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When I did excited state calculations with Gaussian 09, I was asking me on how those values are normalized because a simple sum over the squared of the coefficients did not yield 1.

So ... how are those coefficients normalized then?

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1 Answer 1

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I made a few quick calculations with very high printing levels. (I don't think that the molecule is important for this, but my calculations have been done on 2,3,6,7-tetranitronaphthalene with RPM6/UPM6 and 10 excitated states.)

The default printing level is IOp(9/40=1) which yields excitations with $c \ge 0.1$

Excited State   1:      Singlet-A      2.4747 eV  501.01 nm  f=0.0040  <S**2>=0.000
      41 -> 59        -0.10248
      42 -> 60         0.15085
      47 -> 57         0.15074
      55 -> 57         0.41262
      56 -> 58         0.46353

The sum of the squared coefficients is 0.44110, which is absolutely not 1.

The unrestricted version* for the same excitation includes a lot more coefficients:

Excited State   1:  3.000-A      -0.6305 eV    -1966.38 nm  f=-0.0000  <S**2>=2.000
     37A -> 69A       -0.16535
     41A -> 59A       -0.18261
     41A -> 68A       -0.19102
     42A -> 60A        0.28385
     42A -> 64A        0.19591
     47A -> 57A        0.18349
     48A -> 58A        0.10140
     55A -> 57A        0.46982
     55A -> 65A        0.11894
     56A -> 58A        0.65048
     56A -> 61A       -0.11129
     37B -> 69B        0.16535
     41B -> 59B        0.18261
     41B -> 68B        0.19102
     42B -> 60B       -0.28385
     42B -> 64B       -0.19591
     47B -> 57B       -0.18349
     48B -> 58B       -0.10140
     55B -> 57B       -0.46982
     55B -> 65B       -0.11894
     56B -> 58B       -0.65048
     56B -> 61B        0.11129
     37A <- 69A       -0.12501
     41A <- 59A       -0.13192
     41A <- 68A       -0.14182
     42A <- 60A        0.20466
     42A <- 64A        0.14232
     47A <- 57A        0.13721
     55A <- 57A        0.33618
     56A <- 58A        0.43137
     37B <- 69B        0.12501
     41B <- 59B        0.13192
     41B <- 68B        0.14182
     42B <- 60B       -0.20466
     42B <- 64B       -0.14232
     47B <- 57B       -0.13721
     55B <- 57B       -0.33618
     56B <- 58B       -0.43137

Now taking the sum over those squared coefficients gives 2.72735, which is also obviously not 1. But the subtraction of the de-excitations (<-) from the excitations (->) seems to be appropriate, as it yields 0.99451.

Seeing this, the sum for the restricted case somehow looks near 0.5 ... why not doubling the squared coefficients? This leads to 0.88220 which is acceptable close to 1. Maybe the printing level is too low for the restricted calculation?

Rising the printing level with IOp(9/40=2) ($c \ge 0.01$) yields this output for the first excitation:

 Excited State   1:      Singlet-A      2.4747 eV  501.01 nm  f=0.0040  <S**2>=0.000
      17 -> 66         0.01120
      18 -> 92         0.01037
      23 -> 82         0.01204
      24 -> 73        -0.01804
      24 -> 80        -0.01353
      25 -> 62        -0.01457
      25 -> 74         0.01303
      26 -> 57        -0.01038
      26 -> 69        -0.01748
      27 -> 68        -0.01840
      28 -> 75        -0.01002
      33 -> 73        -0.02580
      33 -> 80        -0.01986
      34 -> 76         0.01612
      34 -> 89        -0.02096
      35 -> 67         0.01410
      35 -> 75        -0.03109
      35 -> 84         0.01033
      35 -> 88         0.01154
      35 -> 91        -0.01028
      36 -> 81        -0.01389
      36 -> 90         0.01911
      37 -> 57         0.02399
      37 -> 63         0.01245
      37 -> 65         0.03441
      37 -> 69        -0.08550
      38 -> 72         0.03176
      38 -> 75         0.01264
      38 -> 88         0.01466
      38 -> 91        -0.01049
      39 -> 63         0.01742
      39 -> 65        -0.01751
      39 -> 69         0.01804
      39 -> 71         0.03883
      39 -> 90         0.01569
      40 -> 61         0.01120
      40 -> 66        -0.01695
      40 -> 70        -0.05758
      40 -> 73        -0.03491
      40 -> 77         0.01693
      41 -> 59        -0.10248
      41 -> 62         0.01194
      41 -> 68        -0.09264
      42 -> 60         0.15085
      42 -> 64         0.09639
      45 -> 57        -0.01773
      47 -> 57         0.15074
      47 -> 63         0.01564
      47 -> 65         0.04802
      47 -> 69        -0.02597
      48 -> 58         0.05950
      48 -> 61        -0.01527
      49 -> 60        -0.05724
      49 -> 64        -0.02758
      50 -> 59         0.04110
      50 -> 68         0.02528
      51 -> 58         0.01641
      51 -> 66        -0.02468
      51 -> 70        -0.02435
      51 -> 82         0.01341
      52 -> 60         0.02192
      52 -> 64         0.01863
      52 -> 67         0.02426
      52 -> 72         0.01922
      52 -> 75        -0.01609
      53 -> 62        -0.03163
      53 -> 74         0.01216
      54 -> 57         0.02946
      54 -> 63         0.03823
      54 -> 65        -0.01008
      54 -> 71         0.02057
      55 -> 57         0.41262
      55 -> 63         0.02503
      55 -> 65         0.09254
      55 -> 69        -0.03194
      56 -> 58         0.46353
      56 -> 61        -0.06155
      33 <- 73        -0.01421
      33 <- 80        -0.01130
      34 <- 89        -0.01340
      35 <- 75        -0.01702
      36 <- 90         0.01197
      37 <- 57         0.01598
      37 <- 69        -0.01657
      38 <- 72         0.01731
      39 <- 63         0.01032
      39 <- 71         0.02069
      40 <- 70        -0.02983
      40 <- 73        -0.01759
      41 <- 59        -0.01636
      41 <- 68        -0.01551
      42 <- 60         0.03039
      42 <- 64         0.01865
      47 <- 57         0.03857
      47 <- 65         0.01219
      48 <- 58         0.01179
      51 <- 70        -0.01217
      54 <- 71         0.01054
      55 <- 57         0.08529
      55 <- 65         0.02862
      55 <- 69        -0.01486
      56 <- 58         0.06624
      56 <- 61        -0.01293

Again, calculating the square of sums yield 0.53927 (doubled: 1.07853) which is a little bit more than 0.5. De-excitation values subtracted from excitation values now give reasonable 0.49889, which is quite as good as the unrestricted case if doubled (0.99778).

Rising the printing level to IOp(9/40=5) and calculating the sum of squared of the coefficients (de-exc. minus exc.) gives 1 for unrestricted case and 0.5 for the restricted case.

Based on that, Gaussian's normalization for excitation coefficients is most probably: $$ \begin{cases} 2 \left( \sum_i c_{i,\text{exc}}^2 - \sum_i c_{i,\text{de-exc}}^2 \right) =1 & \text{for restricted methods}\\ \sum_i c_{i,\text{exc}}^2 - \sum_i c_{i,\text{de-exc}}^2 =1 & \text{for unrestricted methods} \end{cases} $$


* The same geometry as for the restricted calculation was used.

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