Kinectic Isotope Effects (KIE)#
动力学同位素效应(KIE)

Kinetic isotope effects (KIE) are changes in reaction rates caused by a change in a given isotope in a molecule. That is caused due to the difference in mass, that affects vibrational frequencies and/or tunneling probabilities.
动力学同位素效应（KIE）是由于分子中某一特定同位素的变化而导致的反应速率变化。这是由于质量差异影响了振动频率和/或隧穿概率所引起的。

It is an effect often exploited to investigate reaction mechanisms, when there is bond breaking or forming to an atom that can be substituted by an isotope. These KIE are predictable, depending on the mass ratios, and thus can be used to validate proposed mechanisms. The origin of the KIE is related to a change on the Zero Point Energy (ZPE) of the system, that is affected when the vibrational frequencies are lowered or increased due to the mass change. For the heavier deuterium the vibrational frequencies are lowered, thus lowering the ZPE and altering the free energy barrier.
这是一种常用于研究反应机理的现象，当存在键断裂或形成于可被同位素替代的原子时。这些同位素效应是可预测的，取决于质量比，因此可用于验证提出的机理。同位素效应的起源与系统零点能（ZPE）的变化有关，这种变化受振动频率因质量变化而降低或升高的影响。对于较重的氘，振动频率降低，从而降低 ZPE 并改变自由能势垒。

../_images/kie.png — Figure: Schematic origin of the KIE.
图：KIE 的示意图起源。#

Typically, the KIE is quantified by the ratio of the reaction rates of both reactions $\frac{k_{H}}{k_{D}}$ . This can be calculated from the free energy reaction barriers of both reactions according to:
通常，KIE 通过两种反应速率之比来量化 $\frac{k_{H}}{k_{D}}$ 。这可以根据两种反应的自由能反应势垒计算得出：

\frac{k_{H}}{k_{D}} = e^{- (Δ G_{H}^{‡} - Δ G_{D}^{‡}) / R T}

To obtain these barriers, we can use the usual transition state search and optimization algorithms in ORCA, we just have to adjust the mass of the respective atoms to simulate the corresponding isotopes. This can be easily done via the geometry input within the ORCA input file:
为了获得这些势垒，我们可以使用 ORCA 中通常的过渡态搜索和优化算法，只需将相应原子的质量调整为模拟相应的同位素。这可以通过 ORCA 输入文件中的几何输入轻松实现：

!wB97X-3c

* XYZ 0 1
  H     -2.411352    1.128722    0.029849		<= Hydrogen
  H     -1.342943    1.352019   -0.097229 M 2.00141	<= Deuterium
*

Here, we adjusted the mass of H to 2.00141 amu which corresponds to deuterium.
在此，我们将 H 的质量调整为 2.00141 amu，对应于氘。

Example 1: H-atom Abstraction#
示例 1：H 原子抽取

In this example, we use ORCA to make a quantum-mechanical prediction of the KIE for the H-atom abstraction from methane by the hydroxyl radical. The respective KIE ratio is known from highly accurate gas-phase experiments ( $6.75 \pm 0.83$ ).[Tully1993]
在此例中，我们运用 ORCA 进行量子力学预测，计算羟基自由基从甲烷中提取氢原子的同位素效应（KIE）。该 KIE 比值已通过高度精确的气相实验测得（ $6.75 \pm 0.83$ ）。[Tully1993]

First, we searched for the transition states of the respective reactions with the NEB-TS method. In this case, we employed the ωB97X-3c composite method. It is based on a range-separated hybrid functional that is specifically suited to compute transition states and reaction barriers at a reasonable cost. Note, that we also adjusted the temperature for the thermochemistry calculation to obtain free energies at the same conditions as in the experiment (T = 293 K). We further optimize the isolated reactants for both cases. The respective inputs can be found in the tabs below:
首先，我们使用 NEB-TS 方法搜索了各自反应的过渡态。在此过程中，我们采用了ωB97X-3c 复合方法。该方法基于一种范围分离的杂化泛函，特别适合以合理的成本计算过渡态和反应能垒。请注意，我们还调整了热化学计算的温度，以获得与实验条件（T = 293 K）相同的自由能。此外，我们对两种情况下的孤立反应物进行了优化。相应的输入可在下方的标签中找到：

NEB-TS CH₄

!wB97X-3c NEB-TS FREQ

%FREQ   
 TEMP 293     
END

%NEB 
 NEB_END_XYZFILE "products.xyz"
 PREOPT_ENDS true
END

*XYZ 0 2
C     -2.42007     1.14475     0.00503
H     -1.37176     1.42602    -0.24888
H     -2.95035     0.87504    -0.97240
H     -2.97696     2.02414     0.50780
H     -2.39461     0.26127     0.72431
O      1.11295     1.79481    -0.12688
H      0.85814     0.90892     0.95813
*

NEB-TS CD₄

!wB97X-3c NEB-TS FREQ

%FREQ   
 TEMP 293     
END

%NEB 
 NEB_END_XYZFILE "products-deuterated.xyz"
 PREOPT_ENDS true
END

*XYZ 0 2
C     -2.42007     1.14475     0.00503
H     -1.37176     1.42602    -0.24888 M 2.00141
H     -2.95035     0.87504    -0.97240 M 2.00141
H     -2.97696     2.02414     0.50780 M 2.00141
H     -2.39461     0.26127     0.72431 M 2.00141
O      1.11295     1.79481    -0.12688
H      0.85814     0.90892     0.95813
*

Opt OH 光 OH

!wB97X-3c OPT FREQ

%FREQ   
 TEMP 293 
END

*XYZ 0 2
O      1.11295     1.79481    -0.12688
H      0.85814     0.90892     0.95813
*

Opt CH₄ 选择 CH ₄

!wB97X-3c OPT FREQ

%FREQ   
 TEMP 293 
END

* XYZ 0 1
C     -2.42007     1.14475     0.00503
H     -1.37176     1.42602    -0.24888
H     -2.95035     0.87504    -0.97240
H     -2.97696     2.02414     0.50780
H     -2.39461     0.26127     0.72431
*

Opt CD₄ 光盘 CD ₄

!wB97X-3c OPT FREQ

%FREQ 
 TEMP 293 
END

* XYZ 0 1
C     -2.42007     1.14475     0.00503
H     -1.37176     1.42602    -0.24888 M 2.00141
H     -2.95035     0.87504    -0.97240 M 2.00141
H     -2.97696     2.02414     0.50780 M 2.00141
H     -2.39461     0.26127     0.72431 M 2.00141
*

After the successful NEB-TS calculations, we obtain the respective optimized transition states:
在成功完成 NEB-TS 计算后，我们获得了各自优化的过渡态：

../_images/kie-ts.png — Figure: Transition states of the H-atom abstraction from methane and deuterated methane.
图：甲烷和氘代甲烷中氢原子抽提的过渡态。#

Single imaginary frequencies of -1312.67 $c m^{- 1}$ for the hydrogen case and -978.70 $c m^{- 1}$ for the deuterated transition state verify the structures to be true transition states and already reflect the KIE.
氢情况下单虚频为-1312.67 $c m^{- 1}$ ，氘代过渡态为-978.70 $c m^{- 1}$ ，验证了这些结构为真实过渡态，并已反映出 KIE。

The activation barriers $Δ G_{H}^{‡}$ are now calculated as the difference of the free energy of the transition state and the sum of the free energies of the isolated reactants. For our example, we obtain 47.63 and 53.51 $k J \cdot m o l^{- 1}$ , respectively. From these values, a $\frac{k_{H}}{k_{D}}$ ration of 7.17 is calculated. This value is in excellent agreement with the experimental value of $6.75 \pm 0.83$ .
活化能 $Δ G_{H}^{‡}$ 现计算为过渡态自由能与孤立反应物自由能之和的差值。对于我们的例子，分别得到 47.63 和 53.51 $k J \cdot m o l^{- 1}$ 。根据这些数值，计算出 $\frac{k_{H}}{k_{D}}$ 比率为 7.17。该值与实验值 $6.75 \pm 0.83$ 非常吻合。

Important 重要

By simply computing the KIE in terms of changes on the free energy barriers like that, we are not considering any tunneling effects. Those can be quite important, and sometimes need to be included using one of various different approaches, e.g. [Aquilanti2019].
仅仅通过计算自由能垒的变化来确定 KIE，我们并未考虑任何隧穿效应。这些效应可能相当重要，有时需要采用不同的方法来纳入考虑，例如[Aquilanti2019]中所述。

Kinectic Isotope Effects (KIE)#动力学同位素效应(KIE)

Example 1: H-atom Abstraction#示例 1：H 原子抽取

Structures# 结构

Kinectic Isotope Effects (KIE)#
动力学同位素效应(KIE)

Example 1: H-atom Abstraction#
示例 1：H 原子抽取