---
abstract: 'The Jahn-Teller problems of C$_{60}$ anions involving $t_{1g}$ next lowest unoccupied molecular orbital (NLUMO) were theoretically investigated. The orbital vibronic coupling parameters for the $t_{1g}$ orbitals were derived from the Kohn-Sham orbital levels with hybrid B3LYP functional by the frozen phonon approach. With the use of these coupling parameters, the vibronic states of the first excited C$_{60}^-$ were calculated, and were analyzed. The dynamical Jahn-Teller stabilization energy of the first excited C$_{60}^-$ is stronger than that of the ground electronic states, resulting in two times larger splitting of vibronic levels than those of the ground state C$_{60}^-$. The present coupling parameters prompt us to understand more about the excited C$_{60}$.'
author:
- Zhishuo Huang
- Dan Liu
title: 'Dynamical Jahn-Teller effect in the first excited C$_{60}^-$'
---

Introduction
============

Highly symmetric C$_{60}$ exhibits complex Jahn-Teller dynamics characterized by orbital-vibration entanglement in various charged and excited states [@Chancey1997; @Bersuker2006; @Dunn2015]. Among these states, negatively charged C$_{60}$ is one of the most interesting cases because it often serves as a building brick of materials [@Gunnarsson2004; @Capone; @Alloul; @Kamaras; @Takabayashi; @Nomura2016; @Otsuka]. In order to comprehend thoroughly the role of the building brick, many properties of negatively charged C$_{60}$ should be understood clearly, especially about JT effect involved properties. Though JT effect, including dynamic JT effect, of C$_{60}$ anions have been intensively investigated [@Auerbach1994; @Manini1994; @Yu1994; @Dunn1995; @Gunnarsson1995; @OBrien1996; @Tosatti1996; @Yu1997; @Manini1998; @Sookhun2003; @Dunn2005; @Tomita2005; @Hands2008; @Frederiksen2008; @Iwahara2010; @Dunn2012; @Klupp2012; @Stochkel2013; @Ponzellini; @Kundu2015; @Iwahara2018; @Liu2018a; @Liu2018b; @Matsuda2018], 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, which showed the importance of dynamic JT effect [@Liu2018a; @Liu2018b].

So far, the works about the dynamic JT effect in negatively charged C$_{60}$ have been almost always in the ground electronic configuration populating only the lowest unoccupied molecular orbitals, which is the $t_{1u}$ orbital. However, to our knowledge, neither the vibronic coupling parameters for excited electronic configuration, say $t_{1g}$ next lowest unoccupied molecular orbial (NLUMO), nor the relevant JT effect has been theoretically investigated much. While it is believed that the nature of excited C$_{60}$ anions involving the next lowest unoccupied molecular orbital is of fundamental importance to interpret absorption spectra of isolated C$_{60}^-$ [@Kato1991; @Kato1993; @Kodama1994; @Kondo1995; @Kwon2001; @Kwon2002; @Tomita2005; @Stochkel2013; @Watariguchi2016], electron transfer process of fullerene [@ET1; @ET2], and excitation spectra of alkali-doped fullerides [@Knupfer1997; @Chibotaru1999; @Chibotaru2000], and the JT effect involving the NLUMO must be significant in highly alkali doped [@Knupfer1997] and alkali-earth/rare-earth doped fullerides [@Chen1999; @Margadonna2000; @Iwasa2003; @Li2003; @He2005; @Akada2006; @Heguri2010]. Furthermore, it might also be important [@Nava2018] in recently reported light induced superconductivity of alkali-doped fullerides [@Mitrano2016; @Cantaluppi2017]. Recently, bound excited states of C$_{60}^-$ have been theoretically investigated [@EX1; @EX2; @EX3; @EX4; @EX5], and the stability of the first excited ${}^2T_{1g}$ states of C$_{60}^-$ has been confirmed, nevertheless, the vibronic problem has not been investigated.

In this work, we address the dynamical JT effect of first excited C$_{60}^-$ anion populating the $t_{1g}$ NLUMO. The vibronic coupling parameters are derived from the data obtained 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, and are analyzed.

Jahn-Teller Effect
==================

Model Hamiltonian {#Sec:H}
-----------------

The $t_{1g}$ next LUMO of neutral C$_{60}$ with $I_h$ symmetry is triply degenerate and separated from the other orbital levels [@Chancey1997]. According to the selection rule, the $t_{1g}$ orbitals couple to totally symmetric $a_g$ and five-fold degenerate $h_g$ representation as in the case of $t_{1u}$ orbitals [@Jahn1937]: $$\begin{aligned}
 [t_{1g} \otimes t_{1g}] = a_g \oplus h_g.
\label{Eq:selection}\end{aligned}$$ In this work, we take the equilibrium structure of C$_{60}$ as the reference. Therefore, besides the $h_g$ modes, the vibronic couplings to the $a_g$ modes are nonzero. The linear vibronic Hamiltonian of C$_{60}^-$ in the first excited electronic $(t_{1g}^1)$ configuration resembles to that for the ground $t_{1u}^1$ electronic configuration [@OBrien1969; @Auerbach1994; @OBrien1996; @Chancey1997]: $$\begin{aligned}
 H &=& H_a + H_h,
\label{Eq:H}
\\
 H_a &=&
  \frac{1}{2}
  \left(
  p_a^2 + \omega_a^2 q_{a}^2
 \right) +  V_a q_{a},
\label{Eq:Ha}
\\
 H_h &=&
  \sum_{\gamma = \theta, \epsilon, \xi, \eta, \zeta} \frac{1}{2}\left(p_{h\gamma}^2 + \omega_h^2 q_{h \gamma}^2\right)
\nonumber\\
 &&+
 V_h
 \begin{pmatrix}
  \frac{1}{2} q_{h\theta} - \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\zeta} & \frac{\sqrt{3}}{2} q_{h\eta} \\
  \frac{\sqrt{3}}{2} q_{h\zeta} & \frac{1}{2} q_{h\theta} + \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\xi} \\
  \frac{\sqrt{3}}{2} q_{h\eta}  & \frac{\sqrt{3}}{2} q_{h\xi} & -q_{h\theta} \\
 \end{pmatrix}.
\label{Eq:Hh}\end{aligned}$$ Here, $q_{\Gamma\gamma}$ and $p_{\Gamma\gamma}$ ($\gamma = \theta, \epsilon, \xi, \eta, \zeta$ for $\Gamma = h$) are mass-weighted normal coordinates 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 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 in consistent with the original and most used representation [@OBrien1969; @Auerbach1994; @Manini1994; @Obrien1996; @Chancey1997]. The bases are different from those ($Q$) of some previous work [@Dunn1995]. The relation between them are $$\begin{aligned}
\begin{split}
 q_\theta =\sqrt{\frac{3}{8}} Q_\theta + \sqrt{\frac{5}{8}} Q_\epsilon,\\
 q_\epsilon=\sqrt{\frac{3}{8}} Q_\theta - \sqrt{\frac{5}{8}} Q_\epsilon.
\end{split}\end{aligned}$$ 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 for the discussion.

Adiabatic potential energy surface
----------------------------------

The model Hamiltonians for the ground electronic configuration and the first excited configuration are the same. Therefore, many electronic properties of the ground and the first excited electronic configurations are common too. The depth of the adiabatic potential energy surface (APES) with respect to the reference structure is given by [@OBrien1969] $$\begin{aligned}
 U_\text{min} &=& -E_a - E_\text{JT}
\nonumber\\
 &=& -\frac{V_a^2}{2\omega_a^2} - \frac{V_h^2}{2\omega_h^2},
\label{Eq:Umin}\end{aligned}$$ with $$\begin{aligned}
 q_{a,0} = -\frac{V_a}{\omega_a^2}, \quad \left|\bm{q}_{h,0} \right| = \frac{V_h}{\omega_h^2},\end{aligned}$$ where $E_a$ and $E_\text{JT}$ are the first and the second terms in the last expression in Eq. (\[Eq:Umin\]), respectively, and $\bm{q}_{h}$ is the list of $q_{h\gamma}$. The APES has two-dimensional continuous trough [@OBrien1969], suggesting the presence of SO(3) symmetry [@OBrien1971; @Pooler1980].

Vibronic states
---------------

As in the case of the JT problem for the ground electronic configuration [@OBrien1971; @Romestain1971; @Pooler1980], the vibronic angular momenta $\hat{\bm{J}}$ also exist in the first excited state [@Chancey1997]: $$\begin{aligned}
 [\hat{H}_h, \hat{\bm{J}}^2] = [\hat{H}_h, \hat{J}_z] = 0.
\label{Eq:symmetry}\end{aligned}$$ Therefore, the eignestates of $\hat{H}$ (vibronic states) are expressed by $J$, $M_J$, and principal quantum number $\alpha$, $$\begin{aligned}
 \hat{H}_h|\alpha JM_J\rangle &=& E_{\alpha J} |\alpha JM_J\rangle.
\label{Eq:vibronicproblem}\end{aligned}$$ The analytical treatments of the vibronic states in the strong limit of vibronic coupling [@OBrien1969; @OBrien1971; @Auerbach1994; @OBrien1996; @Iwahara2018] and weak coupling limit [@Manini1994] have been discussed much. Nevertheless, for the quantitative description of C$_{60}$ ions, only numerical approach can provide accurate description.

For numerical calculations, it is convenient to expand the vibornic states as $$\begin{aligned}
 |\alpha J M_J \rangle &=& \sum_\gamma \sum_{\bm{n}_h} |T_{1g} \gamma\rangle \otimes |\bm{n} \rangle C_{\gamma \bm{n}; \alpha J M_J}.\end{aligned}$$ 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. (\[Eq:Hh\]). Such an expansion using the direct products of the electronic states and the eigenstates of harmonic oscillator has been developed long time ago [@Longuet-Higgins1958] and has been routinely used to study dynamical JT problems including fullerene anion [@OBrien1971; @Auerbach1994; @Gunnarsson1995; @OBrien1996; @Iwahara2010; @Iwahara2013; @Ponzellini; @Liu2018a].

In the present calculations, the vibrational basis is truncated as $$\begin{aligned}
 0 \le n_{h(\mu)\gamma}, \quad \sum_{\mu\gamma} n_{h(\mu)\gamma} \le 7,\end{aligned}$$ because the dimension of the Hamiltonian matrix rapidly increases. To take account of the eight sets of $h_g$ modes in real C$_{60}$, $\mu$ is added in the condition. For the diagonalization of the vibronic Hamiltonian (\[Eq:Hh\]), Lanczos algorithm was applied [@Pooler1984].

Results
=======

Orbital vibronic coupling parameters
------------------------------------

![ The JT splitting of the NLUMO levels with respect to $q_{h_g(8)\epsilon}$ deformation (in atomic unit). The black points and gray lines indicate the DFT values and model energy, respectively. []{data-label="Fig:V"}](Vh8.eps){width="8cm"}

  ---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------
     $J$                                                                                                  
   $\Gamma$   $\mu$   $\omega_{\Gamma}$   $V_\Gamma$   $g_\Gamma$   $E_\Gamma$   $V_\Gamma$   $g_\Gamma$   $E_\Gamma$
    $a_g$       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
  ---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------

The orbital vibronic coupling parameters are defined by the gradients of the $t_{1g}$ NLUMO level: $$\begin{aligned}
 v_a = \left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_a}\right|_{\bm{q} = \bm{0}},
 \quad v_h = -\left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_{h\theta}}\right|_{\bm{q} = \bm{0}},\end{aligned}$$ 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 $v_\Gamma$: $$\begin{aligned}
 V_\Gamma = v_\Gamma,\end{aligned}$$ 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 calculated in Ref. . The calculations were done using DFT calculations with hybrid B3LYP functional, because, indicated by the previous studies, B3LYP could give closer parameters to the experimental data[@Iwahara2013; @Matsuda2018], and has a good agreement wit GW approximation calculations [@Faber2011]. The derived coupling parameters are listed in Table \[Table:V\], and one of the fittings are shown in Fig. \[Fig:V\] (see for the others Supplemental Materials).

The stabilization energies in the first excited electron configuration are $E_a = 24.04$ and $E_h = 65.58$ meV, which are by 30.7 % and 32.5% larger than the stabilization energies of $a_{g}$ and $h_{g}$ mode for the ground configuration, respectively. Moreover, almost all the vibronic coupling parameters for the $h$ modes in the $T_{1g}$ state are opposite compared with the case for the $T_{1u}$ state. The difference in sign indicates that the relative displacements in the ground and excited electronic states are large, and hence, the vibronic progression under the transition ${}^2T_{1g} \leftarrow {}^2T_{1u}$ tends to be stronger than that under the photoelectron spectra of C$_{60}^-$. This tendency is seen in experimental absorption spectra of C$_{60}^-$ [@Kondo1995; @Tomita2005] and photoelectron spectra of C$_{60}^-$ [@Wang2005; @Huang2014].

Vibronic states
---------------

The ground vibronic levels (termed by $J$=1) for the $T_{1u}$ (LUMO) and $T_{1g}$ (NLUMO) electornic states are $-96.5$ and $-113.8$ meV, respectively. Previous study shows that for LUMO, the contributions from the static and the dynamic JT effect to the ground energy is almost the same[@Liu2018a], but this is the not the same situation for NLUMO, as is shown in Table. \[Table:Energy\_contrib\]. The ratio of the dynamical contribution to the static contribution is smaller in NLUMO case than in LUMO case, which is consistent [@Auerbach1994; @OBrien1996] with the stronger vibronic coupling in NLUMO than in LUMO.

       Orbital       E$_{total}$   E$_{static}$   E$_{dynamic}$   Ratio
  ----------------- ------------- -------------- --------------- -------
        NLUMO         $-113.8$       $-65.6$         $-48.2$      0.74
   LUMO[@Liu2018a]    $-96.46$       $-50.3$         $-46.2$      0.92

  :  Contributions to the ground vibronic energy (E$_{total}$) of NLUMO and LUMO of C$_{60}^{-}$. 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$_{static}$).[]{data-label="Table:Energy_contrib"}

The low-energy vibronic levels are shown in Fig. \[Fig:Vibronic\_E\_level\] (see also Table. \[Table:Energy\_level\]) for NLUMO, and LUMO, and are compared with the vibrational levels of neutral C$_{60}$. For easy comparision, the energies of ground states of LUMO, NLUMO and the lowest vibrational state were shifted to a same level (0 meV). The group of the first excited vibronic levels ($J = 3,2,1$) split more in NLUMO than in LUMO, 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 structure in e.g. high-resolution absorption spectra of C$_{60}^-$.

![Low-lying vibronic levels with respect to NLUMO and LUMO of C$_{60}^{-}$. The numbers next to the energy levels are $J$ with the numbers in the parenthesis are the degeneracy.[]{data-label="Fig:Vibronic_E_level"}](Vibronic_E_level.eps){width="8cm"}

   $J$         NLUMO        LUMO
  ----- --- ------------ -----------
         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       -        $-4.123$
         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       -        $-4.265$
         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       -        $-1.607$
         1   $-46.772$    $-28.549$
         2   $-34.314$    $-14.108$
         3   $-27.326$    $-7.724$
         1   $-60.505$    $-33.554$
         2   $-36.593$    $-14.985$
         3   $-27.281$      $ -$

  :  The vibronic energy levels with respect to NLUMO and LUMO of C$_{60}^{-}$ (meV). The data for LUMO are taken Ref. , and the number in the parentheses indicate $J$.[]{data-label="Table:Energy_level"}

Conclusion
==========

In this work, the vibronic states of the first excited C$_{60}^-$ were calculated based on the orbital vibronic coupling parameters in the $t_{1g}$ orbital, which is compared with that in $t_{1u}$. The results for the $t_{1g}^1$ configuration showed much stronger dynamic JT stabilization than that for the $t_{1u}^-$ configuration, which induces greater splitting group of the first excited vibronic states too. Combining the present vibronic coupling constant and those from Ref. [\[1.3\]]{}, it is possible to investigate the luminescence spectra[@Akimoto2002] involving $t_{1g}$ NLUMO.

The authors thank Naoya Iwahara for his help with numerical calculations. They also gratefully acknowledge funding by the China Scholarship Council.

[75]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**](https://press.princeton.edu/titles/6243.html) (, , ) @noop [**]{} (, , ) [****, ()](\doibase
  http://dx.doi.org/10.1016/j.chemphys.2015.05.016) [**](https://www.worldscientific.com/worldscibooks/10.1142/5404) (, , ) @noop [****,  ()]{} [****,  ()](\doibase 10.1051/epjconf/20122300015) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0953-8984/28/i=15/a=153001) [**** (), 10.3390/cryst8030115](\doibase 10.3390/cryst8030115) [****,  ()](\doibase
  10.1103/PhysRevB.49.12998) [****,  ()](\doibase 10.1103/PhysRevB.49.13008) [****, ()](\doibase 10.1103/PhysRevB.50.5016) [****,  ()](\doibase 10.1103/PhysRevB.52.5996) [****,  ()](\doibase 10.1103/PhysRevLett.74.1875) [****, ()](\doibase 10.1103/PhysRevB.53.3775) [****,  ()](\doibase
  10.1103/PhysRevB.54.17184) [****, ()](\doibase
  https://doi.org/10.1016/S0379-6779(97)81163-0),  [****,  ()](\doibase 10.1103/PhysRevB.58.782) [****,  ()](\doibase
  10.1103/PhysRevB.68.235403) [****,  ()](\doibase
  10.1103/PhysRevB.71.115411) [****,  ()](\doibase
  10.1103/PhysRevLett.94.053002) [****,  ()](\doibase 10.1103/PhysRevB.77.115445) [****,  ()](\doibase
  10.1103/PhysRevB.78.233401) [****,  ()](\doibase
  10.1103/PhysRevB.82.245409) [****,  ()](\doibase
  10.1088/1367-2630/14/8/083038) [****,  ()](\doibase
  10.1038/ncomms1910) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/139/16/10.1063/1.4826097) **, @noop [Master’s thesis]{},  () [****,  ()](\doibase
  10.1021/jp5126409) [****,  ()](\doibase
  10.1103/PhysRevB.97.075413) [****,  ()](\doibase 10.1103/PhysRevB.97.115412) [****,  ()](\doibase 10.1103/PhysRevB.98.035402) [****,  ()](\doibase 10.1103/PhysRevB.98.165410) [****, ()](\doibase
  https://doi.org/10.1016/0009-2614(91)85147-O) [****,  ()](\doibase 10.1016/0009-2614(93)87142-P) [****,  ()](\doibase 10.1021/j100093a002) [****,  ()](\doibase
  https://doi.org/10.1016/0009-2614(95)00287-E) [****,  ()](\doibase 10.1021/jp0043625) [****, ()](\doibase 10.1140/e10053-002-0008-5) [****,  ()](\doibase
  10.2116/analsci.32.463) [****,  ()](\doibase 10.1021/jp983512x) [****,  ()](\doibase
  10.1039/C5CP05254H) [****,  ()](\doibase
  10.1103/PhysRevLett.79.2714) in @noop [**]{} (, ) pp.  [****,  ()](\doibase 10.1063/1.1342459) [****,  ()](\doibase
  10.1103/PhysRevB.60.4351) [****, ()](\doibase 10.1021/cm000330i) [****,  ()](\doibase 10.1088/0953-8984/15/13/202) [****,  ()](\doibase
  10.1103/PhysRevB.68.165417) [****,  ()](\doibase
  10.1103/PhysRevB.71.085404) [****,  ()](\doibase 10.1103/PhysRevB.73.094509) [****, ()](\doibase
  https://doi.org/10.1016/j.cplett.2010.02.069) [****,  ()](\doibase
  doi:10.1038/nphys4288) [****,  ()](https://www.nature.com/articles/nature16522) [****,  ()](\doibase 10.1038/s41567-018-0134-8) [****,  ()](\doibase
  10.1021/jz4018514) [****,  ()](\doibase
  10.1039/C4CP01447B) [****,  ()](\doibase 10.1021/jp412813m) [****,  ()](\doibase
  10.1080/00268976.2015.1060367) [****,  ()](\doibase 10.1039/C6CP00667A) [****,  ()](\doibase 10.1098/rspa.1937.0142) [****,  ()](\doibase
  10.1103/PhysRev.187.407) [****,  ()](\doibase 10.1088/0022-3719/4/16/017) [****,  ()](http://stacks.iop.org/0022-3719/13/i=6/a=013) [****,  ()](\doibase 10.1103/PhysRevB.4.4611) [****, ](\doibase
  http://doi.org/10.1098/rspa.1958.0022) [****,  ()](\doibase
  10.1103/PhysRevLett.111.056401) “,” in @noop [**]{},  (, , ) Chap. , p.  [****, ()](\doibase
  https://doi.org/10.1016/0009-2614(91)90312-W) [****,  ()](\doibase 10.1103/PhysRevB.84.155104) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/123/5/10.1063/1.1998787) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/140/22/10.1063/1.4881421) @noop [ ()]{} [****,  ()](\doibase
  10.1143/JPSJ.71.630)
