Relativistic corrections#
相对论修正

When calculating properties or geometries of systems that contain heavy elements (fourth row and beyond), relativistic effects can have a big impact and should not be ignored. There are even studies that show that for some third row elements, relativistic effects can become significant.
在计算包含重元素（第四行及以后）的系统的性质或几何结构时，相对论效应可能产生重大影响，不应忽视。甚至有研究表明，对于某些第三行元素，相对论效应也可能变得显著。

Currently, there are four methods that can be used in ORCA to include that
目前，ORCA 中可采用四种方法来实现这一点

Effective core potentials (ECPs)
有效的核心势能（ECPs）
Zeroth-Order Regular Approximation (ZORA, [Baerends1996])
零阶正则近似（ZORA, [Baerends1996]）
Douglas-Kroll-Hess (DKH, [Kroll1974] [Hess1985]) Hamiltonians of first and second order
Douglas-Kroll-Hess（DKH，[Kroll1974][Hess1985]）一阶和二阶哈密顿量
spin-free variant of exact two component treatment (X2C, [Weigend2013])
无自旋精确二分量处理（X2C，[Weigend2013]）的自由自旋变体

They are somewhat different and have specific strengths - so one should consider the results carefully - but maybe the most widely used is ZORA and its variants.
它们略有不同且各具特色——因此应仔细考量结果——但或许最广泛使用的还是 ZORA 及其变体。

Effective Core Potentials (ECPs)#
有效芯势(ECPs)

Pseudopotentials or effective core potentials are a very convenient and efficient way to treat relativistic effects. They replace the core electrons with an effective potential that may cover relativistic effects by parameterization. Another benefit of ECPs is the reduced number of electrons that have to be treated explicitly, thus reducing the computation time. One of the most prominent and robust ECP families are the def2-ECPs by Dolg and co-workers.[Dolg2000] These are usually used in combination with the matching def2 basis sets by Ahlrichs.[Ahlrichs2005] Within ORCA these are automatically used if the respective basis set is requested. For example, if the def2-TZVP basis is used for a Hg atom the output will indicate the use of the respective ECP.
赝势或有效芯势是处理相对论效应的一种非常便捷且高效的方法。它们通过参数化将内层电子替换为包含相对论效应的有效势。ECP 的另一优势在于需显式处理的电子数减少，从而缩短计算时间。最为著名且稳健的 ECP 系列之一是由 Dolg 及其同事开发的 def2-ECPs。[Dolg2000] 这些通常与 Ahlrichs 匹配的 def2 基组结合使用。[Ahlrichs2005] 在 ORCA 中，若请求相应的基组，这些 ECP 会自动启用。例如，若对 Hg 原子使用 def2-TZVP 基组，输出将显示相应 ECP 的使用情况。

!wB97X-V def2-TZVP

*XYZ 0 1 
Hg 0 0 0
*

-------------------------
ECP PARAMETER INFORMATION
-------------------------

 Group 1, Type Hg ECP Def2-ECP (replacing 60 core electrons, lmax=3)

ECPs can further be assigned individually in the %basis block
ECPs 可以在 %basis 块中进一步单独分配

%basis
  ECP       "def2-ECP"     # All heavy elements for which the ECP is defined
  NewECP Pt "def2-SD" end  # Different ECP for Pt
end

Here, we defined the def2-ECP to all heavy elements and replaced it by the def2-SD ECP only for platinum.
在此，我们将 def2-ECP 定义应用于所有重元素，并仅对铂元素替换为 def2-SD ECP。

Note 注释

Many different ECP options are available within ORCA. However, the usage of ECPs is still an approximation that has to be made with care. A recent study by Head-Gordon [Head-Gordon2023] and co-workers discusses various ECP families and their strengths and weaknesses.
ORCA 中提供了许多不同的 ECP 选项。然而，ECP 的使用仍需谨慎对待，因其仍是一种近似方法。Head-Gordon 及其同事最近的一项研究[Head-Gordon2023]讨论了各种 ECP 族及其优缺点。

Warning 警告

ECPs should never be used when core properties like NMR chemical shielding tensors are calculated!
在计算诸如 NMR 化学位移张量等核心性质时，绝不应使用 ECPs！

Scalar Relativistic Hamiltonians (ZORA/DKH/X2C)#
标量相对论哈密顿量（ZORA/DKH/X2C）

An explicit relativistic treatment of the core electrons with scalar-relativistic Hamiltonians like ZORA, DKH, or X2C is also possible. These can easily be envoked by the simple input keywords ZORA, DKH, or X2C:
对于内层电子，采用诸如 ZORA、DKH 或 X2C 等标量相对论哈密顿量的显式相对论处理也是可行的。通过简单的输入关键词 ZORA 、 DKH 或 X2C 即可轻松调用这些方法：

!PBE0 D4 ZORA ZORA-DEF2-TZVP

or 或

!PBE0 D4 DKH DKH-DEF2-TZVP

or 或

!PBE0 D4 X2C X2C-TZVPALL

Note that here we are using specific basis for each method, named ZORA- or DKH-DEF2-TZVP, and X2C-TZVPALL. This is necessary because these basis have been specifically designed for these all-electrons calculations, and the relativistic correction should NOT be used together with the regular basis or pseudopotentials. For a detailed description of options, please check the ORCA manual.
请注意，此处我们为每种方法采用了特定的基组，分别命名为 ZORA-或 DKH-DEF2-TZVP，以及 X2C-TZVPALL。这是必要的，因为这些基组是专门为这些全电子计算设计的，且相对论修正不应与常规基组或赝势同时使用。有关选项的详细描述，请查阅 ORCA 手册。

RI and ZORA/DKH/X2C# RI 和 ZORA/DKH/X2C

If you want to use any of the RI methods to accelerate the SCF, another set of special /J basis has to be used:
如果你想利用任何 RI 方法来加速 SCF 计算，必须使用另一组特殊的/J 基组：

!PBE0 D4 ZORA ZORA-DEF2-TZVP RIJDX SARC/J

or 或

!PBE0 D4 DKH DKH-DEF2-TZVP RIJDX SARC/J

or 或

!PBE0 D4 X2C X2C-TZVPALL RIJDX X2C/J

For instance, the SARC/J auxiliary basis can be used for all the ZORA or DKH-DEF2 basis. If no specific basis is available, then one can always use AUTOAUX to automatically generate one.
例如，SARC/J 辅助基组可用于所有 ZORA 或 DKH-DEF2 基组。若无特定基组可用，则可始终使用 AUTOAUX 来自动生成一个。

Important 重要

The SARC/J basis were optimized for RI on the SCF part, not the MP2 or higher-level correlated methods! For correlation specific /C basis consult the ORCA manual and in case of abscence use !AUTOAUX.
SARC/J 基组针对 SCF 部分的 RI 进行了优化，而非 MP2 或更高层次的关联方法！如需特定于关联的 /C 基组，请查阅 ORCA 手册，若无相关内容，请使用 !AUTOAUX。

Example 1: the Hg dimer#
示例 1：汞二聚体#

Let's test the impact of these effects on the Hg dimer, that has an experimental bond length of $3.69 Å$ [Sattler2017].
让我们测试这些效应对 Hg 二聚体的影响，其具有实验测定的键长 $3.69 Å$ [Sattler2017]。

The geometry can be optimized at a regular DFT level using the DEF2-TZVP basis, that makes use of pseudopotentials or using the all-electron Hamiltonians with matching basis sets:
几何结构可以在常规 DFT 水平上进行优化，使用 DEF2-TZVP 基组，该基组采用赝势，或使用匹配基组的全部电子哈密顿量：

!PBE0 D4 DEF2-TZVP OPT

*XYZ 0 1
Hg 0 0 0
Hg 0 0 3
*

!PBE0 D4 ZORA SARC-ZORA-TZVP SARC/J OPT

!PBE0 D4 DKH2 SARC-DKH-TZVP SARC/J OPT

!PBE0 D4 X2C x2c-TZVPall X2C/J OPT

Note 注释

The auxiliary basis for the RIJ approximation used during the relativistic case here was chosen as the appropriate SARC/J.
此处相对论情况下使用的 RIJ 近似辅助基底被选定为适当的 SARC/J。

We can now compare the obtained Hg-Hg bond lengths to the experimental value of 3.69 Å.
我们现在可以将获得的 Hg-Hg 键长与实验值 3.69 Å进行比较。

Relativistic treatment 相对论处理	d(Hg-Hg) / Å
ECP	3.64
ZORA	3.58
DKH2	3.55
X2C	3.49
Exp. 实验。	3.69

In this case, the geometry optimized with the ECP basis set is closest to the experimental value. In general, all relativistic treatments drastically improve the agreement compared to a non-relativistic all-electron calculation. This also shows, that ECPs are in many cases sufficient to optimize geometries.
在这种情况下，使用 ECP 基组优化的几何结构最接近实验值。总体而言，所有相对论处理方法相较于非相对论的全电子计算，都显著提升了符合度。这也表明，在许多情况下，ECP 足以用于几何优化。

Warning 警告

Geometry optimizations using relativistic corrections turn on by default a one-center approximation by default, that changes the energy values. Do not compare single point energies from those you obtain from an !OPT run, these numbers are be incompatible.
使用相对论修正的几何优化默认开启单中心近似，这会改变能量值。请勿将通过!OPT 运行获得的单点能量与这些数值进行比较，它们是不兼容的。

Example 2: ¹¹⁹Sn NMR#
示例 2： ¹¹⁹ 锡核磁共振 #

In this example, we calculate the ¹¹⁹Sn NMR chemical shift with different relativistic treatments. To do so, we calculate the magnetic shielding tensors of the reference molecule SnMe₄ and the investigated teravalent tin compound with the B97M-V functional.
在此示例中，我们采用不同的相对论处理方法计算了 ¹¹⁹ Sn NMR 化学位移。为此，我们使用 B97M-V 泛函计算了参考分子 SnMe ₄ 和所研究的三价锡化合物的磁屏蔽张量。

Respective inputs are: 各自的输入为：

!B97M-V X2C X2C-TZVPALL RIJDX X2C/J

*XYZFILE 0 1 structure.xyz

%eprnmr
  Nuclei = all Sn {shift}
end

!B97M-V ZORA ZORA-DEF2-TZVP RIJDX AUTOAUX

%basis
NewGTO Sn "SARC-ZORA-TZVP" end
end

*XYZFILE 0 1 structure.xyz

%eprnmr
  Nuclei = all Sn {shift}
end

!B97M-V ZORA DKH-DEF2-TZVP RIJDX AUTOAUX

%basis
NewGTO Sn "SARC-DKH-TZVP" end
end

*XYZFILE 0 1 structure.xyz

%eprnmr
  Nuclei = all Sn {shift}
end

Note 注释

Note that if your desired basis set for the nucleus of interest is not available for all elements in the molecule, you can define individual basis sets.
请注意，如果所需的关注核基组对分子中的所有元素不可用，您可以定义单独的基组。

If we now calculate the approximated chemical shift as described in the NMR spectroscopy tutorial, we obtain following values:
如果我们现在按照 NMR 光谱学教程中的描述计算近似的化学位移，我们得到以下数值：

Calculated δ(¹¹⁹Sn) chemical shifts.
计算了δ( ¹¹⁹ Sn)化学位移。#
Relativistic treatment 相对论处理	δ(¹¹⁹Sn) / ppm
ECP	-6.0
ZORA	-137.4
DKH2	-140.5
X2C	-169.8
Exp. 实验。	-115.5

Here, we see that employing an effective core potential (ECP) at the tin nucleus yields very bad results compared to the experiment. All other scalar-relativistic approaches yield reasonable agreement with the experiment considering the lack of spin-orbit coupling and environmental effects in our calculations.
在此，我们观察到，采用锡核的有效核心势（ECP）相较于实验结果产生了极不理想的效果。而所有其他标量相对论方法，在考虑到我们计算中缺乏自旋-轨道耦合及环境效应的情况下，均与实验结果达成了合理的一致性。

Relativistic corrections#相对论修正

Effective Core Potentials (ECPs)#有效芯势(ECPs)

Scalar Relativistic Hamiltonians (ZORA/DKH/X2C)#标量相对论哈密顿量（ZORA/DKH/X2C）

RI and ZORA/DKH/X2C# RI 和 ZORA/DKH/X2C

Example 1: the Hg dimer#示例 1：汞二聚体#

Example 2: 119Sn NMR#示例 2： 119 锡核磁共振 #

Structures Example 2# 结构示例 2

Relativistic corrections#
相对论修正

Effective Core Potentials (ECPs)#
有效芯势(ECPs)

Scalar Relativistic Hamiltonians (ZORA/DKH/X2C)#
标量相对论哈密顿量（ZORA/DKH/X2C）

Example 1: the Hg dimer#
示例 1：汞二聚体#

Example 2: ¹¹⁹Sn NMR#
示例 2： ¹¹⁹ 锡核磁共振 #