In the following, the equilibrium distances and total energies for the states 1σ _{g, u}, 1π _{g, u}, 1δ _{g, u}, and 2σ _g of for 10⁹  Gs≤ B ≤ 4.414 × 10¹³  Gs are given and compared to other results in several tables. In calculations, we fix the parameter u_max = 40 and the number of B-spline basis along the u-axis N_u = 60. So in fact, only two parameters, α in Eq.  (6) and the number of B-spline basis along the v-axis N_v need to be adjusted. The detailed numbers N_u and N_v of the B-spline basis for every state in different magnetic fields have been listed in the corresponding tables. It is worth mentioning that the calculation time is significantly reduced compared to that taken by the one-center method in Ref.  [1]. For example, for the 1σ _g state at B = 4.414 × 10¹³  Gs in Table  1, only 4 minutes are needed in the present paper, while one and a half hours were used in Ref.  [1]. Even for the 2σ _g state at B = 4.414 × 10¹³  Gs in Table  7, the number of the basis N_u × N_v = 60 × 700 is the largest in the present paper, about 40 minutes are needed.

3.1.1 1σ _g state (m = 0, even parity)

Table  1 lists the equilibrium distances R_eq and total energies E_T for the ground state 1σ _g of in different magnetic fields. Compared with the data given by Guan in Refs.  [3] and [24] in fields with strengths below 2.35 × 10¹⁰  Gs, the equilibrium distances and total energies obtained in the present method are in close agreement with the last digit listed by Guan. In Comparison with the results at B = 1, 10, 100, 1000  a.u. given by Vincke in Ref.  [8], which are the most accurate found in the literature, the equilibrium distances are consistent with each other at least 5  s.d. and the total energies at least 11  s.d. Compared with previous results obtained by the one-center method in Ref.  [1], the precision of the total energies is improved obviously, though the computation time is reduced largely. That shows the spheroidal coordinates are more suitable than the spherical coordinates for studying in all the field regions from 10⁹  Gs to 4.414 × 10¹³  Gs. The accuracy of the equilibrium distances is estimated at least about 4  s.d. and that of the total energies about 7∼ 11  s.d. The case is the same as other literatures, the equilibrium distances decrease and the total energies increase as the magnetic field strength increases. This means that the states are increasingly stable as the strength of the magnetic field increases.

Table 1.

Table 1.

Table 1. The equilibrium distances Req and total energies ET for the ground state 1σ g of in different magnetic fields, are here compared to those calculated using other methods. B/Gs Nu Nv 109 60 25 1.9234 − 1.150717897 50 30 1.9234[1] − 1.15071723[1] 1.923[12, 13] − 1.15070[12, 13] 1.924[2] − 1.150718[2] 1.9233[25] − 1.15072[25] 1.9234[24] − 1.150717898[24] 2.35 × 109 60 25 1.7520828 − 0.94997649055 110 85 1.7520[1] − 0.94997644[1] 1.752[12, 13] − 0.94992[12, 13] 1.76[6] − 0.94642[6] 1.7521[3] − 0.94997649[3] 1.7520838[8] − 0.94997649055[8] 1010 60 25 1.2464 1.090302124 50 30 1.2465[1] 1.0903028[1] 1.246[12, 13] 1.09044[12, 13] 1.246[2] 1.090308[2] 1.2464[25] 1.08979[25] 1.2464[24] 1.090302124[24] 2.35 × 1010 60 25 0.9570223 5.650027927964 80 50 0.9570[1] 5.6500282[1] 0.957 ± 0.002[12, 13] 5.65024[12, 13] 0.950[2] 5.65[2] 0.958[26] 5.65[26] 0.957[3] 5.650028[3] 0.9570236[8] 5.650027927964[8] 1011 60 25 0.5933 35.042574 80 50 0.5917[1] 35.042670[1] 0.593 ± 0.001[12, 13] 35.0432[12, 13] 0.593[2] 35.0428[2] 2.35 × 1011 60 50 0.447793 89.707837077 80 50 0.4472[1] 89.707803[1] 0.448 ± 0.001[12, 13] 89.7090[12, 13] 0.446[2] 89.7108[2] 0.448[26] 89.720[26] 0.447794[8] 89.707837077[8] 1012 60 50 0.2834 408.387298 80 50 0.2832[1] 408.38719[1] 0.283 ± 0.001[12, 13] 408.3894[12, 13] 0.28[27] 408.4731[27] 2.35 × 1012 60 100 0.219783 977.21833022 80 50 0.2198[1] 977.21835[1] 0.2197 ± 0.0005[12, 13] 977.2214[12, 13] 0.219783[8] 977.21833022[8] 1013 60 100 0.1472 4219.5588 160 100 0.1472[1] 4219.559[1] 0.1472 ± 0.0002[12, 13] 4219.565[12, 13] 0.15[27] 4219.58[27] 2.35 × 1013 60 200 0.1183 9954.1948 160 100 0.1183[1] 9954.195[1] 0.1183 ± 0.0002[12, 13] 9954.203[12, 13] 4.414 × 1013 60 200 0.1016 18728.4675 160 100 0.1016[1] 18728.468[1] 0.1016 ± 0.0002[12, 13] 18728.477[12, 13]

Table 1. The equilibrium distances Req and total energies ET for the ground state 1σ g of in different magnetic fields, are here compared to those calculated using other methods.

3.1.2 1σ _u state (m = 0, odd parity)

As shown in Table  1, the equilibrium distances R_eq and the total energies E_T for the 1σ _u state in magnetic fields with different strengths are listed in Table  2. Most of the calculations on the 1σ _u state are concentrated on the domain B ≤ 10¹⁰  Gs as shown in Table  2. For the cases of B > 10¹⁰  Gs, there are two calculations, i.e., one by Turbiner^{[12, 13]} and the other by us using the one-center method, ^[1] available. In the same way as in the case of 1σ _g, the calculated equilibrium distances and total energies of the state 1σ _u in the present method are closer to the data listed by Guan^{[3, 24]} for 10⁹  Gs≤ B ≤ 10¹⁰  Gs and Vincke at B = 2.35 × 10⁹  Gs than those obtained in the one-center method.^[1] Results in the present method agree with the last digit given by Guan and Vincke. For 2.35 × 10¹⁰  Gs≤ B ≤ 10¹³  Gs, the results are closer to the data obtained by us in the one-center method^[1] than Turbiner, ^[11] including the equilibrium distance and the total energy at B = 2.35 × 10¹⁰  Gs, where there is a larger difference between them. For the equilibrium distance from B = 2.35 × 10¹³  Gs, our data 2.143 is closer in agreement with 2.134 given by Turbiner than 2.02 by the one-center method. Also for B = 4.414 × 10¹³  Gs, there are no data available using the one-center method. However, the results of the present method are in good agreement with Turbiner’ s; this shows that this method is more suitable for than the one-center method even at magnetic field strengths reaching 4.414 × 10¹³  Gs. The accuracy of the equilibrium distances is estimated at about 2∼ 5  s.d. and the total energies about 7∼ 11  s.d.

Table 2.

Table 2.

Table 2. Equilibrium distances Req and total energies ET for the excited state 1σ u. B/Gs Nu Nv 109 60 25 11.097 − 0.921096088 240 140 11.074[1] − 0.92108946[1] 11.19[12, 13] − 0.92103[12, 13] 11.097[24] − 0.92109608[24] 11.039[25] − 0.92063[25] 2.35 × 109 60 25 9.58768 − 0.66287291463 240 140 9.580[1] − 0.6628674[1] 9.73[12, 13] − 0.66271[12, 13] 9.588[3] − 0.66287292[3] 9.6[6] − 0.66[6] 9.58768[8] − 0.66287291463[8] 1010 60 25 7.1392 1.638944317 240 140 7.1357[1] 1.6389499[1] 7.18[12, 13] 1.63989[12, 13] 7.139[24] 1.63894432[24] 2.35 × 1010 60 25 6.0217 6.5027272 240 120 6.0192[1] 6.5027344[1] 6.336[12, 13] 6.52362[12, 13] 1011 60 50 4.6163 36.83148 280 100 4.6123[1] 36.8315[1] 4.629[12, 13] 36.8367[12, 13] 2.35 × 1011 60 50 4.001 92.41778 320 100 3.996[1] 92.4178[1] 3.976[12, 13] 92.4257[12, 13] 1012 60 100 3.208 413.60514 280 120 3.2108[1] 413.605[1] 3.209[12, 13] 413.6175[12, 13] 2.35 × 1012 60 100 2.848 984.6719 350 120 2.835[1] 984.672[1] 2.862[12, 13] 984.6852[12, 13] 1013 60 200 2.368 4232.5419 350 120 2.344[1] 4232.54[1] 2.360[12, 13] 4232.554[12, 13] 2.35 × 1013 60 400 2.143 9971.7143 240 155 2.02[1] 9971.72[1] 2.134[12, 13] 9971.727[12, 13] 4.414 × 1013 60 400 1.998 18750.0521 2.021[12, 13] 18750.07[12, 13]

Table 2. Equilibrium distances Req and total energies ET for the excited state 1σ u.

Comparing the numbers of the basis (N_u × N_v) for 1σ _u and 1σ _g, the amount of computation for 1σ _u increases obviously; this may refer to the antibonding behavior of the 1σ _u state. It can also explain why less works are found for the 1σ _u state than the 1σ _g state. Table  2 also shows the equilibrium distances R_eq decreasing and total energies E_T of the 1σ _u state increasing as the magnetic field grows from 10⁹  Gs to 4.414 × 10¹³  Gs.

3.1.4 1π _g state (m = − 1, odd parity)

In Table  4, the results of the antibonding state 1π _g in different literatures are listed. The equilibrium distances of the 1π _g were found to be larger than those of the antibonding state 1σ _u. As shown in Table  4, most works focus on the case for B = 2.35 × 10⁹  Gs. Compared with the high-precision results given by Vincke in Ref.  [8], the equilibrium distance is in agreement within 5  s.d. and the total energy within 11  s.d. For B = 10⁹  Gs, the equilibrium distances listed in Ref.  [1] and Refs.  [12] and [13] are 18.08 and 20.10, respectively. The result in the present paper is in good agreement with that using the one-center method in Ref.  [1]. For 10¹⁰  Gs≤ B ≤ 2.35 × 10¹²  Gs, two sets of data, i.e., ours using the one-center method and Turbiner’ s, can be used to compare. The equilibrium distances and the total energies in the present paper are closer to those obtained by the one-center method. The accuracy of the equilibrium distance is estimated at about 3∼ 4  s.d. and the total energy at about 7∼ 8  s.d. For 10¹³  Gs≤ B ≤ 4.414 × 10¹³  Gs, only Turbiner’ s data are available. They are also closely consistent with ours.

Table 3.

Table 3.

Table 3. Equilibrium distances Req and total energies ET for the excited state 1π u. B/Gs Nu Nv 109 60 25 4.9375 − 0.293629861 50 30 4.9378[1] − 0.29362986[1] 4.940[12, 13] − 0.293592[12, 13] 4.9375[24] − 0.293629862[24] 2.35 × 109 60 25 3.677025 − 0.0202355799557 110 85 3.67695[1] − 0.020235580[1] 3.676[12, 13] − 0.020150[12, 13] 3.6769[3] − 0.02023558[3] 3.68[6] − 0.02014[6] 3.677022[8] − 0.020235579956[8] 1010 60 25 2.1271 2.37079164 50 30 2.127[1] 2.3707916[1] 2.130[12, 13] 2.371845[12, 13] 2.35 × 1010 60 25 1.5306 7.29525174 80 50 1.5311[1] 7.2952518[1] 1.526[12, 13] 7.29682[12, 13] 1.510[2] 7.2954[2] 1.510[28] 7.3138[28] 1011 60 25 0.8864 37.6397806 80 50 0.8869[1] 37.639779[1] 0.887[12, 13] 37.6490[12, 13] 2.35 × 1011 60 50 0.6502 93.104752 80 50 0.6500[1] 93.104749[1] 0.651[12, 13] 93.1127[12, 13] 0.645[2] 93.1452[2] 0.645[28] 93.1226[28] 1012 60 50 0.3954 413.62475 80 50 0.3954[1] 413.62476[1] 0.395[12, 13] 413.6306[12, 13] 2.35 × 1012 60 100 0.3002 983.86085 80 50 0.3003[1] 983.86088[1] 0.301[12, 13] 983.874[12, 13] 1013 60 100 0.1946 4229.164 160 100 0.1946[1] 4229.164[1] 0.195[12, 13] 4229.183[12, 13] 2.35 × 1013 60 100 0.1536 9965.896 160 100 0.1536[1] 9965.896[1] 9965.932[12, 13] 4.414 × 1013 60 100 0.1302 18741.878 160 100 0.1302[1] 18741.88[1] 0.130[12, 13] 18741.89[12, 13]

Table 3. Equilibrium distances Req and total energies ET for the excited state 1π u.

3.1.5 1δ _g state (m = − 2, even parity)

In Table  5, the equilibrium distances R_eq and total energies E_T for the excited state 1δ _g in strong magnetic fields are compared with the results in other literatures. Two sets of results, i.e., Turbiner’ s and ours obtained by using the one-centermethod, can be found in various magnetic fields. They agree with each other very well, especially the present results and those obtained by the one-center method, where the equilibrium distances are consistent within 3∼ 5  s.d. and the total energies within 6∼ 9  s.d., except the case for B = 2.35 × 10¹²  Gs. For the equilibrium distance at B = 2.35 × 10¹²  Gs, the present data 0.3533 is closer to Turbiner’ s 0.353 in Refs.  [12] and [13] than 0.3522 obtained by the one-center method in Ref.  [1].

Table 4.

Table 4.

Table 4. Equilibrium distances Req and total energies ET for the excited state 1π g. B/Gs Nu Nv 109 60 25 18.08 − 0.23229265 160 100 18.08[1] − 0.23229266[1] 20.10[12, 13] − 0.232060[12, 13] 2.35 × 109 60 40 13.6573 0.086399699276 160 140 13.657[1] 0.086399692[1] 14.05[12, 13] 0.086868[12, 13] 13.66[3] 0.0863997[3] 13.535[29] 0.101692[29] 13.5[6] 0.0866[6] 13.6572[8] 0.086399699286[8] 1010 60 50 9.256 2.6398330 160 140 9.257[1] 2.6398330[1] 9.370[12, 13] 2.641122[12, 13] 2.35 × 1010 60 50 7.5759 7.7476346 240 120 7.577[1] 7.7476343[1] 7.622[12, 13] 7.749819[12, 13] 1011 60 50 5.605 38.672041 240 120 5.605[1] 38.672041[1] 5.622[12, 13] 38.67642[12, 13] 2.35 × 1011 60 100 4.7808 94.728054 240 140 4.781[1] 94.728053[1] 4.791[12, 13] 94.73386[12, 13] 1012 60 100 3.747 416.927258 240 140 3.749[1] 416.927253[1] 3.767[12, 13] 416.9354[12, 13] 2.35 × 1012 60 200 3.288 988.72012 300 140 3.292[1] 988.72010[1] 3.321[12, 13] 988.7286[12, 13] 1013 60 200 2.6895 4238.030 2.708[12, 13] 4238.038[12, 13] 2.35 × 1013 60 300 2.413 9978.166 2.420[12, 13] 9978.175[12, 13] 4.414 × 1013 60 400 2.219 18757.264 2.237[12, 13] 18757.273[12, 13]

Table 4. Equilibrium distances Req and total energies ET for the excited state 1π g.

We compared the energies of the 1δ _g state to those of the 1σ _u state and 1π _g state. Two energy crossings are found, one is between the states 1δ _g and 1π _g in the range from 2.35 × 10¹⁰  Gs to 10¹¹  Gs, and the other between the states 1δ _g and 1σ _u in the range from 10¹³  Gs to 2.35 × 10¹³  Gs.

3.1.6 1δ _u state (m = − 2, odd parity)

This is another antibonding state of the in the free-field case. The results in different magnetic fields are listed in Table  6. For B = 2.35 × 10⁹  Gs, the present equilibrium distance and the total energy are in prefect agreement with those given by the one-center method in Ref.  [1] and Vincke in Ref.  [8]. It means the present method is also very suitable for the case of the large equilibrium distance. For 10⁹  Gs≤ B ≤ 2.35 × 10¹²  Gs, the present results have better accordance with those obtained by the one-center method in Ref.  [1] than those given by Turbiner in Ref.  [12] and [13]. The equilibrium distance with at least four effective figures and the energy with at least nine effective figures were consistent with the results in Ref.  [1]. However, for 10¹³  Gs≤ B ≤ 4.414 × 10¹³  Gs, the equilibrium distances have large differences from each other. The present results are larger than in another three papers.^{[1, 12, 13]}

Table 5.

Table 5.

Table 5. Equilibrium distances Req and total energies ET for the excited state 1δ g. B/Gs Nu Nv 109 60 25 6.868 − 0.108068323 50 30 6.868[1] − 0.108068323[1] 6.865[12, 13] − 0.107945[12, 13] 2.35 × 109 60 25 4.873327 0.22109797676 110 85 4.8733[1] 0.221097977[1] 4.872[12, 13] 0.221163[12, 13] 4.87[6] 0.22112[6] 4.873330[8] 0.22109797675[8] 1010 60 25 2.696 2.77474027 50 30 2.696[1] 2.77474028[1] 2.694[12, 13] 2.77538[12, 13] 2.35 × 1010 60 25 1.9089 7.84994905 80 50 1.9089[1] 7.84994907[1] 1.907[12, 13] 7.85113[12, 13] 1.880[2] 7.8504[2] 1.880[28] 7.8694[28] 1011 60 50 1.0825 38.581488 80 50 1.0827[1] 38.581488[1] 1.080[12, 13] 38.58470[12, 13] 2.35 × 1011 60 50 0.7857 94.3746395 80 50 0.7858[1] 94.374639[1] 0.782[12, 13] 94.38093[12, 13] 0.778[2] 94.490[2] 0.778[28] 94.3942[28] 1012 60 100 0.4697 415.66604 80 50 0.4695[1] 415.66628[1] 0.470[12, 13] 415.6710[12, 13] 2.35 × 1012 60 100 0.3533 986.50466 80 50 0.3522[1] 986.50928[1] 0.353[12, 13] 986.5119[12, 13] 1013 60 100 0.2254 4233.109 160 100 0.2254[1] 4233.109[1] 0.225[12, 13] 4233.125[12, 13] 2.35 × 1013 60 200 0.1763 9970.781 160 100 0.1763[1] 9970.781[1] 0.176[12, 13] 9970.802[12, 13] 4.414 × 1013 60 200 0.1485 18747.539 160 100 0.1485[1] 18747.540[1] 0.148[12, 13] 18747.572[12, 13]

Table 5. Equilibrium distances Req and total energies ET for the excited state 1δ g.

Table 6.

Table 6.

Table 6. Equilibrium distances Req and total energies ET for the excited state 1δ u. B/Gs Nu Nv 109 60 50 22.073 − 0.068992624 50 30 22.073[1] − 0.068992624[1] 23.902[12, 13] − 0.06873[12, 13] 2.35 × 109 60 50 16.031 0.2935716431 110 85 16.031[1] 0.2935716431[1] 16.377[12, 13] 0.29410[12, 13] 16.0[6] 0.2936[6] 16.0309[8] 0.29357164344[8] 1010 60 50 10.540 2.9681608 120 75 10.540[1] 2.9681608[1] 11.475[12, 13] 2.97742[12, 13] 2.35 × 1010 60 50 8.5226 8.18216588 120 75 8.5225[1] 8.18216589[1] 9.458[12, 13] 8.19892[12, 13] 1011 60 100 6.2026 39.36203771 160 75 6.2024[1] 39.36203771[1] 6.858[12, 13] 39.40596[12, 13] 2.35 × 1011 60 100 5.2485 95.621353 160 100 5.2486[1] 95.621353[1] 5.619[12, 13] 95.69542[12, 13] 1012 60 200 4.0675 418.267017 160 120 4.0672[1] 418.267017[1] 4.071[12, 13] 418.4335[12, 13] 2.35 × 1012 60 200 3.549 990.386839 200 120 3.550[1] 990.386839[1] 3.406[12, 13] 990.6416[12, 13] 1013 60 300 2.88 4240.3588 160 200 2.68[1] 4240.3596[1] 2.625[12, 13] 4240.834[12, 13] 2.35 × 1013 60 300 2.57 9980.9452 160 220 2.042[1] 9980.95742[1] 2.391[12, 13] 9981.587[12, 13] 4.414 × 1013 60 400 2.37 18760.401 200 235 1.68[1] 18760.4473[1] 2.230[12, 13] 18761.18[12, 13]

Table 6. Equilibrium distances Req and total energies ET for the excited state 1δ u.

Comparing the 1δ _u state in Table  4 and the 1π _u state in Table  3 at the same magnetic field strength, the equilibrium distances of the 1δ _u state are larger than the 1π _g state by using the present method. It is different with Turbiner’ s conclusion, where for 10¹³  Gs≤ B ≤ 4.414 × 10¹³  Gs, the equilibrium distances of the 1δ _u state are smaller than those of the 1π _g state.

3.1.7 2σ _g state (m = 0, even parity)

In Table  7, the results of the first excited state 2σ _g for m = 0 and even parity series are listed. The equilibrium distance of the 2σ _g state is larger than that of the ground state 1σ _g. Far less investigation of this state in the 1σ _g can be found in the literature. Only Turbiner and our group used the one-center method for all magnetic fields from 10⁹  Gs to 4.414 × 10¹³  Gs and Kappes for B = 2.35 × 10⁹  Gs to investigate this state. The results obtained using different methods are consistent with each other. For example, for 10⁹  Gs≤ B ≤ 10¹⁰  Gs, the present equilibrium distances coincide exactly with the results obtained by the one-center method in Ref.  [1] and the energies are in agreement with 5∼ 6  s.d. However, present equilibrium distances and total energies at B = 2.35 × 10¹³  Gs and 4.414 × 10¹³  Gs are closer to the data of Turbiner. On the other hand, the one-center method shows weakness while the magnetic field strength is very high, similar to the case of the other antibonding states 1σ _u, 1π _g, and 1δ _u.

Table 7.

Table 7.

Table 7. Equilibrium distances Req and total energies ET for the excited state 2σ g. B/Gs Nu Nv 109 60 25 7.574 − 0.1215213 50 30 7.574[1] − 0.12151935[1] 7.55[12, 13] − 0.121343[12, 13] 2.35 × 109 60 25 6.638 0.34895733 110 85 6.638[1] 0.34895744[1] 6.640[12, 13] 0.34912[12, 13] 6.64[6] 0.34928[6] 1010 60 25 5.199 3.3978627 50 30 5.199[1] 3.3978636[1] 5.2[12, 13] 3.39938[12, 13] 2.35 × 1010 60 50 4.584 9.022256 80 50 4.592[1] 9.0222549[1] 4.6[12, 13] 9.02452[12, 13] 1011 60 50 3.901 41.405316 80 50 3.899[1] 41.405312[1] 3.91[12, 13] 41.4090[12, 13] 2.35 × 1011 60 75 3.653 98.77832 80 50 3.664[1] 98.77835[1] 3.65[12, 13] 98.7822[12, 13] 1012 60 150 3.394 424.2256 80 50 3.382[1] 424.2274[1] 3.40[12, 13] 424.2277[12, 13] 2.35 × 1012 60 400 3.299 998.6609 160 140 3.299[1] 998.6609[1] 3.30[12, 13] 998.6620[12, 13] 1013 60 400 3.188 4253.941 160 220 3.188[1] 4253.941[1] 3.21[12, 13] 4253.937[12, 13] 2.35 × 1013 60 500 3.140 9998.605 160 220 3.119[1] 9998.612[1] 3.145[12, 13] 9998.608[12, 13] 4.414 × 1013 60 700 3.110 18781.572 160 220 3.070[1] 18781.614[1] 3.120[12, 13] 18781.576[12, 13]

Table 7. Equilibrium distances Req and total energies ET for the excited state 2σ g.

We compared the energies of the 2σ _g state in Table  7 to those of the 1δ _u state in Table  6 and the 1δ _g state in Table  5. The two energies were found to cross within the range 10⁹  Gs< B < 2.35 × 10⁹  Gs.

To analyze the influence of the different magnetic fields strengths on the ion, the PECs of the 1σ _{g, u}, 1π _{g, u}, and 1δ _{g, u} states in different fields strengths are compared in Fig.  2#cod#x2013; Fig.  4. For every state, the PECs for B = 1, 10, 100, 1000  a.u. are chosen from bottom to top. There solid lines depict the PECs of the bound states 1σ _g, 1π _u, and 1δ _g in different magnetic fields strengths, and dotted lines depict the antibonding states 1σ _u, 1π _g, and 1δ _u. As shown in Fig.  2#cod#x2013; Fig.  4, with increasing nuclear distance R, there is an overlap of the PECs between the bound state and the matching antibonding state with the same magnetic quantum number m. For example, in Fig.  2, the energies of the states 1σ _g and 1σ _u equal the energies of the state 1s of the hydrogen atom in corresponding fields strengths when the nuclear distance R → ∞ .

Fig.  2.

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	Fig.  2. The Pecs of the states 1σ g, u of the with B = 1, 10, 100, 1000  a.u.

Fig.  3.

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	Fig.  3. The Pecs of the states 1π g, u of the in different magnetic fields strengths.

Fig.  4.

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	Fig.  4. The Pecs of the states 1δ g, u of the in different magnetic fields strengths.

Let us first consider the PECs of the bound states, that is, 1σ _g, 1π _u, and 1δ _g in different magnetic field strengths. A common feature can be easily found. With increasing field strength, the more pronounced potential well located corresponding equilibrium distance is shown. It means stronger attraction exists with increasing magnetic field strength. The depths of the potential well are also given quantitatively by the dissociation energies of Table  8. As expected, for every bound state, the dissociation energies increase with the fields strengths from 1  a.u. to 1000  a.u. If we estimate the influence of the magnetic field on the state according to the ratio of the dissociation energies E_D(B)/E_D(B = 1  a.u.), they are 2.97, 9.43, 25.9 for the state 1σ _g in B = 10, 100, 1000  a.u., respectively. The ratios are 4.24, 15.2, and 45.4 for 1π _u and 4.58, 17.2, and 53.3 for 1δ _g. For the same magnetic field strength, the ratios of the three states 1σ _g, 1π _u, and 1δ _g increase, such as for B = 1000  a.u., 25.9 < 45.4 < 53.3. It shows the influence of the magnetic field increases with | m| from 0 to 2.

Next, let us take a look at the influence of the magnetic field on the antibonding states 1σ _u, 1π _g, and 1δ _u. As depicted in dotted lines in Fig.  2#cod#x2013; Fig.  4, a shallow potential well exists in the PECs. The depths of the potential wells, that is, the dissociation energies of Table  8, are the order of magnitude of 10^{− 4} for B = 1  a.u. and 10^{− 3} for B > 1  a.u. Same as the case of the bound states, dissociation energies of every antibonding state increase with the increasing magnetic field strength from 1  a.u. to 1000  a.u. The ratios of the dissociation energies E_D(B)/E_D(B = 1  a.u.) are 3.13, 4.88, and 6.17 for 1σ _u state in B = 10, 100, 1000  a.u., respectively, and they are 3.74, 5.97, 7.48 for 1π _g and 4.23, 6.96, 8.85 for 1δ _u. In the same way as in the case of the bound states, the ratios increase with increasing | m| for the same magnetic field strength. It means the same magnetic field strength has a greater influence on the state with larger | m| . For the case of the states with the same m and B, the radios are 25.9 and 6.17 for 1σ _g and 1σ _u with B = 1000  a.u., respectively. It is not difficult to find that the ratio of the antibonding state is much smaller than that of the corresponding bound state. The conclusion can be drawn that the same change of magnetic field strength has a smaller influence on the dissociation energies of the antibonding states than the corresponding bound states with the same m.

However, as we all known, existence of the potential well is not a criteria to judge whether a molecular bond is stable with respect to dissociation . Only if a vibrational state can exist in the potential well, the molecular bond is stable. To the best of our knowledge, the vibrational energies are referred to in some works. For example, vibrational energies for 1σ _g with 10⁹  Gs≤ B ≤ 4.414 × 10¹³  Gs are given in Refs.  [12] and [13], and the vibrational frequencies for 1σ _g, 1π _u and 1δ _g with B ≤ 10⁴  a.u. are calculated based on the one-center method in Ref.  [30]. Comparing dissociation energies in Table  8 and the vibrational energies in Refs.  [12], [13], and [30], we can find that the dissociation energies are larger than the corresponding vibrational energies. In every state, the differences between these two energies are larger and larger with the increasing magnetic fields. For example, for the state 1σ _g in B = 1  a.u., dissociation energy 0.28763869708 in Table  8 is larger than the vibrational energy 0.014 in Refs.  [12], [13], and 0.01328 in Ref.  [30]. It means that some vibrational levels can exist in the potential well of the 1σ _g for B = 1  a.u. and the state 1σ _g is stable. For the state 1σ _g for B = 1000  a.u., the dissociation energy is 7.45682330, and the vibrational energy is 0.390 in Refs.  [12], [13], and 0.3588 in Ref.  [30], respectively. Obviously, the difference between these two energies for B = 1000  a.u. is much larger than that for B = 1  a.u. It means the state is more stable in stronger fields. However, no vibrational energy of the antibonding state in different magnetic fields is given because of the high accuracy, which is hard to achieve. The calculation of the vibrational energy is out of this article’ s scope; therefore, the stability of the antibonding states in strong fields cannot efficiently be discussed in the present paper.

Table 8.

Table 8.

Table 8. Dissociation energies ED of the states 1σ g, u, 1π g, u, and 1δ g, u of with B = 1, 10, 100, 1000  a.u. In the calculation, , and ET(H) is the energy of the hydrogen atom in corresponding magnetic field. The data in Ref.  [31] is used in the present paper. in different magnetic fields, are compared here to those calculated using other methods. B/Gs B/a.u. 1σ g 1σ u 1π u 1π g 1δ g 1δ u 2.35 × 109 1 0.28763869708 5.3512116 × 10− 4 0.107041463110 4.06183878 × 10− 4 0.07280597294 3.323066 × 10− 4 2.35 × 1010 10 0.854377744616 1.6785 × 10− 3 0.45390358 1.5207 × 10− 3 0.3336214 1.40457 × 10− 3 2.35 × 1011 100 2.712554451 2.61 × 10− 3 1.625727 2.425 × 10− 3 1.2490260 2.313 × 10− 3 2.35 × 1012 1000 7.45682330 3.3 × 10− 3 4.86231 3.04 × 10− 3 3.88512 2.94 × 10− 3

Table 8. Dissociation energies ED of the states 1σ g, u, 1π g, u, and 1δ g, u of with B = 1, 10, 100, 1000  a.u. In the calculation, , and ET(H) is the energy of the hydrogen atom in corresponding magnetic field. The data in Ref.  [31] is used in the present paper. in different magnetic fields, are compared here to those calculated using other methods.

The EPDDs in the corresponding equilibrium distance R_eq in the x– z plane are plotted. The states from top to bottom are the bound states 1σ _g, 1π _u, and 1δ _g in Fig.  5, and the antibonding states 1σ _u, 1π _g, and 1δ _u in Fig.  6. The positions of the protons are labelled at the interaction points of the dotted lines along the x axis and the z axis. For every state, the density distributions for B = 1, 10, 100, 1000  a.u. are considered and compared from left to right in Fig.  5 and Fig.  6, which can better explain the change of the PECs with the magnetic fields strengths. For B = 100  a.u. and 1000  a.u., drawings of partial enlargement are placed at the upper right corner. We compare the density distributions of every state from left and right in Fig.  5 and Fig.  6, it is easy to find that the cylindrical symmetry is more obviously about the z axis with the increasing magnetic fields, and the electronic distribution is constricted in a smaller space. However, there is also different changing tendency between the bound states and the antibonding states.

First, let us discuss the change of the density distributions of the bound state 1σ _g with the magnetic fields strengths. For B = 1  a.u. and 10  a.u., the electron is found with a great probability in the region around the two nuclei. When the | z | value between the two nuclei gets small, the probability becomes less and less and a saddle point appears at z = 0. When B is added to 100  a.u. and 1000  a.u., the saddle point disappears and a large density distribution of the electron occurs in the region between the two nuclei. The density distributions of the states 1π _u and 1δ _g are similar. A symmetrical density distribution occurs on both sides of the z axis for B = 1  a.u. With the magnetic field increasing from 1  a.u. to 1000  a.u., the distance of two regions gets closer to the internuclear axis. So for 1π _u at B = 1000  a.u., the two regions come together. The electron can occur in the z axis with some probability. The density distribution is closer to the z axis and the increased probability between the two nuclei will screen and reduce the nucleus– nucleus repulsion. Thus, the bond length of the bound states of becomes shorter and shorter with the increasing magnetic fields, and the stronger attraction implies that the depth of the potential well will increase, as shown in the corresponding PECs.

Fig.  5.

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	Fig.  5. The electronic probability density distributions of the bound states in the plane x– z in different magnetic fields. States from top to bottom are 1σ g, 1π u, and 1δ g. The drawings of partial enlargement are placed at the upper right corner.

Fig.  6.

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	Fig.  6. The electronic probability density distributions of the antibonding states in the plane x– z in different magnetic fields. States from top to bottom are 1σ u, 1π g, and 1δ u. The drawings of partial enlargement are placed at the upper right corner.

The EPDDs of the antibonding states are different from those of the bound states. For example, the density distribution for 1σ _u in B = 1  a.u., mainly concentrate in the range of approximately − 1 < x < 1 and 4 < | z | < 6. There is a long range defined by − 4 < z < 4 between the two nuclei. No electron is found. The symmetrical distribution of the electron about the positions z = ± R_eq/2 shows the influence on the electron from the farther nucleus is a lot smaller than the nearer nucleus. With the magnetic field increasing, the distance of the two nuclei becomes shorter. Obviously the compression is the result of the effect of the magnetic field being greater than the repulsion between the two nuclei. It also shows that a stronger attraction occurs with the increasing magnetic field. However for B = 1000  a.u., the electronic distribution of positions z = ± R_eq/2 becomes asymmetric. It reflects that for B = 1000  a.u., the effect on the electron from the nucleus farther away cannot be ignored when the distance of the two nuclei is 2.848  a.u. The density distributions of the 1π _g and 1δ _u are similar. The electron mainly occurs in four regions of both sides of the z axis. With the magnetic field increasing, the region of the distribution is shrinking along the direction of parallel and perpendicular fields. For the state 1π _g in B = 1000  a.u., these four regions of distribution merge into two around the z axis. Similar to the case of the 1σ _u in B = 1000  a.u., the symmetry about the positions z = ± R_eq/2 is also destroyed because of the effect from the nucleus farther away. The similar effect is also found for the state 1δ _g in B = 100  a.u. and 1000  a.u. The more significant asymmetry of the density distribution for the states 1σ _u, 1π _g, and 1δ _u for B = 1000  a.u. shows the relative effect of the other nucleus farther away increases with larger | m| from 0 to 2.