In this work, the PECs of six Λ-S electronic states of NH and PH are computed with the icMRCI + Q method. The first three low-lying electronic states ( X 3 Σ − , a 1 Δ and b 1 Σ + ) of NH and PH have the same electronic configuration σ²π². The electronic configurations of the excited states A 3 Π and c 1 Π are σ¹π³, which could be considered as involving a pσ → pπ transition within the N/P atom. The electronic configuration of the repulsive state Σ − 5 is σ¹π²σ^∗. The PECs of six Λ-S electronic states of NH and PH are depicted in Figures 1 and 2, respectively. As seen in Figures 1 and 2, the X 3 Σ − and Σ − 5 states of NH and PH correlate to the lowest neutral atomic N/PH ( S 4 ) + H ( S 2 ) limit, the a 1 Δ , A 3 Π and c 1 Π states correlate adiabatically to the N/PH ( D 2 ) + H ( S 2 ) limit, and the b 1 Σ + state corresponds to the N/PH ( P 2 ) + H ( S 2 ) limit. Since the spectroscopic constants of the X 3 Σ − and A 3 Π states have been measured in experiment for NH and PH, comparing with the available experimental measurements could give an indicator of the accuracy and reliability of our computations. Our calculated spectroscopic constants of five Λ-S states for NH and PH are tabulated in Tables 1 and 2, respectively, comparing with previous experimental and theoretical values.

As seen in Table 1, for the ground state X 3 Σ − of NH, our computed R _e , ω _e and ω _e χ _e values (1.035 Å, 3,283.98 and 82.46 cm⁻¹) reproduce the experimental data (1.0362 Å, 3,282.27 and 78.35 cm⁻¹) very well (Huber and Herzberg, 1979). It is also encouraging to see that our calculated D _e value of 3.6091 eV for the X 3 Σ − state of NH is in excellent agreement with the experimental result of 3.601 eV (Huber and Herzberg, 1979). Concerning the first excited state a 1 Δ of NH, our computed T _e , ω _e and ω _e χ _e values are 12,537.40, 3,191.72 and 68.05 cm⁻¹, respectively, which are in excellent accordance with the experimental data (12,566, 3,188 and 68.00 cm⁻¹) (Huber and Herzberg, 1979) and much improved compared with the previous calculations (12,529.37, 3,336.04 and 68.18 cm⁻¹) (Song et al., 2016). The calculated R _e and B _e values (1.034 Å and 16.47 cm⁻¹) of the a 1 Δ state are in excellent accordance with the measurements (1.0341 Å and 16.439 cm⁻¹) (Huber and Herzberg, 1979). Next in energy is the b 1 Σ + state of NH. Our calculated T _e value of the b 1 Σ + state (21,216.85 cm⁻¹) is in excellent agreement with the experimental data (21,202 cm⁻¹) (Huber and Herzberg, 1979) and theoretical value (21,196.42 cm⁻¹) (Song et al., 2016). The R _e and ω _e values of the b 1 Σ + state computed by us (1.034 Å and 3,354.35 cm⁻¹) are much closer to the experimental results (1.036 Å and 3352.4 cm⁻¹) compared with the previous theoretical values (1.0322 Å and 3,371.33 cm⁻¹). Besides, our computed ω _e χ _e , D _e and B _e values of the b 1 Σ + state agree well with the experimental results. The experimental T _e value of the A 3 Π state of NH is (29,818.01 cm⁻¹) (Lents, 1973), whereas our calculated T _e value is 29,824.42 cm⁻¹, which is better than the previous computational value (29,794.77 cm⁻¹) (Song et al., 2016). The R _e , ω _e , ω _e χ _e and D _e values of the A 3 Π state computed by us (1.036 Å, 3,234.88 cm⁻¹, 98.68 cm⁻¹ and 2.2989 eV) agree very well with the corresponding experimental data (1.037 Å, 3,231.2 cm⁻¹, 98.60 cm⁻¹ and 2.2875 eV) (Huber and Herzberg, 1979). For the c 1 Π state of NH, the excitation energy is calculated to be 43,783.62 cm⁻¹, noticeably higher than that obtained in the previous calculations (43,468.49 cm⁻¹) (Song et al., 2016), and thus in much better agreement with the measured value of 43,744 cm⁻¹(Huber and Herzberg, 1979). The calculated R _e , ω _e and D _e values of the c 1 Π state of NH are 1.10 Å, 2,124.40 cm⁻¹ and 0.7286 eV, respectively, which agree excellently with the experimental results (1.1106 Å, 2,122.64 cm⁻¹ and 0.7126 eV). In Figure 1, for the c 1 Π state of NH, the bump of the PEC may result from an avoided crossing between the c 1 Π state and a higher Π 1 state. The resultant potential barrier is 1,293.26 cm⁻¹ at 1.80 Å relative to the dissociation limit in this work, which is in very good agreement with the value of 1,292.12 cm⁻¹ calculated by Song et al. (2016)

In Table 2, our calculated R _e and B _e values of the X 3 Σ − state of PH are 1.422 Å and 8.5256 cm⁻¹, respectively, which agree perfectly with the experimental measurements (1.4223 Å and 8.5371 cm⁻¹) (Huber and Herzberg, 1979). The present calculated ω _e and ω _e χ _e values of the X 3 Σ − state are 2,389.89 cm⁻¹ and 46.88 cm⁻¹, respectively, which are in very good agreement with the previous theoretical results (2,392.51 cm⁻¹ and 47.5 cm⁻¹) (Gao and Gao, 2014). For the a 1 Δ state of PH, our calculated T _e value (7,326.99 cm⁻¹) is much closer to the experimental value (7,660 cm⁻¹) (Huber and Herzberg, 1979) than the old one (7,140 cm⁻¹) (Gao and Gao, 2014). The R _e , ω _e , ω _e χ _e , D _e and B _e values of the a 1 Δ state are computed to be 1.422 Å, 2,391.75 cm⁻¹, 41.48 cm⁻¹, 3.6511 eV and 8.5476 cm⁻¹, respectively, which agree very well with the corresponding theoretical results (1.422 Å, 2,390.2 cm⁻¹, 42.5 cm⁻¹, 3.65 eV and 8.5348 cm⁻¹) (Gao and Gao, 2014). The excitation energy of the present work for the b 1 Σ + state of PH is computed to be 14,223.05 cm⁻¹, which is much closer to the experimental result of 14,325.5 ± 0.1 cm⁻¹ (Droege and Engelking, 1984) than the previous calculation (14,160.5 cm⁻¹) (Gao and Gao, 2014). It is also encouraging to see that the present values of R _e and ω _e values for the b 1 Σ + state are 1.420 Å and 2,408.88 cm⁻¹, respectively, which are in excellent agreement with those derived experimentally, 1.4178 ± 0.0004 Å and 2,403.0 ± 0.1 cm⁻¹ (Droege and Engelking, 1984). In addition, the calculated value (41.15 cm⁻¹) for ω _e χ _e of the b 1 Σ + state agrees very well with the experimental value of 42.0 ± 0.1 cm⁻¹ (Droege and Engelking, 1984). Besides, the computed D _e and B _e values of the b 1 Σ + state (3.7250 eV and 8.5679 cm⁻¹) are in very good agreement with the theoretical results (3.73 eV and 8.5668 cm⁻¹) (Gao and Gao, 2014). The experimental excitation energy to the A 3 Π state of PH is 29,484 cm⁻¹ (Rostas et al., 1974), while the present value is 29,528.42 cm⁻¹, which is much improved compared with the previous theoretical value 29,348.15 cm⁻¹ (Gao and Gao, 2014). For the A 3 Π state, the agreement between our computed R _e , D _e and B _e values (1.445 Å, 0.9441 eV and 8.2883 cm⁻¹) and the theoretical data (1.448 Å, 0.92 eV and 8.2539 cm⁻¹) (Gao and Gao, 2014) is very good. There are some deviations between the calculational and experimental (Rostas et al., 1974) results for the ω _e and ω _e χ _e values of the A 3 Π state, although the experimental values were estimated based on the isotopic relation, and may have large uncertainties (Rostas et al., 1974). The experimental T _e value of the c 1 Π state of PH is 37,500 cm⁻¹(Di Stefano et al., 1978), whereas our calculated T _e value is 37,452.45 cm⁻¹, which is much better than the previous computational value of 37,267 cm⁻¹. (Gao and Gao, 2014).

The six Λ-S states X 3 Σ − , a 1 Δ , b 1 Σ + , A 3 Π , c 1 Π and Σ − 5 of NH and PH split into 12 Ω states when the SOC effects are taken into account, including three states with Ω = 0 + ( X 3 Σ 0 + − , b 1 Σ 0 + + and A 3 Π 0 + ), two states with Ω = 0 − ( A 3 Π 0 − and ⁵ Σ 0 − − ), four states with Ω = 1 ( X 3 Σ 1 − , A 3 Π 1 , c 1 Π 1 and ⁵ Σ 1 − ), and three states with Ω = 2 ( a 1 Δ 2 , A 3 Π 2 and ⁵ Σ 2 − ). The PECs of 12 Ω states of NH and PH are depicted in Figures 3 and 4, respectively. The spectroscopic constants of the 9 Ω states of NH and PH including the X 3 Σ 0 + − , X 3 Σ 1 − , a 1 Δ 2 , b 1 Σ 0 + + , A 3 Π 0 + , A 3 Π 0 − , A 3 Π 1 , A 3 Π 2 and c 1 Π 1 states are displayed in Tables 3 and 4, respectively. As seen in Table 3, the spectroscopic constants T _e , R _e , ω _e , ω _e χ _e , D _e and B _e values of the four Λ-S states X 3 Σ − , a 1 Δ , b 1 Σ + and c 1 Π of NH are nearly same to those of the corresponding Ω states. For the four Λ-S states of NH, the energy difference between the four Λ-S states and the corresponding Ω states is less than 1 cm⁻¹. While the SO splitting values of the A 3 Π 1 − A 3 Π 2 , A 3 Π 0 − − A 3 Π 1 and A 3 Π 0 + − A 3 Π 0 − states are 34.04, 34.22 and 0.17 cm⁻¹, respectively, which are in excellent accordance with the computational values (the splitting values of the A 3 Π 1 − A 3 Π 2 and A 3 Π 0 − − A 3 Π 1 states are 34.06 and 34.00 cm⁻¹, respectively) (Yan et al., 2021). In Table 4, the energy difference between the four Λ-S states ( X 3 Σ − , a 1 Δ , b 1 Σ + and c 1 Π ) and the corresponding Ω states of PH is less than 6 cm⁻¹, whereas the SO splitting values of the A 3 Π 1 − A 3 Π 2 , A 3 Π 0 − − A 3 Π 1 and A 3 Π 0 + − A 3 Π 0 − states are 100.32, 102.83 and 1.16 cm⁻¹, respectively. In view of the above, the SOC effects should be taken into account for the spectroscopic study of excited states for NH and PH and thus are important for laser cooling of NH and PH.

Accurate determination of T _e is very important for evaluating the pump and repump wavelengths in laser-cooling cycles, and our computed T _e values, which agree very well with the corresponding experimental ones, give us confidence in the subsequent investigation on molecular laser cooling of NH and PH.

Here, we discuss the effects of the extra electronic states on direct laser cooling of NH and PH. An amplified view of crossing regions of PECs of the A 3 Π and Σ − 5 states for NH and PH is depicted in Figure 5. We can see that the dissociation energies of the A 3 Π state of NH and PH are 18,541.92 and 7,614.34 (Di Stefano et al., 1978)cm⁻¹, respectively. The A 3 Π and Σ − 5 states of NH and PH have a crossing point, which can lead to nonradiative transition (Wu et al., 2019), and may cause predissociation. In the polyatomic molecule cases, this kind of electronic state crossing in a diatomic molecule will become potential energy surface intersections including multiple electronic states (Liu et al., 2003; Zhao et al., 2006). We find that the locations of crossing point between the A 3 Π and Σ − 5 states of NH and PH are higher than the corresponding ν′ = 2 vibrational levels of the A 3 Π state (4,163 and 989 cm⁻¹, respectively) indicating that the crossings with higher electronic states would not affect laser cooling. The large f 00 values of the A 3 Π 2 → X 3 Σ 1 − transition for NH and PH (NH: 0.9994; PH: 0.9675) suggest that the two molecules are promising candidates for efficient and rapid laser cooling. This conclusion can be backed up by experimentalists, since the (1, 1) band of the A 3 Π → X 3 Σ − transition for NH and PH has been observed (Funke, 1935; Fitzpatrick et al., 2002). Generally speaking, a larger atomic mass difference for the diatomic candidate is desirable by experimentalists, and in this respect, PH is a better laser cooling candidate than NH.

It should be noted that the transitions between the singlet and triplet states are allowed when the SOC effects are considered. The effects of the intermediate electronic states ( a 1 Δ 2   and   b 1 Σ 0 + + ) of NH and PH on laser cooling are discussed below. There are two intermediate electronic states a 1 Δ 2 and b 1 Σ 0 + + in the constructed laser cooling schemes for NH/PH based on the A 3 Π 2 → X 3 Σ 1 − transition, where NH/PH molecules are excited from the X 3 Σ 1 − (v = 0) state to the A 3 Π 2 (v′ = 0) state, then they may decay to the X 3 Σ 1 − or a 1 Δ 2 state rather than the b 1 Σ 0 + + state since the A 3 Π 2 → b 1 Σ 0 + + transition is forbidden according to the selection rules. So the intermediate electronic state b 1 Σ 0 + + does not interfere with the laser-cooling. In addition, the absolute transition dipole moments (TDMs) of the A 3 Π 2 → a 1 Δ 2 transition for NH and PH are shown in Supplementary Figure S1. As seen, the TDMs values of the A 3 Π 2 → a 1 Δ 2 transition for NH and PH are 0.000495 and 0.000793 debye (0.082% and 0.1169% of the corresponding A 3 Π 2 → X 3 Σ 1 − transition) at corresponding R _e . The vibrational branching loss ratios ( η i ) [ η i = A A 3 Π 2 → a 1 Δ 2 / ( A A 3 Π 2 ( v ′ = 0 ) → X 3 Σ 1 − ( v ) + A A 3 Π 2 → a 1 Δ 2 ) ] of the A 3 Π 2 → a 1 Δ 2 ( η 1 ) transition for NH and PH are extremely small (NH: 1.81 × 10^–8; PH: 1.08 × 10^–6), and much smaller than the experimental value of YO ( η (YO) < 4 × 10^–4) (Hummon et al., 2013). The extremely small vibrational branching loss ratios of the A 3 Π 2 → a 1 Δ 2 transition for NH and PH indicate that the a 1 Δ 2 intermediate electronic state will not interfere with the laser-cooling. Hence, we will construct feasible three-laser cooling schemes for NH and PH on the basis of the A 3 Π 2 → X 3 Σ 1 − transition in the next section, which satisfy all known criteria including the fourth one proposed in our recent work (Li et al., 2020).

Since the SOC effects are important as shown above, we construct the schemes for laser cooling of NH and PH using the spin-orbit states A 3 Π 2 and X 3 Σ 1 − . We find that only the A 3 Π 2 → X 3 Σ 1 − transition can ensure a closed-loop cooling cycles in the six possible transitions ( A 3 Π 2 → X 3 Σ 1 − , A 3 Π 1 → X 3 Σ 1 − , A 3 Π 1 → X 3 Σ 0 + − , A 3 Π 0 + → X 3 Σ 1 − , A 3 Π 0 + → X 3 Σ 0 + − and A 3 Π 0 − → X 3 Σ 1 − ) from the A 3 Π Ω . The A 3 Π 2 → X 3 Σ 0 + − and A 3 Π 0 − → X 3 Σ 0 + − transitions for NH and PH are forbidden according to the selection rules of transitions between the Ω states. In addition, the A 3 Π 2 state of NH and PH is the energetically lowest-lying state in the 4 Ω states ( A 3 Π 0 + , A 3 Π 0 − , A 3 Π 1 and A 3 Π 2 ), which can avoid the interference from the other A 3 Π Ω states ( A 3 Π 0 + , A 3 Π 0 − and A 3 Π 1 ) and ensure a closed-loop cooling cycles. In the constructed laser cooling schemes for NH/PH molecules based on the A 3 Π 2 → X 3 Σ 1 − transition, NH/PH molecules are excited from the X 3 Σ 1 − (v = 0) state to the A 3 Π 2 (v′ = 0) state, then they will decay to the X 3 Σ 1 − state rather than the X 3 Σ 0 + − state according to the selection rules, and the ultracold NH/PH will be produced through the constructed schemes when the process of cooling cycles repeats constantly. Consequently, the A 3 Π 2 (v') → X 3 Σ 1 − (v) transition of NH and PH is used to establish corresponding laser cooling schemes in this work.

The permanent dipole moments (PDMs) and TDMs for the A 3 Π 2 → X 3 Σ 1 − transition of NH and PH at the icMRCI + Q level are shown in Supplementary Figure S2. The TDMs of NH and PH decrease with the increasing interatomic distance and are 0.6059 and 0.6788 debye, respectively, at corresponding R _e . The FCFs ( f ν ′ ν ) values of the A 3 Π 2 → X 3 Σ 1 − transition for NH and PH are computed and plotted in Figures 6 and 7, respectively. We can clearly see that the f ν ′ ν values of Δ ν = 0 vibrational levels of the A 3 Π 2 → X 3 Σ 1 − transition for NH and PH are remarkably higher than those for the off-diagonal terms. The f 00 values of the A 3 Π 2 → X 3 Σ 1 − transition for NH (0.9994) and PH (0.9675) are so large that the spontaneous decays to ν = 1, 2 vibrational levels of the corresponding X 3 Σ 1 − state are highly restricted. We will use the v' = 0, 1 levels of the corresponding A 3 Π 2 state of NH and PH with three lasers to establish laser cooling cycles on the basis of the A 3 Π 2 → X 3 Σ 1 − transition. Owing to the relative strengths of the photon loss pathways are more directly related to the vibrational branching ratios R ν ′ ν than the f ν ′ ν in the laser cooling cycle, the Einstein spontaneous emission coefficient A ν ′ ν and R ν ′ ν of the A 3 Π 2 → X 3 Σ 1 − transition for NH and PH are calculated and presented in Tables 5 and 6, respectively. As seen, a very large A 00 (NH: 2.10×10⁶ s⁻¹, PH: 1.90×10⁶ s⁻¹) and very low scattering probabilities into off-diagonal bands of NH and PH contribute to a desirable condition for efficient and rapid optical cycles.

The R ν ′ ν are assessed using the following expression: R ν ′ ν = A ν ′ ν ∑ v A ν ′ ν (3)

In addition, the Doppler temperatures ( T D o p p l e r = h / ( 4 k B π τ ) , where h is Planck’s constant, k _B is Boltzmann’s constant, and τ is the radiative lifetime) of the A 3 Π 2 (ν′ = 0) → X 3 Σ 1 − (ν = 0) transition of NH and PH are 8.06 and 7.27 µK, respectively, the radiative lifetimes ( τ ν ′ ) for main cooling transition of NH and PH are 474 and 526 ns, respectively, and the recoil temperatures ( T r e c o i l = h 2 / ( m k B λ 2 ) ,   where   λ   is   the   laser   wavelength ) for main cooling transition of NH and PH are 1.13 and 5.12 µK, respectively.

The constructed laser-cooling schemes for the production of ultracold NH and PH are presented in Figures 8 and 9, respectively. As seen in Figure 8, the laser for the main cycling may drive the X 3 Σ 1 − (ν = 0, J = 1) → A 3 Π 2 (ν′ = 0, J′ = 0) transition of NH at the wavelength λ 00 of 336.1 nm (here J represents the rotational quantum number). According to the angular momentum and parity selection rules, the A 3 Π 2 (J′ = 0) state can only decays to the initial X 3 Σ 1 − (J = 1) state, leading to the elimination of the rotational branching. In addition, another two lasers of 382.8 and 382.6 nm are used to recover the molecules falling to the X 3 Σ 1 − (ν = 1, 2) states of NH, further reducing the vibrational branching loss. So quasi-closed optical cycling can be achieved by using the scheme shown in Figure 8. Similarly, in Figure 9, the constructed scheme for PH take the X 3 Σ 1 − (ν = 0, J = 1) → A 3 Π 2 (ν′ = 0, J′ = 0) transition as the main pump, the X 3 Σ 1 − (v = 1) → A 3 Π 2 (v′ = 0) and X 3 Σ 1 − (v = 2) → A 3 Π 2 (ν′ = 1) transitions as the first and second vibrational repump, respectively. The computed pump and repump wavelengths λ 00 , λ 01 and λ 12 are 341.9, 370.8 and 375.4 nm, respectively, which are all in the range of ultraviolet A (320 ∼ 400 nm) and can be produced with the frequency doubled Ti: sapphire semiconductor laser (Xing et al., 2018). The large R 00 values of NH (0.9952) and PH (0.9977) suggest that the A 3 Π 2 (ν′ = 0) → X 3 Σ 1 − (ν = 0) transition of NH and PH has the largest possibilities, and the vibrational branching loss can be addressed through a reasonable laser cooling cycle process. The off-diagonal R ν ′ ν of NH and PH have also been computed, and we use R 03 + (here 3 + means ν ≥ 3) to evaluate the possibilities of unwanted decay channels for NH and PH. The negligible values of 9.64 × 10^–6 (NH) and 1.20 × 10^–7 (PH) mean that NH and PH can scatter at least 1.04 × 10⁵ (NH) and 8.32 × 10⁶ (PH) photons on average using the present schemes, respectively, which are enough to decelerate NH and PH in a cryogenic beam, in principle (Shuman et al., 2010).

After initial cooling and trapping stages, evaporative cooling is often used to bring molecules to quantum degeneracy or Bose-Einstein condensation. The possibility of evaporative cooling of NH has been investigated (Janssen et al., 2011; Janssen et al., 2013), however, recent accurate quantum calculations (Janssen et al., 2013) indicate that chemical reactions can cause more trap loss than inelastic NH + NH collisions, and evaporative cooling is not favorable for NH. As mentioned above, the laser cooling scheme constructed here allows for 1.04 × 10⁵ photons scattered for NH, which are sufficient for cooling to µK temperatures. In addition, PH seems to be a better candidate than NH for laser cooling. So the present work indicates that the direct laser cooling method can be used to produce magnetically trapped ultracold NH/PH molecules, and it is expected that the subsequent evaporative cooling can be avoided.