准量子方法研究NO（X）-He体系的转动非弹性散射过程
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摘要
转动非弹性散射过程是自然界中普遍存在的物理化学现象，也是碰撞诱导能量及动量转移的最基本过程，它在大气、燃烧、热学，超冷分子等许多领域都有重要意义，因此近几十年来一直是一个备受关注的研究课题。开壳层分子的非弹性散射过程具有许多闭壳层分子所没有的动力学现象，如自旋轨道态以及Λ-双重简并态之间的能级跃迁，无论从实验方面还是理论方面都引起了研究小组的广泛兴趣。一氧化氮分子与惰性气体原子的非弹性碰撞是此领域具有代表性的研究体系，不仅一氧化氮分子在实验中对激光探测具有较高的灵敏度，同时它的研究为我们认识日益严重的大气环境污染及全球暖化问题起到了关键作用。自上世纪七十年代以来，科学家在理论和实验上均取得了一系列重大发现。随着速度成像，激光光谱以及交叉分子束等实验技术的飞速发展，使得探测量子态-态微分散射截面(differential cross section, DCS)成为可能。通过研究碰撞产物的角度分布，可以获得碰撞传能过程中丰富的动力学信息。与此同时，散射理论的研究也取得了长足的发展，各种量子散射理论，经典轨线理论，以及本工作所采用的准量子理论都得到了很好的结果。通过求解微分和积分散射截面(integral cross section, ICS)，可以获得碰撞物体之间相互作用的形式和它们的内部结构。实验测量的产物信息可以帮助我们确定理论研究中所用势能面的精度，而精确的理论则能对实验所观测的现象给出可靠的解释。
Stolte等人发展的准量子计算Quasi-Quantum Treatment (QQT)(Gijsbertsen et al.,JACS2006,128,8777)理论，用来处理双原子分子与惰性气体原子的非弹性散射问题，具有物理含义清晰，计算时间短等优点，一直受到科研工作者的广泛关注。这套理论的基本思想是：①.采用硬壳近似。用体系碰撞能量所对应的等势面， V_(sum)(R,γ_R) E_(col)，取代全量子力学中的完整势能面。对于我们的研究体系NO(X)-He，它的最大势阱深度是25cm~(-1),远小于我们研究的碰撞能量508cm~(-1)，因此QQT近似认为体系相互作用的吸引势可以忽略，仅考虑排斥势。②.采用全新的apse坐标系。认为碰撞过程中总角动量j在动力学apse(kinematic apse) a方向上的投影近似守恒，m'_a=m_a，这使得计算在很大程度上得以简化。这里的a描述碰撞前后动量的变化情况， a≡μ/h (v'_(rel)-v_(rel))=k'-k，其中k和k'分别代表入射和出射波矢量。对于硬壳来说， a的方向与法线n方向相同，能量只沿a方向发生变化，一部分平动能E_(col)=h~2k~2/2μ通过转动激发被转化为内能△E_(j'←j)~(ROT)≡E_(ROT)(j')-E_(ROT)(j)。而沿着硬壳切线方向的动量保持不变， k_(||)=k'_(||)。③.采用角度变量研究散射问题。不像传统的量子力学或准经典方法，QQT不需要对轨道角动量l或碰撞参数b做积分。散射问题完全由a与入射波矢量k间的夹角及最终转动态所决定。④.引入量子干涉效应。碰撞体系的转动状态具有德布罗意波的波动性质，因此分子碰撞过程中的量子干涉效应，是粒子波之间的干涉。QQT理论应用费曼路径积分的方法，认为只有具有相同转动初态，相同散射角，并且散射进入同一个转动末态的轨迹才能产生干涉。对于各向异性势，这些轨迹长度由分子轴在碰撞瞬间的取向所决定。QQT用相位来标定每个散射轨迹的相对路径长度，初末态转动波方程作为散射的几率振幅，最后对所有取向的分子路径做重叠积分，即可获得散射振幅，从而得到碰撞的微分散射截面。
QQT理论取代了分波近似求解大量耦合方程的全量子方法，能更直观的理解散射的内在物理过程。并成功的解释了多年来实验上一直无法解释的“宇称对（parity pair）”现象，即激发到相邻转动态且具有相同宇称的微分散射截面具有相似的角度分布，以及空间不对称性(steric asymmetry)随转动量子数的奇偶变化做正负周期振荡的现象。同时应用QQT理论计算得到的NO-He, NO-Ar, NO-Ne, NO-D_2等体系的微分散射截面与全量子力学和实验结果均符合的很好。但由于目前的QQT理论并未考虑量子衍射效应；另外，QQT理论采用的是硬壳近似，仅限制在经典允许的范围，而且只考虑了自旋轨道态守恒跃迁。这些问题都不利于全面深入的理解非弹性碰撞的动力学过程。因此本论文开展了一系列工作，对QQT理论进行了扩展和优化，并首次发现了微分散射截面的能量标度定律。工作主要创新点如下：
1.将QQT理论扩展到了自旋轨道态变换跃迁。在前面的工作中，QQT理论的研究主要局限在自旋轨道态守恒跃迁。我们以NO(X)-He体系的碰撞诱导转动跃为例，自旋轨道态守恒跃迁NO(~2∏_(1/2), j=0.5)+He→NO(~2∏_(1/2), j')+He的势能面由V_(sum)(R,γ)势所决定,此时QQT硬壳势相对于NO分子轴具有柱对称性，散射平面（由a，k和k'组成的平面）与分子平面（由NO分子轴r_(NO)和R组成的平面，这里R代表体系质心指向碰撞作用点的矢量）重合。本论文中我们首次将QQT理论扩展到了自旋轨道态变换跃迁NO(~2∏_(1/2), j=0.5)+He→NO(~2∏_(3/2), j)+He，这时体系的相互作用势表示为V (R, γ_R,Ⅹ_R)=V_(sum)(R,γ_R)-V_(dif)(R,γ_R).cos (2Ⅹ_R)。我们发现在旋轨态变换跃迁中，需要引入一个空间方位角R，用来表示NO分子的Π轨道相对于NO-He三原子平面的对称性，当Π轨道垂直于三原子平面时，Ⅹ_R=Ⅹ/2；当Π轨道平行于三原子平面时，Ⅹ_R=0。我们可以看到相互作用势从旋轨态守恒跃迁的柱对称壳变到了旋轨态变换跃迁的各向异性壳。通过一系列理论推导，我们得到了旋轨态变换跃迁下的非弹性散射微分和积分截面，并将我们的计算与量子力学的结果做了对比，结果符合的很好。使得QQT理论的适用性得到了扩展。
2.对QQT理论的硬壳近似做了改进。传统的QQT理论采用硬壳近似，将体系的相互作用势看作是排斥势而不考虑吸引势，用碰撞能量所对应的等势面V_(sum)(R,γ_R) E_(col)来近似表示体系的势能面。QQT理论认为等势线以内的势能为碰撞能量，等势线以外的势能为零。但是全量子力学采用的是从头算(ab initio)方法得到的完整势能面，同时包括排斥势和吸引势，是软壳势。这使得QQT理论算得的DCS在角度上整体要比全量子计算得到的DCS向后偏移。也就是说，相同散射角下QQT理论得到的DCS比全量子计算得到的DCS要小。为了改进QQT理论，我们不再使用单一的硬壳势能面，而是将等势线与apse散射角β联系起来，将碰撞能量在apse方向的投影分量所对应的等势面V_(sum)(R,γ_R)=E_(col).cos~2β做为散射角β处的势能面。我们可以将它理解为一个与β相关的梯度势能面，即当β=90o时，我们采用V_(sum_(R,γ_R)=0cm~(-1)等势面，而当β=180o时，我们采用V_(sum)(R,γ_R)=E_(col)等势面。应用改进的势能面，我们应用NO(X~2Π)–He碰撞体系，计算了碰撞能量为508cm~(-1)时的微分散射截面DCS和积分散射截面ICS,并与传统的QQT理论和全量子理论做了比较。结果发现，经过势能面优化的QQT方法算得的DCS要比传统QQT结果有所增大。这种变化在宇称变换跃迁和低转动能级小散射角处尤为明显，但是与全量子结果还尚有差距。这与NO(X)–He体系的排斥势占主导地位的性质有关。我们预期改进的QQT理论会有更明显的优势当应用到具有更软的势能壳的体系，如NO(X)-碱金属原子的非弹性碰撞体系。
3.将QQT理论扩展到经典禁区，并在QQT理论的基础上发现了微分散射截面的能量标度定律。
传统的QQT理论不考虑量子隧穿效应，当碰撞能量小于转动跃迁所需的能量时，禁止发生跃迁。由于采用硬壳近似，认为只有入射波矢k平行于a方向的分量k_1=kcosβ对能量转移起贡献，这里的β与散射角θ存在一一对应关系，因此我们设对应经典禁区的小散射角处的DCS为零。本工作首次将QQT理论扩展到经典禁区，此时的费曼路径可以穿过硬壳势能面，出射波矢k'_L指向a的相反方向。将QQT扩展到经典禁区可以完整的与量子DCS结果进行比对。同时为我们之后推导能量标度定律起到了关键作用。
能量标度定律是建立在QQT理论的基础上的，通过一系列公式推导，我们发现低碰撞能量下DCS的结构信息完全包括在高碰撞能量的DCS内，即高碰撞能量下的DCS通过应用一个标度因子SF_(QQT)=E_(col)~H/E_(col)~l，对其apse散射角和振幅进行调制，可以完全与低碰撞能量下的DCS相吻合。值得注意的是，此定律需要在apse坐标系下才能对角度β进行调制，且只对硬壳的排斥势能面严格适用。
由于QQT仅是一种近似理论，我们希望将能量标度定律扩展到更准确的全量子计算中，以此检测标度定律在处理软壳势能面上的能力。前人已经获得了NO(X)-He体系在碰撞能量分别为E_(col)~H=147meV和E_(col)~L=63meV的实验测量与量子理论计算的DCS结果。因此我们以NO(X)-He体系为例，将能量标度定律应用在碰撞能量分别为E_(col)~H=147meV和E_(col)~L=63meV的量子DCS中。量子理论和QQT理论处理非弹性碰撞问题的最大不同是量子理论采用碰撞坐标系(collision frame)，量子轴Z沿着波矢k方向，散射截面由球角（θ，φ）确定，反映的是出射波矢k相对于入射波矢k的取向；而QQT理论采用apse坐标系，量子轴Za沿着动力学apse a方向，散射截面由（β，α）确定，反映的是apse矢量a相对于入射波矢k的取向。我们将能量标度定律分别应用在闭壳层近似以及开壳层NO(X)分子的自旋轨道守恒跃迁和自旋轨道变化跃迁的量子DCS中。我们首先将高能量E_(col)~H=147meV下的量子DCS从碰撞坐标系转化到apse坐标系，在apse坐标系下应用QQT能量标度定律对DCS进行调制。与QQT不同的是，能量标度定律在量子计算中并非是完全精确的，因此我们引入了一个矫正因子(Calibration Factor) CF_(QM)(j', f←j,i)，意在使能量标度后的高能@E_(col)~H=147meVDCS的积分截面σ_(scale)~(EH)与直接计算得到的低能@E_(col)~L=63meVDCS的积分截面σ_(dir)~(EL)相同，即σ_(scale)~(EH).SF.CF=σ(dir)~(EL)，而QQT理论中却并不需要这个矫正因子，仅通过使用标度因子就可以实现高能到低能微分散射截面的转换，σ_(scale)~En.SF=σ_(dir)~(Er)。最后再将调制后的高能DCS从apse坐标系转换回碰撞坐标系。我们发现调制后的高能DCS@E_(col)~H=147meV与直接计算得到的低能DCS@E_(col)~L=63meV依然符合的很好。甚至小散射角处的衍射振荡区都被准确的标度出来了。
有了能量标度定律，我们可以轻松推导出NO(X)-He体系在能量介于@E_(col)~L=63meV和@E_(col)~H=147meV之间的任何碰撞能量下的的DCS,而无需再耗费大量时间解繁琐的耦合通道方程，同时也为实验上定量的判断测量相应能量下的散射结果的准确性提供了有效的工具。另外，微分散射截面与分子势能面存在密切关系，能量标度定律是建立在QQT理论的基础上推导出来的，该定律在硬壳势能面上是严格适用的，因此通过此定律在量子DCS上的应用，可以帮助我们了解势能面的信息，也就是排斥势占主导地位的分子体系，该定律就会满足的越好，反之，可以证明碰撞体系的吸引势对结果起决定作用。能量标度定律在NO(X)-He体系上的成功应用，更进一步证实了NO(X)-He体系的转动非弹性DCS对势能面中排斥势的强依赖关系。
Rotationally inelastic scattering is one of the fundamental collision-induced energy transferprocesses that underlie intermolecular energy flow in chemically reacting systems such asatmospheric, combustion, thermal, cold and ultra-cold chemistry. The measurement of the fullyquantum state-to-state resolved angular distribution of the scattered products, characterized asthe differential cross sections (DCSs), represents one of the most fundamental quantities ofinelastic scattering. It provides valuable information with different angular distributioncorresponding to regimes of different dynamics. The collision of the NO(X) molecule onto a Heatom is particularly interesting; due to the unpaired electron in the open-shell NO moleculeallows the energy transfer into the excited rotational levels of its upper spin-orbit state.Moreover such a collision may also alter its Λ-doublet and hyperfine states. The NO(X)-raregas atom collision system is a paradigm for what happens in a molecular collision involvingmore than one Born-Oppenheimer potential energy surface (PES).
The quasi-quantum treatment (QQT)(Gijsbertsen et al., JACS2006,128,8777) employs aFeynman path integration over angular variables in the kinematic apse frame to avoid thetraditional quantum mechanical (QM) partial wave expansion approach, while requiring muchless computational effort. QQT provides a physically compelling framework for the evaluationof rotationally inelastic scattering, including both DCSs and integral cross sections (ICSs). Inthis thesis, we have made a series of efforts to improve and extend QQT method. The mainaccomplishments are:
1. The QQT theory has been extended from spin-orbit state conserving transitions to thespin-orbit state changing transitions. This became possible by the introduction of anadditional angular variable, the dihedral angle ⅹ_R, between the plane of the NO-axis and its1Π open shell orbital lobe with respect to the plane of the NO axis and the He atom. Subsequently,the regular two dimensional PES V_(sum)(R,γ_R)was extended to a effective three dimensionalPES V (R,γ_R,Ⅹ_R). The quantum state-to-state resolved DCSs and ICSs for collision of NO(X)with He, obtained from QQT theory and CC QM methods have been investigated employingboth in both spin-orbit conserving and changing transitions. This extension of the QQT modelpromises to provide a tool acquire insight into the underlying mechanism that brings aboutthe spin orbit excitation or purely rotationally inelastic excitation and as such offers a focusfor future work.
2. A modified Quasi-Quantum Treatment (QQT) of the rotationally inelastic scattering hasbeen developed to study the integral and differential cross sections of NO (X~2) with Heat the collision energy of508cm-1, which are compared with previous regular QQT andrigorous Quantum Mechanical (QM) calculations. The rigid shell potential energy surface(PES) defined as the contour V_(sum)(R, γ_R) Ecol, a basic assumption of regular QQT, results amore backwards scattered rotationally inelastic differential cross sections than thoseobtained by the exact QM solution on the complete ab initio PES. To improve on the rigidhard shell approximation, this paper presents the modified QQT, in which treatment, themodified hard shell PES contour V_(sum)(R,γ_R) E_(col). cos~2β is set equal to the fraction of thecollision energy provided by the incoming momentum vector directed anti-parallel to thesurface normal. In our example of He-NO(X), indeed modified QQT results more forwardscattering especially for its parity changing rotationally inelastic transitions, compared toregular QQT but still less compared to exact QM. Tentatively this may be ascribed to thesteep repulsive core of the He-NO(X) V_(sum)PES. A more distinct forward trend formodified QQT is possible for collision systems with an appreciable softer core, as forexample the Alkali atom-NO(X) collision systems.
3. QQT framework is extended to treat the DCS in the classically forbidden region as well asthe classically allowed region. Moreover, the QQT is applied to the collision energydependence of the angular distributions of these DCSs, and a new analytical formalism isderived that reveals a scaling relationship between the DCS calculated at a particular collision energy and the DCS at other collision energies. This scaling is shown to be exactalso for QM calculated or experimental DCSs if the magnitude of scattering amplitudedepends solely on the projection of the incoming momentum vector onto the kinematicapse vector. The QM DCSs of the NO(X)-He collision system were found to obey thisscaling law nearly perfectly for energies above63meV, that is, the DCSs at a collisionenergy of E_(col)~L=63meV can be accurately reconstructed from the correspondingNO(X)+He DCSs at E_(col)~H=147meV. This implies that there is no need to carry outadditional expensive close-coupled calculations to obtain the scattering angle dependenceof the quantum state resolved DCSs at collision energies between E_(col)~L=63meV andE_(col)~H=147meV. The mathematical derivation is accompanied by mechanistic description ofthe Feynman paths that contribute to the scattering amplitude in the classically allowed andforbidden regions, and the nature of the momentum transfer during the collision process.The successful application of the QQT collision energy scaling formalism reinforces theevidence that the He-NO rotationally inelastic DCSs depend very sensitive on theanisotropy of the repulsive part of the PES. This scaling relationship highlights the natureof (and limits to) the information that is obtainable from the collision-energy dependence ofthe DCS, and allows a description of the relevant angular range of the DCSs that embodiesthis information.
Stolte等人发展的准量子计算Quasi-Quantum Treatment (QQT)(Gijsbertsen et al.,JACS2006,128,8777)理论，用来处理双原子分子与惰性气体原子的非弹性散射问题，具有物理含义清晰，计算时间短等优点，一直受到科研工作者的广泛关注。这套理论的基本思想是：①.采用硬壳近似。用体系碰撞能量所对应的等势面， V_(sum)(R,γ_R) E_(col)，取代全量子力学中的完整势能面。对于我们的研究体系NO(X)-He，它的最大势阱深度是25cm~(-1),远小于我们研究的碰撞能量508cm~(-1)，因此QQT近似认为体系相互作用的吸引势可以忽略，仅考虑排斥势。②.采用全新的apse坐标系。认为碰撞过程中总角动量j在动力学apse(kinematic apse) a方向上的投影近似守恒，m'_a=m_a，这使得计算在很大程度上得以简化。这里的a描述碰撞前后动量的变化情况， a≡μ/h (v'_(rel)-v_(rel))=k'-k，其中k和k'分别代表入射和出射波矢量。对于硬壳来说， a的方向与法线n方向相同，能量只沿a方向发生变化，一部分平动能E_(col)=h~2k~2/2μ通过转动激发被转化为内能△E_(j'←j)~(ROT)≡E_(ROT)(j')-E_(ROT)(j)。而沿着硬壳切线方向的动量保持不变， k_(||)=k'_(||)。③.采用角度变量研究散射问题。不像传统的量子力学或准经典方法，QQT不需要对轨道角动量l或碰撞参数b做积分。散射问题完全由a与入射波矢量k间的夹角及最终转动态所决定。④.引入量子干涉效应。碰撞体系的转动状态具有德布罗意波的波动性质，因此分子碰撞过程中的量子干涉效应，是粒子波之间的干涉。QQT理论应用费曼路径积分的方法，认为只有具有相同转动初态，相同散射角，并且散射进入同一个转动末态的轨迹才能产生干涉。对于各向异性势，这些轨迹长度由分子轴在碰撞瞬间的取向所决定。QQT用相位来标定每个散射轨迹的相对路径长度，初末态转动波方程作为散射的几率振幅，最后对所有取向的分子路径做重叠积分，即可获得散射振幅，从而得到碰撞的微分散射截面。
QQT理论取代了分波近似求解大量耦合方程的全量子方法，能更直观的理解散射的内在物理过程。并成功的解释了多年来实验上一直无法解释的“宇称对（parity pair）”现象，即激发到相邻转动态且具有相同宇称的微分散射截面具有相似的角度分布，以及空间不对称性(steric asymmetry)随转动量子数的奇偶变化做正负周期振荡的现象。同时应用QQT理论计算得到的NO-He, NO-Ar, NO-Ne, NO-D_2等体系的微分散射截面与全量子力学和实验结果均符合的很好。但由于目前的QQT理论并未考虑量子衍射效应；另外，QQT理论采用的是硬壳近似，仅限制在经典允许的范围，而且只考虑了自旋轨道态守恒跃迁。这些问题都不利于全面深入的理解非弹性碰撞的动力学过程。因此本论文开展了一系列工作，对QQT理论进行了扩展和优化，并首次发现了微分散射截面的能量标度定律。工作主要创新点如下：
1.将QQT理论扩展到了自旋轨道态变换跃迁。在前面的工作中，QQT理论的研究主要局限在自旋轨道态守恒跃迁。我们以NO(X)-He体系的碰撞诱导转动跃为例，自旋轨道态守恒跃迁NO(~2∏_(1/2), j=0.5)+He→NO(~2∏_(1/2), j')+He的势能面由V_(sum)(R,γ)势所决定,此时QQT硬壳势相对于NO分子轴具有柱对称性，散射平面（由a，k和k'组成的平面）与分子平面（由NO分子轴r_(NO)和R组成的平面，这里R代表体系质心指向碰撞作用点的矢量）重合。本论文中我们首次将QQT理论扩展到了自旋轨道态变换跃迁NO(~2∏_(1/2), j=0.5)+He→NO(~2∏_(3/2), j)+He，这时体系的相互作用势表示为V (R, γ_R,Ⅹ_R)=V_(sum)(R,γ_R)-V_(dif)(R,γ_R).cos (2Ⅹ_R)。我们发现在旋轨态变换跃迁中，需要引入一个空间方位角R，用来表示NO分子的Π轨道相对于NO-He三原子平面的对称性，当Π轨道垂直于三原子平面时，Ⅹ_R=Ⅹ/2；当Π轨道平行于三原子平面时，Ⅹ_R=0。我们可以看到相互作用势从旋轨态守恒跃迁的柱对称壳变到了旋轨态变换跃迁的各向异性壳。通过一系列理论推导，我们得到了旋轨态变换跃迁下的非弹性散射微分和积分截面，并将我们的计算与量子力学的结果做了对比，结果符合的很好。使得QQT理论的适用性得到了扩展。
2.对QQT理论的硬壳近似做了改进。传统的QQT理论采用硬壳近似，将体系的相互作用势看作是排斥势而不考虑吸引势，用碰撞能量所对应的等势面V_(sum)(R,γ_R) E_(col)来近似表示体系的势能面。QQT理论认为等势线以内的势能为碰撞能量，等势线以外的势能为零。但是全量子力学采用的是从头算(ab initio)方法得到的完整势能面，同时包括排斥势和吸引势，是软壳势。这使得QQT理论算得的DCS在角度上整体要比全量子计算得到的DCS向后偏移。也就是说，相同散射角下QQT理论得到的DCS比全量子计算得到的DCS要小。为了改进QQT理论，我们不再使用单一的硬壳势能面，而是将等势线与apse散射角β联系起来，将碰撞能量在apse方向的投影分量所对应的等势面V_(sum)(R,γ_R)=E_(col).cos~2β做为散射角β处的势能面。我们可以将它理解为一个与β相关的梯度势能面，即当β=90o时，我们采用V_(sum_(R,γ_R)=0cm~(-1)等势面，而当β=180o时，我们采用V_(sum)(R,γ_R)=E_(col)等势面。应用改进的势能面，我们应用NO(X~2Π)–He碰撞体系，计算了碰撞能量为508cm~(-1)时的微分散射截面DCS和积分散射截面ICS,并与传统的QQT理论和全量子理论做了比较。结果发现，经过势能面优化的QQT方法算得的DCS要比传统QQT结果有所增大。这种变化在宇称变换跃迁和低转动能级小散射角处尤为明显，但是与全量子结果还尚有差距。这与NO(X)–He体系的排斥势占主导地位的性质有关。我们预期改进的QQT理论会有更明显的优势当应用到具有更软的势能壳的体系，如NO(X)-碱金属原子的非弹性碰撞体系。
3.将QQT理论扩展到经典禁区，并在QQT理论的基础上发现了微分散射截面的能量标度定律。
传统的QQT理论不考虑量子隧穿效应，当碰撞能量小于转动跃迁所需的能量时，禁止发生跃迁。由于采用硬壳近似，认为只有入射波矢k平行于a方向的分量k_1=kcosβ对能量转移起贡献，这里的β与散射角θ存在一一对应关系，因此我们设对应经典禁区的小散射角处的DCS为零。本工作首次将QQT理论扩展到经典禁区，此时的费曼路径可以穿过硬壳势能面，出射波矢k'_L指向a的相反方向。将QQT扩展到经典禁区可以完整的与量子DCS结果进行比对。同时为我们之后推导能量标度定律起到了关键作用。
能量标度定律是建立在QQT理论的基础上的，通过一系列公式推导，我们发现低碰撞能量下DCS的结构信息完全包括在高碰撞能量的DCS内，即高碰撞能量下的DCS通过应用一个标度因子SF_(QQT)=E_(col)~H/E_(col)~l，对其apse散射角和振幅进行调制，可以完全与低碰撞能量下的DCS相吻合。值得注意的是，此定律需要在apse坐标系下才能对角度β进行调制，且只对硬壳的排斥势能面严格适用。
由于QQT仅是一种近似理论，我们希望将能量标度定律扩展到更准确的全量子计算中，以此检测标度定律在处理软壳势能面上的能力。前人已经获得了NO(X)-He体系在碰撞能量分别为E_(col)~H=147meV和E_(col)~L=63meV的实验测量与量子理论计算的DCS结果。因此我们以NO(X)-He体系为例，将能量标度定律应用在碰撞能量分别为E_(col)~H=147meV和E_(col)~L=63meV的量子DCS中。量子理论和QQT理论处理非弹性碰撞问题的最大不同是量子理论采用碰撞坐标系(collision frame)，量子轴Z沿着波矢k方向，散射截面由球角（θ，φ）确定，反映的是出射波矢k相对于入射波矢k的取向；而QQT理论采用apse坐标系，量子轴Za沿着动力学apse a方向，散射截面由（β，α）确定，反映的是apse矢量a相对于入射波矢k的取向。我们将能量标度定律分别应用在闭壳层近似以及开壳层NO(X)分子的自旋轨道守恒跃迁和自旋轨道变化跃迁的量子DCS中。我们首先将高能量E_(col)~H=147meV下的量子DCS从碰撞坐标系转化到apse坐标系，在apse坐标系下应用QQT能量标度定律对DCS进行调制。与QQT不同的是，能量标度定律在量子计算中并非是完全精确的，因此我们引入了一个矫正因子(Calibration Factor) CF_(QM)(j', f←j,i)，意在使能量标度后的高能@E_(col)~H=147meVDCS的积分截面σ_(scale)~(EH)与直接计算得到的低能@E_(col)~L=63meVDCS的积分截面σ_(dir)~(EL)相同，即σ_(scale)~(EH).SF.CF=σ(dir)~(EL)，而QQT理论中却并不需要这个矫正因子，仅通过使用标度因子就可以实现高能到低能微分散射截面的转换，σ_(scale)~En.SF=σ_(dir)~(Er)。最后再将调制后的高能DCS从apse坐标系转换回碰撞坐标系。我们发现调制后的高能DCS@E_(col)~H=147meV与直接计算得到的低能DCS@E_(col)~L=63meV依然符合的很好。甚至小散射角处的衍射振荡区都被准确的标度出来了。
有了能量标度定律，我们可以轻松推导出NO(X)-He体系在能量介于@E_(col)~L=63meV和@E_(col)~H=147meV之间的任何碰撞能量下的的DCS,而无需再耗费大量时间解繁琐的耦合通道方程，同时也为实验上定量的判断测量相应能量下的散射结果的准确性提供了有效的工具。另外，微分散射截面与分子势能面存在密切关系，能量标度定律是建立在QQT理论的基础上推导出来的，该定律在硬壳势能面上是严格适用的，因此通过此定律在量子DCS上的应用，可以帮助我们了解势能面的信息，也就是排斥势占主导地位的分子体系，该定律就会满足的越好，反之，可以证明碰撞体系的吸引势对结果起决定作用。能量标度定律在NO(X)-He体系上的成功应用，更进一步证实了NO(X)-He体系的转动非弹性DCS对势能面中排斥势的强依赖关系。
Rotationally inelastic scattering is one of the fundamental collision-induced energy transferprocesses that underlie intermolecular energy flow in chemically reacting systems such asatmospheric, combustion, thermal, cold and ultra-cold chemistry. The measurement of the fullyquantum state-to-state resolved angular distribution of the scattered products, characterized asthe differential cross sections (DCSs), represents one of the most fundamental quantities ofinelastic scattering. It provides valuable information with different angular distributioncorresponding to regimes of different dynamics. The collision of the NO(X) molecule onto a Heatom is particularly interesting; due to the unpaired electron in the open-shell NO moleculeallows the energy transfer into the excited rotational levels of its upper spin-orbit state.Moreover such a collision may also alter its Λ-doublet and hyperfine states. The NO(X)-raregas atom collision system is a paradigm for what happens in a molecular collision involvingmore than one Born-Oppenheimer potential energy surface (PES).
The quasi-quantum treatment (QQT)(Gijsbertsen et al., JACS2006,128,8777) employs aFeynman path integration over angular variables in the kinematic apse frame to avoid thetraditional quantum mechanical (QM) partial wave expansion approach, while requiring muchless computational effort. QQT provides a physically compelling framework for the evaluationof rotationally inelastic scattering, including both DCSs and integral cross sections (ICSs). Inthis thesis, we have made a series of efforts to improve and extend QQT method. The mainaccomplishments are:
1. The QQT theory has been extended from spin-orbit state conserving transitions to thespin-orbit state changing transitions. This became possible by the introduction of anadditional angular variable, the dihedral angle ⅹ_R, between the plane of the NO-axis and its1Π open shell orbital lobe with respect to the plane of the NO axis and the He atom. Subsequently,the regular two dimensional PES V_(sum)(R,γ_R)was extended to a effective three dimensionalPES V (R,γ_R,Ⅹ_R). The quantum state-to-state resolved DCSs and ICSs for collision of NO(X)with He, obtained from QQT theory and CC QM methods have been investigated employingboth in both spin-orbit conserving and changing transitions. This extension of the QQT modelpromises to provide a tool acquire insight into the underlying mechanism that brings aboutthe spin orbit excitation or purely rotationally inelastic excitation and as such offers a focusfor future work.
2. A modified Quasi-Quantum Treatment (QQT) of the rotationally inelastic scattering hasbeen developed to study the integral and differential cross sections of NO (X~2) with Heat the collision energy of508cm-1, which are compared with previous regular QQT andrigorous Quantum Mechanical (QM) calculations. The rigid shell potential energy surface(PES) defined as the contour V_(sum)(R, γ_R) Ecol, a basic assumption of regular QQT, results amore backwards scattered rotationally inelastic differential cross sections than thoseobtained by the exact QM solution on the complete ab initio PES. To improve on the rigidhard shell approximation, this paper presents the modified QQT, in which treatment, themodified hard shell PES contour V_(sum)(R,γ_R) E_(col). cos~2β is set equal to the fraction of thecollision energy provided by the incoming momentum vector directed anti-parallel to thesurface normal. In our example of He-NO(X), indeed modified QQT results more forwardscattering especially for its parity changing rotationally inelastic transitions, compared toregular QQT but still less compared to exact QM. Tentatively this may be ascribed to thesteep repulsive core of the He-NO(X) V_(sum)PES. A more distinct forward trend formodified QQT is possible for collision systems with an appreciable softer core, as forexample the Alkali atom-NO(X) collision systems.
3. QQT framework is extended to treat the DCS in the classically forbidden region as well asthe classically allowed region. Moreover, the QQT is applied to the collision energydependence of the angular distributions of these DCSs, and a new analytical formalism isderived that reveals a scaling relationship between the DCS calculated at a particular collision energy and the DCS at other collision energies. This scaling is shown to be exactalso for QM calculated or experimental DCSs if the magnitude of scattering amplitudedepends solely on the projection of the incoming momentum vector onto the kinematicapse vector. The QM DCSs of the NO(X)-He collision system were found to obey thisscaling law nearly perfectly for energies above63meV, that is, the DCSs at a collisionenergy of E_(col)~L=63meV can be accurately reconstructed from the correspondingNO(X)+He DCSs at E_(col)~H=147meV. This implies that there is no need to carry outadditional expensive close-coupled calculations to obtain the scattering angle dependenceof the quantum state resolved DCSs at collision energies between E_(col)~L=63meV andE_(col)~H=147meV. The mathematical derivation is accompanied by mechanistic description ofthe Feynman paths that contribute to the scattering amplitude in the classically allowed andforbidden regions, and the nature of the momentum transfer during the collision process.The successful application of the QQT collision energy scaling formalism reinforces theevidence that the He-NO rotationally inelastic DCSs depend very sensitive on theanisotropy of the repulsive part of the PES. This scaling relationship highlights the natureof (and limits to) the information that is obtainable from the collision-energy dependence ofthe DCS, and allows a description of the relevant angular range of the DCSs that embodiesthis information.
引文
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