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Study notes：ΔSCF

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By Bane Dysta
11 min read
Study notes：ΔSCF

关于ΔSCF，引用一下sob老师在乱谈激发态的计算方法中的描述：

ΔSCF：这不是某个具体计算激发态的理论方法，而只是一种结合HF/DFT计算激发态的技术。也就是通过调整初猜，将基态时占据轨道的一个电子挪到一个与之不可约表示不同的虚轨道上，于是经过SCF迭代，就会得到与原先对称性不同的电子态，即激发态。这种计算激发态的好处在于计算耗时和计算基态完全一样，故优化、计算频率很方便，而且考虑了轨道的弛豫，故能够得到激发态的轨道。所有支持对称性的量化程序原理上都能做ΔSCF。然而，此方法有以下明显缺点，导致很少被用于实际研究： （1）调换占据数的那两个轨道必须有不同不可约表示，否则经过SCF又会坍塌到基态。但这点很难满足，实际体系更是往往连对称性都没有。（不过值得一提的是，Q-Chem中有个Maximum Overlap Method方法，可以实现强制SCF收敛到指定的占据态而不受轨道对称性的限制，这将ΔSCF适用范围加宽了） （2）难以计算多个激发态，除非每个激发态都有不同的不可约表示，这个也很难满足。 （3）没法直接给出振子强度。虽然这在理论上也能算，但程序并不直接给出。

ΔSCF是种不太常见的激发态计算方法，全网搜能找到的资料都很少，但这个方法当前又偏偏有不可取代的用途，只好自己研究。

Gaussian

Gaussian使用guess=alter实现ΔSCF，但如sob老师所说要求基态与想算的那个激发态不可约表示不同，否则会坍缩回基态。本例以芴正离子为例。

优化结构略过。使用优化后的结构进行td-dft计算，检查输出中布居分析和激发分析部分：

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**********************************************************************

            Population analysis using the SCF density.

 **********************************************************************

 Orbital symmetries:
       Occupied  (A1) (A1) (B2) (A1) (B2) (B2) (A1) (B2) (A1) (B2)
                 (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B2) (A1)
                 (A1) (B2) (A1) (B2) (A1) (A1) (B2) (A1) (B2) (B2)
                 (A1) (B2) (B1) (A1) (A1) (A2) (B2) (A1) (B2) (B1)
                 (B1) (A2) (A2)
       Virtual   (B1) (B1) (A2) (A2) (B1) (A1) (B2) (A1) (A1) (B2)
                 (B1) (A1) (B2) (A1) (A2) (B2) (A1) (B2) (A1) (B2)
                ... // 不重要的虚轨道略

                 (A1) (A1) (B2) (B2) (A1) (B2) (B2) (A1) (A1) (B2)
                 (A1) (B2) (A1) (B2)
 The electronic state is 1-A1.
... // 其他输出略

 Excited State   1:      Singlet-B2     0.8587 eV 1443.87 nm  f=0.0003  <S**2>=0.000
      43 -> 44         0.70835
 This state for optimization and/or second-order correction.
 Total Energy, E(TD-HF/TD-DFT) =  -499.636266514    
 Copying the excited state density for this state as the 1-particle RhoCI density.

从输出可以看到，芴正离子的基态不可约表示为A1，S1为B2，因此可以使用ΔSCF计算S1。贡献S1的跃迁轨道对是43与44，那么我们可以如此构造S1电子态：

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%chk=deltascf_flourene.chk
#p guess=alter uPBE1PBE/def2svp

deltascf

1 1
C 0.000000 2.509223 1.232265
C 0.000000 3.414161 0.169025
C 0.000000 2.988308 -1.183859
... //坐标略

H -0.000000 -4.485503 0.380499
H -0.000000 -2.861163 2.264922

43,44

// 此处两行空行有可能不够，建议多敲

观察输出：

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Alpha Orbitals:
       Occupied  (B2) (A1) (B2) (A1) (B2) (A1) (B2) (A1) (A1) (B2)
                 (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B2) (A1)
                 (A1) (B2) (A1) (B2) (A1) (A1) (B2) (A1) (B2) (B2)
                 (A1) (B2) (A1) (B1) (A1) (B2) (A2) (A1) (B2) (B1)
                 (B1) (A2) (B1)
       Virtual   (A2) (B1) (A2) (A2) (B1) (B2) (A1) (B1) (A1) (A2)
                 (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B2) (A1) (B2)
                ...

                 (A1) (A1) (B2) (B2) (A1) (B2) (B2) (A1) (A1) (B2)
                 (A1) (B2) (A1) (B2)
 Beta  Orbitals:
       Occupied  (B2) (A1) (B2) (A1) (B2) (A1) (B2) (A1) (A1) (B2)
                 (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B2) (A1)
                 (A1) (B2) (A1) (B2) (A1) (A1) (B2) (A1) (B2) (B2)
                 (A1) (B2) (A1) (B1) (A1) (B2) (A2) (A1) (B2) (B1)
                 (B1) (A2) (A2)
       Virtual   (B1) (B1) (A2) (A2) (B1) (B2) (A1) (B1) (A1) (A2)
                 (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B2) (A1) (B2)
                ...

                 (A1) (A1) (B2) (B2) (A1) (B2) (B2) (A1) (A1) (B2)
                 (A1) (B2) (A1) (B2)
 The electronic state of the initial guess is 1-B2.

我们应当看到Alpha列的HOMO(A2点群)与LUMO(B1点群)已经对调位置，而Beta列则维持不变，此计算成功的希望很大。接下来，我们还要检查chk文件中，目标轨道是否真的调换过来了。

即使有合适的对称性，这个计算也是有可能失败的。

未能维持S1的对称性

某体系S1跃迁轨道对为95 → 96。对调95、96，构造不可约表示为B的S1态：

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 Pairs of Alpha orbitals switched:
   95  96
 No Beta  orbitals switched.
 Initial guess orbital symmetries:
 Alpha Orbitals:
       Occupied  (A) (B) (B) (A) (A) (B) (B) (A) (A) (B) (A) (B)
                 (A) (B) (A) (B) (A) (B) (A) (A) (B) (A) (B) (A)
                ...

                 (B) (B) (B) (A) (B) (A) (B) (B) (A) (B) (A) (B)
                 (A) (B) (A) (B) (B) (A) (B) (A) (A) (B) (B)
       Virtual   (A) (B) (B) (A) (A) (A) (B) (A) (A) (B) (A) (A)
                 (B) (B) (B) (A) (B) (B) (B) (A) (A) (A) (A) (B)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (A) (B) (B) (A) (B)
                 (B) (A)
 Beta  Orbitals:
       Occupied  (A) (B) (B) (A) (A) (B) (B) (A) (A) (B) (A) (B)
                 (A) (B) (A) (B) (A) (B) (A) (A) (B) (A) (B) (A)
                ...

                 (B) (B) (B) (A) (B) (A) (B) (B) (A) (B) (A) (B)
                 (A) (B) (A) (B) (B) (A) (B) (A) (A) (B) (A)
       Virtual   (B) (B) (B) (A) (A) (A) (B) (A) (A) (B) (A) (A)
                 (B) (B) (B) (A) (B) (B) (B) (A) (A) (A) (A) (B)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (A) (B) (B) (A) (B)
                 (B) (A)
 The electronic state of the initial guess is 1-B.

然而最终输出：

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 **********************************************************************

            Population analysis using the SCF density.

 **********************************************************************

 Orbital symmetries:
 Alpha Orbitals:
       Occupied  (A) (B) (B) (A) (B) (A) (B) (A) (A) (A) (B) (B)
                 (A) (A) (B) (B) (A) (A) (B) (A) (A) (B) (A) (A)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (B) (B) (B) (B) (A)
                 (A) (B) (A) (B) (A) (B) (B) (A) (B) (A) (A)
       Virtual   (B) (B) (B) (A) (A) (A) (B) (A) (B) (A) (A) (B)
                 (A) (B) (A) (B) (A) (B) (A) (B) (A) (B) (B) (A)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (A) (B) (B) (A) (B)
                 (B) (A)
 Beta  Orbitals:
       Occupied  (A) (B) (B) (A) (B) (A) (B) (A) (A) (A) (B) (B)
                 (A) (A) (B) (B) (A) (A) (B) (A) (A) (B) (A) (A)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (B) (B) (B) (B) (A)
                 (A) (B) (A) (B) (A) (B) (B) (A) (B) (A) (A)
       Virtual   (B) (B) (B) (A) (A) (A) (B) (A) (B) (A) (A) (B)
                 (A) (B) (A) (B) (A) (B) (A) (B) (A) (B) (B) (A)
                ...

                 (A) (B) (B) (A) (B) (A) (A) (A) (B) (B) (A) (B)
                 (B) (A)
 The electronic state is 1-A.

原本构造的(A) (B) (B) | (A) (B) (B)对称性的态在收敛后变成了(B) (A) (A) | (B) (B) (B)，对称性也变成了A，说明未能收敛到S1，估计是歪到了其他激发态。

应对方法：

  • SCF=QC：使用二次收敛法，力求迅速收敛到构造的电子态附近；
  • SCF=conver=6：放宽收敛限，有可能在轨道顺序乱掉之前达到收敛。这个精度对于单点计算已经足够精确；
  • 读取初猜：读取其他程序的收敛MOM波函数作为初猜。

SCF不收敛

这里开了SCF=QC，出现l508报错。不开QC时也可能出现l502报错。

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 Search did not lower the energy significantly.
 No lower point found -- run aborted.
 Error termination via Lnk1e in /work/home/gaus15/apprepo/gaussian/16-hy/scripts/../app/g16/l508.exe at Wed Apr  9 22:21:04 2025.
 Job cpu time:       0 days 21 hours  8 minutes 12.1 seconds.
 Elapsed time:       0 days  0 hours 39 minutes 56.8 seconds.
 File lengths (MBytes):  RWF=    214 Int=      0 D2E=      0 Chk=     18 Scr=      1

应对方法：按照SCF不收敛问题的一般解决方法去尝试，但有些方法会干扰构造的电子态。

ORCA

截至6.0.1版本，orca已经提供了Maximum Overlap Method (MOM)方法来强行收敛到对应电子态。但我测试了一下，有些情况alpha列正确换过去了，但beta列的轨道会发生变形，没法维持对称性。

note
note excited state
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