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JOURNAL OF CHEMICAL PHYSICS VOLUME 108 NUMBER 11 15 MARCH 1998 Hydrogen bonding described through diatomicsinionicsystems The HF dimer B L Grigorenko and A V Nemukhin Department of Chemistry Moscow State University Moscow 119899 Russian Federation V A Apkarian Department of Chemistry University of California Irvine California 926972025 Received 6 October 199739 accepted 15 December 1997 With the proper inclusion of ionpair con gurations the diatomicsin molecules formalism can be used to accurately describe hydrogen bonding This is demonstrated for the well characterized prototype the HF dimer the structure and entire potential energy surface of which is reproduced within its known accuracy At the stationary points potential minimum and saddle points energies and bond lengths are reproduced with an accuracy of N l and the soft hydrogen bond angles are determined to within N5 This is accomplished through a minimal basis Hamiltoniani l9dimensional matrix to describe the planar complexgconstructed with analytic ts to accurately known or determined pair potentials The construct includes the HIF ionpair states of the HF monomer units The threebody nature of the inductive ionpair interactions with neutrals is preserved in the spirit of diatomicin ionicsystems Based on ab initio estimates in the limited range of interest a Gaussian function describes the mixing between ionic and neutral states The amplitude of this function is the only adjustable parameter in the model The ionicity anisotropy and nonadditivity of interactions responsible for the structure of the HF dimer result naturally from m1x1ng between ionic and neutral surfaces 50021960698024118 I INTRODUCTION The semiempirical diatomics inmolecules DIM theory1 originally developed to construct potential energy surfaces of polyatomics based solely on fragment diatomic interactions as input is a well established formalism which as been successfully implemented in a variety of applications1 3 The formalism is particularly suited for re active dynamics calculations over ground and excited elec tronic surfaces where ef cient evaluation of realistic global potential energy surfaces are crucial392393 and has been success ful in applications to molecular structure determination4 A rather complete citation of the early work can be found in the review article by Kuntz3 More recently DIM has found utility in the description of intermolecular interactions to describe anisotropies and nonadditivities in such with the expressed purpose of developing a systematic approach to the description of multibody interactions such as encoun tered in condensed media5 8 An illustrative example of the power and promise of the approach in such applications can be found in the recent simulations of nonadiabatic dynamics of I2 and I in condensed media simulations which included the full manifold of electronic states that correlate with I2P122P32 limits9 In DIM the choice of atomic con gurations to be in cluded determines the set of polyatomic basis functions PBF and their decomposition in terms of diatomic poten tials There is not a rm prescription for such a choice and hence for its partitioning The guidance and constraints are provided by the aims of using the minimal basis set that 0021960698108114413131500 4413 1998 American Institute of Physics retains the chemical insights at the desired level of rigor and accuracy Through stringent requirements of spin coupling schemes and spatial transformations of directional atomic bases DIM Hamiltonians reproduce the effects of valence bond directionality and con guration interaction1 3 When in addition to neutrals ionic con gurations are included then con guration interaction between ionic and neutral states of the same symmetry produce the effects of polariza tion or of bond ionicity This aspect has been well recog nized in the past A good case study is the construction of the OH2 reactive surface where it has been demonstrated that incorporation of the OiH and H7O7H con gurations are crucial in generating realistic surfaces that contain the proper physics10 Moreover once this important con gura tion is included much of the excited states can be ignored and DIM matrices of manageable dimension can be pro duced to describe the global surfaces Very closely related to the present study are DIM treatments of the HF211 FHF123913 and FF214 reactive surfaces in which ionic contributions from Hir and F fragments have been included These treatments however suffer from the limitation that the inclusion of ionic states is made with strict adherence to DIM treating all interactions as pairwise The closely related formalism of diatomicsinionicsystems DIIS15 evolved from the consideration of condensed phase systems such as the charge transfer states of a Cl atom isolated in the polar izable lattice of Xe11 3911b A main difference in these treat ments is the recognition that ionic interactions and in par ticular induction terms are vectorial multibody terms that 1998 American Institute of Physics Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcplcopyrightjsp 4414 J Chem Phys Vol 108 No 11 15 March 1998 have to be incorporated as such In essence D118 is not strictly a diatomics based method15 While the need for a consistent treatment of electrostatics in extended ionic sys tems is rather obvious more subtle are the nonbonded interactions of small systems such as HF Ar6a and ClZ Ar6c in which dispersion may be expected to be the dominant force Yet we have shown that the known nonad ditivity of interactions in these cases can be retrieved through DIM matrices of reduced dimensionality by ignoring much of the covalent electronic manifold while retaining in the constmct a proper account of the energetics of the lowest ionpair con gurations HTF and CITCIT in these particu lar cases6a 6 As in DIIS the threebody correction for an ionpair interacting with a polarizable neutral was included in these treatments The obvious implication of that analysis is that polarization plays a major role in the nonadditivity of pair potentials in these van der Waals complexes and that these effects can be rather directly reproduced through con guration interaction between ionic and neutral states A natural extension of the above concepts and analysis is the consideration of hydrogen bonding Given the ubiquity of hydrogen bonds and their importance in nature a simple accurate and insightful method of describing them should be valuable We focus on the HF dimer which as the prototype of H bonding has been extensively scrutinized by experi ment and by ab initio and semiempirical theoriesl 25 At present the multidimensional potential energy surface of HF2 is one of the best characterized therefore well suited as a test case We give a systematic construct of the DIM potential surface and compare it with other determinations While previously we had been satis ed with qualitative de scriptions of nonbonded interactions6 in the present case we show that quantitatively accurate results are obtained We surmise that this will not be limited to the case at hand Having established the accuracy of the reproduced PES we dissect various terms in the DIM matrix to provide an intui tive understanding of the subtleties of structure and cou plings in this system It should also be recognized that the surfaces by construct are global in nature valid in regions where experiments have not yet probed Accordingly we provide the necessary details to lend the constructed matrix useful for dynamical calculations on this system The organization of this paper is as follows Section II gives the essentials of the theory in its application to HF2 Speci cation of the blocks for the DIM matrices is clari ed in Appendix A Diatomic fragment data are discussed in Sec III and the corresponding analytical functions are collected in Appendix B The principles of neutralionic mixing ap plied here are discussed in Sec IV The results of the present calculations are given and discussed in Sec V where we compare the derived potential energy surface of HF2 with the benchmark Quack Suhm SQSBDE t18 We close with concluding remarks in Sec VI II THE NEUTRALIONIC MODEL OF HF2 The designations and geometric parameters of the HF2 complex are shown in Fig 1 We distinguish the bound Fbe and free Ffo monomer subunits and the choice of angles Grigorenko Nemukhin and Apkarian xii FIG 1 Geometry of HF2 for inplane rotations 6162 as indicated in Fig 1 Although the matrices are initially constructed in full dimensionality since the most important stationary points of the surface are contained in the planar geometry we will restrict the evalu ations to this plane The DIM formalism is nicely documented in many pa pers we follow the matrix formulation originally given by Tully2 Quite generally the polyatomic Hamiltonian operator may be partitioned into diatomic and monatomic parts1 H E Ha Nez H 1 bgta With the selected set of PBF s this Hamiltonian is converted into the matrix form and the total Hamiltonian matrix whose eigenvalues give estimates of the polyatomic energies is de composed into diatomic and monatomic matrices as pre scribed by Eq 1 The fragment matrices are constructed from the monatomic Va and diatomic Val energies which are assumed to be known The creation of monoatomic matrices requires only a knowledge of atomic energy levels ioniza tion potentials and electron af nities The construction of the diatomic part is more cumbersome For each diatomic constituent the initial basis set should be transformed in such a manner that the speci c electronic states with respect to the spatial and spin quantum numbers are recognized This is accomplished through a set of transformation matrices Heb Rib Rib T317133 VabBabTaszbRab 2 in which ij represents the spatial rotation matrix that ro tates the atomic quantization axis to the ab direction Tab is the spin transformation matrix of the fragment and Bab are the matrices which mix diatomic states Depending on the diatomic states concerned Bab will serve two different purposes In the case of the heteronuclear HF fragments the Bab matrix connects states of the same spatial and spin symmetry In our limited bases the mixing in HF is only between two states the 12 ground state cor relating with the lowest energy dissociation limit of neutral Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpjcpaip0rgjcpcopyrightjsp J Chem Phys Vol 108 No 11 15 March 1998 Grigorenko Nemukhin and Apkarian 4415 TABLE 1 Atomic states contributing to polyatomic basis inctions and corresponding diatomic potentials Diatomic potentials Atomic states HF HZ Fz FZPHZSFZPHZS lv32HF1v3139IHF quot32Hz X 122 2gt32u 31391g31391u3Ag 18 compositions 13 32 32 Kl39lgfl39l KA all states of the Fz fragment F SHFZPHZS 2HF 22mm 22Fgi ZHFgi FZPHZSF SH 22HF ZHHF F SHZSFZPH l2HF 1 IHF FZPHF SHZS 12 compositions HF SHF S 2HF at were 1 term 1r 1r 7 1r otherwise atoms F2PH2S and the ionic 12 state correlating with the F 1SHJr limit Accordingly the needed 2X2 blocks bHF of the unitary matrix BHF are sinBr cos Br bHF icostZF sin WE 39 3 The mixing parameter E which depends on the HiF dis tance serves as the only adjustable parameter of the present model Its evaluation will be described in Sec IV We should note here that the power of the DIM method is to a large extent grounded in the ambiguity of mixing parameters which are allowed to be adjusted at some reference points on the polyatomic potential energy surface and then used for predictions elsewhere The ambiguity arises from the fact that invariably truncated bases are used for the decomposi tion The strategy is however workable as long as the de nitions are soundly grounded in theory The choice of the elements of BHF will be based on results of quantum chem istry calculations and the known experimental dissociation energy of HF In the case of the homonuclear fragments H2 and F2 the matrix Bab unmixes the u and g states of the diatomic ie states symmetrized with respect to inversion within the hom onuclear fragment Table I shows the overall construct of the polyatomic basis functions and lists the electronic states of diatomic fragments HF HF HF HF F2 F used to ll in the diagonal matrices V017 In Appendix A we provide more explicit details of the matrix and its block factorization inA and A symmetries when the treatment is limited to the pla nar geometry Each of the basis functions shown in Table I is written as a product of atomic S functions or the Cartesian components of P functions multiplied by the proper spin factors In the case of homonuclear diatomics these functions are trans formed to relate them to molecular states of either g or u symmetry This transformation is trivial for the S combina tions required in the case of the H fragment In the case of P combinations required for F2 and F fragments the pro cedure is direct and has previously been given explicitly by Gersonde and Gabriel in their treatment of the singlet states of diatomic halogens26 The same applies for the triplet states The corresponding blocks of the matrices B21VAZ BAZ where A2 stands for F2 F2 are given in Appen dix A and should be compared to the closely related matri ces of HF2 and FF2 given by Duggan and Grice113914 As it is clear from Table I which shows only S and Ptype atomic functions the rotation matrices R217 are easily constructed from the unit blocks and the 2X2 blocks of planar rotations with angles deduced from geometric consid erations see Fig 1 Spin transformation matrices Tab can also be obtained unambiguously to generate singlet HF2 from the de nite spin states singlet or triplet of monomer subunits Fbe and Ffo The required 2 X2 blocks of the T matrices are also given in Appendix A The nal expressions for the diatomic Hamiltonian ma trices are Riffa 61 t ngv beBHFREfa 61 H Riff 62 t BI VHfFfBHFREffFa a HHfF TngR Fb 63 iv beREfa 63 THbe 4 H bFf TngfREFf 63 iv b fREfa 63 Tim H b f T1231 VHZBHZTH HFbFf TEZBEZIVFZBFZTFZ gt in which the angles 63 and 64 are those between the z axis and the HbiFf and Hbe directions respectively The dimension of the DIM matrix in the selected basis is 31 This would allow the consideration of arbitrary geom etries of the complex Limiting ourselves to the planar ge Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaip0rgjcpcopyrightjsp 4416 J Chem Phys Vol 108No11 15 March 1998 ometry and the ground electronic state of 1A symmetry it is suf cient to consider the l9gtlt l9 block of the full matrix The energy surfaces of HF2 to be discussed below are obtained as the lowest eigenvalue of this l9 dimensional Hamiltonian matrix The structure of the matrix with com ponents of the basis set arranged in the order neutral mixed neutralionic and pure ionic is schematically illustrated in Appendix A The coupling between the corresponding blocks of the total matrix is governed by the mixing param eter B If 30 the energy of HF2 comes from the 10 X 10 block of neutral states if B 900 then pure ionic struc ture is assumed Intermediate values of B produce neutral ionic mixing We deviate from a strictly DIM implementation by re specting the threebody vectorial nature of an ionpair in teracting with a neutral As in our previous applications we make this correction for the leading electrostatic term of charge induced dipoles only6 Noting that this f4 term is contained in ionneutral pair potentials which are used as scalar functions we correct for the ionpair interactions with neutral centers by adding the crosspolarization term In the present application this correction term rF HrHH rF PrHE AVionaH 3 3 F 3 3 5 rF HrHtH rF FrHtF is added to the diagonal elements of the neutralionic part of the Hamiltonian matrix a stands for polarizability The ef fect of this correction is most signi cant for con gurations FgngFfo and F17H17FJHJr where represents a separation large compared to the bondlength in the bound or free unit Note the inclusion of Eq 5 is what distinguishes the present formulation from a strictly DIM treatment since in effect triatomics are included in the partitioning To be more precise the treatment should be quali ed as DIIS III THE FRAGMENT ENERGIES The constituent fragment energies of the diatomic Hamiltonian matrices are obtained through high level ab ini o quantum chemistry calculations with appropriate empiri cal corrections From pilot DIM estimates we determine which of the diatomic contributions play the most important role in predictions of the HF2 energies within the required limits of polyatomic coordinates and pay special attention to the speci cation of these potential curves Accurate solutions of the adiabatic electronic equations are known for the required states 22 22 of Hg and 12 3E of H228 These ab initio energy points 28 are used in conjunction with the known longrange C 4 param eter and the parameters B in the exchange energy ex pression Vexch B r exp7 r for the H terms29 For the diatomic potentials of F2 F2 HF HF and HF species the same level of accuracy cannot be expected from ab ini o data and appropriate empirical corrections are required to construct the potential curves We use experi mental values for the ionization potential of H 136 eV and electron af nity of F 340 eV29 to describe dissociation lim Grigorenko Nemukhin and Apkarian its For F2 and Fi diatomic fragments the ab inilio results in given in Refs 30 and 31 are used to obtain the functional ts given in Appendix B In spite of the known good quality ab initio data for the HF and HF 7 states we recompute these potentials because the results described in the literature were given only in graphical form zas For the ground state X 12 potential of HF we combine the experimental RKR curve given for in ternuclear distances 085 AltrHFltl A36 with our calcula tions and x the dissociation energy with the experimental value of 612 eV37 Almost all HF potential curves have been obtained using largescale con guration interaction CI cal culations with the help of the GAMESS program suite38 using the AO basis set given explicitly by Segal and Wolf32 For each electronic state rst the solutions of the MCSCF equa tions the full optimized reaction space FORS option of GAMESS have been obtained by including the orbitals 207 4a lrr 27139 in the active space and keeping the lsF orbital in the core Then the CI calculations were performed by using the secondorder CI option of GAMESS with the FORS optimized molecular orbitals and FORS reference con gura tions total number of con guration state functions actually used in CI is 75 0007125 000 To some extent this proce dure is equivalent to the MRDCI treatment of Refs 34 35 In the 12 block of the HF problem we needed two lowest roots of the secular equation the rst corresponded to the X 12 the second to the ionpair HIFF potential The same approach has been applied for the calculations of theX 2H state ofHF The computed pictures of all these curves are found to be consistent with those described in the previous calculations32 35 We did not nd references on the calculations of the 22 state of HF The curve computed by us has a peculiar shape with a barrier separating the potential well from the dissociation limit The nal energy values used in the tting procedure were computed with even a greater effort namely using the QCISDT option of GAUSSIAN94 in conjunction with the AUGccpVTZ basis set39 The computed param eters of this curve the potential well and barrier are consis tent with the results of recent photoelectron spectral studies of HF40 In order to construct the X 22 potential of HF we employ the results of Ref 35 together with those of our calculations As has been concluded in Ref 35 after cross ing the X 12HF curve when moving from the dissocia tion limit HF the most stable form of HF is a free electron HFe state possessing a potential curve which parallels quite closely that of the neutral ground state In fact for these distances both curves should coincide completely in the basis set limit when the number of diffuse atomic orbitals is suf ciently large to describe the unbound electron Therefore to represent this anionic potential we combined two curves at rHFltl4 A which corresponds to a crossing point according to Ref 35 we took the curve of the ground state of neutral HF and at rHFgtl4 we use a curve calcu lated by us with the QCISDT option of GAUSSIAN94 using the AUGccpVTZ basis set The t to all potentials of ionic nature HF HF given in Appendix B takes into account the experimental Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcpcopyrightjsp J Chem Phys Vol 108 No 11 15 March 1998 data for the polarizabilities of the corresponding neutral part ners as well as other longrange coef cients for the 12 states of HF29 IV MIXING PARAMETER FOR NEUTRAL AND IONIC STATES OF HF In this application as in our previous treatments6 we consider the mixing coef cient Br as an adjustable param eter Thus while the construct forms a qualitatively correct framework for energy estimates quantitative results are ob tained by this semiempirical adjustment Note B determines the ionicity of the ground state surface as a function of co ordinates in the HF2 complex It appears explicitly in the HF fragment matrix In principle both intramolecular ion pair states and intermolecular charge transfer states could be included However Br has a strong r dependence scaling as the overlap between electronihole wave functions As such the dominant contribution of mixing between ionic and neutral states can be expected to arise from the intramolecu lar coupling Accordingly we only consider mixing within the Fbe and Ffo monomer units and ignore it for the cross pairs Fbe and F be Note however that although the mix ing coef cients appear only in the monomer units intermo lecular charge transfer con gurations appear through the ex change interactions implicit in the H2 and F fragment matrices An initial estimate of Br is derived from the analysis of the variationally obtained coef cients C k of con guration state functions in the conventional con guration interaction CI method however built on the atomlocalized orbitals More speci cally we apply the transformation from the ca nonical molecular orbitals calculated by the HartreeiFock method to the natural atomic orbitals NAO within Wein hold s natural bond orbital analysis41 The NAO allow a clear interpretation of ionicity In the present case for the HF molecule we insert into the CI expansions a set of atom localized orbitals labeled as core lsF lone pair orbitals 2px Zpy 2pz of F and lsH instead of the canonical 107317 17139 orbitals Construction of CI is carried out in a manner consistent with the full optimized reaction space or complete active space option by distributing 8 valence electrons of the HF molecule over F2s2p and Hls orbitals With the NAO it is possible to recognize the con guration state functions that correspond to the HIF ionic electronic con guration When taking the ratio CizmCizonJrEC em where Cm and Cnem are the variational coef cients of the ground state CI wave function we obtain the required estimate of the weight of the ionic con guration In essence we analyze the ratios of coef cients in valencebondtype wave functions The quantum chemistry calculations described above have been carried out using the triplezeta basis set for the ground state of HF In the relevant range of HiF distances 09 Agtrgt 32 A the computed dependence of sin2 B on r ts the Gaussian form sin2 BA exp7Br7C2 6 Grigorenko Nemukhin and Apkarian 4417 with parameters A 0538 B 148 A and C092 A Accordingly for the crosspairs Fbe where rgtl8 A at equilibrium and Fbe where rgt 32 A complete neglect of mixing is justi ed Thus we include mixing only in the monomer units HfFf and Hbe and while we retain the form suggested by Eq 6 we treat coef cientA of the equa tion as the single adjustable parameter of the model The adjustment is made at a single point on the multidimensional PBS at the equilibrium structure of the dimer to reproduce the experimental dissociation energy of the hydrogen bond D2 1561 cm 7125 This is accomplished by varying A until the minimum energy of the DIIS surface as a function of all coordinates RFF 61 62 RHbe RHfFfiagrees with the reference value The nal value ofA 0383 is used in the rest of the calculations The adjusted mixing parameter B which re ects the ion icity of the HF bond is noticeably different in different treat ments of polyatomic systems where HF enters as a diatomic fragment The value of B reported by Duggan and Grice in their treatment of HFZ11 is approximately twice smaller than the value found optimal in our analysis of HF Similarly in our previous analysis of AriHF6a the value determined for B was considerably smaller than in the present These differences do not re ect serious inconsistencies but rather demonstrate the sensitivity of the results to the diatomic in put information used in these semiempirical analysis Dif ferent procedures for extracting B and different pair poten tials are used in these analyses Nevertheless in all cases acceptable representations of DIM polyatomic surfaces are erive V RESULTS AND DISCUSSION The equilibrium structure of HF2 is rmly established through a variety of experimental an theoretical analysesm zs The structure is illustrated in Fig l and deter mined by the parameters RFF 272 A 61 10 62 63 The equilibrium values of 61 62 which are fairly represen tative of such intermolecular complexes42 is nontrivial to rationalize Quite clearly a subtle balance of forces occurs and this should be possible to generalize in hydrogen bond ing systems In the particular case of the HF dimer a re ned sixdimensional potential energy surface has been con structed by Quack and Suhm18 This socalled SQSBDE sur face is given analytically and in addition to optimizing the surface in full dimensionality to reproduce high resolution vibrationirotation spectra ab inmo quantum chemistry points21 and hydrogen bond dissociation energies are used to re ne parameters of the surface We will use this surface as a reference in discussing the DIIS results The DIIS energies for planar HF2 are obtained as the lowest root of the l9gtlt l9 Hamiltonian matrix described in Sec II using the diatomic inputs described in Sec III and the pair functions collected in Appendix B The predicted surface is illustrated as a contour map in Fig 2 juxtaposed with the reference surface of Quack and Suhm18 These plots represent the topology of the surfaces as a function of 61 and 62 at the equilibrium FiF distance same as Fig 7 of Ref 18 The two surfaces are remarkably similar in all features Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcpcopyrightjsp 4418 J Chem Phys Vol 108 No 11 15 March 1998 92 deg A 61 deg FIG 2 2D cuts of the DIM left panel and SQSBDE Ref 18 right panel potential surfaces along 61 and 62 at the equilibrium FiF distance contour lines are drawn every 100 cm 1 starting from 1600 cmil In Table II we compare the computed parameters of the three most important stationary points the potential minimum the C2 saddle point and the Cool saddle point with those of the reference surface as well as experimental and other theoretical determinations It should be evident that in all TABLE II Parameters of the stationary points of HF2 comparisons the predictions are within the known accuracy of the features of this well determined surface At the CS equilibrium geometry of the dimer the devia tion between our values and those of the reference surface or experiment is within the same limits as those computed with Grigorenko Nemukhin and Apkarian 61 deg 396 39I39I39I39III396OI39I39 1209060 30 O 30 6O 90120150180210240 120906O 30 0 30 60 90120150180210240 distances in A angles in degrees energy in cmil Stationary point rHFb ram RFF 61 62 AEa C5 minimum This work 0921 0922 272 15 64 1560 Quack amp Suhm Ref 18 0923 0921 2722 90 6413 15593 Peterson amp Dunning Ref 19 0922 0920 273 7 69 1610 Necoechea amp Truhlar Ref 20 0923 0921 2723 99 6547 15401 Bunker et 611 Ref 21 0922 0922 2749 71 617 1530 Collins et 611 Ref 22 0923 0921 2742 733 6954 1655 Experiment Ref 23 272r 003 10i6 63i6 Experiment Ref 24 7i3 60i2 Experimentb 1561 CM saddle This work 0921 0921 2640 62 118 332 Quack and Suhm Ref 18 0922 0922 2629 5492 12508 3515 Necoechea Truhlar Ref 20 0922 0922 2629 5577 12423 2297 Bunker et 611 Ref 21 0923 0922 2722 561 1239 332 CW saddle This work 0920 0921 283 0 0 297 Quack and Suhm Ref 18 0923 0920 2815 333 Bunker et 611 Ref 21 0922 0921 2866 345 aAE for the minimum is the dissociation energy to two monomers HFHF AE for the saddle points are given with respect to the corresponding minimum energies bThe experimental binding energies are derived from the D0 of Bohac et al Ref 25 and the zeropoint vibrational corrections of Quack and Suhm Ref 18 Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpjcpaip0rgjcpcopyrightjsp J Chem Phys Vol 108 No 11 15 March 1998 the help of large ab initio calculations Within deviations of 01 the D118 values of the H F distances show the effects of stretching from the original value of 0917 A in the un complexed molecule with a slight asymmetry between bound and free units of the dimer The F F distance and 62 fall well within the uncertainties within which these quanti ties are known The value of 61 showing deviation from the linear hydrogen bond axis seems slightly overestimated in our results This occurs in a very shallow basin the contours of which seem in quite acceptable agreement with the refer ence surface see Fig 2 Astonishing is the accuracy of the D118 determination of the saddle points of the C2 and Cool symmetry particularly in view of the fact that the single adjustable parameter was xed by the energy at the equilibrium geometry The predic tions for both geometry and energy of the saddle points are of comparative accuracy to the ab initio calculations The determined bond lengths and energies at these points show the proper trends and remain within accuracy bars of 1 Somewhat larger is the discrepancy of 61 and 62 at the C2 saddle point when comparing the D118 results with the other theoretical values Again these are soft coordinates and the agreement should be regarded as surprisingly good Quite clearly the present construct is capable in repro ducing the subtle interplay of forces that determine the struc ture energetics and therefore presumably the nature of hy dro gen bonding Once the laborious task of constructing the D118 matrix is completed a major advantage of the formalism is that through the solution of algebraic equations entire surfaces are predicted In particular the scheme should lead to a cor rect description for every dissociation channel since by con stmction all diatomic fragments dissociate into the correct limits and therefore we know a priori that the energetics in asymptotic regions are true We illustrate this feature by showing in Fig 3 plots of the same surface for three increas ing F F distances at 27 4 and 6 A These surfaces then represent the adiabatic dissociation path of the complex into two monomer units The constructed matrix should be quite useful in dynamical calculations Given the successful reproduction of structure and ener getics the investigation of contributions in our Hamiltonian that determine the nal balance of interactions that dictate the HF dimer geometry seems valuable For example the roles of neutral ionic and mixed neutralionic terms in the Hamiltonian can be investigated by simply considering sur faces generated with the mixing parameter 8 set to 0 90 and an intermediate value respectively We offer the follow ing analysis The purely neutral model leads to the linear structure Fb Hb Ff Hf with an exaggerated F F distance extended to 37 A and a binding energy of only 60 cmil Evidently the classical picture of a linear hydrogen bond arises from purely neutral contributions and yields a binding energy characteristic of strictly dispersion interactions Deviations from linearity which result from the anisot ropy of intermolecular interactions can only be explained when neutralionic mixing is taken into account 0lt 8 lt90 The data of Table II and Figs 2 and 3 are obtained 4419 Grigorenko Nemukhin and Apkarian 02 deg 60 I I I I 39 l 39 120 90 60 30 0 30 60 90 120 150 180 210 240 300 270 240 210 180 s 150 120 90 6quot Q 30 62 deg 6deg I 60 er 120 90 60 30 0 30 60 90 120 150 180 210 240 61 deg FIG 3 2D cuts of PBS along 191 and 192 at a RFF272 A contour lines are drawn every 100 cmil starting from 1600 cmil b RFF4 A con tour lines are drawn every 35 cm 1 starting from 7525 cmil c RFF 6 A contour lines are drawn every 20 cm 1 starting from 7 140 cmil for 838 at which value a signi cant admixture between ionic and neutral surfaces occurs Therefore in the present scheme the asymmetry of the complex is entirely due to neutralionic mixing The mixed neutralionic block may be further decom posed to clarify the roles of speci c diatomic potentials To this end the contour plots of Fig 4 are informative In these plots the mixing parameter and the F F distance are xed at the equilibrium values of the full DIIS matrix at 38 and 27A respectively The upper panel refers to the case when the ionic contributions from H and F diatomics have been eliminated from the ionic block and the corrections for the vectorial summation of the remaining ionic contributions Eq 5 have been omitted In sharp contrast with the linear arrangement of the purely neutral representation the com plex is now L shaped with mutually orthogonal axes between bound and free monomers This geometry is mainly Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpjcpaip0rgjcpcopyrightjsp 4420 J Chem Phys Vol 108 No 11 15 March 1998 62 deg 62 deg r 39 I I 0 so 60 90 120150180 210 240 61deg 60 120 90 60 3 FIG 4 2D cuts for PBS along 191 and 192 at RFF 272 A contour lines are drawn every 100 cmil and equilibrium structure computed with various contributions to the total potential see the text determined by the contributions from the 2H and 22 T states of HF and from 22 T state of HF Upon correcting for the three body ionpair atom interactions by including the con tribution from Eq 5 the contour plot of the lower panel is obtained This surface already shows the main features of the full treatment and the predicted equilibrium structure is es sentially the same as the nal We may therefore conclude that the delicate CS equilibrium geometry of HF2 is deter mined mainly by the 2H and 22 states of HF and 22 state of HF and the threebody corrections to the ionatom interactions All other diatomic potentials including inter molecular charge transfer con gurations play minor roles but are necessary in obtaining the quantitative surface VI CONCLUSIONS By virtue of the accuracy with which the HFdimer po tential energy surface is known it serves as a rigorous test for the adequacy of the DIIS method in quantitative treat ments of hydrogen bonding The success of the present treat ment is highly encouraging Given the construct of this semi empirical Hamiltonian based on accurate pair potentials as input and a single adjustable parameter we have little reason to doubt the generality of the result Indeed the approach used here is inspired by our prior success in treating HF Ar and ClZ Ar and ClZ He complexes by the same method6 That work already indicated that a minimal DIIS basis of covalent states augmented by a single ionic con guration in Grigorenko Nemukhin and Apkarian which threebody induction terms were properly included was suf cient to reproduce the known nonadditivity of pair potentials The necessity of including excited ionic con gu rations in DIIS matrices has a long history with the main body of that work relying on coding the ionic contributions on a pairwise basis in strict adherence with the DIM formal ism Neither in the case of the HF dimer nor in the van der Waals clusters considered by us would it be possible to account for the structural details and energetics based on pairwise interactions alone Thus as in the DIIS extension of DIM15 proper treatment of the electrostatic forces in the ionic states we regard as one the most important conclusions of the present quantitative test The peculiar structure of the HF dimer is nontrivial to rationalize by standard arguments It arises from a subtle interplay between intermolecular forces The structure arises naturally in our treatment as a result of mixing between the anisotropic ionic excited state and the ground state That the hydrogen bond is dominated by polarization is perhaps not too surprising That the effect can be quantitatively repro duced by inclusion of the threebody ionpair induced polar ization of the upper state namely the leading electrostatic term in the ionic manifold is an important nding It sug gests a general means for rationalizing such interactions and a relatively direct method for constructing accurate surfaces The required input for constructing such surfaces consists of a small set of diatomic pair potentials atomic polarizabil ities and the mixing function between ionic and covalent potentials The latter is treated as an adjustable parameter in the present Given the important role the mixing function plays in these constructs its systematics and in particular its scaling with basis sets used deserves more careful attention Clearly estimates of the mixing parameters can be obtained either from theory or from known polarities of bonds and at that level already yield qualitatively correct surfaces We should note that the model does not assume a xed dipole on the HF monomers Since the polarity of this bond is pro duced indirectly through mixing with the ionic surface it is a exible function of the geometry of the complex This ex ibility will appear as nonadditivity in intermolecular poten tials created by electrostatic expansions Finally we should admit that the accuracy with which the HFdimer surface was reproduced in particular the struc ture and energetics of the saddle points far from the mini mum where the calibration was made was surprising Indeed the procedure involved the adjustment of one parameter therefore remains semiempirical ACKNOWLEDGMENTS We thank Dr M Suhm for sending us the FORTRAN codes of the SQSBDE surface A MATHCAD PLUS 60 le for evaluation of the DIIS surfaces of HF dimer with all necessary input parameters is available from the authors upon request This research was supported in part by the Russian Basic Research Foundation Grant No 9603 32284 and in part through the US Air Force Of ce of Scienti c Research Grant No F49620l0251 Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpjcpaip0rgjcpcopyrightjsp J Chem Phys Vol 108 No 11 15 March 1998 Grigorenko Nemukhin and Apkarian 4421 TABLE III The decomposition of the polyatomic basis set in terms of atomic functions numbering of the PBF and corresponding diatomic potentials that enter the Vab matrices for the A symmetry block Atomic functions Diatomic potentials N H5 F5 Hf Ff VH5Fb VHFf VH5Ff VHbe VHHb 1 HUS FZPX HUS FZPX H H H H 2 2 HUS FUP HUS FUPX 3H 3H 3H 3H 32 3 HUS FZPX HUS FZPZ H 2 2 H 2 4 HUS FZPX HUS FZPZ 3H 32 32 3H 32 5 HUS FZPy HUS FZPy H H H H 2 6 HUS FZPy HUS FZPy 3H 3H 3H 3H 32 7 HUS FZPZ HUS FUP 2 H H 2 2 8 HUS FZPZ HUS FUP 32 3H 3H 32 32 9 HUS FZPZ HUS FZPZ 2 2 2 2 2 10 HUS FZPZ HUS FZPZ 32 32 32 32 32 11 Ht 1S HUS FUP 2HF H ZHHF 22HF 12 HUS F39 S Ht FUP 22HF ZHHF H 2HF 13 Ht F39 S HUS FZPZ 2HF 2 Z2HF 22HF 14 HUS F39 S Ht FZPZ 22HF Z2HF HtF 15 Ht FUP HUS F S ZHHF 22HF 2HF H 16 HUS FUP Ht F S H 2HF 22HF ZHHF 17 Ht FZPZ HUS F39 S Z2HF 22HF 2HF 2 18 HUS FZPZ Ht F S 2 2HF 22HF Z2HF 19 Ht F39 S Ht F S 2HF 2HF 71r 71r lr APPENDIX A Table 1 of the text shows the constituents of the poly atomic basis set of 31 functions used in the DHS matrix Tables 1H and 1V give a more detailed description of the basis set and the diatomic inputs There we also number the PBF to describe the structure of the matrix If we restrict ourselves to planar symmetry then the 31 X 31 matrix can be blocked out into a 19gtlt 19 matrix of 1A symmetry and a 12gtlt 12 matrix of 1A symmetry The 1A matrix consists of a 10gtlt 10 8 X 8 and 1 X 1 blocks of neutrals neutralionics and ionic functions Similarly the 1A block consists of 8 X 8 and 4 X4 blocks of neutrals and neutralionics This is also the ordering of the PBF vector in Tables 1H and IV The 2X2 spin transformation matrices Tab of Eq 2 enter the neutrals block along the diagonal for the H2 F2 HbFf and Hbe fragments They are 1 13 2 2 IHbFfIHbe V3 1 gt A1 7 2 1 13 2 7 IHZIFZ V3 1 A2 7 2 The F2 fragment matrices appear in the neutrals as over lapping singlet and triplet manifolds 5 X5 for 1A and 4 X4 for 1A Explicitly TABLE IV The decomposition of the polyatomic basis set in terms of atomic Jnctions numbering ofthe PBF and corresponding diatomic potentials that enter the Vab matrices for the 1A symmetry block Atomic Jnctions Diatomic potentials N H5 F Hf Ff VHbe VHfFf VHFf VHFb VHbe 20 HUS FZPX HUS FZPy H H H H 2 21 HUS FUP HUS FZPy 3H 3H 3H 3H 32 22 HUS FZPy HUS FUP H H H H 2 23 HUS FUP HUS FZPX 3H 3H 3H 3H 32 24 HUS FZPy HUS FZPZ H 2 2 H 2 25 HUS FZPy HUS FZPZ 3H 32 32 3H 32 26 HUS FZPZ HUS FZPy 2 H H 2 2 27 HUS FZPZ HUS FZPy 32 3H 3H 32 32 28 H F39 S HUS FZPy 2HF H ZHHF 22HF 29 HUS F39 S Ht FZPy Z2HF ZHHF H 2HF 30 Ht FZPy HUS F S ZHHF 22HF 2HF H 31 HUS FZPy H F S H 2HF 22HF ZHHF Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcpcopyrightjsp 4422 J Chem Phys Vol 108 No 11 15 March 1998 Grigorenko Nemukhin and Apkarian 2 12g 1 Ag 0 2 12g 1 Ag 0 0 0 Ingglnu 0 Inuglng 0 1 1 1 1 1G 2 2g Ag 0 2 2g Ag 0 0 A3 0 Inuilng 0 1Hglnu 0 2 2 0 0 0 0 X12 1Ag12 lAgilz 0 0 2 2 lAgilz 1Ag121I 0 0 1G 2 2 A4 0 1Hglnu Inuilng 2 2 0 Inuilng 1Hglnu 2 2 SAME 0 MHz 0 0 2 2 0 3122911 0 snugsng 0 3G 3Au732g 0 3Au32gi 0 0 A5 2 2 0 3Hf3ng 0 3Hg3r1u 0 2 2 0 0 0 0 132 232mm 23min 0 0 2 2 25pm 2323Au 0 0 3G 2 2 3H 311 3H 3H A6 0 0 E 2 2 0 0 3Hf3ng 3Hg3r1u 2 2 Where the 1G 1G 3G and 3G matrices act on the PBF vectors 1 3 5 7 9 21 23 25 27 2 4 6 8 10 20 22 24 26 respectively The H2 Hr fragment matrices enter the neutralionic blocks as 2 X2 matrices Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaip0rgjcpcopyrightjsp J Chem Phys Vol 108 No 11 15 March 1998 22pm 22322 2 2 22522 22pm 2 2 Grigorenko Nemukhin and Apkarian 4423 along the diagonal and the following combinations of the F electronic states I w 0 0 0 2 2H 211 0 0 0 0 lzgglzu 0 0 0 0 2 2G Hsz 0 0 0 2 2H 72H 0 2 g 0 0 2 2 0 0 Egz 2 0 22 722 0 0 2 I 39an nu 0 2H Jng 0 2 2 0 znggznu 0 2H 2H 2G H Jng 0 2Hg2flu 0 2 2 0 imaging 0 2 APPENDIX B Analytic formulas used for diatomic potentials distances in A energies in eV H2 12H21272673213050707417476723911v07074 32HZ3838 934 22Hg6447 e 3 667339 1 g 759620712384r 4795 7375r2gt 7373542171 890447 7 znuizng A7 I 2 0 0 0 2117211 0 2 g 0 0 2 2 0 0 2g 2 0 2 0 0 Zggizg 2 A8 2H 211 gt 0 0 0 2H 211 0 2 0 0 0 ZEgHEu 0 2 2 2 0 0 0 2 I A9 1 2 H5258 431 7 206527 24667 r u 2 r 3785re 1 8904quot7 4795 16902 r2 74 r 125172 32e 3 ozglril 411 16 675056390 1411 1HuF238621086 415244 1HgF2322727846 424254 lAgFZ1242653e 3 6701quot Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcpcopyrightjsp 4424 J Chem Phys Vol 108 No 11 15 March 1998 2 1EgF21278363e 736539 1211721779913273813quot 3HuF202465e 5 7277971 We 03977 ei3 4547vr1 88 3HgF22050589 e 4 097 13mm14748922739829quot 32F2113292e 36788quot 3A1F21757222e 3 7951quot 2 32072 1682331 e 73759 Grigorenko Nemukhin and Apkarian 22F 75648e 1 6745071 411M 6317 6 73349071 411 22mg 70854271 150W 411M 8381 6 723013071 411 2T141727 708352 e71 4383071 411M 3298 672 8766vr1411 2may 71338e lzg g39l 6162 72 5738071 411 e HF r709169 8464x2710755x39301x477046x53444x67612x7 085ltrHFlt1A 12HP 3 763738e 3339592758872109777 rHFgt1A 12973 11192 10911 14396 12HF71692e 1619quot 1716252r91906r2714649r3if r2 7 r4 7 r 32HF118693e quot 1HHF45312e 71quot 3HHF5212e 856quot 806 22HF4327e 1 59quot120806e 5mquot7 806 2HHF5159e 3318quot12187844724649r22315189e 6191quot7 22HF 13998e 252quotquot1 994r7789r2171r381725976e 1 797quot 1F O Ellison J Am Chem Soc 85 3540 1963 2J C Tully in Modern Theoretical Chemistry Semiempir39ical Methods of Electronic Structure Calculations edited by G A Segal Plenum New York 1977 Vol 7A Chap 6 J C Tully in Potential Energy Surfaces edited by K P Lawley Wiley New York 1980 pp 637112 3a P J Kuntz in Dynamics of Molecular Collisions edited by W H Miller Part B Plenum New York 1976 p 53 b P J Kuntz in AtomiMolecule Collision TheoryiA Guide for the E42wimentalist ed ited by R B Bernstein Plenum New York 1979 Chap 3 c P J Kuntz in Theoretical Models of Chemical Bonding Part 2 The Concept of the Chemical Bond edited by Z B Maksic Springer Berlin 1980 pp 3217376 4a A V Nemukhin and N F Stepanov Int J Quantum Chem 15 49 1979 b S V Ljudkovskii A V Nemukhin and N F Stepanov J Mol Struct THEOCHEM 104 403 1983 5W G Lawrence and V A A karian J Chem Phys 101 1820 1994 A V Danilychev and V A Apkarian ibid 100 5556 1994 6a B L Grigorenko A V Nemukhin and V A Apkarian J Chem Phys 104 5510 1996 b B L Grigorenko A V Nemukhin A A Buchachenko N F Stefanov and S Ya Umanskii ibid 106 4575 1997 c B L Grigorenko A V Nemukhin and V A Apkarian Chem Phys 19 161 1997 d A V Nemukhin B L Grigorenko and A V Savin Chem Phys Lett 250 226 1996 7A A Buchachenko and N F Stepanov J Chem Phys 104 9913 1996 9547 8F Yu Naumkin P J Knowles and J N Murrell Chem Phys 193 27 1995 F Yu Naumkin and P J Knowles J Chem Phys 103 3392 1995 F Yu Naumkin Mol Phys 90 875 1997 9V S Batista and D F Coker J Chem Phys 105 4033 1996 106 7102 6923 1997 1 P A Berg J J Sloan and P J Kuntz J Chem Phys 95 8038 1991 P J Kuntz B I Neifer and J J Sloan ibid 88 3629 1988 R Polak I Paidarova and P J Kuntz ibid 87 2663 1987 R Polak I Paidarova and P J Kuntz ibid 82 2352 1985 11J J Duggan and R Grice J Chem Phys 78 3842 1983 121 Last Chem Phys 69 193 1982 13J J Duggan and R Grice J Chem Soc Faraday Trans 80 739 1983 MJ J Duggan and R Grice J Chem Soc Faraday Trans 80 795 1984 15a 1 Last and T F George J Chem Phys 87 1183 1987 b 1 Last T F George M E Fajardo an pkarian ibid 87 5917 1987 c 1 Last and T F George ibid 93 8925 1990 98 6406 1993 6D G Truhlar in Proceedings of the NATO Workshop on the dynamics of polyatomic van der Waals Complexes NATO Ser B 227 159 1990 17M M A Suhm in Conceptual Perspectives in Quantum Chemistry edited by JL Calais and E S Kryachko Vol III of Concep tual Trends in Quantum Chemistry 417 Kluwer Dordrecht 1997 18M Quack and M A Suhm J Chem Phys 95 28 1991 19K A Peterson and T H Dunning J Chem Phys 102 2032 1995 Znw c Necoechea and D G Truhlar Chem Phys Lett 248 182 1996 Downloaded 12 Feb 2004 to 1282004719 Redistribution subject to AIP license or copyright see httpljcpaiporgjcpcopyrightjsp

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