1Σu+→X1Σg+ transition moments for the hydrogen molecule

Journal of Molecular Spectroscopy 217 (2003) 181–185 www.elsevier.com/locate/jms

1

q 1 þ Rþ u ! X Rg transition moments for the hydrogen molecule

L. Wolniewicz* and G. Staszewska Institute of Physics, Nicholas Copernicus University, Torun 87-100, Poland Received 18 June 2002; in revised form 25 October 2002

Abstract Electronic moments are given for internuclear distances, 0:5 6 R 6 50:0 bohr, for transitions connecting the six lowest 1 Rþ u states and the 1 Rþ g ground state of the hydrogen molecule. Except for the two lowest transitions, the computed moments show—due to avoided crossings—very strong R-dependence; especially at R below 0.7 bohr. This effect is also clearly visible in the adiabatic corrections that are computed for 0:5 6 R 6 0:8. Ó 2003 Elsevier Science (USA). All rights reserved.

1. Introduction

2. Results and discussion

Dipole moments for transitions involving the electronic ground state of the hydrogen molecule are of considerable interest in astrophysical applications. To compute accurate transition probabilities between discrete levels, and between discrete levels and the vibrational continuum, one needs reliable electronic moments at a wide range of internuclear distances. At the present time such results exist for the B–X , C–X [1], and B0 –X [2] only. In a relatively recent paper, Drira [3] published transition moments between many excited states and the ground state. Unfortunately, only absolute values of the moments are given in [3] which makes an accurate evaluation of the transition probabilities impossible. Also, the computations are not very accurate and we believe that there are some misassignments in [3]. Therefore a more accurate recomputation of the moments seems desirable. In a recent paper [4] we have published results of energy computations, with quite flexible variational functions, for the six lowest 1 Rþ u states. In this work we use these functions and the ground state function from [6] to determine the transition moments. Unless otherwise explicitly stated, atomic units are used throughout.

2.1. Results of computations

q Supplementary data for this article are available on ScienceDirect. * Corresponding author. E-mail address: [email protected] (L. Wolniewicz).

Using the molecular wavefunctions from [4,6] we have computed the matrix elements Mz ðRÞ ¼ hWu jz1 þ z2 jWg i 1

ð1Þ Rþ u

1

Rþ g

between the six lowest states and the X ground state for 237 internuclear distances in the interval [0.5, 50.0]. Below we will use the notation n1 Rþ u , n ¼ 1; . . . ; 6 or B1; . . . ; B6 to denote different Ru states. We present the results graphically in Figs. 1 and 2 and partly in Table 1. The complete set of our results is available in the form of an ASCII file as supplementary data for this article and from the authors through Internet [8]. The phases of the wavefunctions are defined by the overlaps with jn; l; m ¼ 0i Heitler–London functions. For the latter functions we adopt the same phase convention as in [5]. We list the overlaps in Table 2 together with the asymptotic quantum numbers n, l1 predicted by Stephens and Dalgarno [7]. For the sixth state l1 is not known because no corresponding data are given in [7] and, at R ¼ 50, this state is still predominantly ionic. We note in passing that, for n < 4, one can almost reproduce Table 2 while using solely the asymptotic results of Stephens and Dalgarno. At R ¼ 50, the first and second order corrections given in [7] are similar in

0022-2852/03/$ - see front matter Ó 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-2852(02)00047-4

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L. Wolniewicz, G. Staszewska / Journal of Molecular Spectroscopy 217 (2003) 181–185

1 þ Fig. 1. 1 Rþ u ! X Rg transition moments in H2 .

Fig. 2. B3–X and B4–X transition moments at avoided crossing.

magnitude. Therefore one should [9] add the first and second order corrections to get the secular equation for the zeroth order function. The resulting amplitudes agree in absolute value with Table 2 to within 0.01 and differ in sign by a factor ð1Þl . This factor must be due to a different phase convention adopted in [7]. We find this agreement remarkable. The rapid changes of the transition moments visualized in Fig. 2 reflect the changes of electronic configurations. For 1:0 6 R 6 1:5 the B3–B6 states have (1s4p), (1s4f), (1s5p), and (1s5f) configurations, respectively [4]. However, this must change for smaller internuclear distances because in the united atom limit the f configurations lie below p configurations (see, e.g., [10]). As is seen in Fig. 2, the configurations change between R ¼ 0:5 and R ¼ 0:6. Such a change affects strongly the adiabatic corrections. Therefore we have extended the computation [4] of these corrections to smaller internuclear distances. In Table 3 we give just as an illustration the expectation values of the nuclear Laplacian, hWu jDR jWu i, for the third and fourth states.

Table 1 1 Ru –X 1 Rþ g transition moments in H2 R 0.500 0.540 0.560 0.562 0.564 0.565 0.566 0.567 0.568 0.570 0.574 0.580 0.600 0.700 0.800 0.900 1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

B–X

B0 –X

B00 B–X

41 Ru –X

51 Ru –X

61 Ru –X

0.54144 0.55672 0.56455 0.56534 0.56613 0.56653 0.56692 0.56732 0.56772 0.56851 0.57010 0.57250 0.58058 0.62282 0.66783 0.71536 0.76517 0.87071 0.98203 1.09578 1.20777 1.31328 1.40763 1.48680 1.54770 1.58833 1.60785 1.60651 1.58572 1.54793 1.49639 1.43479 1.36682 1.29582 1.22453 1.15504

0.25783 0.26385 0.26691 0.26722 0.26753 0.26768 0.26784 0.26799 0.26815 0.26845 0.26907 0.27000 0.27313 0.28901 0.30525 0.32169 0.33807 0.36961 0.39759 0.41946 0.43267 0.43516 0.42581 0.40446 0.37178 0.32881 0.27661 0.21614 0.14843 0.07488 )0.00232 )0.08018 )0.15536 )0.22496 )0.28715 )0.34124

0.00048 0.00139 0.00887 0.01840 0.13909 0.16360 0.16529 0.16575 0.16598 0.16627 0.16668 0.16724 0.16905 0.17810 0.18721 0.19626 0.20509 0.22135 0.23446 0.24279 0.24484 0.23954 0.22649 0.20591 0.17839 0.14458 0.10501 0.06010 0.01054 )0.04207 )0.09477 )0.14350 )0.18446 )0.21556 )0.23672 )0.24900

0.16022 0.16370 0.16523 0.16462 0.09030 0.02763 0.01544 0.01071 0.00817 0.00555 0.00338 0.00214 0.00096 0.00023 0.00007 )0.00005 )0.00016 )0.00043 )0.00081 )0.00132 )0.00200 )0.00290 )0.00405 )0.00551 )0.00734 )0.00961 )0.01242 )0.01588 )0.02017 )0.02552 )0.03232 )0.04119 )0.05332 )0.07099 )0.09884 )0.14505

0.00021 0.00071 0.00296 0.00407 0.00647 0.00910 0.01517 0.03691 0.11056 0.11620 0.11672 0.11712 0.11834 0.12442 0.13050 0.13647 0.14223 0.15258 0.16043 0.16467 0.16430 0.15865 0.14758 0.13134 0.11041 0.08524 0.05615 0.02338 )0.01252 )0.05009 )0.08654 )0.11815 )0.14180 )0.15609 )0.16101 )0.15649

0.11237 0.11473 0.11589 0.11598 0.11599 0.11587 0.11529 0.11034 0.03644 0.00870 0.00343 0.00183 0.00080 0.00021 0.00007 )0.00003 )0.00013 )0.00034 )0.00069 )0.00114 )0.00175 )0.00254 )0.00355 )0.00483 )0.00644 )0.00843 )0.01091 )0.01399 )0.01783 )0.02267 )0.02893 )0.03733 )0.04932 )0.06820 )0.10153 )0.15892

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Table 1 (continued) R

B–X

B0 –X

B00 B–X

41 Ru –X

51 Ru –X

61 Ru –X

5.0 5.2 5.4 5.6 5.8 6.0 6.4 6.8 7.2 7.6 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 11.0 12.0 13.0 14.0 15.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 35.0 36.0 37.0 38.0 40.0 42.0 44.0 46.0 50.0

1.08876 1.02660 0.96902 0.91621 0.86814 0.82465 0.75048 0.69160 0.64595 0.61176 0.58766 0.57907 0.57266 0.56835 0.56608 0.56583 0.56754 0.57121 0.57680 0.58428 0.59363 0.66489 0.75806 0.84596 0.92549 0.99211 1.03122 1.05216 1.05418 1.05424 1.05411 1.05399 1.05390 1.05383 1.05377 1.05373 1.05371 1.05369 1.05368 1.05366 1.05364 1.05362 1.05360 1.05359 1.05357

)0.38747 )0.42659 )0.45956 )0.48735 )0.51086 )0.53085 )0.56263 )0.58643 )0.60467 )0.61888 )0.63004 )0.63468 )0.63877 )0.64236 )0.64548 )0.64815 )0.65038 )0.65219 )0.65357 )0.65451 )0.65501 )0.64985 )0.62707 )0.57503 )0.47909 )0.34403 )0.21361 )0.07087 )0.02665 )0.01282 )0.00757 )0.00504 )0.00357 )0.00263 )0.00199 )0.00154 )0.00136 )0.00121 )0.00107 )0.00096 )0.00078 )0.00063 )0.00052 )0.00043 )0.00031

)0.25323 )0.24784 )0.22050 )0.05800 0.35359 0.45431 0.54581 0.60405 0.64422 0.67046 0.68543 0.68936 0.69120 0.69112 0.68926 0.68572 0.68056 0.67382 0.66549 0.65558 0.64403 0.56055 0.44094 0.32044 0.22955 0.16898 0.12915 0.08344 0.05967 0.04590 0.03738 0.03204 0.02901 0.02822 0.03091 0.04424 0.07012 0.19815 0.30793 0.34320 0.40403 0.41883 0.42080 0.42112 0.42122

)0.21673 )0.30428 )0.39308 )0.49768 )0.41408 )0.36515 )0.34898 )0.35118 )0.35570 )0.35852 )0.35890 )0.35838 )0.35754 )0.35649 )0.35533 )0.35410 )0.35284 )0.35154 )0.35020 )0.34877 )0.34723 )0.33672 )0.32303 )0.31172 )0.30502 )0.30194 )0.30093 )0.30117 )0.30181 )0.30229 )0.30282 )0.30368 )0.30517 )0.30748 )0.31028 )0.31107 )0.30821 )0.29984 )0.28058 )0.24215 )0.11765 )0.04250 )0.01517 )0.00405 0.00474

)0.14210 )0.12271 )0.11213 )0.10973 )0.10399 )0.09590 )0.08476 )0.08174 )0.08234 )0.08269 )0.08053 )0.07819 )0.07494 )0.07078 )0.06572 )0.05978 )0.05300 )0.04543 )0.03713 )0.02817 )0.01868 0.03155 0.07159 0.09303 0.10011 0.09883 0.09338 0.07893 0.06600 0.05624 0.04920 0.04406 0.04019 0.03722 0.03480 0.03278 0.03188 0.03103 0.03021 0.02943 0.02793 0.02656 0.02537 0.02437 0.02271

)0.22113 )0.25275 )0.25631 )0.23899 )0.21514 )0.20227 )0.20434 )0.20805 )0.19400 )0.16760 )0.13770 )0.12297 )0.10876 )0.09511 )0.08203 )0.06944 )0.05726 )0.04540 )0.03375 )0.02224 )0.01078 0.04743 0.10793 0.16588 0.21405 0.24890 0.27152 0.29234 0.29705 0.29622 0.29401 0.29164 0.28911 0.28607 0.28248 0.27978 0.27768 0.21880 0.05937 0.02614 0.00964 0.00473 0.00262 0.00154 0.00059

Table 2 Overlaps hn; ljWi of Heitler–London jn; l; 0i and the present variational molecular functions at R ¼ 50:0 State

n

l1 a

hn; 0jWi

hn; 1jWi

hn; 2jWi

hn; 3jWi

X B B0 B00 B 41 Ru 51 Ru 61 Ru

1 2 2 3 3 3 4

0 1 0 1 2 0 ?

1.0000 )0.0003 )1.0000 )0.0303 )0.7051 0.7084 )0.0116

1.0000 )0.0003 0.9985 0.0112 0.0538 0.0171

0.0459 )0.7090 )0.7037 )0.0153

0.0090

a

l1 is the asymptotic limit according to [7].

These data are also displayed in Fig. 3 to demonstrate the exchange of characters between the two states. At the point of avoided crossing the correction be-

comes huge amounting to 108 cm1 at R ¼ 0:564. A similar picture is obtained in the case of the B5 and B6 pair of states.

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Table 3 Expectation values, hWu j  DR jWu i, for the third and fourth 1 Ru states in a.u R

B00 B

41 Ru

0.500 0.550 0.560 0.562 0.564 0.566 0.568 0.570 0.580 0.590 0.600

48.2 41.0 234.9 3559.8 1024154.9 1724.7 140.8 34.6 6.7 6.0 5.8

8.2 8.0 203.2 3528.9 1024167.7 1755.3 171.7 65.3 36.4 34.7 33.5

Fig. 3. Expectation values, hWu jDR jWu i, at avoided crossing.

Obviously, there arises the question concerning accuracy of the computed moments. In various test computations with slightly less accurate functions—leading to energy shifts of about 0:05 cm1 —we found very small changes in some of the moments. However, these changes did not exceed 3 105 . In this sense our results are stable but we are unable to estimate the actual error. 2.2. Comparison with earlier results The best published results for the B–X transition are those of [1] where transition moments are given for 0 6 R 6 12. Naturally, the wavefunctions used in [1] were definitely less accurate than those used in the present work. Still, except for R ¼ 12, where the error is larger, the difference between the old and present results never exceeds 2 in the last decimal figure given in [1], i.e., 2 104 . The best existing B0 –X transition moments were given by Ford et al. [2] almost thirty years ago. The authors estimated their accuracy at about a few percent. And indeed their results differ from the present results by not more than a few percent. The next three higher transitions were studied relatively recently by Drira [3]. These results have nonuniform accuracy and in some cases are, in our opinion, incorrect. This is probably due to an erroneous assignment of the asymptotic configurations of the electronic states. We compare our results with those of [3] in Table 4. For the B00 B–X transition the agreement is fairly good for R 6 30. For R ¼ 45 and 50 the numbers given in [3] are incorrect. It seems that the B00 B–X transition moment was assigned to 41 Ru –X and vice versa. Similarly, the 41 Ru –X moments for R ¼ 1 and 1.5 given in [3] are

Table 4 Comparison of present results with [3] B00 B–X

41 Ru –X

51 Ru –X

R

jMz ja

Mz ðDÞb

Dc

jMz j

Mz ðDÞ

D

jMz j

Mz ðDÞ

D

1.0 1.5 1.8 2.0 3.0 4.0 6.0 10.0 15.0 20.0 25.0 30.0 45.0 50.0

0.2051 0.2393 0.2448 0.2395 0.1050 0.1435 0.4543 0.6440 0.1690 0.0597 0.0344 0.0282 0.4210 0.4212

0.2046 0.2388 0.2444 0.2394 0.1053 0.1434 0.4529 0.6426 0.1686 0.0602 0.0350 0.0295 0.0078 0.0023

0.0005 0.0005 0.0004 0.0001 0.0003 0.0001 0.0014 0.0014 0.0004 0.0005 0.0006 0.0013 0.4132 0.4189

0.0002 0.0010 0.0020 0.0029 0.0124 0.0412 0.3651 0.3472 0.3019 0.3018 0.3032 0.3075 0.0085 0.0047

0.1426 0.1636 0.0049 0.0068 0.0206 0.0521 0.3657 0.3476 0.3175 0.2699 0.2635 0.2876 0.4217 0.4218

0.1424 0.1626 0.0029 0.0039 0.0082 0.0109 0.0006 0.0004 0.0156 0.0319 0.0397 0.0199 0.4132 0.4171

0.1422 0.1631 0.1643 0.1587 0.0561 0.1182 0.0959 0.0187 0.0988 0.0660 0.0464 0.0372 0.0249 0.0227

0.1115 0.1224 0.1624 0.1572 0.0555 0.1192 0.0995 0.0244 0.0307 0.1618 0.2138 0.2957 0.0019 0.0012

0.0307 0.0407 0.0019 0.0015 0.0006 0.0010 0.0036 0.0057 0.0681 0.0958 0.1674 0.2585 0.0230 0.0215

Asd

1s3p

1s3s

1s3d

1s3p

1s3s

1s3d

a

Present results, absolute value. b Ref. [3]. c Absolute value of the difference. d Asymptotic configuration at dissociation limit.

L. Wolniewicz, G. Staszewska / Journal of Molecular Spectroscopy 217 (2003) 181–185

certainly incorrect but extremely close to our results for the 51 Ru –X transition. In this connection we would like to point out that in our computations the eigenvectors are computed in one diagonalization process. Therefore there is no ambiguity in the assignment of the sequence of states.

Acknowledgments We are indebted to M. Bylicki and E. Bednarz for drawing our attention to [10]. This work was supported by a Polish KBN Grant No. 7 T11F 05 20.

185

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