Isotope dependence of the c 3Πu−b 3Σ u + and D 1Π u + −B′ 1Σ u + predissociation rates of molecular hydrogen

The state-specific predissociation rates of the c 3 Π u (v,N,J) state by b 3 Σ u + and the D 1 Π u + (v,J) state by the B ′ 1 Σ u + continuum of various isotopologues of molecular hydrogen have been calculated from accurate ab initio potential energy curves and electronic coupling matrix elements. Lifetimes and predissociation rates of the c 3 Π u (v,N,J) and D 1 Π u + (v,J) levels and accurate energies of the c 3 Π u − (v,N) and D 1 Π u − (v,J) levels of the isotopologues have been obtained. Significant isotope dependence of state specific predissociation rate has been found even after adjustment for Franck-Condon factors and reduced mass. The use of average electronic matrix elements of H2 for other isotopologues underestimates the c 3 Π u + − b 3 Σ u + predissociation rates of the HD, HT, D2, DT and T2 molecules by ∼12%, ∼16%, ∼30%, ∼40%, ∼52%, respectively, and the D 1 Π u + − B ′ 1 Σ u + rates of the HD, HT, D2, DT and T2 molecules by ∼10%, ∼15%, ∼26%, ∼35% and ∼45%, respectively. When compared at similar rotation, vibration and kinetic energies, the underestimation is nearly independent of the kinetic energy. The absolute value of the average electronic coupling matrix element increases with the reduced mass while that of the vibrational overlap integral decreases with the reduced mass. This accidental substantial cancellation in the D 1 Π u + − B ′ 1 Σ u + system is responsible for the experimental observation that the relative D 1 Π u + predissociation rate of two isotopologues approximately equals the squared inverse reduced mass ratio. The origin and implications of the isotope dependence of the averaged electronic coupling matrix elements are discussed.


Introduction
Predissociation takes place when a molecule is excited to a state which is bound in zeroth order but becomes quasi-bound because of coupling with a continuum state. A predissociation, in effect, is an internal conversion from a bound state to a (quasi) continuum state. Like dissociation, predissociation can be viewed as the second half of a full collision in chemical dynamics. Because some quasi-bound levels are sufficiently narrow, predissociation can offer opportunities to experimentally investigate the second half collision at the statespecific level. The predissociation rate at a vibrational level can also be enhanced or suppressed by the interference between two multi-photon processes when each is driven by a different laser beam (Bandrauk et al 1992, Shapiro andBrumer 2003). Many physical properties of a few low electronic states of H 2 have been accurately calculated and serve to benchmark theoretical models. The small mass of molecular hydrogen 1 helps one to experimentally resolve ro-vibronic transitions and accurately measure spectroscopic parameters such as spectral shape, center frequency and line width at the rotational state-specific level. Isotope substitution in H 2 Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 In the present paper, the phrases 'hydrogen molecule' or 'molecular hydrogen' refer to H 2 and its isotopologues, such as HT and D 2 .
Similarly, 'atomic hydrogen' or 'hydrogen atom' refers to H, D or T atoms. A specific isotopic molecule or atom is identified by chemical symbols such as T 2 and H. The symbol AB is sometimes used to denote hydrogen molecule with different nuclei. The convention applied here is that the mass of nucleus A (m A ) is smaller than that of nucleus B (m B ) so that the asymmetric reduced mass, m m m m a A B B A m = -( ), is always positive. results in a very large relative change in the molecular mass, making it an ideal candidate for the investigation of isotope dependence on photon or electron excitation and dissociation rates. This paper reports on an investigation of the isotope dependence of the predissociation rates of the D u 1 P + state by the coupling of the B¢ u 1 S + continuum and the c u 3 P state by the b u 3 S + state at (v, N, J) state-specific level. Two frequently used isotope approximations introduce large errors in the predissociation rates of the D u 1 P + − B¢ u 1 S + and the c u 3 P −b u 3 S + systems. Figure 1 shows that the decay of H 2 excited to the D u 1 P and c u 3 P states takes place by very different mechanisms, depending on the orbit symmetry. The D u 1 Pstate can only be weakly predissociated by the C u 1 Pcontinuum and weakly autoionized by the X g 2 S + ionization continuum. The primary decay mechanism for the D u 1 Plevels and the D u 1 P + levels below the H(1s)+H(2s) dissociation limit is the electric dipole (E1) transition to the X g 1 S + ground state. The D u 1 P + levels above the H(1s)+H(2s) dissociation limit are strongly predissociated by electron-rotation coupling with the B¢ u 1 S + continuum. Likewise, c u 3 P + is rapidly predissociated by b u 3 S + .The c u 3 Pstate, however, can only be weakly predissociated by spin-orbit and spinspin interactions with the b u 3 S + state. Consequently, the spontaneous emissions of the c u 3 Pstate to a g 3 S + by E1, and to b u 3 S + by magnetic dipole (M1) and electric quadrupole (E2) transitions are competitive processes. The c u 3 P -(v=0) state lies below the a g 3 S + (v=0) state. As a result, the c u 3 P -(0) level is a meta-stable state, which can only decay to the b u 3 S + state by M1 and E2 transitions (Lichten 1960, Johnson 1972, Bhattacharyya and Chiu 1977, Berg and Ottinger 1994. The predissociation of the D u 1 P + state by the B¢ u 1 S + continuum is the prototype of electronic predissociation and has been investigated extensively both experimentally and theoretically. Beutler et al (1935) first found that some D X u g 1 1 P -S + absorption lines are broad and suggested that predissociation of D u 1 P is responsible. Monfils (1961) subsequently found (confirmed by Namioka 1964) that only some P and R branch lines of the D X u g 1 1 P -S + transitions are broadened while the Q branch lines are sharp. He concluded that predissociation by a nearby u 1 S + state is responsible. Comes and Schumpe (1971) found that the line shape of and D u 1 P states. All BO potentials and adiabatic corrections are based on the calculations by Staszewska andWolniewicz (1999, 2002) and Staszewska (2003a, 2003b). Energy values are relative to the v=0 and N=0 level of the X g 1 S + state. Numerical values in cm −1 refer to appropriate asymptotic limits. The asymptotic limit of the a g 3 S + , c u 3 P , B u 1 S + and C u 1 P states is H(1s)+H(2p) while that of B¢ u 1 S + is H(1s)+H(2s). The dissociation limits of both the D u 1 P and the B B u 1 ¢ S ¢ + state are H(1s)+H(3ℓ). For this reason, less precise values are given to the H(1s)+H(2ℓ) as well as the H(1s)+H(3ℓ) dissociation limits. Note the break in the vertical axis between 6.5 and 10.5 eV.
broadened H 2 and D 2 D u 1 P + − X g 1 S + transitions has an asymmetric Beulter-Fano profile, and that predissociation widths decrease with J and the mass of the molecule. Julienne (1971) and Fiquet-Fayard and Gallais (1971) demonstrated that theD u 1 P + state was predissociated by the B¢ u 1 S + continuum. The initial large disagreement between the widths calculated by those two investigations was subsequently attributed to missing factor of 4 in the Julienne (1971) calculation (Fiquet-Fayard and Gallais 1971). The measurement of Glass-Maujean et al (1979) obtained a much better agreement with the early calculations for the low v levels. Beswick and Glass-Maujean (1987), who considered the B B u 1 ¢ S ¢ + , D u 1 P + and B¢ u 1 S + coupling, and Mrugala (1988), who considered the coupling of the D u 1 P + state to the B u 1 S + , C u 1 P + and B¢ u 1 S + states, achieved better agreement with the observed width and asymmetry parameter. Dickenson et al (2010Dickenson et al ( , 2011, Dickenson and Ubachs (2012), and Glass-Maujean et al (2012c) have recently provided accurate experimental measurements of the profiles of the D X u g 1 1 P -S + transitions of H 2 , D 2 and HD using high-resolution absorption and Lyman-α dissociative emission spectroscopy. Good agreement with the measured D 2 widths and asymmetry parameters has been achieved in the D u 1 P + − B¢ u 1 S + single channel coupling calculation of Dickenson et al (2011). The multichannel quantum defect theory (MQDT) calculation of Gao et al (1993) showed that the D u 1 P + -C u 1 P + coupling is two orders of magnitude smaller than its D u 1 P + −B¢ u 1 S + counterpart. The D u 1 P + -B u 1 S + coupling is at least another order of magnitude weaker than the D u 1 P + -C u 1 P + coupling. The MQDT method, in principle, automatically includes the interaction of the D u 1 P + state with other npσ and npπ Rydberg states (Glass-Maujean et al 2012c, 2017) as well as the interference between predissociation and autoionization (Mezei et al 2014). However, all investigations have confirmed that the predissociation of the D u 1 P + state is dominated by the B¢ u 1 S + continuum. Non-adiabatic perturbations of other nearby states have an insignificant impact on the predissociation width (Glass-Maujean et al 2012c).
The couplings of the c u 3 P states to other nearby triplet-ungerade states such as e u 3 S + , f u 3 S + , and d u 3 P states are not well-known and are presumed to be unimportant. Comtet and De Bruijn (1985) calculated the c u 3 P + lifetimes of the N=1 and 2 levels for H 2 and the N=1 levels for HD and D 2 . Martin and Borondo (1988) calculated the lifetimes for the N=1 and 2 levels of the c u 3 P + (v=0-16) state of H 2 . De Bruijn et al (1984) also measured the predissociation lifetimes of several low (v, N) levels of c u 3 P + state through the accuracy of their measurement was probably over-estimated. The predissociation rates of the low N levels of the c u 3 P + (v=0) state were estimated by Chiu and Bhattacharyya (1979) and LaFleur and Chiu (1986). Kiyoshima et al (1999) investigated the rates of the predissociation of the e u 3 S + state by the b u 3 S + state and the d u 3 Pstate by the c u 3 Pcontinuum. Kiyoshima et al (2003) also calculated predissociation and autoionization of the k u 3 Pstate of H 2 and D 2 .
The two principal theoretical methods used to calculate the structures of excited electronic states of molecular hydrogen are traditional ab initio, usually for a few coupled electronic states that are fairly wellseparated, and MQDT, for the whole family of Rydberg states. Since the pioneering work of Kolos and Wolniewicz (1965), many researchers, including Kolos and Rychelwski (1990a, 1990b, 1995, Orlikowski et al (1999), Staszewska andWolniewicz (1999, 2002), Staszewska (2003a, 2003b), and Clementi (2009a, 2009b), have carried out extensive ab initio calculations of H 2 potential energy curves. Based on a large velocity difference between electrons and nuclei, these calculations attempt to obtain accurate Born-Oppenheimer (BO) potentials, adiabatic corrections, electronic dipole transition moment, and occasionally, the non-adiabatic coupling matrix elements. The X g 1 S + state, which is well isolated from other states, is a good example of the ab initio method. Its energies and nuclear wave functions can be accurately obtained from a single potential energy curve. The effect of the coupling with other states is treated as a small non-adiabatic correction (Wolniewicz 1993, 1995, Pachucki and Komasa 2015. When relativistic (Puchalski et al 2017) and radiative (Piszczatowski et al 2009, Puchalski et al 2016 corrections are considered, the calculated and measured X g 1 S + energies agree within ∼0.001 cm −1 for low (v, J) levels and ∼0.005 cm −1 for high (v, J) levels (Komasa et al 2011, Pachucki and Komasa 2010, Niu et al 2014, Trivikram et al 2016, Cozijn et al 2018. Since excited states are not well-separated, the coupled Schrödinger equation method, usually for a few close low-lying states, is used (Senn et al 1988). Abgrall et al (1993aAbgrall et al ( , 1993bAbgrall et al ( , 1993cAbgrall et al ( , 1994Abgrall et al ( , 1999Abgrall et al ( , 2000, Abgrall and Roueff (2006) and Roudjane et al (2006Roudjane et al ( , 2007Roudjane et al ( , 2008 have all used coupled equations to study the B u 1 S + , C u 1 P , B¢ u 1 S + and D X u g 1 1 P -S + band systems. Wolniewicz et al (2006) have performed a large scale coupled channel calculation that includes the first six u 1 S + and first four u 1 P states. As the excited Rydberg electron gets further away from the ion core in the high n states, the velocity difference between the Rydberg electron and nuclei becomes smaller and the adiabatic approximation becomes less appropriate while MQDT becomes more so. MQDT often uses accurate potential energy curves of the low n states obtained from traditional ab initio calculations to extract quantities such as R-dependent quantum defect. When calculating spectral intensities, MQDT also uses the accurate electronic transition moments obtained by the ab initio technique and extends them to high n Rydberg states , Glass-Maujean and Jungen 2009, Glass-Maujean et al 2010, 2011a. Ross and Jungen (1994a, 1994b, 1994c investigations of the triplet structure of H 2 . MQDT has been extensively utilized by Glass-Maujean and coworkers with a great deal of success in the interpretation of H 2 and D 2 high resolution spectral transitions between the X g 1 S + and the singlet-ungerade Rydberg states (Glass-Maujean et al 2012a, 2012b, 2012c, 2013a, 2013b, 2013c, 2015a, 2015b, 2016a, 2016b, Glass-Maujean and Schmoranzer 2018. Glass-Maujean et al (2011b) have also examined the relative accuracy of coupled Schrödinger equations and MQDT methods for H 2 and D 2 . In the case of triplet transitions, accurate calculations of triplet potential energy curves and electronic transition moments have been performed by Staszewska and Wolniewicz (1999), Staszewska (2001) and Spielfiedel et al (2004a).

Theory
In the present work, the subscript index i is used to denote the appropriate quantum numbers of the c u 3 P and D u 1 P state. The index k refers to either the b u 3 S + or B¢ u 1 S + states. As the kinetic energy of the out-going hydrogen atoms, E k , is related to the energy of the b u 3 S + or B¢ u 1 S + state by a constant offset, V E , k k ¥ ( ) also often represents the continuum energy levels of those two states. The electron spin interactions of the triplet states are generally very small (Lichten et al 1979, Spielfiedel et al 2004b. The rotational levels of the c u 3 P state are well-described by Hund's case (b) and are labeled by vibrational and rotational quantum numbers (v, N), where N is the sum of nuclear rotation and electronic orbit angular momentum. Franck-Condon factors (FCFs) and spontaneous transition probabilities do not depend on the orientation of electron spin. Predissociation of the c u 3 Pstate takes place via direct and indirect spin-spin and spin-orbit coupling with the repulsive b u 3 S + state. The rate depends strongly on the particular fine structure component of a given (v, N) level. The three fine structure components are referred to as F 1 , F 2 , and F 3 , which correspond to J=N+1, J=N, and J=N−1, respectively. The fine structure levels in the present paper will collectively be referred as (v, N, J) or (v, N) plus a specific F-component.

c u
3 P + and D u 1 P + predissociation rate Predissociation, by definition, is caused by coupling between two electronic states. In the absence of an external field, non-adiabatic, spin-orbit, spin-rotation, spin-spin and even hyperfine interaction can cause predissociation. In the presence of electric or magnetic fields, coupling can also take place via the electric dipole or magnetic moment of the molecule (Kato and Baba 1995). A common feature of non-adiabatic coupling is that its Hamiltonian is inversely proportional to μ, the reduced mass of the nuclei (Wolniewicz et al 2006). Kato and Baba (1995) considered the electron-rotation coupling as a distinct predissociation mechanism, even though it is non-adiabatic by nature.
The predissociation of the c u 3 P + state by the repulsive b u 3 S + state and the D u 1 P + state by B¢ u 1 S + continuum both take place by heterogeneous coupling between the rotation and electronic motion (which is also known as L-uncoupling and is responsible for Λ-doubling). The predissociation of the c u 3 P + is strongly dominated by the b u 3 S + coupling. As mentioned in section 1, except for the high v levels of the D u 1 P + state, the D u 1 P + −B¢ u 1 S + coupling alone can satisfactorily reproduce the observed predissociation widths. Thus, the examination of the isotope dependence of predissociation will only be made within the frame of coupling between a single discrete state and a single continuum state.
The predissociation width, v N , i i G , and rate, W v N , , are given as (Fano 1961, Kato and Baba 1995, Lefebvre-Brion and Field 2004 : ), which corresponds to the maximum amount of the releasable kinetic energy for a given (v i , N i ) level. For the b u 3 S + and B¢ u 1 S + states, V k ¥ ( )corresponds to the potential energy of the two hydrogen atoms in s s 1 1 + and s s 1 2 + dissociation limits, respectively. The matrix element is non-vanishing only when N i =N k . Similar equations for the D u 1 P + and B¢ u 1 S + continuum are obtained by replacing the appropriate quantum numbers in the equations. H p formally includes all nonadiabatic coupling between the c u 3 P and b u 3 S + states or between D u 1 P + and B¢ u 1 S + states. Because the electron spin interaction, responsible for the weak predissociation of the c u 3 Pstate, is at least three orders of magnitude smaller than the electron-rotation coupling, spin interaction can be safely neglected for the predissociation of thec u 3 P + state.
In the notation of Wolniewicz et al (2006), the non-adiabatic coupling Hamiltonian, H p , is The L + electronic matrix element for D u 1 P + −B¢ u 1 S + coupling from R=0.5-80 a 0 has been accurately calculated by Wolniewicz et al (2006). For the c u 3 P + −b u 3 S + coupling, the i iL k , , x y u á P S ñ + | | matrix element by Martin and Borondo (1988) is one of the most accurate published calculations. The matrix element is available from 0 to 10 a 0 . However, as the asymptotic form of the matrix element is known (Borondo et al 1987), it can be easily extrapolated to R > 10 a 0 . Note that the c iL b , , ux y u á P S ñ + | | given by Martin and Borondo (1988) is 1 2 Wolniewicz et al (2006).

Approximate isotope relation of predissociation rate
When the predissociation rates of two isotopologues such as XH and XD are measured, the objective is to identify the Hamiltonian H p responsible for the predissociation and/or the electronic symmetry of the predissociating state. The goal is usually achieved by comparing the measured relative predissociation rate to certain simple approximate isotope relations. This section presents two frequently used isotope relations whose accuracy will be examined. The factor N N 1 + ( )in equation (4) introduces a trivial N(N+1) factor in both predissociation width and rate in equations (1) and (2). The reduced rate and width, are introduced to eliminate the trivial factor and to make it convenient to compare the rate and width between different N levels. The matrix element in equations (1) and (2), will be referred to as isotope approximation I. It is normally a direct consequence of expressing the electron-rotation coupling as H B N L N L p = + + --+ ( ), where the bound state rotational constant B sometimes is also written as B v to emphasize its vibrational dependence (Kato and Baba 1995). The connection to the original form of equations (3) and (4) is that the expectation value of 1/2μ R 2 over the vibrational wave function,  (8) will be referred to as isotope approximation II. It predicts that the predissociation width and rate ratios of H 2 to D 2 and diatomic hydride XH to XD, when atom X is much heavier than H, are both ∼4.

Potential energy curves
The potential energy curves in the present investigation consist of BO potentials (V BO (R)) plus adiabatic (V adi (R)), relativistic (V rel (R)), radiative (V rad (R)), and empirical non-adiabatic (V nad (R)) corrections: The V BO (R) and V adi (R) of the a g 3 S + , b u 3 S + , and c u 3 P states are taken from the ab initio calculation for H 2 by Staszewska and Wolniewicz (1999). The corresponding V rel (R), V rad (R), and V nad (R) are taken from Liu et al (2017b), with V nad (R) setting to zero for the b u 3 S + state. The a g 3 S + , b u 3 S + , and c u 3 P state calculations by Staszewska and Wolniewicz (1999) are limited to R44a 0 . Accurate calculation of some high (v, N) levels of the a g 3 S + and c u 3 P states of the heavy isotopologues such as DT and T 2 requires the potential up to 600 to 1000 a 0 . The three term extrapolation based on the last 3 points of the potential around 44 a 0 used in our previous work (Liu et al 2012(Liu et al , 2016(Liu et al , 2017a) is insufficient. The H 2 b u 3 S + long range potential of Ovsiannikov and Mitroy (2006) and the a g 3 S + , c u 3 P , B¢ u 1 S + and D u 1 P long range potentials of Vrinceanu and Dalgarno (2008) are used to extend V(R) to the required ranges of R (see also Liu et al 2017b).
For the B¢ u 1 S + state, BO and adiabatic potentials calculated by Staszewska and Wolniewicz (2002) and Wolniewicz and Staszewska (2003b) are used. In our earlier work , an R-independent relativistic correction of −1.92 cm −1 and radiative correction of 0.308 cm −1 were used. In the present work, the V rel (R) is taken as the sum of that of the X g 2 S + H 2 + calculated by Howells and Kennedy (1990) and that of H(2s), −0.456 cm −1 . The radiative correction, V rad (R), of the B¢ u 1 S + states were set to be V rad (R) of the H 2 + X g 2 S + state calculated by Bukowski et al (1992). Since many levels of the B¢ u 1 S + state are strongly coupled to their counterparts of the B u 1 S + , C u 1 P + and D u 1 P + states, the empirical non-adiabatic term cannot account for the localized perturbations. Thus V nad (R) is set to zero for the B¢ u 1 S + state. The BO and adiabatic potentials of Wolniewicz and Staszewska (2003a) along the V rad (R) of the H 2 + X g 2 S + state are used for the D u 1 P state. The dominant configuration of the D u 1 P state is s p 1 3 p for small R. For large R, however, the D u 1 P state correlates to H(1s)+H ( d 3 ) dissociation limit. V rel (R) is therefore taken as the sum of the relativistic correction of the H 2 + X g 2 S + state and that of H(3p), −0.090 cm −1 . The modified empirical non-adiabatic correction is given as (Liu et al 2017b): where n e is the number of electrons and equals to 2 for the hydrogen molecule and 1 for the hydrogen molecular ion. m e is the electron mass and R e is the equilibrium internuclear distance of the electronic state, which is also the position of the minimum of the potential energy curve; β is a parameter. For a diatomic molecule AB, 1/μ=1/m A +1/m B , and 1/μ a =1/m A −1/m B . When the molecule is homonuclear, a m = ¥ and equation (10) becomes the original equation of Alijah and Hinze (2006). The derivative, dV(R)/dR, for the D u 1 P , c u 3 P and a g 3 S + states is obtained in the same way as that described in Liu et al (2017b). When one or two H atoms are replaced by D or T atoms, the adiabatic correction, V adi (R), is reduced by a factor of H 2 m m. The empirical non-adiabatic term, V nad (R), will also change (see Liu et al 2017b) via μ and μ a .
However, since V nad (R) of H 2 itself is only a few cm −1 to a fraction of 1 cm −1 , the change in the V nad (R) due to isotope substitution is typically only a fraction of 1 cm −1 . Other terms are identical to those of H 2 . Figure 1 shows H 2 potential energy curves of several states relevant to the present investigation. The potential energy curves constructed so far neglected the gerade−ungerade (g−u) symmetry-breaking Hamiltonian in the mixed isotopologues such as HT, HD and DT. The b u 3 S + , c u 3 P , B¢ u 1 S + and D u 1 P states of H 2 asymptotically correlate to the H(1s)+H(1s), H(1s)+H(2p), H(1s)+H(2s), and H(1s)+H(3d) dissociation limits. For the mixed isotope molecule AB, there are two possible limits for the c u 3 P , B¢ u 1 S + and D u 1 P states. In the case of the B¢ u 1 S + and c u 3 P states, the two possible asymptotic limits are A(1s)+B(2s/2p) and A(2s/2p)+B(1s). These two limits are separated by (1 2 2 --)/2μ a or 3/8μ a au. For HD, HT and DT, the energy differences are 22.400, 29.851, and 7.450 cm −1 , respectively. The asymptotic value of the g−u symmetrybreaking Hamiltonian is half of the energy gap or 3/16μ a for the c u 3 P and B u 1 S + states and 4/9μ a for the D u 1 P state.
When the effect of the g−u symmetry-breaking Hamiltonian is neglected, the constructed B¢ u 1 S + , c u 3 P , and D u 1 P potentials have the middle points of the A(1s)+B(2s) and A(2s)+B(1s); A(1s)+B(2p) and A(2p)+B (1s); and A(1s)+B(3d) and A(3d)+B(1s) energies as their asymptotic limits, respectively. When the g−u symmetry-breaking term is taken into account, the asymptotic limit of the B¢ u 1 S + state will likely turn out to be the A(1s)+B(2s) limit, while that of its interacting partner, EF g 1 S + , is likely to be the A(2s)+B(1s) limit. Likewise, the asymptotic limits of the c u 3 P and D u 1 P states will likely be the A(2p)+B(1s) and A(1s)+B(3d), respectively. For large R, the potentials used in the present work for the HD c u 3 P and B¢ u 1 S + states are about 11.200 cm −1 higher and lower than the exact values, respectively. The eigenvalue and eigenfunction are mostly determined by the magnitude and shape of the potential within ∼14 a 0 . Unless the level is close to the asymptotic limit, the effect of the g−u symmetry-breaking Hamiltonian on the calculated energies is negligible.
In the predissociation of the D u 1 P + state, the shape and the value of the B¢ u 1 S + continuum in for large R may be important. To assess the effect of the g−u symmetry-breaking on the predissociation of the HD D u 1 P + state, an empirical B¢ u 1 S + potential energy curve for HD is constructed. First, the g−u symmetry-breaking Hamiltonian is assumed to take a constant value of 11.200 cm −1 for R18a 0 and zero for R < 18a 0 . Then, the EF g 1 S + and B¢ u 1 S + state long range potentials that include the V R  ¥ ( ) offset of HD are constructed for R18a 0 by using the coefficients given by Vrinceanu and Dalgarno (2008). The upper and lower potential energies are obtained by diagonalizing a 2 by 2 matrix (De Lange et al 2002). The ab initio B¢ u 1 S + potential is smoothly merged with the upper potential between R=18.00 and 19.90 a 0 . The empirical merged B¢ u 1 S + potential thus consists of the HD ab initio potential from R=0.500 to 18.00 a 0 , the upper potential for R > 19.90 a 0 and the mixture of the two between 18.00 and 19.90 a 0 . Since the b u 3 S + state correlates to the A(1s)+B(1s) dissociation limit, the g−u symmetry-breaking term does not have an impact on the c u 3 P −b u 3 S + predissociation of the levels significantly below the c u 3 P dissociation limit. In the united-atom limit (R 0), the b u 3 S + and c u 3 P states of H 2 both correlate to the s p P 1 2 3 state of the He atom (Corongiu and Clementi 2009a). In the separate-atom limit (R  ¥), they asymptotically correlate to the H(1s)+H(1s) and H(1s)+H(2p) limits, respectively. For R of a few a 0 , the dominant configurations are (1sσ) (2pσ) and (1sσ)(2pπ), respectively. Likewise, the H 2 B¢ u 1 S + and D u 1 P states both correlate to the s p P 1 3 1 states of the He atom in the united-atom limit (Corongiu and Clementi 2009b) and asymptotically to the H(1s)+H( s 2 ) and H(1s)+H ( d 3 ) configurations, respectively, in the separate-atom limit. For R of a few a 0 , the dominant electron configurations of the B¢ u 1 S + and D u 1 P states are (1sσ)(3pσ) and (1sσ)(3pπ), respectively. One consequence of having the same united-atom states is that the two pairs of the states, b u 3 S + and c u 3 P as well as B¢ u 1 S + and D u 1 P , both obey the pure precession relation (Mulliken 1964), which means that the electronic matrix elements, c L b á ñ + | | | | and D L B á ¢ñ + | | | |, approach 2 as R 0. In the separate-atom limit, Borondo et al (1987) The other consequence is a large difference in the adiabatic correction between b u 3 S + and c u 3 P and between B¢ u 1 S + and D u 1 P at small internuclear distances. At R=0.5a 0 , the difference between the H 2 B¢ u 1 S + and D u 1 P is ∼957 cm −1 while it is only ∼8 cm −1 at large R.
2.4. Energies, transition probabilities and c u 3 P -(v J , ) predissociation rates and lifetimes. To obtain the lifetimes of the c u 3 P -(v, J) state, spontaneous transition probabilities of three types of transition are needed in the present work. The first is the discrete-discrete c u 3 P -a g 3 S + E1 transition. The second and third are the discrete-continuous c u 3 P -−b u 3 S + M1 and E2 transitions. Except for the small difference in the V (R) of the a g 3 S + , b u 3 S + and c u 3 P states, the calculation of the three types of transition probabilities and FCFs is identical to those in Liu et al (2017a). As noted in Liu et al (2017b), small changes in V rad (R) and V rel (R) and the addition of the V nad (R) introduce negligible changes (0.1%∼1.0%) in transition probabilities, FCFs and predissociation rates from those of Liu et al (2017a). Other than the small V(R) changes noted above, the predissociation rates of the HT, DT and T 2 c u 3 P -(v, N, J) levels are also calculated in the same ways as in the earlier work (Liu et al 2017a).

Results and discussion
Even though the primary focus of the present investigation is assessing the accuracy of two frequently used approximate isotope relations in the predissociation rate, we start by comparing the calculated energies and predissociation widths against the measured values. The comparison helps establish the accuracy of wave functions and the calculated predissociation matrix elements.

Energies
Unlike Abgrall et al (1993aAbgrall et al ( , 1993bAbgrall et al ( , 1993cAbgrall et al ( , 1994Abgrall et al ( , 1999, the present study obtains eigenfunctions and eigenvalues by solving uncoupled Schrödinger equations of the b u 3 S + , c u 3 P , B¢ u 1 S + and D u 1 P states. Consequently, the c u 3 P + and the c u 3 Pstates as well as the D u 1 P + and the D u 1 Pstates are degenerate. As discussed later, the empirical non-adiabatic coupling V nad (R) only attempts to account for the coupling of the remote states. Since the spin interaction either vanishes or is negligibly small (Lichten et al 1979, Spielfiedel et al 2004b, the u Pstate can only homogeneously couple to another u Pstate or heterogeneously to a u Sstate, which is possible only for a doubly excited configuration. For both the c u 3 Pand the D u 1 Pstates, u u P -P -coupling is negligible, except for the levels close to the dissociation limit. Tables 1-3 compare the measured and calculated D u 1 P -(v, J) energies of H 2 , D 2 and HD molecules. For the D u 1 P state, the adjustable parameter β=0.115 is used for all three molecules. In table 1, the first row of each vlevel lists the experimentally determined energies of the H 2 D u 1 P -(v, J) of Abgrall et al (1994) (see also Roncin and Launay 1994). The second row of each v-level lists two sets of residuals. The first set is the difference between the measured value and the one calculated by the present work, while the second one, enclosed in parentheses, is that between the measured value and the one calculated by Abgrall et al (1994). Where the observed values are not available, the presently calculated values are listed in parentheses. Except for eight (v, J) levels, the residuals of the other observed 134 levels listed in table 1 are all less than 1.0 cm −1 . For those eight (v, J) levels, both sets of residuals tend to be large. The two sets of calculated energy values also agree well with each other. Overall, the calculated energies of Abgrall et al (1994) are slightly closer to the observed values than those calculated in the present work. This is perhaps a consequence of the potentials of the H 2 B u 1 S + , B¢ u 1 S + , C u 1 P , and D u 1 P states in Abgrall et al (1994) having been adjusted slightly so that the calculated energy levels of the J=0 levels of the B u 1 S + and B¢ u 1 S + states and the J=1 levels of the C u 1 Pand D u 1 Pstates agree with observed H 2 energy levels. Table 2 compares measured D u 1 P -(v, J) energies of D 2 with those calculated by Abgrall et al (1998) and by the present study. Most of the measured values listed in the table are from the experimental investigation of Roudjane et al (2006). Some of the values are based on the Q-branch measurement of Dickenson et al (2011). Those entries are identified by a superscript b. In addition, three energy levels of Roudjane et al (2006) have been corrected by Glass-Maujean et al (2011b) and are marked with a superscript c. Once again, the first number of each of the second v-row refers to the difference between the measured and presently calculated energies while the second number, in parentheses, is the difference between the measured energy and that calculated by Abgrall et al (1998). As can be seen, only 4 out of the 172 listed observed levels have their residuals greater than 1.0 cm −1 . Both the present and Abgrall et al (1998) calculations are in significantly better agreement with the experimental values than the coupled channel calculation of Roudjane et al (2006). Once again, the Abgrall et al (1998) calculation is slightly better overall than the present one, largely because of their fit of the B u 1 S + , B¢ u 1 S + , C u 1 P , and D u 1 P potentials to the observed D 2 J=0 levels of the B u 1 S + and B¢ u 1 S + states and the J=1 levels of the C u 1 Pand D u 1 Pstates. Table 3 lists similar comparison for the HD molecule. The experimental values in the table are derived from the Q-branch frequencies of the D u 1 P -(v)− X g 1 S + (0) bands measured by Dickenson and Ubachs (2012) and the calculated X g 1 S + (v, J) energies of Pachucki and Komasa (2010). The O C1 column refers to the difference between the measured and presently calculated energies. The O C2 column refers to the difference between the measured value and that calculated by Abgrall and Roueff (2006) unless the observed value is not available, in which case the current calculated energy is listed in the parenthesis and the O C1 column refers to the difference between the current value and that of Abgrall and Roueff (2006). Unlike H 2 and D 2 , Abgrall and Roueff (2006) did not make any adjustment to the HD B u 1 S + , B¢ u 1 S + , C u 1 P and D u 1 P potentials. Consequently, the present calculation is in a significantly better agreement with the observation than their calculation. The comparison of the observed and presently calculated values also suggests possible perturbations at the (J=2, v=1), and (J=3, v=4 & 5) levels, probably due to the u−g symmetry-breaking interaction. It should be mentioned that many Q-branch frequencies observed by Dickenson and Ubachs (2012) differ significantly (by −7.23 to 5.29 cm −1 ) from those of early observations by Monfil (1965), and Dehmer and Chupka (1983). The good agreement in table 3 thus confirmed the accuracy of the measurement and assignment of Dickenson and Ubachs (2012).
It should be noted that neither the present calculation nor that of Abgrall et al gives good energy values for the high v-levels of H 2 , D 2 and HD. The observed energy values can be derived from the measured frequencies of the Q-branch transitions for the D u 1 P -(v)− X g 1 S + (0) bands of H 2 , HD and D 2 obtained by Dickenson et al (2010Dickenson et al ( , 2011 and Dickenson and Ubachs (2012) as well as the highly accurately calculated X g 1 S + (v, J) energies of Wolniewicz (1995), Komasa (2009, 2010) and Komasa et al (2011). The calculated D u 1 P -(v, J) energies for v=14-17 of H 2 , v=22-23 of D 2 and v=16-18 of HD are all significantly higher than their observed counterparts. For the J=1 levels of the v=14-17 states of H 2 , the residuals (Obs-Cal) of the present calculation are −1.54, −2.51, −3.91 and −4.20 cm −1 , respectively. The corresponding residuals of the Abgrall et al (1994) calculation are −0.84, −1.76, −3.59, and −4.57 cm −1 , respectively. For J=2 levels of the v=22 and 23 states of D 2 , the present (Abgrall et al (1998)) residuals are −2.23 (−1.91), and −2.21(−2.36) cm −1 , respectively. For J=1 levels of the HD v=16-18 states, table 3 shows that the present energies are 1.03, 1.35 and 2.41 cm −1 higher than the observed values. The large residuals for the high v levels are not due to the inaccuracy of the D u 1 P potential energy curve, but are caused by non-adiabatic coupling with the higher D¢ u 1 Pand V u 1 Pstates which becomes significant at high v-levels (i.e. large R) of the D u 1 Pstate. Wolniewicz et al (2006) have calculated the H 2 D u 1 P -(v, J) energies by solved the coupled C u 1 P -, D u 1 P -, D¢ u 1 Pand V u 1 P -Schrödinger equations. If their H 2 D u 1 P -(v, J) energies are combined with the present empirical non-adiabatic correction, the corresponding residuals for the J=1 levels of the v=14-17 states of H 2 become negligible at 0.00, −0.05, −0.14, and −0.37 cm −1 , respectively. The first row of each v lists the observed energies while the second row of each v displays the two sets of the corresponding residuals (obs-cal). The first one is the residual of the present calculation while the second one, enclosed in a parenthesis, is a residual of the Abgrall et al (1998) calculation. Unless noted otherwise, all observed are from Roudjane et al (2006). When the observed value is not available, the presently calculated value is listed in parentheses. The calculated c u 3 P -(v, N) energies of H 2 , D 2 and HD with β=0.055, compared against the experimental values, have been presented elsewhere (Liu et al 2017b). The agreement with experimental values is comparable to those of the D u 1 Pstate discussed earlier. The calculation for HT, DT and T 2 uses the same β values. β is 0.19 for the a g 3 S + state potential energy curves used to calculate the c u 3 P --a g 3 S + transition probabilities. Note that the only change in V(R) for a given state of the various isotopologues is the change in mass of nuclei, which changes the corresponding μ and μ a , which in turn alter V adi (R) and V nad (R). In other words, if the various terms in the V(R) potential of the H 2 are known, the corresponding V(R) for other isotopologues can easily be constructed solely based on the value of nuclei mass. In this sense, all the energies and wave functions of isotopologues can be considered as they were calculated from a common set of potential energy curves.
As noted in Liu et al (2017b), V nad (R) in equation (10) accounts for 91.6% of non-adiabatic shifts of ∼4130 (v, J) levels of the X g 1 S + state of H 2 , D 2 , T 2 , HD, HT, and DT and the X g 2 S + state of H 2 + , and HD + by using a single β value of 0.1211. So, the β value of the D u 1 Pstate, 0.115, is fairly close to that of the X g 1 S + state. However, the empirical non-adiabatic correction is not exact but is rather a convenient approximation that works well when localized perturbation is absent and the coupling state is sufficiently far away. In other words, V nad (R) approximates the overall contribution of electronic states that are remotely coupled with the state being investigated. When localized coupling is present, such as that between the i g 3 Pand j g 3 Dstates or among the C u 1 P -, D u 1 P -, D¢ u 1 P -andV u 1 Pstates, the empirical correction can be applied after the localized perturbation has been taken into account. However, even when the two conditions are met, equation (10) fails to yield a correct asymptotic value of the non-adiabatic correction at large R. As R increases, dV(R)/dR can be shown to decline much faster than 1/R. Thus, as R approaches infinity, V nad (R) in equation (10) au for AB in a state having A(1s)+B(nℓ) as the dissociation limit. The asymptotic value of H 2 is −0.0651 cm −1 for the X g 1 S + and b u 3 S + states, and −0.0407 cm −1 for the a g 3 S + and c u 3 P states. Because the high v levels, which are close to the dissociation limit, are predominately affected by the potential of large R, equation (10) also tends to significantly underestimate their non-adiabatic shifts. However, for levels significantly below the dissociation limit, the energies are primarily determined by the potential within several Å of R e . Tables 1-3 for D u 1 Pand the corresponding tables of c u 3 Pin Liu et al (2017b) show that equation (10) approximates well the non-adiabatic corrections for these levels.  Dickenson and Ubachs (2012). The present calculation enclosed in parenthes is sometimes listed when the derived value is not available. O C1 always refers to the difference between the observation and the present calculation, while O C2 is that between the observation and that by Abgrall and Roueff (2006). When the observed value is not available, O C2 refers to the difference between the present calculation and that of Abgrall and Roueff (2006). b Derived from blended Q-branch transition.

c u
3 P + (v N , ) predissociation rates Although there are many experimental measurements of the D u 1 P + (v, J) predissociation width, the c u 3 P + (v, N) predissociation rate appears not to have been measured so far. There are two other calculations of the c u 3 P + predissociation rate. Martin and Borondo (1988) calculated the lifetimes for the N=1 and 2 levels of the c u 3 P + (v=0-16) state of H 2 with values almost identical to those obtained in the present work. Comtet and de Bruijn (1985) also calculated the c u 3 P + lifetimes of the N=1 and 2 levels for H 2 and the N=1 levels for HD and D 2 that are roughly twice as the presently calculated lifetimes. Martin and Borondo (1988) explained the factor of 2 discrepancy by Comtet and de Bruijn's undercount of the density of states. Figure 2 shows the presently calculated predissociation lifetimes of the c u 3 P + (v, N=1) levels of the H 2 , HD, HT, D 2 , DT and T 2 molecules as a function of the kinetic energy released in the predissociation. For the c u 3 P + −b u 3 S + predissociation, the maximum E k released per hydrogen molecule is ∼10.2 eV. From ∼7.2 to ∼10 eV, the c u 3 P + predissociation lifetimes monotonically decrease with E k (or v i ). Beyond ∼10 eV, the predissociation lifetime increases rapidly with E k (or v i ). As will be shown below, the trends are largely due to the variation of the FCFs with E k (or v i ). The figure also shows that the predissociation lifetimes, at comparable E k , increase rapidly with the reduced mass (note the logarithmic scale). For the (v=0, N=1) level, the predissociation lifetime increases from 3.16 ns for H 2 to 12.0 μs for T 2 . Once again, the decrease of the FCFs with μ is primarily responsible for the strong isotope dependence of the predissociation lifetime and rate. Figure 3 compares the c u 3 P −b u 3 S + FCFs of the N=1 levels of various isotopologues. The FCFs are in units of per milli-hartree and are shown on a logarithmic scale. All FCFs increase with E k from ∼7.2 to ∼10 eV, which is predominantly responsible for the increase in predissociation rate and the corresponding decrease in lifetime noted earlier. Between ∼10 and ∼10.2 eV, the decrease of the FCFs with E k is responsible for the increase in predissociation lifetime with E k seen in figure 2. Figure 3 also shows very large decreases of the FCFs with the reduced mass of the isotopologue. For the v=0 and N=1 levels, the FCFs of HD, HT, D 2 , DT and T 2 drop by factors of ∼3.9, ∼7.1, ∼38, ∼128 and ∼655, respectively, from that of H 2 . This large reduction in FCFs is responsible for the drastic increase in the predissociation lifetime of the heavy isotopes. The c u 3 P −b u 3 S + system of molecular hydrogen is a case in which the FCFs are highly dependent on the mass of the isotopologues. Application of any isotope approximation rule to such systems without accounting for the change in vibrational overlap integral will produce very large errors.

D u
1 P + (v, J) predissociation width The spectral profile of the D u 1 P + (v, J) transition is typically measured either by absorption or by the hydrogen Ly-α dissociative emission method. The observed profile is usually a convolution of the predissociation, Doppler and instrumental profiles. While the predissociation profile is normally represented by the Fano line shape formula, Mezei et al (2014) has suggested that the formula cannot accurately represent the resonant profile for the v6 levels of H 2 . Each Fano profile is described by its center frequency, width (Γ) and asymmetry (q) parameters. Mezei et al (2015) have recently shown that the parameters extracted from the profiles of the D u 1 P + (3)− X g 1 S + (0) transitions do not necessarily coincide with their original meanings given by Fano even The ns unit of the lifetime is nanosecond and its scale is logarithmic. Note the nearly factor of 4,000 increase in the c u 3 P + (v=0, N=1) level lifetimes from H 2 (3.2 ns) to T 2 (12 μs). From left to right, the data points for H 2 , HD, HT, D 2 , DT, and T 2 traces correspond to v=0-21, 0-22, 0-26, 0-29, 0-34, and 0-39, respectively. though the observed profiles can be accurately represented by the Fano formula. Furthermore, some D u 1 P + (v, J) transitions can also overlap with other npσ and npπ Rydberg transitions. Consequently, the measured width parameter often has a non-negligible error.  Julienne (1971) and Fiquet-Fayard and Gallais (1971, 1972, the present calculation shows a smooth decrease of the predissociation width with v. The better agreement with the measured value is presumably because of the use of more accurate B¢ u 1 S + and D u 1 P potential energy curves and B¢ u 1 S + -D u 1 P electronic coupling matrix elements in the present work. Both sets of measured values show erratic oscillations around the present smooth magenta line. Oscillations are also seen in the calculated values of Glass-Maujean et al (2012c) and Mezei et al (2014). The deviations of the measured and calculated rates from the smooth magenta cross line (beyond their respective errors) are primarily due to the non-adiabatic couplings of the D u 1 P + levels to other np u 1 s S + and np u 1 p P + levels. For other isotopologues, those local perturbations will occur at different (v, J) levels with different strength. These differences and their impact on the predissociation rates can not be predicted from a simple isotope relation.    Dickenson et al (2011) and the J=1 level of HD measured by Dickenson and Ubachs (2012) with those calculated in the present work. The D 2 width of the present result is virtually identical to that calculated by Dickenson et al (2011). Two sets of HD predissociation widths have been calculated. One set uses the ab initio B¢ u 1 S + potential energy curve with the u−g symmetry-breaking neglected. The other set uses the merged potential energy curve described in section 2.3. 2 Only small differences between the two sets of calculated widths were found for the levels very close to the threshold of predissociation. For the v=3 and J=3 level, the ab initio B¢ u 1 S + potential without the u−g symmetry-breaking term gives 14.08 meV and 13.41 cm −1 for E k and Γ, respectively. The corresponding values given by the merged potential are 12.69 meV and 13.21 cm −1 . No difference in Γ is found for any v4 levels. Because the two potential energy curves are different for R18a 0 , negligible differences in width are expected as the D u 1 P + −B¢ u 1 S + predissociation primarily takes place via coupling at small R. The good agreement of the perturbative D u 1 P + −B¢ u 1 S + coupling model with the observed width in figures 4 and 5 shows that the interaction with the B¢ u 1 S + continuum dominates over other couplings in the dissociation of the D u 1 P + state. Except for very high v levels, where the D u 1 P + −B¢ u 1 S + FCFs are very small, autoionization is not an efficient channel in the dissociation of the D u 1 P state in spite of the 3 order of magnitude difference between the mass of electron and hydrogen atom. For predissociation at higher n states, non-adiabatic coupling among Rydberg states is stronger and more frequent. The perturbative predissociation model of Fano will be less appropriate. Likewise, the perturbative model is expected to be somewhat more accurate for the c u 3 P + −b u 3 S + predissociation, which takes place by coupling between n=1 and 2 states. Figure 6 suggests that the variations of the width, whether observed or calculated, is due to the decrease of the D u 1 P + −B¢ u 1 S + FCF with E k , partially offset by the increase of the averaged electronic matrix element with E k (discussed later in section 3.4). The reduction of the predissociation width in going from H 2 to HD and D 2 , shown in figures 4 and 5, is also qualitatively consistent with the decrease of the FCFs with μ at comparable E k value. Furthermore, the very large difference between predissociation rates of the c u 3 P + and D u 1 P + states are primarily caused by the differences between the FCFs shown in figures 3 and 6. Note that the FCFs are in units of (milli-hartree) −1 and are shown on a logarithmic scale in figure 3. By contrast, FCFs are in units of (hartree) −1 and are shown on a linear scale in figure 6. Finally, while the D u 1 P −B¢ u 1 S + FCFs at comparable E k values are roughly the same order of magnitude for various isotopologues in figure 6, the low vc u 3 P −b u 3 S + FCFs of H 2 and T 2 at comparable E k in figure 3 differ by about 3 orders of magnitude.
3.4. Isotope dependence of the c u 3 P + (v N , ) and D u 1 P + (v J , ) predissociation rate Since isotope approximation II (Equation (8)) is based on isotope approximation I (equation (7)) with an additional assumption that FCFs are nearly invariant with isotope substitution, approximation I is normally expected to be more accurate than approximation II. Given the extreme dependence of the c u 3 P −b u 3 S + FCFs Figure 5. Comparison of selected measured and calculated predissociation widths for the D u 1 P + state of D 2 and HD molecules. The measured widths for D 2 and DH molecules are from Dickenson et al (2011), andUbachs (2012).
2 v=11 is believed to be the highest discrete vibrational level of the HD B¢ u 1 S + state. When non-adiabatic coupling is neglected, the ab initio potential energy curve that neglected the u−g symmetry-breaking term gives 118661.25 and 118675.92 cm −1 as the energies for J=0 of the v=10 and 11 states, respectively. The corresponding values given by the merged potential energy curve are 118661.34 and 118683.96 cm −1 , respectively. The experimental values of J=0 for the v=10 and 11 levels given by Dabrowski and Herzberg (1976) are 118658.94 and (tentatively) 118683.6 cm −1 , respectively. | of the D u 1 P + − B¢ u 1 S + system shown in figure 8 linearly increases with E k with a similar slope. Moreover, m | H p ē 2 | at a comparable E k value also increases with μ. At a given E k , the ratio of a pair of the lines corresponds to the deviation introduced by approximation I, which is somewhat comparable to the deviation of its c u 3 P + −b u 3 S + counterpart. However, as figure 6 shows, the FCFs for D u 1 P + − B¢ u 1 S + also decrease significantly with μ (though much less drastically than those of c u 3 P + −b u 3 S + system). This slow decrease in the FCFs, which approximately offsets the increase in m | H p ē 2 | , gives the applicability of approximation II to the D u 1 P + −B¢ u 1 S + predissociation rate. The D u 1 P + −B¢ u 1 S + system is a case in which,  -17, 4-20, 4-22, 4-25, 5-28, and 5-31, respectively. 3 In the present investigation, this ratio is actually the squared ratio of the average L R 2 2 + electronic matrix elements between the non-H 2 isotopologue and H 2 . because of a fortuitous partial cancellation of errors, the crude approximation II produces more accurate results than a refined approximation. While the m | H p ē 2 | in figures 7 and 8 show similar trends with respect to both μ and E k , the magnitude of the c u 3 P + −b u 3 S + system is a factor of 4∼5 greater than that of D u 1 P + −B¢ u 1 S + . The difference is primarily caused by the distinct R dependence of c L b á ñ + | | and D L B á ¢ñ + | | electronic matrix elements. First, c L b á ñ + | | remains positive and is always greater than 1 for all R. In contrast, the D L B á ¢ñ + | | matrix element is negative in all but R=10.4-14.5 a 0 (Wolniewicz et al 2006). Further, c L b á ñ + | | for R10a 0 is directly proportional to R (Borondo et al 1987), while D L B á ¢ñ + | | is range bound between −0.76 and −0.41 for R=17 to 80 a 0 . The different behavior of the two matrix elements is mainly caused by the D u 1 P state switching from the s p 1 3 p configuration at small R into the s d 1 3 + configuration at large R. The factor of ∼4-5 difference in m | H p ē 2 | , however, is more than offset by ∼3-6 orders of magnitude difference in FCFs, which makes the predissociation rate of D u 1 P + much faster than c u 3 P + . The break-down of isotope approximation I is primarily caused by the extremely non-linear behavior of the L R R 2 + ( ) term at short internuclear distances. As mentioned, in the united-atom limit the b u 3 S + and c u 3 P states correlate to the s p P 1 2 3 state of the He atom while the B¢ u 1 S + and D u 1 P states to the s p P 1 3 1 state. The absolute values of L R + ( ) electronic matrix element of both systems approach the pure precession value, 2 . The predissociation Hamiltonian H p is thus very large at for small R. When H 2 is substituted by one or two D or T atoms, both explicit and implicit changes occur in the nuclear Schrödinger equation. The change in μ in the A higher v level of the c u 3 P (D u 1 P + ) state has both of its inner turning points and vibrational wave functions starting at a higher energy and a smaller R for both c u 3 P (D u 1 P + ) and b u 3 S + (B¢ u 1 S + ) states, which leads a more sampling of the large L R 2 + value for small R. It is primarily for this reason that the m | H p ē 2 | shown in both figure 7 and 8 increases with E k (and v).
To summarize, the use of the averaged predissociation electronic matrix element of H 2 for other isotopologues (i.e. approximation I) underestimates the c u 3 P + −b u 3 S + predissociation rates of HD, HT, D 2 , DT and T 2 by ∼12%, ∼16%, ∼30%, ∼40% and ∼52%, respectively, and underestimates D u 1 P + − B¢ u 1 S + by ∼10%, ∼15%, ∼26%, ∼35% and ∼45%. 4 The large deviation is primarily caused by the behavior of the L R R 2 + ( ) at small R. In a system where L R + ( ) decreases rapidly as R approaches 0, the expected deviation will be smaller. The molecular hydrogen isotopologues represent very drastic variation of μ. The only comparable case in other molecules is the deuteration of a diatomic hydride XH, which can result in nearly a factor of two increase in μ. The present investigation shows that approximation I can introduce up to 26∼30% error. The use of approximation I for HD introduces about 12% error. The variation in μ caused by an isotope substitution of a non-hydrogen nucleus is much smaller than the 33% change in HD. Unless L R + ( ) increases rapidly as R approaches 0, the expected error caused by the use of approximation I is at most a few percent for isotope substitution of non-hydrogen nucleus. The predissociation Hamiltonian of homogeneous non-adiabatic coupling contains the term B ik (R)d/dR (Wolniewicz et al 2006 and cause large changes in the predissociation rates. The predissociation of the e u 3 S + state by the b u 3 S + state and the d u 3 Pstate by the c u 3 Pcontinuum take place by homogeneous non-adiabatic coupling. Kiyoshima et al (1999) found the predissociation rates of e u 3 S + (v) state and d u 3 P -(v4) of H 2 increase rapidly with v; however, the D 2 predissociation rates of the both e u 3 S + and d u 3 Pstates are ∼1000 times slower than their H 2 counterparts. The autoionization rate of the D 2 k u 3 Pstate was found to be at least 18-64 times slower than that of H 2 (Kiyoshima et al 2003). The dependence of the e u 3 S + and d u 3 Ppredissociation rates on the reduced mass is much larger than that of the c u 3 P + state, shown in figure 2 and table 4. Isotope approximations I and II are unlikely to be applicable to predissociation of the e u 3 S + and d u 3 P state. The deviation in predissociation rate from using the isotope approximations I and II in the present investigation is primarily caused by the change of vibrational wave functions. In a discrete-discrete coupling, the local perturbation is not only sensitive to the change of vibrational wave functions and magnitude of non- Figure 10. Comparison of the differential of the ro-vibrational overlap integrals, , of the H 2 D u 1 P + (v=5, J=1) level (red line) and the T 2 D u 1 P + (v=0, J=2) level (blue line). Since the FCF ratio of the (v=5, J=1) level of H 2 to the (v=9, J=2) level of T 2 is 1.304, the green line representing the differential ro-vibrational overlap integral of T 2 has been scaled up by a factor 1.142. The green trace of T 2 shows clearly that it weighs the small R more than the red trace of H 2 , where L R R 4 2 + ( ) , shown in black, has largest values. It is primarily the greater weighting factor for small R in the heavier isotope that leads to its larger reduced mass scaled average electronic D u 1 P + − B¢ u 1 S + predissociation matrix element. Note that the rotation and vibration energies of the (v=5, J=1) level of H 2 are comparable to those of the (v=9, J=2) level of T 2 . The E k of these two levels are 0.547 and 0.542 eV, respectively. 4 The average percentage number of D 2 , DT and T 2 given here have been approximately corrected for small non-linearly deviations in the levels with E k <7.8 eV shown in figure 7. Consequently, they differ slightly from those given in table 4 for the low N levels of the c u 3 P + (v=0) state.
adiabatic coupling Hamiltonian (i.e. reduced mass) but also to the energy separation of the two interacting levels. Consequently, it is possible for the local perturbation of a heavy isotopologue to be stronger than that of a light isotopologue. Glass-Maujean et al (2015b) have identified a number of the np u 1 p Plevels where the coupling of D 2 is stronger than that of H 2 . The predissociation rate of the D¢ u 1 Pstate of D 2 , for example, is found to be stronger than its H 2 counterpart because the indirect local coupling with the predissociating D u 1 P levels in the former is stronger than in the latter. While the heavier isotope and corresponding higher density of states lead to more localized coupling, a large number of interacting vibrational channels is required to explain the stronger than expected D 2 predissociation rate (Glass-Maujean et al 2015b).
3.5. c u 3 P -(v N J , , ) predissociation rate and lifetime The predissociation of c u 3 P -(v, J, N) arises from coupling of a b u 3 S + level of the same J but different N. The coupling takes place by two mechanisms. The first is the direct coupling of the c u 3 Pstate to the b u 3 S + state by spin-orbit and spin-spin interactions. Those two interactions can only couple the N level of the c u 3 Pstate to the N±1 levels of the b u 3 S + state (Liu et al 2017a). In a triplet state, at most three N levels can form a J value, namely, J−1, J, and J+1, corresponding to the F 1 , F 2 , and F 3 fine structure components. For the c u 3 P state, both J and N must be greater than or equal to 1. Consequently, the F 1 and F 3 components of the rotational level N of the c u 3 Pstate can only couple to the F 2 component of the (N+1) and (N−1) rotational levels of the b u 3 S + state, respectively. In contrast, the F 2 component of the rotational level N of the c u 3 Pstate can couple to both F 1 of the (N−1) level and F 3 of the (N+1) level of the b u 3 S + state. The second mechanism of c u 3 Ppredissociation is an indirect and second order process. The spin interaction first mixes an N level of the c u 3 Pstate with the N±1 levels of the c u 3 P + state. The N±1 levels of the c u 3 P + state, in turn, are predissociated by the c u 3 P + −b u 3 S + electronic-rotational coupling (section 2.1). The N and J levels of the c u 3 Pand b u 3 S + states involved in the coupling are the same in both the direct and indirect processes. Because the predissociation rate of c u 3 P + is very fast, the indirect second order process is comparable to and even dominates the direct first order process.
Although the c u 3 Ppredissociation rate is the square of the sum of the first and second order components, it is possible to discuss the effect of isotope substitution on the first and second order rates individually (that is by assuming the other order vanishes). For the same (N, J) level, the ratio of first order predissociation rates of two isotopologues is related to the FCF ratio: ) . Table 5 shows the c u 3 Ppredissociation rates of the fine structure levels of the v=0-4 and J=1−9 states for HT, DT and T 2 molecules, calculated using the same method as that of Liu et al (2017a). The F 2 component has a larger rate than the F 1 component, which, in turn, is greater than the F 3 component. There are two reasons for this result. First, the c u 3 P -F 2 component of the level N can couple to both the F 1 of the N−1 level and the F 3 of the N+1 level of the b u 3 S + state. In contrast, the F 1 and F 3 components of the N level can only couple to the F 2 component of the N+1 and N−1 levels, respectively. Second and more importantly, the spin-orbit and spin-spin interactions add in the F 2 component but cancel in the F 1 and F 3 components (Liu et al 2017a). The overall predissociation rate also decreases with the reduced mass, which is a consequence of the strong μ dependence of the FCFs and the c u 3 P + −b u 3 S + coupling via the second order process. The c u 3 P -a g 3 S + electronic transition moment for R=0.6 to 20 a 0 , calculated by Staszewska and Wolniewicz (1999), is used to compute the Q branch c u 3 P --a g 3 S + transition probabilities. The values of the E1 transition moment at R=19.5 and 20 a 0 are used to extrapolate to the asymptotic value of R  ¥. As mentioned, the c u 3 P -(v=0)-a g 3 S + (v=0) radiative decay is impossible. The c u 3 P -−b u 3 S + magnetic dipole and electric quadrupole transition probabilities are calculated using the M1 and E2 transition moments calculated by Chiu and Lafleur (1988). As discussed in Liu et al (2017a), the M1 and E2 transition moments listed for R=1.23 to 10.02 a 0 do not cover a sufficient range of R, which restricts their use to the low v and low N levels; there is also a question of the accuracy of these values themselves. Liu et al (2017a) found that good agreement of with both the measured lifetimes of the meta-stable H 2 , HD and D 2 by Johnson (1972) and the measured H 2 c u 3 P -(v, N, J) lifetimes of Berg and Ottinger (1994) can be achieved if a constant offset of 370 s −1 is added to all the calculated total M1 and E2 transition probabilities of H 2 , HD and D 2 . This constant offset is also added for the HT, DT and T 2 molecules in this paper. Table 6 lists the calculated lifetimes of the first five vibrational levels of HT, DT, and T 2 . Since the lifetimes of the three fine structure components of DT and T 2 are very close, their average lifetimes based on the (2J+1) statistical weight of the decay rate are also given in the table. The corresponding lifetimes for H 2 , HD and D 2 are given in Liu et al (2017a).
For the meta-stable c u 3 P -(0) levels, the lifetimes for all fine structure components of the D 2 , DT, T 2 molecules and the F 1 and F 3 components of the H 2 , HD, and HT molecules are comparable. This is because the predissociation rates of these fine structure levels are negligible and lifetimes are essentially determined by the M1 and E2 transition probabilities. The predissociation rates for the F 2 components of the H 2 , HD, and HT molecules, however, are appreciable and decrease significantly with reduced mass, which makes the F 2 lifetimes different among the 6 isotopologues and from their F 1 and F 3 counterparts.
In addition to the constant offset of 370 s −1 added to the calculated total M1 and E2 transition probabilities, other approximations were made to obtain the predissociation rates and lifetimes of the c u 3 Pstate(Liu et al 2017a). Very good agreement in the fine structure component lifetimes of the N=1-5 and v=0-3 levels of H 2 has been achieved between the Berg and Ottinger (1994) measurement and the Liu et al (2017a) calculation. Similar accuracy is expected in the lifetimes of the fine structure components of the c u 3 P -(v=0-4) levels for the other 5 isotopologues. More accurate M1 and E2 c u 3 P −b u 3 S + transition moments are required to remove the constant offset of 370 s −1 . Accurate ab initio c u 3 P −b u 3 S + electronic matrix elements of the spin-orbit A −1 (R) and spin-spin B −1 (R) constants and the corresponding c u 3 P electronic matrix elements of the A 0 , B 0 and B 2 (R) constants are also required to remove the other assumptions and extend the calculation to higher v levels.
In conclusion, (v, N, J) state-specific predissociation rates of the c u 3 P and D u 1 P + states have been calculated for various isotopologues of molecular hydrogen. The isotope approximation based on the reduced mass ratio alone, while coincidentally giving a fairly good result for the relative predissociation rate of the D u 1 P + state, completely breaks down for the c u 3 P + state. The approximation that uses the average electronic predissociation matrix element of H 2 for other isotopologues produces 10%∼52% of error, with larger errors for isotopologues with heavier reduced masses. The error arises from a preferential sampling of the average electronic predissociation matrix element at small R by isotopologues with a large reduced mass. At similar E k , the deviations from H 2 are roughly independent of E k . In addition to the c u 3 P + state, predissociation rates and lifetimes of the low v levels of the c u 3 Pstate have been obtained. The calculated D u 1 P -(v, J) and c u 3 P -(v, N) energies agree well with measured values. This investigation strongly suggests that approximate isotope relations should be applied with caution to the predissociation rate.