Very accurate, rigorous and fully variational, all-particle, non-Born–Oppenheimer calculations of the vibrational spectrum of the H2 molecule have been performed. Very high accuracy has been achieved by expanding the wave functions in terms of explicitly correlated Gaussian functions with preexponential powers of the internuclear distance. An indicator of the high accuracy of the calculations is the new upper bound for the H2 nonrelativistic nonadiabatic ground state energy equal to 21.164 025 030 0 hartre

Adamowicz, Ludwik

Bubin, Sergiy

Nazarbayev University Repository

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 7 15 FEBRUARY 2003Variational calculations of excited states with zero total angular momentumvibrational spectrum of H2 without use of the Born–OppenheimerapproximationSergiy Bubina)Department of Physics and Department of Chemistry, University of Arizona, Tucson, Arizona 85721Ludwik Adamowiczb)IRSAMC, Laboratoire Physique Quantique, UMR 5626 Universite Paul Sabatier 118, Route de Narbonne,31062 Toulouse Cedex, France~Received 24 September 2002; accepted 21 November 2002!Very accurate, rigorous and fully variational, all-particle, non-Born–Oppenheimer calculations ofthe vibrational spectrum of the H2 molecule have been performed. Very high accuracy has beenachieved by expanding the wave functions in terms of explicitly correlated Gaussian functions withpreexponential powers of the internuclear distance. An indicator of the high accuracy of thecalculations is the new upper bound for the H2 nonrelativistic nonadiabatic ground state energyequal to 21.164 025 030 0 hartree. © 2003 American Institute of Physics.@DOI: 10.1063/1.1537719#We recently introduced a new correlated Gaussian basisset suitable for high accuracy nonadiabatic calculations ondiatomic molecules1 and showed that the basis is capable ofproviding the most accurate ground-state nonadiabatic ener-gies of H2 ~Ref. 2! and LiH.3 In this article we continue thevalidation of our approach by presenting highly accurate,variational, non-Born–Oppenheimer ~non-BO! nonrelativis-tic calculations of the ‘‘vibrational spectrum’’ of the H2 mol-ecule. Although we use the term ‘‘vibrational spectrum,’’ thestates we have calculated can be better characterized as stateswith the zero total angular momentum that is the sum of theangular momenta of the electrons and the nuclei. We calcu-late these states by using totally spherically symmetric wavefunctions dependent on the coordinates of both nuclei andelectrons in an internal coordinate system that excludes thethree coordinates describing the motion of the center of massof the molecule. By describing our method as ‘‘variational’’we mean that the energies of the ground and excited states inour calculations are obtained by direct diagonalization of theHamiltonian matrix.Such direct calculations of the whole vibrational spec-trum have not previously been reported in the literature. Sofar the only direct nonadiabatic calculation of H2 vibrationallevels has been the early work of Wolniewicz,4 which onlyconsidered the ground and the first excited state.A by-product of the calculations is a new variationalupper-bound for the ground state energy of the H2 molecule.This new upper-bound, as well as the variational upper-bounds to the excited vibrational energies, should provide avery accurate reference for future nonadiabatic molecularcalculations and for evaluating the quality of methodologieswhere the nonadiabatic effects are determined as ‘‘correc-a!Electronic mail: bubin@email.arizona.edub!Permanent address: Department of Chemistry, University of Arizona, Tuc-son, AZ 85721.3070021-9606/2003/118(7)/3079/4/$20.00Downloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP tions’’ to the BO approximation. The present high-accuracyresults may also be used in the future to estimate the relativ-istic effects in the vibrational spectrum of H2 as a differenceof the transition energies and the experimental energies.In the nonadiabatic molecular approach that does notinvoke the BO approximation regarding the separability ofthe electronic and nuclear motions, all particles ~nuclei andelectrons! are treated equally. Without invoking any approxi-mations, the total nonrelativistic Hamiltonian can be sepa-rated into an operator representing the translational motionof the center of mass and an operator representing the inter-nal energy. For H2 we perform this separation by makingtransformation to an internal reference frame with the originat one of the two nuclei,r5F r1r2r3G5FR22R1R32R1R42R1G , ~1!where the Ri are the original Cartesian coordinates in thelaboratory coordinate system. This transformation to the in-ternal coordinates together with the conjugate momentumtransformation yields the nonadiabatic Hamiltonian for theinternal energy of H2 expressed in terms of coordinates ofthree pseudoparticles asH5212 S (i513 1m i„ i21(iÞ j3 1M 1„ i8„jD 1(i513 q0qiri1(i, j3 qiq jri j, ~2!where m i5M 1M i /(M 11M i) is the reduced mass associatedwith the first and ith particles, and M 1 is the mass of particle1 ~the reference nucleus!. The potential energy is the same as9 © 2003 American Institute of Physicslicense or copyright; see http://jcp.aip.org/about/rights_and_permissions3080 J. Chem. Phys., Vol. 118, No. 7, 15 February 2003 S. Bubin and L. Adamowiczin the total Hamiltonian but is now written using internaldistance coordinates. The charges are mapped from the origi-nal particles as $Q1 ,Q2 ,Q3 ,Q4%°$q0 ,q1 ,q2 ,q3%. In the in-ternal coordinates, the interparticle distances are: ri j5uri2rju5uRi112Rj11u with r j5urju5uRj112R1u. More in-formation on the nonadiabatic Hamiltonian and eliminatingthe coordinates of the center of mass motion can be found inRef. 1.Explicitly correlated spherical Gaussians multiplied bypowers of the internuclear distance were used as basis func-tions in the present calculations. As we showed before,1,2 thepreexponential powers of the internuclear distance allow usto describe the nuclei correlation effects. They are also im-portant in generating nodes in the wave functions of the vi-brational excited states. The general form of the basis func-tion is ~the prime represents vector/matrix transposition and^ is the Kronecker product symbol!fk5r1mk exp@2r8~LkLk8^ I3!r# , ~3!where for H2 r is a 931 vector of the internal Cartesiancoordinates of the three pseudoparticles, Lk is 333 rank 3lower triangular matrix of nonlinear variation parameters,and I3 is the 333 identity matrix. To ensure the proper per-mutational symmetry of the two identical nuclei and the twoelectrons, the appropriate symmetry projections were appliedto the basis functions. More details regarding the form of thesymmetry projection operators can be found in Refs. 1 and 2.Using only spherical Gaussians we can only describe stateswith the zero total angular momentum. In order to considerrotationally excited states we would need to include in thebasis functions angular factors. Such factors have been con-sidered in recent works by Suzuki and Varga ~see Ref. 7!.Our calculations were performed for the proton–electronmass ratio of 1836.152 667 5 taken from the CODATA 98.5We used the atomic units except where otherwise noted.Thus, \51, me51, energies are in hartrees, and distancesare in bohrs.The wave functions for the ground state and the excitedstates were obtained using the variational method by mini-mizing the energy of each state obtained through diagonal-ization in a separate calculation. The minimization was donewith respect to the linear expansion coefficients, ck , andwith respect to the nonlinear parameters of the basis func-tions, i.e., the basis set exponent matrices, Lk , and the pow-ers of the internuclear distance, mk .In our previous non-BO calculations of the ground statesof H2 and LiH2,3 we employed a procedure where we di-rectly optimized the Rayleigh quotient with respect to boththe linear and the nonlinear parameters,E~L,m,c!5 min$L,m,c%c8H~L,m!cc8S~L,m!c ,where H~L,m! and S~L,m! are the Hamiltonian and overlapmatrices, respectively. Both are functions of the nonlinearparameters of the basis functions. In the optimization weused the analytically calculated gradients of the energy withrespect to the linear and exponential nonlinear parameters.1The powers, mk , were optimized separately.Downloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP In the present calculations we used a mixed approach.First we optimized a wave function for each state in the basisof 128 functions ~175 functions for the highest, fourteenth,excited state! using the procedure based on analytical gradi-ents to obtain a good first approximation. Then we continuedthe optimization using the stochastic variational method.6,7 Inthis step we enlarged the basis set for each state by includingadditional basis functions with randomly selected nonlinearparameters and powers of the internuclear distance, and bysubsequently optimizing these parameters of the functions inone-dimensional optimizations. After each addition of 100–200 new functions to the basis we carried out a cyclic one-dimensional optimization of each nonlinear parameter of ev-ery function in the basis set. Although the approach, unlikethe approach based on the analytical gradient, does not usu-ally provide a very well optimized set of parameters, it al-lows one to compensate for this by significantly increasingthe basis set size. In the present calculations we used 3000basis functions for each of the 15 calculated states. As will beshown next, with 3000 basis functions we noticeably low-ered our previous variational upper-bound of the ground stateenergy of H2 .2 This indicated to us that the stochastic varia-tional approach is very effective in the energy minimization.The non-BO wave functions of different excited stateshave to differ from each other by the number of ‘‘nodes’’ interms of the internuclear distance, r1 . To accurately describethe node structures in all 15 states considered in the presentcalculations, a wide range of powers, mk , had to be used.While in the previous calculations of the H2 ground state2 thepower range was 0–40, in the present calculations it wasextended to 0–250. Also, as was the case in the previouscalculations, we only used even powers.All optimizations performed in this work have been car-ried out using the double precision ~64 bit!. A peril of sig-nificantly increasing number of terms in the expansions ofthe wave functions in such calculations is the numerical in-stability. To verify this point, all results presented in thiswork were recalculated with the quadruple precision ~128bit!. No numerical inaccuracies have been detected.The high degree of parallelism of the algorithm that un-derlies the stochastic variational method allowed develop-ment of a parallel implementation of the procedure. The cal-culations described here have been carried out using MPI~message passing interface! on several multiprocessor com-puter platforms. They included our ‘‘Beowulf’’ 16-processorSUN ULTRA 10 cluster, an IBM SP3 computer, and SGIORIGIN 2000 super-computer. The calculations have beencarried out continuously for over four months.Table I contains total variational energies of the lowest15 states corresponding to the rotational ground state (J50) calculated with 3000 basis functions each. We alsoshow the expectation values of the internuclear distance andits square calculated as average values using the optimizedwave function of each state (^r1&,^r12&). The energy for theground state of 21.164 025 030 0 hartree is noticeably lowerthan our previously reported upper-bound2 of21.164 025 023 2 hartree. We are certain that for at least afew lowest excited states the quality of the results is verysimilar as for the ground state. However, for the highestlicense or copyright; see http://jcp.aip.org/about/rights_and_permissions3081J. Chem. Phys., Vol. 118, No. 7, 15 February 2003 Excited vibrational states of H2DowTABLE I. Nonadiabatic variational energies for 15 states of the H2 molecule with zero total angular momentum~the ground rotational states! obtained with 3000 basis functions for each state and expectation values of theinternuclear distance and the square of the internuclear distance, ^r1&, ^r12& . Also, the nonrelativistic energies ofWolniewicz are presented for comparison. All quantities in atomic units.v E ^r1& ^r12& E ~from Ref. 11!0 21.164 025 030 0 1.448 738 0 2.127 045 9 21.164 025 018 51 21.145 065 367 6 1.545 349 5 2.473 996 7 21.145 065 362 92 21.127 177 915 2 1.646 057 9 2.856 817 2 21.127 177 932 43 21.110 340 442 9 1.751 708 2 3.281 414 3 21.110 340 485 54 21.094 539 118 7 1.863 424 5 3.755 699 5 21.094 539 194 05 21.079 769 321 7 1.982 733 2 4.290 541 7 21.079 769 480 36 21.066 037 073 7 2.111 758 7 4.901 320 7 21.066 037 284 97 21.053 360 489 0 2.253 534 9 5.610 516 3 21.053 360 825 88 21.041 772 695 0 2.412 595 2 6.452 556 7 21.041 773 113 99 21.031 324 945 4 2.595 894 0 7.482 387 6 21.031 325 470 810 21.022 091 784 9 2.814 949 0 8.794 679 6 21.022 092 487 611 21.014 178 260 1 3.090 179 8 10.566 922 21.014 179 153 612 21.007 730 195 1 3.462 701 0 13.181 814 21.007 731 197 913 21.002 949 375 8 4.034 237 3 17.680 148 21.002 950 463 314 21.000 115 048 2 5.211 018 1 28.919 890 21.000 115 976 2a 20.999 455 679 4aNonrelativistic threshold.states, where the number of nodes in the wave function ismuch higher, the quality of the calculations decreases, but webelieve that it still allows determination of the transition en-ergies, discussed next, with the accuracy similar to the ex-perimental uncertainty, if not higher.In Table I we also compare our total variational energieswith the energies obtained by Wolniewicz. In his calculationsWolniewicz employed an approach where in the zeroth orderthe adiabatic approximation for the wave function was used~i.e., the wave function is a product of the ground state elec-tronic wave function and a vibrational wave function! and hecalculated the nonadiabatic effects as corrections.8,9 In gen-eral the agreement between our results and the results ofWolniewicz is very good. However, one notices that theagreement is much better for the lower energies than for thehigher ones. While for the two lowest states our energies arelower than those obtained by Wolniewicz, the energies forthe higher states are progressively higher.We found it interesting how the distribution of powers ofr1 in the basis set changes with the increased excitationlevel. We demonstrate this change in Fig. 1 by showing thepower distributions optimized for the v50, v51, and v514 states. As expected, only one maximum in the powerdistribution appeared for the ground-state wave function,where there are no nodes and the wave function peaks nearthe ground-state equilibrium distance (^r1&51.449 bohr).For the first excited state, the distribution was somewhatwider, which seems to be a result of two overlapping peaksand reflects the presence of a maximum and a minimum inthe wave function of this state. For the v514 state, whosewave function has 14 ‘‘nodes,’’ the power distribution be-came almost uniform in the 0–250 range.Included in Table II is the comparison of the transitionfrequencies calculated from the energies obtained in thiswork with the experimental transition frequencies ofDabrowski.10 To convert theoretical frequencies into wavenumbers we used the factor of 1 hartreenloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP FIG. 1. The distribution of r1 powers for the states with v50 ~a!, v51 ~b!,and v514 ~c!.license or copyright; see http://jcp.aip.org/about/rights_and_permissions3082 J. Chem. Phys., Vol. 118, No. 7, 15 February 2003 S. Bubin and L. AdamowiczDowTABLE II. Comparison of vibrational frequencies Ev112Ev ~in cm21) of H2 calculated from our non-Born–Oppenheimer energies with the experimental values of Dabrowski and with the results of Wolniewicz obtainedusing the conventional approach based on the potential energy curve. Differences between the calculated andthe experimental results are shown in parentheses.v Experimenta This work ~Diff.! Wolniewiczb ~Diff.! Wolniewiczc ~Diff.!0 4161.14 4161.165 (10.025) 4161.163 (10.023) 4161.167 (10.027)1 3925.79 3925.842 (10.052) 3925.837 (10.047) 3925.836 (10.046)2 3695.43 3695.398 (20.032) 3695.392 (20.038) 3695.389 (20.041)3 3467.95 3467.990 (10.040) 3467.983 (10.033) 3467.976 (10.026)4 3241.61 3241.596 (20.014) 3241.577 (20.033) 3241.564 (20.046)5 3013.86 3013.880 (10.020) 3013.869 (10.009) 3013.851 (20.009)6 2782.13 2782.189 (10.059) 2782.161 (10.031) 2782.136 (10.006)7 2543.25 2543.227 (20.023) 2543.209 (20.041) 2543.175 (20.075)8 2292.93 2293.016 (10.086) 2292.993 (10.063) 2292.950 (10.020)9 2026.38 2026.445 (10.064) 2026.406 (10.026) 2026.351 (20.029)10 1736.66 1736.818 (10.158) 1736.776 (10.116) 1736.707 (10.047)11 1415.07 1415.187 (10.117) 1415.163 (10.093) 1415.076 (10.006)12 1049.16 1049.269 (10.109) 1049.250 (10.090) 1049.139 (20.021)13 622.02 622.063 (10.043) 622.098 (10.078) 621.956 (20.064)aReference 10.bReference 11 ~nonrelativistic values!.cFrom Ref. 8 ~includes relativistic and radiative corrections!.5219 474.63137 cm21 from CODATA 98.5 For all the fre-quencies our results are either within or very close to theexperimental error bracket of 0.1 cm21. We hope that thepresent calculations will inspire a remeasurement of the vi-brational spectrum of H2 with the accuracy lower than0.1 cm21. With such high-precision results it would be pos-sible to verify whether the larger differences between thecalculated and the experimental frequencies for higher exci-tation levels, that now appear, are due to the relativistic andradiative effects.Finally, in Table II we also compare our results for thetransition energies with the results obtained by Wolniewiczin Refs. 8 and 9. Wolniewicz also calculated transition ener-gies corrected for the relativistic effects and these results arealso shown in Table II. Upon comparing the results one no-tices that our transition energies are, in general, very similarto Wolniewicz’s nonrelativistic results. Both sets of resultsshow higher positive discrepancies in comparison with theexperimental values for the higher excitation levels. Thesediscrepancies decrease somewhat when the relativistic ef-nloaded 27 Mar 2012 to 129.59.117.132. Redistribution subject to AIP fects are included. However, the insufficient accuracy of theexperiment, as well as perhaps that of some of the theoreticalresults, does not allow one to carry out a more detailedanalysis of the remaining discrepancies.This work was supported by the National Science Foun-dation. We would like to thank Professor Wolniewicz foruseful discussions and for making his results available to us.1 D. B. Kinghorn and L. Adamowicz, J. Chem. Phys. 110, 7166 ~1999!.2 D. B. Kinghorn and L. Adamowicz, Phys. Rev. Lett. 83, 2541 ~1999!.3 C. E. Scheu, D. B. Kinghorn, and L. Adamowicz, J. Chem. Phys. 114,3393 ~2001!.4 W. Kolos and L. Wolniewicz, J. Chem. Phys. 41, 3674 ~1964!.5 See CODATA database of fundamental constants at http://physics.nist.gov/cuu6 V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. ~France! 3, 795 ~1977!.7 Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems, Lecture Notes in Physics ~Springer, Ber-lin, 1998!.8 L. Wolniewicz, J. Chem. Phys. 103, 1792 ~1995!.9 L. Wolniewicz ~private communication!.10 I. Dabrowski, Can. J. Phys. 62, 1639 ~1984!.license or copyright; see http://jcp.aip.org/about/rights_and_permissions

Variational calculations of excited states with zero total angular momentum  vibrational spectrum  of H2 without use of the Born–Oppenheimer approximation

Sergiy Bubin

Ludwik Adamowicz

Kukulin V. I.

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Variational calculations of excited states with zero total angular momentum (vibrational spectrum) of H2 without use of the Born–Oppenheimer approximation

Leo Tolstoy

Virginia Smith

Image & Multimedia Collections

Anna Karenina

http://nur.nu.edu.kz/bitstream/handle/123456789/1047/bubin_jcp_118_3079_2003.pdf?sequence=1&isAllowed=y

Variational calculations of excited states with zero total angular momentum vibrational spectrum of H2 without use of the Born–Oppenheimer approximation

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