Nonadiabatic interactions between the ground and low-lying excited electronic states: Vibronic states of the Cl HCl complex

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 13 1 OCTOBER 2001 Nonadiabatic interactions between the ground and low-lying excited electronic states: Vibronic states of the Cl HCl complex Petra Žďánska J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Czech Republic and Center for Complex Molecular Systems and Biomolecules, Dolejškova 3, Prague 8, Czech Republic and Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Dana Nachtigallová J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Czech Republic and Center for Complex Molecular Systems and Biomolecules, Dolejškova 3, Prague 8, Czech Republic Petr Nachtigall J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Czech Republic and Center for Complex Molecular Systems and Biomolecules, Dolejškova 3, Prague 8, Czech Republic and University of Pardubice, Faculty of Chemical Technology, Čs. legií 565, Pardubice, Czech Republic Pavel Jungwirth a) J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Czech Republic and Center for Complex Molecular Systems and Biomolecules, Dolejškova 3, Prague 8, Czech Republic Received 29 March 2001; accepted 18 June 2001 The Cl HCl radical complex is investigated by a combination of accurate ab initio quantum chemical methods for the evaluation of the three lowest electronic potential energy surfaces and nonadiabatic couplings between them, and quantum evaluation of vibronic states using wave function propagation in imaginary time within a close coupling scheme. The sensitivity of the vibronic energies on the quality of the potential surfaces is clearly demonstrated. Moreover, it is shown that nonadiabatic couplings between the three lowest electronic states play an important role, especially for highly excited vibronic states. Since under experimental conditions the complex is prepared in a superposition of excited vibronic states close to the dissociation limit, the inclusion of nonadiabatic effects is crucial for a quantitative interpretation of future higher resolution spectroscopic experiments American Institute of Physics. DOI: / I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: jungwirt@jh-inst.cas.cz Nonadiabatic interactions between electronically excited states 1,2 play a prominent role in many photochemical processes, whenever situations like avoided curve crossings or conical intersections occur. 3 5 In contrast to that, ground electronic state structures and properties can be, as a rule, adequately described within the usual Born Oppenheimer separation of electronic and nuclear degrees of freedom. This is due to the fact that the ground electronic state is usually well energetically separated from the lowest excited state, at least in the vicinity of the minimal structure, spanned by the nuclear vibrational wave functions. However, there exist systems that do not satisfy the above energy condition, such as certain open-shell radicals and radical complexes. Especially in cases when such systems exhibit large amplitude nuclear motions such as libration or hindered rotation, the validity of the Born Oppenheimer approximation is no more obvious. Here we address the question of nonadiabatic couplings between low-lying electronic states and their role in an accurate description of intermolecular vibrational states within a case study of a Cl HCl complex, in which a chlorine radical is weakly bound to a hydrogen chloride molecule. The ClHCl species has been studied both experimentally and theoretically by numerous groups in the last decade. Photoelectron spectroscopy experiments and vibrational calculations have been pursued for the electron photodetachment process from the ClHCl anionic precursor The photodetachment experiment directly accesses the transition state of the symmetric hydrogen exchange reaction ClHCl HClCl, which is an important prototype of a reaction of a light atom between two heavy atoms. 11,12 This process, together with the strongly exothermic Cl 2 H HClCl exchange reaction, has been intensely investigated computationally On the experimental side, the ClHCl collision has been probed by high resolution IR laser dopplerimetry. 22,23 Most relevant to the present study is the fact that the Cl HCl complex has been recently directly prepared independently in two laboratories. 24,25 The experiments employ either an IR-UV double resonance 24 or a Doppler-selected time-of-flight technique 25 to prepare Cl HCl by a bondspecific photodissociation of the hydrogen chloride dimer /2001/115(13)/5974/10/$ American Institute of Physics

2 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Vibronic states of Cl HCl 5975 These experiments lead to a nascent complex in a coherent superposition of excited intermolecular vibrational states on the three lowest electronic surfaces, qualitatively corresponding to three possible orientations of the singly occupied p-orbital of the chlorine radical. 24 As recognized in earlier calculations, 26 the Cl HCl complex exhibits a rather complex vibrational structure on a potential energy surface, which is shaped primarily by electrostatic and spin orbit interactions. As a result, a large amplitude bending librational motion, which is coupled to the intermolecular stretch vibration, occurs. Recently, a similar behavior has been observed also in analogous F HF and Br HBr radical complexes. 27,28 In our previous study, we have exploited unusual properties connected with the quantum delocalization of hydrogen in a shallow bending potential of the Cl HCl complex for suggesting a novel photochemical control scheme. 29 Namely, we have shown that the quantum yield of the production of molecular chlorine via UV photodissociation of the Cl HCl complex can be efficiently controlled by an IR preexcitation of the hydrogen libration. The principal goal of the present study is to revisit the vibrational states of Cl HCl with a twofold aim. First, we base now vibrational calculations on accurate intermolecular potentials obtained by reliable ab initio methods. We acknowledge here the ingenuity of previous investigators who have constructed amazingly good potential surfaces for this complex, given their empirical approach based purely on physical intuition. Nevertheless, the authors recognize that this potential will be superseded once high-quality ab initio calculations tailored for the long-range region become available. 26 It is our ambition to provide here such calculations. Second, we construct from accurate many-electron wave functions also ab initio surfaces of nonadiabatic couplings between the three lowest electronic surfaces and investigate the effect of nonadiabatic couplings on vibronic energies. Since this effect is expected to be most important for highly excited vibrational states of the complex, it becomes particularly relevant in view of the fact that the Cl HCl system is prepared experimentally with large amounts of internal energy. 24 The paper is organized as follows: In Sec. II we describe the system under investigation. Sections III and IV provide the details of the electronic structure and vibronic calculations. In Sec. V we present and discuss the results, while a brief summary is given in Sec. VI. II. SYSTEM AND GEOMETRY The complex under investigation, consisting of a chlorine radical weakly bound to hydrogen chloride, is held together primarily by electrostatic forces, with secondary dispersion and induction contributions. The interactions between the quadrupole of the chlorine radical and the dipole and quadrupole of the hydrogen chloride molecule shape the three low-lying adiabatic potential energy surfaces, connected with the three possible orientations of the singly occupied p-orbital of the chlorine radical. Further reshaping and splitting of these potential surfaces is due to the spin orbit interaction, originating again from the chlorine radical. 26 A quantitative description of these three surfaces and a detailed characterization of relevant stationary points FIG. 1. A general geometry of the Cl HCl complex and definition of coordinates. T is the center of mass of the whole system, while t is the center of mass cms of the HCl molecule. R and r are the vibrational coordinates of the Cl cmshcl and HCl subsystems, respectively. and are rotational coordinates of the Cl cmshcl subsystem in the space-fixed frame, while and are rotational coordinates of the HCl diatomics in the body-fixed frame. The corresponding Jacobi coordinates are r, R, and. minima and saddles is presented in Sec. V. Here, it suffices to mention that the global minimum on the lowest potential energy surface corresponds to a collinear structure with hydrogen pointing towards the chlorine radical. However, all three surfaces also possess other minima. The vibrational motions of the triatomic Cl HCl complex are conveniently described in Jacobi coordinates r, R, and, denoting the HCl bond distance, the distance between the chlorine radical and the HCl center of mass cms, and the angle between these two coordinates. The Jacobi coordinates are depicted in Fig. 1 for a general geometry of the Cl HCl complex. In the same picture we also show space fixed Cartesian coordinates with origin in the center of mass of the whole complex together with rotational angles characterizing the R coordinate in this frame. The above coordinates are employed in the vibronic calculations described in Sec. IV. III. CALCULATION OF POTENTIAL SURFACES AND NONADIABATIC COUPLINGS The ab initio calculations of low-lying potential energy surfaces and nonadiabatic couplings of the Cl HCl system have been carried out within the C s symmetry. The H Cl distance has been kept frozen at the equilibrium bond length of HCl molecule of Å. The Cl Cl distance and Cl Cl H angle were calculated in the range Å and 0 180, respectively. The potential energy surface of the system for the lowest two A and the lowest A states have been calculated using the multireference coupled pair functional method MRACPF, employing an augmented correlation-consistent polarization valence-triple-zeta augcc-pvtz basis set. 34 In this way, effects of both static and dynamic correlations have been accounted for in a satisfactory way, and the present calculations provide relative energies that are close to convergence. Electrons in 1s orbitals of Cl atoms have been kept frozen and the basis set superposition error has been included using the standard counterpoise technique. 35 For construction

3 5976 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Žďánská et al. the chlorine radical is not diagonal in the basis of the adiabatic electronic states. Therefore, the spin orbit term further reshapes the potential energy surfaces. 26,37,38 In this study, we account for the effect of the spin-orbit interaction similarly as in Ref. 26. However, we employ a computationally convenient representation of the adiabatic electronic states within the double group formalism 39 for details see Appendix A. IV. CALCULATION OF VIBRONIC STATES The total Hamiltonian Ĥ for the Cl HCl complex in Jacobi coordinates can be written as FIG. 2. Orientations of the singly occupied p-orbital of the chlorine radical in the Cl HCl complex. a Space-fixed primitive diabatic basis p x, p y, and p z, and b p-orbital orientations corresponding to the three spin-free adiabatic states. of MCSCF wave functions only electrons in 2p lone-pair orbitals of Cl atoms have been included into the active space. Thus, these calculations have been carried out with 5 active electrons in 3 orbitals. Molecular orbitals have been optimized in state-averaged CASSCF calculations. Nonadiabatic coupling matrix elements between the two A states have been calculated by finite differences for MCSCF wave functions. A first-order algorithm has been used in which the wave functions has been calculated at two slightly displaced geometries. All ab initio calculations have been performed using the MOLPRO program package. 36 The three low-lying adiabatic states correspond in a oneelectron picture to the three possible orientations of the singly occupied p-orbital of the chlorine radical. These orientations are dictated by the electrostatic interactions between the Cl and HCl species. For a general bending angle in the Cl HCl complex, a simple electrostatic analysis, as well as ab initio calculations, show that the singly occupied p-orbital is perpendicular parallel to the HCl axis for the lowest first excited state of the A symmetry. 29 Finally, the out-of-plane perpendicular orientation of the p-orbital for the A state is fully determined by symmetry considerations. The orbitals corresponding to the three adiabatic states are depicted in Fig. 2b, while Fig. 2a shows a space-fixed primitive diabatic basis p x, p y, and p z. 26 The above one-electron adiabatic basis is employed for including the effect of spin orbit interactions. The spin orbit splitting between the J3/2 and J1/2 states of the chlorine radical amounts to cm 1 and is, therefore, comparable to actually larger than the binding energy of the Cl HCl complex. 26 Since for relevant geometries of the complex there is no sizable charge transfer between the HCl and Cl subsystems, the above spin orbit constant of the chlorine radical can be directly employed for the study of the Cl HCl system. 26 Nevertheless, the matrix representation of the spin orbit coupling operator lˆ ŝ (lˆ and ŝ are angular momentum and spin operators of the unpaired p-electron of 2 2 Ĵ 2 Ĥ R 1 2 ClHCl R 2 R 2 ClHCl R r 2 HCl r 2 r ĵ 2 2 HCl r 2 Ĥel. 1 Here, ClHCl m Cl m HCl /(m Cl m HCl ) is the reduced mass of the Cl cmshcl subsystem, Ĵ 2 is the square of the angular momentum operator corresponding to the end-over-end rotation, HCl m H m Cl /(m H m Cl ) is the reduced mass of the HCl subsystem, ĵ 2 is the square of angular momentum operator for the HCl rotation see also Fig. 1, and Ĥ el is the electronic Hamiltonian including the spin orbit term. The total wave function (r e,r,,,r,,,) is dependent on electronic degrees of freedom r e, the Cl HCl radius R, the Cl HCl angular coordinates and, the H Cl distance r, and HCl angular coordinates, see Fig. 1. The meaning of the time variable is clarified below. The total wave function is first expanded in the basis of three electronic wave functions n, parametrically dependent on nuclear coordinates r, R, and. These electronic wave functions correspond to the three spin orbit free adiabatic electronic states n as obtained from ab initio calculations described in the previous section. The index signs the irreducible representation of the spin double group, to which the spin orbit symmetrized electronic basis belongs. The spin functions are included using the one-electron approximation, which is discussed in detail in Appendix A. In the next step, the nuclear wave function on each electronic state is expanded within a close coupling scheme into a dual rotor basis JMjm in the uncoupled representation, 40 where J,M and j,m are rotational quantum numbers corresponding to the end-over-end rotation in angles and ) and the HCl rotation in angles and ), respectively. We note here that this expansion is different from that employed in Ref. 26, albeit in both cases the expansion becomes numerically exact, when a sufficiently large number of rotational functions is employed. While the representation in Ref. 26 is tailored to the use of empirical diabatic potential surfaces, the present one is more suitable for adiabatic potentials, as obtained from the ab initio calculations. Finally, the total wave function (r e,r,,,r,,,) is expressed as

4 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Vibronic states of Cl HCl 5977 r e,r,,,r,,, r R n,j,m, j,m njm jm n,j,m, j,m R,. As in Ref. 26, we have decoupled the HCl vibrational function (r) from the rest of the problem. This is well justified, since the characteristic frequency of the HCl vibration is nearly two orders of magnitude larger than the inverse period of any other motion in the complex. Under the relevant experimental conditions, 24 HCl remains in its ground vibrational state and this degree of freedom can be safely omitted from our discussion. The Cl cmshcl vibrational functions n,j,m, j,m (R,) are represented on an equidistant spatial grid. The stationary quantum vibronic problem is solved by propagating numerically the total wave function in imaginary time it, 41 i.e., by solving the time-dependent Schrödinger equation, which transforms into a diffusion equation after the implementation of the imaginary time variable, r e,r,,,r,,, Ĥr e,r,,,r,,,. The working equations are obtained by substituting the close coupling expansion on three electronic surfaces of the total vibronic wave function see Eq. 2 into the above Schrödinger equation. In the actual calculations, we neglect the fine effect of the end-over-end rotation characterized by quantum numbers J and M). In other words, this means neglecting any dependence of the Hamiltonian on angles and. This approximation is based on the small value of the corresponding rotational constant amounting only to cm It can also be seen in Table II of Ref. 26, that the resulting splitting of energy levels due to this fine effect is of the order of 0.1 cm 1. In the next section we show that in the present calculation the splittings of low-lying vibronic levels are of the order of 5 10 cm 1, while those for levels close to dissociation still amount to cm 1. Therefore, the present approximation is justified, at least until an experiment with resolution below 1 cm 1 is performed. The neglect of the effect of the end-over-end rotation not only simplifies the present calculation but also grossly reduces the number of states to be considered. As a result, we are then able to construct vibronic states all the way to the dissociation limit of the complex. All the arising matrix terms, originating from the adiabatic potentials, nonadiabatic couplings, and rotational terms are presented and discussed in detail in Appendix B. When the Hamiltonian matrix is large and a relatively small portion of eigenvalues and eigenfunctions located low on the energy scale is to be evaluated, the propagation of the time-dependent Schrödinger equation in imaginary time turns out to be a simple and practical method. The underlying idea is to apply the operator exp((/)ĥ) on an arbitrary initial wave function. The overlap elements of the resultant wave function with the Hamiltonian eigenstates depend exponentially on their corresponding eigenenergies. 2 3 Therefore, the ground state wave function and energy are obtained by a sufficiently long imaginary time propagation. 41 Excited states can be extracted in a similar way one after another in the order of ascending energy by projecting out all the lower states. 41 This procedure requires a long propagation for each state, especially for near-degenerate states. Therefore, we employ here an improved procedure, which allows for significantly shorter propagation times. The idea is to construct a small basis set by the unconverged imaginary time propagation and, subsequently, to employ a diagonalization scheme. The procedure starts with generating a linearly independent set of N functions, where N is larger than or equal to the number of calculated states. In the next step, these N functions undergo a short imaginary time propagation. This provides a basis set for diagonalization. The accuracy of the result of the diagonalization depends on the length on the previous propagation. In other words, a minimal time of propagation providing converged results must be established. In practice, this is done by performing subsequent short time propagations followed by diagonalizations until a convergence is reached. The optimal propagation time differs from state to state and, thus, the converged states are gradually excluded from the propagation. The coefficient which plays a key role in the efficiency of the diagonalization method is the optimal ratio between the size of the basis set N and the number of calculated states. A detailed description of the procedure is provided elsewhere. 42 Rovibronic wave functions are evaluated using a grid of 30 equidistant points for the vibrational coordinate R ranging from 2.64 Å to 8.12 Å. The HCl distance is kept fixed at Å. Rotational coordinates of the HCl diatomic fragment are treated by an expansion into spherical harmonics with quantum numbers j ranging from 0 to 15 and m ranging from 7 to 7. Three lowest electronic states are included into the calculation. Eigenstates ranging from the ground state all the way to the dissociation limit are calculated by propagation in imaginary time, employing a basis set of N 100 functions for diagonalization. The Chebychev method 43 is employed for imaginary time propagation. Depending on the particular state, an imaginary time step between 6 and 20 fs has been used. In order to obtain converged energies with accuracy better than 10 8 cm 1, the employed total length of propagation has been 1 5 imaginary picoseconds for different vibronic state. V. RESULTS AND DISCUSSION A. Potential surfaces and nonadiabatic couplings The three low-lying potential energy surfaces of the Cl HCl complex, both without and with the inclusion of the effect of spin orbit couplings, are depicted in Fig. 3. The energy is plotted as a function of the Jacobi coordinates R and see Fig. 1, with zero value corresponding to dissociation into Cl and HCl fragments. Note that for the highest state this dissociation limit moves upwards by the Cl spin orbit splitting of cm 1 with respect to the two lower states upon accounting for spin orbit interactions. As can be

5 5978 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Žďánská et al. FIG. 3. The low lying potential energy surfaces of the Cl HCl complex. R and are Jacobi coordinates, defined in Fig. 1. Zero energy corresponds to the dissociation of the complex into Cl and HCl species. On the left panel a,b,c the spin-free potential energy surfaces are displayed: a Ground state, b excited state of the A symmetry, c excited state of the A symmetry. On the right panel d,e,f the potential energy surfaces after the inclusion of spin orbit interactions are displayed: d Ground state, e excited state with the same dissociation limit as the ground state, f excited state with the dissociation limit cmz 1 above the dissociation limit of the ground state. seen from Fig. 3, the spin-orbit effect on reshaping the potential energy surfaces is very strong, leading even to the appearance of new stationary points. The lowest electronic adiabatic surface of the A symmetry Fig. 3a possesses two minima, a global collinear structure and a secondary one around 90 with the Cl cmshcl distance significantly shortened by 0.5 Å. The spin orbit effect strongly reduces the barrier between the two minima, making the second minimum very shallow see Fig. 3d. The second surface of the A symmetry Fig. 3b has only one shallow minimum, which corresponds to a collinear geometry with hydrogen pointing away from the chlorine radical. Upon inclusion of the spin orbit interactions a second collinear minimum, with hydrogen flipped between the two heavy atoms, develops see Fig. 3e. The third electronic surface, which is of the A symmetry, possesses two collinear minima see Fig. 3c. Since for collinear geometries the A state coincides with the lowest A state, these minima are the same as the collinear stationary points on the latter potential energy surface compare Figs. 3c and 3a. Note, however, that the secondary minimum on the A surface is a saddle point on the lowest A surface. When spin orbit interactions are accounted for, both minima become much more shallow and the originally lower minimum moves towards a noncollinear geometry see Fig. 3f. Present calculations can be readily compared with earlier empirical potential surfaces for the Cl HCl complex, 26 constructed from the experimentally well-known Ar HCl sur-

6 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Vibronic states of Cl HCl 5979 FIG. 4. The surface of nonadiabatic derivative couplings in rad 1 with respect to the angle between the two low-lying A states of the Cl HCl complex. R and are Jacobi coordinates, defined in Fig. 1. face by adding electrostatic and spin orbit interactions. We start by noting that the authors of Ref. 26 did a very good job and their simple empirical surfaces are qualitatively correct. However, there are nonnegligible quantitative differences which have, as is demonstrated below, a significant effect on vibronic energies. For example, the secondary noncollinear minimum on the lowest A surface is much deeper in our calculations, so that it survives the inclusion of spin orbit interactions. On the other hand, the present global collinear minimum turns out to be more shallow and squeezed in the R-coordinate than that in Ref. 26. Similar quantitative differences occur also on the two other surfaces. As an example, the secondary minima on these surfaces including spin orbit interactions come out more shallow in the present calculations. A methodologically novel aspect of the present study is the inclusion of nonadiabatic interactions between the investigated adiabatic potential energy surfaces within the ab initio many-electron model. Since the upper state is of a different symmetry than the lower two states, only kinetic couplings between the two A surfaces have to be considered. More precisely, this is also due to the fact, that in the present calculations the spin orbit splitting constant is assumed to be that of the chlorine radical independently from the geometry of the complex. Therefore, no additional derivative couplings arise from the spin orbit part of the Hamiltonian. Moreover, derivative couplings with respect to the angle are more than an order of magnitude larger than those with respect to the R-coordinate. Therefore, we have neglected the effect of the latter. This difference in magnitude is quite understandable, since the electronic wave function which can be qualitatively represented by the orientation of the singly occupied p-orbital of the chlorine radical changes much more dramatically with than with R. 29 The surface of first-order derivative couplings with respect to between the two A surfaces is depicted in Fig. 4. First, note extended regions of significant nonadiabatic couplings around 60 and close to 180. These correlate very well with the crossing points of the primitive diabats p x and p z for R fixed at 3.9 Å, as depicted in Fig. 5 of Ref. 26. Second, Fig. 4 clearly shows that the surface of nonadiabatic couplings is not separable in the and R coordinates. In other words, for different Cl cmshcl separations the nonadiabatic couplings acquire different values and generally peak at different angles. In conclusion, a simple cut for a fixed value of R would not be sufficient and the complete two-dimensional surface has to be constructed, in order to quantitatively account for nonadiabatic effects in the Cl HCl complex. B. Vibronic states Energy levels of the vibronic states of the Cl HCl complex are primarily determined by the shape of the adiabatic potential energy surfaces including spin orbit interactions. Important are also the effects of rotation of the HCl fragment and mixings due to nonadiabatic electronic couplings. Finally, the fine effect of the end-over-end rotations is ne-

7 5980 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Žďánská et al. FIG. 5. Vibrational energy levels of the Cl-HCl complex from the ground state all the way to the dissociation limit a including, b excluding the nonadiabatic interactions. glected in the present discussion, mainly in order to keep the number of vibronic states considered within reasonable margins. The progression of vibronic states of the Cl-HCl complex from the ground state all the way to the dissociation limit of the complex is depicted in Fig. 5. The energies of all these 72 states can also be obtained in a tabulated as a supplementary material form through EPAPS. 44 In Fig. 5 we present results of present calculations, both with and without the inclusion of nonadiabatic couplings. In addition, Fig. 6 contrasts the ground vibronic wave functions with hydrogen preferentially between the heavy atoms to the third excited eigenfunction, where hydrogen points away from the heavy atoms. This demonstrates the effect of quantum delocalization of the light hydrogen atom in a shallow intermolecular bending potential, allowing for a flip between isomers by a low-energy vibrational excitation. 29 Analyzing the energy levels presented in Fig. 5, we first note that our results for the lowest states can be directly compared to calculations presented in Ref. 26. We immediately see that the energy of the ground vibronic state shifts by more than 40 cm 1 towards higher energies in the present calculation. About half of this shift is due to the fact that the depth of the global minimum is smaller on the present ab initio potential surface than on the empirical surface in Ref. 26. The second half of the difference is due to the fact that the portion of the potential energy surface adjacent to the global minimum is more steeply rising in the present model, which results in an increase in the zero point energy. Also the energies of the next low-lying states, where comparison with previous calculations is possible, are significantly shifted upwards. However, this effect is not uniform, which shows that the present potential energy surfaces differ from the previous FIG. 6. Vibronic wave functions of the Cl-HCl complex corresponding to the a ground state, b fourth eigenstate. We plot the modulus squared of the vibronic wave functions integrated over all degrees of freedom except for the angular coordinates and of the HCl fragment. Dark regions, which correspond to high values of the plotted function, illustrate the hydrogen delocalization in rotational coordinates. ones in a rather complicated way, which cannot be reduced to a simple potential shift. A most remarkable effect is the nonnegligible influence of nonadiabatic couplings between the two potential energy surfaces of the A symmetry on the vibronic energies. Nonadiabatic effects are common for dissociative VIS or UV excited electronic states accompanied by highly energetic nuclear motions. However, the present system is in its ground or low-lying excited electronic states with a very modest energy content. The fact that nonadiabatic effects are, nevertheless, observed is due to the presence of a highly delocalized light hydrogen atom. Interestingly, the nonadiabatic effect is observable already for low-lying vibronic states where it amounts up to 1 2 cm 1. For higher bending/ stretching states, where hydrogen effectively moves with higher velocity, the nonadiabatic effect is more important, both because its absolute value increases and since the shift due to nonadiabatic couplings becomes larger than the energy difference between adjacent vibronic states at these energies. Thus, inclusion of nonadiabatic couplings will be crucial for spectral assignment in future experiments, which will prepare the Cl HCl complex with higher spectral resolution than that of the present measurements. 24,25

8 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Vibronic states of Cl HCl 5981 VI. CONCLUSIONS The Cl HCl radical complex has been reinvestigated with focus on two issues. First, the three lowest electronic potential energy surfaces have been constructed by accurate ab initio calculations and the effect of spin orbit interactions has been included using an adequate one-electron scheme. Second, nonadiabatic couplings between the ground and low-lying excited surfaces have been evaluated using ab initio techniques. Calculations of vibronic states of the complex show that the refinement of the potential energy surfaces has a nonnegligible effect on their energies. Moreover, inclusion of nonadiabatic couplings further shifts the rovibronic energies, especially for experimentally relevant highly excited states close to the dissociation limit of the complex. ACKNOWLEDGMENTS The authors are grateful to Nimrod Moiseyev, Jeremy Hutson, and Eva Hudečková for valuable discussions. This work has been supported by the Volkswagen Foundation via Grant No. I/ Support from the Czech Ministry of Education to the Center for Complex Molecular Systems and Biomolecules Grant No. LN00A032 is gratefully acknowledged. APPENDIX A: SPIN ORBIT INTERACTION The spin orbit coupling is accounted for within a oneelectron approximation. 30,31 Within this picture, the three adiabatic spin-free electronic states correspond to the three possible orientations of the unpaired p-electron of the chlorine radical, depicted in Fig. 2b. In the body-fixed representation the transformation from the primitive diabatic basis pp x,p y,p z ] see Fig. 2a to the basis of adiabatic states na GS,A ES,A] where GS stands for ground and ES for excited state of the A symmetry is accomplished by a transformation matrix Acos sin sin cos 0. A1 In a space-fixed representation, the adiabatic states are obtained by an appropriate transformation from the bodyfixed representation. Such a transformation can be found, e.g., in Ref. 33 and the final expression for the adiabatic states in the space-fixed representation may be written as npba, where BR R R and sin 0 R cos sin cos , cos 0 sin R sin 0 cos, A2 A3 R cos sin 0 sin cos A4 The spin orbit Hamiltonian of the unpaired p-electron is equal to 30,31 Ĥ SO 2 3 lˆ ŝ, A5 where is the spin orbit splitting between the 2 P 3/2 and 2 P 1/2 states of the chlorine radical. In the next step, the spinorbitals are symmetry adapted according to the,* representations of the C s (M) 2 spin double group, 32 p p *]p])t, where the matrix A6 i i 0 0 T 1 0 i A i i i performs the symmetry adaption. In the absence of magnetic field, it is possible to restrict the calculation to only one function of each of the three degenerate Kramer s pairs. The spin orbit Hamiltonian is calculated using the multiple transformation, n Ĥ SO n A B p 3lˆ ŝp 2 BA, A8 where the index denotes the double group, and we employ the relation 3 0 i i p 3lˆ ŝp 2 1 i 0 1 A9 i 1 0. APPENDIX B: CLOSE COUPLING SCHEME The close coupling expansion of the total wave function (r e,r,,,r,,,) defined in Eq. 2 leads to the following Hamiltonian acting on functions n,j,m, j,m (R,): 2 ĤVRK R R 2. B1 The matrix elements of the potential term V(R) and the kinetic term K R, which are indexed according to the electronic basis functions n and the rotational basis functions JMjm of a dual rotor, are listed explicitly in Table I. The inherent dependence of the electronic wave function on the angle see Fig. 1 is reflected in coupling of the diatomic rotational number j by the potential term 1a and the nonadiabatic term 1b in Table I. The rotational operators in the above Hamiltonian lead to centrifugal terms 1c 1d and to the nonadiabatic term. All functions are also coupled due to the presence of the spin orbit interaction term 1e. Neglecting the end-over-end rotation leads to the following simpli-

9 5982 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Žďánská et al. TABLE I. Matrix Matrix element (njmjm,njmjm) Electronic Integrals V(R) jme n,n jm J Mm,JMm 1a En Ĥ el Ĥ SO n 2 2 HCl n,n r jm f 2 2 cot jm J Mm,JMm 1b f n n 2 2 HCl r j j1 2 njm jm,njmjm 1c 2 2 ClHCl R JJ1 2 njm jm,njmjm 1d JMjmH SO n,n jmjm 1e H SO n Ĥ SO n K R 2 2 ClHCl n JM jm,njmjm 2 fications: The centrifugal term 1d is omitted, the spin orbit term 1e is replaced by a term jmh SO n,n jm and the term B in Eq. A3 reduces to the cylidrical rotation over the complex axis, BR. B2 Equations A5 and A6 lead now to the following matrix terms: n Ĥ SO n 3H 1 SO 0 2H 1 SO 1 H SO 1, B3 where 0 0 i cos H SO i sin B4 i cos i sin 0 and i sin 2 i1cos 2 2 sin B5 For the term 1e in Table I we thus obtain the following matrix elements: 3 jmh 0 SO n,n jm m,m i H SO 1 e i1cos 2 i sin 2 2 cos 6 jm1h 1 SO n,n jm m,m1 6 jm1h 1 SO n,n jm m,m1. B6 Numerical evaluation of matrix elements in Table I boils down to the evaluation of integrals of the type jm fjm, jm1e i fjm, and jm f (2(/)cot ) jm, where f is a real function of the angle. General numerically feasible formulas are derived which expand the function f in the basis of associated Legendre polynomials P k l and employ analytical integrals of spherical harmonics, jm fjm1 m 2 j12 j1 j j j l j l l1 a 2 jlj mp 0 l f, B7 j l j a jlj m, m 0 m jm1e i fjm 1 m 2 j12 j1 j j j l j l l1 b 2 jlj mp 1 l f, b jlj m 1 ll1 j l j m 1 m1, jm f 2 cot jm j j 1 m 2 j12 j1 l1 c jlj mp l 1 f, B8 j l j l1 2

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