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Toward Spin−Orbit Coupled Diabatic Potential Energy Surfaces for Methyl Iodide Using Effective Relativistic Coupling by Asymptotic Representation Nils Wittenbrink, Hameth Ndome, and Wolfgang Eisfeld* Theoretische Chemie, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany S Supporting Information *

ABSTRACT: The theoretical treatment of state−state interactions and the development of coupled multidimensional potential energy surfaces (PESs) is of fundamental importance for the theoretical investigation of nonadiabatic processes. Usually, only derivative or vibronic coupling is considered, but the presence of heavy atoms in a system can render spin− orbit (SO) coupling important as well. In the present study, we apply a new method recently developed by us (J. Chem. Phys. 2012, 136, 034103, and J. Chem. Phys. 2012, 137, 064101) to generate SO coupled diabatic PESs along the C−I dissociation coordinate for methyl iodide (CH3I). This is the first and mandatory step toward the development of fully coupled fulldimensional PESs to describe the multistate photodynamics of this benchmark system. The method we use here is based on the diabatic asymptotic representation of the molecular fine structure states and an effective relativistic coupling operator. It therefore is called effective relativistic coupling by asymptotic representation (ERCAR). This approach allows the efficient and accurate generation of fully coupled PESs including derivative and SO coupling based on high-level ab initio calculations. In this study we develop a specific ERCAR model for CH3I that so far accounts only for the C−I bond cleavage. Details of the diabatization and the accuracy of the results are investigated in comparison to reference ab initio calculations and experiments. The energies of the adiabatic fine structure states are reproduced in excellent agreement with ab initio SO−CI data. The model is also compared to available literature data, and its performance is evaluated critically. This shows that the new method is very promising for the construction of fully coupled full-dimensional PESs for CH3I to be used in future quantum dynamics studies.

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INTRODUCTION The detailed study of reactive processes at the molecular level is of fundamental interest for understanding chemistry. The tremendous progress in experimental techniques provides a plethora of data, and the great advance in theory can deliver the corresponding interpretations to gain deep insight. However, the complexity of the experimental data and the effort for a theoretical treatment of a reactive process increase extremely rapidly with the size of the system. One of the main theoretical bottlenecks for systems beyond three atoms is the development of accurate analytical potential energy surfaces (PESs), which are essential for a theoretical (quantum) reaction dynamics investigation. At least for adiabatic ground states, two strategies have proven feasible for developing higher-dimensional PESs with good accuracy. One possibility is based on local interpolation techniques1−5 while the other utilizes invariant theory and permutation symmetry of indistinguishable nuclei.6 Unfortunately, an extension of these methods to excited state PESs is not straightforward because of the difficulty to account for the state−state interactions. So far, this has been attempted by the modified Shepard interpolation7−9 and very recently by using invariant polynomials and complete nuclear permutation−inversion (CNPI) symmetry.10 For excited states it © 2013 American Chemical Society

generally is of great advantage to use a quasi-diabatic representation for the coupled electronic states.11 Most attempts to develop coupled PESs for a manifold of states are based on some kind of quasi-diabatic representation. This simplifies the quantum dynamics treatment because the diabatization removes the singularities of the nonadiabatic derivative couplings. Furthermore, the matrix elements of the electronic Hamiltonian become particularly simple functions in a quasi-diabatic basis, making it easier to find analytical expressions for them. While any appropriate diabatization will account for the derivative or vibronic couplings, all relativistic, particularly spin−orbit (SO), couplings require a different treatment. Great progress has been made in the ab initio determination of SO coupling elements and energies of fine structure states by relativistic quantum chemistry.12,13 However, only very few relativistically coupled PESs have been developed to date. One Special Issue: Joel M. Bowman Festschrift Received: February 8, 2013 Revised: April 16, 2013 Published: April 16, 2013 7408

dx.doi.org/10.1021/jp401438x | J. Phys. Chem. A 2013, 117, 7408−7420

The Journal of Physical Chemistry A

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An example for a more complicated system of high relevance is methyl iodide (CH3I) for which particularly the photodissociation dynamics is of great interest (see below). The additional complexity originates from the fact that the strong interaction (Franck−Condon) region is highly relevant, the iodine introduces strong relativistic effects, the reaction takes place in multiple excited states, and the dissociation fragments are both open-shell. Consequently, the geometry dependence of the SO coupling is of greater importance and the number of fine structure states is much larger compared to the F/Cl + H2 reaction. Nevertheless, Amatatsu et al. developed fully coupled PESs for a subset of the fine structure states which are relevant for the photodissociation process.22,23 Their approach is based on ab initio configuration interaction calculations including an effective one-electron spin−orbit Hamiltonian,24 which provide adiabatic energies for the fine structure states of interest. The authors developed a quasi-diabatic potential model that accounts for the permutation invariance of the three protons and that was capable of reproducing the ab initio data very well. This pioneering work provided coupled full-dimensional (9D) PESs of a reactive system at a time when accurate, higher than three-dimensional PESs even for a single electronic state were still very scarce. Therefore, these surfaces were used immediately after publication for several quantum dynamics studies.25−28 However, these studies revealed considerable problems with the PES model that were partially improved in a later version.29 Remaining fundamental errors in the quasidiabatic model were discussed very recently in the latest fulldimensional quantum dynamics study using these updated PESs.30 Beyond this, a conceptual weakness of the method by Amatatsu et al. is that two different kinds of state−state couplings are modeled by a single quasi-diabatic matrix in a fairly empirical way. The vibronic coupling of the Jahn−Teller active excited states of CH3I is well understood, and quasidiabatic models are well established.11 This is much less the case for molecular SO coupling. However, since relativistic effects in a molecular system are strongly atom based, it is clear that the geometry dependence of the SO coupling is very different from that of the vibronic coupling. Thus, it is unlikely that the physics of these two types of couplings can be properly represented by just one average coupling matrix. We recently have been developing a new method to overcome these problems and to generate accurate SO coupled quasi-diabatic PESs for molecular systems.31,32 This approach is based on the idea that the adiabatic molecular states can be expanded in an asymptotic quasi-diabatic basis where the asymptote corresponds to the removal of the relativistic atom of interest. Therefore, the method is called effective relativistic coupling by asymptotic representation (ERCAR). The quasidiabatic basis states are expressed as direct products of a fragment and an atom state. In this basis it is straightforward to evaluate an effective relativistic coupling operator like for SO coupling, which results in a constant coupling matrix. The geometry dependence of the coupling is accounted for by the geometry dependent expansion coefficients corresponding to quasi-diabatic basis states of the adiabatic wave functions. This approach has been shown to yield very accurate potentials for hydrogen iodide (HI), and it does not require the very demanding ab initio computation of fine structure states and energies. The SO coupling is treated separately from the vibronic coupling and is purely based on physical considerations. Thus, the method should be ideal for the generation of accurate high-dimensional PESs for polyatomic systems like

of the main reasons, besides the increased complexity of treating both excited states and relativistic couplings, may be that it is not straightforward to find a common quasi-diabatic basis to represent all the effects. Standard electronic structure techniques only can determine the relativistic effects in an adiabatic basis, which is not ideal for an analytic representation of the coupled PESs. Furthermore, the accurate ab initio computation of SO couplings is fairly demanding computationally, limiting the amount of data points that can be determined and used to develop the PESs. Therefore, in early approaches the ab initio treatment of the SO coupling problem has been circumvented. Tully introduced his version of the diatomics-in-molecules (DIM) method in 1973, which was a breakthrough in the development of polyatomic PESs.14 In a second paper from the same year he introduced nonadiabatic and SO couplings in the DIM framework.15 The approach is based on decomposing the molecular SO interaction into atomic terms and using the known experimental data for atomic SO couplings in terms of the Russel−Saunders (L̂ ·Ŝ) coupling scheme. This is indeed a viable approach for systems for which the DIM method yields reasonable results (like rare gas clusters). One year later, Cohen and Schneider proposed a method to determine SO coupled potential functions for diatomics, which is based on uncoupled adiabatic state energies from high-level ab initio calculations and a constant SO coupling matrix derived from atom states. Again the coupling parameters are taken from experiment. This method yields excellent results in the asymptotic region, which is relevant for many scattering treatments, and is still widely used. However, it is clear that the geometry dependence of the SO coupling is not properly accounted for since in the strong interaction region the adiabatic wave functions differ considerably from the atomic states. It is also obvious that this approach will fail in the presence of state crossings. Similar approximations have been used by Rebentrost and Lester in their studies of F (2P) + H2 (1Σ+g ) collisions.16 Namely, they assume that L̂ ·Ŝ coupling is an appropriate approximation and that the corresponding coupling constant is independent of geometry. In contrast to Cohen and Schneider they used ab initio based quasi-diabatic potential energy surfaces to represent the electronic Hamiltonian17 and added the SO coupling afterward for the evaluation of the matrix elements used in coupled-channel calculations. Both methods are similar in the treatment of the SO coupling and thus show the same limitations when SO effects become relevant in the strong interaction region. Therefore, Alexander et al. extended these ideas to account better for the geometry dependence of the SO effects for their studies of the F (2P) + H2 → HF + H (2S) reaction.18,19 They developed a quasidiabatic potential model based on the highly accurate analytical PES by Stark and Werner20 in which the direction of the p hole of the fluorine atom is well-defined. Then they added an analytic SO coupling matrix to this quasi-diabatic model, again assuming the validity of the L̂ ·Ŝ coupling (but expressed in the usual terms of a linear molecule), and made the coupling matrix elements geometry dependent. The corresponding function parameters were obtained by fitting the complete model with respect to ab initio results for the fine structure energies. This approach yielded excellent results and later was extended to the Cl (2P) + H2 → HCl + H (2S) reaction.21 However, this method may become quite cumbersome for more complex systems, especially if performing a large number of accurate SO ab initio calculations becomes unfeasible. 7409



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by inclusion of the further coordinates. The aim of the present work is to test this model with respect to accurate available data particularly from high-level ab initio calculations.

CH3I. The only roughly comparable method known to us has been suggested by Angeli and Persico.33 They employ a wave function based direct diabatization scheme34 with respect to a set of asymptotic reference states and a single electron approximation within an ab initio code to directly compute quasi-diabatic and fine structure states and energies. Unfortunately, little is known about the accuracy of this approach and how to generate an analytical PES model from it. However, this approach was also adopted by other groups for the study of diatomics.35,36 A similar and fairly recent ab initio approach by Valero and Truhlar does not improve or solve the issues discussed above.37 The aim of the present work is to extend our new ERCAR method beyond diatomics to a real polyatomic system. We chose to study CH3I, which is an ideal benchmark system. The multistate photodissociation dynamics of CH3I has been investigated experimentally for decades because it shows a number of fundamental phenomena. The literature on this topic is exuberant, and the interested reader may find very good overviews over the existing research.29,30,38−40 Depending on the excitation wavelength, several excited states are populated simultaneously of which the lower ones undergo direct dissociation. Two distinct channels are observed yielding either ground state or SO excited state iodine atoms. The I/I* ratio is indicative of nonadiabatic processes induced by the presence of a conical intersection among two of the excited states. This conical intersection is entirely due to the very strong SO coupling caused by the heavy iodine atom. The methyl fragment is produced in its electronic ground state, which has a planar equilibrium geometry. Therefore, excitation of the umbrella motion is observed while excitation of other vibrational modes may or may not take place depending on the experimental conditions. A further issue is the rotational excitation of the methyl fragment, which is colder in the I* and hotter in the I channel. This can be explained by the e rocking mode of the methyl fragment being an important coupling mode between the two states forming the conical intersection. Deformation of this mode promotes the nonadiabatic transition to the lower state and after dissociation turns into methyl rotation perpendicular to the C3 axis. A good overview over the physics involved was given by Eppink and Parker39 and Evenhuis and Manthe.30 Because of the interesting and fundamental processes taking place in the multistate photodissociation of this seemingly simple molecule, this is also a very interesting subject for theoretical investigations. However, amazingly few ab initio electronic structure studies have been carried out on the electronic states and the SO splittings.22−24,41−46 The majority of the theoretical studies are devoted to the dynamics of the photodissociation process.25−30,47−63 All the earlier studies used simple empirical model potentials of low dimensionality while most studies after 1991 applied the surfaces by Amatatsu et al.22,23,29 Unfortunately, even the latest version of these surfaces still seems to be in disagreement with current experiments.30 Therefore, it is highly desirable to develop more accurate PESs based on state-of-the-art ab initio calculations and accounting for all relevant coupling effects. Our goal is to use our newly developed ERCAR method to generate a set of full-dimensional fully coupled quasi-diabatic PESs for the description of the photodissociation process. We start out by developing the ERCAR model for the dissociation coordinate, which is the subject of the present study. This will define the SO coupling and a basic quasi-diabatic model, which then can be extended

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THEORY Formal Background of the ERCAR Method. A description of the basic theory has been presented in detail in our previous papers,31,32 and we will give only a brief outline here. The basic idea is to represent the effective relativistic coupling in a basis of asymptotic atomic states because relativistic effects are strongly atom based. The most important relativistic coupling is spin−orbit coupling, and our effective SO operator for a given atom A reads (A) Ĥ SO =

̂ ̂ ∑ a(A) j Pj ∑ lk · sk̂ j

k

(1)

The SO interaction is described by the sum over singleelectron operators lk̂ and ŝk, corresponding to orbital and spin angular momenta for electron k, respectively. This operator is scaled by a state and atom specific coupling constant a(A) and j projected onto the corresponding states by the projection operator P̂ j. The atomic spin-space states |ψatj ⟩ are expressed as configuration state functions (CSFs), and the matrix elements of Ĥ (A) SO in this basis yield an atomic SO coupling matrix, which is composed of blocks corresponding to atom states j.64 The necessary coupling constants can be computed ab initio or taken from experiment. The molecular states are expanded in a quasi-diabatic basis that corresponds to direct products of the states of the relativistic atom and the remaining fragment, respectively. The reference states for this diabatization correspond to the supermolecular wave functions of the corresponding dissociation asymptote. The molecular spin-space states for any given geometry Q are expanded as |ψi mol(Q )⟩ =

∑ uji(Q )|ψjfrag(Q )⟩|ψjat(Q 0)⟩ j

(2)

where Q0 denotes the asymptotic reference geometry. In this representation of the adiabatic molecular states, the evaluation of the effective relativistic operator becomes trivial. The matrix elements Hdjk of the quasi-diabatic electronic Hamiltonian, including the SO operator eq 1, simply are H jkd = ⟨ψjat|⟨ψ jfrag|H0 + HSO|ψkfrag⟩|ψkat⟩ = wjkd (Q ) + ⟨ψjat|HSO|ψkat⟩

(3)

wdjk(Q)

Here, are the geometry dependent matrix elements of the quasi-diabatic potential matrix, which account for the Coulomb interaction and the scalar relativistic effects. The remaining term corresponds to an element of the known atomic SO coupling matrix and is independent of geometry. Obviously, the SO coupling among the quasi-diabatic states is independent of geometry, which is not the case for the adiabatic molecular states. The reason is the geometry dependent adiabatic-diabatic transformation, which is expressed by the matrix U(Q) in eq 2. The geometry dependence of the SO coupling that is observed in the adiabatic fine structure states is modeled properly by the diabatization within the ERCAR approach. This gives rise to a formally correct description of the influence of the geometry on the SO coupling. Diabatization. A central aspect of the ERCAR method is the adiabatic−diabatic transformation in eq 2. The adiabatic 7410



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(MRCI) data obtained by the Molpro code. 75 The implemented internally contracted MRCI method76,77 is very efficient and also allows the computation of SO coupling and fine structure states.78 Heavy elements like the iodine in CH3I require a large state basis to converge the SO−CI calculations, and it becomes extremely cumbersome to keep this basis consistent throughout nuclear configuration space. Due to the specific diabatization approach explained in the previous section, the ERCAR method requires the calculation of far fewer electronic states and therefore makes the ab initio calculations much easier and more efficient. For each data point we first compute a reference wave function by state-averaged CASSCF. All scalar relativistic effects are accounted for by a relativistic effective core potential (RECP), replacing the 28 inner electrons of iodine.79 The remaining electrons are represented by the corresponding augmented correlation-consistent AO basis set of triple-ζ quality (AVTZ).80,81 The reference MOs and states for the MRCI calculations are generated in the following way. In a first CASSCF calculation, a minimal active space of the 5p iodine orbitals and the 2pz carbon orbital is used, and the stateaveraged wave function is optimized for the following set of states: 1A1, 1E, 3E, and 3A1. In a second CASSCF calculation all occupied MOs from the previous CASSCF are frozen, the 6s iodine orbital is made active, and the 5A2 and 5E states are computed. The natural orbitals from this calculation are then diabatized with respect to a reference calculation at the asymptote. These diabatic orbitals and reference states are then used in the following MRCI calculations, which yield the adiabatic energies (including Davidson correction) used for fitting the ERCAR model and the SO−CI calculations. The final step is a SO−CI calculation with a basis of 7 singlet, 9 triplet, and 3 quintet states to obtain the fine structure states and energies. Within C3v symmetry, these states are as follows: 2 1A1, 2 1E and 1A2, 3 3E and 3 3A1, 5A2 and 5E. This last step is not required for the ERCAR approach and is only needed to generate the reference data for comparison. For further comparison, we also carried out a set of reference calculations similar to those of Alekseyev et al.44 In these calculations the large core RECP by LaJohn et al.82 was combined with an AO basis developed by Glukhovtsev et al.83 This combination was shown to yield particularly accurate SO splittings for atomic iodine.

molecular states obtained by ab initio calculations need to be diabatized appropriately.11 At this point it should be mentioned that it has been shown that a strictly diabatic representation does not exist in general.65,66 But it is always possible to define a quasi-diabatic basis by the requirement that the character of each quasi-diabatic state is preserved as much as possible with respect to a reference state when changing the nuclear geometry. For simplicity, we will call this a diabatic state from here on. Our diabatization approach is based on diabatic orbitals, which are smoothly varying throughout nuclear configuration space and maintain the character of the reference orbitals as much as possible. As reference point the asymptote is chosen and thus the reference orbitals are atomic orbitals (AOs) for the relativistic atom and molecular orbitals (MOs) for the remaining fragment. The adiabatic MOs are transformed to diabatic orbitals using a maximum overlap criterion with respect to the reference orbitals. When using a complete active space self-consistent field (CASSCF) wave function, this leaves the CASSCF energies invariant.67 The corresponding CI vectors are transformed into the basis of diabatic orbitals, and the leading CI coefficients could be used directly for a diabatization.67−69 Such a diabatization can be performed on any set of multideterminantal wave functions based on CASSCF orbitals. The diabatization method used in the ERCAR approach is a hybrid between the long established “diabatization by ansatz”70 and the block-diagonalization of diabatic CI coefficients.67−69 It is optimized to reproduce a small set of low-lying adiabatic energies, wiia (Q), with high accuracy and simultaneously accounting for high-lying states relevant for the SO coupling. The diabatic matrix elements wdij(Q) are simple parametrized functions of the internal coordinates Q, which define the potential matrix ansatz. The diagonalization of the diabatic potential matrix by U†W dU = W a = diag(wiia)

(4)

is required to optimally reproduce the selected adiabatic energies, which yields the transformation U between adiabatic and diabatic states. This ensures the high accuracy of the final PESs without SO coupling. The second condition is that the difference between the diabatic CI vectors of the selected states in the subspace of the relevant configurations with respect to the corresponding eigenvectors of Wd has to be minimized. This ensures that the state composition in terms of the asymptotic diabatic basis states is well reproduced and is responsible for an accurate treatment of SO coupling. The diabatic CI vectors are readily available from calculations of the low-lying states at no additional computational cost. Typical vibronic coupling approaches using an ansatz are restricted to a very limited region, which would be insufficient for our purpose. An extension to larger regions is possible by the use of more flexible functions for the diabatic matrix elements as we have shown in recent work.71−74 The best possible approximation for a given diabatic matrix is found by adjusting the free parameters in the matrix elements by leastsquares fitting. The only technical difficulty in this approach is that the fitting is nonlinear due to the diagonalization of the diabatic matrix. Electronic Structure Calculations. The electronic structure data required by ERCAR is obtained from standard ab initio methods. For the present study, we use CASSCF and multiconfiguration−reference configuration interaction

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RESULTS AND DISCUSSION The ERCAR methodology outlined in the previous section is now applied to the CH3I system. The focus will be on the lowlying fine structure states, which correlate asymptotically with the CH3 (2A″2 ) + I (2P3/2) ground state dissociation asymptote and the CH3 (2A2″) + I (2P1/2) spin−orbit excited dissociation channel. Higher fine structure states do not play any role in the dynamics studied by most of the experiments. In the following, we will define the diabatic and SO model specific for CH3I, analyze the accuracy of the resulting fine structure states, and compare the results to other theoretical and experimental studies. ERCAR Model for CH3I. In our previous studies we developed the ERCAR method using hydrogen iodide (HI) as a proof-of-principle example.31,32 HI is particularly simple because there is only one valence coordinate and the electronic structure of the hydrogen atom is fairly simple. By contrast, CH3I is a polyatomic molecule with nine internal degrees of freedom and several bonds to break. We choose the 7411



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because for each spin symmetry the diabatic matrix block is again block-diagonal corresponding to the irreducible representations of the diabatic states in C3v, which are A1, A2, and E. Once the other internal coordinates are included into the model, these blocks will be coupled to each other by distortions along symmetry breaking coordinates. These vibronic coupling effects like the well-known Jahn−Teller coupling will then be described by already known coupling blocks in the diabatic matrix.11,70−74 The benefit of staying in C3v symmetry is that the matrix blocks corresponding to different symmetry can be set up separately from each other. Each diabatic matrix element is expressed by a simple function with a few parameters, in the present case either of the following types:

fragmentation into a molecular methyl radical and an iodine atom as the reference for the diabatization required by the ERCAR approach. The main difference to our previous test case, HI, is that the CH3 fragment is not a simple atom anymore although in certain ways it behaves quite similarly to an H atom. All low-lying adiabatic electronic states of interest can be derived either from the methyl (2A2″) ground state and the first few low-lying atomic doublet and quartet states of iodine or from ionic states CH3+ (1A1) and CH3− (1A1) and corresponding iodine ion states. Excited methyl states do not contribute to states of our interest. The low-lying states of prime interest are all due to the iodine ground state, 2P, with a 5s25p5 configuration. In a first step, the diabatic CI vectors are analyzed along the dissociation coordinate to decide which basis states have to be included in the diabatic model. CASSCF rather than MRCI state vectors need to be used for this purpose in order to avoid spurious results as was discussed before.32 All basis states with corresponding absolute CI coefficients greater than 0.35 in the state vectors of interest anywhere along the dissociation coordinate are considered important for our diabatic model. That means that all states are omitted in our model, which have a maximum weight of less than 10% in the adiabatic wave function. Thus, effects caused by these states should be negligible. The result of this analysis is that the following basis states, which give rise to SO coupling, are selected:

4

w(b)(r ) = p0 +

∑ pn [1 − e−α(r− r )]n 0

− pn

n=2

6

w(r)(r ) = p0 + p1 e−αr +

∑ n=2

(5)

pn r 2n

(6)

The function w(b)(r) is an expansion in Morse type coordinates and thus particularly suitable for the representation of a bound potential. By contrast, w(r)(r) is a combination of a repulsive exponential term and an expansion in inverse distances, which is more flexible and easily can model both bound and repulsive potentials. All off-diagonal elements are represented by w(r)-functions while for the diagonal elements the function type depends on the character of the diabatic state. In general, all of these coupling matrices are full matrices with coupling functions among all the diabatic basis states. To reduce the number of coupling functions and parameters, the CASSCF state vectors are analyzed and coupling elements for states with only a negligible interaction are set to be zero. This reduces the complexity of the fitting procedure considerably. The resulting coupling model for the 1A1 states reads

I (12 P) ⊗ CH3 (2A″2) 1 ′ I+(3P) ⊗ CH3− (A 1)

I (22 P) ⊗ CH3 (2A″2) I (4 P) ⊗ CH3 (2A″2)

In addition, a number of states are found relevant and are selected, which do not contribute to SO coupling or for which the SO coupling is neglected. Nevertheless, these states are important in the ERCAR framework because their contributions to the wave functions tune the effective SO coupling depending on geometry. These relevant noncoupling states in our diabatic basis are

⎛ w (r) w (r) w (r) 0 ⎞ 12 13 ⎟ ⎜ 11 ⎜ (r) (b) (r) ⎟ 0 w24 ⎟ ⎜ w12 w22 W (r , 1 A1) = ⎜ ⎟ (r) (b) 0 ⎟ ⎜ w13 0 w33 ⎟ ⎜⎜ (r) (b) ⎟ ⎝ 0 w24 0 w44 ⎠

1 ′ I− (1S) ⊗ CH3+ (A 1) 1 ′ I+ (1S/1D) ⊗ CH3− (A 1)

(7)

The optimal coupling model for the 1E states is found to be

I (2S/2 D) ⊗ CH3 (2A″1 )

⎛ w (r) w (r) 12 ⎜ 11 ⎜ (r) (b) ⎜ w12 w22 ⎜ (r) (r) 1 W (r , E) = ⎜ w13 w23 ⎜ (r) ⎜ w14 0 ⎜⎜ (r) ⎝ w15 0

1 ′ I− (1P) ⊗ CH3+ (A 1)

1 ′ I− (3P) ⊗ CH3+ (A 1)

This selection of relevant CI configurations and corresponding asymptotic states defines the diabatic basis of our model. The two open-shell fragments can combine to singlet, triplet, and quintet states, which in the absence of SO coupling do not mix. Therefore, the diabatic potential matrix will be blockdiagonal with blocks corresponding to each total spin. Without any further symmetry, these matrix blocks would be full matrices without further structure. This will be the case in general for the full-dimensional potential matrix of a polyatomic system like CH3I. However, we only treat the C−I bond dissociation coordinate in the present study and thus the system maintains C3v symmetry. We utilize this symmetry

(r) (r) (r) ⎞ w13 w14 w15 ⎟ ⎟ (r) w23 0 0 ⎟ ⎟ (r) (r) w33 w34 0 ⎟ ⎟ (r) (b) w34 w44 0 ⎟ ⎟ (r) ⎟ 0 0 w55 ⎠

(8)

The triplet coupling matrices for A1 and E states are shown in the following two equations: ⎛ w (r) w (r) ⎞ 11 12 ⎟ W (r , A1) = ⎜ ⎜ (r) (b)⎟ ⎝ w12 w22 ⎠ 3

7412

(9)


The Journal of Physical Chemistry A ⎛ w (r) w (r) 12 ⎜ 11 ⎜ (r) (b) ⎜ w12 w22 ⎜ (r) ⎜ w13 0 3 W (r , E) = ⎜ (r) ⎜ w14 0 ⎜ ⎜ 0 0 ⎜ ⎜ 0 ⎝ 0

(r) (r) w13 w14

0

0

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0 0

(r) (r) w33 w34

0

(r) (b) w34 w44

0

0

0

(b) w55

(r) w36

0

(r) w56

0 ⎞⎟ ⎟ 0 ⎟ (r) ⎟ w36 ⎟ ⎟ 0 ⎟ ⎟ (r) ⎟ w56 ⎟ (r) ⎟ w66 ⎠

(10)

Singlet and triplet A2 states are both uncoupled. Thus, no coupling matrix is needed for them and the potentials are represented by one w(r)-type function for each of those states. In general, we represent ionic diabatic basis states by matrix elements w(b) ii because due to the strong Coulomb attraction these states are usually bound and form covalent bonds at short distance. Neutral diabatic states may or may not be bound and are represented by matrix elements of type w(r) ii in our model. For example, the major diabatic components of the adiabatic CH3I ground state are I (1 2P) ⊗ CH3 (2A″2 ), I+ (1S/1D) ⊗ CH3− (1A1″), and I− (1S) ⊗ CH+3 (1A1′ ). For long distances, the repulsive spin−orbit active state dominates, whereas for short distances the bound ionic states become more important, which do not contribute to SO coupling. This also explains the quenching of the SO effect depending on the nuclear geometry. The weights of the SO active states decrease, and so does the SO coupling. Of course, the effect of the SO splitting is also reduced by the increasing energetic separation of the diabatic potentials corresponding to any of the SO active basis states. Both effects together ensure a correct description of the geometry dependence of the SO coupling. As noted earlier, major diabatic components for some higherlying adiabatic states are also included in this model and fulfill two needs. First of all, they help to improve the accuracy of the low-lying adiabatic state energies, which have to be reproduced quantitatively. Second, at least a qualitative description of higher-lying adiabatic states must be ensured because at short distance they form intersections with states that are relevant for the SO model. However, it is not necessary to model these higher-lying states with high accuracy. Finally, the presence of the I (4P) ⊗ CH3 (2A″2 ) basis state gives rise to additional highlying 3/5A and 5E states. The energies of these states are very high and do not matter much for the accuracy of the ERCAR model. Therefore, these states are simply represented by uncoupled functions of the b- or r-type. The corresponding function parameters are determined by fitting with respect to energy data from very approximate model calculations. All these diabatic matrices for the different symmetry blocks define the “ansatz” of our diabatic model, and it remains to determine the free parameters. This is achieved by a nonlinear least-squares fitting procedure by which the parameters are optimized to accurately reproduce ab initio reference data at 30 points for CH3−I distances from 1.6 to 20 Å. Singlet and triplet states are treated separately, so that only the two lowest eigenvalues of the diabatic matrices for each total spin are fitted to the corresponding ab initio energies. Additionally, the squared elements of the first two eigenvectors are fitted with respect to the weights of the corresponding diabatic basis states in the diabatic CASSCF CI vectors. To avoid artifactual intruder states, additional ab initio energies are included for higher-lying states with very low fitting weights. These energies

Figure 1. Schematic coupling model with SO-coupling blocks (blue squares) and diabatic coupling elements (red dots). Diagonal elements represent diabatic energies.

do not need to be reproduced with high accuracy. As described earlier, accurate data from high-level calculations are only needed for the low-lying states, which makes the ERCAR approach very efficient. The optimized parameters for the present model are listed in the Supporting Information. In a final step the diabatic model is combined with the SO coupling blocks of the asymptotic basis states. Each spin−orbit active diabatic basis state corresponds to an atomic spin−orbit coupling matrix which can be derived analytically. The basis states are direct products of an atom (iodine) and molecular fragment (methyl) state. If both states are of spin S > 0, the direct product basis functions are not necessarily eigenfunctions of the Ŝ2 operator. Thus, the direct product basis and corresponding SO matrices have to be transformed to a spin eigenbasis before they can be used together with the diabatic model. For example, the I (1 2P) ⊗ CH3 (2A″2 ) basis state is a product of six components of the 2P state of iodine and two components of the (2A2″) state of methyl. This gives a total of 12 fine structure basis states, which after transformation to the spin eigenbasis results in one set of three singlet and one set of nine triplet states. The resulting 12 × 12 coupling block is complemented by the corresponding diabatic matrix elements corresponding to the appropriate 1A1, 1E, 3A1, and 3E diabatic states. The third basis state, I (2 2P) ⊗ CH3 (2A″2 ) uses the same type of SO coupling block but differs in the diabatic basis states that enter the final coupling matrix. Different coupling blocks are necessary for the I+ (3P) ⊗ CH3− (1A1′ ) and the I (4P) ⊗ CH3 (2A″2 ) states. The former one of dimension 9 × 9 simply is the coupling matrix of a 3 P state with 5p 4 configuration. The latter one is a transformed block of dimensions 22 × 22 because the doublet and quartet spin of the fragments can couple to either triplet or quintet total spin. The quintet part is irrelevant in the present case, which allows us to remove the two fine structure components with total spin projections MS = ±(5/2). All these matrices are equivalent to those for HI and were already discussed in our previous work.31,32 Finally, all SO coupling matrices are combined in one, complemented with the states that are not SO active, and the diabatic coupling elements are added to the appropriate 7413



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85 × 85 total coupling matrix. A scheme of this matrix and the coupling pattern is shown in Figure 1. Accuracy of Potential Functions. The performance of the ERCAR approach and the accuracy of the potential energy surfaces it yields in comparison to ab initio reference data are discussed in the following. Although we focus on the behavior of the dissociation coordinate, the umbrella angle is vital for a description of the photodissociation. Therefore, the same model was fitted twice with slightly different geometries of the methyl fragment. These geometries are mainly distinguished by the CH3 umbrella angle, which is defined as the unique angle between each of the C−H bonds with a line through the carbon atom called the trisector. For C3v symmetry this simply corresponds to any of the three H−C−I angles. One calculation is performed for planar CH3 geometry, corresponding to an umbrella angle of 90°, and with a C−H distance of 1.085 Å. This is referred to as “planar geometry” in the following. The second calculation was carried out for an umbrella angle of 110° with a C−H distance of 1.130 Å and is referred to as “bent geometry” from here on. We focus on the four lowest lying adiabatic states, because they all correlate with the iodine 2P ground state. The results from the diabatic model for these states and the corresponding ab initio data for planar and bent geometry are presented in Figure 2 and Figure 3, respectively. Apparently, the diabatic model reproduces the ab initio energies very well for both geometries. The total root-meansquare (rms) error for all states up to 30000 cm−1 above the asymptote (20 Å) is only 10 cm−1 for planar geometry and 14 cm−1 for bent geometry. This is quite small if one takes into account that this corresponds to a total energy range of the surface of more than 50000 cm−1 . As expected, the ground state shows a typical Morse-like bound potential and the excited states are repulsive. At higher energies, further bound states can be found which are not shown on the scale of the figures. On a closer view, additional weak van der Waals minima are found at around 4.25 Å with a well depth of about 200 cm−1 for the 3A1 state and roughly 100 cm−1 for the 3E state. The excited 1E state only shows a very weak minimum of 40 cm−1 at about 4.5 Å. The well depths are given for bent geometry but are similar for the planar configuration. Additionally, a conical intersection between the 3A1 and the 3 E state at approximately 3.6 Å can be observed. These features are also reproduced by the diabatic model (see Figure 4) demonstrating the high accuracy that can be achieved. The eigenvectors of the diabatic model provide interesting information as well and deserve a closer inspection. Since the diagonalization of the diabatic matrix yields the diabatic/ adiabatic transformation, the eigenvectors show the composition of the adiabatic wave functions in terms of the diabatic basis states. At the potential minimum of the bent surface the 1 A1 ground state is composed of 64% I (1 2P) ⊗ CH3 (2A″2 ), 18% I− (1S) ⊗ CH3+ (1A′1), and 18% I+ (1S/1D) ⊗ CH3− (1A′1). This is still a quite high contribution of the repulsive first basis state. Apparently, a 36% fraction of strongly bound components is enough to form a deep potential well. The excited singlet 1E state is repulsive and is composed of the repulsive I (1 2P) ⊗ CH3 (2A″2 ) basis state (70%) and only one bound component, the I+ (1S/1D) ⊗ CH3− (1A1′ ) basis state (27%). The I− (1S) ⊗ CH3+ (1A1′ ) basis state obviously is vital to form the bond, which might be explained by the lower corresponding diabatic energy compared to the I+ (1S/1D) ⊗ CH3− (1A′1) state. The first triplet state (3E) is dominated by the repulsive I (1 2P) ⊗ CH3 (2A2″) basis state (68%) with a contribution of 31% of the

Figure 2. Eigenvalues of the diabatic model (lines) and ab initio MRCI energies (symbols) for the low-lying states of CH3I with planar CH3 geometry.

Figure 3. Eigenvalues of the diabatic model (lines) and ab initio MRCI energies (symbols) for the low-lying states of CH3I with bent CH3 geometry.

Figure 4. Eigenvalues of the diabatic model (lines) and ab initio MRCI energies (symbols) for the low-lying states of CH3I with planar CH3 geometry in the van der Waals region.

places in this matrix. The combination of the spin−orbit matrices and additional states with the diabatic model yields an 7414



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I+ (3P) ⊗ CH3− (1A′1) basis state. As expected, this is quite similar to the 1E adiabatic state because the difference between these states is mainly the spin coupling of the two unpaired electrons. This kind of analysis can be performed for all states at every point of the nuclear configuration space and results in insights into the change of electronic structure during processes like bond formation and bond breaking. Up to here the effect of SO coupling is not taken into account. In the next step of the ERCAR approach the diabatic model from the previous section is combined with the SO coupling matrix. After the combined coupling matrix has been set up, diagonalization yields the fine structure eigenstates and energies. The potential curves for the low-lying fine structure states are displayed in Figure 5 and Figure 6 for planar and bent geometries, respectively.

which can be interpreted physically. The effect of the SO coupling is that basis states of different symmetry as well as different spin multiplicities are mixed. If the mixing is particularly strong, a characterization of a fine structure state by space symmetry and spin multiplicity becomes impossible and pointless. In CH3I such mixing of space and spin symmetries is significant but not overly strong and the fine structure states can be characterized by the dominating basis states. For example, at the minimum of the bent geometry, the second excited fine structure state, commonly known as 3Q1, is mainly composed of 3E basis states with a contribution of 9% of 1 E basis states. The computed 1E contribution is similar but slightly less than that obtained from the ab initio calculation (15%). The second optically active state, 3Q0, is also mainly composed of 3E states and has only negligible contributions from other states ( HCl+H Reaction. Science 2002, 296, 715−718. (22) Amatatsu, Y.; Morokuma, K.; Yabushita, S. Ab initio PotentialEnergy Surfaces and Trajectory Studies of A-Band Photodissociation Dynamics: CH3I*→CH3+I and CH3+I*. J. Chem. Phys. 1991, 94, 4858−4876. (23) Amatatsu, Y.; Yabushita, S.; Morokuma, K. Full NineDimensional Ab Initio Potential Energy Surfaces and Trajectory Studies of A-Band Photodissociation Dynamics: CH3I*→CH3+I, CH3+I*, and CD3I*→CD3+I, CD3+I*. J. Chem. Phys. 1996, 104, 9783−9794. (24) Yabushita, S.; Morokuma, K. Potential-Energy Surfaces for Rotational-Excitation of CH3 Product in Photodissociation of CH3I. Chem. Phys. Lett. 1988, 153, 517−521.

photophysics and photochemistry of CH3I. In this context, some features of the excited fine structure states are of high relevance, which need to be properly reproduced by the model. Besides the vertical excitation energies, these are the energy gradients, the nonadiabatic couplings, the SO couplings, and, of utmost importance, the location of the conical intersection between the 3Q0 and 1Q1 states. None of these properties can be obtained directly from experiment, but they all influence the photodynamics. Therefore, we compare the results from our ERCAR model with high-level ab initio calculations and a potential model by Xie et al.,29 which is known to have several shortcomings. In this comparison these shortcomings clearly become apparent. Though the excitation energies are found to be quite reliable, the position and energy of the conical intersection as well as the gradients at the Franck−Condon point are far away from our results. All these effects may explain the reported problems observed in recent quantum dynamics calculations. The present study is only the first but necessary step toward a full-dimensional set of coupled PESs for CH3I. The onedimensional results clearly show that a high accuracy can be obtained by the ERCAR approach. Furthermore, this method does not require the computationally demanding and cumbersome ab initio calculation of fine structure states and energies and thus is highly efficient with regard to the data acquisition for the multidimensional PESs. The extension of the present model to account for additional internal coordinates is more or less straightforward because this only affects the diabatic model. The main effort in the development will be to find appropriate functional forms for the diabatic matrix elements and determine the corresponding parameters that yield sufficiently accurate adiabatic energies compared with reference ab initio data. Work in this direction is currently in progress.

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ASSOCIATED CONTENT

* Supporting Information S

The parameter sets corresponding to the 40 unique matrix elements for each treated geometry. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the Deutsche Forschungsgemeinschaft (DFG) for generous financial support through SFB 613.

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REFERENCES

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