Consider the $\mathrm{H_2O}$ molecule with $C_{2v}$ symmetry, where the potential energy surface of its highly excited vibrational states exhibits significant anharmonicity and vibration-rotation coupling. Considering higher-order perturbations (such as Darling-Dennison resonance, Fermi resonance, Coriolis coupling) and isotopic substitution, select the statement that does not contain a correct description.

1. Under this symmetry, the symmetric stretching $\nu_1$ and antisymmetric stretching $\nu_3$ both belong to the $A_1$ irreducible representation, so their Fermi resonance must satisfy $\Delta v=0$.
2. The $v=2$ energy level of the bending vibration $\nu_2$ can be coupled with $\nu_1+\nu_3$ through generalized Darling-Dennison resonance, which needs to satisfy the rotational selection rule of $\Delta K_a=\pm2$.
3. The substitution of the $\mathrm{H}$ atom with a $\mathrm{D}$ atom will significantly reduce the frequency of $\nu_3$, but the frequency change of $\nu_2$ is mainly determined by anharmonicity corrections.
4. In this molecule, there is no rotation-induced Coriolis coupling interaction between $\nu_1$ and $\nu_3$.
5. In the rovibrational spectrum of $\mathrm{H_2{^{18}O}}$, the $Q$ branch of the $\nu_3$ band is almost completely forbidden due to Coriolis coupling, but the $Q$ branch appears in $\mathrm{HD {^{18}}O}$ obtained after isotopic substitution due to the reduction in symmetry.
6. If $\nu_1$ and $2\nu_2$ undergo Fermi resonance, its resonance intensity is proportional to the cubic anharmonic coefficient $k_{122}$. In $\mathrm{D_2O}$, the vibrational frequency of $\nu_2$ decreases, and the resonance disappears.
7. Higher-order centrifugal distortion terms (such as $H_{JK}$) lead to $K_a-$dependent splitting in the rotational energy levels of $\mathrm{H_2O}$, and its sign may be opposite to that of $\mathrm{D_2O}$ due to the different mass dependence of rotational constants $B$ and $C$.
8. In the $\nu_2$ excited state $v2=1$ of $\mathrm{H_2O}$, the energy gap $\Delta E$ of $l-$type doubling $(l=\pm1)$ can be estimated by second-order perturbation theory: $\Delta E\propto \zeta^2_2(J+1)/B$, where $\zeta_2$ is the Coriolis coupling coefficient.
9. The hyperfine structure of $\mathrm{H_2{^{17}}O}$ originates from nuclear spin-rotation coupling, and the $F-$branch splitting interval of its $\nu_3$ band is linearly related to $J$.
10. In the $\nu_1+\nu_3$ combination band of $\mathrm{H_2O}$, the rotational transition intensity of $\Delta K_a=\pm1$ is modulated by the Herman-Wallis factor, and its functional form is $(1+aK_a^2)$.
考虑对称群为 $C_{2v}$ 的 $\mathrm{H_2O}$ 分子，其高激发振动态的势能面存在显著的非谐性和振动-转动耦合。在考虑高阶微扰（如 Darling-Dennison 共振、Fermi 共振、Coriolis 耦合）和同位素取代的情况下，选出不含正确叙述的一项。

1. 此对称性下，对称伸缩 $\nu_1$ 和反对称伸缩 $\nu_3$ 均属于 $A_1$ 不可约表示，因此它们的Fermi共振需满足 $\Delta v=0$。
2. 弯曲振动 $\nu_2$ 的 $v=2$ 能级可通过广义Darling-Dennison共振与 $\nu_1+\nu_3$ 耦合，需要满足 $\Delta K_a=\pm2$ 的转动选择定则。
3. $\mathrm{D}$ 原子对 $\mathrm{H}$ 原子的取代会显著降低 $\nu_3$ 的频率，但 $\nu_2$ 的频率变化主要由非谐性修正决定。
4. 在此分子中 $\nu_1$ 和 $\nu_3$ 之间无转动诱导的Coriolis耦合相互作用。
5. 在$\mathrm{H_2{^{18}O}}$的振转光谱中，$\nu_3$ 带的 $Q$ 支因 Coriolis 耦合几乎完全禁阻，但同位素取代后得到的 $\mathrm{HD {^{18}}O}$ 因对称性降低而出现 $Q$ 支。
6. 若 $\nu_1$和 $2\nu_2$ 发生 Fermi 共振，其共振强度正比于立方非谐系数 $k_{122}$。在 $\mathrm{D_2O}$ 中 $\nu_2$ 振动频率降低，共振消失。
7. 高阶离心畸变项（如$H_{JK}$）在 $\mathrm{H_2O}$ 的转动能级中导致 $K_a-$ 依赖的分裂，其符号可能与 $\mathrm{D_2O}$ 相反，因转动常数 $B$ 和 $C$ 的质量依赖性不同。
8. 在 $\mathrm{H_2O}$ 的 $\nu_2$ 激发态 $v2=1$ 中，$l-$型双重分裂 $(l=\pm1)$ 的能隙 $\Delta E$ 可由二阶微扰理论估算：$\Delta E\propto \zeta^2_2(J+1)/B$，其中 $\zeta_2$ 是 Coriolis 耦合系数。
9. $\mathrm{H_2{^{17}}O}$ 的超精细结构源于核自旋-转动耦合，其 $\nu_3$ 带的 $F-$支分裂间距与 $J$ 成线性关系。
10. 在 $\mathrm{H_2O}$ 的 $\nu_1+\nu_3$组合带中，$\Delta K_a=\pm1$ 的转动跃迁强度受 Herman-Wallis 因子调制，其函数形式为 $(1+aK_a^2)$。
The remaining options do not meet the criteria.
1, 2, 3, 4, 5
1, 3, 5, 8, 10
2, 4, 5, 6, 7
3, 6, 7, 8, 9
1, 3, 4, 6, 9
4, 7, 8 ,9 ,10
2, 5, 7, 9, 10
其余选项均不符合条件
1, 2, 3, 4, 5
1, 3, 5, 8, 10
2, 4, 5, 6, 7
3, 6, 7, 8, 9
1, 3, 4, 6, 9
4, 7, 8 ,9 ,10
2, 5, 7, 9, 10
The $\mathrm{H_2O}$ molecule with $C_{2v}$ symmetry has three vibrational modes: the symmetric stretching vibration $\nu_1$ and the bending vibration $\nu_2$ both belong to the $A_1$ irreducible representation, and the antisymmetric stretching vibration $\nu_3$ belongs to the $B_2$ irreducible representation. Furthermore, $\Delta\nu=0$ is not a condition for Fermi resonance. Statement 1 is incorrect.

<Checkpoint>[Vibration $\nu_3$ belongs to the $B_2$ irreducible representation, and $\Delta\nu=0$ is not a condition for Fermi resonance. Statement 1 is incorrect.](0.5)</Checkpoint>

Under $C_{2v}$ symmetry, $2\nu_1$ belongs to $A_1\otimes A_1=A_1$, and $\nu_1+\nu_3$ belongs to $A_1\otimes B_2=B_2$. The symmetries of the two do not match. For the $H_2O$ molecule, $K_a$ is the rotational quantum number around the $C_2$ axis. When it is even, the symmetry of the rotational wave function is $A_1$ or $A_2$, and the eigenvalue for the $C_2$ operation is $+1$. When it is odd, the symmetry of the rotational wave function is $B_1$ or $B_2$, and the eigenvalue for the $C_2$ operation is $-1$. When a rotational transition of $\Delta K_a=\pm2$ occurs, the same parity of $\Gamma_{rotation}$ is maintained. At this time, the symmetry of the vibrational state becomes $A_1\otimes\Gamma_{rotation}$ and $B_2\otimes\Gamma^\prime_{rotation}$, introducing a non-zero vibration-rotation cross term, making the transition allowed. Statement 2 is correct.

<Checkpoint>[Rotational transitions of $\Delta K_a=\pm2$ introduce a non-zero vibration-rotation cross term, making the transition allowed. Statement 2 is correct.](0.5)</Checkpoint>

The basic formula for vibrational frequency is $\nu=\dfrac1{2\pi}\sqrt{\dfrac k\mu}$. The substitution of the $\mathrm{D}$ atom for the $\mathrm{H}$ atom increases the reduced mass $\mu=\dfrac{m_{\mathrm O}m_{\mathrm H}}{m_{\mathrm O}+m_{\mathrm H}}$ by approximately 89%, and the $\nu_3$ frequency decreases by approximately 27%. For the bending vibration $\nu_2$, it is obtained through constructing a geometric model that $\mu \approx \dfrac{m_{\mathrm H}m_{\mathrm O}}{2(M_{\mathrm O}+2M_{\mathrm H})}\times r^2 \times \cos^2\dfrac{\theta}2$. Therefore, the substitution of the $D$ atom for the $H$ atom increases the reduced mass by approximately 81%, and the $\nu_2$ frequency decreases by approximately 26%. From this, it can be seen that the frequency changes of $\nu_2$ and $\nu_3$ are mainly caused by changes in the reduced mass, rather than nonlinear corrections. Statement 3 is incorrect.

<Checkpoint>[The frequency changes of $\nu_2$ and $\nu_3$ are mainly caused by changes in the reduced mass, rather than nonlinear corrections. Statement 3 is incorrect.](0.5)</Checkpoint>

Coriolis coupling requires that the direct product of the two vibrational modes contains the symmetry of the angular momentum component, i.e., $ \Gamma(\hat J_a)\subset\Gamma(\psi_i)\otimes\Gamma(\psi_j)$. For the two vibrational states $\nu_1$ and $\nu_3$, we have $A_1\otimes B_2=B_2$, and the rotation operator $\hat J_Y $ belongs to $B_2$ symmetry, so $\nu_1$ and $\nu_3$ can be coupled through $\hat J_Y $. Statement 4 is incorrect.

<Checkpoint>[ $\nu_1$ and $\nu_3$ can be coupled through $\hat J_Y $. Statement 4 is incorrect.](0.5)</Checkpoint>

The antisymmetric stretching vibration $\nu_3$ has intrinsic angular momentum, and Coriolis coupling causes an increase in vibrational angular momentum, so the rotational angular momentum must decrease accordingly. However, the $Q$ branch requires that the rotational angular momentum remains unchanged, i.e., $\Delta J=0$, so it is almost completely forbidden. The symmetry of $\mathrm{HD {^{18}}O}$ obtained after isotope substitution is reduced from $C_{2v}$ to $C_s$, and all vibrational modes become $A^\prime$. The symmetry constraint is greatly reduced, so the $Q$ branch can be observed. Statement 5 is correct.

<Checkpoint>[ The $Q$ branch is almost completely forbidden because it requires the rotational angular momentum to remain unchanged; the symmetry of $\mathrm{HD {^{18}}O}$ is reduced, allowing it. Statement 5 is correct.](0.5)</Checkpoint>

The anharmonic Hamiltonian is $\hat H_{非谐}=\sum k_{i,j,k}\times q_iq_jq_k$, so the coupling matrix element of $\nu_1$ and $2\nu_2$ is $W=\langle \nu_1|\hat H_{非谐}|\nu_2 \rangle=\langle \nu_1|K_{122}q_1q_2^2|\nu_2 \rangle$, which is proportional to $k_{122}$. In $\mathrm{D_2O}$, the isotope mass effect causes the frequencies of $\nu_1$ and $\nu_2$ to decrease almost proportionally, and the resonance is weakened due to the increase in the energy difference, but it still exists. Statement 6 is incorrect.

<Checkpoint>[Coupling matrix element $W=\langle \nu_1|K_{122}q_1q_2^2|\nu_2 \rangle\propto k_{122}$. In $\mathrm{D_2O}$, the resonance is weakened due to the increase in the energy difference, but it still exists. Statement 6 is incorrect.](0.5)</Checkpoint>

The rotational Hamiltonian is in the form of $\hat H_{rot} =A\hat J_a^2+B\hat J_b^2 + C\hat J_c^2 - D_J \hat J^4 - D_{JK}\hat J^2\hat J_a^2 - D_{K}\hat J_a^4 +H_{JK}\hat J^2\hat J_a^2+...$, where for the $H_{JK}$ term we have $H_{JK}\hat J^4K_a^2$. Similarly, processing other terms, the relationship between the rotational energy level and $K_a$ can be expressed as $E(J,K_a)=BJ(J+1)+(C-B)K_a^2-D_JJ^2(J+1)^2-D_KK_a^4+H_{JK}K_a^2J^4+...$, where higher-order centrifugal distortion terms such as $D_J,D_K,H_{JK}$ all introduce $K_a-$ dependent energy level splitting. The dependence of the rotational constant on mass is expressed through the moment of inertia, i.e., $B=\dfrac{\hbar^2}{2I_b},C=\dfrac{\hbar^2}{2I_c}$. Considering the sixth-order dominant term $H_{JK}\propto \dfrac{(B-C)^3}{\omega^2}$, which depends on the rotational constants $B,C$, the mass effect brought about by isotope substitution causes a change in the magnitude of the moment of inertia, so $B-C$ may change sign, and the corresponding $H_{JK}$ changes sign accordingly. Statement 7 is correct.

<Checkpoint>[Sixth-order dominant term $H_{JK}\propto(B-C)^3$, the mass effect causes a change in the magnitude of the moment of inertia, $B-C$ may change sign, and $H_{JK}$ changes sign accordingly. Statement 7 is correct.](0.5)</Checkpoint>

In the excited state of the bending vibration $\nu_2$, the $l-$type doubling $(l=\pm1)$ originates from Coriolis coupling. Rotational energy level mixing in the second-order perturbation theory introduces the intermediate state $|v_2=0,J^\prime\rangle$, obtaining $\Delta E\approx2\sum_{J^\prime}\dfrac{\langle v_2=1,l=+1∣\hat H_{\text{Coriolis}}|v_2=0,J^\prime\rangle\langle v_2=0,J^\prime∣\hat H_{\text{Coriolis}}|v_2=1,l=-1\rangle}{E_{v=1}-E_{v=0}}$. Where $\hat H_{\text{Coriolis}}=-2\sum_{\alpha=a,b,c} B_\alpha\zeta_\alpha^{(\Gamma)}J_\alpha l_\alpha$. For $\nu_2$ belonging to the $A_1$ irreducible representation, only $\zeta_b^{(A_1)}\propto\zeta_2\neq0$, obtaining $\hat H_{\text{Coriolis}}=-2 B_b\zeta_2J_b l_b$. Substituting $E_{v=1}-E_{v=0}\approx \hbar \omega_2,J^\prime=\pm1$ and the quantization condition, we get $\Delta E\propto \dfrac{(2B_b\zeta_2J_b*)^2}{\hbar\omega_2}\propto\dfrac{\zeta_2^2(J+1)}{B}$. Statement 8 is correct.

<Checkpoint>[Introducing the intermediate state $|v_2=0,J^\prime\rangle$, substituting the energy gap and the quantization condition, we get $\Delta E\propto\dfrac{\zeta_2^2(J+1)}{B}$. Statement 8 is correct.](0.5)</Checkpoint>

The nuclear spin of $^{17}O$ is $I = 5/2$, which couples with the angular momentum $J$ of the molecular rotation to form a total angular momentum $F = J + I$, leading to hyperfine splitting of the rotational energy levels. The main term of the hyperfine Hamiltonian is $H_Q \propto 3(I \cdot J)^2 - I^2J^2$, expanding $(I \cdot J)^2 = \dfrac14(F^2-I^2-J^2)$, substituting the energy level correction $\Delta E_Q \propto \dfrac34 K^2-I(I+1)J(J+1)$, where $K = F(F+1) - I(I+1) - J(J+1)$, and $F$ is the total angular momentum quantum number. Therefore, the splitting spacing is not linearly related to $J$. Statement 9 is incorrect.

<Checkpoint>[Energy level correction $\Delta E_Q \propto \dfrac34 K^2-I(I+1)J(J+1)$, therefore the splitting spacing is not linearly related to $J$. Statement 9 is incorrect.](0.5)</Checkpoint>

The Herman-Wallis factor describes the change in rotational transition intensity with $K_a$. The sign and magnitude of the parameter $a$ are related to the rotational constant, vibration-rotation coupling, etc. of the molecule. For transitions of $\Delta K_a=\pm1$, let the rotation-vibration wave function of the molecule be $|v, J, K_a\rangle$, the transition dipole moment element $M = \langle v', J', K_a'|\mu|v, J, K_a\rangle$. Considering the Herman-Wallis effect, the vibrational wave function can be expanded as $|v\rangle = |v_0\rangle + \alpha K_a^2 |v_0\rangle$, so the transition dipole moment element is expanded as $M \approx \langle v_0', J', K_a'|\mu|v_0, J, K_a\rangle + \alpha K_a^2 \langle v_0', J', K_a'|\mu|v_0, J, K_a\rangle$. The transition intensity $I \propto |M|^2 \approx I_0(1 + 2\alpha K_a^2)$. Therefore, the Herman-Wallis factor is $(1+aK_a^2)$, where $a=2\alpha$. Statement 10 is correct.

<Checkpoint>[Vibrational wave function expansion $|v\rangle = |v_0\rangle + \alpha K_a^2 |v_0\rangle$, transition intensity $I \propto |M|^2 \approx I_0(1 + 2\alpha K_a^2)$. The Herman-Wallis factor is $(1+aK_a^2)$, where $a=2\alpha$. Statement 10 is correct.](0.5)</Checkpoint>

The incorrect statements are 1, 3, 4, 6, 9, so choose F.

<Checkpoint>[The incorrect statements are 1, 3, 4, 6, 9, so choose F.](0.5)</Checkpoint>
对称群为 $C_{2v}$ 的 $\mathrm{H_2O}$ 分子有三种振动模：对称伸缩振动 $\nu_1$ 和弯曲振动 $\nu_2$ 都属于 $A_1$ 不可约表示，反对称伸缩振动 $\nu_3$ 属于 $B_2$ 不可约表示。此外$\Delta\nu=0$并不是Fermi共振的条件。叙述1错误。

<Checkpoint>[振动 $\nu_3$ 属于 $B_2$ 不可约表示， $\Delta\nu=0$并不是Fermi共振的条件。叙述1错误。](0.5)</Checkpoint>

在 $C_{2v}$ 对称性下， $2\nu_1$ 属于 $A_1\otimes A_1=A_1$， $\nu_1+\nu_3$ 属于 $A_1\otimes B_2=B_2$，二者对称性不匹配。对 $H_2O$ 分子而言 $K_a$ 是绕 $C_2$ 轴的转动量子数，为偶数时转动波函数对称性为 $A_1$ 或 $A_2$， 对 $C_2$ 操作本征值为 $+1$，为奇数时转动波函数对称性为 $B_1$ 或 $B_2$， 对 $C_2$ 操作本征值为 $-1$。当发生 $\Delta K_a=\pm2$ 的转动跃迁时，保持了 $\Gamma_{转动}$ 的相同奇偶性 。此时振动态的对称性变为 $A_1\otimes\Gamma_{转动}$ 和 $B_2\otimes\Gamma^\prime_{转动}$ ，引入了非零的振动-转动交叉项，使跃迁变得允许。叙述2正确。

<Checkpoint>[ $\Delta K_a=\pm2$ 的转动跃迁引入了非零的振动-转动交叉项，使跃迁变得允许。叙述2正确。](0.5)</Checkpoint>

振动频率基本公式 $\nu=\dfrac1{2\pi}\sqrt{\dfrac k\mu}$，$\mathrm{D}$ 原子对 $\mathrm{H}$ 原子的取代使约化质量 $\mu=\dfrac{m_{\mathrm O}m_{\mathrm H}}{m_{\mathrm O}+m_{\mathrm H}}$ 增大约89%， $\nu_3$ 频率下降约27%。对于弯曲振动 $\nu_2$，通过构建几何模型得到 $\mu \approx \dfrac{m_{\mathrm H}m_{\mathrm O}}{2(m_{\mathrm O}+2m_{\mathrm H})}\times r^2 \times \cos^2\dfrac{\theta}2$，因此 $D$ 原子对 $H$ 原子的取代使约化质量增大约81%， $\nu_2$ 频率下降约26%。由此可知 $\nu_2$ 和 $\nu_3$ 频率变化都主要由约化质量变化引起，而不是非线性修正。叙述3错误。

<Checkpoint>[ $\nu_2$ 和 $\nu_3$ 频率变化都主要由约化质量变化引起，而不是非线性修正。叙述3错误。](0.5)</Checkpoint>

Coriolis 耦合要求两个振动模的直积包含角动量分量的对称性，即 $ \Gamma(\hat J_a)\subset\Gamma(\psi_i)\otimes\Gamma(\psi_j)$。对 $\nu_1$ 和 $\nu_3$ 两个振动态有 $A_1\otimes B_2=B_2$，而转动算符 $\hat J_Y $ 属于 $B_2$对称性，故 $\nu_1$ 和 $\nu_3$ 可以通过 $\hat J_Y $ 耦合。叙述4错误。

<Checkpoint>[  $\nu_1$ 和 $\nu_3$ 可以通过 $\hat J_Y $ 耦合。叙述4错误。](0.5)</Checkpoint>

反对称伸缩振动 $\nu_3$ 具有内禀角动量，而Coriolis耦合造成了振动角动量增加，相应地转动角动量必须减少。但 $Q$ 支要求转动角动量不变，即 $\Delta J=0$，因此几乎完全禁阻。同位素取代后得到的 $\mathrm{HD {^{18}}O}$ 对称性从 $C_{2v}$ 降低到 $C_s$，所有振动模式均变为 $A^\prime$，对称性约束大大减弱，因此可以观测到 $Q$ 支。叙述5正确。

<Checkpoint>[  $Q$ 支因要求转动角动量不变几乎完全禁阻；$\mathrm{HD {^{18}}O}$ 对称性降低而允许。叙述5正确。](0.5)</Checkpoint>

非谐哈密顿量 $\hat H_{非谐}=\sum k_{i,j,k}\times q_iq_jq_k$，因此 $\nu_1$和 $2\nu_2$ 的耦合矩阵元 $W=\langle \nu_1|\hat H_{非谐}|\nu_2 \rangle=\langle \nu_1|K_{122}q_1q_2^2|\nu_2 \rangle$，正比于 $k_{122}$。在 $\mathrm{D_2O}$ 中，同位素质量效应造成$\nu_1$与 $\nu_2$ 频率几乎同比例降低，共振因能量差变大而减弱，但仍存在。叙述6错误。

<Checkpoint>[耦合矩阵元 $W=\langle \nu_1|K_{122}q_1q_2^2|\nu_2 \rangle\propto k_{122}$。 $\mathrm{D_2O}$ 中共振因能量差变大而减弱但仍存在。叙述6错误。](0.5)</Checkpoint>

转动哈密顿量形如 $\hat H_{rot} =A\hat J_a^2+B\hat J_b^2 + C\hat J_c^2 - D_J \hat J^4 - D_{JK}\hat J^2\hat J_a^2 - D_{K}\hat J_a^4 +H_{JK}\hat J^2\hat J_a^2+...$，其中对 $H_{JK}$项有 $H_{JK}\hat J^4K_a^2$。同样处理其他项，则转动能级与 $K_a$ 的关系可表示为 $E(J,K_a)=BJ(J+1)+(C-B)K_a^2-D_JJ^2(J+1)^2-D_KK_a^4+H_{JK}K_a^2J^4+...$，其中 $D_J,D_K,H_{JK}$ 等高阶离心畸变项都引入了 $K_a-$ 依赖的能级分裂。转动常数对质量的依赖通过转动惯量表达，即 $B=\dfrac{\hbar^2}{2I_b},C=\dfrac{\hbar^2}{2I_c}$，考虑六阶主导项 $H_{JK}\propto \dfrac{(B-C)^3}{\omega^2}$，其依赖于转动常数 $B,C$，故同位素取代带来的质量效应造成转动惯量大小改变，从而 $B-C$ 可能变号，对应的 $H_{JK}$ 随之变号。叙述7正确。

<Checkpoint>[六阶主导项 $H_{JK}\propto(B-C)^3$，质量效应造成转动惯量大小改变，$B-C$ 可能变号，$H_{JK}$ 随之变号。叙述7正确。](0.5)</Checkpoint>

弯曲振动 $\nu_2$ 激发态中，$l-$型双重分裂 $(l=\pm1)$ 源于Coriolis耦合。二阶微扰理论中转动能级混合，引入中间态$|v_2=0,J^\prime\rangle$，得到 $\Delta E\approx2\sum_{J^\prime}\dfrac{\langle v_2=1,l=+1∣\hat H_{\text{Coriolis}}|v_2=0,J^\prime\rangle\langle v_2=0,J^\prime∣\hat H_{\text{Coriolis}}|v_2=1,l=-1\rangle}{E_{v=1}-E_{v=0}}$。其中 $\hat H_{\text{Coriolis}}=-2\sum_{\alpha=a,b,c} B_\alpha\zeta_\alpha^{(\Gamma)}J_\alpha l_\alpha$。对属于 $A_1$ 不可约表示的 $\nu_2$ ，仅 $\zeta_b^{(A_1)}\propto\zeta_2\neq0$，得到 $\hat H_{\text{Coriolis}}=-2 B_b\zeta_2J_b l_b$。代入 $E_{v=1}-E_{v=0}\approx \hbar \omega_2,J^\prime=\pm1$以及量子化条件得到$\Delta E\propto \dfrac{(2B_b\zeta_2J_b*)^2}{\hbar\omega_2}\propto\dfrac{\zeta_2^2(J+1)}{B}$。叙述8正确。

<Checkpoint>[引入中间态$|v_2=0,J^\prime\rangle$，代入能隙以及量子化条件得到$\Delta E\propto\dfrac{\zeta_2^2(J+1)}{B}$。叙述8正确。](0.5)</Checkpoint>

$\mathrm{^{17}O}$ 的核自旋 $I = 5/2$，与分子转动的角动量 $J$ 耦合，形成总角动量 $F = J + I$，导致转动能级的超精细分裂。超精细哈密顿量主项 $H_Q \propto 3(I \cdot J)^2 - I^2J^2$ ，展开 $(I \cdot J)^2 = \dfrac14(F^2-I^2-J^2)$ ，代入能级修正 $\Delta E_Q \propto \dfrac34 K^2-I(I+1)J(J+1)$，其中 $K = F(F+1) - I(I+1) - J(J+1)$，$F$ 为总角动量量子数。因此分裂间距与 $J$ 不成线性关系。叙述9错误。

<Checkpoint>[能级修正 $\Delta E_Q \propto \dfrac34 K^2-I(I+1)J(J+1)$，因此分裂间距与 $J$ 不成线性关系。叙述9错误。](0.5)</Checkpoint>

Herman-Wallis因子描述了转动跃迁强度随 $K_a$ 的变化，参数 $a$ 的符号和大小与分子的转动常数、振动-转动耦合等有关。对于 $\Delta K_a=\pm1$ 的跃迁，设分子的转动-振动波函数为 $|v, J, K_a\rangle$，跃迁偶极矩元 $M = \langle v', J', K_a'|\mu|v, J, K_a\rangle$，考虑 Herman-Wallis 效应则振动波函数可展开为 $|v\rangle = |v_0\rangle + \alpha K_a^2 |v_0\rangle$，因此跃迁偶极矩元展开为 $M \approx \langle v_0', J', K_a'|\mu|v_0, J, K_a\rangle + \alpha K_a^2 \langle v_0', J', K_a'|\mu|v_0, J, K_a\rangle$。跃迁强度 $I \propto |M|^2 \approx I_0(1 + 2\alpha K_a^2)$。因此 Herman-Wallis 因子为 $(1+aK_a^2)$，其中 $a=2\alpha$。叙述10正确。

<Checkpoint>[振动波函数展开 $|v\rangle = |v_0\rangle + \alpha K_a^2 |v_0\rangle$，跃迁强度 $I \propto |M|^2 \approx I_0(1 + 2\alpha K_a^2)$。Herman-Wallis 因子为 $(1+aK_a^2)$，其中 $a=2\alpha$。叙述10正确。](0.5)</Checkpoint>

错误叙述有1, 3, 4, 6, 9，故选F。

<Checkpoint>[错误叙述有1, 3, 4, 6, 9，故选F。](0.5)</Checkpoint>