Excited-State Polarizabilities: A Combined Density Functional Theory and Information-Theoretic Approach Study

Abstract

Accurate and efficient determination of excited-state polarizabilities (α) is an open problem both experimentally and computationally. Following our previous work, (Phys. Chem. Chem. Phys. 2023, 25, 2131−2141), in which we employed simple ground-state (S₀) density-related functions from the information-theoretic approach (ITA) to accurately and efficiently evaluate the macromolecular polarizabilities, in this work we aimed to predict the lowest excited-state (S₁) polarizabilities. The philosophy is to use density-based functions to depict excited-state polarizabilities. As a proof-of-principle application, employing 2-(2′-hydroxyphenyl)benzimidazole (HBI), its substituents, and some other commonly used ESIPT (excited-state intramolecular proton transfer) fluorophores as model systems, we verified that either with S₀ or S₁ densities as an input, ITA quantities can be strongly correlated with the excited-state polarizabilities. When transition densities are considered, both S₀ and S₁ polarizabilities are in good relationships with some ITA quantities. The transferability of the linear regression model is further verified for a series of molecules with little or no similarity to those molecules in the training set. Furthermore, the excitation energies can be predicted based on multivariant linear regression equations of ITA quantities. This study also found that the nature of both the ground-state and excited-state polarizabilities of these species are due to the spatial delocalization of the electron density.

1. Introduction

Molecular polarizability, especially static electric dipole polarizability (α), is a fundamental physicochemical property. It reflects the change of a molecule’s dipole moment in a linear-response manner resulting from an external electric field perturbation [1]. It is a fundamental dielectric property important in many disciplines, especially in materials science [2,3]. Experimental determination of excited-state electrostatic properties is mainly determined using Stark spectroscopy, the electronic absorption/emission [4,5] method, and the flash photolysis time-resolved microwave-conductivity (FP-TMRC) [6,7] technique.

In classical physics, polarizability can be approximately obtained in terms of the volume of a system [8,9]. For example, many strong correlations have been observed for both atoms and molecules [10,11,12,13,14,15,16,17,18]. Tkatchenko and Scheffler (TS) [19] proposed use of atomic volumes and atomic polarizabilities to predict the ground-state polarizabilities for small molecules. Recent progress can be found in [20]. However, performance for excited-state systems has not been reported.

In quantum mechanics, polarizability can be obtained by iteratively solving the coupled-perturbed Hartree–Fock (CPHF) equation [21,22] or its Kohn–Sham DFT (density functional theory) counterpart [23]. This protocol requires a sufficiently large basis set with polarization and diffuse functions and huge computational costs, but the computational barriers can be partly overcome by using some linear-scaling methods [24,25,26]. In addition, machine learning (ML)-based [27,28,29] methods and a regression-based [30] model have been applied to predict the S₀ polarizabilities. Polarizability can be related to the band gap of HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) in an inverse manner [31,32,33].

In the literature, only a few studies [34,35,36,37,38,39,40] have reported excited-state polarizabilities. This is likely because accurate predictions of excited-state geometries and molecular properties of large molecules are difficult, especially when there are perturbations such as external fields.

Following our previous work, used information-theoretic approach (ITA) quantities to predict the S₀ polarizabilities of both small and large molecules [41,42]. Here, we aim to predict the S₁ polarizabilities of 2-(2′-hydroxyphenyl)benzimidazole (HBI, 1) and its derivatives as shown in Scheme 1. For 1, it is well-documented [43] that the S₀ (Scheme 1a) intramolecular proton transfer (IPT) reaction takes place with difficulty, whereas the S₁ (Scheme 1b) or T₁ (triplet, not shown) intramolecular proton transfer (ESIPT) process can easily happen. Thus, in this work, only S₀ and S₁ were considered to reduce computational cost without compromising the results. We have found that with S₀ or S₁ electron densities as inputs, ITA quantities had good correlations with the excited-state polarizabilities. When transition densities were considered, both S₀ and S₁ polarizabilities had good relationships with ITA quantities. Furthermore, excitation energies could be predicted based on multiple linear regression equations of ITA quantities. For the first time, we applied the ITA quantities to predict excited-state polarizabilities. It is anticipated that this protocol can be readily applied to condensed-phase systems.

2. Results

Shown in

Table 1. Correlation coefficient (R²) between the isotropic molecular polarizability (α_iso, in Bohr³) and ITA quantities (in a.u.), molecular volume (Vol, in Bohr³/mol), and the isotropic quadrupole moment (Θ_iso, in a.u.) at S₀ for all the systems in Scheme 1 and Scheme 2.

Index	αiso	SS	IF	SGBP	rR2	rR3	IG	G1	G2	G3	Vol	Θiso
1	175.68	96.35	4343.89	746.75	112.64	117.78	1.38	−35.34	24.07	139.78	1715.75	−87.39
2	200.87	20.66	13,967.27	973.91	146.61	151.68	1.36	−34.93	23.70	141.74	2073.70	−103.29
3	237.32	142.90	5692.80	1016.86	154.03	161.91	2.10	−50.98	35.00	196.05	2441.40	−115.43
4	182.97	97.74	4793.46	801.56	120.76	126.07	1.45	−34.44	22.49	147.78	1855.73	−91.79
5	197.76	107.94	5045.63	855.45	129.06	135.01	1.60	−39.10	25.75	159.82	1886.74	−95.32
6	175.11	92.94	4917.98	801.81	120.62	125.71	1.37	−34.55	22.69	144.27	1765.67	−93.87
7	188.87	81.70	6524.29	855.22	128.61	133.70	1.37	−34.93	22.13	142.27	1959.26	−100.99
8	197.51	20.60	13,967.06	973.87	146.60	151.67	1.36	−34.95	23.67	141.84	1850.09	−101.48
9	194.07	102.37	4928.42	829.28	124.78	130.18	1.46	−36.68	23.12	151.22	1897.41	−106.85
10	189.36	106.46	4595.48	800.72	120.93	126.64	1.53	−39.13	25.97	150.98	1931.31	−93.18
11	194.81	107.96	5045.77	855.48	129.06	135.02	1.60	−39.13	23.71	159.81	1994.38	−95.26
12	191.68	81.72	6524.35	855.23	128.60	133.66	1.36	−34.98	22.67	142.29	1957.16	−105.11
13	200.29	20.66	13,967.37	973.93	146.60	151.67	1.36	−35.01	22.86	141.85	1980.91	−111.96
14	198.88	102.42	4928.71	829.32	124.77	130.16	1.45	−36.79	23.62	151.19	1696.49	−112.11
15	190.18	106.46	4595.43	800.71	120.93	126.64	1.53	−39.12	24.61	151.02	1842.17	−94.41
16	191.22	96.42	6318.70	965.73	144.80	150.24	1.46	−37.81	23.40	165.13	2064.71	−117.92
17	255.95	135.18	5826.22	1017.73	153.69	160.90	1.92	−52.71	36.65	192.16	2401.28	−125.79
18	246.58	135.11	5826.00	1017.67	153.68	160.90	1.92	−52.74	36.93	192.52	2494.60	−119.81
19	277.67	146.79	6528.00	1126.45	170.10	178.13	2.14	−56.48	38.53	212.18	2819.11	−137.35
20	295.24	153.19	7225.20	1222.09	184.32	192.76	2.25	−58.90	39.85	226.76	2761.18	−154.13
21	268.01	146.71	6527.76	1126.38	170.10	178.12	2.14	−56.47	39.11	212.53	2467.93	−128.04
22	283.05	153.12	7225.00	1222.03	184.32	192.76	2.25	−58.93	39.71	227.00	2913.80	−139.45
23	205.46	91.83	6775.95	909.20	136.89	142.52	1.51	−38.70	26.44	153.26	1963.41	−110.94
24	205.03	106.53	6570.28	1019.70	153.09	159.11	1.61	−41.55	26.57	176.20	2201.13	−123.83
25	327.20	173.93	7308.31	1288.65	194.73	204.01	2.46	−70.08	50.17	244.74	3066.91	−156.33
26	320.10	190.84	7330.19	1328.47	201.30	211.70	2.76	−74.61	52.80	255.30	3332.26	−160.67
27	175.19	82.72	6640.05	966.86	144.50	149.36	1.31	−32.94	20.03	158.50	3218.41	−108.98
28	173.51	96.75	4012.72	706.25	106.66	111.88	1.39	−37.76	26.11	130.76	1856.23	−84.26
29	169.68	91.91	4453.45	747.21	112.55	117.54	1.33	−36.02	24.66	137.43	1800.53	−87.68
30	186.83	80.69	5927.71	800.75	120.52	125.45	1.31	−36.59	25.86	133.70	2002.44	−95.13
31	163.90	90.16	4002.93	692.24	104.49	109.37	1.29	−36.26	25.07	129.50	1582.16	−82.37
32	176.33	103.65	4822.91	815.19	123.03	128.93	1.58	−41.53	28.16	148.35	1952.13	−95.39
33	243.77	128.34	5902.42	1004.58	151.72	158.98	1.94	−49.38	32.40	187.62	2418.28	−113.49
34	204.56	113.06	5648.71	937.42	141.25	147.59	1.70	−44.88	30.55	170.08	2071.58	−114.39
35	116.08	68.78	3451.98	571.00	85.89	89.56	0.99	−21.64	13.14	104.71	1404.36	−66.08
36	135.67	74.85	4147.84	666.78	100.16	104.28	1.15	−22.46	12.48	118.31	1405.54	−78.67
R2		0.560	0.068	0.890	0.897	0.908	0.922	0.936	0.912	0.938	0.705	0.893

are the correlation coefficients (R²) between the S₀ polarizabilities (α_iso) and ITA quantities, molecular volumes, and quadrupole moments, which were obtained at the CAM-B3LYP/6-311+G(d) level for all the molecular systems shown in Scheme 1 and Scheme 2. The linear regression equations can be found in Table S1. It is clear from Table 1 that the Ghosh−Berkowitz−Parr (GBP) entropy (S_GBP), 2nd and 3rd relative Rényi entropy (^rR₂ and ^rR₃), information gain (I_G), G₁, G₂, and G₃, and quadrupole moments (Θ_iso) have strong linear relationships with α_iso, with R² > 0.8. Of note, the G₃ data were shown to be strongly correlated with α_iso for various systems, among which were 30 planar or quasi-planar bases [41], and 20/40/8000 amino acids, dipeptides and tripeptides [41,42]. The Θ_iso values had a good relationship with α_iso, the theoretical rationale for which can be found in [42]. However, a solid and sound theoretical verification between G₃ and α_iso was lacking. Additionally, molecular volumes had only reasonably good correlation with α_iso, with R² = 0.705, indicating that it is not a good descriptor of α_iso. For Shannon entropy and Fisher information, there exist three exceptions for 2, 8, and 13. It was found that these three molecules are Br-containing, indicating that there may be some regions for heavy atoms where the electron density and/or its gradients are numerically inconsistent. Taking the three outliers away, much stronger correlations were found for Shannon entropy (better than Fisher information). Here, the nature of ground-state polarizabilities was due to the spatial delocalization of the electron density. Overall, the strong correlations indicate that our computational results are sound.

Table 2. Correlation coefficients (R²) between the isotropic molecular polarizability (α_iso, in Bohr³) and ITA quantities (in a.u.), molecular volume (Vol, in Bohr³/mol), and the isotropic quadrupole moment (Θ_iso, in a.u.) at S₁ for all the systems in Scheme 1 and Scheme 2.

Index	αiso	SS	IF	rR2	rR3	G2	G3	Vol	Θiso
1	166.08	96.94	4345.68	112.73	118.02	23.88	138.16	1797.33	−89.93
2	189.81	21.28	13,969.19	146.67	151.87	23.61	140.08	2016.04	−114.26
3	224.23	143.51	5694.61	154.12	162.16	34.15	194.24	2493.81	−126.71
4	174.81	98.36	4795.23	120.83	126.27	22.33	146.07	1709.27	−96.38
5	187.84	108.57	5047.40	129.14	135.22	25.53	158.13	1977.08	−101.96
6	166.88	93.55	4919.65	120.68	125.87	22.56	142.61	1732.69	−97.69
7	182.88	82.29	6525.95	128.66	133.83	23.33	140.53	1814.22	−105.58
8	188.20	21.19	13,968.88	146.65	151.80	22.97	140.28	1849.59	−110.07
9	182.03	103.04	4930.44	124.83	130.34	24.17	149.51	1973.71	−110.22
10	178.88	107.05	4597.05	121.02	126.88	26.58	149.31	1997.79	−96.99
11	182.81	108.54	5047.21	129.14	135.22	25.57	158.15	1950.36	−99.73
12	178.54	82.29	6526.00	128.69	133.90	23.14	140.43	1956.40	−103.35
13	186.41	21.24	13,968.99	146.69	151.92	23.40	139.90	1829.44	−105.81
14	185.73	102.96	4930.10	124.85	130.37	24.33	149.38	1967.61	−111.23
15	179.03	107.06	4597.29	121.01	126.87	26.78	149.25	1905.76	−94.93
16	178.64	96.95	6320.18	144.88	150.45	23.97	163.21	1970.62	−111.79
17	240.42	135.77	5827.95	153.76	161.09	36.81	190.49	2259.46	−117.96
18	253.40	135.66	5827.52	153.75	161.08	36.75	190.81	2360.15	−120.97
19	260.31	147.38	6529.77	170.17	178.31	39.03	210.40	2749.56	−124.40
20	280.48	153.76	7226.84	184.39	192.94	40.08	225.02	2771.27	−136.33
21	267.99	147.29	6529.43	170.17	178.33	38.50	210.79	2513.32	−128.47
22	327.99	153.50	7225.08	184.32	192.78	39.42	226.35	2727.21	−138.46
23	190.90	92.40	6777.37	136.99	142.77	25.46	151.82	1979.58	−110.61
24	191.30	107.05	6571.54	153.18	159.32	26.69	174.50	2133.33	−119.08
25	331.10	174.16	7307.99	194.72	204.02	48.71	244.76	3175.10	−168.67
26	309.58	191.50	7332.09	201.39	211.92	53.05	253.29	3230.21	−164.60
27	190.21	83.28	6640.35	144.47	149.29	19.82	157.57	1816.70	−115.25
28	218.02	97.17	4012.63	106.64	111.81	26.47	129.86	1682.95	−83.82
29	212.42	92.41	4453.25	112.53	117.45	23.89	136.29	1811.12	−88.93
30	173.32	81.32	5929.80	120.61	125.71	25.00	132.03	1756.26	−92.75
31	170.34	90.61	4002.95	104.46	109.29	25.03	128.51	1684.98	−83.54
32	178.67	104.22	4824.82	123.06	129.03	27.87	146.61	1942.93	−94.35
33	301.59	128.75	5902.73	151.65	158.79	32.41	186.62	2379.78	−111.36
34	202.06	113.48	5649.74	141.23	147.54	29.60	169.10	2107.86	−113.38
35	148.12	69.41	3452.63	85.88	89.58	13.17	103.81	1270.25	−67.10
36	129.10	74.99	4148.18	100.12	104.16	12.54	117.98	1486.28	−83.28
R2		0.552	0.025	0.729	0.740	0.821	0.824	0.818	0.672

shows S₁ polarizabilities, S₁ ITA quantities, including Shannon entropy (S_S), Fisher information (I_F), 2nd and 3rd relative Rényi entropy (^rR₂ and ^rR₃), G₂ and G₃, molecular volumes (Vol), and quadrupole moments (Θ_iso), obtained at the TD-CAM-B3LYP/6-311+G(d) level. Table 2 also shows the correlation coefficients (R²) between the S₁ polarizabilities and other quantities at S₁. The linear regression equations can be found in Table S2. Note that some ITA quantities, well-defined at S₀, are numerically inconsistent (for example, negative occupancy) at S_1, and are thus missing. One can see that G₂, G₃, and Vol have good correlations with α_iso at S₁, with R² > 0.8. It is intriguing that at S₁ and G₃ still have good correlations with α_iso. This is the first time such a phenomenon has been observed. However, admittedly, the theoretical foundation lags behind the numerical evidence found in this work. Moreover, we have found that for Vol, the R² value was much larger at S₁ (0.818) than that at S₀ (0.705). One possible reason is that the excited-state relaxation expands the volume and polarizability space. Finally, in Columns 3 and 4 of S_S and I_F, 2, 8, and 13 seem to have abnormal values compared with the others. They are all Br-containing, indicating that there may be some regions for heavy atoms where density gradients are numerically inconsistent at S₁. Overall, we showed that excited-state densities and molecular polarizabilities α_iso are mutually entangled.

After showing that ITA quantities can be correlated with α_iso either at S₀ or S₁, it is important to ask if one can use ITA quantities at S₀ to predict α_iso at S₁. The answer is definitely yes.

Table 3. Correlation coefficients (R²) between the α_iso@S₁ (in Bohr³), and α_iso@S₀ (in Bohr³), ITA quantities@S₀ (in a.u.), molecular volume Vol@S₀ (in Bohr³/mol), and the isotropic quadrupole moment Θ_iso@S₀ (in a.u.) for all the systems in Scheme 1 and Scheme 2.

	αiso@S0	SS	IF	SGBP	rR2	rR3
R2	0.845	0.553	0.025	0.723	0.730	0.742
	IG	G1	G2	G3	Vol	Θiso
R2	0.819	0.844	0.827	0.820	0.674	0.689

shows the correlation coefficients (R²) between the α_iso values at S₁ and ITA quantities, molecular volumes, and quadrupole moments at S₀ introduced in Table 1. More details can be found in Table S3. The linear regression equations can be found in Table S4. One can see that the α_iso@S₀, I_G, G₁, G₂, and G₃ have good relationships with α_iso@S₁, with R² > 0.8. Moving forward, we ask if one can use the transition density (matrix) as an input for ITA quantities to correlate with α_iso either at S₀ or S₁. The answer again is yes.

Table 4. Correlation coefficients (R²) between the ground-state (α_iso@S₀, in Bohr³) or the first excited-state (α_iso@S₁, in Bohr³) molecular polarizability, and molecular volume (Vol, in Bohr³/mol), the isotropic quadrupole moment (Θ_iso, in a.u.), and ITA quantities (in a.u.) based on the transition density matrix as an input for all the systems in Scheme 1 and Scheme 2.

R2	SS	IF	SGBP	rR2	rR3	IG	G1	G2	G3	Vol	Θiso
αiso@S0	0.560	0.068	0.890	0.897	0.909	0.925	0.938	0.912	0.940	0.923	0.887
αiso@S1	0.553	0.025	0.723	0.730	0.743	0.825	0.843	0.824	0.821	0.818	0.667

shows strong correlations between α_iso either at S₀ or S₁ and ITA quantities with the transition density matrix as an input. The linear regression equations can be found in Table S5. Except for Shannon entropy (S_S) and Fisher information (I_F), the other ITA quantities have reasonably good or strong correlations with α_iso, either at S₀ or S₁, with R² ranging from 0.72 to 0.94. The implication of this is that electron-density-based quantities can be used to predict excited-state properties, such as molecular polarizabilities.

To further showcase the transferability of the linear regression models, we compared both the S₀ and S₁ polarizabilities for all systems in Scheme 3, as shown in

Table 5. Comparison of molecular polarizabilities (α_iso, in Bohr³/mol) at S₀ predicted with the linear regression equations in Table S1 with conventional data as a reference for all the systems in Scheme 3.

Index	αiso	SGBP	rR2	rR3	IG	G1	G2	G3	Vol	Θiso
37	105.36	109.32	109.77	110.45	143.92	146.72	154.26	120.04	137.25	115.53
38	155.68	151.14	151.26	151.64	169.53	185.83	187.63	158.87	163.68	147.97
39	216.91	196.27	196.48	197.01	207.58	229.76	236.35	203.48	205.75	184.62
40	101.70	113.02	113.08	113.47	140.59	136.98	143.60	120.15	136.21	100.86
41	166.55	150.90	151.41	152.11	174.87	185.60	194.70	166.02	168.11	144.96
42	111.10	102.29	102.60	102.89	130.30	116.57	125.36	97.52	125.61	106.82
43	196.65	193.10	192.90	193.28	200.66	216.50	224.05	196.26	199.29	253.81
MUE a (%)		−2.38	−2.22	−1.88	14.82	17.96	22.97	2.17	11.36	0.18
MSE b (%)		6.64	6.61	6.56	16.05	17.96	22.97	7.58	12.83	10.89

^a MUE: mean unsigned error. ^b MSE: mean signed error.

and

Table 6. Comparison of molecular polarizabilities (α_iso, in Bohr³/mol) at S₁ predicted with the linear regression equations in Table S2 with conventional data as a reference for all the systems in Scheme 3.

Index	αiso	rR2	rR3	G2	G3	Vol	Θiso
37	118.81	113.52	113.85	152.35	120.93	116.81	114.30
38	192.21	153.15	153.33	189.82	159.90	158.02	132.22
39	290.76	196.25	196.61	235.54	204.05	179.41	170.00
40	117.99	116.65	116.65	139.41	121.02	119.35	100.33
41	148.92	153.43	154.11	194.56	165.60	167.43	159.88
42	111.22	106.39	105.78	122.68	97.10	128.12	107.88
43	205.36	192.73	192.65	215.87	196.92	197.54	227.59
MUE a (%)		−9.41	−9.36	10.32	−6.84	−4.69	−10.90
MSE b (%)		10.27	10.35	16.10	11.28	12.91	16.10

^a MUE: mean unsigned error. ^b MSE: mean signed error.

, respectively. Since Shannon entropy and Fisher information have exceptions for the linear regressions, they were not used for predicting the polarizabilities of all systems in Scheme 3. From Table 5, one can see that, S_GBP, ^rR₂, ^rR₃, and G₃ perform well in reproducing the conventional S₀ polarizabilities, as evidenced by the mean unsigned errors (MUEs) and mean signed errors (MSEs). The other five quantities show reasonably good predictions. From Table 6, one can observe that the predicted S₁ polarizabilities are in reasonably good agreement with the conventional data. All the quantities show comparable performance. Taken together, we have shown that the transferability of the linear regression models is convincing, as the systems in Scheme 3 show little or low similarity to those in Scheme 1 and Scheme 2.

Next, we compared the α_iso data (either at S₀ or S₁) predicted by the TS formulas with conventional results as a reference, as shown in

Table 7. Comparison of molecular polarizabilities (α_iso, in Bohr³/mol) at S₀/S₁ predicted by the original and the new TS formula with conventional data as a reference for all the systems in Scheme 1 and Scheme 2.

	Ground-State (S0)			Excited-State (S1)
	Becke	Hirshfeld	Avg.	Becke	Hirshfeld	Avg.
Based on the original TS formula
MUE (%) a	−26.44	7.92	−9.26	−24.94	8.93	−8.01
MSE (%) b	28.87	9.03	14.40	28.59	12.76	15.53
Based on the new TS formula
MUE (%) a	−29.02	7.22	−10.90	−27.31	8.31	−9.50
MSE (%) b	31.45	8.33	16.04	30.95	12.61	17.02

^a MUE: mean unsigned error. ^b MSE: mean signed error.

. The predicted α_iso results are shown in Tables S6 and S7. Employing the original Tkatchenko–Scheffler (TS) formula [19] with Becke [44] or Hirshfeld [45] partitions, the α_iso data (either at S₀/S₁) were either strongly underestimated or overestimated, with MUE(%) up to −26.44/−24.94 and 7.92/8.93, respectively. It was found that a mean value could reduce the MSE(%) to 14.40/15.53. Moreover, with the new TS formula [20], the results were not improved but worsened, as shown in Table 7. Taken together, we found that the TS formulas have room to improve in predicting S₁ polarizabilities.

Finally, we found that excitation energies can be predicted in addition to multivariant linear regression equations of ITA quantities. For example, one can use the S₀ density matrix as an input for ITA quantities to correlate with excitation energies. Similarly, if the transition density matrix is used for ITA quantities, the excitation energies can also be predicted. Based on the two regression equations constructed from all the systems in Scheme 1 and Scheme 2, we collected predicted excitation energies, and those from conventional calculations, for all the systems in Scheme 3, as shown in

Table 8. Comparison of the excitation energies (in eV), as predicted with the multivariant linear regression equations for all the systems in Scheme 3, with conventional data as a reference.

Index	Excitation Energies
	Conv.	Mod_1 c	Mod_2 d
37	4.28	2.24	2.25
38	3.52	2.67	2.55
39	3.53	2.64	2.51
40	4.55	3.00	3.45
41	4.12	3.13	3.33
42	3.71	3.30	NA e
43	4.28	5.14	5.65
MUE a		−0.84	−0.76
MSE b		1.08	1.21

^a MUE: mean unsigned error. ^b MSE: mean signed error. ^c Based on the S₀ ITA quantities with the S₀ molecular wavefunction as an input for all the systems in Scheme 1 and Scheme 2. The multivariant linear regression equation is: λ_fit = 0.306 × S_S + 0.00304 × I_F + 0.661 × S_GBP − 4.210 × ^rR₂ − 0.363 × ^rR₃ + 0.102 × G₁ − 0.0499 × G₂ − 0.114 × G₃ + 3.490 × I_G. ^d Base on the S₁ ITA quantities with the transition density matrix as an input for all the systems in Scheme 1 and Scheme 2. The multivariant linear regression equation is: λ_fit = 0.290 × S_S + 0.00317 × I_F + 1.037 × S_GBP − 14.075 × ^rR₂ + 6.914 × ^rR₃ + 0.104 × G₁ − 0.0494 × G₂ − 0.109 × G₃ − 16.163 × I_G. ^e NA means not available. S_GBP, G₁, and I_G are not numerically obtained.

. The MUEs and MSEs were −0.84/1.08 eV and −0.76/1.21 eV, respectively, for the excitation energies. This indicates that the inaccuracy of this protocol is comparable to that of underlying approximations of DFT [46,47].

3. Discussion

To accurately and efficiently predict excited-state polarizabilities is an ongoing issue. Solving standard CPHF/CPKS equations is computationally intensive, and the computational costs can be excessive for macromolecular systems. Other algorithms and models available in the literature are normally concerned with ground-state polarizabilities. Within this context, we applied density-based ITA quantities to correlate with α_iso at S₀/S₁. This was inspired by our previous work on predicting the ground-state polarizabilities for small and macromolecular systems. Our tentative results show that the protocol is a promising theoretical tool. More systems along this line need to be considered to make this protocol more robust and applicable. We have to point out that when the system under study increases in size, the molecular wavefunctions (thus electron density) become difficult to assess and sometimes computationally unfeasible. Under these circumstances, we have to resort to linear-scaling electronic structure methods, such as GEBF (generalized energy-based fragmentation method) [48,49,50,51], where only small subsystems of a few atoms or groups are treated.

Next, we adopted the TS method, as mentioned previously. We had found previously that, based upon the Hirshfeld or Becke partition scheme, the original TS formula had unsatisfactory performance either by overestimating or underestimating the S₁ polarizabilities. An apparent reduction of the deviations can be obtained by averaging the two sets of results. The reason behind this is unclear at the moment. From the original formula,

$${\alpha }_{\mathrm{m}\mathrm{o}\mathrm{l}}^{\mathrm{T}\mathrm{S}-\mathrm{o}\mathrm{l}\mathrm{d}}=\sum _{\mathrm{A}}{\alpha }_{\mathrm{A}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\sum _{\mathrm{A}}{\alpha }_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\left(\frac{V_{\mathrm{A}}^{\mathrm{e}\mathrm{f}\mathrm{f}}}{V_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}\right)$$

one can argue that the weights $\left(\frac{V_{\mathrm{A}}^{\mathrm{e}\mathrm{f}\mathrm{f}}}{V_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}\right)$ may be the root cause of its poor performance, mainly because the atomic polarizabilities ${\alpha }_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ are experimentally determined and computationally verified, as summarized in ref [52]. In the same way, a revised TS formula,

$${\alpha }_{\mathrm{m}\mathrm{o}\mathrm{l}}^{\mathrm{T}\mathrm{S}-\mathrm{n}\mathrm{e}\mathrm{w}}=\sum _{\mathrm{A}}{\alpha }_{\mathrm{A}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\sum _{\mathrm{A}}{\alpha }_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}{\left(\frac{V_{\mathrm{A}}^{\mathrm{e}\mathrm{f}\mathrm{f}}}{V_{\mathrm{A}}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}\right)}^{4/3}$$

improved performance of predicting S₀ polarizabilities. However, its predicting power was shown to be far from satisfactory for macromolecules. In this work, we further found that both the original and new TS formulas fail to give a satisfactory description of the S₁ polarizability. This indicates that the two formulas are oversimplified and may be system-dependent. Overall, the volume-based TS equations are inferior to the density-based ITA regression equations.

4. Materials and Methods

4.1. Information-Theoretic Approach Quantities

Shannon entropy S_S [53] and Fisher information I_F [54] are two cornerstone quantities in information theory. They are defined as Equations (1) and (2), respectively:

$$S_{\mathrm{S}}=-\int \rho \left(\mathbf{r}\right)\mathrm{l}\mathrm{n}\rho \left(\mathbf{r}\right)d\mathbf{r}$$

(1)

$$I_{\mathrm{F}}=\int \frac{{|\nabla \rho (\mathbf{r}\left)\right|}^2}{\rho \left(\mathbf{r}\right)}d\mathbf{r}$$

(2)

where $\rho \left(\mathbf{r}\right)$ is the electron density and $\nabla \rho \left(\mathbf{r}\right)$ is the density gradient. The physical picture of $S_{\mathrm{S}}$ and $I_{\mathrm{F}}$ is clear; the former measures the spatial delocalization of the electron density and the latter gauges the sharpness or localization of the same.

Apart from total density, components such as kinetic-energy density can be used to define an ITA quantity. With both electron density and the kinetic energy density, Ghosh, Berkowitz, and Parr developed a formula for entropy (S_GBP) [55]:

$$S_{\mathrm{G}\mathrm{B}\mathrm{P}}=-\int \frac{3}{2}k\rho \left(\mathbf{r}\right)\left[c+\mathrm{l}\mathrm{n}\frac{t(\mathbf{r};\rho )}{t_{\mathrm{T}\mathrm{F}}(\mathbf{r};\rho )}\right]d\mathbf{r}$$

(3)

where t(r; ρ) and t_TF(r; ρ) represent the non-interacting and Thomas–Fermi (TF) kinetic energy density, respectively. Here k, c, and c_K are three constants (k, the Boltzmann constant, c = (5/3) + ln(4πc_K/3), and c_K = (3/10)(3π²)^2/3). Full integration of kinetic energy density t(r; ρ) leads to the total kinetic energy T_S via

$$\int t\left(\mathbf{r};\rho \right)d\mathbf{r}=T_{\mathrm{S}}$$

(4)

while t(r; ρ) can be obtained from the canonical orbital densities,

$$t\left(\mathbf{r};\rho \right)=\sum _i\frac{1}{8}\frac{\nabla {\rho }_i·\nabla {\rho }_i}{{\rho }_i}-\frac{1}{8}{\nabla }^2\rho$$

(5)

and t_TF(r; ρ) is simply cast in terms of $\rho \left(\mathbf{r}\right)$,

$$t_{\mathrm{T}\mathrm{F}}(\mathbf{r};\rho )=c_{\mathrm{K}}{\rho }^{5/3}\left(\mathbf{r}\right)$$

(6)

Of note, the kinetic-energy density can differ in its form, and can be used in different contexts [56,57,58,59,60,61,62,63]. However, S_GBP satisfies the maximum-entropy requirement from a mathematical viewpoint [55].

Moving forward, some ITA quantities were introduced for chemical reactions. As new reactivity descriptors in conceptual density functional theory (CDFT) [64,65,66,67], one example is relative Rényi entropy [68] of order n

$$R_n^r=\frac{1}{n-1}\mathrm{l}\mathrm{n}\left[\int \frac{{\rho }^n\left(\mathbf{r}\right)}{{\rho }_0^{n-1}\left(\mathbf{r}\right)}d\mathbf{r}\right]$$

(7)

Information gain [69] (also called Kullback−Leibler divergence or relative Shannon entropy) I_G is expressed as follows,

$$I_{\mathrm{G}}=\int \rho \left(\mathbf{r}\right)\mathrm{l}\mathrm{n}\frac{\rho \left(\mathbf{r}\right)}{{\rho }_0\left(\mathbf{r}\right)}d\mathbf{r}$$

(8)

In Equations (7) and (8), ρ₀(r) and ρ(r) satisfy the same normalization condition, and ρ₀(r) denotes the reference-state density.

Recently, in [70] one of the present authors proposed three functions G₁, G₂, and G₃, at both atomic and molecular levels. These are defined as below:

$$G_3=\sum _{\mathrm{A}}\int {\rho }_{\mathrm{A}}\left(\mathbf{r}\right)[\nabla \mathrm{l}\mathrm{n}\frac{{\rho }_{\mathrm{A}}\left(\mathbf{r}\right)}{{\rho }_{\mathrm{A}}^0\left(\mathbf{r}\right)}{]}^{2}d\mathbf{r}$$

(9)

$$G_1=\sum _{\mathrm{A}}\int {{\nabla }^2\rho }_{\mathrm{A}}\left(\mathbf{r}\right)\frac{{\rho }_{\mathrm{A}}\left(\mathbf{r}\right)}{{\rho }_{\mathrm{A}}^0\left(\mathbf{r}\right)}d\mathbf{r}$$

(10)

$$G_2=\sum _{\mathrm{A}}\int {\rho }_{\mathrm{A}}\left(\mathbf{r}\right)\left[\frac{{{\nabla }^2\rho }_{\mathrm{A}}\left(\mathbf{r}\right)}{{\rho }_{\mathrm{A}}\left(\mathbf{r}\right)}-\frac{{\nabla }^2{\rho }_{\mathrm{A}}^0\left(\mathbf{r}\right)}{{\rho }_{\mathrm{A}}^0\left(\mathbf{r}\right)}\right]d\mathbf{r}$$

(11)

Equations (9)–(11) have been theoretically derived, numerically verified, and used in many applications, as in refs [41,42,70,71]. As one of our major achievements over the past decade, we aimed to combine density functional theory and information theory in a seamless manner, since the two theories can have electron density as an input. Our recent progress along this line can be found in two reviews [72,73]. As another prominent example, we have applied the ITA to assess homochirality [74,75].

Finally, the Hirshfeld’s stockholder approach [45,76,77,78,79] is often introduced to partition atoms in a molecule in the literature, as defined in Equation (12),

$${\rho }_{\mathrm{A}}\left(\mathbf{r}\right)={\omega }_{\mathrm{A}}\left(\mathbf{r}\right)\rho \left(\mathbf{r}\right)=\frac{{\rho }_{\mathrm{A}}^0\left(\mathbf{r}\right)(\mathbf{r}-{\mathbf{R}}_{\mathrm{A}})}{\sum _{\mathrm{B}}{\rho }_{\mathrm{B}}^0\left(\mathbf{r}-{\mathbf{R}}_{\mathrm{A}}\right)}\rho \left(\mathbf{r}\right)$$

(12)

Here, ${\rho }_{\mathrm{A}}\left(\mathbf{r}\right)$ is the atomic Hirshfeld density, ${\omega }_{\mathrm{A}}\left(\mathbf{r}\right)$ is a sharing function, and ${\rho }_{\mathrm{B}}^0\left(\mathbf{r}-{\mathbf{R}}_{\mathrm{A}}\right)$ is the atomic density of B centered at R_A. The sum over all the free atom densities, typically spherically averaged S₀ atomic densities, is usually termed the promolecular density. The Stockholder approach is used in the context of ITA because it is also based on information-theoretic arguments. Alternative partitioning schemes include Becke’s fuzzy atom approach [44] and Bader’s zero-flux atoms-in-molecules (AIM) method [80].

4.2. Computational Details

All density functional theory (DFT) calculations were performed with the Gaussian 16 [81] package. Default options included ultrafine integration grids and tight self-consistent field convergence, which were adopted to eliminate numerical noises. The ground- and excited-state structural relaxation was carried out at the CAM-B3LYP/6-311+G(d) [82,83] and TD-CAM-B3LYP/6-311+G(d) [82,83,84,85,86] level, respectively, for 36 molecular systems, as shown in Scheme 1, Scheme 2 and Scheme 3. Optimized atomic Cartesian coordinates are supplied in the Supplementary Materials. Harmonic vibrational frequency calculations were executed at the same level, and no imaginary frequencies were observed by direct visual inspection. Isotropic polarizabilities (α_iso = (α_xx + α_yy + α_zz)/3) and isotropic quadrupole moments (Θ_iso = (Θ_xx + Θ_yy + Θ_zz)/3), molecular volumes (at 0.001 e/Bohr³ contour surface of electronic density), and molecular wavefunctions, were obtained at the CAM-B3LYP/6-311+G(d) level. The Multiwfn 3.8 [87] program was utilized to calculate all ITA quantities at S₀ and S₁ using the molecular wavefunction file as an input. Of note, except for the S₀ state, the keyword “density=current” in Gaussian 16 was used to dump the density matrix in a wavefunction file. The stockholder Hirshfeld partition scheme of atoms in molecules was employed when atomic contributions were concerned. The reference-state density was the neutral atom calculated at the same level of theory as molecules.

5. Conclusions

In this work, we extended the information-theoretic approach (ITA) to treat excited-state polarizabilities. With the ground-state, the first excited-state, or the transition density matrix as an input for ITA quantities, strong linear correlations were observed between molecular polarizabilities and ITA quantities. Based upon these regression equations for molecules in the training set, one can further predict the molecular polarizabilities for other systems in the test set. We verified that good accuracy and transferability of the linear regression models can be obtained. In addition, we found that the nature of both ground-state and excited-state polarizabilities is due to the electron delocalization of the electron density. Finally, we revealed that excitation energies can be predicted on top of a multivariant linear regression equation of ITA quantities. We mention in passing that the ITA quantities were applied to ground-state macromolecules in our previous work and excited-state systems in this work. More work along this line is underway for condensed-phase systems, and the results will be presented elsewhere.