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A. Appendix: Electron-phonon coefficients

The electron-phonon coefficients g are defined as

g$\scriptstyle \bf q$ν($\displaystyle \bf k$, i, j) = $\displaystyle \left(\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right.$$\displaystyle {\hbar\over 2M\omega_{{\bf q}\nu}}$$\displaystyle \left.\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right)^{{1/2}}_{}$ψi,$\scriptstyle \bf k$|$\displaystyle {dV_{SCF}\over d {\hat u}_{{\bf q}\nu}}$$\displaystyle \hat{\epsilon}_{{{\bf q}\nu}}^{}$| ψj,$\scriptstyle \bf k$+$\scriptstyle \bf q$〉. (1)
The phonon linewidth γ$\scriptstyle \bf q$ν is defined by

γ$\scriptstyle \bf q$ν = 2πω$\scriptstyle \bf q$ν$\displaystyle \sum_{{ij}}^{}$$\displaystyle \int$$\displaystyle {d^3k\over \Omega_{BZ}}$| g$\scriptstyle \bf q$ν($\displaystyle \bf k$, i, j)|2δ(e$\scriptstyle \bf q$, i - eF)δ(e$\scriptstyle \bf k$+$\scriptstyle \bf q$, j - eF), (2)
while the electron-phonon coupling constant λ$\scriptstyle \bf q$ν for mode ν at wavevector $\bf q$ is defined as

λ$\scriptstyle \bf q$ν = $\displaystyle {\gamma_{{\bf q}\nu} \over \pi\hbar N(e_F)\omega^2_{{\bf q}\nu}}$ (3)
where N(eF) is the DOS at the Fermi level. The spectral function is defined as

α2F(ω) = $\displaystyle {1\over 2\pi N(e_F)}$$\displaystyle \sum_{{{\bf q}\nu}}^{}$δ(ω - ω$\scriptstyle \bf q$ν)$\displaystyle {\gamma_{{\bf q}\nu}\over\hbar\omega_{{\bf q}\nu}}$. (4)
The electron-phonon mass enhancement parameter λ can also be defined as the first reciprocal momentum of the spectral function:

λ = $\displaystyle \sum_{{{\bf q}\nu}}^{}$λ$\scriptstyle \bf q$ν = 2$\displaystyle \int$$\displaystyle {\alpha^2F(\omega) \over \omega}$. (5)

Note that a factor M-1/2 is hidden in the definition of normal modes as used in the code.

McMillan:

Tc = $\displaystyle {\Theta_D \over 1.45}$exp$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$ (6)
or (better?)

Tc = $\displaystyle {\omega_{log}\over 1.2}$exp$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$ (7)
where

ωlog = exp$\displaystyle \left[\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right.$$\displaystyle {2\over\lambda}$$\displaystyle \int$$\displaystyle {d\omega\over\omega}$α2F(ω)logω$\displaystyle \left.\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right]$ (8)


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