Fractional occupation density (FOD)#
分数占据密度 (FOD)

Most conventional DFT and many wave-function-theory (WFT) methods are single-reference methods and thus do not cover static electron correlation effects. Cases with strong static electron correlation typically can not be described by a single Slater determinant and are thus often referred to as "multi-reference" (MR) cases. For example, transition metal complexes and specifically those including 3d metals, are known to be prone to static electron correlation and therefore single-reference methods may not be suitable to describe them correctly. There are many MR diagnostics described in the literature with individual strengths and weaknesses. Nevertheless, the fractional occupation density (FOD) [Grimme2015] is a very simple yet powerful analytic tool to even visualize potential MR situations that is available in ORCA.
大多数传统的密度泛函理论（DFT）和许多波函数理论（WFT）方法都是单参考方法，因此无法涵盖静态电子相关效应。具有强静态电子相关的体系通常无法用单一的 Slater 行列式描述，因此常被称为“多参考”（MR）情况。例如，过渡金属配合物，特别是包含 3d 金属的配合物，已知容易受到静态电子相关的影响，因此单参考方法可能不适合准确描述它们。文献中描述了许多多参考诊断方法，各有优缺点。然而，分数占据密度（FOD）[Grimme2015]是一种非常简单但强大的分析工具，甚至可以在 ORCA 中可视化潜在的多参考情况。

The default FOD analysis in ORCA employs the TPSS/def2-TZVP method with a electronic temperature of 5000K to "smear" static correlation prone electrons. It is simply envoked by:
ORCA 中默认的 FOD 分析采用 TPSS/def2-TZVP 方法，并设置电子温度为 5000K，以“涂抹”易受静态关联影响的电子。只需简单调用即可实现：

!FOD

* XYZFILE 0 1 structure.xyz

Note 注释

In principle, the FOD analysis can be performed with any functional. Nevertheless, the electronic temperature has to be adjusted to match the functional, and specifically the amount of Fock-exchange included if a hybrid is used. How to change the method and the electronic temperature is documented in the ORCA manual.
原则上，FOD 分析可使用任何泛函进行。然而，电子温度需调整以匹配该泛函，特别是当使用杂化泛函时，需调整包含的 Fock 交换量。如何更改方法及电子温度，在 ORCA 手册中有详细记载。

Example: Ni(bis-dithiolene)#
示例：Ni(双二硫烯)

Let us do the FOD analysis for the potential electrocatalyst NiBDT.
让我们对潜在电催化剂 NiBDT 进行 FOD 分析。

!FOD

* XYZFILE 0 1 nibdt.xyz

After a successful calculation, ORCA will write the N_FOD value to the output. This value is a first indication for MR situations. In this case N_FOD is very high for a normal closed-shell complex:
成功计算后，ORCA 会将 N _FOD 值写入输出文件。该值是 MR 情况的首个指示。在此情况下，对于一个正常的闭壳层复合物，N _FOD 值异常高：

N_FOD =  1.877046

Note 注释

Note that the N_FOD value is not size-consistent and may be devided by the number of electrons if complexes of different sizes are compared.
注意，N _FOD 值不具有尺寸一致性，在比较不同尺寸的复合物时，可能需要除以电子数。

Further, ORCA stores the FOD density in the basename.densities container which can be visualized with the orca_plot tool that is delivered with ORCA. Its interactive interface can be used via the command line input:
此外，ORCA 将 FOD 密度存储在 basename.densities 容器中，该容器可以通过随 ORCA 提供的 orca_plot 工具进行可视化。其交互界面可通过命令行输入使用：

orca_plot basename.gbw -i

PlotType       ... MO-PLOT
MO/Operator    ... 0 0
Output file    ... (null)
Format         ... Grid3D/Binary
Resolution     ... 40 40 40
Boundaries     ...   -22.359392    16.311484 (x direction)
                     -12.628859    13.574504 (y direction)
                      -7.001812     7.002006 (z direction)

       1 - Enter type of plot
       2 - Enter no of orbital to plot
       3 - Enter operator of orbital (0=alpha,1=beta)
       4 - Enter number of grid intervals
       5 - Select output file format
       6 - Plot CIS/TD-DFT difference densities
       7 - Plot CIS/TD-DFT transition densities
       8 - Set AO(=1) vs MO(=0) to plot
       9 - List all available densities
      10 - Perform Density Algebraic Operations

      11 - Generate the plot
      12 - exit this program
Enter a number:

Now we enter 1 to choose the type of plot we need:
现在我们进入 1 以选择所需的绘图类型：

     1 -   molecular orbitals                          
     2 -   (scf) electron density                              ......  (scfp                     )  => AVAILABLE
     3 -   (scf) spin density                                  ......  (scfr                     )  - NOT AVAILABLE
     4 -   natural orbitals                               
     5 -   corresponding orbitals                         
     6 -   atomic orbitals                                
     7 -   mdci electron density                               ......  (mdcip                    )  - NOT AVAILABLE
     8 -   mdci spin density                                   ......  (mdcir                    )  - NOT AVAILABLE
[...]
    42 -   LFT QDPT unrelaxed transition AO density            ......  (Tdens-LFTQDSOC           )  - NOT AVAILABLE
Enter Type:

Here, we choose 2 for (scf) electron density. The interface will now ask, which density should be plotted. The default electron density file is basename.scfp but as we want to plot basename.scfp_fod we change the chosen density file.
在此，我们选择 2 用于 (scf) electron density 。界面将询问应绘制哪种密度。默认的电子密度文件是 basename.scfp ，但由于我们希望绘制 basename.scfp_fod ，因此我们更改了所选的密度文件。

The default name of the density would be: basename.scfp
Is this the one you want (y/n)? n
Enter the FileName: basename.scfp_fod

Now, we can choose the desired output format via 5 - Select output file format, e.g., Gaussian cube, and finally generate the file basename.eldens.cube to process with the visualization program of choice via 11 - Generate the plot. To visualize the FOD, an iso surface value of 0.005 a.u. is recommended. For our example, NiBDT, the FOD plot looks like:
现在，我们可以通过 5 - Select output file format 选择所需的输出格式，例如高斯立方体，并最终生成文件 basename.eldens.cube ，以便通过 11 - Generate the plot 与所选的可视化程序进行处理。为了可视化 FOD，建议使用 0.005 a.u.的等值面值。对于我们的示例 NiBDT，FOD 图如下所示：

../_images/fod.png — Figure: FOD plot for NiBDT at an isosurface value of 0.005 a.u.
图：NiBDT 在等值面值为 0.005 a.u.时的 FOD 图#

The plot shows a significant delocalization of FOD over the whole molecule indicating strong static electron correlation.
该图显示了 FOD 在整个分子中的显著离域化，表明存在强烈的静态电子相关性。

For very large structures or high-throughput screening it may be further useful to use the xtb-based FOD anaylsis via ORCA. To do so, you can call xtb to do the FOD analysis via ORCAs interface to xtb:
对于非常大的结构或高通量筛选，进一步使用基于 xtb 的 FOD 分析通过 ORCA 可能更为有效。为此，您可以通过 ORCA 的接口调用 xtb 进行 FOD 分析：

!XTB2

%xtb
  XTBINPUTSTRING  "--fod" # string passed on to XTB call
  ETemp           5000
end

%output 
  print[p_xtb_prop] 1 
end

* XYZFILE 0 1 nibdt.xyz

For our NiBDT example, this will print the N_FOD value to the ORCA printout and will write a plottable fod.cub file in Gaussian cube format.
对于我们的 NiBDT 示例，这将把 N _FOD 值打印到 ORCA 输出中，并生成一个可绘制的 fod.cub 文件，格式为 Gaussian 立方体。

NFOD :     2.4882

Warning 警告

Even though the FOD is intuitive and easy-to-use, it is not a 100% safe diagnostic for MR character and should only be used as a first indication. We recommend to always combine the FOD analysis with more sophisticated multi-reference diagnostics.
尽管 FOD 直观且易于使用，但它并非 100%安全的 MR 特征诊断工具，仅应作为初步指示。我们建议始终将 FOD 分析与更为复杂的多参考诊断相结合。

Fractional occupation density (FOD)#分数占据密度 (FOD)

Example: Ni(bis-dithiolene)#示例：Ni(双二硫烯)

Structures# 结构

Fractional occupation density (FOD)#
分数占据密度 (FOD)

Example: Ni(bis-dithiolene)#
示例：Ni(双二硫烯)