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Vibronic coupling and symmetry breaking in core electron ionization
Chemical Physics 25 (1977) 189-196 0 North-Holland Publishing Company
VIBRONIC COUPLING AND SYMMETRY BREAKING IN CORE ELECTRON IONIZATION W. DOMCKE and L.S. CEDERBAUM Fakultiir jiir Physik der Universitiit Freiburg. D-78 Freiburg. Germany Received 13 April 1977
It is shown that for highly symmetric molecules the ionization of a core electron leads quiee generally to a lowering of the symmetry. The breaking of the symmetry is a consequence of the viironic coupling between nearly degenerate core orbitals of different symmetry. The viironic coupling leads to strong excitation of non-totally symmetric vibrational modes in addition to the usually observed excitation of totally symmetric modes.As an example,the viirational structure of the 01s line of the CO2 molecule is computed on the one-particle level.
The aspects of electron-vibrational interactions in polyatomic molecules have been extensively discussed in the literature [l-3] _The vibrational structure in electronic spectra such as photoelectron spectra [4] or optical absorption and emission spectra [l] is usually due to the excitation of totally symmetric modes. It has been shown that the observed vibrational intensity distribution can be well reproduced considering only the linear coupling between the electronic and vibrational motions [5,6]. For non-totally symmetric vibrational modes and non-degenerate electronic states linear coupling is symmetry-forbidden within the adiabatic approximation. The vibrational excitation due to quadratic, quartic or higher order coupling is usually found to be very weak [l, 51. Very strong excitation of non-totally symmetric modes is observed, however, when the electronic transition is accompanied by a Lowering of the symmetry of the molecule [l]. In this paper we consider inner-shell ionization processes. The excitation of totally symmetric modes due to core electron ionization of small molecules has been experimentally observed by Cellus et al. [7]. In refs. [8,9] it has been shown that the observed band shapes can be quantitatively reproduced with ab initio methods. Relaxation effects have been found to influence con$derably the eledtron-vibrational coupling for core electrons [8]- Here we want to point out that the coupling to non-totally symmetric vibrational modes accompanied by a lowering of the symmetry is a com-
mon pLnomenon in inner-shell ionization processes of highly symmetric molecules. The general hamiltonian describing the electronic and vibrational motions and the interaction between them has been derived in refs. [5,6]. To simplify the discussion we restrict ourselves to the Iiartree-Fock one-particle approximation for the electronic motion and the harmonic approximation for the vibrational motion in fhe electronic ground state. The hamiltonian then reads [6]
+
C ei(Q)bf(Q)ai(Ql i
-"iI
,
(1)
where the b, and bf are annihilation and creak:: operators for vibrational quanta of the sth normal mode. They are related to the normal coordinate * Qs according to b, = 2-‘12(Qs + WQ,)
,
b,’= 2-1/2(~~ - a/aQ,) .
(2)
The ws are the corresponding hanonic vibrational frequencies. Q stands for the set of Mnormal coordi* See ref. [lo]. The Qs used here are dimensionless normal coordinates, obtained by multiplying the normal coordinates as defined by Wilson et al. by cfi-’ ws)‘/y_
i9d
.-._
iv. Domcke, L.S. CederbaumfCore electron ionization
n&s of the molecule. The ai(Q) and a:(Q) are annihilatiqn.and creation opera& for electrons in the &&&ticle orbitals k&j)) with orbital energies ci(Q)_. nidenotes the occupation number of the orbital I~i) in the electronic ground state, i.e. ni = l(0) for orbitals occupied (unoccupied) in the electronic ground state. E. is the electronic ground state energy at the equilibrium geometry. The electronic-vibrational coupling is a consequence of the dependence of Ei and $Q~ on the vibrational coordinates Q,. The single-particle energies ej may be arbitrary functions of Q. Therefore we do not restrict
ourselves to the harmonic approximation as far as the ionic states are concerned. The ei(Q) are obtained by diagonalizing the electronic one-particle hamiltonian at every nuclear configuration Q. This prescription implies that the one-particle wavefunctions ~piand the electronic creation and annihilation operators a:, ai are also functions of $. The %-dependence of the latter is given by [6]
In the adiabatic approximation [I l] the dependence of aj, a: on QS is not explicitly considered [6]. Expanding ei(Q) up to first order in QS we obtain in aJle adiabatic approximation
where HeI denotes-the hamiltonian of the molecule. (in the oneiparticle approximation) excluding the nuclear kinetic energy and the internuclear repulsion, we fmd
a
(%@)l
aQ,
I P(Q)) = I
(IpktQ)ia~el/aQ,l~io> E~‘(Q)-~~(Q)
’
(7)
Combining eqs. (5) and (7) we see that the derivative of the electronic operators aj, U: with respect to $ may become large or even singular if the two orbital energies are close to each other or degenerate- A wellknown example where this happens is the Jahn-Teller effect [12,13]. In this case one can fmd a linear transformation of the electronic orbit& which eliminates the singular derivatives of the electronic operators at the expense of introducing non-diagonal linear electronic-vibrational coupling terms into the bamiltonian 16,131. We are now in a position to discuss vibronic coupling effects in core electron ionization spectra. Let us consider the most simple representative of a polyatomic molecule, namely a linear symmetric triatomic molecule such as CO, belonging to the b,, symmetry group. The extension of the treatment to any other symmetry group is straightforward. In particular, we are interested in the ionization from the 10s and lu, orbitals composed of the 01s atomic core orbitals. From eq. (7) it follows that the matrix element <~P,U$!$3Q,]~,.,u>, where Q, denotes the antisymmetric stretching coordinate, may indeed become large due to the very near degeneracy of the lug and luu core orbitals. Nondiagonal matrix elements of a/3$; on the other hand, where $ denotes the symmetric stretching coordinate, are of no importance, since @las(0)l(aHe~/aQg)o
For non-totally symmetric modes the derivative &/aQ,)O is identically zero. Within the adiabatic approximation there is thus no linear coupling to nontotally symmetric vibrational modes. To include non-adiabatic effects we have to consider the dependence of aFaf on Qs. It follows from eq. (3)
Using the fact that the [IFi
* It may seem,at the fii
ghxe,
that a
fiite order Taylor
expansion of the electronic,operators is completkly meaningless, since, according to eqs. (5) and (7), the coefficient of the fust order terra is very large or even divergent. However,
the quantity, which is actually expand+, is the hamiltonian H(Q), and the fust order expamion coefficients are of the form [~j(o) - Ek(o)](lPk(o)la/aQsl~~))o and thus fiite. I It can be shown that this expansion is completely equivalent to performing a linear transformation from the ag(Q).au(Q) to the ag(0), au(O) as ia the Jaha~T~ller case mentioned
obey the Scbrodinger
equation
H,l(Q)lipi(e)>=~i(Q)l~i(Q)>,
Irpl,,u(0)) = 0 and all other molecular orbitah are energetically well separated from the lug, 1uu core orbitals. Expanding the orbital energies up to first order in Qa and the electronic creation and annihilation operators up to first order in Qu *we obtain
(6)
above.
W. Domcke; L.S. CederbaumlCoreelectron ionization
I11 I
\
+h(Q;Q,+Q;Qg)(b, + b;) ,
(8)
with Kg = -
2-112(&a/aQ,>,
Ku-= -
2-112(&,/aQ$o,
x = 242
,
@a)
c~~(o)l(aH,,/aQ~,,),l~“(o)) -
@b)
The electronic operators Q, a+ in the hamiltonian (8) are those at the equilibrium geometry and thus independent of Q. They therefore commute with the boson operators b, b+. From now on, only such Q-independent electronic operators are considered. The hamiltonian (8) represents a typical vibronic coupling problem. Model problems related to the above hamiltonian have been considered, for example, by Fulton and Gouterman [14] and by Gregory et al. [I 51. A considerable simplification occurs in the present core ionization p:oblem due to the fact that ea (0) = eu (0) = e(0) , fcg=/L,=/c,
(IO)
to a very good approximation. The splitting of the 1~~ and la, core orbital energies of CO,, for example, is 0.0015 eV [16]. Introducing new electronic creation and annihilation operators according to Qg =
2-lj2 (9 + Qz) ,
Q, = 2-1!2(Q1 the
- Q2),
(11)
hamiltonian (8) takes the form
H= w&b, +
++) f Ou(b;b,
++) + ‘@)(Q;Ql
+QJjQ2)
K (Q~Q; + a2Q$) (bg + bi)
+ X(f+zl
- Q&2)
(b,
+ bi).
(12)
diagonal electron-vibrational coupling terms appear in the transformed hamiltonian. This is a consequence of relation (10) and not true in the general case represented by the hamiltonian (8). Only
\I \
:\
191
b)
Fig. 1. Potential curves of the ground and the ionic state in the antisymnetric stretching direction. (a) The potential energy curves resulting when the hamiltonian (8) is diagonalized in the electronic coordinates and the energy split eg(0) - ~~(0) is neglected. There are two separate anharmonic ionic curves which are degenerate at Qu = 0. (b) The potential energy curves corresponding to the hamiltonian (12). The two ionic state curves do not possess the symmetry of the initial state.
It is instructive to consider the ionic state electronic potential energy surfaces associated with the hamiltonians (8) and (I 2). Fig. 1 shows the potential energy curves of the ground state and the ionic state along the antisymmetric stretching coordinate Qu. With the hamiltonian (8), as it stands, we cannot associate an adiabatic ionic state potential energy surface, since it contains a coupling term which is non-diagonal in the electronic coordinates. However, it is possible to diagonalize the hamiltonian (8) in the electronic coordinates. Then the potential surfaces depicted in fig. la are obtained. These are two separate adiabatic potential energy surfaces as indicated by the full and the broken line. Actually the two surfaces are split by = 10U3 eV at Qu = 0 and possess a vanishing derivative at Qu = 0 instead of a tip. The splitting is, however, more than hundred times smaller than the vibrational frequency wu and has a negligible influence on the spectrum. Due to the smallness of the splitting the lower ionic state potential energy curve has doubleminimum character even for very smah values of the coupling constant h. Obviously both surfaces are symmetric with respect to the plane Q, = 0 and possess, therefore, the full symmetry of the initial state. One has to keep in mind that the electronic wavefunctions corresponding to these two ionic state potential energy
- W.Domcke.L.S. CederbaumfCore electronionization
132
surfaces are strongly non-adiabatically coupled near Q, = 0, leading to a complete break-down of the Franck-Condon principle [ 14,151. Fig. 1b shows the potential energy surfaces to be associated with the transformed hamiltonian (12). There are two separate potential energy surfaces which are degenerate at Qu = 0, but none of these possesses the initial symmetry of the molecule. Through the transformstionto the non-symmetry-adapted electronic operators aI, a2 we have been able to generate a harndltonian which explicitly describes the breaking of the initial state molecular symmetry. Starting from the hamiltonian (12) the photoelectron spectrum can be calculated exactly. The transition probability per unit time and unit energy at energy w is given by [5] P(w) = $ dt eiwr W,,I~T(t)j+n), __ where r(r) = eZ’T e-at ator
(13)
and T is the transition oper-
T= Crioi,
(14)
i
P(W) =21r~2~dtei[W-~(o)~r~0,0~e~~0,0),
T = Tgag + ?,a, .
(15)
Changing according to eq. (11) to the transformed or: bital basis we have +rza2,
IO, 0) denotes the vibrational ground state of the molecule, i.e. b,jO,O) = 0 for s = g, u. In eq. (17) the calculation of the spectrum P(o) has been reduced toa pure boson problem. The interesting aspect is that H, which describes the vibrational motion in the ionic state, is the hamiltonian for a shifted harmonic oscihator with shifts both in the symmetric stretching as well as in the antisymmetric stretching direction. The shift in the antisymmetric coordinate is a consequence of the vibronic interaction of the lo,, and lus core orbitals. The remaining steps in the calcula$on ofP(o) are very simple. It is easy to show that His diagonalized by the canonical transformation U= expL--(a/w&s
- 6:) - 0&J@,
P(o)=21212~dre
twrW&+t
(t)&J
,
(16)
where we have used the fact that IT, 12 = I?#
= JA
and
OPOla~al(f)l Q, = (9, IaTa2(t)I*,> = 0 .
Us@ the hamiltonian (12), the adiabatic approximation [ 1 1] for I eo> and the fact that the electronic
- bZ)l , (1%
Inserting now the complete set of vibrational states 15, r.2,) into eq. (17) we arrive at the fmal result exp [--uz P(w) = 21~1~ C % “g
lfu)]
~~ ng! n,! (21)
x6bJ-
and
(17)
with
i.e.
ri being the matrix element for photoionization of the orbital i. lUO>denotes the ground state wavefunction of the molecule. In the present core electron ionization problem
T=rIaI
operators commute tith the boson operators, we obtam
40) +fpg
ffuwu
-
ngwg_- nuwu] ,
with
fg = (hJg)2.fu
=
WJ2 -
(22)
The vibrational intensity distribution is thus given by the product of two Poisson distributions. The “coupling parameters” feand fudetermine the strength of the excitation of the symmetric and the antisymmetric stretching mode, respectively. So far we have neglected, without further justification, the bending mode of the linear triatomic molecule. Since the bending mode is of n symmetry, it can induce in first order vibroqic coupling between a u and
193
W. Domcke, L.S. CederbaumfCore el&tron ionization
_a n orbital, but not between two CJor two ‘IIorbitals. The second order coupling is of the two components of a n orbital is known as the Renner-Teller effect [13]. Since there are no n orbitals in the core region, it is clear that in the one-particle approximation discussed here there can be no vibronic coupling involving the bending mode and thus no breaking of the linearity of a molecule. A breaking of the linear symmetry might, in principle, occur due to the change of correlation energy upon core ionization. For a quantitative’application of the above consideration we have to compute the coupling constants defined in eq. (9). For this purpose SCF calculations on CO, have been performed [16] employing a basis set of 9s-type and Sp-type Cartesian gaussian functions, contracted to 4%type and Zp-type functions. The coupling constant for the symmetric stretching vibration is, according to eq. (9a), simply given by the derivative of the 01s orbital energy with respect to Qa. It would be rather difficult to determine h from eq. (9b). Instead we consider the transformed hamiltonian (12), from which it is seen &at h governs the linear splitting of the 01s orbital energies with the antisymmetric coordinate Q,. More precisely, A = 2-‘/2(&,/aQu),-,
= - 2-‘/2(&2/aQu)I,,
(23)
where EI and e2 denote the orbital energies corresponding to the transformed orbitals 1 and 2. Eq. (23) reveals the close analogy between the coupling constant X and the coupling constant K for the totally symmetric vibration given by eq. (9a). Note that the sign of h is of no signjficance, while the sign of K determines whether the orbital in question is bonding or antibonding with respect to the totally symmetric vibration. We point out that we do not make use of the so-called ‘%rude adiabatic approximation” in the ab initio determination of the coupling constants K and h. It has been argued that this approximation gives poor results when used for the calculation of vibronic coupling matrix elements [26]. It suffices to perform, in addition to the SCF calculation at the ground state equilibrium geometry, a SCF calculation for a slightly increased (or decreased) C-O distance as well as a SCF calculation at a slightly nonsymmetric nuclear configuration in order to obtain the derivatives of the orbital energies and thus K and h. The result is K = 0.098 eV ,
X = 0.288 eV,
(24)
leading to coupling parameters
f,=O.32, f,=O.98.
(25)
The calculation thus predicts a weak excitation of the symmetric stretching mode and a fairly strong excitation of the antisymmetric stretching mode. Since K is positive, the 01s orbitals are antibonding with respect to the symmetric stretching vibration, which means that the CO distance decreases upon 01s ionization. The strong excitation of the antisymmetric stretching vibration indicates that the two C-O bond lengths differ considerably in the core-ionized species. The geometry shift in the antisymmetric stretching direction is given by AS, = (2/P,P@/W”) where
>
(26)
s, = 2-1/2(rr - r3_)! and I/fi=l]mO
+2/m,.
rl and r2 denote the two C-O distances. p is the re-
duced mass corresponding to the antisymmetric stretching vibration. With X from eq. (24) it is found that Arl =-Ar2=o.12A.
(27)
The above determination of the coupling constants serves merely as an illustration and should not be considered as an accurate ab initio calculation of’the core ionization spectrum. It is well known that the Hartme-Fock one-particle approximation (Koopmans’ approximation ]17]), on which the hamiltonian (1) is based, is not appropriate for determining the ionization potentials of core electrons due to the pronounced reorganization of the valence shell upon core electron ionization. Indeed it has been shown in ref. [8] for CO and for the totally symmetric vibration of CH4 that reorganization effects influence considerably the vibrational coupling constants of core electrons. The reorganization effects can be simply taken into account by a “renormalization” of the vibrational coupling constants [6,8] _The renormalized coupling constant K is obtained by replacing -e(QJ in eq. (9a) by the true ionization potential IE’(Qe). IP(Qa) may be approximately obtained, for example, by a Green function calculation [6,8] or, especially in the core region, by a ASCF calculation. From eq. (23) it is clear that A can be renormalized in the same way. Whilst this work W~.Salready written up, we became
194
W. Domcke. LA. Cederbaum/Core electron ionization
aware of direct SCF hole state calculations on the core and valence ionized states of CO, performed by Clark and Mii!ler [18]. By calculating a potential energy surface for the 01s hole state they found that CO2 is indeed non-symmetric in this state. Clark and Miiller alSOcalculated the vibrational structure of the 01s line. They found a very weak coupling to the symmetric stretching mode and a rather strong coupling to the antisymmetric stretching mode [18]. In contrast to the coupling parameters (24) the results of Clark and Miiller include reorganization effects. From a comparison of their data with the present ones it is seen that reorganization effects act to reduce the coupling for the symmetric stretching mode, in analogy to the results found for the Cls orbitals in CH4 and CO previously [8]. With respect to the excitation of the antisymmetric stretching vibration, on the other hand, the present results are in close agreement with those obtained by CIark and Miiller. In table 1 the FranckCondon factors for the transitions (0,O) + (n,,O) obtained from eqs. (21,25) are compared with the Franck-Condon factors reported by Clark and Miiller. The agreement seems to indicate that reorganization effects do not significantly influence the coupling constant A. The above discussed effect of symmetry breaking due to viironic coupling of core orbitals bears some relationship to the well-known problem of the “localization” of a core hole [19,20] _When there are several equivalent atomic sites in a molecule (such as the two oxygen nuciei in CO,), the question arises whether the core hole, which remains after the ionization process, should be thought of as localized at one of the centers or as delocalized over the equivalent centers. Direct SCF hole state calculations have shown that a Table 1 Franck-Condon factorsfor the excitation of the antisymmetric stretchingvibration in the 01s band of CO2as obtained in the present study (within the one-particle approximation) and by Clark and MiiUer(from direct hole state SCF calculations) nu
Thiswork
0 1 2 3 4
0.377 0.368 0.180 0.058 0.014
Clark and Mtiller[18]
0.390 0.365 0.175 ,0.055 0.015
considerable lowering of the total ionic SCF energy is obtained when the core hole is chosen to be localized [ 18,20-221. Therefore, core holes are generally considered as being localized. It has been shown recently [23] that the tendency of the core hole to localize can be understood as a correlation effect. When delocalized (symmetry-adapted) orbitals are used, correlation effects are of the same order of magnitude as reorganization effects and therefore ASCF calculations, which do not account for correlation effects, give poor.results. When localized (non-symmetry-adapted) orbitah are used, correlation effects are much smaller than correlation effects, explaining the good results obtained with ASCF calculations employing localized orbitah. An analogous behavior is found in the vibronic coupling problem. The hamiltonian (8), which is formulated in the summetry-adapted (delocalized) orbital basis, contains a coupling term which is non-diagonal in the electronic operators. Therefore one cannot associate an adiabatic electronic potential energy surface with this hamiltonian. Changing to a localized (nonsymmetry-adapted) orbital basis by the transformation (1 l), the hamiltonian (12) is obtained, which now contams only diagonal electron-vibrational coupling terms (as long as eq. (10) is fulfilled). Therefore, a potential energy surface can be associated with the transformed hamiltonian. This surface reflects directly the lowering of the molecular symmetry (see fig. lb). Vibronic coupling effects can thus be eliminated in the same way as correlation effects by the transformation to a non-symmetry-adapted orbital basis. It should be kept in mind that the “localization” of the core hole due to correlation effects does nor lower the symmetry of the electronic wavefunction. The total ionic wavefunction has still the full symmetry of the point group [20,23]. The argument [18] that due to the localized character of the core hole the symmetry of CO3, for example, is reduced from Dmh to C,, and that therefore linear coupling to the antisymmetric stretching vibration is symmetry-allowed, is, strictly speaking, incorrect. To understand the excitation of the antisymmetric stretching vibration and the lowering of the symmetry of the molecule one has to. consider the vlbronic coupling of the degenerate 1 IJ,, and lug orbitals as outlined above *. It is noteworthy that this vibronic coupling mechanism is already pres*For footnote see next page.
W. Domcke. L.S. Cec?erbaum/Core electron ionization
ent on the one-particle level, whereas the localization of the core hole is a pure many-body effect. The vibronic coupling problem represented by the hamiltonian (8) could only be solved exactly because the splitting of the u and g orbital energies at Q,, = 0 tis neglected. In the more general case that e,(O) # err(0) (and possibly ~g # K") we have to deal with a very complicated dynamical problem. It is no longer possible to separate the electronic and the vibrational motion in the ionic state and the Franck-Condon principle is inapplicable. The 20~ and 3ug deep valence orbitals of CO, represent an example where this situation occurs. In conclusion, we would like to stress that the vibronic coupling mechanism discussed in an exemplary manner for CO, above is a quite general phenomenon in inner-shell ionization of symmetric molecules. The mechanism is, of course, also if importance in X-ray absorption processes involving core orbitals, which have recently been discussed by Gel’mukhanov et al_ [24]. The benzene molecule may serve as an example to illustrate the very complex dynamical processes occurring upon core electron ionization- The six Cls electrons of benzene populate the four very nearly degenerate orbitals alg, b,,, elur e2a [25]. From the above discussion it is clear that in the Cls-ionized state the alg and bIu orbitals are strongly non-adiabatically coupled by the two bI, modes. The eIu and e2g orbitals experience a Jahn-Teller splitting by the four e2z modes- As is well-known 1131, the Jahn-Teller lnteraction involves both diagonal and non-diagonal cou* Sincethe transition operator descriiing the interaction of the molecule with the photon field is independent of the vibrational coordinates, the ion is created in a vibrational state which has the samesymmetry as the initial vibrational
state. (It has been shownabove,by employingthe transformation (11). that the electronicand vibrationalmotions in the ionic state can be separated in the same way as in the nondegenerate initial state. Therefore it is possiile to speak in terms of ionic state viirational wavefunctions.) It is clear that each viirational level in the ionic state is doubly degenerate, having associated with it a vibrational wavefunction of g as well as of u symmetry_ According to the FranckCondon principle, only the vibrational states of g symmetry can be reached from the initial state vibrational ground state. Nevertheless, the molecule is non-symmetric in the ionic state. The symmetry of the ionic state vibrational wavefunction merely represents the fact that we do not know which of the two equivalent C-O distances becomes shorter or longer, respectively.
195
plings within the degenerate orbital The alg and b 1n orbitals interact with the elu and ezg orbitals via the vibrations ofelu and e2z symmetry and the el, and e2g orbitah interact with one another via the b,,, b2, and elu vibrations. The two totally symmetric vibrations can, of oourse, couple in the usual adiabatic manner to each of the four core orbitals. Thus twenty of the thirty normal modes of benzene [lo] may be excited, in principle, upon ionization of a Cls electron_ These twenty modes comprise aU in-plane normal modes with the exception of the one of a2g symmetry. There is no vibronic coupiing involving the nine outof-plane vibrational modes, since no n orbitals are present in the core region. Considering the results obtained for CO, above, it is not expected that aU of the symmetry-allowed vibrational couplings are small. Due to the very near degeneracy of the core levels a breaking of the symmstry occurs even for very small values of the non-diagonal coupling constants. The Cls line of benzene should thus exhibit a very complex vibrational structure. ‘The symmetry of benzene is reduced from Dsh to C2v upon core ionization.
References [l] G. Her&erg, Molecular spectra and molecular structure, III. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand, New York, 1966). [2] T.E. Sharp and H.M. Rosenstock, J. Chem. Phys. 41 (1964) 3453. [3] M.D. Frank-Kamenetskii and A. Lukashii, Soviet Phys. Usp. 18 (1976) 391. [4] D.W. Turner, C. Baker, A.D. Baker and C.R. Bundle, Molecuhr photoelectron spectroscopy (Wiley, New York, 1970). [S] L.S. Cederbaum and W. Domcke, J. Chem. Phys. 64 (1976) 603,612. [6] L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. [7] U. Gelius, S. Svensson, H. Siegbahn. E. Basilier, A. Fa_xBlv and K. Siegbahn, Chem. Phys. Letters 28 (1974) 1. [8] W. Domcke and L.S. Cederbaum, Chem. Phys. Letters 31 (1975) 582. [9] D.T. Clark and J. Miiller, Theoret. Chim. Acta 41 (1976) 193. [lo] E.B. Wilson, J-C. Decius and P.C. Cross, Molecular vihrations (McGraw-Hill,New York, 1955). [ 111M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford Univ. Press, London, 1954) Appendix VII. [12] H.A. Jahn and E. Teller, Proc. Roy. Sot. A161 (1937) 220.
196.
W.~Ij&k& L.S. C&baum/Cor.6 ele&ph iotiizatihi~
[ 131 H.C.-Longuet-Hi&.$ Adv& Spetry; 2 (1961) 429; -R. EngJmau, The Jahn-Teller effect Oyiley, New York, 1972). .~. .‘.. [lb] R-L. Fuhon, J.Che& Phys. 56 (1972) 1210; M. Gouterr&, J. Cberr&Phys. 42 (iyf35j351, andref; erences therein; [ 151 A.R. Gregory, W.H. Henneker, W. Siebrand and M.Z. Zgierski;J. Chein. Phys. 65 (1976) 2071, and references therein. -. [la] W. von Nielsen, private communication. [17] T. Koopmans, Physica l(1933) 104. [18] D.T. Clarkand J. MiiUer,Chem. Phys. 23 (1977) 429. [19] L.C. Snyder, J. Chem. Phys. 55 (1971) 95. [20] P.S. Bagus% and H.F. Schaefer IN, J. Chem. Phys. 56 (1972) 224. [21] L.A. Curtiss and P.W. Deutsch, J. Electr. Spectry. 10 (1977) 193.
:..._ ; : ‘_; .., ‘--:
..-.
___
_(1977)5B*4_.. :..::: :-;;- ..-; -: .,. --..;. ,... -1.. [24] F K:-&&n&anov- L N. Ma&Jov AV Niiolaev ..’ A\-_ J&&atenko, ;_;;* Sm*; $J_~W&&nd”~ .: A.P..Sado&kii, DokJ. Akad. NaukSSSR 225 (i975) 597; F.K~~Gel’mukhanov,L.N;J;Mriz&ov’andA;V.~K&dratenko, ChenxPhy& Letters 46 (1977) 133.~ ---:-1 .. [25] WC. E&ler andC.W. K&n,J. Chem: Phya 58 (1973) 3458; :. W. vorr Niessen, L.S. Cederbaum and W.P. Kraemer; J. Chem. Phys. 65 (1976) 1378 :. [26] W.C. Johnson Jr. and 0-E;Weigang Jr.; J. Chem. Phys. 63 (1975) 2135; 0. Atabel?, A. Hard&on &rd R. Lef;bwe, Chem. Phys. Letters 20 (1973) 40.
VIBRONIC COUPLING AND SYMMETRY BREAKING IN CORE ELECTRON IONIZATION W. DOMCKE and L.S. CEDERBAUM Fakultiir jiir Physik der Universitiit Freiburg. D-78 Freiburg. Germany Received 13 April 1977
It is shown that for highly symmetric molecules the ionization of a core electron leads quiee generally to a lowering of the symmetry. The breaking of the symmetry is a consequence of the viironic coupling between nearly degenerate core orbitals of different symmetry. The viironic coupling leads to strong excitation of non-totally symmetric vibrational modes in addition to the usually observed excitation of totally symmetric modes.As an example,the viirational structure of the 01s line of the CO2 molecule is computed on the one-particle level.
The aspects of electron-vibrational interactions in polyatomic molecules have been extensively discussed in the literature [l-3] _The vibrational structure in electronic spectra such as photoelectron spectra [4] or optical absorption and emission spectra [l] is usually due to the excitation of totally symmetric modes. It has been shown that the observed vibrational intensity distribution can be well reproduced considering only the linear coupling between the electronic and vibrational motions [5,6]. For non-totally symmetric vibrational modes and non-degenerate electronic states linear coupling is symmetry-forbidden within the adiabatic approximation. The vibrational excitation due to quadratic, quartic or higher order coupling is usually found to be very weak [l, 51. Very strong excitation of non-totally symmetric modes is observed, however, when the electronic transition is accompanied by a Lowering of the symmetry of the molecule [l]. In this paper we consider inner-shell ionization processes. The excitation of totally symmetric modes due to core electron ionization of small molecules has been experimentally observed by Cellus et al. [7]. In refs. [8,9] it has been shown that the observed band shapes can be quantitatively reproduced with ab initio methods. Relaxation effects have been found to influence con$derably the eledtron-vibrational coupling for core electrons [8]- Here we want to point out that the coupling to non-totally symmetric vibrational modes accompanied by a lowering of the symmetry is a com-
mon pLnomenon in inner-shell ionization processes of highly symmetric molecules. The general hamiltonian describing the electronic and vibrational motions and the interaction between them has been derived in refs. [5,6]. To simplify the discussion we restrict ourselves to the Iiartree-Fock one-particle approximation for the electronic motion and the harmonic approximation for the vibrational motion in fhe electronic ground state. The hamiltonian then reads [6]
+
C ei(Q)bf(Q)ai(Ql i
-"iI
,
(1)
where the b, and bf are annihilation and creak:: operators for vibrational quanta of the sth normal mode. They are related to the normal coordinate * Qs according to b, = 2-‘12(Qs + WQ,)
,
b,’= 2-1/2(~~ - a/aQ,) .
(2)
The ws are the corresponding hanonic vibrational frequencies. Q stands for the set of Mnormal coordi* See ref. [lo]. The Qs used here are dimensionless normal coordinates, obtained by multiplying the normal coordinates as defined by Wilson et al. by cfi-’ ws)‘/y_
i9d
.-._
iv. Domcke, L.S. CederbaumfCore electron ionization
n&s of the molecule. The ai(Q) and a:(Q) are annihilatiqn.and creation opera& for electrons in the &&&ticle orbitals k&j)) with orbital energies ci(Q)_. nidenotes the occupation number of the orbital I~i) in the electronic ground state, i.e. ni = l(0) for orbitals occupied (unoccupied) in the electronic ground state. E. is the electronic ground state energy at the equilibrium geometry. The electronic-vibrational coupling is a consequence of the dependence of Ei and $Q~ on the vibrational coordinates Q,. The single-particle energies ej may be arbitrary functions of Q. Therefore we do not restrict
ourselves to the harmonic approximation as far as the ionic states are concerned. The ei(Q) are obtained by diagonalizing the electronic one-particle hamiltonian at every nuclear configuration Q. This prescription implies that the one-particle wavefunctions ~piand the electronic creation and annihilation operators a:, ai are also functions of $. The %-dependence of the latter is given by [6]
In the adiabatic approximation [I l] the dependence of aj, a: on QS is not explicitly considered [6]. Expanding ei(Q) up to first order in QS we obtain in aJle adiabatic approximation
where HeI denotes-the hamiltonian of the molecule. (in the oneiparticle approximation) excluding the nuclear kinetic energy and the internuclear repulsion, we fmd
a
(%@)l
aQ,
I P(Q)) = I
(IpktQ)ia~el/aQ,l~io> E~‘(Q)-~~(Q)
’
(7)
Combining eqs. (5) and (7) we see that the derivative of the electronic operators aj, U: with respect to $ may become large or even singular if the two orbital energies are close to each other or degenerate- A wellknown example where this happens is the Jahn-Teller effect [12,13]. In this case one can fmd a linear transformation of the electronic orbit& which eliminates the singular derivatives of the electronic operators at the expense of introducing non-diagonal linear electronic-vibrational coupling terms into the bamiltonian 16,131. We are now in a position to discuss vibronic coupling effects in core electron ionization spectra. Let us consider the most simple representative of a polyatomic molecule, namely a linear symmetric triatomic molecule such as CO, belonging to the b,, symmetry group. The extension of the treatment to any other symmetry group is straightforward. In particular, we are interested in the ionization from the 10s and lu, orbitals composed of the 01s atomic core orbitals. From eq. (7) it follows that the matrix element <~P,U$!$3Q,]~,.,u>, where Q, denotes the antisymmetric stretching coordinate, may indeed become large due to the very near degeneracy of the lug and luu core orbitals. Nondiagonal matrix elements of a/3$; on the other hand, where $ denotes the symmetric stretching coordinate, are of no importance, since @las(0)l(aHe~/aQg)o
For non-totally symmetric modes the derivative &/aQ,)O is identically zero. Within the adiabatic approximation there is thus no linear coupling to nontotally symmetric vibrational modes. To include non-adiabatic effects we have to consider the dependence of aFaf on Qs. It follows from eq. (3)
Using the fact that the [IFi
* It may seem,at the fii
ghxe,
that a
fiite order Taylor
expansion of the electronic,operators is completkly meaningless, since, according to eqs. (5) and (7), the coefficient of the fust order terra is very large or even divergent. However,
the quantity, which is actually expand+, is the hamiltonian H(Q), and the fust order expamion coefficients are of the form [~j(o) - Ek(o)](lPk(o)la/aQsl~~))o and thus fiite. I It can be shown that this expansion is completely equivalent to performing a linear transformation from the ag(Q).au(Q) to the ag(0), au(O) as ia the Jaha~T~ller case mentioned
obey the Scbrodinger
equation
H,l(Q)lipi(e)>=~i(Q)l~i(Q)>,
Irpl,,u(0)) = 0 and all other molecular orbitah are energetically well separated from the lug, 1uu core orbitals. Expanding the orbital energies up to first order in Qa and the electronic creation and annihilation operators up to first order in Qu *we obtain
(6)
above.
W. Domcke; L.S. CederbaumlCoreelectron ionization
I11 I
\
+h(Q;Q,+Q;Qg)(b, + b;) ,
(8)
with Kg = -
2-112(&a/aQ,>,
Ku-= -
2-112(&,/aQ$o,
x = 242
,
@a)
c~~(o)l(aH,,/aQ~,,),l~“(o)) -
@b)
The electronic operators Q, a+ in the hamiltonian (8) are those at the equilibrium geometry and thus independent of Q. They therefore commute with the boson operators b, b+. From now on, only such Q-independent electronic operators are considered. The hamiltonian (8) represents a typical vibronic coupling problem. Model problems related to the above hamiltonian have been considered, for example, by Fulton and Gouterman [14] and by Gregory et al. [I 51. A considerable simplification occurs in the present core ionization p:oblem due to the fact that ea (0) = eu (0) = e(0) , fcg=/L,=/c,
(IO)
to a very good approximation. The splitting of the 1~~ and la, core orbital energies of CO,, for example, is 0.0015 eV [16]. Introducing new electronic creation and annihilation operators according to Qg =
2-lj2 (9 + Qz) ,
Q, = 2-1!2(Q1 the
- Q2),
(11)
hamiltonian (8) takes the form
H= w&b, +
++) f Ou(b;b,
++) + ‘@)(Q;Ql
+QJjQ2)
K (Q~Q; + a2Q$) (bg + bi)
+ X(f+zl
- Q&2)
(b,
+ bi).
(12)
diagonal electron-vibrational coupling terms appear in the transformed hamiltonian. This is a consequence of relation (10) and not true in the general case represented by the hamiltonian (8). Only
\I \
:\
191
b)
Fig. 1. Potential curves of the ground and the ionic state in the antisymnetric stretching direction. (a) The potential energy curves resulting when the hamiltonian (8) is diagonalized in the electronic coordinates and the energy split eg(0) - ~~(0) is neglected. There are two separate anharmonic ionic curves which are degenerate at Qu = 0. (b) The potential energy curves corresponding to the hamiltonian (12). The two ionic state curves do not possess the symmetry of the initial state.
It is instructive to consider the ionic state electronic potential energy surfaces associated with the hamiltonians (8) and (I 2). Fig. 1 shows the potential energy curves of the ground state and the ionic state along the antisymmetric stretching coordinate Qu. With the hamiltonian (8), as it stands, we cannot associate an adiabatic ionic state potential energy surface, since it contains a coupling term which is non-diagonal in the electronic coordinates. However, it is possible to diagonalize the hamiltonian (8) in the electronic coordinates. Then the potential surfaces depicted in fig. la are obtained. These are two separate adiabatic potential energy surfaces as indicated by the full and the broken line. Actually the two surfaces are split by = 10U3 eV at Qu = 0 and possess a vanishing derivative at Qu = 0 instead of a tip. The splitting is, however, more than hundred times smaller than the vibrational frequency wu and has a negligible influence on the spectrum. Due to the smallness of the splitting the lower ionic state potential energy curve has doubleminimum character even for very smah values of the coupling constant h. Obviously both surfaces are symmetric with respect to the plane Q, = 0 and possess, therefore, the full symmetry of the initial state. One has to keep in mind that the electronic wavefunctions corresponding to these two ionic state potential energy
- W.Domcke.L.S. CederbaumfCore electronionization
132
surfaces are strongly non-adiabatically coupled near Q, = 0, leading to a complete break-down of the Franck-Condon principle [ 14,151. Fig. 1b shows the potential energy surfaces to be associated with the transformed hamiltonian (12). There are two separate potential energy surfaces which are degenerate at Qu = 0, but none of these possesses the initial symmetry of the molecule. Through the transformstionto the non-symmetry-adapted electronic operators aI, a2 we have been able to generate a harndltonian which explicitly describes the breaking of the initial state molecular symmetry. Starting from the hamiltonian (12) the photoelectron spectrum can be calculated exactly. The transition probability per unit time and unit energy at energy w is given by [5] P(w) = $ dt eiwr W,,I~T(t)j+n), __ where r(r) = eZ’T e-at ator
(13)
and T is the transition oper-
T= Crioi,
(14)
i
P(W) =21r~2~dtei[W-~(o)~r~0,0~e~~0,0),
T = Tgag + ?,a, .
(15)
Changing according to eq. (11) to the transformed or: bital basis we have +rza2,
IO, 0) denotes the vibrational ground state of the molecule, i.e. b,jO,O) = 0 for s = g, u. In eq. (17) the calculation of the spectrum P(o) has been reduced toa pure boson problem. The interesting aspect is that H, which describes the vibrational motion in the ionic state, is the hamiltonian for a shifted harmonic oscihator with shifts both in the symmetric stretching as well as in the antisymmetric stretching direction. The shift in the antisymmetric coordinate is a consequence of the vibronic interaction of the lo,, and lus core orbitals. The remaining steps in the calcula$on ofP(o) are very simple. It is easy to show that His diagonalized by the canonical transformation U= expL--(a/w&s
- 6:) - 0&J@,
P(o)=21212~dre
twrW&+t
(t)&J
,
(16)
where we have used the fact that IT, 12 = I?#
= JA
and
OPOla~al(f)l Q, = (9, IaTa2(t)I*,> = 0 .
Us@ the hamiltonian (12), the adiabatic approximation [ 1 1] for I eo> and the fact that the electronic
- bZ)l , (1%
Inserting now the complete set of vibrational states 15, r.2,) into eq. (17) we arrive at the fmal result exp [--uz P(w) = 21~1~ C % “g
lfu)]
~~ ng! n,! (21)
x6bJ-
and
(17)
with
i.e.
ri being the matrix element for photoionization of the orbital i. lUO>denotes the ground state wavefunction of the molecule. In the present core electron ionization problem
T=rIaI
operators commute tith the boson operators, we obtam
40) +fpg
ffuwu
-
ngwg_- nuwu] ,
with
fg = (hJg)2.fu
=
WJ2 -
(22)
The vibrational intensity distribution is thus given by the product of two Poisson distributions. The “coupling parameters” feand fudetermine the strength of the excitation of the symmetric and the antisymmetric stretching mode, respectively. So far we have neglected, without further justification, the bending mode of the linear triatomic molecule. Since the bending mode is of n symmetry, it can induce in first order vibroqic coupling between a u and
193
W. Domcke, L.S. CederbaumfCore el&tron ionization
_a n orbital, but not between two CJor two ‘IIorbitals. The second order coupling is of the two components of a n orbital is known as the Renner-Teller effect [13]. Since there are no n orbitals in the core region, it is clear that in the one-particle approximation discussed here there can be no vibronic coupling involving the bending mode and thus no breaking of the linearity of a molecule. A breaking of the linear symmetry might, in principle, occur due to the change of correlation energy upon core ionization. For a quantitative’application of the above consideration we have to compute the coupling constants defined in eq. (9). For this purpose SCF calculations on CO, have been performed [16] employing a basis set of 9s-type and Sp-type Cartesian gaussian functions, contracted to 4%type and Zp-type functions. The coupling constant for the symmetric stretching vibration is, according to eq. (9a), simply given by the derivative of the 01s orbital energy with respect to Qa. It would be rather difficult to determine h from eq. (9b). Instead we consider the transformed hamiltonian (12), from which it is seen &at h governs the linear splitting of the 01s orbital energies with the antisymmetric coordinate Q,. More precisely, A = 2-‘/2(&,/aQu),-,
= - 2-‘/2(&2/aQu)I,,
(23)
where EI and e2 denote the orbital energies corresponding to the transformed orbitals 1 and 2. Eq. (23) reveals the close analogy between the coupling constant X and the coupling constant K for the totally symmetric vibration given by eq. (9a). Note that the sign of h is of no signjficance, while the sign of K determines whether the orbital in question is bonding or antibonding with respect to the totally symmetric vibration. We point out that we do not make use of the so-called ‘%rude adiabatic approximation” in the ab initio determination of the coupling constants K and h. It has been argued that this approximation gives poor results when used for the calculation of vibronic coupling matrix elements [26]. It suffices to perform, in addition to the SCF calculation at the ground state equilibrium geometry, a SCF calculation for a slightly increased (or decreased) C-O distance as well as a SCF calculation at a slightly nonsymmetric nuclear configuration in order to obtain the derivatives of the orbital energies and thus K and h. The result is K = 0.098 eV ,
X = 0.288 eV,
(24)
leading to coupling parameters
f,=O.32, f,=O.98.
(25)
The calculation thus predicts a weak excitation of the symmetric stretching mode and a fairly strong excitation of the antisymmetric stretching mode. Since K is positive, the 01s orbitals are antibonding with respect to the symmetric stretching vibration, which means that the CO distance decreases upon 01s ionization. The strong excitation of the antisymmetric stretching vibration indicates that the two C-O bond lengths differ considerably in the core-ionized species. The geometry shift in the antisymmetric stretching direction is given by AS, = (2/P,P@/W”) where
>
(26)
s, = 2-1/2(rr - r3_)! and I/fi=l]mO
+2/m,.
rl and r2 denote the two C-O distances. p is the re-
duced mass corresponding to the antisymmetric stretching vibration. With X from eq. (24) it is found that Arl =-Ar2=o.12A.
(27)
The above determination of the coupling constants serves merely as an illustration and should not be considered as an accurate ab initio calculation of’the core ionization spectrum. It is well known that the Hartme-Fock one-particle approximation (Koopmans’ approximation ]17]), on which the hamiltonian (1) is based, is not appropriate for determining the ionization potentials of core electrons due to the pronounced reorganization of the valence shell upon core electron ionization. Indeed it has been shown in ref. [8] for CO and for the totally symmetric vibration of CH4 that reorganization effects influence considerably the vibrational coupling constants of core electrons. The reorganization effects can be simply taken into account by a “renormalization” of the vibrational coupling constants [6,8] _The renormalized coupling constant K is obtained by replacing -e(QJ in eq. (9a) by the true ionization potential IE’(Qe). IP(Qa) may be approximately obtained, for example, by a Green function calculation [6,8] or, especially in the core region, by a ASCF calculation. From eq. (23) it is clear that A can be renormalized in the same way. Whilst this work W~.Salready written up, we became
194
W. Domcke. LA. Cederbaum/Core electron ionization
aware of direct SCF hole state calculations on the core and valence ionized states of CO, performed by Clark and Mii!ler [18]. By calculating a potential energy surface for the 01s hole state they found that CO2 is indeed non-symmetric in this state. Clark and Miiller alSOcalculated the vibrational structure of the 01s line. They found a very weak coupling to the symmetric stretching mode and a rather strong coupling to the antisymmetric stretching mode [18]. In contrast to the coupling parameters (24) the results of Clark and Miiller include reorganization effects. From a comparison of their data with the present ones it is seen that reorganization effects act to reduce the coupling for the symmetric stretching mode, in analogy to the results found for the Cls orbitals in CH4 and CO previously [8]. With respect to the excitation of the antisymmetric stretching vibration, on the other hand, the present results are in close agreement with those obtained by CIark and Miiller. In table 1 the FranckCondon factors for the transitions (0,O) + (n,,O) obtained from eqs. (21,25) are compared with the Franck-Condon factors reported by Clark and Miiller. The agreement seems to indicate that reorganization effects do not significantly influence the coupling constant A. The above discussed effect of symmetry breaking due to viironic coupling of core orbitals bears some relationship to the well-known problem of the “localization” of a core hole [19,20] _When there are several equivalent atomic sites in a molecule (such as the two oxygen nuciei in CO,), the question arises whether the core hole, which remains after the ionization process, should be thought of as localized at one of the centers or as delocalized over the equivalent centers. Direct SCF hole state calculations have shown that a Table 1 Franck-Condon factorsfor the excitation of the antisymmetric stretchingvibration in the 01s band of CO2as obtained in the present study (within the one-particle approximation) and by Clark and MiiUer(from direct hole state SCF calculations) nu
Thiswork
0 1 2 3 4
0.377 0.368 0.180 0.058 0.014
Clark and Mtiller[18]
0.390 0.365 0.175 ,0.055 0.015
considerable lowering of the total ionic SCF energy is obtained when the core hole is chosen to be localized [ 18,20-221. Therefore, core holes are generally considered as being localized. It has been shown recently [23] that the tendency of the core hole to localize can be understood as a correlation effect. When delocalized (symmetry-adapted) orbitals are used, correlation effects are of the same order of magnitude as reorganization effects and therefore ASCF calculations, which do not account for correlation effects, give poor.results. When localized (non-symmetry-adapted) orbitah are used, correlation effects are much smaller than correlation effects, explaining the good results obtained with ASCF calculations employing localized orbitah. An analogous behavior is found in the vibronic coupling problem. The hamiltonian (8), which is formulated in the summetry-adapted (delocalized) orbital basis, contains a coupling term which is non-diagonal in the electronic operators. Therefore one cannot associate an adiabatic electronic potential energy surface with this hamiltonian. Changing to a localized (nonsymmetry-adapted) orbital basis by the transformation (1 l), the hamiltonian (12) is obtained, which now contams only diagonal electron-vibrational coupling terms (as long as eq. (10) is fulfilled). Therefore, a potential energy surface can be associated with the transformed hamiltonian. This surface reflects directly the lowering of the molecular symmetry (see fig. lb). Vibronic coupling effects can thus be eliminated in the same way as correlation effects by the transformation to a non-symmetry-adapted orbital basis. It should be kept in mind that the “localization” of the core hole due to correlation effects does nor lower the symmetry of the electronic wavefunction. The total ionic wavefunction has still the full symmetry of the point group [20,23]. The argument [18] that due to the localized character of the core hole the symmetry of CO3, for example, is reduced from Dmh to C,, and that therefore linear coupling to the antisymmetric stretching vibration is symmetry-allowed, is, strictly speaking, incorrect. To understand the excitation of the antisymmetric stretching vibration and the lowering of the symmetry of the molecule one has to. consider the vlbronic coupling of the degenerate 1 IJ,, and lug orbitals as outlined above *. It is noteworthy that this vibronic coupling mechanism is already pres*For footnote see next page.
W. Domcke. L.S. Cec?erbaum/Core electron ionization
ent on the one-particle level, whereas the localization of the core hole is a pure many-body effect. The vibronic coupling problem represented by the hamiltonian (8) could only be solved exactly because the splitting of the u and g orbital energies at Q,, = 0 tis neglected. In the more general case that e,(O) # err(0) (and possibly ~g # K") we have to deal with a very complicated dynamical problem. It is no longer possible to separate the electronic and the vibrational motion in the ionic state and the Franck-Condon principle is inapplicable. The 20~ and 3ug deep valence orbitals of CO, represent an example where this situation occurs. In conclusion, we would like to stress that the vibronic coupling mechanism discussed in an exemplary manner for CO, above is a quite general phenomenon in inner-shell ionization of symmetric molecules. The mechanism is, of course, also if importance in X-ray absorption processes involving core orbitals, which have recently been discussed by Gel’mukhanov et al_ [24]. The benzene molecule may serve as an example to illustrate the very complex dynamical processes occurring upon core electron ionization- The six Cls electrons of benzene populate the four very nearly degenerate orbitals alg, b,,, elur e2a [25]. From the above discussion it is clear that in the Cls-ionized state the alg and bIu orbitals are strongly non-adiabatically coupled by the two bI, modes. The eIu and e2g orbitals experience a Jahn-Teller splitting by the four e2z modes- As is well-known 1131, the Jahn-Teller lnteraction involves both diagonal and non-diagonal cou* Sincethe transition operator descriiing the interaction of the molecule with the photon field is independent of the vibrational coordinates, the ion is created in a vibrational state which has the samesymmetry as the initial vibrational
state. (It has been shownabove,by employingthe transformation (11). that the electronicand vibrationalmotions in the ionic state can be separated in the same way as in the nondegenerate initial state. Therefore it is possiile to speak in terms of ionic state viirational wavefunctions.) It is clear that each viirational level in the ionic state is doubly degenerate, having associated with it a vibrational wavefunction of g as well as of u symmetry_ According to the FranckCondon principle, only the vibrational states of g symmetry can be reached from the initial state vibrational ground state. Nevertheless, the molecule is non-symmetric in the ionic state. The symmetry of the ionic state vibrational wavefunction merely represents the fact that we do not know which of the two equivalent C-O distances becomes shorter or longer, respectively.
195
plings within the degenerate orbital The alg and b 1n orbitals interact with the elu and ezg orbitals via the vibrations ofelu and e2z symmetry and the el, and e2g orbitah interact with one another via the b,,, b2, and elu vibrations. The two totally symmetric vibrations can, of oourse, couple in the usual adiabatic manner to each of the four core orbitals. Thus twenty of the thirty normal modes of benzene [lo] may be excited, in principle, upon ionization of a Cls electron_ These twenty modes comprise aU in-plane normal modes with the exception of the one of a2g symmetry. There is no vibronic coupiing involving the nine outof-plane vibrational modes, since no n orbitals are present in the core region. Considering the results obtained for CO, above, it is not expected that aU of the symmetry-allowed vibrational couplings are small. Due to the very near degeneracy of the core levels a breaking of the symmstry occurs even for very small values of the non-diagonal coupling constants. The Cls line of benzene should thus exhibit a very complex vibrational structure. ‘The symmetry of benzene is reduced from Dsh to C2v upon core ionization.
References [l] G. Her&erg, Molecular spectra and molecular structure, III. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand, New York, 1966). [2] T.E. Sharp and H.M. Rosenstock, J. Chem. Phys. 41 (1964) 3453. [3] M.D. Frank-Kamenetskii and A. Lukashii, Soviet Phys. Usp. 18 (1976) 391. [4] D.W. Turner, C. Baker, A.D. Baker and C.R. Bundle, Molecuhr photoelectron spectroscopy (Wiley, New York, 1970). [S] L.S. Cederbaum and W. Domcke, J. Chem. Phys. 64 (1976) 603,612. [6] L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. [7] U. Gelius, S. Svensson, H. Siegbahn. E. Basilier, A. Fa_xBlv and K. Siegbahn, Chem. Phys. Letters 28 (1974) 1. [8] W. Domcke and L.S. Cederbaum, Chem. Phys. Letters 31 (1975) 582. [9] D.T. Clark and J. Miiller, Theoret. Chim. Acta 41 (1976) 193. [lo] E.B. Wilson, J-C. Decius and P.C. Cross, Molecular vihrations (McGraw-Hill,New York, 1955). [ 111M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford Univ. Press, London, 1954) Appendix VII. [12] H.A. Jahn and E. Teller, Proc. Roy. Sot. A161 (1937) 220.
196.
W.~Ij&k& L.S. C&baum/Cor.6 ele&ph iotiizatihi~
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___
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