Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

Vþ ¼ V0 þ V2 and V ¼ V0 V2 (actually, effective operators acting onto functions of r and f), corresponding to the zeroth-order vibronic functions of the form cos ðy fÞ and sin ðy fÞ, respectively. PL–H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electronic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electronic state was assumed to be of quartic order in r, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and l ¼ K ) to keep the problem tractable by means of simple perturbation theory; they are, however, of no direct concern to us. By deriving the parameters entering their model from the experimental data for unperturbed K ¼ 0 vibronic levels, PL–H succeeded in achieving a near coincidence between theoretical and experimental results for all vibronic species. They concluded their paper with the statement that their theory was ‘‘limited by a number of severe approximations.’’ Note that Brown and Jørgensen’s opinion about this model is not very favorable, but for other reasons [12].

At this place, we make a chronological jump to comment on the paper by Dixon [35], representing a direct continuation of the story about the NH2 spectrum. This work was motivated by new experimental results (unpublished at that time) by Ramsay et al. for low-lying vibronic levels of the upper electronic state. They indicated, particularly in connection with the predictions based on the Walsh’s rules [70], that the upper state could have a slightly nonlinear equilibrium geometry. Thus Dixon allowed both bending curves to have a double minima. These potentials were approximated by a combination of quadratic functions and Gaussian functions. The model was otherwise equivalent to that of PL–H (both of them neglect the coupling of the bending vibrations with the stretching modes and end-over-end rotations, as well as the spin–orbit coupling, and employ the zeroth-order kinetic energy operator). The vibronic problem was solved by diagonalization of a truncated infinite Hamiltonian matrix using a computer. It was found that the molecule is in the upper electronic state quasilinear (equilibrium bond angle of 144 ), with only a single K ¼ 0 level lying below the barrier to linearity. Dixon’s results significantly improved the agreement between theory and experiment.

The first theoretical handling of the weak R–T combined with the spin–orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL–H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator corresponding to values ¼ 1=2. The spin–orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the

vibronic quantum number K and the spin quantum number are conserved. The main conclusions following from this study were: the positions of K ¼ 0 levels are shifted with respect to the case when the spin–orbit coupling is neglected and their þ= classification ceases to be precise; the lowest K ¼6 0 levels are due to the spin–orbit coupling split into pairs of levels, separated from each other by the value of the spin–orbit coupling constant; all other K ¼6 0 vibronic levels are only slightly split through the spin–orbit interaction. Pople’s study concerns doublet electronic states, but it can be modified straightforwardly to other multiplicities. Unfortunately, formulas (3.7) and (3.8) in the original work are erroneous as observed by Hougen, who corrected them in a subsequent study [72] (see also [5]).

The expressions for the rotational energy levels (i.e., also involving the end- over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the ‘‘isomorfic Hamiltonian’’ was introduced, which has later been widely used in treating linear molecules (see, e.g., [55]).

The first handling of the R–T effect in a electronic state of a triatomic was carried out experimentally and theoretically by Merer and Travis [73]. It concerned the A2 state of CCN. These authors derived the second-order perturbative formulas for the combined effect of the weak vibronic and spin–orbit couplings. The splitting of the bending potential curves due to the R–T interaction was assumed to involve a single term being of fourth order in the bending coordinate; on the other hand, the mean adiabatic potential was assumed to be harmonic. Curiously enough, also in this case the perturbative formulas printed in the original reference were not correct (caused by a trivial error of a factor of 4 concerning the norm of the basis functions used, see [12]).

E.Pragmatic Models

We shall now comment in some detail on the approach developed by Barrow, Dixon, and Duxbury (BDD) [74], historically the first one of those we classify in the category of ‘‘pragmatic’’ ones. BDD extracted from the complete vibration– rotation Hamiltonian the terms describing the bending vibrations and the z-axis rotations. They employed the operator derived by Freed and Lombardi (FL) [75], differing from Eqs. (35)–(37) in the choice of the molecule-bound coordinate system. In the FL’s Hamiltonian, the axes of the moving system are attached to the instantaneous principal moments of inertia of the molecule, being the optimal choice for handling the molecular rotations. In the case of symmetric triatomics (ABA) undergoing infinitesimal stretching vibrations, the axes of the HC’s molecule-bound frame [48,50] coincide with those preferred by FL (note that the last term in the FL’s operator H3 should be multiplied by 2 to become equal to the correspond term in the HC Hamiltonian).

The first form of the BDD Hamiltonian corresponds to the bending plus z-rotation part of the FL Hamiltionian for symmetric (ABA) molecules, supposed to undergo infinitesimal stretching vibrations. BDD corrected this operator following the idea by FL that the high-frequency stretching vibrations and the low-frequency bending can be handled in a way analogous to the treatment of electronic and vibrational motions in the framework of the usual BO approximation. Thus they carried out integration over the stretching coordinates (assuming parametric dependence of the stretching vibrational frequencies on the bending coordinate r) and incorporated the leading part of the stretch–bend coupling, taken in the second-order perturbation theory, into the bending Hamiltonian. This operator has nearly the same form as the zerothorder bending Hamiltonian by HBJ [60]. In handling the R–T effect, BDD carried out a contact transformation, chosen to diagonalize all of the 2 2 matrix, representing the effective bending operator, except the nuclear kinetic energy operator. This ansatz unifies the ‘‘bent’’ and ‘‘linear’’ ones described above: for small values of r the BDD secular problem reduces to (31), at large r values to (24.). The bending potential energy curves were assumed to be that of a harmonic oscillator perturbed by a Lorenzian hump, and the system of coupled R–T equations was solved by the Cooley–Numerov numerical integration technique [76]. The spin–orbit part of the Hamiltonian was assumed in the phenomenological form (16). The end-over-end rotations were handled separately. The B rotational constant was computed as an average value of the expression involving the reciprocal value of the instantaneous principal moment of inertia corresponding to the bisector of the valence angle. An accurate calculation of the C rotational constant is somewhat more complex; because of the Coriolis interaction, the factor multiplying the operator Jy2 in the expression (37) does not reduce to 1=ð2Iyy0 Þ even if symmetric triatomics undergoing infinitesimal stretching vibrations are considered. This becomes, however, the case when the adiabatic transformation analogous to that described above (for taking into account the stretch–bend interaction) is applied [75]; for a more refined treatment see the original [74].

The BDD approach has been applied in a number of studies that employ the parameters derived from the experimental findings [77–85]. The approach has been extended by Duxbury an co-workers, particularly Alijah; in its present version, involving the new stretch-bender Hamiltonian [84,85], which follows the idea by HBJ [60], it approaches the methods we tentatively call ‘‘benchmark.’’

The approach developed by Jungen and Merer (JM) [24] is of a similar level of sophistication. The main difference is that JM prefer to remove the coupling between the electronic states by a transformation of the Hamiltonian matrix (i.e., vibronic energy matrix), rather that of the Hamiltonian itself. They first calculate the large amplitude bending functions for one of the adiabatic potentials, as if it belonged to a electronic state. These functions are used as

620 miljenko peric´ and sigrid d. peyerimhoff

the basis for matrix representation of the secular problem given by Eq. (31). In the next step, the off-diagonal elements of the Hamiltonian matrix are minimized by a similarity transformation. The resulting matrix is finally diagonalized.

JM employ the semirigid bender Hamiltonain by Bunker and Landsberg [63]. They also neglect the x, y rotations in handling the vibronic problem, that is, they assume K to be a good quantum number. The spin–orbit operator is taken in the phenomenological form involving only z components of the angular momenta. On the other hand, their approach allows considering the dependence of the mean value of Lz on the bending coordinate. JM applied this approach to calculate the vibronic spectra in the X2B1, A2A1 (2 u) state of NH2 and H2Oþ [25,26] and the A1 u state of C3 [27]. They used the potential energy curves derived by fitting of experimentally determined positions of K ¼ 0 levels, not undergoing R–T coupling. The results of these calculations impressively demonstrated that their approach was able to reproduce reliably not only the positions of all K ¼6 0 vibronic levels measured, but also very fine effects like erratic pattern of the spin–orbit splitting of these levels and the variation of the rotational constants from level to level.

The approach having been employed by the present authors in their ab initio handling of the R–T effect in a series of triatomic molecules (these results are reviewed in [16,17,21,86]) is not very different from the two described above. We have employed the same kind of the kinetic energy operator for large amplitude bending and the same form of the spin–orbit operator as DBB and JM. The vibronic energy levels and wave functions have been computed variationally, by employing as basis functions either the eigenfunctions of a suitably chosen 2D harmonic oscillator [21], or Fourier series in r [86]. All matrix elements appearing in the vibronic secular equations are computed by using simple recurrence formulas. Although our program package allows for handling of the R–T effect along both the ‘‘linear’’ and ‘‘bent’’ formalisms, the great majority of the calculations have been carried out in the framework of the first one.

The use of ab initio computed potentials and other relevant quantities has its advantages, as well as its drawbacks. The greatest advantage is that there are no problems like those that are caused by a shortage or insufficient quality of experimental data. Further, some quantities that are difficult to extract from the experimental findings, like variation of the bond lengths or the mean value for Lz upon bending, are easy to compute. The greatest drawback of a pure ab initio handling of the R–T effect is the limited accuracy of quantities entering the model Hamiltonan, particularly of potential energy surfaces. However, the high accuracy achieved in the handling the R–T effect by using the potential curves and structural parameters derived by fitting the experimental data can be deceptive, because it can be based on cancellation of the errors in the potential

and kinetic energy part of the model Hamiltonian. An example is the X2B1, A2A1 (2 u) system of H2Oþ: The calculations of JM [25,26] excellently reproduced all the experimental findings, although the potentials they employed were based on an incorrect numbering of the vibronic levels observed, as it was shown in later ab initio studies [87,88]. Let us also note that the discrepancy between the ab initio computed bending potential curves and their counterparts derived by fitting the experimentally observed features must not be automatically ascribed to the inaccuracy of the former. The ‘‘experimental’’ potentials correspond to expressions with a certain number of free parameters that are chosen so that the eigenvalues of the 1D model Hamiltonian (into which these parameters enter) match experimentally observed vibronic levels. In this way, they effectively incorporate all kinds of coupling with the other degrees of freedom and the other electronic states. On the other hand, the ab initio potential curves are well-defined 1D sections of the three-dimensional (3D) potential surfaces computed in the framework of the BO approximation. A part of the coupling with the other modes and states can be indirectly incorporated, but this always represents an approximation. Thus both sets of curves do not represent exactly the same quantity; they obtain the same meaning only if the 1D approach is realistic. This matter has been discussed in detail in [17].

The essential equivalence of all three approaches for handling the R–T effect presented in this section have been demonstrated through the computations in which the same input data have been used, that is, in [78] BDD and JM are compared and in [21] and [86] our approach with these two. The result of the latter two studies showed that JM had exaggerated claiming that for a direct diagonalization of the vibronic matrix ‘‘it would be necessary to chose an enormous basis in order to avoid truncation errors’’ [24]. We were able to reproduce their results by diagonalizing matrices of dimensions < 100. With this observation we do not want to question the general utility of the Hamiltonianor matrix transformations implemented in the approaches by BDD and JM; in the approaches tailored to lean on experimental findings such a subtle handling is of much more relevance than in the ab initio calculations where in some steps the brute force philosophy can be applied without undesirable consequences.

F.Benchmark Handling

Let us first stress that the program of a ‘‘benchmark’’ handling of the R–T effect, as presented in Table I, represents an idealization; in none of the studies that have been published thus far has it been realized in all points.

The most consequent and the most straightforward realization of such a concept has been carried out by Handy, Carter, and Rosmus (HCR) and their coworkers. The final form of the vibration–rotation Hamiltonian and the handling of the corresponding Schro¨dinger equation in the absence of the vibronic

coupling is a result of a short but exciting discussion between HC and Sutcliffe [48,49,89,90]. In variational handling of the R–T effect in singlet electronic states, HC employ the bending–electronic–rotational basis functions of the form [50]

(in our notation). Note again that W denotes the bond angle, and not the coordinate conjugate to Lz. The functions Pðcos WÞ are associated Legendre polynomials of degree n2 ¼ 2u2 jlj, where jlj ¼ jK j. The functions e are the electronic species of type (26), and DJoK ðb; gÞ are the M ¼ 0 (end-over-end) rotational wave functions depending on the Euler angles b and g. The bond stretching expansion functions are expressed in terms of Morse oscillator basis functions. In handling doublet electronic states, this basis is completed by appropriately chosen spin functions [51]. The matrix elements of the Hamiltonian (35)–(37) in this basis are obtained partly analytically and partly by a numerical integration. The only good quantum number assumed in this treatment of the R–T effect is J, corresponding to the total angular momentum of the molecule.

Handy and co-workers are certainly right in claiming that they use ‘‘probably the most appropriate general Hamiltonian’’ [48]. However, in praxis they solve the corresponding Schro¨dinger equation by making several approximations, some of them being avoided in another treatments: They neglect the geometrical dependence of the mean value of Lz, use the spin–orbit operator in the phenomenological form (16), and assume the spin–orbit coupling constant to be really a constant.

HCR and co-workers carried out a number of studies by employing 3D potential energy surfaces calculated by means of highly sophisticated ab initio approaches [88,91–101]. The results of these computations are in impressive agreement with the corresponding experimental findings. The discrepancies in the order of 100 wavenumbers, as in early ab initio studies [16,17], have been reduced in the HCR studies to only a few wavenumbers. In conclusion of their paper on the X2B1, A2A1 (2 u) system of NH2, Gabriel et al. state: ‘‘We believe that the results presented in this paper are near as possible definitive from the theoretician. It has been a major challenge to us for 15 years to be able to compute these properties of NH2 to such accuracy. . .’’ [97].

The excellent agreement of the results of HCR ab initio studies with the corresponding experimental findings clearly shows that the strongest influence on the numerical accuracy of the vibronic levels have effects outside of the R–T effect, that is, primarly the replacement of the effective bending approaches employed in previous works by a full 3D treatment of the vibrational motions (for an analysis of this matter see, e.g., [17]). Let us note, however, that such a

high level of accuracy seems never to have been achieved without a slight modification of the ab initio computed potential surfaces (typically, they have been shifted by 100 wavenumbers). This is at least partially caused by neglect of some fine effects, like for, example, non-adiabatic corrections of the potential surfaces. On this basis, it can be concluded that the HCR’ predictions concerning the yet unobserved spectra are somewhat less reliable.

An alternative ‘‘benchmark’’ approach for handling the R–T effect in triatomic molecules has been developed by Jensen and Bunker (JB) and coworkers. It is based on the use of ‘‘MORBID’’ Hamiltonian [66,67,102], a very sophisticated variant of the above described approaches that handle the bending motion in a different way than their stretching counterparts. This method is described in great detail in a recent book [2], so that we restrict ourselves here only to a small comment. It might look anachronistic (this approach postpones that of HCR) to develop a very ambitious approach not employing ‘‘probably the most appropriate general Hamiltonian.’’ A justification is given by JB in their book [15]: ‘‘However, one disadvantage (. . .of the approaches like HC’s. . .) is the fact that in practice, many (if not most) interactions between molecular basis states are weak and could be successfully treated by perturbation theory in the form of a contact transformation. In the variational approaches, these weak interactions are treated by direct matrix diagonalization at a high cost of computer time and memory.’’ We cannot judge if this sentence is relevant in the case of triatomics, but it certainly gains weight when the larger molecules are to be handled.

In several papers [102–105] JB presented the results of their calculations on CH2, CHþ2 , and BH2. We find their study of the X2A1, A2B1 state of BH2 particularly interesting [104], because this system was treated also by Brommer and HCR [94]. The results of JB et al. are of comparable accuracy with those by HCR; like the latter authors, JB were forced to modify their original ab initio potentials slightly to improve the agreement with the available experimental data. An attempt was undertaken to make a direct comparison of both approaches, but it did not lead to a final conclusion, because of difficulties in transforming the HCR potentials into the form they enter within the JB algorithms. Let us note that both works confirmed the conclusion of our old ab initio study [33] that the assignment of the bands observed in the of A2B1 X2A1 absorption spectrum, made by Herzberg and Johns [37], was not correct.

G.Effective Hamiltonians

Another group of approaches for handling the R–T effect are those that employ various forms of ‘‘effective Hamiltonians.’’ By applying perturbation theory, it is possible to absorb all relevant interactions into an effective Hamiltonian, which for a particular (e.g., vibronic) molecular level depends on several parameters whose values are determined by fitting available experimental data. These Hamiltonians are widely used to extract from high-resolution [e.g.,

electron spin resonance (ESR)] spectra precise values for molecular parameters [19,106–111]. Since the main subject of this chapter is ab initio handling of the R–T effect, we refrain from description of the effective Hamiltonian approaches and refer instead to the excellent review published recently by Brown [3].

H.Beyond the Two-State Renner–Teller Effect

In many cases, the two electronic states building an R–T pair are energetically well separated from all other electronic species and the two-state model expressed by the ansatz (10) is quite reliable. However, there are also many exceptions from this rule. Even when a spatially degenerate electronic species represents the ground state of the molecule, its interactions with the other species, possibly at nuclear arrangements differing considerably from the equilibrium geometry of the molecule (as found, e.g., in the series of closely related molecules NH2 ½20&, PH2 ½112&, and SHþ2 ½113&), can take place. When the R–T state is an excited electronic species, the interactions with the other electronic states represents a normal situation. The energetic vicinity of neighboring species has dramatic effects on the vibronic structure within the R–T state. This topic has been investigated by many authors, particulary by Ko¨ppel, Domcke, and Cederbaum [13,14,114]. We present here only an example.

The C2H radical, a species of great astrophysical–chemical interest and an important intermediate in many chemical reactions, has a 2 ground electronic state and an extremely low-lying 2 excited species. At the equilibrium geometry of the ground state, the energy difference between them is4000 cm 1, but it diminishes drastically upon C C stretching, so that at larger C C bond lengths the ordering of the states is reversed [115]. A consequence of these facts is a peculiar structure of the vibronic spectrum in both electronic states, which can be understood only in the framework of a coupled threestate electronic problem. This was realized already in early experimental studies carried out by Curl et al. [116]; for an exhaustive literature survey up to 1992 the reader is referred to [117]. The controversies concerning the X2 , A2 spectrum of C2H have motivated a series of ab initio studies on this three-state system, involving computations of its vibronic, spin–orbit and magnetic hyperfine structure [117–124]. The results of these studies contributed to elucidation of a number of measured spectral features and have been used in later experimental works to help in the assignment of the measured data [125–129]. On the other hand, relative simplicity of the theoretical treatment and low computational efforts caused restricted numerical accuracy of the results. Ten years later, Carter et al. [130] published a really impressive ab initio study, which was able to reproduce all available experimental finding quantitatively.

IV. TETRAATOMIC MOLECULES

A.Theoretical Treatment

1.General Remarks

The first evidence of the R–T effect in a tetraatomic molecule was reported by Herzberg in 1963 [131]. Contrary to the situation with triatomics, the first theoretical model (Petelin and Kiselev (PK) [132]) appeared almost 10 years after Herzberg’s observation. The theory of the R–T effect in tetraatomic molecules is much more complicated than in triatomics, because of the existence of two bending modes. PK elaborated a perturbative approach for singlet electronic states, which described several special coupling cases, the majority of them concerning the situation in which only one bending vibration is excited. Thus most of the perturbative formulas derived were effectively equivalent to their counterparts in the framework of the classical Renner’s theory for triatomics. The equations derived by PK have been used by Colin et al. [133] for an analysis of the high-resolution absorption spectrum of acetylene in the

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1205–1255-A region and by several other authors who have studied the structure of spectra of highly exited (Rydberg) states of acetylene [134,135] and the ground state of C2Hþ2 [136–138]. This approach was extended by Tang and Saito [139] and applied to analyze the ground state of HCCS.

The idea of PK was employed by the present authors [140] who developed a variational approach for an ab initio treatment of the Renner–Teller effect in tetraatomic molecules. The approach by PK was extended to handle bothand electronic states and to treat the bending vibrations beyond the harmonic approximation. The main practical advantage of this method was of course that it enabled us to obtain the term values and the wave functions for all (bending) vibronic levels of interest. It was applied to compute the vibronic structure of two Rydberg-type electronic states of acetylene. Unfortunately, lack of corresponding experimental findings made it impossible to check the reliability of these results. The approach was later extended to take into account the interplay between the vibronic, spin–orbit [141], magnetic hyperfine [142,143] couplings, and the effects of noninfinitesimal bending vibrations [144,145].

For a long time after Herzberg’s observation [131], the experimental information on the R–T effect in tetraatomic molecules has been very scarce. The situation has changed, however, in the last decade in which a series of experimental studies on the Rydberg states of acetylene [134,135,146], and of the ground state of the acetylene ion [136–138,147] has been published. This made it possible for us to judge the validity of the ab initio approach proposed [140]. The results of the ab initio computations enabled a very reliable interpretation of all available experimental findings concerning the ground state