Janne Pesonen

Zickendraht W.

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Gradients of vibrational coordinates from the variation of coordinates along the path of a particle

Pesonen, Janne

Helsingin yliopiston digitaalinen arkisto

This document is downloaded fromHELDA - The Digital Repository ofUniversity of Helsinki.Title Gradients of vibrational coordinates from the variation ofcoordinates along the path of a particleAuthor(s) Pesonen, JanneCitation Journal of Chemical Physics -- September 1, 2001 --Volume 115, Issue 9, pp. 4402-4403Date 2001-09-01URL http://link.aip.org/link/?jcp/115/4402;http://hdl.handle.net/10138/1076Rights Copyright (2001) American Institute of Physics. This articlemay be downloaded for personal use only. Any other userequires prior permission of the author and the AmericanInstitute of Physics.NOTESGradients of vibrational coordinates from the variation of coordinatesalong the path of a particleJanne Pesonena)Laboratory of Physical Chemistry, P.O. Box 55 (A. I. Virtasen aukio 1),FIN-00014 University of Helsinki, Finland~Received 26 December 2000; accepted 14 June 2001!© 2001 American Institute of Physics. @DOI: 10.1063/1.1389849#The elements of the reciprocal metric tensor @g# , whichappear in the exact internal kinetic energy operator,1,2Tˆ (int)5(i jF2 \22 S ]]qi 1 1J ]J]qiD g (qiq j) ]]q j212 g(LiL j) lˆ i lˆ j2\2 g(Liq j) lˆ i]]q j2\2 S ]]qi 1 1J ]J]qiD g (qiL j) lˆ jG ~1!of a polyatomic molecule can, in principle, be written as themass-weighted sum of the inner products of the measuringvectors ea(qi) associated with the coordinates qi ~or, in thecase of rotational degrees of freedom, to the body-framecomponents lˆ i5ui8 lˆ of the angular momentum operator lˆ),and to the nuclei of the molecule asg (qiq j)5(aN 1maea(qi)ea(q j) , ~2!where N is the number of the particles, and ma is the mass ofthe particle a. The J in Eq. ~1! is the ~absolute value of the!Jacobian, the indices i and j run from 1 to 3N26 for theshape coordinates, and from 1 to 3 for the angular momen-tum components, and to be precise, it is supposed that theintegrations are performed using the volume-element dt5Jdq1dq2fl . The values of the measuring vectors ea(qi) atthe reference configuration are the familiar Wilson’ss-vectors.3 In the case of vibrational degrees of freedom, themeasuring vectors are the gradients of the vibrational coor-dinates,ea(qi)5„aqi . ~3!It has been shown in Ref. 1 that the gradients can be directlyand easily obtained for any geometrically defined shape co-ordinates by the vectorial differentiation. However, there isin some cases a more practical way to obtain these gradientsfrom the variation of the coordinate qi along the path xa(t)of the particle a as ~p. 108 of Ref. 4!q˙ i5(aNx˙a~ t !„aqi , ~4!where I use the short-hand notation F˙ 5 dF/dt , and t issome scalar parameter. It should be mentioned, that all therules of the scalar differentiation apply, with an exception ofthe commutativity. For example, the product rule reads asddt ~FG !5F˙ G1FG˙ , ~5!because generally F˙ GÞGF˙ unless one of them is a scalar~or a trivector!. In order to give the reader a chance to com-pare the present method to other methods, I rederive theknown gradients for the bond stretching.1,3Example 1: The most basic of all possible internal coor-dinates is the bond length rab5urabu5(rabrab)1/2, whererab5xb2xa . When both sides of the equation,rab2 5rabrab ~6!are differentiated with respect to some scalar parameter t ,the result2rabr˙ab52rab r˙ab ~7!follows. When the terms are rearranged, one obtainsr˙ab5rab r˙abrab5rab~ x˙b2 x˙a!rab, ~8!where the coordinate gradients are easily picked with thehelp of Eq. (4) as„brab5rabrab52„arab . ~9!The present approach can often be used effectively tofind the gradients of the shape coordinates, which are givenas implicit functions of the nuclear positions, and would bea!Electronic mail: Janne.Pesonen@Helsinki.fiJOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 9 1 SEPTEMBER 2001LETTERS TO THE EDITORThe Letters to the Editor section is divided into three categories entitled Notes, Comments, and Errata. Letters to the Editor arelimited to one and three-fourths journal pages as described in the Announcement in the 1 July 2001 issue.44020021-9606/2001/115(9)/4402/2/$18.00 © 2001 American Institute of PhysicsDownloaded 16 Aug 2007 to 128.214.3.188. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspvery difficult ~or impossible! to obtain by other methods. Asan example of this kind, I derive the gradients of the eigen-values l1 , l2 , and l3 of the moment tensorM~v!5(amayayav, ~10!where ya is the internal ~center-of-the-mass! positionya5xa2X ~11!andX5(aNmaxaM ~12!is the center-of-mass of the molecule ~M5(aNma is the massof the molecule!. Incidentally, the eigenvectors u18 , u28 , andu38 of the moment tensor are the principal axes of the mol-ecule, because the moment tensor is related to the inertiatensor I(v) asI~v!5(aNmaya2 v2M~v!. ~13!The eigenvalues of moment tensor ~or related coordinates,such as principal moments of inertia! have been used suc-cessfully to describe many of the most interesting features ofreactive scattering involving three and four particle systems~such as atom and diatom reaction!; see, e.g., Refs. 5–8. Tomy knowledge, the gradients of the eigenvalues of momenttensor have not been presented before in the literature.Example 2: To find out the gradients of the eigenvaluesof the moment tensor, we differentiate both sides of theeigenequation,M~ui8!5l iui8 . ~14!By substituting u˙i85v3ui8 ~p. 100, Ref. 4! to the result,which in turn is dotted with ui8 , an equation(aNmaui8~ y˙ayaui81yay˙aui81yayav3ui8!5l˙ i~15!follows. Becauseui8M~v3ui8!5ui8MS (j3v juj83ui8D5ui8M~2v juk81vkuj8!50 ~16!~where vk5vuk8 , and the indices i , j ,k are in cyclic orderin the second last term in the right-hand side) as seen by thedirect substitution of Eq. (14), one concludes thatl˙ i5(aN2mayaui8ui8 y˙a5(aN2mayaui8ui8S x˙a2(bNmbx˙bM D ~17!and the coordinate gradients are extracted as„al i52maS 12 maM D yaui8ui82 (bÞaN2mbmaM ybui8ui85(bN2mbS dab2 maM D ybui8ui8 , ~18!where dab is the Kronecker’s delta ~dab51 if a5b , andzero otherwise!. However, because (bNmbyb50, the sumover the index b may be omitted, and the result can be sim-plified to„al i52mayaui8ui8 . ~19!It should be emphasized that Eq. ~19! applies to a gen-eral N-body system without any restriction to number of par-ticles. Because the eigenvectors ui8 can be calculated rela-tively straightforwardly as functions of ya ~see Ref. 4!, itshould be simple to obtain the explicit expressions for thekinetic energy operator of an arbitrary N-body system ~atleast if it is simple to express yayb in terms of the coordi-nates used!. On the other hand, if the reciprocal metric tensorg (qiq j) is obtained as the inverse of covariant metric tensorgqiq j ~as in Ref. 8!, the expressions depend crucially on thenumber of nuclei, and in practice one needs to invert gqiq jseparately for the systems of different size.The method presented here also has the advantage thatthe gradients of the implicitly defined coordinates can oftenbe obtained without the need to represent the coordinate inquestion explicitly in terms of the nuclear positions. Further-more, the scalar differentiation used in this method is com-mutative operation, unlike the differentiation with respect tothe position vector xa , i.e., (d/dt) (CF)5CF˙ , where C isany multivector independent of t . Thus, the reader who is notfamiliar with geometric algebra may find the current methodmore readily adopted than the method presented in Ref. 1.I wish to thank both The Academy of Finland and TheEuropean Commission ~Contract No. HPRN-CT-1999-00005! for financial support. I wish to thank Professor V.Aquilanti, Dr. Simonetta Cavalli, and Dr. Andrea Beddonifor kind discussions during my visit to the University ofPerugia. I am also grateful to Professor L. Halonen for usefuldiscussions and valuable comments on the manuscript.1 J. Pesonen, J. Chem. Phys. 112, 3121 ~2000!.2 J. Pesonen, J. Chem. Phys. 114, 10598 ~2001!.3 E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations ~Dover,New York, 1980!.4 D. Hestenes, New Foundations for Classical Mechanics, 2nd. ed. ~KluwerAcademic, Dordrecht, 1999!.5 W. Zickendraht, J. Math. Phys. 12, 1663 ~1970!.6 B. R. Johnson, J. Chem. Phys. 73, 5051 ~1980!.7 R. G. Littlejohn and M. Reinsch, Phys. Rev. A 52, 2035 ~1995!.8 V. Aquilanti and S. Cavalli, J. Chem. Soc., Faraday Trans. 93, 801 ~1997!.4403J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Gradients of vibrational coordinatesDownloaded 16 Aug 2007 to 128.214.3.188. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Gradients of vibrational coordinates from the variation of coordinates along the path of a particle

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Helsingin yliopiston digitaalinen arkisto