D matrix analysis of the Renner-Teller effect: An accurate three-state diabatization for NH2

D matrix analysis of the Renner-Teller effect: An accurate three-state diabatization for NH2 G. J. Halász Department of Information Technology, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary Á. Vibók Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-40410 Debrecen, Hungary R. Baera兲 Department of Physical Chemistry and The Lise Meitner Center for Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel M. Baerb兲 The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel 共Received 17 May 2006; accepted 17 July 2006; published online 1 September 2006兲 Some time ago we published our first article on the Renner-Teller 共RT兲model to treat the electronic interaction for a triatomic molecule 关J. Chem. Phys. 124, 081106 共2006兲兴. The main purpose of that Communication was to suggest considering the RT phenomenon as a topological effect, just like the Jahn-Teller phenomenon. However, whereas in the first publication we just summarized a few basic features to support that idea, here in the present article, we extend the topological approach and show that all the expected features that characterize a three 共multi兲state RT-type’3 system of a triatomic molecule can be studied and analyzed within the framework of that approach. This, among other things, enables us to employ the topological Dmatrix 关Phys. Rev. A 62, 032506 共2000兲兴 to determine, a priori, under what conditions a three-state system can be diabatized. The theoretical presentation is accompanied by a detailed numerical study as carried out for the HNH system. The D-matrix analysis shows that the two original electronic states 2A1and 2B1共evolving from the collinear degenerate ⌸doublet兲, frequently used to study this Renner-Teller-type system, are insufficient for diabatization. This is true, in particular, for the stable ground-state configurations of the HNH molecule. However, by including just one additional electronic state—a Bstate 共originating from a collinear ⌺state兲—it is found that a rigorous, meaningful three-state diabatization can be carried out for large regions of configuration space, particularly for those, near the stable configuration of NH2. This opens the way for an accurate study of this important molecule even where the electronic angular momentum deviates significantly from an integer value. © 2006 American Institute of Physics.关DOI: 10.1063/1.2336219兴 I. INTRODUCTION In 1934 Renner published a detailed study of a linear polyatomic molecule, which possesses a single adiabatic potential energy surface but then, upon bending, splits into two surfaces.1 As long as a 共triatomic兲molecule is limited to its linear configuration we expect the electronic orbital angular momentum quantum number ⌳ប 共defined with respect to the molecular axis兲to serve as a good quantum number 共i.e., to be conserved兲. Moving slightly away from linearity introduces another quantum number, namely, the nuclear angular momentum quantum number, ᐉប. As long as the deviation is small enough both ⌳and ᐉserve as good quantum numbers.1–5 This deviation from collinearity has one additional effect on the molecular structure. All electronic states 共with a given energy兲, except for the ⌺state, split into two states with different energies. Renner was the first to study this splitting and the corresponding physical implications 共for a ⌸state兲 and therefore the whole phenomenon, which is also well supported by numerous experiments, bears his name. Introducing the radial coordinate q共which measures the deviation from collinearity兲, Renner assumed that for q⬎0 the ⌳state splits into two states, i.e., E⌳± ␴ 共q兲, where ␴ 共q兲= ␴ 0q2. Renner showed that as long as ␴ 0is small enough ⌳and ᐉare, as mentioned earlier, good quantum numbers, but once it becomes large the two quantum numbers lose their relevance as such, and K, defined as K =⌳±ᐉ, becomes the good quantum number.1–5In this way the spectrum of a triatomic molecule which for a weak coupling is characterized by the quantum numbers 共v共±兲,⌳,ᐉ =±1兲turns, for a strong coupling, into a new spectrum of states, characterized by the quantum numbers 共v共±兲,K =⌳±1兲. In fact, the more relevant part 共to our issue兲in Renner’s treatment is the nonadiabatic coupling term 共NACT兲that a兲Electronic mail: [email protected] b兲Electronic mail: [email protected] THE JOURNAL OF CHEMICAL PHYSICS 125, 094102 共2006兲 0021-9606/2006/125共9兲/094102/9/$23.00 © 2006 American Institute of Physics125, 094102-1 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp couples the two adiabatic states: E⌳± ␴ 共q兲. Expressing this coupling in terms of polar coordinates our main interest is in the angular component: 共1/q兲p ˜ ⌳ ␸ ±ᐉ共s兲where p ˜ ⌳ ␸ ±ᐉ共s兲is given in the form1–13 p ˜ ⌳ ␸ ±ᐉ共s兲= 冓 ␨ ⌳+ᐉ共se兩s兲兩 ⳵ ⳵␸␨ ⌳−ᐉ共se兩s兲 冔 .共1兲 Here, ␨ ⌳±ᐉ共se兩s兲are the two electronic eigenfunctions, seand sare the electronic and the nuclear coordinates, respectively, and ␸ is the corresponding 共nuclear兲polar coordinate associated with ᐉ. This kind of interaction term is also expected from the Born-Oppenheimer 共BO兲treatment.14,15 Comment: It is important to realize that ␸ is not a regular internal body-fixed coordinate but an out-of-plane coordinate associated with a rigid rotation of the molecule around an axis located in the plane formed by the three atoms 关see, i.e., Fig. 1. Applying his model, Renner assumed that p ˜ ⌳ ␸ ±ᐉ共s兲is a constant independent of sand established that in the vicinity of the collinear axis p ˜ ⌳ ␸ ±ᐉ共s兲is equal to ⌳, which is an integer. In case of a ⌸state we have ⌳=1 and therefore p ˜ ⌳ ␸ ±ᐉ共s兲=⌳=1. 共2兲 In the earlier years until the end of the 1950s this value for p ˜ ⌳ ␸ ±ᐉ共s兲was employed in numerous applications. Later, efforts were made to calculate this value as a function of q and to apply the actual value 共which was found to differ from 1 or from any other integer兲.6–13 However, this 共noninteger兲 calculated value cannot be used in a straightforward way due to difficulties related to the diabatic potentials, which become multivalued.13 To avoid this difficulty the value of p ˜ ⌳ ␸ ±ᐉ共s兲is assumed to be ⌳=1 in those expressions that are directly related to the diabatic potentials, whereas in other instances the calculated value is used.6,7,9–13 It seems that all the numerical treatments related to the Renner-Teller 共RT兲 interaction are based on the two-state approximation and in no situation was a three-state diabatization carried out. Our main interest in the Renner-Teller-type molecules is that due to the vibronic coupling, their electronic state of a given energy splits into two 共or more兲states 共with different energies兲, and that this splitting disappears once q=0, namely, when the molecule becomes linear. This situation is reminiscent of the Jahn-Teller 共JT兲interaction3,16–28 which stems from the fact that two 共or more兲electronic states become degenerate. The similarity between the two interactions is even more pronounced because just like the points of degeneracy within the JT framework that arrange themselves along seams,29共a兲the degeneracy points within the RT framework are located along the collinear axis which for all practical purposes can be considered as a seam. To continue, we switch from Renner’s notation for the NACT 关presented in terms of p ˜ ⌳ ␸ ±ᐉ共s兲兴 to our notation that has been used, now for several years, while treating the JT effect. We consider the NACT, ␶ jk共s兲共known to be a vectorial matrix element兲, which couples the two adiabatic states jand k, and is given as a function of the coordinates of the nuclei. These NACTs are of the form ␶ jk共s兲=具 ␨ j共se兩s兲兩ⵜ ␨ k共se兩s兲典,共3兲 where ␨ j共se兩s兲and ␨ k共se兩s兲are the corresponding two 共electronic兲eigenfunctions and ⵜis the grad operator. Recently we published a Communication30 in which we discussed the RT interaction and how it is related to the corresponding RT NACTs and the degeneracy points. To be more specific we considered the HNH molecule and employed an ab initio package to calculate p ˜ ⌳ ␸ ±ᐉ共s兲for the case ᐉ=⌳=1. In what follows this particular magnitude is designated ␶ ␸ 12共s兲and is written in the form FIG. 1. Ab initio RT nonadiabatic coupling terms, ␶ ␸ 12共q兩z兲and ␶ ␸ 23共q兩z兲, as calculated for the collinear and planar HNH molecule. The results are presented as a function of q—the vertical distance of the rotating atom from the fixed 共collinear兲axis—for different configurations: 共a兲the rotating atom is nitrogen and the calculations are done for z=1.95 a.u. 共the C2vsymmetry兲;共b兲the rotating atom is nitrogen and the calculations are done for z=1.0 a.u.; 共c兲the rotating atom is hydrogen and the calculations are done for z=3.9 a.u. 共—兲 ␶ ␸ 12共q兩z兲;共¯兲 ␶ ␸ 23共q兩z兲. 094102-2 Halász et al. J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp ␶ ␸ 12共s兲= 冓 ␨ 1共se兩s兲兩 ⳵ ⳵␸␨ 2共se兩s兲 冔 .共4兲 To carry out this calculation we assume that two out of the three atoms are clamped at their positions and only one atom is allowed to rotate around the molecular axis. Consequently ␶ ␸ 12共s兲described in terms of three cylindrical coordinates related to this atom, namely, s=共 ␸ ,q,z兲共see Fig. 1兲. Locating the origin at the position of the left hand side clamped hydrogen and assuming the zaxis to be along the molecular axis, we get that 共1兲the coordinate zmeasures, for the collinear configuration, the distance of the rotating atom from the origin 共2兲the radius qmeasures its vertical distance from the 共molecule兲axis; and 共3兲the angle ␸ is the corresponding 共out-of-plane兲cylindrical angular coordinate around the axis 关see Fig. 1兴. It is important to emphasize that such calculations were carried out not only for the two ⌸states 共designated as states 1 and 2兲but for any two states included in the numerical study. Preliminary results along these lines 共for the two ⌸ states兲were published in that Communication.30 For completeness they are presented once again 共following an improved convergence兲. Thus, we consider three different configurations: two configurations refer to the case that nitrogen is rotating around the fixed HH axis 关see panels 共a兲and 共b兲兴 and one configuration refers to the case that hydrogen is rotating around the fixed HN axis 关see panel 共c兲兴. All calculations are done for the case that at the collinear arrangement the distance RHH, between the two hydrogens, is RHH =3.9 a.u. In panel 共a兲are presented results for which the nitrogen is located along the symmetry line 共the C2vconfiguration兲and therefore z=1.95 a.u.: in panel 共b兲are presented results for the case where the nitrogen is shifted away from the symmetry line, namely, for z=1.0 a.u. 共thus, the nitrogen is much closer to one of the hydrogens兲. In panel 共c兲we show results for the case where a hydrogen and the 共single兲 nitrogen are clamped whereas the second hydrogen is allowed to rotate around the axis 共in this case the HN axis兲. The two main features to be noted are 共1兲the value ␶ ␸ 12共q⬃0.0,z兲⬃1.0 for all three cases; this implies 共in “our” language兲that the RT NACTs, in the slightly bent HNH system, are quantized and that their value is 1 共and not, e.g., 1/2 like in the cases of JT NACTs兲. This result is expected, as elaborated in the literature.1–5共2兲However, it is also noted that ␶ ␸ 12共q,z兲decreases monotonically as qincreases 共i.e., as the molecule becomes more and more bent兲; but the rate of decrease is zdependent. This feature too is mentioned in the literature6,7,9–14 关mainly for the C2vconfiguration presented in panel 共a兲兴. It is well known that the deviation of ␶ ␸ 12共q,z兲from 1 leads to multivalued diabatic potentials.6–12 However, it is not clear if a rigorous treatment was ever done to resolve the multivaluedness of the diabatic potentials. We intend to show that our theory that supplies the means to rigorously resolve such difficulties in case of the JT interaction, applies also for the RT interaction. The present article contains the following sections: In Sec. II is given a brief account of the theory as developed for the JT intersections and their NACTs. Certain aspects of this theory are then applied for the RT NACTs. In Sec. III are given numerical results as obtained for the HNH molecule: We present potential energy curves and the RT NACTs as a function of q, for different zvalues, related to the three lower states of this molecule, namely, the two ⌸states and the state that evolves from the ⌺state. A discussion and the conclusions are given in Sec. IV in which we also briefly relate to JT cis located in the same region of configuration space. II. THEORY A. Background comments As mentioned earlier the theoretical treatment applies to the NACTs due to both the RT interaction and the JT interaction. The RT interaction, like the JT interaction, can be traced back to the BO treatment14,15 but what really makes the two types similar is the fact that both are caused by the degeneracy of eigenfunctions at certain points in configuration space. Moreover, in both cases, these points have the tendency to arrange themselves along continuous, usually infinite, lines defined in configuration space. These lines are known, in case of the JT interaction, as seams and we suggest applying the same name to the lines in case of the RT interaction. The main difference between the two kinds is that the JT seams are usually accidental and therefore have to be exposed numerically by trial and error, whereas the RT seams are a result of bending modes and therefore run along well defined geometrical lines such as the axes of triatomic molecules, etc. The theory we intend to present is based on closed contours that surround the seams.20共b兲,29共b兲The main outcome of this theory is that it supplies the conditions for a group of states to yield, in a given region in configuration space, single-valued diabatic potentials. This theory is successfully applied in case of the JT interaction and we intend to show that it applies also in case of the RT interaction. The difference in treating these two kinds of interaction is the positions of the seams with regard to the plane formed by the three atoms 共A, B, and C兲. In case of the JT interaction the seam intersects that plane at one point 共or at most, at a few isolated points兲and therefore the contour that surrounds the seam can be assumed, without detracting from generality, to be fully located in that plane. In case of the RT interaction the seam is always located in the plane and therefore the 共closed兲contour that surrounds it has to be located outside that plane. The simplest choice is to assume it to be in a plane perpendicular to the seam 共see Fig. 1兲, although any other contour 共that surrounds the seam兲is expected to yield similar results. Although the seams are located in different positions 共with regard to the triatom plane兲the contours are constructed by clamping two atoms and permitting the third atom, that is atom A, to move freely and form the 共closed兲 contour. In both cases we assume the contour to be in a plane and therefore if the origin of the system of coordinates is in that plane the position of the free atom is determined by two coordinates 共usually polar coordinates兲. This description can be easily generalized to more elaborate contours. 094102-3 Analysis of the Renner-Teller effect J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp In both cases the free atom 共atom A in our case兲is used to examine the values of the various NACTs, ␶ jk共s兲;j,k =兵1,N其where Nis the number of states included in the manifold and sis a nuclear coordinate in the region of interest. According to our way of presenting the NACTs the point sis written as s=共 ␸ ,q兩z兲where 共 ␸ ,q兲are the two polar coordinates mentioned earlier and zis an indirect coordinate 共or describes numerous indirect coordinates兲to characterize different contours. In the present study we consider systems with two or three states, thus N=2,3, and treat only more significant NACTs, namely, ␶ 12共s兲and ␶ 23共s兲关the third NACT, i.e., ␶ 13共s兲, is usually negligibly small and therefore is ignored in our study兴. Before we start elaborating on the theoretical tools to be applied in this study we mention another issue, namely, the Hilbert subspace.20共b兲,29共b兲A Hilbert subspace, usually in a given region in configuration space, is defined as a group of states which approximately are not coupled 共or at most are weakly coupled兲to states outside the group. Thus if ␶ jk共s兲is a NACT that couples states jand kwhere state jbelongs to the subspace and state kdoes not, then for all practical purposes all its components are negligibly small. The importance of the subspace stems from the fact that the corresponding NACTs 关see Eq. 共3兲兴can form single-valued diabatic potentials. In others words if a group of states does not form a Hilbert subspace the corresponding NACTs will deliver multivalued diabatic potentials which cannot be applied in any further studies. In the next subsections we briefly discuss the connection between the ability to form single-valued diabatic states and the features of the nonadiabatic coupling matrix 共NACM兲. This was done on numerous occasions in case of the JT NACM and here, among other things, we emphasize the friendly features of the RT NACM. B. The NACM and the diabatic potential matrix Having the BO 共diagonal兲adiabatic potential matrix, u共s兲, the diabatic potential matrix W共s兲is obtained following the adiabatic-diabatic transformation 共ADT兲matrix A共s兲:20共b兲,29共c兲,31 W共s兲=A†共s兲u共s兲A共s兲.共5兲 The ADT matrix A共s兲can be shown to be an orthogonal 共unitary兲matrix that fulfills the following first order differential 共vector兲equation31 ⵜA共s兲+ ␶ 共s兲A共s兲=0,共6兲 where ␶ 共s兲is the NACM with the elements as defined in Eq. 共3兲. The solution of this equation can be written as an exponentiated line integral, A共s兩s0,⌫兲=㜷exp 冉 − 冕 s0 s ds· ␶ 共s兩⌫兲 冊 A共s0,⌫兲,共7兲 where 㜷is the ordering operator, s0is the initial point of integration, ⌫is the contour along which Eq. 共6兲is required to be solved, the dot stands for a scalar product, and A共s0,⌫兲 is the initial value of A共s兲on ⌫. In what follows A共s0,⌫兲is assumed to be the unit matrix. It is well noticed that the only component of ␶ 共s兲that affects the above line integral is the tangential component along the contour ⌫. Another matrix of interest is the topological matrix D共⌫兲 which is identical to the Amatrix but is calculated for a closed contour.20共b兲,23共b兲,23共c兲,29共b兲,29共c兲Thus D共⌫兲=A共s0兩s0,⌫兲=㜷exp 冉 − 内 ⌫ ds· ␶ 共s兩⌫兲 冊 .共8兲 It can be shown that in order for the diabatic potential matrix W共s兲to be single valued in the region of interest the D matrix, for any chosen closed contour ⌫in the region, has to be diagonal. Since D关just like A共s兲兴 is unitary its elements are expected to be20共b兲,23共b兲,23共c兲,29共b兲,29共c兲 Djk共⌫兲= ␦ jk exp共i ␪ j共⌫兲兲,j=兵1,N其,共9a兲 where ␪ j共⌫兲;j=兵1,N其are real phases. In case of real eigenfunctions the phases become integer multiples of ␲ so that the D-matrix elements are Djk共⌫兲=± ␦ jk,j=兵1,N其.共9b兲 Next we briefly analyze what happens in case ⌫is chosen to be a circle defined by the position of its center and the relevant radius q. In this situation the ADT matrix can be written as A共 ␸ 兩q,⌫兲=㜷exp 冉 − 冕 0 ␸ d ␸ ␶ ␸ 共 ␸ 兩q,⌫兲 冊 ,共10兲 where we identify 共1/q兲 ␶ ␸ 共 ␸ 兩q兲as the angular component of ␶ . In the same way the topological matrix Dtakes the form D共q,⌫兲=㜷exp 冉 − 冕 0 2 ␲ d ␸ ␶ ␸ 共 ␸ 兩q,⌫兲 冊 .共11兲 Earlier we said 共without proof兲that the condition for the diabatic potentials to be single valued is the fulfillment of Eqs. 共9兲. Moreover, it is seen that the single valuedness is solely determined by the NACM. Therefore in order to find out if a group of Nstates is capable of yielding single-valued diabatic potentials all that is needed is to calculate the corresponding N⫻NDmatrix and see to what extent it is diagonal. Comment: Equation 共9b兲presents two possibilities for the various diagonal D-matrix elements, namely, ±1. These two possibilities are typical for the JT NACTs. However, in case of the RT NACTs we encounter one sign only that can be assumed to be +1. C. The Renner-Teller topological matrix Assuming the center of the circular contour ⌫to be located on the collinear axis, we find that the NACM is independent of the polar coordinate ␸ so that the Dmatrix as given in Eq. 共10兲takes the form D共q兩z兲= exp共−2 ␲ ␶ 共q兩z兲兲,共12兲 where the integration over ␸ was done trivially because the NACTs, just like the potentials, do not depend on out-ofplane coordinates. To continue we replaced ⌫by zto specify 094102-4 Halász et al. J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp the circle that is employed 共this circle is also characterized by q兲. In what follows we discuss two cases, namely, the two-state Hilbert subspace and the three-state Hilbert subspace. Comment: In Eq. 共12兲we dropped the subscript ␸ because, from now on, we refer only to the angular components of both the NACM and its NACTs and no confusion is expected. Thus ␶ and ␶ jk stand for ␶ ␸ and ␶ ␸ jk, respectively.兴 1. The two-state Hilbert subspace Since the 2⫻2 ␶ matrix is of the form ␶ 共q兩z兲= 冉 0 ␶ 12共q兩z兲 − ␶ 12共q兩z兲0 冊 ,共13兲 substituting Eq. 共13兲in Eq. 共12兲can be shown to yield the following Dmatrix:20共b兲 D共2兲共q,z兲= 冉 cos共2 ␲ ␶ 12兲sin共2 ␲ ␶ 12兲 − sin共2 ␲ ␶ 12兲cos共2 ␲ ␶ 12兲 冊 .共14兲 In order for the Dmatrix to be diagonal the NACT, ␶ 12共q兩z兲, has to be ␶ 12共q,z兲= 再 n;共a兲 共2n+1兲/2; 共b兲 冎 共15兲 where nis an integer. In other words, the two states under consideration form a Hilbert subspace in a region if and only if for each point 共q,z兲in that region nis an integer. Later, in the numerical part, we show that case 共a兲applies to the RT interaction. 2. The three-state Hilbert subspace To treat the three-state case we assume the 3⫻3 ␶ matrix to be of the form ␶ 共q兩z兲= 冢 0 ␶ 12共q兩z兲0 − ␶ 12共q兩z兲0 ␶ 23共q兩z兲 0− ␶ 23共q兩z兲0 冣 ,共16兲 where ␶ 13 is assumed to be negligibly small and therefore is ignored. Moreover here it can be shown that substituting Eq. 共16兲in Eq. 共12兲yields, for the Dmatrix, the following result:23共b兲,23共c兲 D共3兲共q,z兲= ␻ −2 冢 ␶ 23 2+ ␶ 12 2C ␶ 12 ␻ S ␶ 12 ␶ 23共1−C兲 ␶ 12 ␻ S ␻ 2C− ␶ 23 ␻ S ␶ 12 ␶ 23共1−C兲 ␶ 23 ␻ S ␶ 12 2+ ␶ 23 2C 冣 , 共17兲 where C= cos共2 ␲␻ 兲, 共18兲 S= sin共2 ␲␻ 兲, ␻ =冑 ␶ 12 2+ ␶ 23 2, and we recall that ␶ jk⬅ ␶ jk共q兩z兲. It is well noticed that the D共3兲matrix in Eq. 共17兲becomes diagonal if and only if ␻ =nwhere nis an integer. It is interesting to mention that the eigenvalues of ␶ 共q兩z兲are 兵i ␻ ,−i ␻ ,0其. Since our main concern is the diagonal elements of the D共3兲matrix we calculate them employing the following expressions: 共D11,D22,D33兲= 冉 ␶ 12 2C+ ␶ 23 2 ␶ 12 2+ ␶ 23 2,C, ␶ 12 2+ ␶ 23 2C ␶ 12 2+ ␶ 23 2 冊 .共19兲 Summary: In order for the three states to form single-valued diabatic potentials 共in the region of interest兲each of the three matrix elements in Eq. 共19兲has to be equal to 1 at each point in that region. Among other things this implies that the eigenvalues of ␶ 共q兩z兲at each such a point are 兵i,−i,0其. III. NUMERICAL RESULTS Three kinds of results are discussed in the article: the 共two兲tridiagonal elements of the NACM, the corresponding 共three兲potential energy surfaces, and the diagonal elements of the topological Dmatrix. These results are presented as a function of qfor the three configurations already given in Fig. 1关details related to Fig. 1are given in the paragraph that follows Eq. 共4兲兴. The calculation of the 共angular兲NACTs was carried out at the state-average completed active space selfconsistent-field 共CASSCF兲level employing the following basis functions: For the nitrogen we applied s,p,d, and f functions and for the hydrogens we employed s,p, and d functions all from the aug-cc-pVTZ set. We used the active space, including all seven valence electrons distributed on 10 orbitals. Six electronic states, including the three states specifically studied, were computed by the state-average CASSCF method with equal weights. In certain cases these calculations were repeated with seven states to check for convergence. The relevant 共angular兲NACTs, namely, ␶ 12共q兩z兲and ␶ 23共q兩z兲as well as the corresponding potential energy surfaces Ej共q兩z兲;j=1,2,3 were calculated employing the MOLPRO program.32 In what follows we discuss three adiabatic states of the HNH system, namely, the two ⌸states that evolve to become 2A1and 2B1states and a third state which evolves from the 共collinear兲⌺state to become a second Bstate. In what follows we designate that second Bstate as B⌺. While doing that we show that as long as the region of interest is close enough to the axis of the molecule 共for any zvalue and any geometry兲the 2A1and 2B1states are strongly coupled to each other but only loosely coupled to other states of the molecule and therefore form, by themselves, a 共two-state兲Hilbert subspace. However, once this region is increased to include sections further away from the axis, these two states get coupled to other states and therefore fail to form a Hilbert subspace in the enlarged region. We intend to show that adding a third 094102-5 Analysis of the Renner-Teller effect J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp state, which in this case is the above mentioned B⌺state, leads to a Hilbert subspace formed by these three states. The main purpose of this article is to supply the numerical evidence for these assertions. A. Potential energy curves In Fig. 2are presented the three potential energy curves under consideration as a function of qfor the three different configurations. It turns out that the two 共⌸兲states 2A1and 2B1are the two lower 共planar兲states and that the 2B1is the lowest of the two.33 It is also well noticed that in all three configurations these two states are close to each other as long as qis rather small and eventually become degenerate once q=0. As for the B⌺state, it is seen to be the highest among the three and that as long as qis small enough it is well separated from any state. As qincreases it seems to approach the 2A1state. B. Nonadiabatic coupling terms In Fig. 1are presented the two major 共angular兲NACTs, namely, ␶ 12 which couples states 2A1and 2B1and ␶ 23 which couples the 2A1state and the B⌺state. It is well noticed that along the HNH axis 共where q⬃0兲the values of ␶ 12⬃1 but then as qincreases the values of ␶ 12 decrease. The rate of decrease is uniform but zdependent. A different situation is encountered for ␶ 23. Along the axis 共where q⬃0兲the values of ␶ 23 are very small 共and, in fact, are expected to become zero, once q=0兲but they increase as a function of q. The rate of increase depends on zand is not always uniform. Of all three configurations the C2vcase 共namely, when z=1.95 a.u.兲requires more attention, in particular, the second NACT, ␶ 23共q兩z=1.95 a.u.兲. As long as 兵0⬍q ⬍1.1 a.u.其 ␶ 23 is well defined as it couples the second state 共that is, the state 2A1兲and, energetically, the third state, which is the B⌺state. Unexpected things happen once q ⬎1.1 a.u. If again we assume ␶ 23共q兩z=1.95 a.u.兲to be the NACT between 2A1and the third 共energetically兲state we find that ␶ 23共q⬎1.1 a.u.兩z=1.95 a.u.兲⬅0. The reason is that the third state, in that interval, is not the expected B⌺state but an Astate. This, among other things explains why this NACT is identically zero. Continuing in the same manner we could establish that in along this interval the B⌺state is, energetically, only the fifth state. In other words it is ␶ 25共q ⬎1.1 a.u.兩z=1.95 a.u.兲which, along the interval q ⬎1.1 a.u., is the 共natural兲continuation of ␶ 23共q艋1.1 a.u.兩z =1.95 a.u.兲. More details on these and similar issues will be presented in a forthcoming publication. The behavior of the RT NACTs is reminiscent on what is encountered in case of the JT NACTs. We found that at a given JT point of degeneracy all NACTs are zero except those that are formed at that particular point.29共a兲Assuming the same to happen in the present case implies that since the 共1,2兲NACTs, ␶ 12, are formed at points located along the HNH axis, the 共2,3兲NACTs, ␶ 23, are expected to be zero along this axis. This simply means that the source for the ␶ 23 is located at other region共s兲in configuration space but then is forced to decay to zero while approaching the molecular axis. C. The topological D-matrix and the single-valued diabatic potentials In Figs. 3and 4are presented the diagonal elements of the Dmatrix calculated once for the two-state case 关thus D共2兲兴and once for the three-state case 关thus D共3兲兴. The main issue in the present section is the comparison between the diagonal elements of these two matrices. In both cases we expect them 共as a function of qbut for various zvalues兲to be FIG. 2. Energy curves related to three electronic states: the 2B1state the 2A1state 共both evolving from the two degenerate ⌸states兲and the state which evolves from the 共collinear兲⌺state to become the B⌺. These three states are the lower ones for the collinear arrangement and at regions close to it: 共¯兲EB共q兩z兲;共—兲 EA共q兩z兲;共---兲EB⌺共q兩z兲. 094102-6 Halász et al. J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 1. Whereas when they are close enough to the molecular collinear axis they are practically equal to 1, we start noticing deteriorations as qincreases. Since the two diagonal elements of D共2兲, namely, D11 共2兲共q兩z兲and D22 共2兲共q兩z兲, are always identical 关see Eq. 共14兲兴. we show, in Fig. 3, only one of them. A different situation is encountered in case of D共3兲. Here the three diagonal elements, Djj 共3兲共q兩z兲;j=1,2,3, are not necessarily identical and therefore the three of them are presented in Fig. 4. Comparing the curves in Figs. 3and 4one notices that the unit value for the various D-matrix elements is better preserved 共as a function of q兲in case of the three-state case. A more direct comparison between the two-state case and the three-state case is given in Fig. 3where a second curve, related to the three-state case, is presented. The values of this curve are defined as the geometric average of the three diagonal elements of D共3兲, namely, D ¯ 共3兲共q兩z兲=共D11 共3兲D22 共3兲D33 共3兲兲共1/3兲.共20兲 It is obvious that for the cases that Djj 共3兲共q兩z兲=1; j=1,2,3 we also have D ¯ 共3兲共q兩z兲=1. From Fig. 3it can be seen that as long as the calculations are done close enough to the molecular axis the D11 共2兲共q兩z兲 FIG. 3. The two-state and the three-state D-matrix elements. Two curves are shown. Oone represents the 共1,1兲element of the D共2兲matrix and the second the average value, D ¯ 共3兲共q兩z兲, of the three diagonal elements of the D共3兲matrix 关see Eq. 共20兲兴:共¯兲D11 共2兲共q兩z兲;共—兲D ¯ 共3兲共q兩z兲. FIG. 4. The three-state D-matrix elements: 共¯兲D11 共3兲共q兩z兲;共—兲D22 共3兲共q兩z兲;共---兲D33 共3兲共q兩z兲. 094102-7 Analysis of the Renner-Teller effect J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp values are indeed close to 1. However, when the calculations are performed further away from the axis 共i.e., when qbecomes larger兲D11 共2兲共q兩z兲values deteriorate, sometimes even quite rapidly 共except for the C2vconfiguration where the deterioration is slower兲. A significant different situation is encountered for the D ¯ 共3兲共q兩z兲values. They are not only close to 1 for small qvalues but remain 1 for a much larger qinterval. As mentioned earlier the diagonal elements of the Dmatrix contain information regarding the single valuedness of the diabatic potentials. Thus if the N⫻NDmatrix is diagonal then the corresponding N⫻Ndiabatic potential matrix is single valued in the relevant region. In general, we prefer Nto be as small as possible. Therefore as long the region is close enough to the molecular axis the two-state ADT 关see Eq. 共5兲兴is expected to yield a single valued 2⫻2 diabatic potential matrix.29共d兲However, as the region extends to larger qvalues, D共2兲deviates significantly from the unit matrix while the three-state diabatization remains single valued 共sometimes even for relative large q values兲because D共3兲is either equal to the unit matrix or very close to being a unit matrix. In other words if dynamic calculations have to be extended to such regions we have to employ the 3⫻3 ADT matrix in order to form the required single-valued diabatic potentials. In the present study the three states are the two states 2A1and 2B1that evolve from the two degenerate ⌸states and the B⌺state that evolves from the ⌺state. IV. ANALYSIS AND CONCLUSIONS In this study we showed for the first time that the theory developed in recent years to study the topological effects formed by the JT degeneracy applies also in the case of the topological effects formed by the RT degeneracy. In order to achieve that, we had to define the molecular axis as the corresponding seam and refer to contours that surround this axis. In the present article we employed circular contours perpendicular to the molecular axis and centered at the axis; however, we expect the results to be unaffected if different contours are applied 共as long as they surround the molecular axis, close enough to the axis兲. Whereas the idea to use the ab initio treatment to calculate the terms that couple the two ⌸states has been known for some time,13 the way to incorporate the effect of the interaction due to other states seems to be new. The ability to incorporate the effect of RT coupling terms related to higher states opens up possibilities to predict the effect of indirect coupling terms due to remote states on processes that take place at lower states. An example for such a study is given here where we show how the interaction due to the “isolated” B⌺state 共which evolves from the collinear ⌺state兲can be included. One major issue has not been mentioned so far, namely, the possibility that the RT topological effects are affected by JT interactions. It is not obvious that these two interactions are connected because they are formed by two orthogonal angular motions, one is an in-plane 共angular兲motion and the other is an out-of-plane 共angular兲motion. In Fig. 5are shown the position of four degeneracy points responsible for the formation of the JT cis 共recently we reported on such a study for the NHH configuration34兲: two of them, namely, ␶ ␾ 12 JT 共 ␾ , ␳ 兩q,z兲and ␶ ␾ 23 JT 共 ␾ , ␳ 兩q,z兲, that couple the two lower 2Astates 共i.e., 2A1and 2A2兲and the two upper 2Astates 共i.e., 2A2and 2A3兲are located along the C2v line and the other two form two ci-twins of the type ␶ ␾ 23 JT 共 ␾ , ␳ 兩q,z兲located on both sides of the C2vline and couple the two upper 2Astates35 共the two corresponding NACTs are of opposite signs兲. Here ␾ is the angular coordinate and ␳ is the radial coordinate of a point with respect to the position of the considered 共JT兲ci. Near each of the ci points is given the value of the corresponding topological 共Berry兲phase calculated according to the well known line integral which in case of a circular contour becomes29共c兲 ␣ jj+1 JT 共 ␳ 兩q,z兲= 冕 0 2 ␲ d ␾␶ ␾ jj+1 JT 共 ␾ , ␳ 兩q,z兲,j=1,2. 共21兲 It is noticed that in all four cases the ␣ phases are close to ± ␲ 共and not 2 ␲ 兲indicating that indeed the NACTs are formed by JT cis. A similar study, although for a spatial configuration determined by a different fixed RHH distances, supports the existence of four similar JT cis.36 It is also noticed that the region that contains the four cis is the region where the RT interaction is relatively strong. Nevertheless the JT topological phases do not seem to be affected. FIG. 5. The position of the four JT cis in configuration space as found for RHH=3.95 a.u.: 共⽧兲 ␶ 12 JT共q兩z兲;共⫻兲 ␶ 23 JT共q兩z兲.TheC2v共1,2兲ci is located at a distance q=2.27 a.u. from the collinear axis. The C2v共2,3兲ci is located at a distance q=1.08 a.u. from the collinear axis. The two 共2,3兲ci twins are located at a distance q=2.26 a.u. from the collinear axis and at a distance of 0.867 a.u. from the symmetry line. The numerical value near each ci stands for the topological phase ␣ 共 ␳ 兩q,z兲as calculated for ␳ =0.01 a.u. 关see Eq. 共20兲兴. 094102-8 Halász et al. J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp From the present study it is not clear yet whether the JT interaction interferes with the RT coupling 共and vice versa兲 to the extent that it affects, for instance, the RT D-matrix elements. Although we have indications that these two types of interactions are transparent to one another still this issue has to be studied in more detail which will be done in forthcoming publications. To summarize the findings of the present article we say the following: 共1兲The RT NACTs are similar to the JT ones; they are a result of a line of degeneracy points—seam—that yields a line of poles 共called兲.共2兲Their behavior and their effect on the diabatization process is well treated by the theory that originally was developed to study the JT NACTs. Before concluding this article we would like to call attention to one important issue; while deriving the BO Schrödinger equation we anticipate it to contain both JT and RT singular coupling terms. Therefore the two have to be eliminated simultaneously in order to permit solving the resulting equations. This procedure, to the best of our knowledge, has never really been carried out. ACKNOWLEDGMENTS Three of the authors 共M.B., Á.V., and G.J.H兲acknowledge the US-Israel Bi-national Science Foundation for partly supporting this study. One of the authors 共Á.V.兲acknowledges the OTKA Grant Nos. T037994 and M041537 and the computational resources provided by the John-von-Neumann Institute, Research Centre Juelich 共Project ID ehu01兲. 1E. Renner, Z. Phys. 92, 172 共1934兲. 2See, G. Herzberg, Molecular Spectra and Molecular Structure 共Krieger, Malabar, 1991兲, Vol. III. 3H. C. Longuet-Higgins, Adv. Spectrosc. 共N.Y.兲2, 429 共1961兲. 4K. Dressler and D. A. Ramsay, J. Chem. Phys. 27, 971 共1957兲; Philos. Trans. R. Soc. London, Ser. A 251,553共1958兲. 5J. A. Pople and H. C. Longuet-Higgins, Mol. Phys. 1, 372 共1958兲. 6Ch. Jungen and A. J. Merer, Mol. Phys. 40,1共1980兲;40,95共1980兲;Ch. Jungen and A. J. Merer, in Molecular Spectroscopy, Modern Research, edited by K. N. Rao 共Academic, NY, 1977兲, Vol. 2, p. 127 7J. M. Brown and F. Jorgenson, Adv. Chem. Phys. 52,117共1983兲. 8H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 53,117 共1986兲. 9J. M. Brown, in Computational Molecular Spectroscopy, edited by P. Jensen and R. J. Bunker 共Wiley, New York, 2000兲, p. 517. 10 S. Carter and N. C. 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Billing, Phys. Rev. A 62, 032506 共2000兲. 24 D. R. Yarkony, J. Chem. Phys. 105, 10456 共1996兲. 25 A. Kuppermann, in Dynamics of Molecules and Chemical Reactions, edited by R. E. Wyatt and J. Z. H. Zhang 共Dekker, NY, 1996兲,p.411;R. Abrol, A. Shaw, and A. Kuppermann, J. Chem. Phys. 116, 1035 共2002兲. 26 A. M. Mebel, G. J. Halász, Á. Vibók, A. Alijah, and M. Baer, J. Chem. Phys. 117, 991 共2002兲; G. J. Halász, Á Vibók, A. M. Mebel, and M. Baer, ibid. 118, 3052 共2003兲; M. Baer, T. Vertsi, G. J. Halász, Á. Vibók, and S. Suhai, Faraday Discuss. 127, 337 共2004兲. 27 R. Baer, J. Chem. Phys. 117, 7405 共2002兲; I. Ryb and R. Baer, ibid. 121, 10370 共2004兲. 28 B. Sarkar and S. Adhikari, J. Chem. Phys. 124, 074101 共2006兲; P. Puzari, B. Sarkar, and S. Adhikari, ibid. 121, 707 共2004兲; E. S. Kryachko, Adv. Quantum Chem. 44,119共2003兲; P. Barragan, L. F. Errea, A. Macias, L. Mendez, A. Riera, J. M. Lucas, and A. Aguilar, J. Chem. Phys. 121, 11629 共2004兲. 29 共a兲共Ref. 19兲, Chap. VI. 共b兲共Ref. 19兲, Chap. I; 共c兲共Ref. 19兲, Chap. II; 共d兲 共Ref. 19兲, Chap. III. 30 G. J. Halász, Á. Vibók, R. Baer, and M. Baer, J. Chem. Phys. 124, 081106 共2006兲. 31 M. Baer, Chem. Phys. Lett. 35,112共1974兲; Chem. Phys. Lett. 40,1011 共1980兲. 32 H.-J. Werner, P. J. Knowles, J. Almlöf et al. MOLPRO a package of ab initio programs. 33 C. F. Bender and H. F. Schaefer III, J. Chem. Phys. 33, 4798 共1971兲. 34 Á. Vibók, G. J. Halász, S. Suhai, D. K. Hoffman, D. J. Kouri, and M. Baer, J. Chem. Phys. 33, 024312 共2006兲. 35 ci twins 共possessing opposite signs兲were encountered before while studying the C2H system. A. M. Mebel, M. Baer, and S. H. Lin, J. Chem. Phys. 114,5109共2001兲. 36 Z.-W. Qu, H. Zhu, R. Schinke, L. Adam, and W. Hack, Mol. Phys. 122, 204313 共2005兲. This study was carried out for a much smaller RHH value and they found three 共1,2兲JT cis 关instead of our one 共1,2兲ci兴and one 共2,3兲JT ci 关instead of our three 共2,3ci兲兴. The swap between 共1,2兲and 共2,3兲cis, as encountered here 共while varying RHH兲is known to exist in other systems. See, e.g., H+H2system. G. J. Halász, Á. Vibók, A. M. Mebel, and M. Baer, J. Chem. Phys. 118, 3052 共2003兲. 094102-9 Analysis of the Renner-Teller effect J. Chem. Phys. 125, 094102 共2006兲 Downloaded 10 Jul 2008 to 193.174.59.126. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp