Very accurate, rigorous, variational, non-Born-Oppenheimer  non-BO  calculations have been performed for the fully symmetric, bound states of the LiH+ ion. These states correspond to the ground and excited vibrational states of LiH+ in the ground 2 + electronic state. The non-BO wave functions of the states have been expanded in terms of spherical N-particle explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance and 5600 Gaussians were used for each state. The calculations that, to our knowledge, are the most accurate ever performed for a diatomic system with three electrons have yielded six bound states. Average interparticle
distances and nucleus-nucleus correlation function plots are presente

Adamowicz, Ludwik

Bubin, Sergiy

Nazarbayev University Repository

THE JOURNAL OF CHEMICAL PHYSICS 125, 064309 2006Non-Born-Oppenheimer variational calculation of the ground-statevibrational spectrum of LiH+Sergiy BubinaDepartment of Physics, University of Arizona, Tucson, Arizona 85721and Department of Chemistry, University of Arizona, Tucson, Arizona 85721Ludwik AdamowiczDepartment of Chemistry, University of Arizona, Tucson, Arizona 85721and Department of Physics, University of Arizona, Tucson, Arizona 85721Received 2 June 2006; accepted 6 July 2006; published online 11 August 2006Very accurate, rigorous, variational, non-Born-Oppenheimer non-BO calculations have beenperformed for the fully symmetric, bound states of the LiH+ ion. These states correspond to theground and excited vibrational states of LiH+ in the ground 2+ electronic state. The non-BO wavefunctions of the states have been expanded in terms of spherical N-particle explicitly correlatedGaussian functions multiplied by even powers of the internuclear distance and 5600 Gaussians wereused for each state. The calculations that, to our knowledge, are the most accurate ever performedfor a diatomic system with three electrons have yielded six bound states. Average interparticledistances and nucleus-nucleus correlation function plots are presented. © 2006 American Instituteof Physics. DOI: 10.1063/1.2244563I. INTRODUCTIONOur previous works concerning non-Born-Oppenheimernon-Bo variational calculation of the vibrational spectrumof the H2 molecule1,2where we used 3000 and 5000 explic-itly correlated Gaussian functions ECGFs, respectively, foreach state yielded a very similar accuracy for all 15 vibra-tional transition energies as that achieved a decade ago byWolniewicz.3 Since, unlike Wolniewicz’s, our approach isnot restricted to two-electron systems, we decided to under-take a series of rigorous, variational, non-BO calculations ofall vibrational states of a three-electron system in its groundelectronic state. The vibrational spectrum of the system wehave chosen, the LiH+ ion, has been studied before usingmethods based on the BO approximation.4,5 The present cal-culations on LiH+ are not only the first non-BO calculationson this system but also the first non-BO calculations of thevibrational spectrum ever performed for a three-electron di-atomic molecule.In a non-BO calculation one does not assume the sepa-rability of the electronic and nuclear motions and the wavefunction depends explicitly on the coordinates of all particlesinvolved in the system. In such a situation the correlationeffects, which in the BO calculations are restricted to elec-trons, also appear in the relative nucleus-nucleus motion, aswell as in the relative motion involving the nuclei and theelectrons. For this reason, non-BO calculations are signifi-cantly more complicated and time consuming than BO cal-culations. Moreover, if a high precision of non-BO calcula-tions is required the basis functions used have to explicitlydepend on the distances between particles this is why ex-plicitly correlated Gaussians have been used in the presentwork, and not only long expansions of the wave function inaElectronic mail: bubin@email.arizona.edu0021-9606/2006/1256/064309/4/$23.00 125, 0643Downloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP licterms of basis functions must be used but also linear andnonlinear parameters of the basis function have to be exten-sively optimized.The non-BO calculations presented in this work wereperformed using the method we have been developing in thelast few years. For a description of the method, we refer thereader to our recent reviews.6,7 In the next section we willbriefly describe the method and its application to calculatethe vibrational states of LiH+. We will also briefly describethe procedure used to optimize the linear and nonlinear pa-rameters of the wave functions. It should be noted that theterm “vibrational states” can only approximately describestates with zero total angular momentum considered in thiswork. This is because if the BO approximation is not as-sumed, the vibrational motion is coupled with the electronicmotion and, strictly speaking, the vibrational quantum num-ber is not a good quantum number.II. THE NON-BO APPROACHIn the non-BO calculation we start with the nonrelativ-istic N-particle Hamiltonian in the laboratory Cartesian coor-dinate system. The particles are electrons and nuclei:Hˆ tot = − i=1N 12MiRi2 + ij=1N QiQjRij. 1The masses, charges, and positions of the particles in 1 aredenoted as Mi, Qi, and Ri, respectively. Next we rigorouslyseparate out from Hˆ tot the motion of the center of mass. Thisis accomplished by a coordinate transformation. The newcoordinate system comprises three laboratory coordinates ofthe center of mass and 3N−3 internal Cartesian coordinates.The center of the internal coordinate system is placed at oneof the particles usually the heaviest one called the reference© 2006 American Institute of Physics09-1ense or copyright; see http://jcp.aip.org/about/rights_and_permissions064309-2 S. Bubin and L. Adamowicz J. Chem. Phys. 125, 064309 2006particle and the internal coordinate axes are made parallel tothe axes of the laboratory coordinate frame. In the internalcoordinate system the particles other than the reference par-ticle are referred to the reference particle using the positionvectors ri, i=1, . . . ,n, where n=N−1. After the coordinatetransformation the laboratory Hamiltonian Hˆ tot separates intothe Hamiltonian describing the kinetic energy of the center-of-mass motion and the following internal Hamiltonian Hˆ :Hˆ = −12i=1n 1miri2 + ij=1n 1M1rirj+ i=1nq0qiri+ jinqiqjrij. 2In 2 q0=Q1 is the charge of the reference particle,qi=Qi+1 i=1, . . . ,n are the charges of the other n particles,rij = r j −ri, and the prime   is used to indicate vector-matrix transposition. The internal Hamiltonian 2 describesn pseudoparticles with charges qi and reduced masses mi=M1Mi+1 / M1+Mi+1 moving in the spherical potential ofcharge q0. In this work, particle 1, the reference particle, wasthe Li nucleus, particle 2 was the proton, and particles 3, 4,and 5 were the electrons.The key element of the non-BO calculation is the selec-tion of the basis set for the expansion of the wave function.Since the internal Hamiltonian Hˆ is spherically symmetrici.e., isotropic with respect to any rotation about the center ofthe internal coordinate system, the basis functions for de-scribing states corresponding to zero angular momentumi.e., the vibrational states have to be spherically symmetricas well. In our works concerning non-BO calculations onlight diatomic molecular systems see, for example, Refs. 1,2, and 8 and references in Refs. 6 and 7, we have shown thatthe explicitly correlated Gaussians ECGs involving func-tions with preexponential multipliers consisting of the inter-nuclear distance r1 raised to a non-negative even power mk,k = r1mk exp− rAk  I3r = r1mk exp− rA¯ kr , 3are very effective in describing such states. In 3 Ak is asymmetric, positive definite, nn matrix, and A¯ k is aKronecker product of Ak and I3, A¯ k=Ak I3, where I3 is the33 identity matrix.In accordance with the Pauli principle, the basis func-tions 3 must have the proper permutational symmetry soTABLE I. Non-BO total energies of LiH+ vibrational states obtained with 32v 3200 3600 40000 −7.783 246 42 −7.783 246 60 −7.783 246 72 −1 −7.781 627 05 −7.781 627 57 −7.781 627 95 −2 −7.780 433 17 −7.780 434 26 −7.780 434 97 −3 −7.779 655 89 −7.779 658 02 −7.779 659 30 −4 −7.779 243 63 −7.779 247 15 −7.779 249 15 −5 −7.779 038 64 −7.779 075 42 −7.779 084 56 −that when the spatial and spin parts are combined the result-Downloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP licing function is antisymmetric with respect to interchangingthe electron labels. In our calculations the antisymmetriza-tion was implemented using the standard approach based onYoung operators, as described in Ref. 9.In the non-BO calculations we use the variationalmethod and we minimize the energy functional the RayleighquotientE	ck,	mk,	Ak = mincH	mk,	AkccS	mk,	Akc4with respect to the linear expansion coefficients ck, the basisfunction exponential parameters 	Ak, and the preexponentialpowers 	mk. The optimization is done using an algorithmbased on analytical derivatives of the functional 4 with re-spect to the exponential parameters. The optimization of thebasis set was performed separately for each state and thebasis was grown from a small size to a set of 5600 functions.We believe that with this many functions the total energiesare converged to about seventh decimal figure. The conver-gence is certainly better for the lower lying states than for thehigher states due to increasingly more complicated structurelarger number of nodes in the wave functions of the highervibrational states. The preexponential powers 	mk in our cal-culations ranged from 0 to 250 and they were partially opti-mized for each state. More details on the Hamiltonian trans-formation, proper symmetrization of the basis functions, andselection of nonlinear parameters for calculations withECGFs can be found in Refs. 6 and 7. The masses of the00, 4000, 4400, 4800, 5200, and 5600 basis functions. All values are in a.u.400 4800 5200 56003 246 79 −7.783 246 84 −7.783 246 88 −7.783 246 911 628 20 −7.781 628 45 −7.781 628 66 −7.781 628 850 435 46 −7.780 435 80 −7.780 436 03 −7.780 436 199 660 16 −7.779 660 73 −7.779 661 13 −7.779 661 409 250 34 −7.779 251 07 −7.779 251 56 −7.779 251 919 087 92 −7.779 089 50 −7.779 090 42 −7.779 090 93FIG. 1. Convergence of the total energy, Ev, for states with v=0–5. K is the00, 3647.787.787.787.777.777.77number of basis functions.ense or copyright; see http://jcp.aip.org/about/rights_and_permissions064309-3 Calculation of spectrum of LiH+ J. Chem. Phys. 125, 064309 2006Li 7Li isotop and H nuclei used in the calculations weremLi=12 786.3933me and mp=1836.152 672 61me, where mestands for the mass of the electron.III. RESULTS AND DISCUSSIONThe previous calculations of Berriche and Gadea5 of thevibrational spectrum in the ground electronic state of theLiH+ system predicted the existence of seven bound vibra-tional states. Those calculations were done within the Born-Oppenheimer approximations and the electronic energy wasdetermined with a method that utilized an effective core po-tential. Thus, they can be expected to be significantly lessaccurate than the calculations presented in this work. Westart the presentation of the results by showing the conver-gence of the total energies of the ground and excited vibra-tional states as the function of the size of the basis set. Thenumber of states whose energies we have managed to con-verge below the dissociation threshold was 6. An attemptmade to converge the seventh state below the dissociationthreshold failed. With nearly 5000 functions in the basis setthe energy was still above the threshold and the convergencepattern did not indicate a realistic possibility of lowering theenergy below the threshold. The convergence patterns of theenergies of the six vibrational states are shown in Table I.They are also shown on Fig. 1 where, for each state, weplotted the difference between the energy obtained with aparticular number of basis functions and the energy obtainedwith 5600 functions. As mentioned, it takes more basis func-tions to obtain the same quality of the results for higherexcited states than for the lower states because of the largernumber of radial nodes in the wave functions of the higherstates. Figure 1 demonstrates this trend very well. The curvecorresponding to the ground state v=0 is flat, indicatingthat the energy of this state is already very well convergedTABLE II. Transition frequencies for LiH+ vibratio4400, 4800, 5200, and 5600 function basis for each stvalues are in cm−1.v→v 3200 3600 4000 44001→0 355.411 355.338 355.279 355.2412→1 262.027 261.901 261.828 261.7753→2 170.592 170.364 170.240 170.1604→3 90.480 90.176 90.018 89.9455→4 44.990 37.691 36.121 35.6476→5TABLE III. Mean interparticle distances of LiH+ and5600 basis functions for each state. Here, rLi-pr1, rv rLi-p rLi-e rp-e r0 4.329 22 1.829 63 3.415 24 3.201 4.821 88 1.995 62 3.742 09 3.532 5.536 66 2.236 90 4.215 53 4.013 6.681 00 2.622 09 4.973 88 4.784 8.706 93 3.300 29 6.319 15 6.135 12.721 0 4.638 19 8.990 11 8.80Downloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP licwith 4500 basis functions while the energy of v=5 statewith that many functions is still far from the same degree ofconvergence.In the case of slightly lower accuracy of the calculatedenergy value for the v+1 state than for the v state the tran-sition energies, Ev+1−Ev, are usually slightly overestimated.Thus, our calculated transition energies shown in Table IIshould provide upper bounds to the exact results. In Table IIwe not only include the transition energies obtained with thelargest basis set of 5600 functions but we also show how theenergies converge with the number of basis functions. Theconvergence pattern allows us to estimate the accuracy ofour calculations. It is safe to state that this accuracy is of theorder of 0.1 cm−1. Further extension of the basis set sizewould certainly lead to higher accuracy. However, at thispoint, after over six months of continuous calculations, wehave reached the limits of our computer capabilities.In Table II we also compare our transition energies withthose of Berriche and Gadea5 All their energies are higherthan ours by 2–4 cm−1. To what extent this is an effect of notassuming the Born-Oppenheimer approximation in our cal-culations and/or to what extent this results from inaccuraciesof the potential energy curve calculated by Berriche andGadea, it is, at this point, difficult to say.Finally, in Table III we present the expectation values ofthe interparticle distances and their squares for the six boundvibrational states obtained with 5600 basis functions for eachstate. Those include lithium nucleus-proton rLi-p, lithiumnucleus-electron rLi-e, proton-electron rp-e, and electron-electron re-e distances. The results show that the internu-clear distance increases, as expected, with the vibrational ex-citation. It also can be noticed that the average value of theproton-electron distance parallels the average value of theelectron-electron distance. This indicates that when the vi-brational excitation moves the nuclei further apart, two elec-tates obtained in this work with 3200, 3600, 4000,s the frequencies of Berriche and Gadea Ref. 5. All800 5200 5600 Berriche and Gadea5.197 355.158 355.125 357.431.757 261.754 261.757 265.610.107 170.070 170.047 172.969.911 89.889 89.874 92.205.460 35.367 35.331 37.459.22squares of the interparticle distances calculated withr2, rp-er12, and re-er23. All values are in a.u.rLi-p2  rLi-e2  rp-e2  re-e2 18.9229 6.962 33 13.9343 13.940023.8518 8.634 60 17.2254 17.283731.8581 11.355 4 22.5621 22.724446.7935 16.414 0 32.5126 32.840879.7911 27.519 0 54.5023 55.0501170.743 57.953 0 115.130 115.918nal sate v435261783meanLi-ee-e8 638 077 664 556 988 71ense or copyright; see http://jcp.aip.org/about/rights_and_permissionsction064309-4 S. Bubin and L. Adamowicz J. Chem. Phys. 125, 064309 2006trons stay with the Li nucleus and one moves away with theproton. Thus, at the limit LiH+ dissociates into Li+ and H.The increase of the internuclear distance with the vibra-tional excitation is the result of the radial function describingthe system becoming increasingly more diffuse and oscillat-ing more. This effect is shown in Fig. 2 where the nucleus-nucleus correlation function, which is the same as one-pseudoparticle density corresponding to pseudoparticle 1i.e., pseudoproton, is plotted for all vibrational stateswe calculated in this work. This function is defined as seeRef. 10g = rr1 − r , 5where r is the total wave function of the system underconsideration.IV. SUMMARYIn this work we performed very accurate non-BO calcu-lations on fully symmetric bound states of the LiH+ ion andwe found six such states. These states are conventionallycalled vibrational states although, if the BO approximation isnot assumed, the vibrational quantum number is, strictlyspeaking, not a good quantum number due to the couplingbetween the electronic and vibrational motions. We esti-mated that the accuracy of our vibrational transition energiescalculated as differences between the energies of adjacentFIG. 2. Nucleus-nucleus correlation funDownloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP licenergy levels is about 0.1 cm−1 in relation to the exact tran-sitions calculated without including relativistic corrections.This makes, to our knowledge, the present calculations themost accurate ever performed for a diatomic system withthree electrons.ACKNOWLEDGMENTSThis work has been supported in part by the NationalScience Foundation. We would also like to thank Universityof Arizona Center of Computing and Information Technol-ogy for the computer time on HP Alpha GS1280 parallelcomputer system.1 S. Bubin and L. Adamowicz, J. Chem. Phys. 118, 3079 2003.2 M. Stanke, D. Ke¸dziera, S. Bubin, M. Barysz, and L. Adamowicz, J.Chem. Phys. 125, 014318 2006.3 L. Wolniewicz, J. Chem. Phys. 99, 1851 1993.4 O. Grabandt, H. J. Bakker, and C. A. de Langes, Chem. Phys. Lett. 189,291 1992.5 H. Berriche and F. X. Gadea, Chem. Phys. 203, 373 1996.6 M. Cafiero, S. Bubin, and L. Adamowicz, Phys. Chem. Chem. Phys. 5,1491 2003.7 S. Bubin, M. Cafiero, and L. Adamowicz, Adv. Chem. Phys. 131, 3772005.8 S. Bubin, L. Adamowicz, and M. Molski, J. Chem. Phys. 123, 1343102005.9 M. Hamermesh, Group Theory and Its Application to Physical ProblemsDover, New York, 1962.10 S. Bubin and L. Adamowicz, Chem. Phys. Lett. 403, 185 2005.for v=0–5 vibrational states of LiH+.ense or copyright; see http://jcp.aip.org/about/rights_and_permissions

Non-Born-Oppenheimer variational calculation of the ground-state vibrational spectrum of LiH+

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Non-Born-Oppenheimer variational calculation of the ground-state vibrational spectrum of LiH+

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