On the VB adiabatic–diabatic picture of bonding in CO2+

Journal of Molecular Structure (Theochem) 547 (2001) 17±26

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On the VB adiabatic±diabatic picture of bonding in CO 21q Rudolf PolaÂk a,*, JirÏÂõ CÏÂõzÏek a,b a

J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, DolejsÏkova 3, 182 23 Prague 8, Czech Republic b Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ont., Canada, N2L 3G1 Received 13 September 2000; revised 10 October 2000; accepted 20 October 2000

Abstract Simple methods to calculate non-adiabatic coupling elements and to approximately diabatize electronic states of the 1P manifold of a classical VB model for CO 21 [Chem. Phys. 232 (1998) 25] are used to discuss bonding in metastable species. It is shown that the bound character of an adiabatic state is compatible with a system of intersecting repulsive diabatic energy curves and associated diabatic coupling terms. The results contradict the existing interpretations of dication bonding which claim the universality of the two-state crossing model. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Valence bond approach; Dication; Bonding; Non-adiabatic coupling; Diabatizing transformation

1. Introduction The doubly charged molecular cations are important species [1] whose scienti®cally interesting properties are primarily related to the existence of stable or metastable electronic states. Besides application of the best currently available quantum chemical techniques (e.g. Refs. [2,3]), the problem for double charged species was also tackled by a number of thoughtful considerations on the physical background of the origin and nature of metastability. Since the pioneering work of Pauling on the simplest dication He 221 [4], two basic approaches to interpret the bonding features of dications, related to potential wells and barriers have been developed on the dissociation pathway. Whereas one of them, close q

Dedicated to Josef Paldus on the occasion of his 65th birthday, in appreciation of his contribution to the ®eld of molecular physics. * Corresponding author. Tel.: 1420-2-6605-2077; fax: 1420-2858-2307. E-mail address: [email protected] (R. PolaÂk).

to Pauling's view, emphasizes covalent bonding superimposed by Coulomb repulsion [5,6], the other one uses a pseudocrossing picture originating from interaction of symmetrically-charge distributed and charge-transfer structures [7,8]. The latter model undoubtedly shows a large degree of predictive power in the context of estimation of transition structure parameters and kinetic energy release in dication fragmentation. The appropriateness of both approaches to special systems was discussed [9±14] and it was found that the conclusions drawn were not wholly consistent with each other. While Refs. [12,13] argued in favor of the avoided crossing model, Refs. [9±11,14] rejected this. In our previous paper [14], we used a classical valence bond model of CO 21 to discuss the origin of bonding interactions in a protype dication. It appeared that even a rudimentary kind of VB model shows a general qualitative agreement with accurate studies [10,15] and reproduces the topological behavior of the potential curves associated with metastability of the dication obtained by accurate ab initio calculations,

0166-1280/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(01)00456-0

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the closest coincidence being found for the 1P(1) state. The analyses of classical VB wave functions showed that the charge-transfer structure is crucial for the appearance of metastability, however, not in the form of direct coupling between the leading covalent and charge-transfer structures as it is assumed by the avoided crossing model, but rather by mediated interaction through additional states. This wave function analysis was based solely on approximate adiabatic wave functions. The purpose of this paper is to complement our previous work by calculating and examining nonadiabatic coupling matrix elements between all VB model states of the 1P manifold and by converting the adiabatic picture of the problem to the diabatic one. A slight distinction of this task with respect to the majority of other similarly oriented investigations using MO CI approaches consists in that the state functions are expressed in terms of non-orthogonal VB structures instead of orthogonal basis functions. Further, the approach is not restricted to a two-state (or limited number of states) truncation (e.g. Refs. [12,13,16,17]), but uses the whole model space of adiabatic functions in the diabatizing transformation. Emphasis is laid on simplicity and computational convenience, since our ®nal objective is to gain additional insight into the relationship between the wave function character and the topological behavior of the potential energy curve along the dissociation coordinate of the dication, and to examine the extent of correspondence between the VB structures and diabatic functions. 2. Non-adiabatic coupling and the diabatizing transformation The formal and computational development of techniques applicable to non-adiabatic coupling phenomena, including state-of-the-art quantum chemical description of electronic states and application of analytic gradient techniques, reaches almost a routine level and is well documented by a number of review articles [18±21]. Because of the applied ad hoc non-standard approach to the selected case, it is appropriate to provide a few explanatory remarks. The approximate adiabatic wave functions F i corresponding to eigenvalues E iA of a molecular

system expressed in terms of non-orthogonal basis functions C k is given by X i Fi ˆ Ck C k ; …1a† k

which may be rewritten using the notation of Montet et al. [22] in the form

F ˆ CC; C T C ˆ S:

…1b†

This is typically the case with the classical valence bond (VB) approach [23] where the C k's are spin- and space-symmetry adapted VB structures in the form of composite functions [24]. Restricting oneself to diatomic systems, the nonadiabatic ®rst-derivative coupling matrix element (e.g. Ref. [21]) between real state functions F i and F j is   X  j 2 i 2Cl F j ˆ G ij …R† ˆ F i C C k C l k 2R 2R k;l   2C l 1 Cki Clj C k ; 2R

…2†

where R is the nuclear displacement coordinate of the molecule. The two types of terms in the summation involve differentiation of variational expansion coef®cients and VB basis functions. The terms including overlap are, of course, usually missing in these approaches for providing adiabatic state functions which are based on MO theory. Using a ®nite difference approximation [25±27] for the derivatives, the non-adiabatic coupling (NAC) matrix element (2) can be approximated by # " X i DC j 1 i j DkC k uC l l l Gij ˆ kC k uC l l 1 Ck Cl Ck 2 DR DR k;l ˆ dC 1 dS:

…3†

In actual calculations, a satisfactory value for DR appeared to be 10 25 bohr. With this value of DR, the antihermitian property [28] Gij ˆ 2Gji is accurate to ®ve or six signi®cant digits. An additional numerical veri®cation of formula (3) was sought in calculating coupling elements Goij by means of wave functions F i spanned by the LoÈwdin [29] orthonormalized basis set J obtained by transformation of the non-orthogonal

Ï ÂõzÏek / Journal of Molecular Structure (Theochem) 547 (2001) 17±26 R. PolaÂk, J. C

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Fig. 1. Asymptotic dissociation energy limits associated with the composite function basis set of the VB model for the 1P states of CO 21.

VB structures C

J ˆ CS 21=2 :

…4†

The coupling elements, Goij ˆ

X k

cik

Dcjk ; DR

…5†

expressed in terms of the expansion coef®cients c in the orthonormalized basis, c ˆ S1=2 C;

…6†

coincided with the values calculated by formula (3) in four to ®ve signi®cant digits with the exception of very short distances. However, even in such cases the results from both approaches were indistinguishable in graphical representation. Within the Smith±Baer approach [30±32] the nonadiabatic coupling matrix G is a prerequisite for calculating diabatic molecular functions f i by means of a unitary transformation

f ˆ FU

…7†

where U as a function of R is found by integrating the equation ZR G…R†U…R†dR: …8† U…R† ˆ 1 2 1

From the two alternatives to calculate coupling elements for Eq. (8) we use G o, because the orthogonalized VB structures are simply related to the diabatic states by a unitary transformation, still assuring the adiabatic states to coincide with the diabatic ones at in®nite separations. The Euler method was applied to calculate U at a ®nite number of properly selected points. In order to preserve strict unitarity of the transformation matrix during the numerical procedure, at each separation Rn a slight modi®cation of the calculated matrix U 0 (Rn) was performed by transformation U ˆ B 1=2 U 0 ;

…9a†

where B ˆ ‰U 0 U 0T Š21 :

…9b†

The stepwise corrections for preserving unitarity make this procedure approximate. The diagonal elements of the usual electrostatic Hamiltonian in the representation of (approximate) diabatic functions, X 2 A HjjD ˆ Uij Ei ; …10† i

are the (approximate) diabatic energies.

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Fig. 2. Adiabatic potential curves for the 1P states.

3. Results and discussion The above described diabatizing procedure was applied to the manifold of adiabatic 1P states of the VB model [14] which is based on the symmetrically orthogonalized formulation [33] of the Mof®tt second version of the atoms-in-molecules method [34,35]. The polyatomic basis set, comprising C 1´O 1, C 21´O and C´O 21 structures, consists of spin- and spacesymmetry adapted composite functions (CFs) built from approximate atomic eigenfunctions. The state groups contributing to the 20 CFs of the VB model space are displayed in Fig. 1, showing the energy level spacings at the dissociation limit, with the exception of the C( 3Pg)´O 21( 3Pg) asymptote which lies about 0.25 hartree above the highest indicated level. Thus,

the three asymptotes of the charge-transfer structures considered lie 9.40, 13.94 and 20.59 eV above the lowest 1P dissociation limit. The complete collection of PECs corresponding to 1P adiabatic states is shown in Fig. 2. It is physically justi®ed to consider the CFs as diabatic templates [21] and therefore we also give in Fig. 3 the diagonal hamiltonian matrix elements related to the polyatomic basis functions as a function of R. In Fig. 4 the R-dependence of the NAC matrix element G12 calculated using Eq. (3), together with its partitioning into individual contributions dC and dS, and adiabatic energies corresponding to states 1 and 2 are presented. One observes that dS is for the major part of internuclear separations, negligible compared to dC, however it becomes important at

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Fig. 3. Diagonal hamiltonian matrix elements in the CF representation for the 1P states as functions of internuclear separation.

small values of R. Both contributions can attain mutually opposite signs, as it happens in Fig. 4 for R in the region between about 3 and 6 bohr. The o values of the NAC elements G 12 calculated using the orthonormalized basis (cf. Eq. (5)) lie on top of the solid curve representing G12, indicating that both approaches yield very close results. For illustrative purposes, we present in Fig. 5 the NAC curves Go1i …R† and Goi21;i …R† labeled by (1,i) and (i 2 1,i), respectively, for the adiabatic states de®ned by i [ k2; 9l: The selected points used in the calculations are indicated in the top panels of these ®gures. Although the strongest non-adiabatic coupling occurs usually between neighboring states, we notice that substantial coupling can extend over several states [compare, e.g. the values of the coupling between states 1 and 8, (1,8), with the values of (1,2)]. With

respect to the solution of Eq. (8) for calculating the diabatizing transformation, this indicates that a truncation of the `complete' model space to a smaller one has to be submitted for careful consideration. Mostly the position of maxima of the NAC curves can be predicted from the shape of the PECs [e.g. NAC (8,9) and corresponding PECs], but this is not always so. The absolutely largest value of coupling was found between adiabatic states 5 and 6 at R ˆ 5.0 bohr with the NAC curve extremely sharply peaked. Although the majority of NAC curves tend to zero at large separation of atoms, the ®nite difference method had dif®culties in cases when levels become degenerate at in®nite separation. This holds, e.g. for the element (2,3). The reason lies in the fact that ± from the numerical viewpoint ± the corresponding wave functions become large (but ®nite) R unspeci®ed

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Fig. 4. Comparison of non-adiabatic coupling matrix elements pertinent to 1P states 1 and 2 as obtained via ®nite difference approximation for non-orthogonal and orthogonalized VB basis sets.

within a unitary transformation which makes the numerical variation of one of the functions with R dubious. This situation is also manifested by the fact that NAC calculated by formulae (3) and (5) is prone to differ. In such cases, the NAC element curve entering Eq. (8) was adapted to follow the regular asymptotic behavior. The approximate diabatic energy curves calculated according to Eqs. (8)±(10) are shown in Fig. 6. For a more ef®cient comparison with other types of energy curves, all energy graphs are provided with the Coulomb repulsion dependence 1/R. Generally, all types of energy curves become steeper than the Coulomb interaction at a smaller internuclear separation than a certain value of R, including chargetransfer type (i.e. C 21´O or C´O 21) curves. The latter ones, of course, proceed the major part of separations at a much smaller slope than the C 1´O 1 curves. Among the diabatic templates (Fig. 3) and the diabatic potential curves (Fig. 6), one ®nds only one slightly bound curve (by about 0.01 hartree) corresponding to the diagonal hamiltonian matrix element of the C 21( 3Pu)´O( 3Pg) structure (cf. Fig. 3). Despite the remarkable resemblance between the energy curves presented in Figs. 3 and 6, one notices

an interesting difference between the diagonal elements of the hamiltonian matrix in CF representation and the diabatic PECs. There is a group of three low-lying diabatic PECs which do not exceed the steepness of the term 1/R to a much smaller distance than the corresponding diagonal terms. On the contrary, the diabatic state 1 rises faster than the corresponding diagonal matrix element, and crosses a number of diabatic states in a narrow region of separations. The crossing is more robust than that occurring with the corresponding diagonal hamiltonian matrix elements, suggesting the following interpretation of formation of the adiabatic quasibound lowest 1P state within the diabatic view of the classical VB model: the bound character of the adiabatic state comes from curve crossing pertinent to repulsive diabatic energies and the associated electronic interaction energy terms. This bonding picture is different from that found by Kolbuszewski et al. for BF 21 [12] and He 221 [13] in an adiabatic±diabatic analysis of high quality ab initio electronic-structure calculations. Their study corroborates the usefulness of the avoided crossing model which takes into account a pair of diabatic potential curves, one attractive corresponding to the

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Fig. 5. Selected non-adiabatic coupling matrix elements as functions of internuclear separation. (a) Interaction between the ground state and some excited states. (b) Interaction between `neighbor' states.

charge-transfer structure and the other purely repulsive, and the diabatic coupling term. Within this model, the main parameter [7,8,11,12] distinguishing individual cases and describing the interaction between the two corresponding states, is the asymptotic energy separation D between the covalent and 21 charge-transfer structures. For BF 21, He21 2 and CO the values of D are 7.73, 29.82 and 9.40 eV, respectively, which do not signal substantially different behavior for CO 21 compared to the other two species. What might be different, however, are the number and the intrinsic properties of the electronic covalent and the charge-transfer states which have an effect on the shape of the potential energy surface. More

speci®cally, as also appears from Fig. 6, the prediction of the topology of the adiabatic PECs requires a detailed description of the location of crossings and the slopes of the involved diabatic curves, and the corresponding diabatic coupling. Although the VB model offers a plausible way of describing formation of metastable states in doubly charged ions, one has, of course, be aware of the fact that the classical VB approach bears a limited capability of providing single VB structures with binding property based on the fact that it discards optimization of atomic functions with geometry variation, setting the standard con®guration interaction as the dominant effect. Under these circumstances, we

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Ï ÂõzÏek / Journal of Molecular Structure (Theochem) 547 (2001) 17±26 R. PolaÂk, J. C

Fig. 5. (continued)

rather seek an understanding of interesting features about bonding in general, than quantitative description of electronic states. It is satisfying that our ®ndings reinforce the interpretation of bonding in CO 21 provided by Levasseur et al. [10] on the basis of MCSCF data.

of the limitations of the classical VB approach and the speci®city of one electronic state in one diatomic, the model calculation highlights factors which might govern bond formation in cases when the twostate avoided crossing model [7,8] is not applicable. The present ®ndings are in many respects supporting the conclusions reached in earlier papers [9±11,14].

4. Conclusions An adiabatic±diabatic analysis of 1P states obtained within a VB model for CO 21 [14] showed that the formation of a dication quasibound state can be explained in terms of intersecting repulsive diabatic states and electronic diabatic interactions. Regardless

Acknowledgements This article was supported in part by a NSERC of Canada research grant (J.C.) which is gratefully acknowledged.

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Fig. 6. 1P approximate diabatic energy curves.

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