Update documentation

kthpanor · Jun 20, 2024 · e0b0bae · e0b0bae
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diff --git a/_sources/docs/mol_struct/coordinates.ipynb b/_sources/docs/mol_struct/coordinates.ipynb
@@ -175,7 +175,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.8"
+   "version": "3.11.5"
   }
  },
  "nbformat": 4,

diff --git a/_sources/docs/spec_prop/cubic_resp.ipynb b/_sources/docs/spec_prop/cubic_resp.ipynb
diff --git a/_sources/docs/spec_prop/ecd.ipynb b/_sources/docs/spec_prop/ecd.ipynb
@@ -0,0 +1,146 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a467e0ec-ad20-4afb-90bd-48e1479f07a2",
+   "metadata": {},
+   "source": [
+    "# Electronic Circular Dichroism"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fd8345de-68f1-436e-abf2-81ae167d0ce2",
+   "metadata": {},
+   "source": [
+    "The lowest-order absorption process involving mixed\n",
+    "electric and magnetic perturbations, the\n",
+    "absorptive analogue of the optical rotation, is known as \n",
+    "*electronic circular dichroism* (ECD) or just CD for short.\n",
+    "\n",
+    "The differential absorption of circularly polarized light,\n",
+    "corresponding to the difference between the absorptive index of the two\n",
+    "circular components of linearly polarized light, is proportional\n",
+    "to the rotational strength, which is normally calculated as the\n",
+    "residue of the linear response mixed electric dipole-magnetic dipole\n",
+    "polarizability\n",
+    "\n",
+    "$$\n",
+    "\\lim_{\\omega \\rightarrow \\omega_{n0}}\n",
+    "\\hbar (\\omega - \\omega_{n0}) \\langle\\langle  \\hat{\\mu}_{\\alpha};\\hat{m}_{\\beta}  \\rangle\\rangle_{\\omega}\n",
+    "= \\langle  0 \\mid\n",
+    "{\\hat{\\mu}}_{\\alpha} \\mid n\\rangle \\langle n \\mid {\\hat{m}}_{\\beta} \\mid\n",
+    "0\\rangle.\n",
+    "$$\n",
+    "\n",
+    "Since this expression corresponds to the infinite lifetime\n",
+    "approximation for the excited state, only a single number will be\n",
+    "obtained at the frequency of the electronic excitation. \n",
+    "\n",
+    "In general $^nR$, the rotatory strength for the transition $\\mid 0\\rangle  \n",
+    "\\rightarrow \\mid n\\rangle $ includes an electric \n",
+    "dipole-magnetic dipole contribution\n",
+    "\n",
+    "$$\n",
+    "^nR^{\\rm m}_{\\alpha\\beta} = -\\frac{3 i {e^2}}{4 m_e } (\\delta_{\\alpha\\beta}\n",
+    "\\langle  0 \\mid\n",
+    "{\\bf r} \\mid n\\rangle \\langle n \\mid {\\bf{l}^{\\rm T}} \\mid 0\\rangle -\n",
+    "\\langle  0 \\mid\n",
+    "{\\hat{r}}_{\\beta} \\mid n\\rangle \\langle n \\mid {\\hat{l}}_{\\alpha} \\mid\n",
+    "0\\rangle ),\n",
+    "$$\n",
+    "\n",
+    "and an electric dipole--electric quadrupole contribution\n",
+    "\n",
+    "$$\n",
+    "^nR^{\\rm Q}_{\\alpha\\beta} = -\\frac{3 \\omega_{n0} {e^2}}{4}\n",
+    "{\\mbox{${\\varepsilon}_{\\alpha \\gamma \\delta}$}}\n",
+    "\\langle  0 \\mid\n",
+    "{\\hat{r}}_{\\gamma} \\mid n\\rangle \\langle n \\mid {{\\hat{q}}}_{\\delta\\beta} \\mid\n",
+    "0\\rangle. \n",
+    "$$\n",
+    "\n",
+    "For randomly oriented molecules, the averaging leaves only the\n",
+    "electric dipole-magnetic dipole contribution and the scalar\n",
+    "rotatory strength is given by\n",
+    "\n",
+    "$$\n",
+    "^nR = -\\frac{i {e^2}}{2 m_e}\n",
+    "\\langle  0 \\mid\n",
+    "{\\bf r}^{\\rm T} \\mid n\\rangle \\langle n \\mid {\\bf{l}} \\mid 0\\rangle,\n",
+    "$$\n",
+    "\n",
+    "These expressions are given in the length gauge. In the velocity\n",
+    "gauge\n",
+    "\n",
+    "$$\n",
+    "^nR^{\\rm m}_{\\alpha\\beta} = \\frac{3 {e^2}}{4 m_e^2 \\omega_{n0}} (\\delta_{\\alpha\\beta}\n",
+    "\\langle  0 \\mid\n",
+    "{\\bf{p}} \\mid n\\rangle \\langle n \\mid {\\bf{l}^{\\rm T}} \\mid 0\\rangle -\n",
+    "\\langle  0 \\mid\n",
+    "{\\hat{p}}_{\\beta} \\mid n\\rangle \\langle n \\mid {\\hat{l}}_{\\alpha} \\mid\n",
+    "0\\rangle ),\n",
+    "$$\n",
+    "\n",
+    "and\n",
+    "\n",
+    "$$\n",
+    "^nR^{\\rm Q}_{\\alpha\\beta} = \\frac{3 {e^2}}{4 m_e^2  \\omega_{n0}}\n",
+    "{\\mbox{${\\varepsilon}_{\\alpha \\gamma \\delta}$}}\n",
+    "\\langle  0 \\mid\n",
+    "{\\hat{p}}_{\\gamma} \\mid n\\rangle \\langle n \\mid {\\hat{T}}^+_{\\delta\\beta} \\mid\n",
+    "0\\rangle,\n",
+    "$$\n",
+    "\n",
+    "where ${\\mathbf{T}}^+$ indicates the *velocity* form of the\n",
+    "electric quadrupole\n",
+    "\n",
+    "$$\n",
+    "{\\mathbf{T}}^+ = -\\Big( {\\mathbf{r}} {\\mathbf{p}} + {\\mathbf{p}}\n",
+    "{\\mathbf{r}} \\Big).\n",
+    "$$\n",
+    "\n",
+    "This form has an advantage in comparison to the length form.\n",
+    "Although with a translation of the reference frame the magnetic \n",
+    "dipole and electric quadrupole components change, the total tensor \n",
+    "in the velocity gauge is invariant to such a change of origin. For \n",
+    "the length gauge, this invariance depends in addition on the \n",
+    "fulfillment of the hypervirial relation, \n",
+    "\n",
+    "$$\n",
+    " \\langle 0 \\mid \\hat{p}_\\alpha \\mid n \\rangle =\n",
+    "i m_e \\; \\omega_{0n}\\; \\langle 0 \\hat{r}_\\alpha \\mid n \\rangle,\n",
+    "$$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fb43daa2-b04e-43cf-8774-d50dfae37a7e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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+   "language": "python",
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+   "mimetype": "text/x-python",
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+}