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"# One-photon absorption"
]
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"Calculations of one-photon absorption enable the determination of the extinction coefficient $\\varepsilon$,\n",
"describing the attenuation of the intensity of an\n",
"incoming light beam when passing through a sample of absorbing\n",
"species. In the lowest-order perturbative expansion of the\n",
"interaction between light and matter, $\\varepsilon$ is\n",
"related to the excitation energy and to\n",
"$\\left< 0 \\mid {\\mathbf \\mu} \\mid n \\right>$, \n",
"the transition moment between\n",
"the ground $\\ket{0}$ and excited $\\ket{n}$ electronic states.\n",
"The derived quantities are\n",
"the dipole strength,\n",
"\n",
"\\begin{equation*}\n",
"^nD_{\\alpha\\beta} =\n",
"\\left< 0 \\mid\n",
"{\\hat{\\mu}}_{\\alpha} \\mid n\\right>\\left,\n",
"\\end{equation*}\n",
"\n",
"and the oscillator strength. In the length gauge, the latter can be written as\n",
"\n",
"\\begin{equation*}\n",
"f^{\\rm r}_{0n} = \\frac{2 m_e \\omega_{0n}}{3 \\hbar e^2} \\bra{0}\n",
" {\\hat{\\mu}_\\alpha} \\ket{n} \\bra{n} { \\hat{\\mu}_\\alpha} \\ket{0}\n",
" = \\frac{2 m_e}{3 e^2} \\lim_{\\omega \\rightarrow \\omega_{0n}}\n",
" (\\omega - \\omega_{0n}) \\, \\omega \\, \\langle\\langle {\\hat{\\mu}_\\alpha};{\\hat{\\mu}_\\alpha}\n",
"{\\rangle\\rangle}_\\omega.\n",
"\\end{equation*}\n",
"\n",
"Using the hypervirial relationship\n",
"\n",
"\\begin{eqnarray*}\n",
" \\bra {0} {\\hat{p}_\\alpha} \\ket{n} =\n",
"i m_e \\; \\omega_{0n}\\; \\bra{0}{\\hat{r}_\\alpha} \\ket{n},\n",
"\\end{eqnarray*}\n",
"\n",
"we obtain in the velocity gauge\n",
"\n",
"\\begin{equation*}\n",
"f^{\\rm v}_{0n} = \\frac{2}{3 \\hbar m_e \\omega_{0n}} \\bra{0} {\\hat{p}_\\alpha}\n",
" \\ket{n} \\bra{n} { \\hat{p}_\\alpha} \\ket{0}\n",
" = \\frac{2}{3 m_e} \\lim_{\\omega \\rightarrow \\omega_{0n}}\n",
" (\\omega - \\omega_{0n}) \\, \\omega^{-1} \\,\n",
" \\langle\\langle {\\hat{p}_\\alpha};{\\hat{p}_\\alpha} {\\rangle\\rangle}_\\omega,\n",
"\\end{equation*}\n",
"\n",
"and in the mixed length-velocity gauge\n",
"\n",
"\\begin{equation*}\n",
"f^{\\rm rv}_{0n} = - \\frac{2i}{3 \\hbar e} \\bra{0} {\\hat{\\mu}_\\alpha} \\ket{n}\n",
" \\bra{n} { \\hat{p}_\\alpha} \\ket{0}\n",
" = - \\frac{2i}{3 e} \\lim_{\\omega \\rightarrow \\omega_{0n}}\n",
" (\\omega - \\omega_{0n}) \\, \\langle\\langle {\\hat{\\mu}_\\alpha};{\\hat{p}_\\alpha}\n",
" {\\rangle\\rangle}_\\omega,\n",
"\\end{equation*}\n",
"\n",
"respectively. \n",
"\n",
"The Thomas-Reiche-Kuhn (TRK) sum rule, which can be written in \n",
"terms of the dipole oscillator strengths as\n",
"\n",
"\\begin{equation*}\n",
" \\sum_{n} f_{0n} = N_{\\rm el},\n",
"\\end{equation*}\n",
"\n",
"where $N_{\\rm el}$ is the total number of electrons, can be used \n",
"as another test of the accuracy of the calculations.\n",
" \n",
"Of particular concern are Rydberg states,\n",
"characterized by a very diffuse orbital\n",
"occupied by the excited electron.\n",
"DFT in general fails to properly describe Rydberg-excited\n",
"states, as many of the common exchange--correlation\n",
"functionals do not\n",
"display the correct asymptotic behaviour for the $-1/r$\n",
"operator. This deficiency of modern exchange-correlation functionals\n",
"can be partially rectified by introducing the correct asymptotic\n",
"behaviour for the exchange-correlation\n",
"functional. It is worth noting \n",
"that time-dependent Hartree-Fock\n",
"(TDHF) \n",
"theory does not display the same\n",
"problems in describing Rydberg states, due to use of the exact exchange\n",
"operator. However, the lack of electron correlation\n",
"effects in TDHF may\n",
"limit the applicability of this approach for excited states in\n",
"general.\n",
"\n",
"To account for the broadening of the\n",
"experimental spectrum (related not only to the finite\n",
"lifetime of the electronic state, but also to the\n",
"rovibrational structure, collisions and other\n",
"aspects of the interaction between light and matter),\n",
"and in particular to investigate whether\n",
"certain bands may be hidden in the experimental spectrum due to\n",
"overlapping bands,\n",
"simple Lorentz line broadening is often added in the\n",
"form\\index{Lorentz line broadening}\n",
"\n",
"\\begin{equation*}\n",
"L\\left(\\nu\\right)\\approx\\frac{\\tilde{\\Gamma}}{\\pi}\\left[\\frac{1}{\\left(\\nu_0-\\nu\\right)^2\n",
"+ \\tilde{\\Gamma}^2}\\right],\n",
"\\end{equation*}\n",
"\n",
"where $\\tilde{\\Gamma}$ is\n",
"related to the lifetime of the excited state, and $\\nu_0$ is the\n",
"frequency of the electronic excitation. Assuming an isolated\n",
"absorption band, $\\tilde{\\Gamma}$ is equal to half the width of\n",
"the absorption band at half height (the value of $\\tilde{\\Gamma}$ \n",
"is often adjusted in the calculations to fit the\n",
"experimental data). It is common also to\n",
"employ a Gaussian function to reproduce the line broadening\n",
"\n",
"\\begin{equation*}\n",
"L\\left(\\nu\\right) = \\frac{2}{\\Gamma \\sqrt{2 \\pi}} \\;\n",
"e^{-{2(\\nu-\\nu_0)^2}/{\\Gamma^2}},\n",
"\\end{equation*}\n",
"\n",
"and we have $\\tilde{\\Gamma} = \\Gamma \\sqrt({\\rm ln2}/2)$.\n",
"\n",
"Finally, we recall that the transition moment between two excited \n",
"states $\\langle m \\mid A \\mid n\\rangle $ can be obtained\n",
"using only the reference state wave function from a double residue \n",
"of the quadratic response\n",
"function\n",
"(and if $\\mid m\\rangle = \\mid n\\rangle $ we may in this way determine\n",
"the expectation value of $A$ in the excited state)."
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