This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The vibrational nonlinear dynamics of HOBr in the bending and O–Br stretching coordinates with anharmonicity and Fermi 2:1 coupling are studied with dynamical potentials in this article. The result shows that the H–O stretching vibration mode has significantly different effects on the coupling between the O–Br stretching mode and the H–O–Br bending mode under different Polyad numbers. The dynamical potentials and the corresponding phase space trajectories are obtained when the Polyad number is 27, for instance, and the fixed points in the dynamical potentials of HOBr are shown to govern the various quantal environments in which the vibrational states lie. Furthermore, it is also found that the quantal environments could be identified by the numerical values of action integrals, which is consistent with former research.

Atmospheric pollution has caused widespread concern because it has led to not only a lot of diseases but also serious damage to the ecological balance. HOBr is a very unstable oxidizing substance in the atmosphere. Radical generation of Bromine (Br) is caused by the O–Br bond breaking in highly excited vibrations of HOBr molecules. It is known that photolysis of the refrigerator refrigerant Freon produces bromine (Br), which leads to the destruction of atmospheric O_{3}[

The dynamics of bending and OBr stretching vibration of HOBr molecules have already been fully studied, and the main research methods are the first principle calculations or semi-classical methods [

In this work, the dynamic potentials of highly excited vibrational states of HOBr in bending and of O–Br stretching coordinates with anharmonic resonance and Fermi coupling will be shown. The effect of the H–O stretching vibration mode on the O–Br stretching mode and the H–O–Br bending mode under different Polyad numbers will be investigated and the dynamical potentials, including the corresponding phase space trajectories will also be studied, which is helpful to understand the dynamics of highly excited vibrational states.

The dynamical properties of highly excited vibrational states of HOBr in the energy region of 5 × 10^{3}–2.5 × 10^{4} cm^{−1} are abundant and are the focus of this work [

The corresponding coefficients are shown in ^{+}_{2} and _{3} mode in HOBr and is observed in reference [

Besides the conserved quantities of the _{1}, considering the 2:1 Fermi resonance, 2_{2} + _{3} is also a conserved quantity as a whole, which is called the Polyad number (P number). It is known that _{1} ≤ 7,

It is also easy to semi-classify the second quantization Hamiltonian (1), which is useful in the further analysis. The coset space SU(2)/U(1) [_{2}, _{2}) as follows:

With the coordinates (_{3}, _{3}), the Hamiltonian can be written as:

With the semi-classical Hamiltonian, the above-mentioned can further be used to obtain the dynamics potential and, in this way, we can study the dynamic nature of the HOBr’s highly excited vibrational states.

The dynamical potential of _{i}_{i}_{i}_{i}_{i}_{2} and n_{3} are nonnegative. The dynamical potential composed of these maximal and minimal energies as a function of _{i}_{i}_{i}

The dynamical potentials of HOBr (in coordinates _{2}, _{3},) with _{1} = 0,1,2,3 of different P numbers is considered, and their levels can be calculated with the aforementioned Hamiltonian (ground state from the potential energy at the bottom of 2764.0 cm^{−1}). It is found that the dynamical potentials are different for small and large

_{1} is equal to 0,1,2,3, the dynamical potentials in the _{2} coordinates are all simple inverse Morse potentials, which shows that the n_{1} mode has little effect on the dynamical potentials of the _{2} coordinates. In the theory of the dynamical potential [_{1}, which shows that the n_{1} mode has almost no significant effect on the coupling of _{2} and _{3}. In contrast, the dynamical potentials of HOBr in the _{3} coordinate consist of an inverted Morse potential and a positive Morse potential, and with the increase of _{1}, the top of the inverted Morse potential gradually flattens and the positive Morse potential wells become deeper, which shows that the dynamical potential is turning into an almost pure positive Morse potential, especially when _{1} is equal to three. As a result, the stability of the high energy levels becomes worse while the original fixed-point still remains the same. All the above phenomena show that, although the n_{1} mode is not coupled with the _{2} and _{3}, it still has some effect on the resonant coupling of the other two modes, and this impact is reflected in the stability of the stretching mode (when _{1} is equal to three, this impact is particularly evident). Or said differently, the H–O stretching mode has an important influence on the dynamics of the other degrees of freedom. According to qualitative analysis, this is because the intramolecular vibrational relaxation (IVR), caused by the resonance between _{2} and _{3}, which enhances the stability of the energy levels [

For larger

When _{1} is equal to three, the geometrical shape of the dynamical potentials of both _{2} and _{3} becomes increasingly complex and the number of the potential wells also increases. There are two inverted Morse in the _{2} dynamical potentials, yet there are two positive Morse in the _{3} dynamical potentials. This shows that with the increase of the _{1} mode, the dynamical potential system consists of _{2} and n_{3}, which makes it more complex. This conclusion confirms once again that the n_{1} mode has some effect on the dynamical behavior of the other mutually coupled resonance modes and that this effect is more apparent compared with the case of

_{2} dynamical potential with _{1} = 3 is more complex than the one with _{1} = 0,1,2 and that the [_{2}] fixed point appears. This fixed point leads to the vibration modes corresponding to the highest three energy levels in the dynamical potential becoming localized vibrational when _{1} is equal to three (in the two inverted Morse potentials). In addition, the fixed points of the dynamical potentials in _{3} remain the same, but when _{1} is equal to three, the vibration mode corresponding to the low energy level becomes localized vibrational (in the two positive Morse potentials). Compared with the dynamical potentials when P is equal to 15, it is obvious that the effect of n_{1} on _{2} (or _{3}’s resonant coupling) is closely related to the value of the P number. The general conclusion is that n_{1} has little effect on the geometrical shape of the dynamical potential of _{2}, _{3} when the P number is small, however, this effect becomes larger when the

In the above-mentioned study, it was found that although the H–O stretching mode was not coupled with the H–O–Br bending mode and the O–Br stretching mode, it still affected the resonant coupling of the two modes, thereby affecting the dynamics of HOBr. It is elucidated with qualitative analysis that the geometrical shape of the dynamical potentials and the corresponding fixed points are sensitive to the change of the H–O stretching mode in the sense of geometry. However, this phenomenon and the related explanation should be verified in further studies of other molecular systems.

The qualitatively different quantum number n_{1} of the HOBr system corresponding to dynamical potentials and characteristics are discussed in the previous section. Next we will discuss quantitatively the dynamical potentials of the specific _{2} and _{3} coordinates, where the horizontal lines show the energy levels sharing the designated P. The reason why the case of _{1} is equal to three are as follows:

The energy is with respect to the ground state which is 2764.0 cm^{−1} above the bottom of the potential well, and the points in _{i}_{i}

To further quantitatively analyze the features of related energy levels, the representative trajectories of phase space in _{i}_{i}_{2} coordinates constitute a class and the ones of the L0–L1, L2, L3, L4–L6, L7–L13 corresponding levels, respectively, constitute a class in _{3} coordinates. They are partly shown in

In

In

In order to study the quantal environment of each energy level, the calculation of the action integral of the phase space trajectories (quantum number) is needed and the corresponding formula is as follows:

The results are shown in

_{2}, _{2}) and (_{3}, _{3}). From the action integrals calculated from (_{2}, _{2}), the levels can be grouped into two subsets: L0 to L2 and L3 to L13. For the former ones, the action integral increases almost in a constant with the decline of energy levels due to the inverse-Morse well where they are located, and for the latter ones [_{3}, _{3}), the levels can be grouped into two subsets, also with constant increment (decrement) in their action integrals: L0 to L6 and L7 to L13. For the former ones, the action integral increases with the decline of energy levels due to the inverse-Morse well where they are located, but with minor deviation on L2, and for the latter ones, the action integral decreases with the decline of energy levels due to the positive Morse wells where they are located. The constant action integral increment/decrement demonstrates the compatibility of our classical treatment with the quantized levels and also shows that the levels stay in various dynamical environments, which are defined essentially by the classical fixed points though these energies belong to the same Polyad number. In addition, from the view of the (_{2}, _{2}) space, the difference between adjacent energy levels’ action integrals is almost always the same (approximately equal to one, but the L1 is an exception and should be further studied in the future). This value is equal to two if in the view of (_{2}, _{2}) because _{2} + _{3}, which is also shown in _{2}, _{2}), which corresponds exactly with the system of 2:1 Fermi resonance. The dynamics potential is actually the same as the trajectory connecting each fixed point, depicting the restrictive conditions in which individual energy levels are located.

The highly excited vibrational state is hard to study using the quantum mechanical calculation because of its nonlinear interactions, but it is full of important information. The algebraic Hamiltonian, action integral and dynamical potential can bring direct physical pictures into the geometric sense of this research field. It is well known that the effective Hamiltonians with a single resonant interaction only are completely integrable systems and that the quantization can be done by the quantization of action integrals [_{3}, NO_{2}, CS_{2}, C_{2}H_{2} and so on), including their dissociation, isomerization and dynamical symmetry. These conclusions are helpful and significant for us to understand the dynamics of molecular highly excited vibrational dynamics.

This work was supported by the Natural Science Foundation of China (Grant No. 11104156), and the Natural Science Foundation of Jiangxi Province (Grant No. 2009GZW0001), the Open Foundation of Engineering Research Center of Nuclear Technology Application, Ministry of Education (Grant No. HJSJYB2011-05).

The authors declare no conflict of interest.

The dynamical potentials of HOBr (_{1} = 0,1,2,3. The energy levels included in the P number are represented by the lines.

The dynamical potentials of HOBr (_{1} = 0,1,2,3. The energy levels included in the P number are represented by the lines.

The dynamical potentials in _{2} (_{3} (

The trajectories of phase space of L1, L2, L3, L5, L7, L9 (_{2} coordinates).

The trajectories of phase space of L1, L2, L3, L5, L7, L9 (_{3} coordinates).

The coefficient of HOBr molecular vibration Hamiltonian.

Parameter name | Parameter values (cm^{−1}) |
---|---|

_{1} |
3769.5381 |

_{2} |
1187.7106 |

_{3} |
622.3722 |

_{11} |
−70.7877 |

_{12} |
−26.8651 |

_{13} |
10.3315 |

_{22} |
−5.4432 |

_{23} |
−3.7702 |

_{33} |
−3.5507 |

_{113} |
−3.4216 |

_{122} |
−1.1955 |

_{133} |
−0.9150 |

_{223} |
−0.6071 |

_{1133} |
0.2036 |

_{1333} |
0.0121 |

_{2333} |
−0.0015 |

_{3333} |
−0.0006 |

_{11112} |
0.3505 |

_{11122} |
−0.6569 |

_{11222} |
0.2578 |

_{11223} |
0.0326 |

_{12222} |
−0.0202 |

_{22233} |
0.0033 |

_{22333} |
−0.0017 |

_{1} |
0.8717 |

_{2} |
−0.3183 |

_{3} |
−0.1935 |

_{22} |
−0.0170 |

The action integrals of the levels corresponding to

State label | Action integral in(_{2}, _{2}) space |
Difference of the action integrals the neighboring levels | State label | Action integral in(_{3}, _{3}) space |
Difference of the action integrals the neighboring levels |
---|---|---|---|---|---|

L0 | 0.20 | / | L0 | 0.39 | / |

L1 | 0.61 | 0.41 | L1 | 1.21 | 0.82 |

L2 | 1.64 | 1.03 | L2 | 3.27 | 2.06 |

L3 | 10.83 | / | L3 | 5.32 | 2.05 |

L4 | 9.45 | 1.38 | L4 | 8.08 | 2.76 |

L5 | 8.29 | 1.14 | L5 | 10.41 | 2.33 |

L6 | 7.18 | 1.11 | L6 | 12.70 | 2.29 |

L7 | 6.12 | 1.06 | L7 | 12.27 | / |

L8 | 5.13 | 0.99 | L8 | 10.27 | 2.00 |

L9 | 4.19 | 0.94 | L9 | 8.37 | 1.90 |

L10 | 3.19 | 1.00 | L10 | 6.39 | 1.98 |

L11 | 2.19 | 1.00 | L11 | 4.37 | 2.02 |

L12 | 1.18 | 1.01 | L12 | 2.37 | 2.00 |

L13 | 0.19 | 0.99 | L13 | 0.37 | 2.00 |