Faraday Symp. Chem. SOC.,1984 19 63-77 Calculation of Molecular Spectra Electronic Transitions Vibrational Patterns and Radiative Lifetimes of Spin-allowed and Spin-forbidden Transitions BY SIGRID D. F’EYERIMHOFF Lehrstuhl fur Theoretische Chemie Universitat Bonn WegelerstraDe 12 D-5300 Bonn 1 West Germany Received 21st August 1984 The use of large-scale configuration-interaction calculations for a purely ab initio description of the details of molecular spectra is discussed. It is shown that the theoretical tool is applicable to a large range of problems and that it produces quantitative data such as transition energies to within 0.2 eV or better independent of the wavelength region fine-structure effects due to spin-orbit coupling vibronic features in large-amplitude motion and radiative lifetimes of excited states on the to lo3 s timescale.In addition it gives insight into the electronic structure and the origin of the various processes; this is important for an understanding and the prediction of features which occur when for example first-row atoms in molecules are replaced by their second-row analogues or for radiation processes in competition with intersystem crossing. Ab initio calculations have become very efficient and quantitatively reliable over the years and they can now be employed in molecular spectroscopy as an alternative to experimental techniques. Even though they do not predict energies to the same level of accuracy as high-resolution spectroscopy they possess various important advantages.They are completely general and hence are able to treat any wavelength region and any molecule positive or negative ion (as long as the compounds are small i.e. possess not more than six atoms other than hydrogens) in any electronic state regardless of the physical or chemical stability of the system. Furthermore the calculations can give the various electronic states over the entire geometrical range of structural parameters i.e. the entire potential-energy surface which is essential for an understanding of the photochemistry. Various properties of excited states can also be extracted relatively simply from the calculations even though not too much emphasis has been placed on this area so far presumably because there is very little experimental competition in this case.In recent years considerable effort has been made to account theoretically for fine-structure effects in the spectra (primarily those arising from spin-orbit coupling) or to treat relativistic effects in heavy atoms such as XeF or Xe on a purely ab initio basis. In what follows various examples of our work will be given carried out by groups at Bonn and Wuppertal the latter group under the direction of R. J. Buenker. METHOD All studies were carried out using the Bonn-Wuppertal MRD-CI package plus supplementary programs developed over recent years within our laboratories. The various features of our MRD-CI method which is a configuration-driven technique have been described in the literat~re.l-~ The configuration space thereby consists of 63 CALCULATION OF MOLECULAR SPECTRA all single and double excitations with respect to a set of reference configurations (multi-reference set) generally between 10 and 40.Since large A0 basis sets are required for spectral calculations this MRD-CI space expands very fast to lo6 configurations or more a number which is too large to be handled routinely in an economical way on standard general-purpose computers. Hence the MRD-CI space is partitioned into strongly interacting and weakly interacting subspaces based on the energy contribution of each configuration (relative to the energy of the reference set alone) evaluated numerically ; the subspace of strongly interacting configurations (routinely 10000-16000 which contribute more than a predetermined ‘selection threshold’ T = hartree) is treated directly to obtain the lowest-energy solutions while the energy contribution of the more weakly interacting species is accounted for in a perturbative manner based on their numerical values for energy lowering evaluated at the earlier partitioning stage.This partitioning technique is effective because the combined effect of the weakly interacting functions although large in number is quite small. A direct comparison *between the full CI energy of -25.227 6274 hartree obtained for BH in a medium-sized A0 basis,4 and the energies resulting from truncated subspaces show that errors are 0.005208 hartree 0.000285 hartree 0.000 151 hartree and 0.000053 hartree if only 586 10389 13880 or 17049 configurations respectively are selected5 from the total space of 132686 configurations.Hence the extra computational expenditure which is necessary to treat the total space is in no way related to the gain in accuracy one obtains thereby in particular since (a) only energy differences are important for spectroscopy problems and (b)errors introduced by other omissions (such as restrictions in the A0 basis) may be larger. Finally the full CI is estimated by analogy with the formula for the contribution of triple and quadruple excitations given by Davidson6 as E(ful1 CI est.) = E(MRD-CI) +(1 -Zcf) [E(MRD-CI) -E(ref.)] or generally E(ful1 CI est.) = E(MRD-CI) +AAE whereby the sum is taken over all reference configurations. For the factor (1 -ZcX) various other formulae are also found in the literature.la It is important in this connection that the correction AE to the MRD-CI energy is small otherwise the perturbative nature is not justified.In actual calculations it is found that transition energies are essentially the same regardless of whether the MRD-CI or full CI estimates are employed provided the contribution of the reference sets as measured by Xcf is approximately the same (and > 90%) in both states involved,’ i.e.both states are described in a balanced manner. If its magnitude differs in the two states by a few percent the energies from the estimated full CI are generally the more reliable. The MO basis in the MRD-CI calculations consists generally of SCF MO of the same or of different states depending on the application or computer time available; use of only one set is more economical.It has been found that the parent SCF MO are only required if a large core (doubly occupied MO) is maintained simply because in those cases relaxation of the core cannot be accomplished by the CI. In some cases it is convenient to employ natural orbitals in order to achieve a more compact representation of the MRD-CI expansion (fewer reference configurations). These are generated by diagonalizing the first-order density matrix obtained in the first MRD-CI step. Experience shows that calculated transition energies are generally within 0.1 eV or less at the full CI level if different sets of MO or NO are employed which is a good indication that the calculation level is close to the full CI in which the type of orbital transformation within a given A0 basis is immaterial for the results.Finally the A0 basis for excited states is larger than in ground-state calculations S. D. PEYERIMHOFF 65 of the same accuracy. In order to represent the excited states it is often necessary to optimize additional functions for those excited atomic states in which dissociation is possible. For calculations on excited states of MgNa+ for example 3s,4s 3p 4p and 3d functions for magnesium and sodium had to be generated; ArH calculations required the correct description of hydrogen 2s 2p 3s 3p and 3d and argon 4s 5s 4p and 3d states by a number of contracted Gaussians. This procedure is often time- consuming and it would be advantageous if not only basis sets for ground states and polarization functions were available in the literature but also those for the actual excited atomic species.Calculation of Rydberg states requires additional long-range functions but since the nature of a Rydberg state (hydrogen-like) is simple generally one or at most two functions suffice for the description of a Rydberg member. Negative ions require at least semi-diffuse p functions but if the electron affinity must be obtained to an accuracy of 0.2 eV or better semi-diffuse d functions are also needed according to our experience with for example C Si and F. In summary the A0 basis requires careful expansion over the double-zeta plus polarization routine basis generally employed for ground-state calculations.A reduction in size will invariably lead to errors in the overall treatment. In this connection the advantage of the partitioning technique of the MRD-CI space again becomes apparent. It allows the use of large A0 basis sets and hence maintains good overall accuracy but is nevertheless computationally quite feasable; the alternative route to decrease the number of A0 functions or reference configurations in order to save computer time would lead to reduced accuracy in the overall treatment. POTENTIAL-ENERGY SURFACES FOR GROUND AND EXCITED STATES The calculation of potential-energy surfaces is in principle straightforward in the MRD-CI procedure. For simple predominantly vertical transitions the energy AEe between the two electronic states at a given geometry can be related to the absorption (or emission) maximum and deviations from measured values are usually 0.2 eV or less using standard A0 basis sets.For unknown transitions such theoretical values are good guidelines for further experimental work or for the clarification of assignments. Numerous examples are given in the literature. lay 9-11 Of more interest is an investigation of the crossing of potential curves. Experimentally such a situation can be deduced only from perturbations in the spectrum but determination of the character of the possible perturber or the form of its energy surface is involved and tedious. On the other hand the calculations are a powerful tool for obtaining this information. If the interacting states are of different spin or spatial symmetry the calculations (separate for each state) are straightforward.Typical examples in which the calculations were instrumental in explaining the observations are Fi and Cli. In Cli no vibrational quantum numbers could be assigned to the upper state in the A 211,-X 2JIg transition because the vibrational levels of 211upresented 'a chaotic distribution'.12 Calculati~nsl~~ l4 have shown that.. .n4,ni 0 states 2A and "C; lie in the same energy region and cross A ",(. . .xi@) very close to its minimum and must thus be considered to be the prime perturber responsible for the complex pattern. Furthermore while in Cli the crossing point is found to the right (larger bond lengths) of the 211u minimum which allows in principle for large coupling of vibrational states (all three states possess a similar minimum energy) it is placed to the left (repulsive branch) in Ft (the minima of 2X and 2Au lie much lower than that of A TI,)and hence the perturbation together with the smaller spin-orbit coupling in Fi relative to Cli is considerably less in Ft ; this in turn explains why a fairly regular vibrational pattern could be observed in the equivalent 211,-2JIg transitions.The 3 FAR CALCULATION OF MOLECULAR SPECTRA perturbing states have also been analysed in a number of simple hydrideslO7 l5 such as in PH PH+ SiH and SH in which predissociation is an important factor as a result of the interaction. In the predissociation of +Z; in 0; the interacting states 4Zi and 411g are known but ab initio calculations16 were able to determine details of the radiationless transition by studying the spin-orbit interaction responsible for the actual magnitude of interaction.A more complicated interaction theoretically and even less likely to be detected in measurements is the crossing of states of the same symmetry. From a computational point of view care must be taken that the set of reference configurations is balanced with respect to both states. This often requires a smaller MRD-CI partitioning (selection) threshold i.e. a larger MRD-CI subspace which has to be diagonalized. Furthermore the completely different character of interacting states sometimes causes problems for the choice of the MO basis. Experience shows that it is better to use MO which are optimal for neither one nor the other state than the parent MO of one state; alternatively averaged natural orbitals for the two (or more) interacting states are an adequate choice.Finally it is not always clear whether the Born-Oppenheimer approximation on which the CI calculations are based is still valid in this crossing area. However the ratio between the energy AEof closest approach and the vibrational frequency we in the higher potential well can be used as an indication in this regard. In various causes the calculations have shown that what was thought to belong to two different progressions is in reality due to one transition into a double-minimum well of one of the states. A prime examplelo is the B 2Z+ and C 2C+state of SiH which in reality is a double-minimum state 502n2in character at small SiH separations and 50~60at large distances.17 The mixing between Rydberg and valence states is essential for the interpretation of absorption and emission spectra of molecules such as HF,18 HC119-22and Cl,.239 24 In each case the repulsive inner-branch of an intravalence shell (ionic) state with a minimum at large internuclear separation cuts through the entire Rydberg-state manifold possessing minima at small bond lengths and causes various avoided crossings with members of the same symmetry.As a result the otherwise regular pattern of a Rydberg family is heavily perturbed as experimentally observed for HCl and HF without any e~planation~~ (‘lack of theoretical guidance has hampered the analysis of the spectrum’) and various new minima appear with Rydberg-like character at the outer-branch and valence-shell character at the inner- branch of the potential well.The intense progression in C1 absorption between 78000 and 8 1000 cm-l can be explained by such a newly built Rydberg valence well 2 lZ;(a; n34pn-0 $ ou),and the fluorescence between 50000 and 75000 cm-l by the double-minimum in the 1 state with the complementary coefficients of the same configurations. A further example comes from the rare-gas diatomics. In this case the various Rydberg states originating from ou,n, nu and o MO interact since they show different behaviour with internuclear separation excitation out of the strongly antibonding ouinto non-bonding Rydberg orbitals leads to potential minima; Rydberg states originating from bonding o and nulead to a family of repulsive states whereby for Ne, for which an extensive study has been performed,26 the 0,-+ R potential curves are more repulsive than those of Rydberg states originating from nu.The situation with n is more complex. In Ne it is weakly antibonding and the corresponding Rydberg states depopulating ng may be weakly bound. The various interactions are essential for an understanding of the behaviour of Ne in its excited states and can to some extent be generalized to other noble-gas dimers. Even though potential curves have so far only been discussed in connection with electronic spectra knowledge of them is very important for various other related phenomena.Two of them can be seen in fig. 1 which shows potential-energy curves S. D. PEYERIMHOFF CI -1 155.70 \ \\ -j 3.0 4.0 5.0 6.0 7.0 8.0 RcPc,(atomic units) Fig. 1. Calculated MRD-CI potential-energy curves for C1 or C1- removal from CF,Cl,. for CF,Cl as a function of C-Cl removal. First the lowest excited-state curves,' are all repulsive and hence it is clear that photodecomposition of CF,Cl favours release of C1 atoms. This is of course one of the problems of propellant gases like CF,Cl since Cl atoms can undergo chain reactions in the stratosphere and then con- tribute to the destruction of the ozone layer. The second phenomenon apparent from fig. 1 is dissociative electron attachment at almost thermal energies under release of negatively charged chlorine ions.The negative-ion curve crosses that of the neutral system close to the latter's minimum and thus there is the possibility of intersystem crossing from the neutral to the negative-ion system whereby the latter is almost repulsive CF,Cl-Cl-. Indeed in experiments for dissociative electron 29 a C1- peak is found at 0.7 eV with an appearance potential as low as 0.1 eV; this is in perfect agreement with the ab initio work which shows the crossing to be not too far above the CF,Cl minimum. It is of interest that this process must be considered in competition with release of C1 if the balance of the 0 reactions is considered. The CF,Cl energy at small C-Cl separation which lies above that of the neutral system has been calculated by a modified MRD-CI procedure developed for short-lived to electron resonance^.^^ This method has also been applied succe~sfully~~ the 211g(7ctng)and 211n,(7t3,3s2)resonance states of N and is generally applicable to short- lived negative-ion states as is the MRD-CI method; it uses a simple method to stabilize the discrete component of a resonance state by increasing the nuclear charge.A third process for which reliable potential-energy curves are necessary is the ion-molecule (or ion-atom) reaction with charge exchange an example of which is shown in fig. 2. Relatively high accuracy is especially important since the various reaction channels are often very close in energy. The Na (,S)+Mg+ (,S)energy at 2.506 eV is only 0.2 eV CALCULATION OF MOLECULAR SPECTRA 14.0j 12.0 -10.0 -< 8.0: .4-6.0 4.0 -2.01-0.0 - ‘ “ ‘ I V “ A. t 5.0 10.0 15.0 Rlao Fig. 2. calculated potential-energycurves for various statesinNa+ +Mg (NaMg)+and Na +Mg+. The estimated full CI values are given. 2.5-2.0-.3 1.5-4 Q 1.0-0.5 -\ I 0.04 3.0 3:5 4.0 4.5 5.0 RscHIao Fig. 3. Calculated MRD-CI potential-energy curves for the low-lying states in ScH. below that of Na+ (lS)+Mg (3P),while the corresponding calculated data are 2.514 and 0.12 eV. Similar relations hold for the fourth and fifth reaction channels. The (Na++Mg) (”) curve crosses that of 3C+ and a charge-exchange reaction near the crossing point of both curves leads to a Na +Mg+ system.The potential curves agree very nicely with the experimental observation~.~~ S. D. PEYERIMHOFF I I I \ 0.4 I % 0.0 x311 A311 A31T a x" 'x31-\x3r 0.4 SiN Sip X2E+ x2x+ 0.0 CN CP Fig. 4. Calculated energy differences between the lowest states of isovalent compounds always evaluated for the respective equilibrium geometry. The last important feature to be mentioned in this section is the possibility of obtaining more insight into the electronic structure and an understanding of the causes of the behaviour of potential-energy curves; this in turn allows qualitative predictions to be made without resorting to additional calculations. Two examples illustrate this point. In transition-metal hydrides the magnitudes of the structural parameters are sometimes surprising.For example in ScH (fig. 3) the ground state XIC+ curve exhibits entirely different behaviour from the other multiplets of seemingly the same electronic configuration 6a2d17a1. Closer analysis shows that while this is the configuration for 371A(db7a) lII(dn7a) and 3C+(da7a) there is a second quite important occupation 39 7a2possible for the singlet states lC+(da7a and 7a2) with higher bonding properties and mixing of both configurations leads a marked shift of R,(X lC+) to smaller bond lengths and higher values of co,. Note also that the CI is very important in this example since the SCF approach alone places lC+ above %+ and would predict erroneously a 3A ground state for ScH. Ths measured absorption bands are between 17690 and 18350 cm-l (2.19-2.37 eV) and must certainly be attributed to the 2 W-X lC+ transition which we calculate at 2.10 eV.32 Similar situations have been observed in other transition-metal hydride~~~ as long as the dn7a configuration can be decomposed into dn-lda7a.Finally one of the most attractive theoretical studies is a comparison of compounds in which first-row constituents are replaced by those of the second or higher rows. The increased orbital stability of a(pa+pa) relative to n(pn+pn) upon replacement of a first- by a second- (or higher-) row atom causes a predictable trend in the relative stabilities of corresponding electronic states and hence absorption and emission properties. A typical example is presented in fig. 4 for the 2C+(a~4) and 211(a2n3)states of the isovalent series CN CP SIN and SIP in which the state with double population CALCULATION OF MOLECULAR SPECTRA of the 0 MO and only three electrons in n orbitals gains increased stability relative to %+.The equivalent trend is present in the corresponding ionic compounds or in the related C, CSi and Si whereby the first-row species favour X lZ+(n4) the compounds with atoms from both first- and second-row favour 31-I(0n3)and the second- (and presumably higher-) row molecules favour X 3C-(02n2)occupation. In a similar manner the increase in relative stability of n* (and n as its in-plane component) relative to n upon introduction of second-row atoms causes a decrease in the splitting of the n and n ionization potential (i.p.) in H,ABH so that n is the first i.p.in H,CN both i.p. are close together in H,SiNH and H,CPH while in H,SiPH+(2A”) (n-+ a)is the ground state. Equivalent arguments based on n n and n* orbital stability predict the location of Rydberg and (n n*) and (n,n*) states whereby it is of interest that only in the first-row compound H,CNH the 3(n,n*) is the first excited state as known from H,CO for example while already in the mixed compounds as well as in H,SiPH (if the geometrical H,ABH structure is assumed) the transition to 3(n,n*)is the lowest in the vertical region. FINE STRUCTURE The MRD-CI wavefunctions can also be used for the calculation of multiplet splitting. For this purpose the Hamiltonian is written in the form H = HO +H, +H, whereby the spin-orbit and spin-spin operators are taken from the Breit-Pauli formulation as All integrals are evaluated 37 Typical examples are the multiplet splittings for the X 3C-states of S, SO or 0,38 and the atomic states of the Br atom.39 If in SO the spin-orbit coupling between the 3C-state and the four b lX+ A” 3X+ C 311 and lll states is taken into account in the second-order perturbation sum Xi+k( i 1 H, 1 k),/AE a value for the zero-field splitting parameter D between 7.96 and 9.15 cm-l is obtained as long as A0 basis sets of double-zeta quality or better are employed.40 The contribution of H, is much smaller betizeen 0.66 and 0.715 cm-l depending on the A0 basis sets but evaluated in the single-cunfiguration treatment only.The resulting value D,,+D, of ca.9.5 cm-l must be compared with the measured splitting of 10.55 cm-l. It can be assumed that the error is caused by truncating the perturbation sum since a CI in which all symmetry-adapted functions generated by the CI for the lX+ 3Z+ and 3f117 states are coupled by the H, operator leads to an increase in D, of the order of 1-1.5 cm-l and thus places the theoretical D,,+D, in very good agreement with the measured splitting. Another example is the X splitting in OH. Recent MRD-CI calc~lations~~ in a standard DZ-A0 basis find a zero-field splitting parameter averaged over the zeroth vibrational level of A(v,) = -138.543cm-l compared with the measured A(v,) = -139.054cm-l if only the diagonal contribution is considered. Second-order effects (off-diagonal elements) are found to be very small and decrease the value of A by 0.086 cm-l if contributions from the first 2C+,,C-,4C-and 411 states are taken into account.The values differ only slightly from those obtained in a more expanded A0 a finding which is consistent with our experience that spin-orbit matrix S. D. PEYERIMHOFF elements (standard accuracy) do not require any particular A0 basis-set extension40 relative to what is usually employed in excited-state calculations. One definite advantage of the theoretical treatment is that it allows us to determine in a straightforward manner the dependence of the spin-orbit interaction or multiplet splitting as afunction of some geometrical parameter or normal coordinate information which is more difficult to extract experimentally.The considerable dependence of this quantity on the geometry has been demonstrated on various occasionsL6 and in most cases it has been found to be a direct consequence of a change in the character of the states (variation in CI expansion) under consideration. An interaction of spin-orbital momentum and rotation leads to additional splitting of the R+ and R-levels in the X 211 state of OH and is normally referred to as A doubling. This quantity can also be calculated by employing the spin-orbit matrix elements between X and the first excited 2X+ state of OH in addition to the rotational coupling matrix element (211ILJ 2C+). The actual splitting is conventionally given in terms of two parametersp and q.If purely theoretical quantities are employed (ie.the various matrix elements as well as the calculated rotational constant B) one obtains41 values of p = 0.2352 cm-l and q = -0.0381 cm-l compared with the experimental values of p = 0.2357 cm-l and q = -0.0391 cm-l.These examples should suffice to demonstrate that the theoretical approach to the determination of fine-structure effects in molecular spectra is quite successful; it is expected that a considerable number of applications in this area will follow in the future. VIBRATIONAL PATTERNS The total wavefunction for a given state consists of an electronic part treated and a function describing the nuclear motion. The vibrational wavefunctions are usually generated by solving the Schrodinger equation for nuclear motion in which the Born-Oppenheimer energy curves as described above serve as the potential term often approximated to some convenient analytical form.The calculation of vibrational levels in diatomics is very simple; polyatomics are usually treated only in the harmonic approximation and for small bending amplitudes but while this problem is tractable the results are only of limited significance especially if more than the lowest excited states are involved. The general problem is especially complicated as soon as (1) large-amplitude motion (2) coupling of modes and (3) the breakdown of the Born-Oppenheimer approximation because of coupling of electronic and vibrational motion are present. The first two problems have been treated for the CH chromophore in CD,H in a joint experimental and theoretical study.43 The vibrational spectrum has been measured between 900 and 12000 cm-l under high resolution and the vibrational dynamics has been treated by employing an effective spectroscopic Hamilt~nian~~ on a potential-energy surface constructed from > 279 data points.The results show semi-classical zero-point amplitudes of 15" and a bending amplitude of > 45" for the N = 4 and 5 levels of the CH chromophore quantum numbers (N = vs+ivb); they demonstrate clearly to what extent the domain of small-amplitude motion is diverged from and to what extent the Taylor series expansion of the potential is expected to lack convergence in low order. The analysis finds that the tridiagonal Fermi resonance causes strong coupling between the stretching and bending vibrations in the CH chromophore and leads to fast energy redistribution between these two motions; the picture of a pure local-mode stretching motion can thus not be maintained.Another intriguing problem which requires coupling of electronic and vibrational states is the Renner-Teller effect;45 a general survey of the problem has been given CALCULATION OF MOLECULAR SPECTRA by Jungen and Me~er~~ The electronically degenerate electronic states and D~xbury.~’ of linear molecules are split in the course of bending where again the large-amplitude motion is important. Ab initio w~rk~~-~~ has been undertaken for AH systems such as NH, PH, SHZ BH, AlH and SiHz in their X states for CH, NHZ and SiH in their excited ,Ag states and for HNO+ HNF and C,.50 The potential-energy curves show minima at bent geometries for both 211 (,A’ ,A/’)components in NH, PH and SHZ while the upper component is linear in the next three molecules men- tioned above.Two procedures have been employed so far in the ab initio treatments (a) a variational calculation of vibrational levels is carried out for each of the two CI (electronic) energy curves and the molecular Hamiltonian (T,-+H,) is then diagonalized in the basis consisting of products of the electronic and vibrational wavefunctions of both states or (b) the molecular Hamiltonian is diagonalized in a basis built from products of linear combinations of the electronic wavefunctions obtained in the Born-Oppenheimer approximation and eigenfunctions of a two- dimensional harmonic oscillator.It has been shown for NH that both methods give the same numerical One problem in the treatment is the choice of the Hamiltonian for nuclear motion. In the AH systems the most suitable coordinates seem to be polar coordinates p for bending and 4 for the phase angle of the molecular plane with respect to a space-fixed plane (‘rotation’). If coupling of stretching and bending motions is neglected (or treated approximately at a different stage) the most convenient form in ab initio work describing large-amplitude bending seems to be T,-= -qT(1 P) a2/ap2+T,(P) vaP+ &(PI a2/a4,+ T,(P)l 2 whereby the coefficient~~l can be expanded in polynomial form conveniently in the same basis as used for the potential energy.In the limit p -+ 0 this form reduces to the Hamiltonian of a two-dimensional harmonic oscillator Results of such a treatment (in large A0 basis sets includingf-type functions in some instances) give for example T,f,tt values [differences between the lowest Z(K = 0) vibronic level of the upper electronic state and the first lower state level which can combine with it i.e.us = 0 K” = 11 for those species which are known experimentally of 1 1 150 cm-l (v’ = 1 V” = 0) for NH compared with 11 126 cm-l experimentally 4145 cm-l (4194 cm-l if reassignment of vibrational labelling is made) for BH, 18 820 cm-l (18 276 cm-l) for PH and 18 620 cm-1 (1 8 520 cm-l) for SHZ with measured values given in parentheses.Furthermore the character of vibronic levels above the barrier to linearity agrees well with experimentally extracted information in the few cases for which the latter is available; the calculations describe the degree of mixing of wavefunctions while the experimental information catalogues the levels as belonging to the ground- or excited-state potential or as mixed levels. These two examples demonstrate that the description of a complicated vibrational pattern can be tackled for certain systems but it should also be clear that such treatments even though suitable potential-energy surfaces exist or could be generated are far from routine and considerable ground-work still needs to be done in this area. TRANSITION INTENSITIES AND RADIATIVE LIFETIMES The calculation of intensities for spin-allowedelectronic transitions is straightforward and requires only the use of the electronic and vibrational wavefunctions for the states involved.The radiative lifetime ziof an initial excited level is determined by the sum 7;' S. D. PEYERIMHOFF 73 of all transition probabilities (Einstein coefficients) A to all (final) levels of the lower state as = X A = +Z AE:f IRif12. f f If rotational levels are neglected and the wavefunction is written as a product function Y = Yexvof electronic and vibrational terms the matrix element R is Hence the calculation of intensities requires in addition to the knowledge of energy levels and vibrational wavefunctions (which are not even necessary if a Franck-Condon factor of 1 is assumed) evaluation of the electronic transition moment Re,,.between electronic surfaces (in which usually the dipole operator Me = Z,,er summed over all electrons is employed); it reduces to the calculation of the dipole moment (as a function of nuclear coordinates) if the same state is involved and only vibrational transitions are considered.There is furthermore a simple relation between the Einstein coefficients for emission Aif and absorption Bif.An alternative quantity often used is the oscillator strength also based on the electronic transition probability as fe = IRer,,,12AEete-,or fere,lvtv,t if oscillator strengths are evaluated for transitions between vibronic levels. There are numerous calculations for intensities in the literature.la9 52-56 A typical example for the intensity pattern in an infrared spectrum is given in fig.5 which shows the Einstein coefficients for Av = 1 in the X 211ground state of OH. It is obvious that the various state-of-the-art calculation^^^ give similar results and represent the actual situation quite well. Electronic transition moments have mostly been evaluated for diatomics quite often as a function of internuclear separation. Generally the agreement between measured and calculated data is within 10-30% or at least smaller than a factor of two for quantities related to lRe~,7~12. Such deviations are probably not alarming since the errors in the experimental results are in a similar range. In a few cases as for example in the A lll,-X lXi Phillips system of C, the uncertainties are of concern because accurate knowledge of the absorption oscillator strength is a prerequisite for the determination of interstellar C abundances from astronomical observations; similarly these f values enter the analysis of the rotational populating of interstellar C and are needed to determine the contribution of the line opacity of carbon stars.Experimental values for foo vary from 1.41 x to 3.9 x (with much smaller error bars in each instance) while the calculated values of Di~hoeck~~ and of Chabalow~ki~~ are both 2.7 x a value of 2.5 x lop3 is obtained in a less extensive ab initio calc~lation.~~ Even though the theoretical data are within the experimental range of results in this case and the curve as a function of the C-C separation is parallel in the theoretical treatment and most experiments lifetimes are very sensitive quantities so that the A lll,(u' = 4) level lives almost twice as long (ca.1 1 ps) according to two recent experiments than is predicted by the calc~lations.~~~ 53 The source ofthe discrepancies between the various treatments is not known and awaits further explanation. More interesting from a theoretical point of view is the calculation of spin-forbidden transitions. This has recently been undertaken for a number of states in 0 (a lAS b lXi c 'Xi A' 3Au and A "L) for the n2states in S, SO NF and NC1 (a IA and b IZ) the 311ustate in C1 as well as for various states in simple hydrides (NH and ASH). In this case not only the dipole transition operator but also magnetic dipole CALCULATION OF MOLECULAR SPECTRA 50 l3 I i i 40 i 112 I! j! 30 j j!//f I v) 1 .i .T .I I.f: 20 10 0 rIIr,,,j 12345678 V' Fig. 5. Einstein coefficients for the Av = 1 transitions in the X ground state of the OH molecule obtained from different treatments. The MC-SCF and CEPA results are taken from ref. (57). 1 MRD-CI ; 2 MC-SCF; 3 MC-SCF/CI ; 4 CEPA2; 5 experimental. and (sometimes) electric quadrupole operators are necessary so that the Einstein coefficients for spontaneous emission can be written in the form A = ga3(AE)3(l(flRI i)12+$a2I (flL+2SI i) 12+&a2(AE)2I (flR x RI i),) where a is the fine-structure constant. It is important to consider both terms in the magnetic dipole moment operator i.e.the interaction of the radiation field with the orbital angular momentum as well as with the spin. The total electronic wavefunction can be written in a perturbation expansion as @rn=#rn+ ak(Pk k#m whereby all zero-order solutions #k are large-scale MRD-CI wavefunctions. The contributions of the perturbing states of different symmetries arise form the spin-orbit interaction and the corresponding coefficients akare evaluated as ak = ((Pk Iffsol 4m)/ (&rn-&k). For the X "; a Idgand b lZ; states in 0 and S the following perturbing states have been for X "c the states 'Z; and 3v111g for a lAg the state 311g and for b 'Cg the states "c and 311g. The results are contained in table 1.First good agreement between the observed data for 0 and those obtained theoretically is seen. All values reported for S are predictions. Secondly in contrast to measurements the calculations allow an analysis of the transition mechanisms. It is found that the dominant term in the b 1Cg+-3Cg transition moment is the spin contribution (spin flip in ";) to the magnetic dipole operator b1Cg-3C; (m,= _+ 1) transitions corresponding to a much larger lifetime of 6 x lo6 s. The major term determining the lifetime of a lAg is the orbital angular moment term in the magnetic dipole operator and arises from lAg-lIIS and X3C,-311g transitions in the perturbed X3Cg and a lAS S. D. PEYERIMHOFF 75 Table 1. Calculated lifetime~~~-~l (in s) for the first two excited states in various molecules possessing the same electronic configuration.Experimental values whenever available are given in parentheses. 02 s2 so NF NC1 b lZ+ 11.65 11.0" 3.4 0.013 0.018 0.002 (12) -(0.007) (0.015 0.022) (0.0006) a lA 5270,4330" 350 0.45 3.13 0.8 (3900) -(1 5.6) (0.002) b lP-u lA 720 780 450 -428 (400 within a --factor of two) " Two states of each perturbing symmetry are included in a slightly different A0 basis and type of treatment. wavefunctions. The lifetimes in S are smaller since the spin-orbit matrix elements are considerably larger and hence the perturbers to the zero-order wavefunctions are more important. The b lC,'-a lAS transitions arise from quadrupole interaction and are thus independent of spin-orbit effects i.e.show approximately the same intensity in 0 and S (table 1). In the heteronuclear SO molecule two states of 39111 3Z-and lC+ symmetry have been included in the perturbation expansion. The lifetimes are considerably shorter because the absence of the inversion symmetry favours electric (rather than magnetic) dipole processes. The b lC+-X 3C-transition borrows its intensity predominantly from terms connecting b lZ+-2 lC+ and X 3Z-2 3Z- which occur as perturbers in the pure-spin wavefunctions. For a lA the calculations predict intensity borrowing from A 311-X 3Z-and C 311-X 3C-as well as lA-lll and lA-2 lII dipole transitions. Mechanisms in the other two heteroatomic systems which have been calculated in a comparable manner,61 are equivalent.Finally an interesting study6 is the absorption intensity of the 2 lIIUand 2 311u states in Cl,. All states up to 90000 cm-l have been allowed to couple via the spin-orbit process with the 31Tu state but it was found that only the 2 llTU state couples to any significant extent. The oscillator strength for the singlet X Xi-2 lIIu transition evaluated in the usual way as indicated is 0.07; the triplet is found to be weaker only by a factor of 8 in good accord with absorption experiments which estimate a factor between 5 and 9. In summary it is seen that calculations are able to predict lifetimes of excited states with good accuracy over a timescale of twelve powers of ten from to > lo3 s. This possibility together with the energy predictions makes modern CI-type calcu- lations a very valuable tool for spectroscopic studies with a very wide field of application.The author wishes to thank all her colleagues and associates who have helped obtain the results reported in this work in particular Prof. R. J. Buenker (Wuppertal) Dr M. PeriC (Beograd) and Drs P. J. Bruna C. M. Marian and R. Klotz (Bonn). 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