We consider a model which includes the electronic ground (excited) state of the molecule |

*g*〉 (|

*e*〉). The electron on the molecule interacts with the molecular vibrations and the surface plasmons. The Hamiltonian of the system is

$\begin{array}{ll}\phantom{\rule{1em}{0ex}}H=& \sum _{m=g,e}{\mathit{\u03f5}}_{m}{c}_{m}^{\u2020}{c}_{m}+\hslash {\omega}_{0}{b}^{\u2020}b+\hslash {\omega}_{p}{a}^{\u2020}a\phantom{\rule{2em}{0ex}}\\ +\sum _{\beta}\hslash {\omega}_{\beta}{b}_{\beta}^{\u2020}{b}_{\beta}+M{Q}_{b}{c}_{e}^{\u2020}{c}_{e}\phantom{\rule{2em}{0ex}}\\ +V\left(a{c}_{e}^{\u2020}{c}_{g}+\mathrm{H.c.}\right)+\sum _{\beta}{U}_{\beta}{Q}_{b}{Q}_{\beta},\phantom{\rule{2em}{0ex}}\end{array}$

(1)

where ${c}_{m}^{\u2020}$ and *c*_{
m
}(*m* = *e*, *g*) are creation and annihilation operators for an electron with energy *ϵ*_{
m
} in state |*m*〉. Operators *b*^{†} and *b* are boson creation and annihilation operators for a molecular vibrational mode with energy $\hslash {\omega}_{0}$; *a*^{†} and *a* are for a surface plasmon mode with energy $\hslash {\omega}_{p}$, and ${b}_{\beta}^{\u2020}$ and *b*_{
β
} are for a phonon mode in the thermal phonon bath, with *Q*_{
b
} = *b* + *b*^{†} and ${Q}_{\beta}={b}_{\beta}+{b}_{\beta}^{\u2020}$. The energy parameters *M*, *V*, and *U*_{
β
} correspond to the coupling between electronic and vibrational degrees of freedom on the molecule (electron-vibration coupling), the exciton-plasmon coupling, and the coupling between the molecular vibrational mode and a phonon mode in the thermal phonon bath.

By applying the canonical (Lang-Firsov) transformation [

15],

*H* becomes

$\begin{array}{ll}\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}=& {\mathit{\u03f5}}_{g}{c}_{\mathrm{g}}^{\u2020}{c}_{\mathrm{g}}+\stackrel{~}{{\mathit{\u03f5}}_{e}}{c}_{\mathrm{e}}^{\u2020}{c}_{\mathrm{e}}+\hslash {\omega}_{0}{b}^{\u2020}b+\hslash {\omega}_{p}{a}^{\u2020}a\phantom{\rule{2em}{0ex}}\\ +\sum _{\beta}\hslash {\omega}_{\beta}{b}_{\beta}^{\u2020}{b}_{\beta}\phantom{\rule{2em}{0ex}}\\ +V\left(a{X}^{\u2020}{c}_{e}^{\u2020}{c}_{g}+\mathrm{H.c.}\right)+\sum _{\beta}{U}_{\beta}{Q}_{b}{Q}_{\beta},\phantom{\rule{2em}{0ex}}\end{array}$

(2)

where *X* = exp[-*λ*(*b*^{†} - *b*)], $\stackrel{~}{{\mathit{\u03f5}}_{e}}={\mathit{\u03f5}}_{e}-{M}^{2}/\left(\hslash {\omega}_{0}\right)$ and $\lambda =M/\left(\hslash {\omega}_{0}\right)$.

The luminescence spectra of the molecule are expressed in terms of Green’s function of the molecular exciton on the Keldysh contour [

16], which is defined as

$\begin{array}{l}\phantom{\rule{-6.0pt}{0ex}}L(\tau ,{\tau}^{\prime})\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-6.0pt}{0ex}}=\frac{1}{i\hslash}{\u3008{T}_{C}\left\{{c}_{g}^{\u2020}\right(\tau \left){c}_{e}\right(\tau \left){c}_{e}^{\u2020}\right({\tau}^{\prime}\left){c}_{g}\right({\tau}^{\prime}\left)\right\}\u3009}_{H}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-6.0pt}{0ex}}=\frac{1}{i\hslash}{\u3008{T}_{C}\left\{{c}_{g}^{\u2020}\right(\tau \left){c}_{e}\right(\tau \left)X\right(\tau \left){c}_{e}^{\u2020}\right({\tau}^{\prime}\left){c}_{g}\right({\tau}^{\prime}\left){X}^{\u2020}\right({\tau}^{\prime}\left)\right\}\u3009}_{\stackrel{~}{H}},\phantom{\rule{2em}{0ex}}\end{array}$

(3)

where 〈⋯ 〉_{
H
} and ${\u3008\cdots \phantom{\rule{0.3em}{0ex}}\u3009}_{\stackrel{~}{H}}$ denote statistical average in representations by system evolution for *H* and $\stackrel{~}{H}$, respectively. *τ* is the Keldysh contour time variable, and *T*_{
C
} is the time ordering along the Keldysh contour.

By assuming the condition of stationary current, the distribution function

*N*_{pl} of the surface plasmons excited by inelastic tunneling between the tip and the substrate is given by

${N}_{\text{pl}}\left(\omega \right)=\left\{\begin{array}{ll}{T}_{\text{pl}}\left(1-\left|\frac{\hslash \omega}{e{V}_{\text{bias}}}\right|\right),& \left|\hslash \omega \right|<\left|e{V}_{\text{bias}}\right|\\ 0,& \text{others}\end{array}\right.,$

(4)

where

*T*_{pl} is a coefficient related to the current amplitude due to the inelastic tunneling [

17]. We calculate

*L* according to the calculation scheme previously reported by us [

12]. The spectral function and the luminescence spectra of the molecule are defined by the relations,

$\begin{array}{ll}{A}_{L}\left(\omega \right)& =-\frac{1}{\Pi}\mathcal{I}\left[{L}^{r}\left(\omega \right)\right],\phantom{\rule{2em}{0ex}}\end{array}$

(5)

$\begin{array}{ll}{B}_{L}\left(\omega \right)& =-\frac{1}{\Pi}\mathcal{I}\left[{L}^{<}\left(\omega \right)\right],\phantom{\rule{2em}{0ex}}\end{array}$

(6)

where *L*^{
r
} and *L*^{<} are the retarded and lesser projection of *L*.

The parameters are given so that they correspond to the experiment on the STM-LE from TPP molecules on the gold surface [13]: $\left(\stackrel{~}{{\mathit{\u03f5}}_{e}}-{\mathit{\u03f5}}_{g}\right)=1.89\phantom{\rule{1em}{0ex}}\text{eV}$, $\hslash {\omega}_{0}=0.16\phantom{\rule{1em}{0ex}}\text{eV}$, and $\hslash {\omega}_{p}=2.05\phantom{\rule{1em}{0ex}}\text{eV}$. The statistical average is taken for temperature *T* = 80 K [13]. The square of *λ* is reported to be 0.61 on the basis of first-principles calculations [18]. The parameter *U*_{
β
} is given so that the molecular vibrational lifetime due to the coupling to the thermal phonon bath is 13 ps [13]. A Markovian decay is assumed for the surface plasmon so that the plasmon lifetime for *V*=0 eV becomes 4.7 fs [13, 18]. The coefficient *T*_{pl} is set in the range of 10^{-4} to 10^{-2}, where the tunneling current is *I*_{
t
} = 200 pA, and an excitation probability of the surface plasmons per electron tunneling event is considered to be in the range of 10^{-2} to 1.