Electron Paramagnetic Resonance (EPR)#
电子顺磁共振（EPR）

Here we will demonstrate the basics on how to predict EPR properties and spectra using ORCA, that can even include many of the complications involved such as spin-orbit coupling and other relativistic effects.
在此，我们将展示如何使用 ORCA 预测电子顺磁共振（EPR）性质及谱图的基础知识，甚至涵盖自旋-轨道耦合及其他相对论效应等复杂因素。

The CH₃ radical#
CH ₃ 部首 #

The methyl radical is a planar molecule, and its planarity was in part confirmed by EPR spectroscopy. It has a $g$ of $2.0025$ [Gordy1966] and hyperfine couplings (HFC) $A_{x x} = - 31.4, A_{y y} = - 93.7,$ and $A_{z z} = - 60.30$ MHz [Dmitriev2004], let's try to predict that from first principles!
甲基自由基是一个平面分子，其平面性在一定程度上通过 EPR 光谱学得到了证实。它具有 $g$ 的 $2.0025$ [Gordy1966]以及超精细耦合（HFC） $A_{x x} = - 31.4, A_{y y} = - 93.7,$ 和 $A_{z z} = - 60.30$ MHz [Dmitriev2004]，让我们尝试从第一性原理来预测这些参数！

First, the geometry can be optimized using:
首先，可通过以下方式优化几何结构：

!B3LYP DEF2-TZVP OPT
* xyz 0 2
C   3.20276693914010      2.91777006829308     -1.46614904250634
H   4.21682401439677      3.22254500904168     -1.25560882817865
H   2.69109288861722      3.31313693290047     -2.33037576806431
H   2.69057615784592      2.24676798976475     -0.79431636125068
*

and you get the trigonal planar coordinates, as expected:
并且你得到了预期的三角平面坐标：

The next step is then to calculate the EPR properties, and that can be done by setting the %EPRNMR options. The important point here is that these are the only options on the ORCA input that should come after the geometry section, and never before.
下一步是计算 EPR 性质，这可以通过设置%EPRNMR 选项来实现。这里的关键点是，这些是 ORCA 输入中唯一应在几何部分之后出现的选项，绝不应出现在之前。

An input to compute the g tensor and HFCs for the hydrogen atoms, using the B3LYP functional could be:
用于计算氢原子的 g 张量和超精细耦合常数的输入文件，采用 B3LYP 泛函，示例如下：

!B3LYP EPR-II AUTOAUX
* XYZFILE 0 2 CH3_opt.xyz
%EPRNMR
        GTENSOR   TRUE
        NUCLEI    = ALL H {AISO, ADIP, AORB}
END

and it should be given with some comments:
并应附带一些注释：

The GTENSOR flag must be set to TRUE in order to compute it, the default is FALSE.
必须将 GTENSOR 标志设置为 TRUE 才能进行计算，默认值为 FALSE。
The NUCLEI defines the atoms for the HFC calculation. ALL H calculates the HFC on all hydrogens, or use ALL N, ALL C and so on for different atoms. You can also use NUCLEI = 1,5,8, to give one list per atom type with the atom numbering staring from 1.
NUCLEI 定义了用于 HFC 计算的原子。ALL H 计算所有氢原子的 HFC，或者使用 ALL N、ALL C 等来计算不同原子的 HFC。您还可以使用 NUCLEI = 1,5,8，为每种原子类型生成一个列表，原子编号从 1 开始。
The AISO, ADIP and AORB flags request the isotropic part of the HFC, its dipolar part and 2nd order contribution from SOC respectively.
AISO、ADIP 和 AORB 标志分别请求 HFC 的各向同性部分、偶极部分以及来自 SOC 的二阶贡献。
The AORB term is costly, but contributes very little to organic molecules without heavy atoms and could be neglected in those cases.
AORB 术语成本高昂，但在不含重原子的有机分子中贡献甚微，在那些情况下可以忽略不计。
A large basis is recommended for EPR properties. Here we used the special basis EPR-II, for more info one can check the ORCA manual.
建议使用较大的基组以获得 EPR 性质。此处我们采用了专用基组 EPR-II，更多信息可查阅 ORCA 手册。
The spin-orbit coupling integrals are calculated using the SOMF(1X) [Neese2005] method by default, but more efficient options are also available.
自旋-轨道耦合积分默认采用 SOMF(1X) [Neese2005]方法计算，但也有更高效的选择。

Warning 警告

If you are using a list of atoms, like NUCLEI = 1,5,8, there should be one list for each atom type!
如果你使用的是原子列表，例如 NUCLEI = 1,5,8，那么每种原子类型都应该有一个对应的列表！

The output file# 输出文件 #

After a successful single point calculation, the first step is the computation of the integrals used for the SOC treatment:
在成功完成单点计算后，第一步是计算用于自旋-轨道耦合处理的相关积分：

------------------------------------------------------------------------------
                      ORCA SPIN-ORBIT COUPLING CALCULATION
------------------------------------------------------------------------------

GBW file                                    ... epr.gbw
Input density                               ... epr.scfp
Output integrals                            ... epr
Operator type                               ... Mean-field/Effective potential
   One-Electron Terms                       ... 1
   Coulomb Contribution                     ... 2
   Exchange Contribution                    ... 3
   Correlation Contribution                 ... 0
   Maximum number of centers                ... 4

and then comes the:
然后是：

------------------------------------------------------------------------------
                              ORCA EPR/NMR CALCULATION
------------------------------------------------------------------------------

GBWName                      ... epr.gbw
Electron density file        ... epr.scfp
Spin density file            ... epr.scfr
Spin-orbit integrals         ... epr
Origin for angular momentum  ... Center of electronic charge
Coordinates of the origin    ...    6.05064281    5.51881078   -2.76743570 (bohrs)

Please notice that the "Origin for angular momentum" here is "Center of electronic charge", and that means these results are not gauge-invariant and can change depending on your coordinates.
请注意，这里的“角动量原点”是“电子电荷中心”，这意味着这些结果不具有规范不变性，并且会根据坐标系的不同而变化。

Usually, with a sufficiently large basis, the results should be almost invariant, but if one wants the exact results, gauge-invariant atomic orbitals (GIAOs) are needed [Ditchfield1973], [Pulay1990]. These can be constructed from all basis sets, and to use them, just add ORI GIAO to the %EPRNMR section:
通常，在基组足够大的情况下，结果应几乎不变，但若需精确结果，则需使用规范不变原子轨道（GIAOs）[Ditchfield1973], [Pulay1990]。这些轨道可由所有基组构建，使用时只需在%EPRNMR 部分添加 ORI GIAO 即可：

%EPRNMR
        GTENSOR   TRUE
        NUCLEI    = ALL H {AISO, ADIP, AORB}
        ORI       GIAO
END

Note 注释

The calculation of the integrals involving GIAOs is much more costly than the regular ones, so be prepared for longer calculation times.
涉及 GIAOs 的积分计算比常规积分要昂贵得多，因此请做好计算时间更长的准备。

g Tensor# 张量 #

If the following CP-SCF converges, you will get the g tensor matrix:
如果以下 CP-SCF 收敛，您将获得 g 张量矩阵：

-------------------
ELECTRONIC G-MATRIX
-------------------

      The g-matrix:
                   2.0027537    0.0001534    0.0001065
                   0.0001534    2.0024625   -0.0002479
                   0.0001065   -0.0002479    2.0026476

      gel          2.0023193    2.0023193    2.0023193
      gRMC        -0.0001271   -0.0001271   -0.0001271
      gDSO(tot)    0.0000289    0.0000680    0.0000680
      gPSO(tot)    0.0000034    0.0005594    0.0005595
                  ----------   ----------   ----------
      g(tot)       2.0022245    2.0028196    2.0028196 iso=  2.0026213
      Delta-g     -0.0000948    0.0005003    0.0005004 iso=  0.0003020

Here the components of the g tensor are specified (see [Neese2001]) and the total components for the x, y and z directions are printed after g(tot), with the isotropic value afterwards. The Delta-g shown is the difference from the free-electron g value.
此处指定了 g 张量的分量（参见[Neese2001]），并在 g(tot)后打印了 x、y 和 z 方向的总分量，随后是各向同性值。显示的 Delta-g 是从自由电子 g 值的差异。

As you can see, the predicted value is already quite close to the experimental value, considering we completely ignored any matrix effects and its largest deviation from the free-electron comes due to SOC effects, as suggested by Gordy [Gordy1966].
如你所见，考虑到我们完全忽略了任何矩阵效应，且其与自由电子的最大偏差主要源于 SOC 效应，正如 Gordy 所指出的那样[Gordy1966]，预测值已经非常接近实验值。

Hyperfine couplings# 超精细耦合

After the g tensor calculation, comes the HFC for each atom:
在 g 张量计算之后，接下来是每个原子的超精细耦合常数：

-----------------------------------------
ELECTRIC AND MAGNETIC HYPERFINE STRUCTURE
-----------------------------------------

-----------------------------------------------------------
Nucleus   1H : A:ISTP=    1 I=  0.5 P=533.5514 MHz/au**3
               Q:ISTP=    2 I=  1.0 Q=  0.0029 barn
-----------------------------------------------------------
Solving for nuclear perturbation:
Forming RHS of the CP-SCF equations         ...

Here "I=" indicates the spin of that given atom and "P=" is the proportionality factor $μ_{e} μ_{N} β_{e} β_{N}$ used. After that, another CP-SCF solution is requested and the HFC information is printed:
此处，“I=”表示给定原子的自旋，“P=”为所使用的比例因子 $μ_{e} μ_{N} β_{e} β_{N}$ 。随后，请求另一个 CP-SCF 解，并打印出 HFC 信息：

Raw HFC matrix (all values in MHz):
-----------------------------------
              -30.0552              10.8247               7.4998
               10.8247             -77.0049              20.6090
                7.4998              20.6090             -92.4037

A(FC)         -66.5005             -66.5005             -66.5005
A(SD)          41.0835              -0.8524             -40.2311
A(ORB+DIA)      0.0114              -0.0006               0.0271    A(PC) =   0.0126
A(ORB)          0.0065               0.0006               0.0258    A(PC) =   0.0109
A(DIA)          0.0050              -0.0012               0.0013    A(PC) =   0.0017
            ----------           ----------           ----------
A(Tot)        -25.4056             -67.3536            -106.7045    A(iso)=  -66.4879

Again, the individual components are printed and the A(iso) is shown in the end. As you can see, the results are in quite good agreement with the experimental data.
同样，各个部件被打印出来，最终展示了 A(iso)。如您所见，结果与实验数据相当吻合。

At the very end, the ORCA EULER output is printed, with information about the relative angle between the g-tensor and the HFCs, which have an arbitrary relationship from the calculation:
最后，打印出 ORCA EULER 的输出结果，其中包含 g 张量与 HFCs 之间相对角度的信息，这些角度关系在计算中是任意的：

-------------------------------------------------------------------------
          Euler rotation of hyperfine tensor to g-tensor
-------------------------------------------------------------------------

-------------------------------------------------------------------
 Atom  |   Alpha    Beta    Gamma   |   Ax       Ay       Az
       |          [degrees]         |           [MHz]
-------------------------------------------------------------------
  1H      -90.1     42.7     89.9     -67.35   -25.41  -106.70
  2H       90.0     17.4    -90.0     -67.36   -25.41  -106.71
  3H      -90.1     12.7     90.0     -67.36  -106.72   -25.42
-------------------------------------------------------------------

You can use the orca_euler software later if you want to rotate any of these for any specific application.
如果需要为特定应用旋转这些图像，稍后可以使用 orca_euler 软件。

Comparison to experiment#
与实验的比较 #

To summarize our results, here is a table comparing the calculated and predicted data:
总结我们的结果，以下是计算数据与预测数据的对比表：

Comparison of calculated versus experimental EPR properties for CH₃
CH ₃ 计算与实验 EPR 性质的比较#
Property 财产	Exper. 实验。	Theor. 理论。
$g$	2.0025	2.0026
$A_{x x}$	-31.4	-25.4
$A_{y y}$	-93.7	-106.7
$A_{z z}$	-60.3	-67.3
$A_{i s o}$	-61.8	-66.5

With these, one can use tools such as EasySpin, which already has a specific function for reading orca outputs and simulate the EPR spectrum. The comparison between the theory prediction and an EPR spectra taken at 4K in a CH₄ matrix is:
利用这些工具，可以采用诸如 EasySpin 的软件，其已具备读取 orca 输出并模拟 EPR 谱图的专用功能。理论预测结果与在 CH ₄ 基质中 4K 条件下测得的 EPR 谱图的对比情况如下：

Of course differences here are expected due to impurities and matrices effects and the graphic was displaced to match the $g$ .
当然，由于杂质和基质效应，此处存在差异是预料之中的，图形已调整以匹配 $g$ 。

Important 重要

The HFCs are quite sensitive to the geometry of the molecules! Be sure to have a good one before computing them.
HFCs 对分子几何结构非常敏感！在计算之前，务必确保其几何结构良好。

Another example, the naphthalene anion radical#
另一个例子是萘阴离子自由基

In aprotic solvents and in the presence of alkali metals, polyaromatics such as naphthalene can reduce to form radical anions [Garst1971], such as:
在非质子溶剂和碱金属存在下，如萘等聚芳烃可还原形成自由基阴离子[Garst1971]，例如：

These will present nice EPR spectra, with two sets of four protons and thus more complex lines. To predict its spectrum, again one can start by optimizing the geometry of the anion:
这些将呈现出良好的 EPR 谱图，包含两组四个质子，因此线条更为复杂。为预测其谱图，同样可从优化阴离子的几何结构开始：

!RI-B2PLYP DEF2-TZVP DEF2-TZVP/C TIGHTOPT
* XYZFILE -1 2 nap.xyz

We used a double-hybrid functional to add RI-MP2 correlation and get a better density and TIGHTOPT to obtain a good symmetrical geometry.
我们采用双杂化泛函引入 RI-MP2 相关性，以获得更优的密度，并通过 TIGHTOPT 获取良好的对称几何结构。

Now we run an EPR calculation, with the specialized EPR-III basis and RI-SOMF(1X) to speed-up the calculation of the integrals for the SOC part:
现在我们运行一个 EPR 计算，采用专门的 EPR-III 基组和 RI-SOMF(1X)方法来加速 SOC 部分积分的计算：

!RI-B2PLYP EPR-III AUTOAUX RI-SOMF(1X)
* XYZFILE -1 2 nap_optimized.xyz
%eprnmr
        gtensor true
        nuclei = all H {aiso, adip, aorb}
end

One can average the two sets of similar $H_{α}$ and $H_{β}$ couplings (calculated from the MP2 relaxed density ) and use again some software like EasySpin to compute the predicted spectrum and compare it to the experiment. This will look like:
可以将两组相似的 $H_{α}$ 和 $H_{β}$ 耦合常数（由 MP2 松弛密度计算得出）取平均值，然后再次使用 EasySpin 等软件计算预测光谱，并与实验结果进行比较。这将呈现如下形式：

Visualizing the g tensor orientation#
可视化 g 张量方向

You can use Avogadro to visualize the orientation of the g tensor. Here is how it works: ORCA prints a basename.gori.xyz file, where dummy atoms are used to represent these vectors, He for the origin, Ne for the direction of $g_{z}$ , Ar for $g_{y}$ and Kr for $g_{x}$ .
您可以使用 Avogadro 来可视化 g 张量的取向。具体操作如下：ORCA 会输出一个 basename.gori.xyz 文件，其中使用虚拟原子来表示这些矢量，He 代表原点，Ne 表示 $g_{z}$ 的方向，Ar 表示 $g_{y}$ ，Kr 表示 $g_{x}$ 。

Open this file and select the "Align" option in Avogadro, on the right of the "E" used for optimization.
打开此文件并在 Avogadro 中选择“Align”选项，位于用于优化的“E”右侧。
Now select the He-Ne bond and align the x axis and repeat in order for the y and z axis.
现在选择 He-Ne 键并沿 x 轴对齐，然后依次对 y 轴和 z 轴重复此操作。
Delete the noble gas atoms.
删除惰性气体原子。
Click on "Axes", the first option of the "Display Settings". Here it is!
点击“Axes”，即“显示设置”中的第一个选项。在这里！

Note 注释

You can increase the size of the vectors by clicking on the "Axes" options. The order is red for x, green for y and blue for z (remember the RGB sequence).
您可以通过点击“轴”选项来增大向量的大小。顺序为红色代表 x 轴，绿色代表 y 轴，蓝色代表 z 轴（记住 RGB 顺序）。

Zero-field Splitting (ZFS)#
零场分裂 (ZFS)

If you have systems with $S > 1 / 2$ , you might be interested in ZFS values. The ORCA_EPRNMR module can also be used to predict those. Let's try the example of cyclopentadienylidene carbene, which has known D value of $0.408 c m^{- 1}$ and a $E / D$ ratio of $0.046$ [Kirmse1971]. First, the geometry is optimized using:
如果你有包含 $S > 1 / 2$ 的系统，可能会对 ZFS 值感兴趣。ORCA_EPRNMR 模块也可用于预测这些值。让我们以环戊二烯基卡宾为例，其已知的 D 值为 $0.408 c m^{- 1}$ ， $E / D$ 比率为 $0.046$ [Kirmse1971]。首先，使用以下方法优化几何结构：

!B3LYP DEF2-TZVP OPT
* XYZFILE 0 3 carb.xyz

to get the planar carbene:
得到平面型卡宾：

The ZFS can then be computed on simple DFT level using:
然后可以在简单的 DFT 级别上计算 ZFS，使用：

!B3LYP EPR-II AUTOAUX UNO
* XYZFILE 0 3 carb_optimized.xyz
%EPRNMR DTENSOR SSANDSO
        DSS     UNO
        DSOC    CP
END

The DTENSOR flag asks to include the spin-spin and the spin-orbit components of the $D$ (SSANDSO), and the DSS and DSOC control the specific algorithms. For more information, consult the ORCA manual.
DTENSOR 标志请求包含 $D$ 的旋-旋和旋-轨分量（SSANDSO），而 DSS 和 DSOC 控制具体算法。更多信息，请参阅 ORCA 手册。

After the calculation of the SOC-related integrals, one should see the header:
在计算完与 SOC 相关的积分后，应看到以下标题：

------------------------------------------------------------------------------
                                ORCA EPR/NMR CALCULATION
------------------------------------------------------------------------------

GBWName                      ... ZFS.gbw
Electron density file        ... ZFS.scfp
Spin density file            ... ZFS.scfr
Spin-orbit integrals         ... ZFS
Multiplicity                 ... 3
g-tensor                     ... 0
NMR chemical shifts          ... 0
D-tensor                     ... spin-spin and spin-orbit
D(SS)-algorithm              ... spin-density from UNOs
D(SOC)-algorithm             ... CP (=Coupled-perturbed)

explaining the details of the calculation. The actual result is below the CP-SCF:
解释计算细节。实际结果位于 CP-SCF 之下：

D   =     0.425943  cm**-1
E/D =     0.015874

which is again close to the experiment, considering that we are using DFT and a single-reference method.
考虑到我们使用的是 DFT 和单参考方法，这与实验结果再次相当接近。

Important 重要

This was intended to be a simple demonstration. In general, much better results for ZFS can be obtained from multi-reference theories such as CASSCF or MRCI. Nonetheless, it is still quite good and is easily applicable to large systems.
这是一个简单的演示。通常，通过多参考理论如 CASSCF 或 MRCI 可以获得 ZFS 的更好结果。尽管如此，它仍然相当不错，并且易于应用于大型系统。

Electron Paramagnetic Resonance (EPR)#电子顺磁共振（EPR）

The CH3 radical#CH 3 部首 #

The output file# 输出文件 #

g Tensor# 张量 #

Hyperfine couplings# 超精细耦合

Comparison to experiment#与实验的比较 #

Another example, the naphthalene anion radical#另一个例子是萘阴离子自由基

Visualizing the g tensor orientation#可视化 g 张量方向

Zero-field Splitting (ZFS)#零场分裂 (ZFS)

Structures# 结构

Electron Paramagnetic Resonance (EPR)#
电子顺磁共振（EPR）

The CH₃ radical#
CH ₃ 部首 #

Comparison to experiment#
与实验的比较 #

Another example, the naphthalene anion radical#
另一个例子是萘阴离子自由基

Visualizing the g tensor orientation#
可视化 g 张量方向

Zero-field Splitting (ZFS)#
零场分裂 (ZFS)