J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1839-1847 Gradient-line Reaction Paths for 1,2 H Shift Reactions in Phosphinonitrene and Formaldehyde, and H, Elimination from Formaldehyde Ruslan M. Minyaev Institute of Physical and Organic Chemistry, Rostov State University, Stachka Ave., 19413,Rostov-on-Don 344 104, Russia David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge, UK CB2 IEW The reaction pathway for the 1,2 H shift in H,PN may be described as two equivalent gradient lines (double-line reaction path) leading from phosphinonitrene to cis-phosphazene rather than to trans-phosphazene. The gradient-line reaction path does not contain branching points and coincides with the minimum-energy pathway. Two different reactions, 1,2 H shift and H, elimination for formaldehyde, occur along two different gradient lines which both enter the minimum parallel to the softest Hessian eigenvector. 1.Introduction The notion of a transition state and the pathwaylP4 linking it on the Born-Oppenheimer potential-energy surface’ (PES) to both minima, reactant and product, plays a fundamental role in chemistry and biochemistry. Reaction rate calculations6-’ are based upon these concepts, along with our understanding of the reaction mechanism,’-’ the behaviour of non-rigid molecular systems,’ 3-1 ’ and the interpretation of vibrational spectra.16*17 Each of these concepts requires a detailed know- ledge of the PES topology along the whole reaction path from one minimum to another.Understanding enzyme cata- lytic action at the molecular level requires us to characterise the transition state and how it Usually one con- siders the minimum-energy reaction path (MERP)’ 1-26 con-necting reactant and product via the transition state. However, the MERP may bifurcate away from the line of steepest In this case, point group symmetry need not be conserved along the MERP and can change at the branching points.32 This does not contradict pear son'^^^ and Pe~hukas’~~theorems because the latter should be applied only to steepest descent paths. However, symmetry breaking along an MERP creates difficulties in understand- ing the dependence of the symmetry properties of the product on those of the reactant and transition state.Such behaviour also hampers construction of the true topology of the PES. Some unusual properties of reaction pathways are often attributed to bifurcation of the reaction path. For example according to ab initio calculation^^^-^^ the 1,2 H shift reac- tion (Scheme 1) occurs through a non-symmetrical transition state, 2. H H Scheme 1 The appearance of such a transition state is often attributed to the presence of bifurcation points on the PES. The aim of the present paper is to show that there are actually no branching points between minimum 1 and tran- sition state 2 for the reaction (Scheme 1) on the gradient-line reaction path, which is in fact described by Scheme 2. The reaction pathway of Scheme 2 differs significantly from that of Scheme 1.First, Scheme 2 leads to cis-phosphazene H \ P=N ,P=N, H’ HH Scheme 2 (4), rather than to trans-phosphazene. Secondly, the reaction pathway does not contain any bifurcation points and prin- cipally consists of two enantiomeric gradient lines passing through two transition states, 2a and 2b. In fact, there is a gradient line linking minimum 1 to minimum 3 which passes through a critical point of index two (2. = 2). ’It has also been suggested that all reaction pathways in mass-weighted coordinates (or ‘kinematic space’390) enter a minimum tangential to the softest normal mode.24*40*4’ Indeed, as is shown in Section 3, an infinite set of gradient lines do enter a minimum in the direction of the ‘softest’ Hessian eigenvector (eigenvector corresponding to the small- est eigen~alue)~’~ which may not correspond to the softest normal mode.The second aim of the present paper is to illus- trate two gradient-line reaction pathways for formaldehyde that enter the minimum parallel to the softest Hessian eigen- vector which has different symmetry from the softest normal mode. These results testify that the behaviour of the gradient line and intrinsic reaction pathway^^^,^' on the PES is differ- ent. Our considerations are all based upon the gradient-line reaction path (GLRP) which we have introduced in previous and have shown to have some useful properties. The GLRP is defined as the set of gradient lines linking two successive minima.43,44 Gradient lines cannot bifurcate or coalesce at any non-stationary points on the PES, and hence the transition-vector symmetry group4* is conserved along a gradient line.Such methodology has now been applied to Scheme 2. The methods are described in Section 2. Two reac- tion pathways, 1,2 H shift and H, elimination for formalde- hyde, are considered in Section 4. 2. Methods All the stationary-point optimisations and approximate gradient-line calculations in this work were performed by eigenvector-following45 (EF) using analytic first and second derivatives at every step generated by the CADPAC program.46 The particular EF implementation has been developed and discussed in previous ~ork.~~,~* The method enables one to follow a particular Hessian eigenvector uphill, whilst simultaneously minimising the energy in all the conju- gate directions.This procedure is used to find transition states, while minimisation in all directions is used to locate minima. All calculations were conducted in Cartesian coordi- nates using a projection operator to remove overall trans- lation and rotation4* A maximum step length criterion was used to scale the steps, if necessary. Approximations to the gradient lines were found by dis- placing each transition-state (or saddle-point) geometry along the two directions corresponding to the (non-mass-weighted) Hessian eigenvector associated with the (usually) unique ima- ginary frequency. Perturbations consisted of adding/ subtracting of the components of the normalised eigenvector in each case. Except for one case, the pathways were always followed downhill, i.e.by minimising the energy, and no symmetry restrictions were imposed. Calculations were considered converged when the maximum step size was -= a, and the rms gradient was < E, a, 't for two consecutive steps, which usually reduces the 'zero' Hessian eigenvalues to the order of 0.3 cm-'. The basis sets employed were those supplied in the standard CADPAC library;46 the DZP basis is obtained by adding polarisation functions with exponents 1.0 (H), 0.8 (C) and 1.2 (F) to the Dunning double zeta bask4' It is well known that pathways calculated with second- derivative-based algorithms are not quite the same as true steepest-descent or gradient-line paths; in fact, for some pathological surfaces it is possible to converge to the wrong minimum.50 However, we had no reason to expect such diffi- culties in the present case, and the algorithm adopted also means that all the Hessian eigenvectors are available at each step.Hence it was not difficult to follow the evolution of the eigenvalues and eigenvectors along the path using the overlap (dot product) of eigenvectors at successive steps to identify the correlation. However, since some of our results were unexpected we checked several of the paths by simply taking small steps along the negative gradient vector. The eigenvector-following pathways were found to be essentially correct in each case. 3. Behaviour of Gradient Lines in the Local Region of a Minimum The equations which define gradient lines in 3N-6 dimension-al configuration space (where N is the number of nuclei in the molecule) can be written in the form24,27,39b*41 where Qk(k = 1, 2, .. . ,3N-6) are curvilinear coordinates and gkJis a metric tensor.51 The PES E = E(Q)in the local region of a minimum can always be expressed as a Taylor expansion in diagonal form:52,53 1 3N-6( a2E ) @')' E(Q) = 2 (aQ32 min t 1 a, (bohr) ~5.29177x lo-" m. 1 E, (hartree) ~4.35975x lo-'* J. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Such a representation means that in the region of a minimum (or any stationary point) the Euclidian = 6kj (3) can be introduced. Here akj is the Kronecker symbol defined as (4) This local expansion is also called the Hessian eigenvalue rep- re~entation,~~'and corresponds to an orthogonal transform- ation of basis in terms of the Hessian eigenvectors.Taking into account relations (2) and (3),eqn. (1)can be rewritten as a system of ordinary differential equations with constant coefficients: -=-akQk, k = 1, 2, ..., 3N-6 (5)dt where ak = -((:22)min Eqn. (5) has two types of solution for each k: trivial39b Qk= 0 (6) and non-trivia12 9 9b27349 Qk = (Qk)o exP(-ak 1) (7) where (Qk))ois an initial point corresponding to t = 0 and ak is the kth Hessian eigenvalue at the minimum. Solutions with all Qk = 0 except for one particular Qj define the 3N-6 axis lines or directions of principal curvature at the minimum.Solutions with only two Qk non-zero, say (Ii and Qj, are parabolas in the Qi-Qj plane which enter the minimum parallel to whichever of the two vectors has the smallest eigen~alue,~~~ so long as the eigenvalues are non- degenerate (see Fig. 1). Thus, all the gradient lines (GLs) entering the minimum have components satisfying eqn. (6)or (7) in the local region of the minimum. The gradient lines coinciding with coordinate axes (principal directions of curvature) we will call principal gradient lines (pGLs); the rest we will call non-principal GL (npGL). Obviously, there are only 2(3N-6) pGLs (if we consider positive and negative directions for each axis39b) whereas there exists an infinite set of npGLs. In fact, every non-stationary point on the PES lies E 4 h=O Fig.1 Behaviour of principal (solid lines) and non-principal (thin lines) gradient lines in the local region of a minimum J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 on a unique gradient 1ine.34.52-54 Naturally, transition states may lie on either sort of GL, principal or non-principal. If two transition states lie on two different pGLs then the two gradient-line reaction path^^^-^^ enter the minimum perpen- dicular to one another. Transition states may also lie on an npGL. An interesting situation arises when the eigenvector q1 corresponding to the smallest eigenvalue (a1)is invariant under a reflection or two- fold rotation given by operation R. In this case there may be two equivalent gradient lines, npGL and R(npGL), leading to two (possibly enantiomeric) equivalent transition states, TS and R(TS) as shown in Scheme 3. min Scheme 3 The two npGLs, npGL and R(npGL), can lead either to the same minimum or to two different versions of the same minimum.The first case is observed for the 1,2 H shift reac- tion in phosphinonitrene which is considered in Section 4. 4. 1,2 H Shift in Phosphinonitrene According to our ab initio (DZP) calculations, H,PN in the ground electronic state has three different minima 1, 3 and 4, while structure 2 is a true transition state (A = 1)for the 1,2 H shift from P to N and structure 5 is a second index (A = 2) critical point (see Table 1 and Fig. 2). H ,P-N/\ H 5, c, ',l,390 (1.407) 126.3 (128.8) (1.515) H' ld2.2 (98.3) t,c, (h= 0) 3,cs(h= 0) H' (1.437) 2a,C1(h=1) H 5,cs(h= 2) 6,Cs(h= 1) Fig.2 Geometrical parameters of minima 1, 3 and 4, and transition states 2 and 6 for H,PN calculated at the DZP and MP2/6-31G** (numbers in parentheseP) levels. Structure 5 corresponds to a second index critical point. Bond lengths are given in A, bond angles in degrees. The total energies and geometrical parameters for structures 1-5 calculated in the present work at the DZP/SCF level agree with those obtained in previous ab initio calcu- lation~.'~-~*The latter work has shown that the positions of critical points in nuclear configuration space for this molecule do not change significantly when the computational effort is increased from SCF/DZP to MP4/6-311 + +G(df, p)35 (see Table 1).Hence the topology of the H,PN PES is insensitive to the level of the calculation and we therefore employed the DZP/SCF approach throughout. Within the Born-Oppenheimer approximation,' and from a classical point of the reaction path is a line belonging to the PES. In the absence of any external fields, the gradient lines of the PES define the local resultant force at any point. Scheme 4 is, in fact, composed of such gradient lines. H' H 26,C,(A= 1) Scheme 4 The complete GLRP (Scheme 4) consists of two equivalent lines [GLl and R(GLl)] corresponding to the 1,2 H shifts 1+2a e4 and 1 26 e4 passing through transition states 2a and 26, respectively.The gradient line (GL2) connecting minima 1 and 3 passes through second index critical point 5 where it correlates with the Hessian eigenvector with the largest-magnitude negative eigenvalue. Furthermore, minimum 3, trans-phosphazene, is linked to cis-phosphazene, minimum 4, by gradient line GL3 passing through structure 6,which corresponds to the transition state for inversion at nitrogen. Taking EF45 steps with starting displacements along both of the directions defined by the Hessian eigen- vector of TS 2 corresponding to the unique negative eigen- value, we calculated the pathways and energy profiles from TS 2 to minima 1 and 4. The pathway is only an approximate gradient line45v47 since the EF steps are not truly parallel to the gradient vector.50 However, this path is probably close enough to GL1 and R(GL1) for our purposes and has the same symmetry properties, as corroborated by the corre-lation of the Hessian eigenvectors corresponding to the lowest Hessian eigenvalues along the pathway.We also checked these pathways employing a steepest-descent method and obtained the same results: the steepest descent from 2 in one direction leads to minimum 1 and in the other direction to 4 (Fig. 3). The evolution of the five smallest Hessian eigen- values along these paths is shown in Fig. 4. Comparing Fig. 4(a) and (b) we see that the pathway obtained by EF gives a reasonable impression of how the eigenvectors and eigenvalues are correlated. However, since these paths maintain C,symmetry throughout there should be no crossings in Fig.4, barring accidental degeneracies. The Table 1 Total energies (EJ,relative energies (AE), number of imaginary frequencies (A) and values of the imaginary normal mode frequencies (io)calculated for structures 1-6 at different levels of theory EJEh AEjkcal mol -I, io/cm -MP4 MP4 MP4 MP4 structure symmetry DZP" 6-31 ++GLb /6-311 +G(df, p)' DZP" 6-31 ++G*b /6-311 ++G(df, p)' DZP" 6-31 ++G*' /6-311 ++G(df, p)' DZP" 6-31 ++G*b /6-311 ++G(df, p)' 1 c,, -396.22700 -396.213 62 -396.63647 0 0 0 0 0 4 C,(cis) -396.298 77 -396.281 48 --45.0 -42.6 -0 0 3 C,(trans) -396.30001 -396.283 33 -396.70084 -45.8 -43.7 -40.0 0 0 5 c, -396.11862 -396.102 10 -396.56056 68.0 70.0 47.6 2 2 2 c1 -396.181 38 -396.16909 -396.564 04 28.6 27.9 45.4 1 1 6 cs -396.277 60 ---31.7 --1 -~ ,,Present work.'Ref. 36.Ref. 35. Calculations with 3-21G* basis sets.36 Table 2 Total energies (EJ, relative energies (AE), number of imaginary frequencies (A) and values of the imaginary normal mode frequencies (io) calculated for structures 7-10 at different levels of theory EJEh AElkcal mol-' 1 iw/cm -' MP4(SDTQ)/6-3 1 1G** MPqSDTQ)/6-311G** MP4(SDTQ)6-311G** MPqSDTQ)/6-3116** structure symmetry DZP" 6-3 1G*' f/MP2/6-3 1G*' DZP" 6-31G*b //MP2/6-31G*' DZP" 6-31G*b //MP2/6-31G1' DZP" 6-31G*b //MP2/6-31G* c,_____~ ~~ 7 C," -113.895 51 -113.869 74 -1 14.262 63 0 0 0 0 0 0 ---9 C,(tr~n~) -113.81923 -113.79149 -1 14.177 82 47.9 49.1 53.1 0 0 ----~8 c, -113.736 31 -113.709 3 1 -114.177 82 100.0 104.6 81.4 1 1 1 2655.3 2710.0' 10 cs -1 13.726 76 -113.702 41 -114.12595 105.9 105.0 79.6 1 1 1 2307.2 2184.0d -"Present work.'Ref.63(a). Ref. 63(b). 4 wP J. CHEM.SOC. FARADAY TRANS., 1994, VOL. 90 4 transition vector -396.23L-1 0 10 20 30 40 50 60 70 \ steps -396.221 h -396.28 -396.3 0 10 20 30 40 50 60 70 steps N Fig. 3 Energy profile and evolution of the transition vector along the gradient line (steepest-descent path) from TS 2 (C,) to minima 1 (C2J (a) and 4 (C,) (b).On the left is the transition vector for TS 2 which correlates with the smallest eigenvalue for both minima. On the right is the eigenvector corresponding to the smallest positive eigenvalue of TS 2 which correlates with the second smallest positive eigenvalue of both minima.avoided crossings are clear for the steepest-descent paths. Because of the mixing which must occur it is difficult to associate particular eigenvectors of the two minima with the reaction vector. Furthermore, in an EF search for a transition state with a reasonable step size the algorithm might take a 'forbidden' step in the vicinity of an avoided crossing, and cross over to the softest mode. We also note that the EF path has a region with two negative eigenvalues, indicating that it has strayed somewhat from the steepest-descent path. In order to elucidate the topology of the H,PN PES we calculated the gradient lines emanating from 5 along the Hessian eigenvector corresponding to the negative eigenvalue of smaller magnitude.In this case the path was not calculated by simple minimisation but instead EF steps were employed following the Hessian eigenvector corresponding to the most negative eigenvalue uphill. These gradient lines link structure 5 with transition states 2u and 2b as corroborated by the correlation of the Hessian eigenvectors corresponding to the lowest Hessian eigenvalues along the pathway and the evolu- tion of the five smallest Hessian eigenvalues along this path 1843 min 1 TS2 min 4 1.071 (a) 0.8 1 % X X X 2 00 ................':-0.2.~ -0.4 10 ( steps TS 2 min 4~~0.9'Imin 1 0.7 52 0.2-1 00 ..................... -0.1 { -0.2 J 1 ...............................7+---+%-*-q 1 ....... I -60 -40 -20 0 20 40 60 steps Fig.4 (a) Evolution of the five lowest eigenvalues of the (non-mass- weighted) projected Hessian along the approximate DZP gradient-line path from TS 2 to minima 1 and 4, as determined by overlap of the corresponding Hessian eigenvectors. (b) Same as (a), but for steepest descent rather than EF paths. The steps correspond to those in Fig. 3. (see Fig. 5 and 6).The H,PN PES has a topology which can be schematically represented by the two-dimensional surface in Fig. 7. Thus, the GLRP of Scheme 4 consists of two equiv- alent (enantiomeric) npGLs and the reaction is equally likely to proceed along either of them.33b,s5 Thus, we have a 'double-line' reaction pathway which consists of so-called narcissistic reactions as considered by Salems6 and others.29 Note that there are no branching points on GL1 and R(GL1): none of the initially positive Hessian eigenvalues change their signs along the pathways and the GLRP in this case coincides with the MERP.The pGL entering minimum 1 along the Hessian eigenvector corresponding to the small- est Hessian eigenvalue links it to the second index critical point 5. The observed double-line reaction pathway may be quite common in organic, inorganic and other reactions. For example, the 1,2 H shift in ethenes7 and H2PP,3s and inter- nal rotation in dimethyl ether,s7-60 propane,s9 acetones9 and other molecules61-62 containing two equivalent rotors all occur along double-line reaction pathways.1844 (0 c .-& 0.0-Q) C I-% -0.2-aI -0.4-5 TS 2-0.6 0 2 4 6 8 1012141 i I t t/ I t I Fig. 5 Evolution of the five lowest Hessian eigenvalues (top) and two eigenvectors (bottom) along the gradient line from structure 5 (Cs,A = 2) to TS 2 (Cl). The TS 2 transition vector correlates with the Hessian eigenvector of 5 with the most negative eigenvalue. The Hessian eigenvector of TS 2 corresponding to the smallest positive eigenvalue correlates with the Hessian eigenvector of 5 with the nega- tive eigenvalue of smaller magnitude. I .v 1 O.* 1 0.4 i -0.44 min 3 -0.6-, ,0 2 4 6 8 10 12 14 16 18 t t \ Fig.6 Top: Correlation of the five lowest eigenvalues of the (non- mass-weighted) projected Hessian along the gradient line from struc- ture 5 to minimum 3. Bottom: Evolution of the Hessian eigenvector corresponding to the imaginary frequency of greatest magnitude for the same path. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Fig. 7 Schematic three-dimensional representation of the H,PN PES 5. 1,2 H Shift and H, Elimination from Formaldehyde According to our ab initio (DZP) calculations, formaldehyde in the ground electronic state is stable in C,, form 7 with geometrical parameters, given in Fig. 8, which are in agree- ment with previous ab initio calculation^^^*^^-^^ and experi- mental data66,67 (see Table 2 and Fig. 8) The calculated frequencies together with experimental values66 (in parentheses) and the forms of the normal modes of this molecule are given below in Scheme 5.In Scheme 6 are shown the non-mass-weighted Hessian eigenvectors and eigenvalues. As can be seen from comparison of Schemes 5 and 6, the softest eigenvector, el, does not coincide with the softest normal mode and has different symmetry. H, H (1.208) 116.4\ '*I8' (116.5) -1.094(1.116)H 7,cs(h= 0) 9,cs(h= 0) (1.356) H (1.727)1.205 /\ 1.585 H 8,Cs(h= I) l0,Cs (h= 1) Fig. 8 Geometrical parameters for the stationary points of H,CO calculated at the DZP level. Bond lengths are given in A, bond angles in degrees. Values in parentheses for structure 7 are experi- mental data66 and for 8-10 are MP2,/6-31G* results.63 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 J"\ \/c--01(b1)=1337 cm-' 02(b2)=1369 cm-' %(a,)=1659 cm-' (1167) (1251) (1501) 04(al)=2~9cm-' os(a,)=3149 cm-' w6(b2)=3226 cm" (1746) (2766) (2843) Scheme 5 t e,(a1)=0.156 a.u. e2(b2)=0.196 a.u. e3(b,)=0.287 a.u. 't HB A=-/ r' e4(a,)=0.464 a.u. es(b2)=0.977 a.u. e6(a,)=2.135 a.u. Scheme 6 Formaldehyde can undergo a 1,2 H shift from carbon to oxygen (Scheme 7) with formation of a trans-hydroxymethylene 9 and can also undergo an H, elimination reaction (Scheme 8). /\ /H,,c-0 -,c-0 H -c-0 H H' H/ 10, c, Scheme 8 The reaction (Scheme 7) passes through planar transition state 8 and becomes parallel to el in the limit at minimum 7 (see Scheme 6).Taking EF45 steps with starting displace- ments along both of the directions defined by the Hessian eigenvector of TS 8, corresponding to the unique negative eigenvalue, we calculated the pathways and energy profiles from TS 8 to minima 7 and 9. We have also considered the evolution of the Hessian eigenvalues and the transition vector along this GL which are given in Fig. 9 and 10. Repeating these procedures for TS 10 we obtained the pathways and energy profiles from TS 10 to minimum 7 and the dissociated system H, + CO; the evolution of the Hessian eigenvalues and transition vector along this GL are given in Fig. 11 and 12. As can be seen from Fig. 9-12, the reactions of Schemes 7 and 8 evolve along GLs entering minimum 7 in the limit tangentially to el which does not correspond to the softest normal mode.f -0.2 -0.4 TS8 1 min 7-0.6:' r 1 1 -l 1 2 3 4 5 6 7 8 9 10 steps Fig. 9 Evolution of the five lowest eigenvalues of the (non-mass- weighted) projected Hessian along the approximate DZP gradient-line path from TS 8 to minimum 7 for H,CO, as determined by overlap of the corresponding Hessian eigenvectors. 4 c TS 8, C, min 7,CZv Fig. 10 Evolution of two Hessian eigenvectors along the gradient line from TS 8 to minimum 7 for H,CO. Below is the transition vector for TS 8 which correlates with the Hessian eigenvector corre- sponding to the smallest positive eigenvalue of 7. Above is the Hessian eigenvector corresponding to the second smallest positive eigenvalue of 8 which correlates with the second smallest positive eigenvalue of 7.The steps correspond to points 1, 4, 6 and 10 in Fig.9. _-*. . . ---1.o ,----0.84 A' 0.6--S 0.4-lu--. - 4- -. -* -_--. X X M I1 lu P O O D I1 0.231 r r w = + + a [:I ( > a o 0.0- + + + u)3 -0.2- TSlO / min 7 TS 10. C, min 7,Cpv transition vector Fig. 12 Evolution of two Hessian eigenvectors along the gradient line from TS 10 to minimum 7. Below is the transition vector for TS 10 which correlates with the smallest eigenvalue of 7, above is the Hessian eigenvector with the smallest positive eigenvalue for TS 10 which correlates with the second smallest positive eigenvalue of minimum 7.The steps correspond to points 1,4,6 and 10 in Fig. 11. 6. Conclusions If a reaction evolves along a non-principal gradient line then the path enters the minimum tangentially to the softest Hessian eigenvector for which it has a non-zero component. This may not correspond to the softest normal mode, as illus- trated by two different reactions, the 1,2 H shift and H, elimi- nation for formaldehyde. When the softest Hessian eigenvector has at least one element of symmetry the gradient-line reaction path may consist of two equivalent gradient lines, forming so-called narcissistic reaction paths. An example is the pathway for the 1,2 H shift in H,PN which may be described as two equiva- lent gradient lines (double-line reaction path) leading from phosphinonitrene to cis-phosphazene rather than to trans-phosphazene.The GLRP at the SCF/DZP level of theory does not contain branching points and coincides with the minimum-energy pathway. 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