A new density for transition properties within the similarity transformed equation of motion approach

Figures & data

Table 1. Excitation energy (ω in eV) and oscillator strength (f in a.u.) values calculated at the STEOM levels of theory using the full STEOM density, L2 indicating the presence of the correction term.

		ω(STEOM)/f(STEOM)	ω(STEOM)/f(STEOM)	ω(STEOM)/f(STEOM)	ω(STEOM)/f(STEOM)
Molecule	State	$N_R=30$, $N_G=30$	$N_R=30$, $N_G=60$	$N_R=5$, $N_G=60$	$N_R=30$, $N_G=60$, L2
Ethene	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	8.31/0.492	8.31/0.491	8.31/0.512	8.31/0.405
E-butadiene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	6.65/0.973	6.65/0.975	6.65/0.993	6.65/0.816
All-E-hexatriene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	5.64/1.496	5.65/1.472	5.65/1.502	5.65/1.257
All-E-octatetraene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	4.98/2.019	4.99/2.012	4.99/2.050	4.99/1.750
Norbornadiene	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	7.65/0.269	7.65/0.269	7.65/0.275	7.65/0.197
s-Tetrazine	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	7.82/0.539	7.83/0.497	7.83/0.497${}^a$	7.83/0.389
s-Tetrazine	${}^1$B${}_{2u}$ ($\pi \to {\pi }^{\ast }$)	8.64/0.575	8.50/0.483	8.50/0.498${}^a$	8.50/0.364

Notes: $N_R$ is the number of roots requested, and $N_G$ is the number of CIS guess states from which the active space of orbitals is determined. The superscript a refers to a calculation with $N_R=15$, $N_G=60$.

Table 2. Excitation energy (ω in eV) and oscillator strength (f in a.u.) values calculated at the CC3, CIS and STEOM levels of theory for the small molecules in Thiel's test set.

molecule	state	ω(CC3)	f(CC3)	ω(STEOM)	f(CIS)	f(STEOM)	f(STEOM-DD)
Ethene	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	8.37	0.389	8.31	0.607	0.405	0.399
E-butadiene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	6.58	0.726	6.65	1.138	0.816	0.791
All-E-hexatriene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	5.58	1.129	5.65	1.646	1.257	1.210
All-E-octatetraene	${}^1$B${}_u$ ($\pi \to {\pi }^{\ast }$)	4.94	1.549	4.99	2.179	1.750	1.686
Cyclopropene	${}^1$B${}_1$ ($\sigma \to {\pi }^{\ast }$)	6.90	0.001	6.95	0.002	0.001	0.001
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	7.10	0.083	7.09	0.164	0.087	0.087
Cyclopentadiene	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	5.73	0.095	5.70	0.162	0.103	0.099
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	8.69	0.596	8.62	1.005	0.630	0.600
Norbornadien	${}^1$A${}_2$ ($\pi \to {\pi }^{\ast }$)	5.64	0.000	5.55	0.000	0.000	0.000
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	6.49	0.027	6.49	0.064	0.034	0.031
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	7.64	0.185	7.65	0.368	0.197	0.188
	${}^1$A${}_2$ ($\pi \to {\pi }^{\ast }$)	7.71	0.000	7.64	0.000	0.000	0.000
Benzene	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	6.68	0.000	6.72	0.000	0.000	0.000
	${}^1$E${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	7.45	0.630	7.41	1.089	0.690	0.650
Naphthalene	${}^1$B${}_{2u}$ ($\pi \to {\pi }^{\ast }$)	5.03	0.085	5.10	0.184	0.104	0.098
	${}^1$B${}_{3u}$ ($\pi \to {\pi }^{\ast }$)	6.33	1.325	6.30	2.217	1.520	1.439
	${}^1$B${}_{2u}$ ($\pi \to {\pi }^{\ast }$)	6.57	0.239	6.61	0.514	0.304	0.285
	${}^1$B${}_{1g}$ ($\pi \to {\pi }^{\ast }$)	6.79	0.000	6.76	0.000	0.000	0.000
Furan	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	6.60	0.155	6.67	0.241	0.155	0.149
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	8.53	0.450	8.53	0.812	0.504	0.471
Pyrrole	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	6.71	0.167	6.73	0.243	0.155	0.148
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	8.17	0.478	8.17	0.829	0.526	0.485
Imidazole	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	6.58	0.081	6.49	0.064	0.044	0.043
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	6.83	0.004	6.73	0.008	0.005	0.005
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	7.10	0.082	7.04	0.169	0.108	0.103
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	7.94	0.005	7.88	0.007	0.004	0.004
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	8.45	0.410	8.46	0.576	0.413	0.380
Pyridine	${}^1$B${}_1$ ($n\to {\pi }^{\ast }$)	5.06	0.005	5.01	0.007	0.006	0.006
	${}^1$A${}_2$ ($n\to {\pi }^{\ast }$)	5.51	0.000	5.50	0.000	0.000	0.000
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	6.85	0.014	6.91	0.018	0.012	0.011
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	7.59	0.482	7.59	0.915	0.581	0.552
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	7.70	0.526	7.66	0.980	0.613	0.575
	${}^1$B${}_2$ ($n\to R$)	9.11	0.114	9.26	0.100	0.104	0.095
Pyrazine	${}^1$B${}_{3u}$ ($n\to {\pi }^{\ast }$)	4.25	0.007	4.21	0.010	0.008	0.007
	${}^1$A${}_u$ ($n\to {\pi }^{\ast }$)	5.05	0.000	5.08	0.000	0.000	0.000
	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	7.07	0.070	7.17	0.113	0.074	0.073
	${}^1$B${}_{2u}$ ($\pi \to {\pi }^{\ast }$)	8.05	0.376	8.05	0.787	0.489	0.456
	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	8.06	0.407	8.03	0.813	0.476	0.460
Pyrimidine	${}^1$B${}_1$ ($n\to {\pi }^{\ast }$)	4.51	0.006	4.44	0.009	0.007	0.007
	${}^1$A${}_2$ ($n\to {\pi }^{\ast }$)	4.93	0.000	4.85	0.000	0.000	0.000
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	7.06	0.043	7.14	0.055	0.037	0.035
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	7.74	0.391	7.72	0.780	0.509	0.481
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	8.01	0.415	7.95	0.859	0.548	0.512
Pyridazine	${}^1$B${}_1$ ($n\to {\pi }^{\ast }$)	3.93	0.006	3.85	0.010	0.007	0.007
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	6.93	0.012	7.00	0.033	0.019	0.019
	${}^1$B${}_2$ ($\pi \to {\pi }^{\ast }$)	7.55	0.340	7.60	0.817	0.499	0.475
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	7.82	0.433	7.86	0.883	0.534	0.503
s-Triazine	${}^1$A${}_2^{″}$ ($n\to {\pi }^{\ast }$)	4.76	0.016	4.80	0.024	0.019	0.019
	${}^1$A${}_1^{″}$ ($n\to {\pi }^{\ast }$)	4.78	0.000	4.75	0.000	0.000	0.000
	${}^1$E${}^{″}$ ($n\to {\pi }^{\ast }$)	4.82	0.000	4.67	0.000	0.000	0.000
	${}^1$A${}_1^{\prime }$ ($\pi \to {\pi }^{\ast }$)	7.41	0.000	7.49	0.000	0.000	0.000
	${}^1$E${}^{″}$ ($n\to {\pi }^{\ast }$)	7.82	0.000	7.89	0.000	0.000	0.000
	${}^1$E${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	8.04	0.386	8.02	0.754	0.509	0.480
s-Tetrazine	${}^1$B${}_{3u}$ ($n\to {\pi }^{\ast }$)	2.54	0.007	2.47	0.011	0.008	0.009
	${}^1$A${}_u$ ($\pi \to {\pi }^{\ast }$)	3.80	0.000	3.80	0.000	0.000	0.000
	${}^1$A${}_u$ ($n\to {\pi }^{\ast }$)	5.46	0.000	5.40	0.000	0.000	0.000
	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	7.45	0.002	7.62	0.007	0.001	0.001
	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	7.79	0.349	7.83	0.668	0.387	0.373
	${}^1$B${}_{2u}$ ($\pi \to {\pi }^{\ast }$)	8.51	0.307	8.50	0.648	0.364	0.337
Formaldehyde	${}^1$A${}_2$ ($n\to {\pi }^{\ast }$)	3.95	0.000	3.82	0.000	0.000	0.000
	${}^1$B${}_1$ ($\sigma \to {\pi }^{\ast }$)	9.19	0.003	9.02	0.004	0.003	0.003
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	10.45	0.342	10.33	0.366	0.337	0.319
Acetone	${}^1$A${}_2$ ($n\to {\pi }^{\ast }$)	4.40	0.000	4.26	0.000	0.000	0.000
	${}^1$B${}_1$ ($\sigma \to {\pi }^{\ast }$)	9.17	0.000	9.00	0.000	0.000	0.000
	${}^1$A${}_1$ ($\pi \to {\pi }^{\ast }$)	9.65	0.245	9.68	0.309	0.255	0.241
p-Benzoquinone	${}^1$B${}_{3g}$ ($\pi \to {\pi }^{\ast }$)	4.59	0.000	4.70	0.000	0.000	0.000
	${}^1$B${}_{1u}$ ($\pi \to {\pi }^{\ast }$)	5.62	0.485	5.69	0.820	0.594	0.554
	${}^1$B${}_{3g}$ ($\pi \to {\pi }^{\ast }$)	7.28	0.000	7.35	0.000	0.000	0.000
Formamide	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.66	0.001	5.48	0.001	0.001	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	7.24	0.030	7.46	0.011	0.012	0.011
	${}^1$A${}^{\prime }$ ($n\to R$)${}^a$	8.27	0.386	8.34	0.319	0.328	0.303
	${}^1$A${}^{\prime }$ ($n\to R$)	8.47	0.125	8.66	0.123	0.142	0.128
Acetamide	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.70	0.001	5.49	0.001	0.001	0.000
Propanamide	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.72	0.000	5.56	0.001	0.000	0.000

Note: CIS here means the approximate STEOM density discussed in the main text. For almost all calculations, the number of roots $N_R$ is 30 and the number of guess states is $N_G=30$, while the superscript a refers to a calculation with $N_R=20$, $N_G=40$.

Table 3. Excitation energy (ω in eV) and oscillator strength (f in a.u.) values calculated at the CC3, CIS and STEOM levels of theory for the nucleobases in Thiel's test set.

molecule	state	ω(CC3)	f(CC3)	ω(STEOM)	f(CIS)	f(STEOM)	f(STEOM-DD)
Cytosine	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	4.72	0.046	4.48	0.077	0.055	0.052
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.16	0.001	5.12	0.002	0.002	0.002
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.52	0.001	5.66	0.001	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	5.61	0.130	5.57	0.289	0.256	0.235
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.97	0.000	5.97	0.000	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	6.61	0.520	6.40	0.694	0.590	0.549
Thymine	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	4.94	0.000	4.88	0.000	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	5.34	0.172	5.14	0.279	0.212	0.202
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	6.34	0.072	6.42	0.135	0.120	0.108
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	6.59	0.000	6.27	0.000	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)${}^a$	6.71	0.197	6.69	0.339	0.309	0.291
Uracil	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	4.90	0.000	4.89	0.000	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	5.44	0.174	5.21	0.281	0.214	0.197
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	6.29	0.046	6.42	0.109	0.100	0.096
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	6.32	0.000	6.20	0.000	0.000	0.000
	${}^1$A${}^{\prime }$ ($\pi \to {\pi }^{\ast }$)	6.84	0.152	6.84	0.269	0.232	0.213
Adenine	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.34	0.001	5.25	0.001	0.001	0.001
	${}^1$A${}^{″}$ ($n\to {\pi }^{\ast }$)	5.96	0.002	5.86	0.002	0.002	0.002

Note: CIS here means the approximate STEOM density discussed in the main text. For almost all calculations, the number of roots $N_R$ is 30 and the number of guess states is $N_G=30$, while the superscript a refers to a calculation with $N_R=25$, $N_G=50$.

Figure 1. Comparison of oscillator strength values computed at the CIS, STEOM-CCSD and CC3 levels.

Table 4. Linear regression results (y = ax + b) corresponding to Figure 1, Table 2 and Table 3.

x	y	a	b	$R^2$
f(CC3)	f(CIS)	1.5851	0.0162	0.9653
f(CC3)	f(STEOM)	1.1359	0.0041	0.9903
f(CIS)	f(STEOM)	0.7006	−0.0026	0.9805
f(CC3)	f(STEOM-DD)	1.0826	0.0025	0.9908

Figure 2. The spectra of two selected species from Thiel's test set.