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

of C2Hþ2 [141,148]. The method was also applied to predict the vibronic structure of heretofore unobserved spectra of B2H2 (11 g electronic state) [149,150] and B2Hþ2 (X2 u state) [142,143] and to interpret the spectra of HCCO [145,151] and HCCS [152]. The key points of this approach will be described in Section IV.2. For a more detailed description, the reader is referred to the original references and the reviews [18,153,154].

2.Hamiltonian

The complexity of the problem makes it almost imperative to employ every sensible simplification. For this reason, we immediately exclude from consideration the stretching vibrations and the end-over-end rotations. Since we will restrict ourselves to linear and quasilinear tetraatomic molecules (ABCD), this represents a quite acceptable approximation. We are left thus with four nuclear coordinates. We make the following choice: the supplement of the ABC bond angle, r1; the supplement of the BCD bond angle, r2 ðr1 ¼ r2 ¼ 0 at the linear molecular geometry); the angle between the plane ABC and a space-fixed plane with the common z axis (coinciding with the BC bond length), f1; the angle between the plane BCD and the same space-fixed plane, f2. We denote the bond lengths A B, C D and B C by r1, r2, and r3, respectively.

The complete vibration–rotation Hamiltonian for acetylene-like tetraatomic molecules has been derived by Handy et al. by hand [155] and using a computer algebra program [156]. (Note that in both of the mentioned papers there are some minor errors, see also [144,157,158]). Handy uses as bending coordinates W1 and W2, connected with those we prefer by the relations W1 ¼ p r1 and W2 ¼ p r2, and instead of our f1 and f2 the torsional and z-rotational coordinates g and f, respectively. In [144], from this Hamiltonian was extracted the part involving derivatives with respect to the above mentioned coordinates, and the volume element sin W1 sin W2 dqi was replaced by r1 r2 dqi. This resulted in a two-bending/z-rotation Hamiltonian given by Eqs. (6)–(8) of [144], which allows the treatment of the large-amplitude bending vibrations at constant bond lengths. This kinetic energy operator may look oversimplified. Indeed, some more sophisticated variant can be used, for example, by taking into account the bend–stretch coupling in line with the BDD [74] or HBJ [60,63] approach. Note that there already exist several effective bending operators derived to describe reliably large-amplitude bending in quasilinear tetraatomics [159,160]. We find, however, that there are other type of problems, that are more important for the understanding of spectral features and leave the discussion about optimal kinetic energy operator for a future review. In this chapter, we make a further simpification by considering only the small-amplitude form of

are the reduced masses for the trans and cis bending vibrations respectively. m mA, M mB, r A–B, R B–B.

3.Vibronic Problem

The model Hamiltonian we use is of the form

where He represents the electronic operator and T the nuclear kinetic energy operator (46) or (49). The vibronic wave functions describing the situation when two electronic states are coupled with each other are assumed in the same general form as for triatomics,

where c1 and c2 are electronic species, and f1, f2 are functions of nuclear (bending, torsional, and z rotational) coordinates. In [18,153], various possible electronic basis sets for ab initio handling of the R–T effect in tetraatomic molecules were discussed. In the present approach, we employ the basis consisting of the functions

where cþ and c represent the solutions of the electronic Schro¨dinger equation in the framework of the BO approximation (we assume c to be imaginary):

t is a ‘‘rotational angle,’’ which determines the spatial orientation of the adiabatic electronic functions cþ and c . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a state one of the electronic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R–T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)

reduces in triatomics to the angle f between the instantaneous molecular plane and a space-fixed plane with the common z axis. However, at an arbitrary nuclear arrangement, a tetraatomic molecules does not possess any symmetry elements and thus the orientation of its electronic wave functions, that is, the angle t, cannot be determined on the basis of symmetry considerations. The angle t in such cases is a function of all nuclear coordinates involved.

The relationship between the angle t and the nuclear coordinates considered can be derived in the framework of the model analogous to that developed by Pople and Longuet-Higgins [69] for triatomic molecules. Let us consider a tetraatomic molecule in an electronic state that is twofold spatially degenerate at linear nuclear geometry, at which it is described by the Hamiltonian H0. We represent the molecular Hamiltonian at bent geometries as a sum of H0 and an additional (electronic) part H0, disappearing in the linear limit. We assume the basis electronic wave function of the system in the form given by Eq. (28), we denote now j i and j i . We suppose that H0 consists of two terms, the first, V, having only diagonal elements in this basis (being the same for both basis functions), and the second, W, coupling the basis electroinic states. The matrix representation of H0 is thus

h j j i h j j i

The first-order energy correction with respect to the unperturbed problem is then

W W

The quantities V given by Eq. (56) represent the first-order approximation for

the adiabatic bending potentials. If these potentials are known, V can be

630 miljenko peric´ and sigrid d. peyerimhoff

infinitesimal-amplitude bending to (28) with the same meaning of the angle y; we may employ these asymptotic forms in the computation of the matrix elements of the kinetic energy operator and in this way avoid the necessity of carrying out calculations of the derivatives of the electronic wave functions with respect to the nuclear coordinates. The electronic part of the Hamiltonian is represented in the basis (53) by

The matrix elements (60) represent effective operators that still have to act on the functions of nuclear coordinates. The factors expð 2i tÞ determine the selection rules for the matrix elements involving the nuclear basis functions.

In a general case, we assume the potential energy part of the Hamiltonian in the form of an expansion involving the terms rma rnb cos ½kðfb faÞ& (a and b stand for 1 and 2 in the case of ABCD molecules, and for T and C if we consider symmetric ABBA species), subject to certain symmetry constraints (see [144]).

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians Ha0 ¼ Ta0 þ 1=2kar2a and Hb0 ¼ Tb0 þ 1=2kbr2b,

where the functions Ru;l are defined by Eq. (8). It is easy to verify that the model Hamiltonian we use commutes with the projection of the total angular momentum (excluding spin) onto the z axis, Nz,

It follows that the only possible values for la þ lb are K and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator He, diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number l ¼ la þ lb. The off-diagonal elements of He [second and third of Eqs. (60)] connect the basis functions with l ¼ la þ lb and l0 ¼ l0a þ l0b ¼ l 2 .

The spin–orbit coupling in vibronic states is taken into account assuming the spin–orbit part of the Hamiltonian in the phenomenological form is given by Eq. (16). This operator is added to (51) and the total Hamiltonian is diagonalized in the basis of the above basis functions, assumed that the electronic functions are eigenfunctions of the spin operator, with the eigenvalue . The total model Hamiltonian (including spin–orbit operator) commutes with the projection of the total angular momentum on the z axis so that the vibronic secular equation can be solved for each value of the quantum number P ¼ K þ separately. In the lowest order approximation (Aso assumed to be constant) with the electronic basis (53) the spin–orbit contribution to the total model Hamiltonian is diagonal with respect to all quantum numbers ( ; la; ua; lb, and ub) labeling the basis functions.

Although the present approach formally does allow for a treatment of large amplitude bending vibrations (when the corresponding kinetic energy operator and the form of the potentials also involving higher order terms in nuclear coordinates is applied) the restrictions of the model imply a more limited viewpoint. The most critical point is the estimate for the angle t, being related to the low-order approximation of the molecular potentials (this topic is discussed in [145]). Thus we expect that the model in its present form will be reliable at relatively small-amplitude bending vibrations (typically ra; rb 40 ) around linear equilibrium geometry.

In Section IV.A.4, we show what this general model looks like in the case ofelectronic states of symmetric tetraatomic molecules. The situation in states of asymmetric tetraatomics is briefly discussed in Section IV.B, where we present the handling of a concrete case, the X2 u state of the HCCS radical. Forstates the reader is referred to original references [18,149,150,153].

4.Electronic States of ABBA Molecules

In this section, we consider electronic state ð ¼ 1Þ of ABBA type molecules. The additional Hamiltonian H0 is of the form

632 miljenko peric´ and sigrid d. peyerimhoff

As already noted by PK, [132], the term involving VTCðrT; rCÞ exp½ ið2y fT fCÞ& does not appear for molecules belonging to the D1h point group for symmetry reasons.

In the harmonic approximation, V does not involve the cross-term rT rC

where eT and eC are dimensionless parameters analogous to the Renner parameter

First, let us note that the adiabatic potentials Vþ and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles fT and fC; this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate r. The forms of the functions V, VT , and VC are determined by the adiabatic potentials

Thus the function VT represents one-half of the splitting of the adiabatic potentials computed for pure trans bending, and VC one-half of the splitting of the cis-bending curves.

By employing the angle t defined by (68), the perturbative Hamiltonina H0 can be formulated in the form completely analogous to the Pople and LonguetHiggins’ ansatz [69]:

Thus the angle t plays the role analogous to that of the angle f defining the orientation of the instantaneous molecular plane in triatomic molecules. Employing the relations (69) and (59) one obtains

Thus in the lowest order approximation the angle t is eliminated from the offdiagonal matrix elements of He [second and third of Eqs. (60)]; it solely determines the selection rules for matrix elements of He with respect to nuclear basis functions.

B.An Example: X2 Electronic State of HCCS

To illustrate the reliability of the above described model for handling the R–T effect and spin–orbit coupling in tetraatomic molecules, we present here the results of calculation of the spectrum for the X2 u state of the HCCS radical. In spite of its importance from the astrophysical–chemical point of view and due to its relationship to a number of other radicals of interest involving carbon, hydrogen, sulfur, and oxygen, HCCS has been subject to a relatively small number of experimental and theoretical studies, which all have left a number of questions open. An absorption spectrum detected by Krishnamachari and

˚

Venkitachalam [161] in the region 3770 and 4170 A during the flash photolysis of thiophene, and tentatively assigned to C4H3, was latter attributed by Krishnamachari and Ramsay [162] to a 2 –2 electronic transition of HCCS. The complexity of the spectrum, characterized by large spin–orbit splitting in several bands, precluded a definitive determination of the vibrational fundamentals, particularly for the bending modes [163]. The value for the spin–orbit coupling constant of 185 cm 1 was estimated [164]. An extensive study of the R–T effect in the ground electronic states of HCCS and DCCS was undertaken by Tang and Saito [139]. They derived perturbative formulas for the coupling cases

not considered by PK [132] and applied it to interpret the microwave spectra obtained in the frequency range between 160 and 400 GHz. In the simulation of the spectrum they used the Renner parameters ab initio computed by Szalay [165], and concluded that a reliable reproduction of experimental findings was obtained with the value of the spin–orbit coupling constant assumed to be 270 cm 1. This value disagreed with the results of ab initio computations by Goddard ( 360 cm 1) [166] and Szalay and Blaudeau ( 347 cm 1) [167], as well as with the estimate of the previous experimental study by Vrtilek et al. [164].

An extensive ab initio study was thus undertaken with the goal of providing a reliable interpretation of the available experimental findings, and to predict the structure of yet unobserved parts of the HCCS and DCCS long-wavelengths spectra [152]. Potential energy surfaces for the electronic states of the HCCS radical correlating at linear nuclear arrangement with the X2 state were calculated by means of a large scale multireference configuration-interaction approach (for details see [152]). Particular attention was paid to calculate accurate 3D potential surfaces involving variations of two bending and torsional coordinates that play the central role in vibronic interactions, determining, together with the spin–orbit coupling, the structure of spectra of this radical.

In the lowest order (quadratic) approximation for electronic states of asymmetrical (ABCD) tetraatomics, the electronic matrix elements (60) have the forms [18,152,153]:

In this approximation, the angle t does not appear explicitly in the vibronic secular equation. From Eq. (76) it follows that

k1; k2; k12; e01; e02, and e012 from the mean adiabatic potential and the difference of the adiabatic potentials for two components of the electronic state spatially degenerate at linear molecular geometry.

Thus only six parameters are required to determine the potentials entering our model for handling the R–T effect in asymmetric tetraatomic molecules