized in order to obtain a good representation of the spectrum rises very rapidly with k2 if g is non-zero. If g = 0 many fewer basis functions are needed for a given I22 and the matrix may be written in tridiagonal form. For k2 = 14.54, g = 0 the two maxima in the spectrum are separated by CL.37eV. If it is assumed that the 1478.1 cm-l CIi2 de-formation mode [6] is excited in transitions to the 2,” state the coupling parameters are k2 = 3.169, g = 0.2266. In the spectra calculated for .tiese parameters_ (fig. .3a) tici for g = 0 (fig. 3b) t.he,‘s&comJ maximum is reduced to a shoulder ‘.on the high energy side of the band. h-for the -‘i72
b.
-3940123456
the J&n-Teller
ted et vibrational structure for transiO> level of the neutral ground state to distorted 2El state of the positive ion
g=. -llLdLL 0.
k2 c 3.169 1133
-2
-1
0
1
2
3
c
5
b.
9=Q
-_lllL -2 -1
0
1
2
3
L
5
for transiFig. 3. CalctiI ted 6’ vibrational titrucke tfon from the tq,O> level of the neutral ground state to the Jabn-Teller distorted 2Ew &ate of thhepositive ion.
Volume
9,
number 2
CHEXEAL
23’ state the ~troduction of second order coupling does not greatly titer the shape of the band contour. If the vibration frequency were greatly reduced on ionization this shoulder would develop into a separate maximum, but its energy separation from the first maximum would be decreased. The results of the UNDO calculations
for the 23” state show a vertical-adiabatic transition energy difference of 4.8 kcaf for the D3h ~pectes.
Afairly
long
progression
in the totally
symmetric CH2 deformation mode should therefore be observed in the spectrum. In the semiclassical configuration coordinate model 1111 this would bave a width of 0.46 eV. The observed band in the photoekectron spectrum will be a fold of the a’, and e’ progressions and the width of the a’ progression may be expected to obscure any structure specifically a~ribut~le to the
presence
of J&m-Teller
coupling.
The calculations were carried out on the Univac PLO8at the Centre de Caleul of the Universite de Paris-Sud and on the IRM 360/75 at the Centre Inter-discipline Regional de Calcul Electronique of the C.N.R.S., Orsay. I would like to thank Mr. Xavier ~apuisat for the use of his reparameterized version of the CNDO/2 pro-
15 April 19n
PXYslCS LETTERS
gramme written by &gal and dfstributed by the QCPEi Professor Lionel Safem for his hospital-
ity at the Laboratoire de Cltimie ThPorique and the generous provision of compufer time and the Royal Satiety for the award of a European Exchange Programme Fellowship.
REFEZEUZNCES [I)
FLBasch,
M. B. Robin,
and D. W. Turner,
[2] G. Herzberg,
N.A.Kuebler,
f. Chem. Phys.
C, Baker St
(1969) 52.
Electronicspectraof polyatemic
Ecolecules fvan Nostrand, PrincetrJa, L966). f3I R. S. Da.@ andI. B. Leviuson, Opt. Spectry. Suppl. 3, Mol.Spectry. II (.lSSS) 1. (41 J.A.PopIe and G.A. &gal. 3. Chem.Phys. 44 (1966)
3289.
(1958)
141.
151H. Fischer and H. Kollmar. Theoret. Chim.Acta 13 (IQSS) 213. [6] S, J. Cyvin. Spectrochim.Acts 16 (rSS0) 1022. [?I E. Raeelbach. Cbam. Phys. Letters 7 (l970) 428. [8] W. Moffitt and WK. Thorson. Co% Iat. C.N.R.S. 82 19) H. 6. Lo~et-bins, U.&J&. M. K. L. Puce and %A.-ack, %oc. key. E&:A244 (lSS6) -1. ilO] M. C. M.O’Brien. Proc. Phys. Sot. (London) 86 (1965) 847, Ill] M.D. Sturge, Solid Stat-ePhys. 20 (196’1) 91. 1121&Z. Loaguet-Kigeas, Advan. spectry. 2 (Ls6.2) [X3] W.&offftt and A. D. Lie&r. Pbya. Rev. LO6 (1956) 1195. [14J S.Be!l. J. Phys. B (Atom. &Tot. Phys.) 3 (1970) 745.
173