Particle scattering and resonances involving avoided crossing

In molecules, the nonadiabatic couplings between two adiabatic potentials build the avoided crossing (AC) region. The rovibronic resonances in the AC region of two-coupled potentials are very special, since they are not in the bound state region of the adiabatic potentials, and they usually do not play important roles on the scatterings and are less discussed. Exemplified in particle scattering, resonances in the AC region are comprehensively investigated. The effects of resonances in the AC region on the scattering cross sections strongly depend on the nonadiabatic couplings of the system, it can be very significant as sharp peaks, or inconspicuous buried in the background. More importantly, it shows a simple quantity proposed by Zhu and Nakamura (1992 J. Chem. Phys. 97 8497) to classify the coupling strength of nonadiabatic interactions, can be well applied to quantitatively estimate the importance of resonances in the AC region. Example applications of the quantity for real molecules (MgH, CO and OH) can well explain the evolutions of cross sections in the AC region published in the literatures. This work provides a simple and practical way to determine the candidate molecules when studying the resonances in the AC region.


Introduction
Different from bound and continuum states, a resonance is known as a discrete quantum state embedded in and coupled to a continuum [1][2][3][4][5]. It occurs as a short-lived decaying state of the scattering system. Mathematically, resonance can be defined as the pole of S-matrix in the fourth quadrant of the complex energy plane as ε s = ε sr − i 2 Γ si [6], where ε sr and Γ si are the resonance position and width, respectively. The larger of width Γ si , the shorter of resonance lifetime. Physically, the unique existence of resonance is interpreted by Ugo Fano as the results of interference between bound and continuum channels, exemplified with the doubly excited state He (2s2p) [7]. Resonances are very important in many areas related to physics and chemistry [8][9][10][11]. They generally appear in quantum scattering and its subsequent dynamic processes, manifested as abrupt changes in the cross sections of the (electron/photon/ion) energy spectra [6,12,13], including nuclear collisions [14], electron scatterings [9], photon absorption [7,15], heavy ion collisions [16], ultracold atom and molecule physics [17], chemical reaction dynamics [10] and so on.
Generally, resonances are classified as shape and Feshbach resonances from the aspect of the interaction-potentials [3,5]. Taking the particle-collision system as an example, figure 1 shows the typical interaction potential energy curves. The involved energy space can be clearly divided into four regions by the two adiabatic potential energy curves (APECs) V a 11 and V a 22 . If the nonadiabatic electronic couplings between the two APECs can be ignored, regions I and III can be considered as the bound and continuum regions of potential V a 11 , respectively. And in region IV, both the continuum states of potential V a 11 and the bound states of potential V a 22 sit independently. However in region II, quasi-bound states could be temporally built by the potential barrier, these quasi-bound states will decay into continuums through tunneling and are known as the shape resonances. Alternatively, for the system with important nonadiabatic electronic couplings, obviously, region I is still the bound state region mainly pertaining to potential V a 11 , and the quasi-bound states in region II would be altered by the electronic couplings. However in region IV, the bound states of potential V a 22 can not be sustained any more, due to the electronic couplings, they would leak or decay into potential V a 11 as continuum states, these states are known as the Feshbach resonances, or Fano resonances, interpreted as the results of interactions between bound and continuum channels. The left region III, or the avoided crossing (AC) region, becomes into a quite special region by the nonadiabatic interactions. Besides particle scattering, the avoided crossing of potentials is also important to a large number of optical spectra and dynamical processes [16,18,19]. Among these phenomena are the spectroscopic anomalies [20], predissociation [21], quenching [22], energy transfer processes [23], ion pair formation or recombination [24], chemi-ionization [25], collisional excitation [26], and the harpooning mechanism of chemical reactions [27].
Interestingly, there could be resonances in the AC region due to the electronic couplings between the potentials. However, the formation of these resonances are quite complex, since no adiabatic potentials can sustain the 'bound' states following the picture of Fano (see figure 1). An complementary way is to transform the system into the diabatic representation, then the bound and continuum states of the crossing diabatic potential energy curves (DPECs) V d 22 and V d 11 in the AC region could be roughly treated as bound and continuum channels for these resonances, respectively. The issues of representations are broadly and necessarily discussed for dynamics as heavy ion collisions and predissociation, due to the driving of nonadiabatic electronic couplings [16], and also molecular electronic structure calculations to go beyond the Born-Oppenheimer approximation. Many documentations on adiabatic and diabatic representations can be achieved [16,[28][29][30], here we only briefly summarized the framework for the two-level coupled system.
Generally, the total wavefunction of the system could be well built as the product basis set of electronic and nuclear functions, then the electronic levels are represented by the potential energy curves, and the nucleus evolve on these potential energy curves. In the adiabatic representation, the nuclear kinetic energy operator is nondiagonal and the potential energy operator is diagonal, while in the diabatic representation, the nuclear kinetic energy operator is diagonal and potential energy operator is nondiagonal [28]. Explicitly, in the diabatic representation, the total Hamiltonian may be written as where the kinetic T(R) = − 2 2μ d 2 , μ is the reduced mass of the system, R is the internuclear distance, J is the rotational quantum number, V d nn (n = 1, 2) are the DPECs, and the off-diagonal potential energy curves V d 12 = V d 21 . In contrast, the total Hamiltonian in the adiabatic representation may be written as where the diagonal kinetic matrix elements T nn (R) = T(R) − 2μ 2 B nn (R) and its off-diagonal terms T nl (R) = − μ 2 A nl (R) d dR + 2B nl (R) , with the radial coupling matrix elements defined as A nl (R) = ϕ a n d dR ϕ a l r and B nl (R) = ϕ a n d 2 dR 2 ϕ a l r . Subscript r indicates the integration over the electronic coordinates between different adiabatic electronic states |ϕ a n,l , B = A 2 + d dR A is satisfied with a complete basis set, and V a nn (R) are the APECs. More details of the transformations between adiabatic and diabatic representations will be introduced in section 2.
Such a diabatic picture for resonances in AC region could also be applied to the existence of resonances in regions II and IV, i.e., the bound and continuum channels of resonances could be roughly 'interpreted' pertaining to either APECs or DPECs. However, it should be noted that the bound levels of APECs and DPECs differ from each other (especially the ones close to the AC region), and the resonance positions (ε sr ) also differ from these bound levels [31,32]. Resonances in regions II and IV could exhibit as sharp peaks in the scattering cross sections, and resonances in the AC region generally do not contribute much on the scattering and spectra [21,33,34]. The exhibition of resonances in the AC region depend quite strongly on the nonadiabatic electronic couplings, and in few cases, they can also alter the cross sections abruptly as revealed in the photodissociation process of OH molecule [32]. So it would be quite interesting if we could roughly determine the properties of resonances in the AC region, that the importance of such resonances on the scattering cross sections can be roughly estimated by escaping doing scattering calculations [16,[35][36][37][38] or complex scaled structure calculations [39][40][41], and it would become practical to determine the candidate molecules when using resonances in the AC region for molecular manipulations [42][43][44][45].
All the issues of resonances are more or less related to the nonadiabatic electronic couplings, which couple or combine the independent electronic states (potentials) into a unified one. For such a coupled system, it can be equivalently described both in the adiabatic representation and the diabatic representation, which will be exemplified with the two-level system in section 2. Another issue of broad and long interest in the same coupled system is the nonadiabatic electronic transition (NET) [46][47][48][49], studying the nuclear motions on multiple potential energy curves. The Landau-Zener formula [47,48] can well estimate the nonadiabatic transition probability for scatterings. Obviously, the issues of resonances and NET are intrinsically correlated by the nonadiabatic electronic couplings. So one possible way to promote the present investigations of resonances in the AC region could learn from the well-founded theories of NET. Further studies reveal that, a simple quantity proposed by Zhu and Nakamura [46,[50][51][52] to classify the coupling strength of nonadiabatic interactions, can be well applied to quantitatively estimate the importance of resonances in the AC region, and the relevant applications and results will be presented in section 3.

Transformations between adiabatic and diabatic representations
The Hamiltonians H d and H a can be unitarily transformed as H d = C † H a C, where C(R) is the transformation matrix created so that in the diabatic representation, its transformed radial coupling matrix A d = C † AC + C † ∂ ∂R C is zero and in the limit of C(R → ∞) → I, or as, The relevant nuclear wave vectors of equations (1) and (2) satisfy ψ d = C † ψ a . For the present two level-system, the unitary matrix can be explicitly solved as with the diabatic angle defined as . And the potential energy curves satisfy the relations Note that in structure calculations, the potential energy curves and radial coupling matrix elements are generally and firstly calculated in the adiabatic representation within the Born-Oppenheimer approximation; after solving the transformation matrix from equation (3), the diabatic potential energy curves and other parameters such as (transition) dipole moments can be achieved by unitary transformation from the adiabatic representation. Generally, if the nonadiabatic coupling is weak, or the radial coupling matrix A nl (R) is tiny, the system can be well described in the adiabatic representation by ignoring the terms of A and B; while if it is the case of strong nonadiabatic coupling, the molecular orbital evolves rapidly in the AC region, resulting in the sharp variations of A nl (R), then it would be better to treat the system in the diabatic representation that the off-diagonal potential energy curves will also evolve smoothly.
Actually, the values of A nl (R) or V d ij (R) can only partially represent the nonadiabatic coupling strength, which also relates to the size of the AC region. Luckily, the relevant question has been broadly discussed when studying the dynamics of NET. In 1990s, an improved semiclassical theory for NET was documented by Zhu and Nakamura [50][51][52], its formulas can well treat not only the transition probabilities but also the phases induced by the transitions [46,[50][51][52][53]. What is more, it proposed a dimensionless parameter a 2 , which can effectively represent the coupling strength of the system. Zhu-Nakamura parameter a 2 is constructed with the parameters of the diabatic potentials at the crossing point R X in the AC region as [46] where F j (j = 1, 2) is the slope of the jth diabatic potential at the crossing point showing the force strength from the DPECs, with V X = V d 12 (R X ) indicating the diabatic coupling. Note that in the AC region, the DPECs could be roughly achieved by interchanging the APECs (see figure 1) that F j (j = 1, 2) could also be approximately estimated from the APECs, and 2V X roughly equals to the lowest energy gap between the APECs (see equation (5)). The exact expressions of F j (j = 1, 2) and V X in adiabatic representation can refer to reference [46]. For the cases with a 2 1, a 2 1 and a 2 1, Zhu and Nakamura suggested that the system can be well classified as the strong, intermediate and weak nonadiabatic coupling regime in the adiabatic representation (or weak, intermediate and strong diabatic coupling regime in the diabatic representation), respectively [51]. It should be mentioned that M.Child has also proposed a similar parameter in 1970s for the classification of nonadiabatic couplings [54].
Such a simple classification of the coupling strength is quite impressive, since a 2 can well indicate the situations that when the adiabatic representation or the diabatic representation could be employed efficiently. Explicitly, when a 2 1, the system is weakly coupled, all the 'bound' and 'continuum' states of the system could be well estimated from the APECs, and the weak nonadiabatic coupling (A nl ) slightly alters these states; while when a 2 1, everything is reversed, the system is strongly coupled, all the 'bound' and 'continuum' states of the system could be well estimated from the DPECs, and the weak diabatic coupling (V d ij ) slightly alters these states; when a 2 1, there would be no priority to employ either representation. Finally, we realize that the simple quantity a 2 is of crucial importance for the present interest of resonances in the AC region, since a 2 can efficiently and properly suggest us the 'bound' channels of resonances, while the 'bound' channels are quite important to build resonances following the picture of Fano. Our further investigations for different systems approve that a 2 can be well applied to quantitatively estimate the importance of resonances in the AC region, as revealed in the next section, for system with a 2 1 (a 2 1 ), resonance in the AC would abruptly (invisibly) change the particle scattering cross sections, and its contribution is unobtrusive when a 2 1.
The next section presents the applications and discussions. Further theories involved in the present studies are quantum-mechanical molecular orbital close-coupling method (QMOCC) [16,[35][36][37][38] for ion scattering cross sections, and contour deformation method (CDM) [55] for resonance parameters. All these methods have been well documented in the literatures provided and will not be repeated here any more.

Results and discussion
A typical model system of the Mulliken C + type [56] is firstly investigated, to comprehensively represent all the properties of resonances in the particle scattering. The Child-Lefebvre model [57] is of this type and slightly modified as In the studies, the diabatic coupling (V d 12 ) will be altered by changing β e to represent cases with different coupling strength (a 2 ), while the DPECs (V d 11 and V d 22 ) do not change. All the relevant parameters of the  potentials are listed in table 1 with reduced mass μ = 8 AMU, and the DPECs and APECs (transformed with equation (5)) are shown in figure 2 for different coupling cases. As clearly shown in figure 2, the upper APEC (V a 22 ) is a bound one and the lower APEC (V a 11 ) has a substantial barrier; with the increasing of β e , the AC region becomes broader and the energy gap between the APECs becomes larger, corresponding to the decreasing of nonadiabatic couplings between the APECs. In this model, the DPECs are fixed, F j does not change; the diabatic coupling V X and parameter a 2 change with the variations of β e , which is adjusted so that a 2 are 10, 1.0, 0.1 and 0.01, corresponding to values of β e being 575, 1238, 2670 and 5750 cm −1 , respectively.
With the full potential energy curves of figure 2 and supposing the lower state being the initial projecting channel, the elastic particle scattering cross sections are calculated by the advanced QMOCC method [16,[35][36][37][38]. To highlight the resonance effect in the AC region, the typical s-wave Feshbach resonances and elastic cross sections (ECS) are investigated as showcases, unless stated explicitly. Note that higher partial-waves can be studied in the same way without changes of the theory. The calculations have performed with full coupled channels (equation (1)   (n + 1/2)π and nπ, respectively [6,12]. Sharp structures on top of the background modulations are contributed by resonances, and some resonances can greatly destroy the background modulations and even contribute inverted window structures. Note that the phase shift would increase sharply by about π when the scattering energy passes the resonance, and generally decreases with the increasing of scattering energy [6,12], so there would be always the cases that the phase shift could be (n + 1/2)π or nπ, and the background modulations generally would appear accompanying with the resonances.
As also shown in figure 4, for the results between FC-and OC-models, when the scattering energy is below the potential barrier of V a 11 (left side of AC region), both models produce quite similar ECS and resonance structures except in the energy region close to AC region, but such differences become smaller and smaller with the increasing of β e or the decreasing of a 2 . While when the scattering energy is over the potential barrier of V a 11 , besides the significant differences between the background ECS of the two models, the FC-model is still featured by the resonance structures, while no remarkable resonance peaks exist in the OC-model; with the increasing of β e or the decreasing of a 2 , the background modulations of FC-model gradually fade out, and both models produce almost the same ECS except the sharp resonance structures. All these features of similarities and differences of the ECS are directly related to the coupling strength of the system.
When a 2 is far less than 1 (a 2 = 0.01 or β e = 5750 cm −1 , figure 4(d)), the upper channel (V a 22 ) is weakly coupled to the lower one (V a 11 ), and the upper channel can hardly affect the elastic scatterings, then the FCand OC-model would surely produce very similar results in the whole energy region, including the Shape resonances; Feshbach resonances exist once the system is coupled (FC-model) and their positions will move close to the bound states (vertical green-dotted line) of APEC V a 22 , but no resonance peaks can be observed in the AC region. While when a 2 is much larger than 1 (a 2 = 10 or β e = 575 cm −1 , figure 4(a)), the projecting channel (V a 11 ) is strongly coupled to the upper one (V a 22 ), so the differences of the ECS between the two models are quite large above the potential barrier of V a 11 , due to the contributions from the upper channel; the scattering and resonances are also strongly affected by the coupling even below the potential barrier around the AC region, such effects become tiny once the scattering energy is far below the potential barrier; all the resonance positions move to the bound states (vertical purple dash line) of diabatic potential V d 22 , indicating the efficiency and suitability to describe the strongly coupled system in the diabatic representation. In such a strongly coupled system, resonances in the AC region exhibit as sharp peaks (see the peak at ν V d 22 = 18 in figure 4(a)). When a 2 is about 1 (a 2 = 1 or β e = 1238 cm −1 (see figure 4(b)), both channels are coupled intermediately, the evolutions of ECS do not show much differences from that of strongly coupled case, however, Shape resonances below the potential barrier of In a short summary, resonances depend strongly on the electronic couplings and play important roles on the scatterings. For the weakly coupled system (a 2 1), resonances sit around the (quasi-) bound states of APECs, no resonances can be observed in the AC region; while resonances will move close to the bound states of DPECs when the system is strongly coupled (a 2 1), and resonances in the AC region exhibit significantly in the scattering; for the intermediately coupled system (a 2 1), resonances will be around both the bound states of APECs and DPECs, and the tiny roles of resonances in the AC region could also been recorded in the scattering. Surprisingly, all these features agree quite well with the classifications of coupling strength with the dimensionless parameter a 2 suggested by Zhu and Nakamura [51]. One can conclude that the exhibition of resonances in the AC region is directly related to the coupling strength or parameter a 2 , that for system with a 2 1 (a 2 1), resonances in the AC would abruptly (invisibly) change the cross sections, and its contribution is unobtrusive when a 2 1.
As a first application of these results to real molecules, the coupled predissociation system of MgH [58,59] is investigated. The potentials are described as with the reduced mass μ = 0.9672 AMU and other parameters listed in table 2. Resonances of this system have been extensively studied [58][59][60][61][62], here we focus on the ion scattering cross sections around the AC region. Figures 5(a) and (b) show the relevant results for the case of A int = 0.015 a.u., broadly studied in previous literatures [58][59][60][61][62]. Further calculation shows a 2 = 0.018, much smaller than 1, indicating this  predissociation system of MgH being a weakly coupled one. With the estimations from the previous model system, resonances in the AC region of this system should bury in the background modulations, which totally agrees with the ECS shown in figure 5(b) that only the background modulations are observed in the AC region. Resonance parameters from the present CDM [55] and previous CSM methods [58][59][60] are also shown in figure 5(a). Both methods produce the same resonance widths, however, the resonance parameters in the AC region can not be calculated due to its quite broad decay width. It turns invisible to read the broad resonance structures on the ECS ( figure 5(a)), the width of such broad resonance should bury in the ECS modulation background, we would expect the lower limit of the broad resonance width to be ∼ 10 3 cm −1 .
We also reduce the parameter A int to make artificial systems with stronger nonadiabatic couplings. Figures 5(c) and (d) and figures 5(e) and (f) show the relevant results for the cases with a 2 = 1 and a 2 = 10, respectively. As the figures show, resonances in the AC region exhibit significantly when a 2 = 1 ( figure 5(d)), and appear as sharp peaks when a 2 = 10 (figure 5(f)), totally consistent with the estimations of the model system; resonance parameters in the AC region can be calculated, and their widths become smaller with the increasing of coupling strength. Note that the scattering cross sections are proportional to |I − S(ε)| 2 , and resonances are the poles of S-matrix in the fourth quadrant of the complex energy plane as ε s = ε sr − i 2 Γ si [6], so when the scattering energy ε passes over the pole or resonance through ε sr , a peak structure would appear in the cross sections. Such a structure would be very sharp if Γ si is small or the pole is close to the real energy axis; while this structure could never be observed if the resonance width is broad or the pole is far away from the real energy axis. So the exhibitions of resonances in the AC region are essentially related to their widths, and it looks that their widths would be small for strongly coupled system (a 2 1), and quite broad for weakly coupled one (a 2 1) in the AC regions. Note that the well-isolated resonance peaks can also be indicated by the Fano q-factor [5,7], in figure 5 (b) the sharp peak in the lower Figure 6. (a) Resonance parameters calculated for the coupled B 1 Σ + − D 1 Σ + system of CO by CSM [63] (triangle) and present CDM (square); (b) s-wave elastic particle scattering cross section by the present QMOCC, the red-bold line is in the AC region. energy region is q ≈ −0.465, and the two sharp peaks in the higher energy region are q ≈ 3.537 and 1.439, respectively; besides, the well-formed resonance peak in AC region of figure 5(f) is q ≈ −0.551.
The coupled B 1 Σ + − D 1 Σ + states of CO are also illustrated, with all the potentials and parameters from reference [64], and a 2 = 0.158 is far less than 1. Figure 6 shows the resonance parameters and the s-wave ECS from the lower state B 1 Σ + around the AC region. As shown in figure 6(a), both theoretical methods (the present CDM and CSM in the literature [63]) can produce the same resonance parameters, and many resonances are quite broad. The s-wave ECS are dominated by the background modulations ( figure 6(b)), no resonances in the AC region can be observed in the cross sections, consistent with the estimations from the value of a 2 . Note that four sharp resonance peaks can be observed away from the AC region, corresponding to the narrow resonances (Γ si < 90 cm −1 ); other resonances (including the ones in the AC region) are quite broad (Γ si > 600 cm −1 ) and play invisible roles on the cross sections. Figures 5(b) and 6(b) also show that the background modulations are more pronounced in the AC region in CO than in MgH, it could be related to the fact that CO is about 7 times heavier than MgH. CO will experience much more rapid changes of wave numbers and phases than MgH in the AC energy region of ∼ 6 × 10 3 cm −1 (see figures 5(b) and 6(b)), and more modulation structures are expected.
Extended applications to the existing results from the literatures also approve the present studies. Taking the OH molecule as an example, figure 7 of reference [32] shows its APECs are sharply avoided, a 2 ≈ 14. 5 1 can be easily obtained. We will estimate that the resonances in the AC region should play very important roles in such a strongly coupled system. As shown in figures 6(b) and 8 of reference [32], the photodissociation cross sections surely exhibit a very sharp peak in the AC region (at about 10 eV), totally consistent with the determinations from a 2 . The generality and applicability of the present conclusions would be expected.

Conclusions
Resonances in the AC region on particle scattering are comprehensively investigated. The contributions of resonances in the AC region to the cross sections depend strongly on the coupling strength of the system, and the simple quantity a 2 proposed by Zhu and Nakamura can be well applied to quantitatively estimate the importance of resonances in the AC region. For the strongly (weakly) coupled system with a 2 1 (a 2 1), resonances in the AC region would abruptly (invisibly) change the cross sections, and their contributions are unobtrusive when a 2 1. Example applications of the quantity a 2 for different system can well explain the evolutions of cross sections in the AC region. Further analysis shows that the present conclusions are essentially related to the resonance widths, that narrow resonances could exhibit strongly as sharp peaks, and broad ones would bury in the background. Practically, this work provides an easy way to determine the candidate molecules when studying the resonances in the AC region.