# How does Gaussian 09 normalize the excitation coefficients?

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?

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.