Dynamical Jahn-Teller effect in the first excited C$_{60}^-$

TL;DR

This study theoretically investigates the dynamical Jahn-Teller effect in the first excited state of C60 anions, revealing stronger stabilization compared to the ground state through derived vibronic coupling parameters and Hamiltonian diagonalization.

Contribution

It provides a detailed theoretical analysis of the dynamical Jahn-Teller effect in excited C60 anions, including derivation of vibronic coupling parameters and energy stabilization insights.

Findings

Dynamical Jahn-Teller stabilization energy is stronger in the first excited state.

Orbital vibronic coupling parameters were derived from Kohn-Sham orbitals.

Exact diagonalization of the Hamiltonian was used to analyze vibronic states.

Abstract

The Jahn-Teller effect of C$_{60}$ anions in the first electronically excited states was theoretically investigated. The orbital vibronic coupling parameters for the $t_{1g}$ next lowest unoccupied molecular orbitals were derived from the Kohn-Sham orbital levels with hybrid B3LYP functional by using the frozen phonon approach. With the use of these coupling parameters, the vibronic states of the first excited C$_{60}^-$ were derived by exactly diagonalizing the dynamical Jahn-Teller Hamiltonian. The dynamical Jahn-Teller stabilization energy of the first excited C$_{60}^-$ is stronger than that of the ground electronic states.

Tables

Table 1. Table 1: The frequencies ω Γ subscript 𝜔 Γ \omega_{\Gamma} (cm -1 ), vibronic coupling parameters V Γ subscript 𝑉 Γ V_{\Gamma} (10 -4 a.u.), dimensionless vibronic coupling parameters g Γ = V Γ / ℏ ​ ω Γ 3 subscript 𝑔 Γ subscript 𝑉 Γ Planck-constant-over-2-pi superscript subscript 𝜔 Γ 3 g_{\Gamma}=V_{\Gamma}/\sqrt{\hslash\omega_{\Gamma}^{3}} , and stabilization energies E Γ subscript 𝐸 Γ E_{\Gamma} (meV) for the a g subscript 𝑎 𝑔 a_{g} and h g subscript ℎ 𝑔 h_{g} modes. The data for LUMO are taken from Ref. Liu  et al. ( 2018 ) and the frequencies are taken from Ref. Bethune  et al. ( 1991 ) .

$J$			NLUMO			LUMO
$\Gamma$	$\mu$	${\omega }_{\Gamma }$	$V_{\Gamma }$	$g_{\Gamma }$	$E_{\Gamma }$	$V_{\Gamma }$	$g_{\Gamma }$	$E_{\Gamma }$
$a_g$	1	496	$-0.449$	$-0.418$	5.38	$-0.264$	$-0.245$	1.849
	2	1470	$-2.480$	$-0.452$	18.66	$-2.380$	$-0.422$	16.543
$h_g$	1	273	$-0.406$	$-0.926$	14.50	0.192	0.455	3.415
	2	437	$-0.476$	$-0.536$	7.78	0.450	0.503	6.886
	3	710	$-1.061$	$-0.577$	14.64	0.754	0.396	7.069
	4	774	$-0.594$	$-0.284$	3.86	0.554	0.259	3.256
	5	1099	$-0.498$	$-0.141$	1.35	0.766	0.209	3.038
	6	1250	$-1.664$	$-0.387$	11.61	0.578	0.132	1.360
	7	1428	0.125	0.024	0.05	2.099	0.394	13.867
	8	1575	$-2.113$	$-0.348$	11.79	2.043	0.326	10.592

Table 2. Table 2: The vibronic energy levels with respect to the first excited states and the ground states of C − 60 superscript subscript absent 60 {}_{60}^{-} (meV). The data for are taken Ref. Liu  et al. ( 2018 ) , and the number in the parentheses indicate J 𝐽 J .

$J$		NLUMO	LUMO
1	1	$-113.815$	$-96.469$
	2	$-74.589$	$-60.753$
	3	$-57.370$	$-38.126$
	4	$-47.520$	$-29.757$
	5	$-40.736$	$-26.841$
	6	$-31.168$	$-11.395$
	7	$-25.493$	$-8.099$
	8	-	$-5.411$
	9	-	$-4.123$
2	1	$-78.132$	$-61.918$
	2	$-58.955$	$-40.873$
	3	$-46.127$	$-29.004$
	4	$-31.122$	$-12.071$
	5	$-26.748$	$-8.264$
	6	-	$-6.155$
	7	-	$-4.265$
3	1	$-87.854$	$-65.135$
	2	$-63.252$	$-46.786$
	3	$-55.282$	$-31.703$
	4	$-40.633$	$-25.806$
	5	$-33.815$	$-14.620$
	6	$-32.593$	$-12.575$
	7	$-25.581$	$-8.417$
	8	-	$-3.843$
	9	-	$-1.607$
4	1	$-46.772$	$-28.549$
	2	$-34.314$	$-14.108$
	3	$-27.326$	$-7.724$
5	1	$-60.505$	$-33.554$
	2	$-36.593$	$-14.985$
	3	$-27.281$	$-$

Table 3. Table 3: Contributions to the ground vibronic energy (E total ) of NLUMO and LUMO of C − 60 superscript subscript absent 60 {}_{60}^{-} . E static , and E dynamic represent the static JT and dynamic JT stabilization energies. Ratio refers the ratio between E static and E dynamic (E dynamic /E static ).

Orbital	Etotal	Estatic	Edynamic	Ratio
NLUMO	$-113.8$	$-65.6$	$-48.2$	0.74
LUMOLiu, Iwahara, and Chibotaru (2018)	$-96.46$	$-50.3$	$-46.2$	0.92

Equations

$$[t_{1g}\otimes t_{1g}]=a_g\oplus h_g.$$

$$H$$

$$H_a$$

$$H_h$$

$$q_{\theta }=\sqrt{\frac{3}{8}}{Q}_{\theta }+\sqrt{\frac{5}{8}}{Q}_{ϵ},q_{ϵ}=\sqrt{\frac{3}{8}}{Q}_{\theta }-\sqrt{\frac{5}{8}}{Q}_{ϵ}.$$

$$U_{\text{min}}$$

$$q_{a,0}=-\frac{V_a}{{\omega }_a^2},\left|{\mathbf{q}}_{h,0}\right|=\frac{V_h}{{\omega }_h^2},$$

$$[{\hat{H}}_h,{\hat{\mathbf{J}}}^2]=[{\hat{H}}_h,{\hat{J}}_z]=0.$$

$${\hat{H}}_h|\alpha J{M}_J⟩$$

$$|\alpha J{M}_J⟩$$

$$0\le n_{h(\mu )\gamma },\sum _{\mu \gamma }{n}_{h(\mu )\gamma }\le 7,$$

$$v_a={\frac{\partial {ϵ}_{t_{1g}z}}{\partial q_a}|}_{\mathbf{q}=0},v_h=-{\frac{\partial {ϵ}_{t_{1g}z}}{\partial q_{h\theta }}|}_{\mathbf{q}=0},$$

$$V_{\Gamma }=v_{\Gamma },$$

Full text

I Introduction

Highly symmetric C60 Kroto et al. (1985) exhibits complex Jahn-Teller (JT) dynamics characterized by orbital-vibration entanglement in various charged and excited states Chancey and O’Brien (1997); Bersuker (2006); Dunn, Alqannas, and Lakin (2015). Negatively charged C60 has been one of the most investigated because it forms various molecular crystals Gunnarsson (2004); Capone et al. (2009); Alloul, H. (2012); Kamarás and Klupp (2014); Takabayashi and Prassides (2016); Nomura et al. (2016); Otsuka et al. (2018). Since the molecular nature strongly remains in these materials, for the thorough understanding of the JT effect of C60 anions is crucial. Though JT effect, including dynamic JT one, of C60 anions has been intensively investigated Auerbach, Manini, and Tosatti (1994); Manini, Tosatti, and Auerbach (1994); Wang, Bishop, and Yu (1994); Dunn and Bates (1995); Gunnarsson et al. (1995); O’Brien (1996); Tosatti, Manini, and Gunnarsson (1996); Wang et al. (1997); Manini and Tosatti (1998); Sookhun, Dunn, and Bates (2003); Dunn and Li (2005); Tomita et al. (2005); Hands et al. (2008); Frederiksen et al. (2008); Iwahara et al. (2010); Dunn, Lakin, and Hands (2012); Klupp et al. (2012); Støchkel and Andersen (2013); Ponzellini (2014); Kundu et al. (2015); Iwahara (2018); Liu, Iwahara, and Chibotaru (2018); Liu et al. (2018); Matsuda et al. (2018), it is only last years that the actual situation in the ground electronic states of C${}_{60}^{n-}$ molecule $(n=1-5)$ has been established with accurate coupling parameters Liu, Iwahara, and Chibotaru (2018); Liu et al. (2018).

The JT effect is also considered to be important in the excited states of C60 anions. For example, the JT effect in the first excited C60 anion where the $t_{1g}$ next lowest unoccupied molecular orbitals (NLUMOs) is populated is of fundamental importance to interpret absorption spectra of isolated C${}_{60}^{-}$ Kato et al. (1991); Kato, Kodama, and Shida (1993); Kodama et al. (1994); Kondo, Momose, and Shida (1995); Kwon et al. (2001); Kwon, Yoo, and Jang (2002); Tomita et al. (2005); Støchkel and Andersen (2013); Watariguchi et al. (2016), electron transfer process of fullerene Fujitsuka, Luo, and Ito (1999); Fujitsuka, Ohsaka, and Majima (2015), and excitation spectra of alkali-doped fullerides Knupfer and Fink (1997); Chibotaru and Ceulemans (1999, 2000). The importance is also suggested Nava et al. (2018) in recently reported light induced superconductivity of alkali-doped fullerides Mitrano et al. (2016); Cantaluppi et al. (2018). Moreover, the JT effect involving the NLUMO must be significant in highly alkali doped Knupfer and Fink (1997) and alkali-earth/rare-earth doped fullerides Chen, Taga, and Iwasa (1999); Margadonna et al. (2000); Iwasa and Takenobu (2003); Li et al. (2003); He et al. (2005); Akada et al. (2006); Heguri and Kobayashi (2010).

So far, the dynamic JT effect in negatively charged C60 in the ground electronic configuration where only the $t_{1u}$ LUMOs has been mainly investigated. Recently, bound excited states of C${}_{60}^{-}$ have been theoretically investigated Klaiman, Gromov, and Cederbaum (2013, 2014); Zakrzewski, Dolgounitcheva, and Ortiz (2014); Gromov, Klaiman, and Cederbaum (2015, 2016), and the stability of the first excited ${}^{2}T_{1g}$ electronic states of C${}_{60}^{-}$ has been confirmed. Nevertheless, the JT effect in the excited C60 has not been theoretically investigated, and the actual situation in C60 anions remains unclear.

In this work, we address the JT effect of first excited C${}_{60}^{-}$ anion of $t_{1g}^{1}$ configuration. The vibronic coupling parameters are derived from the $t_{1g}$ orbital energy levels calculated by density functional theory (DFT) calculations with hybrid B3LYP exchange-correlation functional. Using these coupling parameters, the vibronic states are obtained by numerically diagonalizing the dynamical JT Hamiltonian matrix. Compared with the case of the ground electronic state of C${}_{60}^{-}$, $t_{1u}^{1}$, the stabilization in the present case is found to be stronger by about 20 %.

II Jahn-Teller Effect

II.1 Model Hamiltonian

The $t_{1g}$ next LUMO of neutral C60 with $I_{h}$ symmetry is triply degenerate and separated from the other orbital levels Chancey and O’Brien (1997). According to the selection rule, the $t_{1g}$ orbitals couple to totally symmetric $a_{g}$ and five-fold degenerate $h_{g}$ normal modes as in the case of $t_{1u}$ orbitals Jahn and Teller (1937):

[TABLE]

Therefore, the linear vibronic Hamiltonian of C${}_{60}^{-}$ in the first excited $t_{1g}^{1}$ electronic configuration is given as in the case of $t_{1u}^{1}$ O’Brien (1969); Auerbach, Manini, and Tosatti (1994); O’Brien (1996); Chancey and O’Brien (1997):

[TABLE]

We take the equilibrium structure of C60 as the reference structure. Here, $q_{\Gamma\gamma}$ and $p_{\Gamma\gamma}$ ($\gamma=\theta,\epsilon,\xi,\eta,\zeta$ for $\Gamma=h$) are mass-weighted normal coordinates Inui, Tanabe, and Onodera (1990) and conjugate momenta, respectively, $\omega_{\Gamma}$ is frequency, and $V_{\Gamma}$ the vibronic coupling parameters. The basis of the marix is in the order of $|T_{1g}x\rangle$, $|T_{1g}y\rangle$, $|T_{1g}z\rangle$. The representation for the normal coordinates and conjugate momenta possess the symmetry of real $d$-type [$(2z^{2}-x^{2}-y^{2})/\sqrt{6}$, $(x^{2}-y^{2})/\sqrt{2}$, $\sqrt{2}yz$, $\sqrt{2}zx$, $\sqrt{2}xy$], as they are consistent with the original and most used representation O’Brien (1969); Auerbach, Manini, and Tosatti (1994); Manini, Tosatti, and Auerbach (1994); O’Brien (1996); Chancey and O’Brien (1997). The bases are different from those ($Q$) of some previous work Dunn and Bates (1995):

[TABLE]

In the above equation, the indices $g$ or $u$ indicating the parity and the indices $\mu$ distinguishing the frequencies are omitted for simplicity. They are added when necessary.

II.2 Adiabatic potential energy surface

The model Hamiltonians, and hence the formulae, for the ground electronic configuration and the first excited configuration have the same structure. The depth of the adiabatic potential energy surface (APES) with respect to the reference structure is given by O’Brien (1969)

[TABLE]

with

[TABLE]

where $E_{a}$ and $E_{\text{JT}}$ are the first and the second terms in the last expression in Eq. (6), respectively, and $\bm{q}_{h}$ is the list of $q_{h\gamma}$. The APES has two-dimensional continuous trough O’Brien (1969), suggesting the presence of SO(3) symmetry O’Brien (1971); Pooler (1980).

II.3 Vibronic states

As in the case of the JT problem for the $t_{1u}^{1}$ configurations O’Brien (1971); Romestain and Merle d’Aubigné (1971); Pooler (1980), the vibronic angular momenta $\hat{\bm{J}}$ exist in the case of $t_{1g}^{1}$ Chancey and O’Brien (1997):

[TABLE]

Therefore, the eignestates of $\hat{H}$ (vibronic states) are expressed by $J$, $M_{J}$, and principal quantum number $\alpha$,

[TABLE]

The analytical treatments of the vibronic states in the strong limit of vibronic coupling O’Brien (1969, 1971); Auerbach, Manini, and Tosatti (1994); O’Brien (1996); Iwahara (2018) and weak coupling limit Manini, Tosatti, and Auerbach (1994) have been discussed much. Nevertheless, for the quantitative description of C60 ions, only numerical approach can provide accurate description.

For numerical calculations, it is convenient to expand the vibornic states as

[TABLE]

Here, $\bm{n}_{h}=(n_{h\theta},n_{h\epsilon},n_{h\xi},n_{h\eta},n_{h\zeta})$ is the set of vibrational quantum numbers of the Harmonic oscillation part of Eq. (4). Such an expansion using the direct products of the electronic states and the eigenstates of harmonic oscillator has been proposed long time ago Longuet-Higgins et al. and has been routinely used as a reliable approach to quantitatively study the dynamical JT systems including fullerene anions O’Brien (1971); Auerbach, Manini, and Tosatti (1994); Gunnarsson et al. (1995); O’Brien (1996); Iwahara et al. (2010); Iwahara and Chibotaru (2013); Ponzellini (2014); Liu, Iwahara, and Chibotaru (2018).

In the present calculations, the vibrational basis is truncated as

[TABLE]

because the dimension of the Hamiltonian matrix rapidly increases. To take account of the eight sets of $h_{g}$ modes in real C60, $\mu$ is shown. For the diagonalization of the vibronic Hamiltonian (4), Lanczos algorithm was employed Pooler (1984).

II.4 Orbital vibronic coupling parameters

The orbital vibronic coupling parameters are defined by the gradients of the $t_{1g}$ NLUMO level:

[TABLE]

where $\bm{q}$ is the set of all normal coordinates. In the present case, the vibronic coupling parameters $V_{\Gamma}$ correspond to the orbital vibronic coupling parameters $v_{\Gamma}$:

[TABLE]

in a good approximation because of the very small mixing of the orbitals under JT deformation.

The vibronic coupling parameters are derived by fitting the model potential to the gradients of NLUMO levels of neutral C60 calculated in Ref. Liu et al., 2018. The derivations were done using the DFT data with hybrid B3LYP functional because in the studies of C60 anions, this functional has been shown to give the coupling parameters close to those derived from experimental data Iwahara et al. (2010) [In the study, high-resolution photoelectron spectra Wang, Woo, and Wang (2005) was used]. The coupling parameters derived from the B3LYP data are in good agreement with those from the gradients of the GW quasiparticle levels Faber et al. (2011). Furthermore, with the use of these parameters, the spin gap of C${}_{60}^{3-}$ was well reproduced Liu, Iwahara, and Chibotaru (2018). The vibronic coupling parameters of C${}_{60}^{-}$ have been derived using local density approximation Manini et al. (2001), generalized gradient approximation Frederiksen et al. (2008), and also by post Hartree-Fock calculations Iwahara et al. (2012). Nevertheless, the former two methods underestimate and the latter one overestimates the vibronic coupling parameters. Therefore, we expect that the vibronic coupling parameters derived from the gradient of the NLUMO levels are accurate enough to reveal the low-energy states of excited C${}_{60}^{-}$.

The derived vibronic coupling parameters are listed in Table 1 and one of the fittings is shown in Fig. 1 (see for the other fittings Supplemental Materials). In the fitting, the $h_{g}\epsilon$ deformation is used instead of the $h_{g}\theta$ with $v_{h}=-(2/\sqrt{3})\partial\epsilon_{x}/\partial q_{h\epsilon}|_{\bm{q}=0}$. The stabilization energies (6) in the $T_{1g}$ electronic states are $E_{a}=24.0$ and $E_{h}=65.6$ meV, which are by 30.7 % and 32.5 % larger than the stabilization energies of for the $a_{g}$ and $h_{g}$ modes in the $T_{1u}$ ground electronic states, respectively.

II.5 Vibronic states

The vibronic Hamiltonian matrix was numerically diagonalized as described in Sec. II.3. The calculated data are listed in Table 2 and the levels are shown in Fig. 2. In the figure, the vibronic levels for the ground C${}_{60}^{-}$ and the vibrational levels of neutral C60 as well as the vibronic levels of the first excited C${}_{60}^{-}$. The ground vibronic and vibrational levels are used as the origin of the energy.

One should note that the distributions of the vibronic states of C60 anions differ much from that of the vibrational levels of neutral C60. The ground vibronic levels with vibronic angular momentum $J$=1 for the $T_{1u}$ ($t_{1u}^{1}$) and $T_{1g}$ ($t_{1g}^{1}$) electornic states are $-96.5$ and $-113.8$ meV, respectively. Previous study shows that for the ground electronic states, the contributions from the static and the dynamic JT effect to the ground energy are almost the sameLiu, Iwahara, and Chibotaru (2018). However, the ratio of the dynamical contribution to the static contribution is smaller in the $t_{1g}^{1}$ case than in the $t_{1u}^{1}$ case (Table 3), which is consistent Auerbach, Manini, and Tosatti (1994); O’Brien (1996) with the stronger orbital vibronic couplings for the $t_{1g}$ NLUMO than for the $t_{1u}$ LUMO.

The difference in the vibronic couplings in the $t_{1u}$ and the $t_{1g}$ levels appear in the excited vibronic levels too (Fig. 2). The group of the first excited vibronic levels ($J=3,2,1$) split more in the case of $t_{1g}^{1}$ than in $t_{1u}^{1}$, as expected from the stronger vibronic coupling in the former: the splitting of the former, 13.3 meV, is about two times larger than that of LUMO (4.4 meV). Such splitting may be observed as fine structures in e.g. high-resolution absorption spectra of C${}_{60}^{-}$.

III Discussion

The dynamical JT effect in the $T_{1g}$ electronic states has been discussed in e.g. Refs. Kondo, Momose, and Shida, 1995; Tomita et al., 2005. Indeed, as found in Sec. II.5, the stabilization by the dynamical JT effect is even stronger than in the ground electronic states (Table 3), suggesting the importance of the vibronic dynamics in the excited electronic states.

The presence of the dynamical JT effect in the excited states can be found in various spectroscopic data. For example, one should note that the signs of almost all the vibronic coupling parameters for the $h_{g}$ modes in the $T_{1g}$ state, and hence the JT deformations, are opposite to those for the $T_{1u}$ state. The difference in the direction of the JT deformations in the $T_{1u}$ and the $T_{1g}$ states indicate that the relative displacements in these electronic states are large, and thus, the vibronic progression in the excitation spectra of ${}^{2}T_{1g}\leftarrow{}^{2}T_{1u}$ tends to be stronger than that in the photoelectron spectra of C${}_{60}^{-}$ because the latter is only related to the vibronic coupling in the $T_{1u}$ state. Indeed, the the peaks in the absorption spectra of C${}_{60}^{-}$ Kondo, Momose, and Shida (1995); Tomita et al. (2005) are stronger than the peaks in the photoelectron spectra of C${}_{60}^{-}$ Wang, Woo, and Wang (2005); Huang et al. (2014). Furthermore, as mentioned in Sec. II.5, the transition to the excited vibronic states may be seen as the fine structure of the spectra.

By combining the present vibronic coupling parameters for the $t_{1g}$ NLUMO with those for the $t_{1u}$ LUMO (Table 1), it is also possible to address diverse problems of C60. For example, the $(T_{1u}\oplus T_{1g})\otimes h_{g}$ product Jahn-Teller problem Ceulemans and Chibotaru (1996) in C${}_{60}^{2-}$ anion $t_{1u}^{1}t_{1g}^{1}$ electron configurations. Since the signs of the orbital vibronic coupling parameters for the $t_{1u}$ and the $t_{1g}$ levels tend to be opposite to each other, the resulting vibornic coupling of the first excited C${}_{60}^{2-}$ must be weak. Besides, the combination of the present vibronic coupling constants and those for C${}_{60}^{+}$ Huang and Liu (2019) enables us to investigate the luminescence spectra Akimoto and Kan’no (2002) involving $t_{1g}$ NLUMO.

IV Conclusion

In this work, the vibronic coupling parameters of the $T_{1g}$ electronic states of C${}_{60}^{-}$ by using the next lowest unoccupied molecular orbital levels at B3LYP level. Based on the obtained parameters, the vibronic states were calculated by exactly diagonalizing the dynamical Jahn-Teller Hamiltonian. The results for the $t_{1g}^{1}$ configuration showed stronger dynamic JT stabilization than that for the $t_{1u}^{-}$ configuration by about 20 %, indicating the importance of the JT effect in the excited states. The presence of the JT dynamics appears in spectroscopic data. Due to the difference in the direction of the JT deformations in the $T_{1u}$ and the $T_{1g}$ states, stronger vibronic progression is seen in the absorption spectra than in the photoelectron spectra of C${}_{60}^{-}$.

Acknowledgements.

The authors thank Naoya Iwahara and Liviu Chibotaru for fruitful discussions. They also gratefully acknowledge funding by the China Scholarship Council (CSC).