Dipole transitions in the bound rotational-vibrational spectrum of the heteronuclear molecular ion HD +
aa r X i v : . [ phy s i c s . a t m - c l u s ] J un Dipole transitions in the boundrotational-vibrational spectrum of the heteronuclearmolecular ion HD + Horacio Olivares Pil´on ‡ and Daniel Baye Physique Quantique C.P. 165/82, and Physique Nucl´eaire Th´eorique et PhysiqueMath´ematique, C.P. 229, Universit´e Libre de Bruxelles (ULB), B 1050 Brussels,BelgiumE-mail: [email protected], [email protected]
Abstract.
The non-relativistic three-body Schr¨odinger equation of the heteronuclearmolecular ion HD + is solved in perimetric coordinates using the Lagrange-meshmethod. Energies and wave functions of the four lowest vibrational bound orquasibound states v = 0 − v = 0 − Submitted to:
J. Phys. B: At. Mol. Opt. Phys.
24 July 2018 ‡ Present address: Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 70-543, 04510 M´exico, DF, Mexico . Introduction
The heteronuclear diatomic molecule HD + is the lightest isotopomer of H +2 . Theelectronic ground state supports 637 rotational-vibrational levels of which 563 are boundand 74 are quasibound levels [1]. As for any three-body system, exact solutions of theSchr¨odinger equation can not be obtained but it is possible to reach a high accuracy forboth energies and wave functions.Theoretical dissociation energies have been investigated for a long time. In 1993,Moss presented a detailed study including radiative and relativistic corrections of mostof the bound and quasibound levels of the electronic ground state [1] (see also severalreferences to previous works). The dissociation energies are presented there in cm − with nine figures. They correspond to total energies with about ten significant digits.Improvements in accuracy have been reached in several papers but for limited numbersof energy levels [2, 3, 4, 5, 6, 7, 8, 9, 10]. In particular, the ground state has been widelystudied, but not always with the same mass values. The CODATA 1986 masses areemployed in [1, 2, 3, 4, 5, 6] while the CODATA 2002 and 2006 values are used in [7, 8]and [9, 10], respectively. To date, the most accurate result is determined with a 25-digitaccuracy by Hijikata et al [9].The molecular ion HD + as well as the other heteronuclear isotopomers HT + andDT + have a permanent electric dipole moment. Hence, electric dipole (E1) transitionsare possible. In 1953, the original work of Bates and Poots [11] included the theoreticalEinstein A coefficients between some of the lowest vibrational-rotational levels. Thelifetimes of 22 vibrational levels for the total orbital angular momentum L = 0 weredetermined in [12]. Recently, Tian et al [10] published oscillator strengths for the sixfirst rotational and six first vibrational levels.In this work, we extend to HD + our previous studies of transition probabilitieswithin the bound spectra of H +2 [13] and D +2 [14]. The calculations are simplified bythe fact that E1 transitions are not forbidden like in H +2 and D +2 but they are madeheavier by the lower symmetry of the three-body wave functions. Accurate energiesare calculated beyond the Born-Oppenheimer approximation with the Lagrange-meshmethod in perimetric coordinates [15, 16, 17] with which the calculation is particularlysimple and very precise. E1 transition probabilities are obtained from the correspondingthree-body wave functions.The Lagrange-mesh method is an approximate variational calculation using a basisof Lagrange functions and the associated Gauss quadrature. It has the high accuracyof a variational approximation and the simplicity of a calculation on a mesh. The moststriking property of the Lagrange-mesh method is that, in spite of its simplicity, theobtained energies and wave functions can be as accurate with the Gauss quadratureapproximation as in the original variational method with an exact calculation of thematrix elements [18, 19].The Lagrange-mesh method also provides analytical approximations for the wavefunctions that lead to very simple expressions for a number of matrix elements when used2ith the corresponding Gauss-Laguerre quadrature. In the same direction, this methodhas been applied to obtain not only energies and quadrupole transitions probabilities ofthe homonuclear systems H +2 [13] and D +2 [14] but also polarizabilities for H +2 [20]. Wekeep here the CODATA1986 masses to allow a full comparison with the complete setof energies of [1]. The transition probabilities are presented with six significant figuresand do not change with later mass conventions.In section 2, the expressions for the transition probabilities are summarized.Lagrange-mesh expressions for the transition matrix elements are presented briefly. Insection 3, energies are given for the lowest four vibrational levels over the full rotationalbands and E1 transition probabilities are tabulated. Concluding remarks are presentedin section 4.
2. Lagrange-mesh calculation of transition probabilities
In the three-body Hamiltonian that we are considering, two nuclei with masses m and m < m and charges q and q and an electron with mass m e = 1 and charge q e = − L andparity π are constants of motion. The dimensionless dipole oscillator strength for anelectric transition between an initial state i with energy E i and a final state f withenergy E f is given by [21, 22] f (1) i → f = 23 ( E i − E f ) |h iL i || d (1) || f L f i| L i + 1 , (1)where d (1) µ is the electric dipole operator defined below. The transition probability pertime unit for E i > E f is given in atomic units by W (1) i → f = 2 α ( E i − E f ) f (1) i → f , (2)where α is the fine-structure constant. Lifetimes can be calculated using τ = (cid:18) X E f 3. Energies and E1 transition probabilities HD + rovibrational spectrum Energies of natutal-parity states are calculated for the four lowest vibrational levels v = 0 − L = 0 − 47. A high accuracy is obtainedfor most states by choosing N x = N y = 40, N z = 14 and h x = h y = 0 . h z = 0 . N x = N y or h x = h y . The totalnumber of basis states is 22400 for each value of K . In most calculations ( L ≥ K max = 2. Then the size of the matrix is 67200.For the sake of comparison with energies of other works and specially with thevery complete results of Moss [1], the 1986 fundamental constants are used everywhere, m p = 1836 . 152 701 and m d = 3670 . 483 014 in atomic units, except in Tables 2 and 6.The exceptions occur for the comparison with the oscillator strengths of [10] where themore recent 2006 values are used.Because of the different masses of the nuclei, two two-body dissociations are possiblewith thresholds E H+D + d = − . 499 727 839 716 and E D+H + d = − . 499 863 815 249 a.u.The small gap between these two thresholds does not make any quantitative differencein the number of bound states. In Table 1, the obtained energies are presented as the5rst line for each L value. The accuracy is estimated from the stability of the digits withrespect to test calculations with smaller and larger numbers of mesh points (see Table3 below). The error is expected to be at most of a few units on the last displayed digit.Literature results, sometimes truncated and rounded, are displayed in the following lines.Except in the low- L or low- v regions where results of other references are mentioned,the literature results are the 10-digit energies obtained by Moss in [1]. Since the energiesof [1] are converted here from cm − into atomic units, their accuracy is about 2-3 unitson the last digit. Table 1: Energies of the four lowest vibrational bound orquasibound levels in the Σ g rotational band of the HD + molecular ion. Quasibound levels are separated frombound levels by a horizontal bar. For each L value, theLagrange-mesh energies obtained with N x = N y = 40, N z = 14 and h x = h y = 0 . h z = 0 . m p = 1836 . 152 701 and m d = 3670 . 483 014, respectively. L v = 0 v = 1 v = 2 v = 30 -0.597 897 968 645 1 -0.589 181 829 652 -0.580 903 700 33 -0.573 050 546 1-0.597 897 968 645 036 a,b -0.597 897 968 644 84 c -0.589 181 829 653 3 c -0.580 903 700 369 c -0.573 050 546 8 c -0.597 897 968 645 0 d -0.589 181 829 653 8 d -0.580 903 700 6 -0.573 050 546 91 -0.597 698 128 231 2 -0.588 991 112 090 -0.580 721 828 23 -0.572 877 276 7-0.597 698 128 231 122 a -0.597 698 128 231 1 d -0.588 991 112 091 6 d -0.580 721 828 1 -0.572 877 277 22 -0.597 299 643 396 5 -0.588 610 829 493 -0.580 359 195 31 -0.572 531 809 9-0.597 299 643 396 469 e -0.597 299 643 4 -0.588 610 829 6 -0.580 359 195 3 -0.572 531 810 43 -0.596 704 882 815 2 -0.588 043 264 274 -0.579 818 002 15 -0.572 016 268 8-0.596 704 882 815 162 e -0.596 704 882 6 -0.588 043 264 2 -0.579 818 002 0 -0.572 016 269 34 -0.595 917 342 277 4 -0.587 291 784 496 -0.579 101 495 70 -0.571 333 785 6-0.595 917 342 277 360 e -0.595 917 342 4 -0.587 291 784 7 -0.579 101 495 9 -0.571 333 786 35 -0.594 941 578 904 4 -0.586 360 780 057 -0.578 213 907 45 -0.570 488 441 9-0.594 941 578 904 345 e -0.594 941 579 1 -0.586 360 780 0 -0.578 213 907 7 -0.570 488 442 6Continued on Next Page. . . 6able 1 – Continuation L v = 0 v = 1 v = 2 v = 36 -0.593 783 127 985 9 -0.585 255 582 110 -0.577 160 375 39 -0.569 485 193 1-0.593 783 127 985 818 e -0.593 783 128 1 -0.585 255 582 1 -0.577 160 375 4 -0.569 485 193 67 -0.592 448 405 873 9 -0.583 982 369 086 -0.575 946 853 12 -0.568 329 780 6-0.592 448 405 873 834 e -0.592 448 405 7 -0.583 982 368 9 -0.575 946 853 0 -0.568 329 781 28 -0.590 944 602 712 5 -0.582 548 063 063 -0.574 580 009 81 -0.567 028 635 2-0.590 944 602 712 443 e -0.590 944 602 8 -0.582 548 062 8 -0.574 580 009 8 -0.567 028 635 99 -0.589 279 568 877 4 -0.580 960 220 249 -0.573 067 124 61 -0.565 588 775 5-0.589 279 568 877 377 e -0.589 279 568 9 -0.580 960 220 1 -0.573 067 124 7 -0.565 588 776 110 -0.587 461 698 867 7 -0.579 226 919 228 -0.571 415 979 14 -0.564 017 704 8-0.587 461 698 867 668 e -0.587 461 698 6 -0.579 226 919 1 -0.571 415 978 9 -0.564 017 705 211 -0.585 499 816 071 2 -0.577 356 650 307 -0.569 634 751 33 -0.562 323 308 5-0.585 499 816 071 126 e -0.585 499 815 8 -0.577 356 650 1 -0.569 634 751 2 -0.562 323 309 412 -0.583 403 061 368 7 -0.575 358 208 792 -0.567 731 913 16 -0.560 513 757 1-0.583 403 061 368 632 e -0.583 403 061 4 -0.575 358 208 6 -0.567 731 913 1 -0.560 513 757 613 -0.581 180 788 002 2 -0.573 240 594 524 -0.565 716 134 74 -0.558 597 414 2-0.581 180 788 2 -0.573 240 594 4 -0.565 716 134 7 -0.558 597 414 914 -0.578 842 464 558 5 -0.571 012 919 429 -0.563 596 196 28 -0.556 582 752 9-0.578 842 464 6 -0.571 012 919 6 -0.563 596 196 3 -0.556 582 753 815 -0.576 397 587 359 8 -0.568 684 324 282 -0.561 380 909 07 -0.554 478 280 9-0.576 397 587 5 -0.568 684 324 3 -0.561 380 909 3 -0.554 478 281 516 -0.573 855 603 031 8 -0.566 263 905 388 -0.559 079 046 12 -0.552 292 474 9-0.573 855 603 0 -0.566 263 905 6 -0.559 079 046 0 -0.552 292 475 817 -0.571 225 841 568 3 -0.563 760 651 451 -0.556 699 282 76 -0.550 033 725 1-0.571 225 841 7 -0.563 760 651 3 -0.556 699 282 6 -0.550 033 725 918 -0.568 517 459 835 2 -0.561 183 390 542 -0.554 250 147 03 -0.547 710 288 8-0.568 517 459 9 -0.561 183 390 4 -0.554 250 147 1 -0.547 710 289 719 -0.565 739 395 161 4 -0.558 540 746 805 -0.551 739 979 43 -0.545 330 253 6-0.565 739 395 3 -0.558 540 746 9 -0.551 739 979 5 -0.545 330 254 220 -0.562 900 328 449 7 -0.555 841 106 348 -0.549 176 901 70 -0.542 901 508 8-0.562 900 328 2 -0.555 841 106 4 -0.549 176 901 5 -0.542 901 509 321 -0.560 008 656 092 -0.553 092 591 62 -0.546 568 793 72 -0.540 431 725 4-0.560 008 656 0 -0.553 092 591 5 -0.546 568 793 7 -0.540 431 725 9Continued on Next Page. . . 7able 1 – Continuation L v = 0 v = 1 v = 2 v = 322 -0.557 072 469 897 -0.550 303 043 53 -0.543 923 278 04 -0.537 928 343 1-0.557 072 469 7 -0.550 303 043 5 -0.543 923 277 9 -0.537 928 343 623 -0.554 099 544 194 -0.547 480 010 55 -0.541 247 711 25 -0.535 398 564 6-0.554 099 544 3 -0.547 480 010 7 -0.541 247 711 3 -0.535 398 565 124 -0.551 097 329 307 -0.544 630 743 97 -0.538 549 181 51 -0.532 849 356 0-0.551 097 329 4 -0.544 630 743 8 -0.538 549 181 6 -0.532 849 356 625 -0.548 072 950 602 -0.541 762 198 76 -0.535 834 511 66 -0.530 287 453 0-0.548 072 950 7 -0.541 762 198 5 -0.535 834 511 6 -0.530 287 453 626 -0.545 033 212 416 -0.538 881 039 26 -0.533 110 267 45 -0.527 719 373 0-0.545 033 212 6 -0.538 881 039 1 -0.533 110 267 4 -0.527 719 373 727 -0.541 984 606 220 -0.535 993 649 26 -0.530 382 770 54 -0.525 151 432 3-0.541 984 606 0 -0.535 993 649 4 -0.530 382 770 4 -0.525 151 432 728 -0.538 933 322 483 -0.533 106 146 10 -0.527 658 116 05 -0.522 589 769 2-0.538 933 322 6 -0.533 106 146 2 -0.527 658 116 0 -0.522 589 769 429 -0.535 885 265 823 -0.530 224 398 43 -0.524 942 194 62 -0.520 040 373 4-0.535 885 265 5 -0.530 224 398 5 -0.524 942 194 4 -0.520 040 373 630 -0.532 846 073 113 -0.527 354 047 63 -0.522 240 719 44 -0.517 509 122 6-0.532 846 072 8 -0.527 354 047 6 -0.522 240 719 5 -0.517 509 123 031 -0.529 821 134 389 -0.524 500 532 93 -0.519 559 258 59 -0.515 001 827 2-0.529 821 134 1 -0.524 500 532 9 -0.519 559 258 3 -0.515 001 827 332 -0.526 815 616 499 -0.521 669 120 58 -0.516 903 274 04 -0.512 524 287 4-0.526 815 616 4 -0.521 669 120 4 -0.516 903 273 9 -0.512 524 287 633 -0.523 834 489 683 -0.518 864 937 83 -0.514 278 168 95 -0.510 082 365 5-0.523 834 489 6 -0.518 864 937 6 -0.514 278 168 7 -0.510 082 365 834 -0.520 882 557 405 -0.516 093 012 69 -0.511 689 346 16 -0.507 682 081 6-0.520 882 557 5 -0.516 093 012 6 -0.511 689 346 0 -0.507 682 081 535 -0.517 964 490 152 -0.513 358 321 48 -0.509 142 282 33 -0.505 329 742 9-0.517 964 490 0 -0.513 358 321 6 -0.509 142 282 2 -0.505 329 743 136 -0.515 084 864 210 -0.510 665 846 79 -0.506 642 625 18 -0.503 032 128 7-0.515 084 864 2 -0.510 665 846 9 -0.506 642 625 2 -0.503 032 128 637 -0.512 248 207 112 -0.508 020 650 51 -0.504 196 326 01 -0.500 796 767 3-0.512 248 207 0 -0.508 020 650 5 -0.504 196 326 0 -0.500 796 767 338 -0.509 459 052 264 -0.505 427 968 98 -0.501 809 829 30 -0.498 632 382 9-0.509 459 052 3 -0.505 427 969 0 -0.501 809 829 2 -0.498 632 382 739 -0.506 722 006 729 -0.502 893 342 59 -0.499 490 360 08 -0.496 549 678 8-0.506 722 006 5 -0.502 893 342 6 -0.499 490 360 2 -0.496 549 678 740 -0.504 041 838 520 -0.500 422 801 25 -0.497 246 392 46 -0.494 562 907 4-0.504 041 838 5 -0.500 422 801 0 -0.497 246 392 5 -0.494 562 907 441 -0.501 423 593 974 -0.498 023 146 52 -0.495 088 491 10 -0.492 693 76Continued on Next Page. . . 8able 1 – Continuation L v = 0 v = 1 v = 2 v = 3-0.501 423 593 7 -0.498 023 146 5 -0.495 088 491 1 -0.492 693 7742 -0.498 872 763 78 -0.495 702 414 99 -0.493 031 052 7 -0.490 987-0.498 872 763 6 -0.495 702 414 9 -0.493 031 052 9 -0.490 9943 -0.496 395 532 49 -0.493 470 721 23 -0.491 096 94-0.496 395 532 4 -0.493 470 721 0 -0.491 097 044 -0.493 999 183 31 -0.491 342 046 6-0.493 999 183 4 -0.491 342 046 745 -0.491 692 825 67 -0.489 339 4-0.491 692 825 8 -0.489 33946 -0.489 488 926 5-0.489 488 92747 -0.487 407 9-0.487 408The energy for the ground state ( L π , v ) = (0 + , 0) has been improved in a series ofpapers [2, 3, 4, 5]. The best known values were determined with about 18 digits by Yan et al [6] and about 25 digits by Hijikata et al [9], with different values for the massesof the proton and deuteron. Our accuracy is about 10 − . For the (0 + , + , + , 3) vibrational excited states, the accuracies are about 10 − , 10 − and 10 − ,respectively.The energy of the lowest L = 1 level is known with about 18 digits [5]. Resultswith the same accuracy (close to 18 digits) are available for L = 2 − 12 and v = 0[6]. Our error for the lowest vibrational energy remains smaller than 10 − for all thesestates. When comparing the rest of our results with those of Moss [1], one observes thatboth works agree very well. The present energies are a little more accurate for v = 0and a little less accurate for v = 2 and 3. But, in addition, the Lagrange-mesh methodprovides easy-to-use accurate wave functions. Fig. 1 shows the obtained spectrum.Energies of four vibrational levels v = 0 − L =0 − m p = 1836 . 152 672 47 and m d =3670 . 482 965 4 are presented in Table 2. Each first line displays Lagrange-mesh resultsobtained using the set of parameters N x = N y = 40, N z = 20 and h x = h y = 0 . h z = 0 . 5. Each second line contains rounded results of Tian et al [10]. Their ground-state energy agrees with the more accurate result of [9]. When comparing the resultsobtained with two different mass conventions in Tables 1 and 2, one observes that thedifference in the energies appears at the tenth digit.Table 3 present convergence tests as a function of the number of mesh points fortwo different sets of initial and final levels. The scaling parameters are h x = h y = 0 . h z = 0 . 4. The initial and final energies as well as transition probabilities are9 E ( H a r t r ee ) LE d υ =3 υ =2 υ =1 υ =0 Figure 1. Four lowest Σ g rotational bands of the HD + molecular ion. The twodissociation energies E H+D + d and E D+H + d can not be distinguished. Table 2. Comparison of energies for six rotational levels L = 0 − v = 0 − 3. The mass conventions for the proton and deuteronare m p = 1836 . m d = 3670 . N x = N y = 40, N z = 20 and h x = h y = 0 . h z = 0 . et al [10]. L v = 0 v = 1 v = 2 v = 30 -0.597 897 968 609 0 -0.589 181 829 557 -0.580 903 700 22 -0.573 050 546 5-0.597 897 968 608 95 -0.589 181 829 556 7 -0.580 903 700 218 -0.573 050 546 551 -0.597 698 128 192 1 -0.588 991 111 992 -0.580 721 828 12 -0.572 877 277 1-0.597 698 128 192 13 -0.588 991 111 991 8 -0.580 721 828 121 -0.572 877 277 092 -0.597 299 643 351 7 -0.588 610 829 390 -0.580 359 195 20 -0.572 531 810 3-0.597 299 643 351 68 -0.588 610 829 389 6 -0.580 359 195 200 -0.572 531 810 333 -0.596 704 882 761 8 -0.588 043 264 163 -0.579 818 002 03 -0.572 016 269 2-0.596 704 882 761 78 -0.588 043 264 162 6 -0.579 818 002 028 -0.572 016 269 234 -0.595 917 342 212 7 -0.587 291 784 374 -0.579 101 495 56 -0.571 333 786 0-0.595 917 342 212 67 -0.587 291 784 373 9 -0.579 101 495 565 -0.571 333 786 055 -0.594 941 578 825 8 -0.586 360 779 923 -0.578 213 907 30 -0.570 488 442 3-0.594 941 578 825 77 -0.586 360 779 922 8 -0.578 213 907 303 -0.570 488 442 33 displayed. The transition probability W is obtained by restricting (10) to κ = 0 while W corresponds to κ ≤ κ = 1 contributions have an importance smaller than 0.02 %. Withrespect to N z , a 12-digit convergence of the energies and a 10-digit convergence of theprobabilities is already obtained for N z = 14. The convergence with respect to N is10 able 3. Convergence of the energies and transition probabilities as a function of thenumbers N x = N y and N z of mesh points. Two cases are shown: (2 + , → (3 − , L f = L i + 1 (upper set) and (40 + , → (39 − , 0) where L f = L i − h x = h y = 0 . 11 and h z = 0 . N x,y N z E i (2 + , E f (3 − , W (10 s − ) W (10 s − )38 14 -0.588 610 829 483 454 -0.596 704 882 814 981 1.119 849 260 1.119 957 46638 20 -0.588 610 829 483 336 -0.596 704 882 814 924 1.119 849 261 1.119 957 46640 14 -0.588 610 829 492 821 -0.596 704 882 815 199 1.119 849 257 1.119 957 46240 20 -0.588 610 829 492 703 -0.596 704 882 815 146 1.119 849 257 1.119 957 46242 14 -0.588 610 829 494 525 -0.596 704 882 815 244 1.119 849 256 1.119 957 46144 14 -0.588 610 829 494 882 -0.596 704 882 815 238 1.119 849 254 1.119 957 45946 14 -0.588 610 829 494 935 -0.596 704 882 815 237 1.119 849 260 1.119 957 465 N x,y N z E i (40 + , E f (39 − , W (10 s − ) W (10 s − )38 14 -0.504 041 838 520 152 -0.506 722 006 729 020 6.993 933 079 6.992 287 32438 20 -0.504 041 838 516 907 -0.506 722 006 726 162 6.993 933 080 6.992 287 32540 14 -0.504 041 838 520 210 -0.506 722 006 729 109 6.993 933 079 6.992 287 32440 20 -0.504 041 838 516 956 -0.506 722 006 726 242 6.993 933 081 6.992 287 32642 14 -0.504 041 838 520 210 -0.506 722 006 729 114 6.993 933 079 6.992 287 32444 14 -0.504 041 838 520 210 -0.506 722 006 729 107 6.993 933 079 6.992 287 32446 14 -0.504 041 838 520 207 -0.506 722 006 729 118 6.993 933 079 6.992 287 324 slower. Since the convergence is exponential, one can estimate that the relative accuracyon W with N x = N y = 40 and N z = 14 is about 10 − for the two cases (2 + , → (3 − , + , → (39 − , K max can be studiedby comparing the results for K max = 2 with results from wave functions truncated at K max = 0 and K max = 1. The relative error when K max = 0 is smaller than 9 % forall considered transitions while the error for K max = 1 is smaller than 5 × − . Byextrapolation, we estimate that the relative error on the present transition probabilitiesobtained with K max = 2 should be smaller than 10 − .Table 4 presents transition probabilities per second within a same rotational band, L f = L i − v f = v i ≤ 3. Some transition probabilities involving quasibound levelsare also included. We limit the number of significant figures to six. The probabilitiesincrease slowly with L with a maximum at L i = 36, 35, 33 and 32 for v i = 0, 1, 2 and 3,respectively. This is due to a maximum of the energy differences around L i = 27. Themaximum of the transition probabilities is shifted toward higher L i values by a steadyincrease of the reduced matrix elements. 11able 4: Dipole transition probabilities per second W for transitions between levels of a same rotational band( v f = v i , L f = L i − L i v i = 0 v i = 1 v i = 2 v i = 31 6.697 53 − − − − 32 6.390 50 − − − − 23 2.287 55 − − − − 14 5.544 41 − − − − 15 1.087 75+0 1.053 69+0 1.016 08+0 9.752 57 − 16 1.867 43+0 1.807 27+0 1.741 18+0 1.669 70+07 2.923 00+0 2.825 80+0 2.719 60+0 2.605 20+08 4.274 11+0 4.127 03+0 3.967 21+0 3.795 82+09 5.930 37+0 5.718 75+0 5.490 12+0 5.246 04+010 7.891 61+0 7.599 17+0 7.285 04+0 6.951 22+011 1.014 86+1 9.757 66+0 9.340 13+0 8.898 44+012 1.268 38+1 1.217 57+1 1.163 60+1 1.106 75+113 1.547 29+1 1.482 81+1 1.414 69+1 1.343 24+114 1.848 58+1 1.768 45+1 1.684 22+1 1.596 24+115 2.168 79+1 2.071 04+1 1.968 75+1 1.862 34+116 2.504 17+1 2.386 84+1 2.264 61+1 2.137 91+117 2.850 78+1 2.712 00+1 2.568 01+1 2.419 26+118 3.204 61+1 3.042 59+1 2.875 12+1 2.702 64+119 3.561 66+1 3.374 73+1 3.182 18+1 2.984 43+120 3.918 04+1 3.704 67+1 3.485 56+1 3.261 11+121 4.270 03+1 4.028 82+1 3.781 82+1 3.529 37+122 4.614 11+1 4.343 83+1 4.067 74+1 3.786 10+123 4.947 04+1 4.646 58+1 4.340 33+1 4.028 46+124 5.265 82+1 4.934 24+1 4.596 88+1 4.253 82+125 5.567 77+1 5.204 24+1 4.834 94+1 4.459 82+126 5.850 49+1 5.454 29+1 5.052 30+1 4.644 32+127 6.111 85+1 5.682 36+1 5.247 00+1 4.805 39+128 6.349 99+1 5.886 67+1 5.417 31+1 4.941 29+129 6.563 34+1 6.065 68+1 5.561 68+1 5.050 45+130 6.750 52+1 6.218 03+1 5.678 73+1 5.131 40+131 6.910 39+1 6.342 55+1 5.767 20+1 5.182 72+132 7.041 98+1 6.438 21+1 5.825 93+1 5.203 03+133 7.144 46+1 6.504 06+1 5.853 79+1 5.190 83+134 7.217 13+1 6.539 23+1 5.849 58+1 5.144 45+135 7.259 38+1 6.542 84+1 5.812 01+1 5.061 89+1Continued on Next Page. . .12able 4 – Continuation L i v i = 0 v i = 1 v i = 2 v i = 336 7.270 62+1 6.513 96+1 5.739 55+1 4.940 53+137 7.250 27+1 6.451 50+1 5.630 25+1 4.776 80+138 7.197 68+1 6.354 10+1 5.481 46+1 4.565 42+139 7.112 03+1 6.219 96+1 5.289 43+1 4.297 96+140 6.992 29+1 6.046 57+1 5.048 41+1 3.959 40+141 6.836 99+1 5.830 20+1 4.748 90+1 3.516 7 +142 6.644 04+1 5.565 01+1 4.373 14+1 2.8 +143 6.410 27+1 5.241 07+1 3.878 +144 6.130 62+1 4.839 0 +145 5.796 43+1 4.307 +146 5.390 55+147 4.86 +1The probabilities per second for other transitions are displayed in Table 5. Thecolumns correspond to transitions between different vibrational levels ( v i → v f ). Foreach L i > 0, the first line corresponds to L f = L i − L f = L i + 1.The strongest transition from each level occurs in general towards the nearestvibrational level ( v f = v i − L f = L i − L i = 23 and L i = 28.Table 5: Dipole transition probabilities per second W for transitions between different vibrational quantumnumbers ( v i = v f ). For L i ≥ 1, each first and second linescorrespond to L f = L i − L f = L i + 1, respectively. L i (1 → 0) (2 → 0) (2 → 1) (3 → 0) (3 → 1) (3 → − − − − − − − − − − − − − − − − − − − − L i (1 → 0) (2 → 0) (2 → 1) (3 → 0) (3 → 1) (3 → − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − L i (1 → 0) (2 → 0) (2 → 1) (3 → 0) (3 → 1) (3 → − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 137 7.244 80 − − − − − − − − − − 138 8.651 51 − − − − − − − − − − 139 1.017 51+0 2.497 11 − − − − − − − − − 140 1.182 79+0 3.906 92 − − − − − − − − − 141 1.363 07+0 5.796 93 − − − − − − − − − 242 1.561 51+0 8.335 38 − − − − − − − − − 343 1.782 63+0 1.175 − − − − 244 2.031 89+01.532 30 − 245 2.306 +0Continued on Next Page. . . 15able 5 – Continuation L i (1 → 0) (2 → 0) (2 → 1) (3 → 0) (3 → 1) (3 → − -9 -8 -7 -6 -5 -4 0 10 20 30 40 f ( ) i → f L i (1 → → → → → → Figure 2. Oscillator strengths for L f = L i + 1 transitions. Oscillator strengths are depicted in Figs. 2 and 3. For the transitions with∆ L = L i − L f = − L . The behaviour of the curves depends on ∆ v = v i − v f . The ∆ v = 1 strengthspresent a shallow minimum at L i = 2 and a maximum near L i = 26. The ∆ v = 2strengths are smaller by more than an order of magnitude and a minimum appears at L i = 34 for v i = 2 → v f = 0 and L i = 33 for v i = 3 → v f = 1. The ∆ v = 3 strengthsare smaller by more than an order of magnitude than the ∆ v = 2 strengths and theminimum appears at L i = 31. The presence of a minimum is due to a minimum aroundthe same L i value in the matrix element appearing in (3) shifted by the monotoneousdecrease of the energy difference. The ∆ L = 1 strengths presented in Fig. 3 display adeep minimum around L i = 27 for ∆ v = 1, L i = 33 for ∆ v = 2 and L i = 38 for ∆ v = 3.These minima occur at increasing L i values with increasing ∆ v and are all due to achange of sign of the matrix element.In [10], Tian et al not only present rovibrational energies for the six lowestvibrational ( v = 0 − 5) and rotational ( L = 0 − 5) levels but also the dipole oscillatorstrengths between these levels with the 2006 mass convention. Comparisons between L f = L i − v f = v i ) are displayed inTable 6. The L → L + 1 strengths of [10] are multiplied by (2 L + 1) / (2 L + 3) to reversethem into L + 1 → L . The Lagrange-mesh calculations are done using the parameters N x = N y = 40, N z = 20 and h x = h y = 0 . h z = 0 . 5. The agreement betweenboth calculations improves with increasing L i . It almost reaches 12 significant figuresfor L i = 4, 5 and v i = 0, 1. Using relation (2), dipole transition probabilities canbe calculated. We find that the six significant figures presented in Tables 4 and 5 are16 -12 -11 -10 -9 -8 -7 -6 -5 0 10 20 30 40 f ( ) i → f L i (1 → → → → → → Figure 3. Oscillator strengths for L f = L i − unchanged. These numbers are independent of later changes of mass convention for theproton and deuteron. Table 6. Comparison of the dipole oscillator strengths between levels of a samerotational band ( v f = v i , L f = L i − m p = 1836 . m d = 3670 . et al [10]. L i v i = 0 v i = 1 v i = 2 v i = 31 5.219 599 338 877-6 5.563 140 035 05-6 5.911 310 070 3-6 6.264 008 727-65.219 599 339 113-6 5.563 140 035 53-6 5.911 310 071 0-6 6.264 008 700-62 1.252 564 912 793-5 1.334 723 278 42-5 1.417 978 139 8-5 1.502 305 542-51.252 564 912 744-5 1.334 723 278 42-5 1.417 978 139 7-5 1.502 305 541-53 2.012 682 620 839-5 2.143 950 651 96-5 2.276 942 668 5-5 2.411 620 728-52.012 682 620 915-5 2.143 950 651 96-5 2.276 942 668 4-5 2.411 620 725-54 2.782 267 297 866-5 2.962 302 573 95-5 3.144 649 633 9-5 3.329 257 006-52.782 267 297 892-5 2.962 302 573 95-5 3.144 649 633 2-5 3.329 257 003-55 3.555 747 344 258-5 3.783 538 364 46-5 4.014 170 682 2-5 4.247 579 949-53.555 747 344 241-5 3.783 538 364 45-5 4.014 170 681 9-5 4.247 579 945-5 For the 22 vibrational levels of the rotationless state L = 0, lifetimes have beencalculated at the Born-Oppenheimer approximation in [12]. The lifetimes 5 . × − s,2 . × − s and 2 . × − s for the first, second and third L = 0 excited vibrationallevels agree respectively with our values 5 . 460 856 × − s, 2 . 932 100 × − s and2 . 102 753 × − s, obtained from Table 5. The lifetimes of all calculated levels aredisplayed in Fig. 4.Because of the presence of a permanent dipole moment, dipole transitions aredominant in the spectrum of HD + and the lifetimes are expected to be much smallerthan for the quadrupole transitions in H +2 and D +2 . Indeed, the longest lifetime is about17 . − , 0) level, while typical lifetimes are of the order of days for H +2 [13] and months for D +2 [14]. Roughly speaking, except for the ground-state rotationalband below L = 13, the v ≤ L and their valuesare located between about 0.01 and 0.05 s. They slightly decrease with increasingvibrational excitation. -2 -1 0 10 20 30 40 τ ( s ) L υ =0 υ =1 υ =2 υ =3 Figure 4. Lifetimes τ in seconds for the first four rotational bands ( v = 0 − 4. Conclusion With the Lagrange-mesh method in perimetric coordinates, the three-body Schr¨odingerequation of the heteronuclear molecular ion HD + is solved with Coulomb potentials.Energies and wave functions are calculated for up to four of the lowest vibrationalbound or quasibound states from L = 0 to 47. Lagrange-mesh results are obtained with40 mesh points for the x and y coordinates and 14 mesh points for the z coordinate.The accuracy is around 12 digits for the lowest vibrational level and slowly decreaseswith vibrational excitation until 9 digits for the third excited vibrational level. Theseaccuracies are maintained along the whole bound rotational bands.With the corresponding wave functions, a simple calculation using the associatedGauss-Laguerre quadrature provides the electric dipole strengths and transitionprobabilities per time unit over the whole rotational bands. Tests with increasingnumbers of mesh points and various truncations on K show that the accuracy on theseprobabilities should reach at least six significant figures, independently of recent orfuture improvements in the proton and deuteron mass values. For low- L transitions,the first 10 or 11 figures of our oscillator strengths agree with those of [10].The dipole transition probabilities of the heteronuclear HD + are much larger thanthe quadrupole transition probabilities of the homonuclear H +2 and D +2 . Hence, lifetimesare much shorter. Except for the lowest levels in the ground-state rotational band18 ≤ 13, all calculated lifetimes have an order of magnitude around 10 − s. The lifetimeof the first rotational excited level is about 2.5 min. Acknowledgments HOP thanks FRS-FNRS (Belgium) for a postdoctoral grant. 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