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- - C1 - - Y - - - Y - - - - - - - σh? - - - N - - - - - - σh? - - - - Y - Dnh - - N - - - - - nσd? - - Y - - - Dnd - - - N - - Dn - - - - Y - Cnh - - - N - - - - - nσv? - - - Y - - - - Cnv - - - N - - - - - S2n? - - - Y - - - S2n - - N - - Cn - Y - - diff --git a/_images/spectrometer.png b/_images/spectrometer.png new file mode 100644 index 000000000..34c1b78d3 Binary files /dev/null and b/_images/spectrometer.png differ diff --git a/_sources/docs/mol_struct/coordinates.ipynb b/_sources/docs/mol_struct/coordinates.ipynb index cbc5a6995..fc1289c17 100644 --- 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 index bb331cc54..60b598749 100644 --- a/_sources/docs/spec_prop/cubic_resp.ipynb +++ b/_sources/docs/spec_prop/cubic_resp.ipynb @@ -259,16 +259,7 @@ "remove_output" ] }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "* Warning * Environment variable OMP_NUM_THREADS not set.\n", - "* Warning * Setting OMP_NUM_THREADS to 4.\n" - ] - } - ], + "outputs": [], "source": [ "import veloxchem as vlx" ] @@ -289,7 +280,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 5, "id": "d06ef0e9-b38f-4802-ae75-02202808926c", "metadata": { "editable": true, @@ -336,7 +327,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 6, "id": "9aeaf9d4-76ca-4f7d-9c5b-db96d20cad0e", "metadata": { "editable": true, @@ -376,68 +367,74 @@ " Exchange-Correlation Functional : B3LYP \n", " Molecular Grid Level : 4 \n", " \n", + "* Info * Using the Libxc library (version 6.2.2). \n", + " \n", + " S. Lehtola, C. Steigemann, M. J.T. Oliveira, and M. A.L. Marques., SoftwareX 7, 1–5 (2018)\n", + " \n", + "* Info * Using the B3LYP functional. \n", + " \n", + " P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch., J. Phys. Chem. 98, 11623 (1994)\n", + " \n", "* Info * Molecular grid with 40720 points generated in 0.02 sec. \n", " \n", - "* Info * Processing Fock builds... (batch size: 5) \n", + "* Info * Processing Fock builds... (batch size: 6) \n", "* Info * batch 1/1 \n", " \n", - "* Info * 2 gerade trial vectors in reduced space \n", + "* Info * 3 gerade trial vectors in reduced space \n", "* Info * 3 ungerade trial vectors in reduced space \n", " \n", - "* Info * 75.47 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.27 GB of memory available for the solver on the master node \n", + "* Info * 78.19 kB of memory used for subspace procedure on the master node \n", + "* Info * 2.61 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 1 * Residuals (Max,Min): 8.43e-01 and 8.28e-01 \n", + " *** Iteration: 1 * Residuals (Max,Min): 8.43e-01 and 8.25e-01 \n", " \n", "* Info * Processing Fock builds... (batch size: 6) \n", "* Info * batch 1/1 \n", " \n", - "* Info * 5 gerade trial vectors in reduced space \n", + "* Info * 6 gerade trial vectors in reduced space \n", "* Info * 6 ungerade trial vectors in reduced space \n", " \n", - "* Info * 91.79 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.27 GB of memory available for the solver on the master node \n", + "* Info * 94.51 kB of memory used for subspace procedure on the master node \n", + "* Info * 2.61 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 2 * Residuals (Max,Min): 9.39e-02 and 9.19e-02 \n", + " *** Iteration: 2 * Residuals (Max,Min): 9.37e-02 and 9.11e-02 \n", " \n", - "* Info * Processing Fock builds... (batch size: 4) \n", + "* Info * Processing Fock builds... (batch size: 5) \n", "* Info * batch 1/1 \n", " \n", - "* Info * 7 gerade trial vectors in reduced space \n", - "* Info * 8 ungerade trial vectors in reduced space \n", + "* Info * 8 gerade trial vectors in reduced space \n", + "* Info * 9 ungerade trial vectors in reduced space \n", " \n", - "* Info * 102.67 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.27 GB of memory available for the solver on the master node \n", + "* Info * 108.11 kB of memory used for subspace procedure on the master node \n", + "* Info * 2.61 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 3 * Residuals (Max,Min): 9.72e-03 and 9.46e-03 \n", + " *** Iteration: 3 * Residuals (Max,Min): 9.50e-03 and 9.17e-03 \n", " \n", - "* Info * Processing Fock builds... (batch size: 6) \n", + "* Info * Processing Fock builds... (batch size: 5) \n", "* Info * batch 1/1 \n", " \n", - "* Info * 10 gerade trial vectors in reduced space \n", + "* Info * 11 gerade trial vectors in reduced space \n", "* Info * 11 ungerade trial vectors in reduced space \n", " \n", - "* Info * 118.99 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.27 GB of memory available for the solver on the master node \n", + "* Info * 121.71 kB of memory used for subspace procedure on the master node \n", + "* Info * 2.62 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 4 * Residuals (Max,Min): 3.69e-04 and 3.61e-04 \n", + " *** Iteration: 4 * Residuals (Max,Min): 3.92e-04 and 3.80e-04 \n", " \n", - "* Info * Processing Fock builds... (batch size: 4) \n", + "* Info * Processing Fock builds... (batch size: 5) \n", "* Info * batch 1/1 \n", " \n", - "* Info * 12 gerade trial vectors in reduced space \n", - "* Info * 13 ungerade trial vectors in reduced space \n", + "* Info * 13 gerade trial vectors in reduced space \n", + "* Info * 14 ungerade trial vectors in reduced space \n", " \n", - "* Info * 129.87 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.27 GB of memory available for the solver on the master node \n", + "* Info * 135.31 kB of memory used for subspace procedure on the master node \n", + "* Info * 2.62 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 5 * Residuals (Max,Min): 1.18e-05 and 1.17e-05 \n", + " *** Iteration: 5 * Residuals (Max,Min): 1.23e-05 and 1.22e-05 \n", " \n", - " *** Complex response converged in 5 iterations. Time: 1.38 sec \n", + " *** Complex response converged in 5 iterations. Time: 1.44 sec \n", " \n", " \n", - "* Info * Molecular grid with 40720 points generated in 0.02 sec. \n", - " \n", " Fock Matrix Computation \n", " ========================= \n", " \n", @@ -447,7 +444,7 @@ "* Info * batch 3/4 \n", "* Info * batch 4/4 \n", " \n", - "* Info * Time spent in Fock matrices: 2.88 sec \n", + "* Info * Time spent in Fock matrices: 3.03 sec \n", " \n", " \n", " Complex Response Solver Setup \n", @@ -461,6 +458,14 @@ " Exchange-Correlation Functional : B3LYP \n", " Molecular Grid Level : 4 \n", " \n", + "* Info * Using the Libxc library (version 6.2.2). \n", + " \n", + " S. Lehtola, C. Steigemann, M. J.T. Oliveira, and M. A.L. Marques., SoftwareX 7, 1–5 (2018)\n", + " \n", + "* Info * Using the B3LYP functional. \n", + " \n", + " P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch., J. Phys. Chem. 98, 11623 (1994)\n", + " \n", "* Info * Molecular grid with 40720 points generated in 0.02 sec. \n", " \n", "* Info * Processing Fock builds... (batch size: 8) \n", @@ -470,9 +475,9 @@ "* Info * 4 ungerade trial vectors in reduced space \n", " \n", "* Info * 67.66 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.26 GB of memory available for the solver on the master node \n", + "* Info * 2.63 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 1 * Residuals (Max,Min): 9.76e-02 and 8.18e-02 \n", + " *** Iteration: 1 * Residuals (Max,Min): 9.79e-02 and 8.00e-02 \n", " \n", "* Info * Processing Fock builds... (batch size: 8) \n", "* Info * batch 1/1 \n", @@ -481,9 +486,9 @@ "* Info * 8 ungerade trial vectors in reduced space \n", " \n", "* Info * 89.42 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.25 GB of memory available for the solver on the master node \n", + "* Info * 2.64 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 2 * Residuals (Max,Min): 6.61e-03 and 5.82e-03 \n", + " *** Iteration: 2 * Residuals (Max,Min): 6.81e-03 and 5.73e-03 \n", " \n", "* Info * Processing Fock builds... (batch size: 8) \n", "* Info * batch 1/1 \n", @@ -492,9 +497,9 @@ "* Info * 12 ungerade trial vectors in reduced space \n", " \n", "* Info * 111.18 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.25 GB of memory available for the solver on the master node \n", + "* Info * 2.65 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 3 * Residuals (Max,Min): 4.85e-04 and 4.28e-04 \n", + " *** Iteration: 3 * Residuals (Max,Min): 5.21e-04 and 4.42e-04 \n", " \n", "* Info * Processing Fock builds... (batch size: 8) \n", "* Info * batch 1/1 \n", @@ -503,9 +508,9 @@ "* Info * 16 ungerade trial vectors in reduced space \n", " \n", "* Info * 132.94 kB of memory used for subspace procedure on the master node \n", - "* Info * 3.25 GB of memory available for the solver on the master node \n", + "* Info * 2.65 GB of memory available for the solver on the master node \n", " \n", - " *** Iteration: 4 * Residuals (Max,Min): 1.67e-05 and 1.55e-05 \n", + " *** Iteration: 4 * Residuals (Max,Min): 1.70e-05 and 1.56e-05 \n", " \n", " *** Complex response converged in 4 iterations. Time: 1.28 sec \n", " \n", @@ -516,7 +521,7 @@ "* Info * Processing Fock builds... (batch size: 4) \n", "* Info * batch 1/1 \n", " \n", - "* Info * Time spent in Fock matrices: 0.62 sec \n", + "* Info * Time spent in Fock matrices: 0.67 sec \n", " \n", " \n", " Cubic response function: << z;z,z,z >> (0.0,0.0,0.0) \n", @@ -524,17 +529,17 @@ " \n", " Real Imaginary \n", " ---------------------------------------------------- \n", - " CRF -998.63439127 0.00000000j \n", + " CRF -998.63354647 -0.00000000j \n", " \n", " \n", - " Cubic response function: << z;z,z,z >> (0.0656,-0.0656,0.0656) \n", - " ================================================================= \n", + " Cubic response function: << z;z,z,z >> (0.0720028,-0.0720028,0.0720028) \n", + " ========================================================================== \n", " \n", " Real Imaginary \n", " ---------------------------------------------------- \n", - " CRF -1152.42872036 -12.04492771j \n", + " CRF -1188.61696054 -13.89780647j \n", " \n", - " *** Time spent in cubic response calculation: 6.25 sec *** \n", + " *** Time spent in cubic response calculation: 6.51 sec *** \n", " \n" ] } @@ -558,12 +563,14 @@ "crf_drv.c_component = \"z\"\n", "crf_drv.d_component = \"z\"\n", "\n", - "crf_drv.b_frequencies = [0.0, 0.0656]\n", - "crf_drv.c_frequencies = [0.0, -0.0656]\n", - "crf_drv.d_frequencies = [0.0, 0.0656]\n", + "crf_drv.b_frequencies = [0.0, 0.07200280]\n", + "crf_drv.c_frequencies = [0.0, -0.07200280]\n", + "crf_drv.d_frequencies = [0.0, 0.07200280]\n", "\n", "crf_drv.damping = 0.004556 # 1000 cm-1\n", "\n", + "crf_drv.print_level = 2\n", + "\n", "crf_results = crf_drv.compute(molecule, basis, scf_results)" ] }, @@ -583,7 +590,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 7, "id": "5b3a6404-8005-4592-9402-b8d4945abc5f", "metadata": { "editable": true, @@ -599,8 +606,8 @@ "text": [ " w1 w2 w3 gamma_zzzz\n", "------------------------------------------------\n", - " 0.0000 0.0000 0.0000 998.6344-0.0000j\n", - " 0.0656 -0.0656 0.0656 1152.4287+12.0449j\n" + " 0.0000 0.0000 0.0000 998.6335+0.0000j\n", + " 0.0720 -0.0720 0.0720 1188.6170+13.8978j\n" ] } ], @@ -644,7 +651,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.6" + "version": "3.11.8" } }, "nbformat": 4, diff --git a/_sources/docs/spec_prop/ecd.ipynb b/_sources/docs/spec_prop/ecd.ipynb new file mode 100644 index 000000000..f56e38727 --- /dev/null +++ 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": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/_sources/docs/spec_prop/more_optact.ipynb b/_sources/docs/spec_prop/more_optact.ipynb new file mode 100644 index 000000000..3f7f76485 --- /dev/null +++ b/_sources/docs/spec_prop/more_optact.ipynb @@ -0,0 +1,490 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "056c2d1a-9be2-4f57-aebc-68d0b94d1365", + "metadata": {}, + "source": [ + "# More birefringences & dichroisms" + ] + }, + { + "cell_type": "markdown", + "id": "b30ecd0d-5b19-4e9b-8578-cf7827b11996", + "metadata": {}, + "source": [ + "## Faraday effect" + ] + }, + { + "cell_type": "markdown", + "id": "52c8b6a1-f738-4f15-bb84-97a5d0be33a4", + "metadata": {}, + "source": [ + "```{figure} ../../img/spec_prop/Diapositiva13.png\n", + "---\n", + "name: Faraday effect\n", + "width: 600px\n", + "align: center\n", + "---\n", + "```" + ] + }, + { + "cell_type": "markdown", + "id": "20fb0873-5889-4cc8-96e4-fbd8a2196e55", + "metadata": {}, + "source": [ + "Observed in 1846, the Faraday effect is the rotation of the plane of polarization\n", + "of a linearly polarized light beam traversing a transparent sample (glass, liquid, gas) occurring \n", + "when a static magnetic field is applied with its direction parallel to that of propagating beam. Historically it was the first ever \n", + "experimental evidence of a double refraction (birefringence) phenomenon, \n", + "and in particular of a circular birefringence. It takes its name from the discoverer\n", + "and it is also known an *magnetic field induced-optical\n", + "rotation*. The rotation is proportional to the optical\n", + "anisotropy, which depends linearly on the\n", + "strength of the applied magnetic induction field. The mechanisms\n", + "governing this effect involve the interaction of the static \n", + "external magnetic induction field with the oscillating electric \n", + "field of the light wave and with the electric multipoles of the \n", + "sample. The term *magnetic optical rotatory dispersion* \n", + "(MORD) is often used to denote the frequency\n", + "dispersion of the Faraday effect, and thus of the corresponding circular \n", + "birefringence.\n", + "\n", + "For a fluid sample, the rotation\n", + "with respect to the molecular frame\n", + "is often given through the so-called Verdet constant\n", + "$V(\\omega)$\n", + "\n", + "$$\n", + "\\phi = \n", + " V(\\omega) B_z l\\; \\propto \\;\n", + " \\omega \\mbox{${\\varepsilon}_{\\alpha \\beta \\gamma}$} \\big({\\alpha^\\prime}^{\\rm B}_{\\alpha\\beta,\\gamma}+\\frac{1}{kT} \\langle 0 |{\\hat{m}}_\\gamma|0 \\rangle \n", + "{\\alpha^\\prime}_{\\alpha\\beta}\\big)\n", + " B_z l,\n", + "$$\n", + "\n", + "where $l$ is the path length and\n", + "${\\alpha^\\prime}^{\\rm B}$ denotes the magnetic field derivative of ${\\alpha^\\prime}$,\n", + "the antisymmetric electric dipole polarizability.\n", + "The second term in the parentheses\n", + "vanishes for closed-shell systems, as the magnetic dipole is\n", + "in that case quenched, $ \\langle 0 |{\\hat{m}}_\\gamma|0 \\rangle = 0$.\n", + "\n", + "The Verdet constant of a closed-shell system\n", + "free of orbital degeneracies both in the ground and in the excited state, \n", + "is therefore obtained from\n", + "the response\n", + "function/\n", + "\n", + "$$\n", + "V(\\omega) \\propto i \\omega {\\varepsilon}_{\\alpha \\beta \\gamma}\n", + "\\langle\\langle \\hat{\\mu}_\\alpha; \\hat{\\mu}_\\beta, \\hat{m}_\\gamma\n", + "\\rangle\\rangle_{\\omega,0}\n", + "$$\n", + "\n", + "This contribution is sometimes referred to as the $B$ term contribution to\n", + "the Verdet constant.\n", + "For molecules containing orbital degeneracies additional contributions \n", + "arise (the $A$ and $C$ terms), in analogy with the closely\n", + "related phenomenon - magnetic circular dichroism, see below.\n", + "\n", + "Unit conversion factors for the Verdet constant are: 1 a.u. (rad$\\ e\\ a_0$\\ $\\hbar^{-1}$) = \n", + "$8.03961 \\times 10^{4}$ rad T$^{-1}$ m$^{-1}$ (SI) = 2.763816 $\\times 10^{2}$ min G$^{-1}$\n", + "cm$^{-1}$ (CGS)." + ] + }, + { + "cell_type": "markdown", + "id": "ad8ef4b0-e3dd-4994-9c08-a5806404edfa", + "metadata": {}, + "source": [ + "## Hypermagnetizabilities, Magnetic double refraction (Cotton-Mouton effect)" + ] + }, + { + "cell_type": "markdown", + "id": "9e55fe9b-0cd8-47f4-99f2-540fec984d93", + "metadata": {}, + "source": [ + "The *Cotton-Mouton effect* (CME) is a magnetic induction \n", + "field-induced linear birefringence observed in the presence of a \n", + "strong magnetic induction field $\\mathbf{B}$ with a component \n", + "perpendicular to the direction of propagation of the beam. It is \n", + "mainly due to the tendency of both the electric field associated \n", + "with the light beam and the magnetic induction field to align the \n", + "molecules exhibiting both an anisotropic electric dipole \n", + "polarizability and an anisotropic magnetizability tensor \n", + "(according to the Langevin-Born theory). The\n", + "rearrangement of the electron density plays nevertheless a part through the \n", + "mixed electric and magnetic\n", + "hyperpolarizabilities, or hypermagnetizabilities.\n", + "\n", + "The experimental quantity connected to the hypermagnetizability is \n", + "the molar Cotton-Mouton constant $_mC$. Semiclassically, for \n", + "fluids composed of\n", + "closed-shell systems\n", + "\n", + "$$\n", + "_mC = \\frac{2\\pi N_A}{27}\\left[\\Delta\\eta +\n", + "Q\\left(T\\right)\\right]\n", + "$$\n", + "\n", + "where $\\Delta\\eta$ is the hypermagnetizability anisotropy\n", + "\n", + "\\begin{eqnarray}\n", + "\\Delta\\eta = \\frac{1}{15}\\left(3\\eta_{\\alpha\\beta,\\alpha\\beta} -\n", + "\\eta_{\\alpha\\alpha,\\beta\\beta}\\right),\n", + "\\end{eqnarray}\n", + "\n", + "and\n", + "\n", + "\\begin{eqnarray}\n", + "Q\\left(T\\right) =\n", + "\\frac{1}{15kT}\\left(3\\alpha_{\\alpha\\beta}\\xi_{\\alpha\\beta} -\n", + "\\alpha_{\\alpha\\alpha}\\xi_{\\beta\\beta}\\right),\n", + "\\end{eqnarray}\n", + "\n", + "where $\\alpha$ is the (frequency-dependent) \n", + "electric polarizability tensor, and $\\xi$ the magnetizability \n", + "tensor. The two contributions to the molar Cotton--Mouton constant \n", + "are therefore a temperature-independent term \n", + "related to the hypermagnetizability \n", + "$\\eta_{\\alpha\\beta,\\gamma\\delta}$ and a temperature-dependent \n", + "molecular orientational part, the *Langevin term*.\n", + "\n", + "When the magnetizability is expressed as a power series in the perturbing\n", + "electric field\n", + "\n", + "$$\n", + "{\\xi}_{\\alpha\\beta}\\left( \\mathbf{F} \\right) = \\xi_{\\alpha\\beta} +\n", + "\\zeta_{\\gamma,\\alpha\\beta} F_\\gamma + \\frac{1}{2} \\eta_{\\gamma\\delta,\\alpha\\beta} F_\\gamma\n", + "F_\\delta + ...,\n", + "$$\n", + "\n", + "the expansion coefficients $\\zeta_{\\gamma,\\alpha\\beta}$ and \n", + "$\\eta_{\\gamma\\delta,\\alpha\\beta}$ define the molecular first and second \n", + "hypermagnetizability. The first indices of $\\zeta$ and $\\eta$ \n", + "refer to the electric field and the last two to the magnetic \n", + "field. In fluids, the experimentally measured property is related \n", + "to the anisotropy of $\\eta$. From the expansion above it can be\n", + "seen that the static hypermagnetizability $\\eta$ corresponds to \n", + "the fourth derivative of the molecular energy\n", + "\n", + "$$\n", + "\\eta_{\\alpha\\beta,\\gamma\\delta} = -\\left. \\frac{\\partial^{4} E\\left( {{\\mathbf F}},{{\\mathbf B}}\\right)}\n", + "{\\partial F_\\alpha\n", + "\\partial F_\\beta\n", + "\\partial B_\\gamma\n", + "\\partial B_\\delta}\n", + "\\right|_{{{\\mathbf F}},{{\\mathbf B}} = {\\bf 0}}\n", + "$$\n", + "\n", + "It can also be expressed in terms of $\\xi \\left( \\mathbf{F} \\right)$, the magnetizability dependent on ${\\mathbf{F}}$\n", + "\n", + "$$\n", + "\\eta_{\\alpha\\beta,\\gamma\\delta} = \\frac{\\partial^2 \\xi_{\\gamma\\delta}}{\\partial F_\\alpha\n", + "\\partial F_\\beta} \\Big|_{{\\mathbf{F}}=0}\n", + "$$\n", + "\n", + "or $\\alpha \\left( {\\mathbf{B}} \\right)$, the polarizability dependent on ${\\mathbf{B}}$\n", + "\n", + "$$\n", + "\\eta_{\\alpha\\beta,\\gamma\\delta} = \\frac{ \\partial^2 \\alpha(-\\omega;\\omega)_{\\alpha\\beta}}{\\partial B_\\gamma \\partial B_\\delta}\n", + "\\Big|_{ {\\mathbf{B}} =0 }\n", + "$$\n", + "\n", + "and we have indicated in the latter equation that it holds also for the frequency-dependent polarizability.\n", + "\n", + "The hypermagnetizability is given as a sum of paramagnetic and diamagnetic\n", + "contributions\n", + "\n", + "\\begin{eqnarray}\n", + "\\eta_{\\alpha\\beta,\\gamma\\delta} & = & \\eta_{\\alpha\\beta,\\gamma\\delta}^{\\rm para}(-\\omega;\\omega,0,0)+\\eta_{\\alpha\\beta,\\gamma\\delta}^{\\rm dia}(-\\omega;\\omega,0)\n", + "\\end{eqnarray}\n", + "\n", + "corresponding to a sum of a cubic response and a quadratic response\\index function\n", + "\n", + "\\begin{eqnarray}\n", + " \\eta_{\\alpha\\beta,\\gamma\\delta}^{\\rm para}(-\\omega;\\omega,0,0) & = & -\\langle\\langle \\hat{\\mu}_\\alpha;\\hat{\\mu}_\\beta,\\hat{m}_\\gamma,\n", + "\\hat{m}_\\delta \\rangle\\rangle_{\\omega,0,0}, \\\\\n", + " \\eta_{\\alpha\\beta,\\gamma\\delta}^{\\rm dia}(-\\omega;\\omega,0) & = & \\langle\\langle \\hat{\\mu}_\\alpha;\\hat{\\mu}_\\beta,\\hat{\\xi}^{{\\rm B,B}}_{\\gamma\\delta}\n", + "\\rangle\\rangle_{\\omega,0}\n", + "\\end{eqnarray}\n", + "\n", + "where the *diamagnetic magnetizability operator* in the second term is\n", + "\n", + "$$\n", + "\\hat{\\xi}^{{\\rm B,B}}_{\\alpha \\beta} = -2 {\\hat{H}^{{\\rm B,B}}}_{\\alpha \\beta}\n", + "$$\n", + "\n", + "We recall here that the magnetic dipole\n", + "moment and \n", + "diamagnetic magnetizability operators, and thus \n", + "the partitioning of the hypermagnetizability into para- and \n", + "diamagnetic components, may depend on the chosen gauge origin.\n", + "\n", + "Finite field approaches were used in studies of magnetic double rifrangence. In these\n", + "calculations $\\eta_{\\alpha\\beta,\\gamma\\delta}$ is obtained combining the available\n", + "analytic linear response methods with the numerical finite \n", + "difference approach. The frequency-dependent electric dipole \n", + "polarizability\n", + "can be computed in the presence of a magnetic \n", + "induction field, and the hypermagnetizability is obtained by \n", + "numerical differentiation. This approach in \n", + "general requires complex algebra as the perturbation is purely \n", + "imaginary, whereas most electronic structure codes use real wave \n", + "functions. In the more often used finite field approach \n", + "one computes the magnetizability $\\xi$ with and \n", + "without an external electric field perturbation. However, in this \n", + "approach only the static value of the required \n", + "hypermagnetizability is obtained. Analytic calculations of the\n", + "quadratic and cubic responses also\n", + "including London orbitals to ensure gauge origin independence can\n", + "nowadays be carried out. The effect of\n", + "the London orbitals on improving the basis set convergence appears\n", + "to be small.\n", + "\n", + "For atoms, choosing the origin of the gauge at the nucleus \n", + "automatically makes the dependence on the gauge vanish, and there \n", + "is no advantage in using LAOs.\n", + "Moreover, the evaluation of $\\Delta\\eta$ may be reduced to the \n", + "calculation of the Cauchy moments $S(-2n)$ and of the \n", + "dipole-dipole-quadrupole hyperpolarizability frequency dispersion\n", + "coefficients.\n", + "\n", + "Unit conversion factors for hypermagnetizability $\\eta$: 1 a.u. ($(e a_0^{2})^2 (e \\hbar/{{m_{\\mathrm e}}})^{2} {E_{\\rm h}}^{-3})$ = 2.98425 $\\times 10^{-52}$ C$^2$ m$^2$ J$^{-1}$ T$^{-2}$ (SI) = 2.68211 $\\times 10^{-44}$ cm$^3$ G$^{-2}$ (CGS)." + ] + }, + { + "cell_type": "markdown", + "id": "db85cff3-84d4-4500-abf9-b09bc287e3a7", + "metadata": {}, + "source": [ + "## Electric-Field-Gradient-Induced Birefringence (Buckingham birefringence)" + ] + }, + { + "cell_type": "markdown", + "id": "8c4487b9-10a0-4db2-94cc-f0fa8082257f", + "metadata": {}, + "source": [ + "As a magnetic induction field can give rise to an optical\n", + "anisotropy through its interaction with the anisotropic\n", + "magnetizability of a molecule, so can an electric field gradient,\n", + "through its coupling with a permanent molecular quadrupole\n", + "moment.\n", + "The corresponding birefringence is \n", + "commonly called *electric-field-gradient-induced \n", + "birefringence*, more recently *Buckingham\n", + "birefringence*,\n", + "and it exhibits a linear dependence on the strength of the \n", + "electric field gradient. Besides the orientational \n", + "Langevin-Born-type effect, acting on quadrupolar \n", + "molecules and due to both the electric field associated with the \n", + "light wave and to the static external field gradient, it also \n", + "involves the rearrangement of the electron density as a consequence of \n", + "the electron interaction with the electric and magnetic wave vectors \n", + "associated with the beam and with the externally applied electric \n", + "field gradient. In this case mixed electric dipole,\n", + "electric quadrupole and magnetic dipole\n", + "hyperpolarizabilities\n", + "play their role in determining the strength of the effect.\n", + "Buckingham birefringence has been often employed for the\n", + "determination of molecular quadrupole\n", + "moments.\n", + "\n", + "For an ideal gas at constant pressure, for light along the $Z$\n", + "axis\n", + "\n", + "\\begin{eqnarray}\n", + "\\Delta n & = & n_X - n_Y = \\frac{N_A \\nabla F}{15 V_m \\epsilon_0}\n", + "s = \\frac{3 \\nabla F}{2 V_m} {_mQ}(\\omega,T), \\\\\n", + "_mQ(\\omega,T) & = & \\frac{2 N_A}{45 \\epsilon_0} s\n", + "\\end{eqnarray}\n", + "\n", + "where $V_m$ is the molar volume,\n", + "$\\nabla F$ is the external electric field gradient, arranged so that $\\nabla F\n", + "=\\nabla F_{XX} =-\\nabla F_{YY}$ and $\\nabla F_{ZZ}=0$.\n", + "T he above equation introduces the Buckingham constant $_mQ$, which depends \n", + "through the quantity $s$ on the\n", + "circular frequency of the light and on the temperature. For non-dipolar\n", + "systems, where the quadrupole moment does not depend on the choice\n", + "of origin,\n", + "\n", + "$$\n", + "s = b\\left(\\omega\\right) +\n", + "\\frac{1}{kT}\\Theta_{\\alpha\\beta}\\alpha_{\\alpha\\beta}\\left(-\\omega;\\omega\\right),\n", + "$$\n", + "\n", + "involving the molecular quadrupole moment $\\Theta_{\\alpha\\beta}$ and\n", + "the frequency-dependent electric dipole\n", + "polarizability\n", + "${{\\alpha}_{\\alpha\\beta}}(-\\omega;\\omega)$. The temperature-independent\n", + "contribution is\n", + "\n", + "$$\n", + "b\\left(\\omega\\right) = B_{\\alpha\\beta,\\alpha\\beta}\\left(-\\omega;\\omega,0\\right) - \\mathcal{B}_{\\alpha,\\alpha\\beta,\\beta}\n", + "\\left(-\\omega;\\omega,0\\right) - \\frac{5}{\\omega}\n", + "\\varepsilon_{\\alpha\\beta\\gamma}\n", + "J^\\prime_{\\alpha,\\beta,\\gamma}\\left(-\\omega;\\omega,0\\right)\n", + "$$\n", + "\n", + "and is a combination of three mixed\n", + "hyperpolarizabilities\n", + "\n", + "\\begin{eqnarray}\n", + "B_{\\alpha\\beta,\\gamma\\delta}\\left(-\\omega;\\omega,0\\right) &=&\n", + "\\langle\\langle\\hat{\\mu}_{\\alpha};\\hat{\\mu}_\\beta,\\hat{\\Theta}_{\\gamma\\delta}\\rangle\\rangle_{\\omega,0},\\\\\n", + "\\mathcal{B}_{\\alpha,\\beta\\gamma,\\delta}\\left(-\\omega;\\omega,0\\right)\n", + "&=&\n", + "\\langle\\langle\\hat{\\mu}_{\\alpha};\\hat{\\Theta}_{\\beta\\gamma},\\hat{\\mu}_\\delta\\rangle\\rangle_{\\omega,0},\\\\\n", + "J'_{\\alpha,\\beta,\\gamma}\\left(-\\omega;\\omega,0\\right) &=& \n", + "i \\langle\\langle\\hat{\\mu}_{\\alpha};\\hat{m}_{\\beta},\\hat{\\mu}_\\gamma\n", + "\\rangle\\rangle_{\\omega,0}.\n", + "\\end{eqnarray}\n", + "\n", + "The expressions are somewhat more complex for dipolar\n", + "fluids, where the quadrupole moment becomes origin\n", + "dependent. The formal expression for $s$ does\n", + "not change provided that we refer to a frequency-dependent origin\n", + "for the quadrupole operator, commonly labelled as the *effective\n", + "quadrupole center*. \n", + "For any other choice of the origin we have\n", + "\n", + "$$\n", + "s = b\\left(\\omega\\right) + \\frac{1}{kT}\\left\\{\\Theta_{\\alpha\\beta}\\alpha_{\\alpha\\beta}\\left(-\\omega;\\omega\\right)-\\mu_\\alpha\\left[A_{\\beta,\\alpha\\beta}\\left(-\\omega;\\omega\\right) + \\frac{5}{\\omega}\n", + "\\varepsilon_{\\alpha\\beta\\gamma}\n", + "G^\\prime_{\\beta\\gamma}\\left(-\\omega;\\omega\\right)\\right]\\right\\}.\n", + "$$\n", + "\n", + "The effective\n", + "quadrupole center is defined as the point in\n", + "space where the last term \n", + "in this equation, involving \n", + "the molecular dipole moment $\\mu_\\alpha$\n", + "and the mixed polarizabilities\n", + "%\n", + "$A_{\\alpha,\\beta\\gamma}\\left(-\\omega;\\omega\\right)$\n", + "and\n", + "$G^\\prime_{\\alpha\\beta}\\left(-\\omega;\\omega\\right)$,\n", + "vanishes." + ] + }, + { + "cell_type": "markdown", + "id": "4a587a43-074d-4494-b6e0-c27c74486ae9", + "metadata": {}, + "source": [ + "## Magnetic circular dichroism" + ] + }, + { + "cell_type": "markdown", + "id": "ed63c442-5ce9-4485-bc4f-2ebe3f9fe38d", + "metadata": {}, + "source": [ + "For a medium that is isotropic in the absence of magnetic fields, \n", + "the Faraday $A$, $B$ and $C$ terms determining the MCD of a transition from \n", + "the electronic state $|0\\rangle$ to the electronic state \n", + "$|n\\rangle$ are defined as\n", + " \n", + "\\begin{eqnarray*}\n", + "A \\left( 0 \\rightarrow n \\right) &=&\n", + " \\frac{1}{2} \\left[ \\left< n \\left| \\mathbf{m} \\right| n \\right> -\n", + "\\left< 0 \\left| \\mathbf{m} \\right| 0 \\right> \\right] \\cdot {\\mathcal Im} \\Big( \\left<\n", + "0 \\left| \\boldsymbol{\\mu} \\right| n \\right> \\times \\left< n \\left|\n", + "{\\boldsymbol{\\mu}} \\right| 0 \\right> \\Big), \n", + "\\\\\n", + "B \\left(0 \\rightarrow n \\right) &=&\n", + " {\\mathcal Im} \\Big\\{ \\sum_{k \\neq 0} \\frac{ \\left< k | \\mathbf{m}| 0 \\right>}\n", + "{\\omega_{k0}} \\cdot \\left< 0 | \\boldsymbol{\\mu} |n \\right> \\times\n", + "\\left< n | \\boldsymbol{\\mu} | k \\right> + \\\\\n", + "&\\ & \\sum_{k \\neq n}\n", + "\\frac{\\left}\n", + "{\\omega_{kn}}\\cdot\\left<0|\\boldsymbol{\\mu}|n\\right>\\times\n", + "\\left\\Big\\},\n", + "\\\\\n", + "C \\left(0\\rightarrow n\\right) &=&\n", + "\\frac{1}{2} \\left< 0 \\left|\\mathbf{m}\\right|0\\right> \\cdot\n", + "{\\mathcal Im} \\Big(\\left< 0\\left|\\boldsymbol{\\mu}\\right|n\\right> \\times\n", + "\\left \\Big),\n", + "\\end{eqnarray*}\n", + "\n", + "where an average over the molecular orientations has been made. \n", + "These tensors correspond to the single residues of the tensors \n", + "that determine the Verdet constant of the\n", + "Faraday effect.\n", + "The $A$ term exists only if either $|0\\rangle$ or $|n\\rangle$ is\n", + "degenerate, whereas the $C$ term is non-vanishing only if\n", + "$|0\\rangle$ is degenerate. \n", + "\n", + "The $B$ terms exist in all cases, independently of the appearance of any\n", + " ground- and excited-state degeneracies. \n", + "We can rearrange the equation for $B$ above\n", + "recalling that $(\\hat{\\mu}_{\\alpha})_{lm}^{\\ast} = (\\hat{\\mu}_{\\alpha})_{lm}$, \n", + "$(\\hat{m}_{\\alpha})_{lm}^{\\ast} = - (\\hat{m}_{\\alpha})_{lm}$\n", + "and $ {\\mathcal Im} (z) = -i (z - z^{\\ast})/2$,\n", + "and we find that\n", + "the $B(0 \\to n)$ term can be written in terms of a single residue of a\n", + "quadratic response function\n", + "as\n", + "\n", + "\\begin{eqnarray}\n", + "{B \\left( 0\\rightarrow n \\right) = }\n", + "i {\\varepsilon}_{\\alpha \\beta \\gamma} \n", + " \\Big( \\lim_{\\omega \\rightarrow \\omega_{n0}} \\hbar (\\omega - \\omega_{n0})\n", + "\\langle\\langle \\hat{\\mu}_{\\gamma};\\hat{m}_{\\beta},\\hat{\\mu}_{\\alpha}\n", + "\\rangle\\rangle_{0, \\omega} \\Big).\n", + "\\end{eqnarray}\n", + "\n", + "By analogy to the spectral representation expression for the \n", + "single residue we may also write \n", + "\n", + "$$\n", + "\\lim_{\\omega \\to \\omega_{n0}} \\hbar (\\omega-\\omega_{n0})\n", + "\\langle\\langle \\hat{\\mu}_{\\gamma}; \\hat{m}_{\\beta}, \\hat{\\mu}_{\\alpha} \\rangle\\rangle_{0,\\omega}\n", + "= M_{\\mu_\\alpha}^{0 \\gets n} M_{m_\\beta \\mu_\\gamma}^{n\\gets 0}(0),\n", + "$$\n", + "\n", + "where $M_{\\mu_\\alpha}^{0\\gets n}$ and $M_{m_\\beta \\mu_\\gamma}^{n\\gets 0}(0)$ \n", + "indicate specific one- and (formally) two-photon\n", + "transition matrix elements \n", + "between state $|0\\rangle$ and state $|n\\rangle$, respectively.\n", + "\n", + "Both the $A$ and $B$ \n", + "terms come out directly from calculations using the complex polarization propagator approach {cite}`hskrscpnjcp128`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f5a718a6-31b9-4131-ac93-e15897862113", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.8" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/_sources/docs/spec_prop/opa.ipynb b/_sources/docs/spec_prop/opa.ipynb index 2ca833d15..7ff89d156 100644 --- a/_sources/docs/spec_prop/opa.ipynb +++ b/_sources/docs/spec_prop/opa.ipynb @@ -2,19 +2,153 @@ "cells": [ { "cell_type": "markdown", - "id": "04c0a70d-f428-4696-a635-df3ee4b90aac", - "metadata": { - "editable": true, - "slideshow": { - "slide_type": "" - }, - "tags": [ - "remove-output" - ] - }, + "id": "ccfc1fdf-0e8b-4332-9d0f-cb17c3c63d12", + "metadata": {}, + "source": [ + "# One-photon absorption" + ] + }, + { + "cell_type": "markdown", + "id": "3bf09d83-93f1-4985-bc48-7cb65db37297", + "metadata": {}, "source": [ - "# UV/vis absorption" + "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)." ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d523be49-c860-4c11-8d67-52b88b5b6f32", + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { @@ -33,7 +167,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.12" + "version": "3.11.7" } }, "nbformat": 4, diff --git a/_sources/docs/spec_prop/birefring.ipynb b/_sources/docs/spec_prop/optact.ipynb similarity index 77% rename from _sources/docs/spec_prop/birefring.ipynb rename to _sources/docs/spec_prop/optact.ipynb index 4734a4e0d..4ddaa0bbb 100644 --- a/_sources/docs/spec_prop/birefring.ipynb +++ b/_sources/docs/spec_prop/optact.ipynb @@ -2,23 +2,31 @@ "cells": [ { "cell_type": "markdown", - "id": "349b6c4d-ddbd-4f51-844e-2dfbf626cbde", - "metadata": { - "editable": true, - "slideshow": { - "slide_type": "" - }, - "tags": [ - "remove-output" - ] - }, + "id": "2489418e-5e35-417c-b806-abc4e278d3fc", + "metadata": {}, + "source": [ + "# Optical Activity" + ] + }, + { + "cell_type": "markdown", + "id": "aea3daab-e9d9-49ba-a358-22bf1e02b152", + "metadata": {}, "source": [ - "# Birefringences and dichroisms" + "Optical activity is the ability of a substance to rotate the plane of polarization of a light beam that traverses it. It is usually expressed in terms of the so-called specific rotation, via an equation relating the angle through which the plane is rotated, the length of the light path through the sample, and the density of the sample, or its concentration for samples in solution. The specific rotation depends upon the temperature and upon the wavelength of the light. Conventionally, the rotation is assigned a positive value if it is clockwise with respect to an observer facing the light source, negative if counterclockwise. A substance displaying a positive specific rotation is described as dextrorotatory and denoted by the prefix $d$ or $(+)$; one with a negative specific rotation is levorotatory, designated by the prefix $l$ or $(-)$. " ] }, { "cell_type": "markdown", - "id": "dbc9eb5d-2031-4607-969b-20c501481643", + "id": "a61af5fc-be65-47f2-ac57-ed1ea520e18a", + "metadata": {}, + "source": [ + "## Birefringences and dichroisms" + ] + }, + { + "cell_type": "markdown", + "id": "453ec69a-1d35-4a17-9a01-130593c386d8", "metadata": {}, "source": [ "Light experienced in everyday life is mostly non-polarized - that is, the\n", @@ -59,7 +67,18 @@ "Note that only molecules possessing no improper or\n", "rotation-reflection axes may be chiral.\n", "\n", - "# Linear birefringence\n", + "In the following we discuss some general issues with specific reference to birefringences.\n", + "The discussion applies also to dichroisms, with the sole difference that refrective indices ($n$, the real part of the general complex refraction index) should ve substituted with extinction coefficients ($\\epsilon$, the imaginari contribution to the complex refraction index). " + ] + }, + { + "cell_type": "markdown", + "id": "1bb69193-aee6-498a-99ec-45a754328457", + "metadata": { + "jp-MarkdownHeadingCollapsed": true + }, + "source": [ + "## Linear birefringence\n", "\n", "Let us consider first an isotropic \n", "liquid sample subject to a strong external static electric \n", @@ -74,7 +93,7 @@ "*linear birefringence*,\n", "\n", "$$\n", - "\\Delta n^{\\rm lin} = n_\\| - n_\\bot,\n", + "\\Delta n^{\\rm lin} = n_\\| - n_\\bot\n", "$$\n", "\n", "and the effect is known as *electric-field-induced optical \n", @@ -97,12 +116,20 @@ "by the first electric-dipole hyperpolarizability\n", "$\\beta$.\n", "Other examples of linear\n", - "birefringences are the *Cotton-Mouton*, the *Buckingham effects, the *Jones* and\n", + "birefringences are the *Cotton-Mouton*, the *Buckingham effects*, the *Jones* and\n", "*magneto-electric* birefringences, the latter occurring when linearly\n", "polarized light goes through an isotropic fluid perpendicularly to\n", - "both electric and magnetic static fields.\n", - "\n", - "# Circular birefringence\n", + "both electric and magnetic static fields." + ] + }, + { + "cell_type": "markdown", + "id": "d4f9174d-65b7-4f04-840c-3a58346dcb8b", + "metadata": { + "jp-MarkdownHeadingCollapsed": true + }, + "source": [ + "## Circular birefringence\n", "\n", "*Circular birefringence* is observed when the two circular components of\n", "a linearly polarized beam propagate with different circular\n", @@ -110,7 +137,7 @@ "refractive indices $n_+$ and $n_-$:\n", "\n", "$$\n", - "\\Delta n^{\\rm circ} = n_+ - n_-.\n", + "\\Delta n^{\\rm circ} = n_+ - n_-\n", "$$\n", "\n", "The net result is a rotation of the plane of polarization. An \n", @@ -123,9 +150,15 @@ "dependence of the corresponding anisotropy on the wavelength \n", "is described by the \n", "*optical rotatory dispersion* (ORD). The *Faraday effect* is another \n", - "well-known example of a circular birefringence.\n", - "\n", - "# Axial birefringence\n", + "well-known example of a circular birefringence." + ] + }, + { + "cell_type": "markdown", + "id": "71ac1234-4406-4932-b5d0-77aa5786eaf6", + "metadata": {}, + "source": [ + "## Axial birefringence\n", "\n", "An *axial birefringence* occurs with *unpolarized*\n", "light, and an example is the so-called *magnetochiral\n", @@ -139,11 +172,19 @@ "to the field, $n_{\\uparrow\\downarrow}$\n", "\n", "$$\n", - "\\Delta n^{\\rm ax} = n_{\\uparrow\\uparrow} - n_{\\uparrow\\downarrow}.\n", + "\\Delta n^{\\rm ax} = n_{\\uparrow\\uparrow} - n_{\\uparrow\\downarrow}\n", "$$\n", "\n", "The corresponding anisotropy has a different sign for the different\n", - "enantiomers.\n", + "enantiomers." + ] + }, + { + "cell_type": "markdown", + "id": "0193a2a1-c129-4253-a023-b8282b66cf0c", + "metadata": {}, + "source": [ + "## Mechanisms governing the effects\n", "\n", "An external field will have an orientational effect on the\n", "molecules of the sample, and this is a mechanism responsible for\n", @@ -154,7 +195,7 @@ "pressure) is usually of the form\n", "\n", "$$\n", - "\\Delta n \\propto A_0 + \\frac{A_1}{T} + \\frac{A_2}{T^2} + \\cdots,\n", + "\\Delta n \\propto A_0 + \\frac{A_1}{T} + \\frac{A_2}{T^2} + \\cdots\n", "$$\n", "\n", "where the term $A_0$ includes all contributions arising from the\n", @@ -181,16 +222,8 @@ "vanishes for dynamic inducing fields, where the birefringence\n", "arises only from the rearrangement of the anisotropy of the\n", "electric dipole polarizability. The contribution resulting from\n", - "the interaction with permanent multipoles averages over time to zero.\n" + "the interaction with permanent multipoles averages over time to zero." ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "061503b1-0397-4b39-a551-42173f9e093e", - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { @@ -209,7 +242,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.5" + "version": "3.11.7" } }, "nbformat": 4, diff --git a/_sources/docs/spec_prop/or.ipynb b/_sources/docs/spec_prop/or.ipynb new file mode 100644 index 000000000..ed8aa3cc3 --- /dev/null +++ b/_sources/docs/spec_prop/or.ipynb @@ -0,0 +1,234 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "6ae4a636-d983-4dad-809f-aba89b805d0d", + "metadata": {}, + "source": [ + "# Optical Rotation" + ] + }, + { + "cell_type": "markdown", + "id": "555e449b-97d9-4a9e-ad31-dab636850311", + "metadata": {}, + "source": [ + "## Linear Optical Rotation\n", + "\n", + "The first contribution to the induced electric dipole\n", + "moment arising from the time dependence of the magnetic field,\n", + "${\\omega^{-1}}G'$, gives rise to two different observable \n", + "properties. In the dispersive region, it determines the\n", + "optical rotatory power, optical rotation (OR) for short, whereas\n", + "in the absorptive region it determines the rotational strength\n", + "observed in electronic circular dichroism (ECD).\n", + "\n", + "Since there are no terms bilinear in the electric and magnetic\n", + "fields in the Hamiltonian, $G^{\\prime}$, the mixed electric\n", + "dipole--magnetic dipole polarizability, \n", + "can be expressed as a linear response\n", + "function\n", + "\n", + "$$\n", + "G^{\\prime}_{\\alpha\\beta}(-\\omega;\\omega)= -\\frac{2\\omega}{\\hbar}\n", + "\\sum_{n \\neq 0} \n", + "{{\\mathcal Im}} \\frac{\\left< 0 \\mid\n", + "{\\hat{\\mu}}_{\\alpha} \\mid n\\right>\\left}{ \\omega_{n}^2-\\omega^2} =\n", + "-{\\mathcal Im}\\left<\\left<\\hat{\\mu}_{\\alpha};\\hat{m}_{\\beta}\\right>\\right>_\\omega.\n", + "$$\n", + "\n", + "$G^{\\prime}$ vanishes if the electromagnetic field is static ($\\omega=0$). \n", + "\n", + "Measurements of optical rotation are almost exclusively carried out on liquid samples \n", + "(though gas-phase measurements is also \n", + "possible {cite}`tmkbwphvjpca104` and we will therefore be \n", + "concerned primarily with the rotational average of the \n", + "$G^{\\prime}$ tensor, which is described by the \n", + "quantity $^{\\rm ORD}\\beta$ defined as\n", + "\n", + "$$\n", + "^{\\rm ORD}\\beta = -\\frac{1}{3\\omega}G^{\\prime}_{\\alpha\\alpha }.\n", + "$$\n", + "\n", + "## Experimental issues\n", + "\n", + "The most common experimental set-up involves the determination of the\n", + "rotation of plane-polarized light as it passes through\n", + "a sample in which there is an excess of one enantiomer. \n", + "The standard optical rotation $\\left[\\alpha\\right]_D^{25}$,\n", + "proportional to $^{\\rm ORD}\\beta$, is \n", + "reported for light with a frequency corresponding to the \n", + "sodium D-line (589.3~nm) at a temperature of $25^\\circ{\\rm C}$.\n", + "\n", + "The dispersion of the optical rotation was for a long time also\n", + "the focus of much experimental attention through optical rotatory\n", + "dispersion measurements. Even after it became customary to\n", + "restrict the optical rotation measurements to a single frequency,\n", + "ORD served as an important tool for determining excitation\n", + "energies in chiral molecules, although it has now been surpassed\n", + "by electronic circular dichroism for these purposes.\n", + "\n", + "At the sodium frequency, the optical rotation is rather small for\n", + "most molecules. However, the individual diagonal elements of the\n", + "mixed electric dipole-magnetic dipole polarizability may be\n", + "fairly large in absolute value, often cancelling each other out in\n", + "the trace. Consequently, \n", + "the optical rotation is highly sensitive to numerical errors in\n", + "the tensor components, because small residual errors in the individual\n", + "tensor components, arising from the solution of the linear\n", + "response equations, may lead to substantial errors in \n", + "$^{\\mathrm {ORD}}\\beta$. Another consequence\n", + "of this cancellation is that $^{\\mathrm {ORD}}\\beta$ \n", + "is very \n", + "sensitive to the choice of molecular geometry, as well as the \n", + "zero-point vibrational effects {cite}`krprtpoacpl337`, since the small changes introduced \n", + "by these effects in the molecular charge distribution, and thus on \n", + "the different diagonal elements of $G^{\\prime}$, can give rather large\n", + "effects on $^{\\mathrm {ORD}}\\beta$.\n", + "\n", + "## Units\n", + " \n", + "Unit conversion factors for electric dipole-magnetic dipole\n", + "polarizability $G^{\\prime}$ :\n", + "\n", + "1 a.u. ($e^{2} a_0^{3} {\\hbar}^{-1}$) equals $3.60702\\times 10^{-35}$ C$^2$m$^3$J$^{-1}$s$^{-1}$ (SI) and $1.08136\\times 10^{-27}$ Fr cm G$^{-1}$ (CGS)\n", + "\n", + "## Gauge Origin Dependence\n", + "\n", + "Since the mixed electric dipole-magnetic dipole polarizability\n", + "involves the magnetic dipole\n", + "operator, in approximate calculations \n", + "$G^{\\prime}$ carries an origin dependence. Indeed, the \n", + "individual tensor elements of $G^{\\prime}$ are origin dependent. \n", + "The trace of $G^{\\prime}$ must be origin independent, \n", + "since the optical rotation is an experimental observable. \n", + "In non-isotropic \n", + "media, contributions to the optical rotation tensor arise from the\n", + "mixed electric dipole-electric quadrupole polarizability\n", + "$A$,\n", + "\n", + "$$\n", + "A_{\\alpha,\\beta\\gamma}\\left(-\\omega;\\omega\\right) =\n", + "-\\langle\\langle\\hat{\\mu}_\\alpha;\\hat{\\Theta}_{\\beta\\gamma}\\rangle\\rangle_\\omega, \n", + "$$\n", + "\n", + "and it is the combination of the $G^{\\prime}$ and $A$\n", + "contributions that is gauge-origin independent for exact wave\n", + "functions. The contribution from $A$ vanishes in isotropic\n", + "samples, since this mixed electric dipole-electric quadrupole \n", + "polarizability is traceless.\n", + "\n", + "For approximate variational wave functions, the origin\n", + "independence of the trace of $G^{\\prime}$ is only achieved in the\n", + "limit of a complete basis set. One way to overcome this problem is\n", + "to introduce local gauge origins by using London atomic\n", + "orbitals {cite}`thkrklbpjjofd99`.\n", + "\n", + "## Conformationally flexible molecules\n", + "\n", + "The sensitivity of the optical rotation is perhaps most clearly\n", + "illustrated in the case of conformationally flexible \n", + "molecules that is molecules that have a significant population \n", + "of multiple stable minima. In this case different molecular conformations \n", + "can have optical rotations that differ by orders of magnitude and \n", + "even in sign. A thorough conformational search is therefore \n", + "mandatory before the absolute sign of the optical rotation (or any \n", + "birefringence or dichroism for that matter) is determined from \n", + "theoretical calculations for conformationally flexible molecules. \n", + "The optical rotation of the flexible molecule can then be \n", + "determined by Boltzmann averaging over the dominant molecular \n", + "conformations.\n", + "\n", + "## Basis sets for optical rotation\n", + "\n", + "In order to ensure reasonably well-converged results for the \n", + "optical rotation, basis sets of polarized valence double-zeta \n", + "quality are required, \n", + "and sets such as the aug-cc-pVDZ of Woon and Dunning {cite}`dewthdjcp100`, \n", + "or the polarized triple-zeta basis of Sadlej {cite}`ajscccc53` have been \n", + "shown to perform well for calculations of optical rotation. Most \n", + "importantly, diffuse $p$ functions in the outer regions of the \n", + "electron density, which in most cases is described by the electron \n", + "density of hydrogen atoms, are required to ensure qualitatively \n", + "correct results {cite}`gzwhjpca108`.\n", + "\n", + "## The effects of electron correlation\n", + "\n", + "Since the optical rotation is very sensitive even to small changes\n", + "in the electron density,\n", + "electron correlation effects should also be taken into account in\n", + "order to get accurate results. Due to the fact that chiral\n", + "molecules in general have very low symmetry, if any symmetry at\n", + "all, the only viable approach for calculating electron correlation\n", + "effects in chiral molecules is currently density functional\n", + "theory {cite}`pjsfjdjrcmjfjpca105,krpjsfjdprtjrcmjfcpl373`.\n", + "Methods such as MP2 and CCSD could also be used to\n", + "describe electron correlation, but due to their non-variational\n", + "nature it is not ensured that gauge-origin independent results can\n", + "be obtained in the conventional length gauge formulation even in\n", + "the limit of a complete basis set. In principle, the use of the dipole\n", + "velocity gauge will ensure that the calculated results are\n", + "independent of the gauge origin. For all but the smallest basis\n", + "sets, the differences between the velocity and the length\n", + "gauges\n", + "are not very large, with the length gauge in general performing\n", + "better. However, in the case of the optical rotation an \n", + "additional complication arises: whereas $G^{\\prime}$ given by\n", + "the equation given at the beginning of this section \n", + "will vanish in the limit of a static field, this \n", + "is not the case for the dipole-velocity analogue of the optical \n", + "rotation. It has been suggested that the much slower basis set \n", + "convergence of the optical rotation compared to other properties \n", + "calculated using the velocity-gauge formulation can be improved by \n", + "subtracting the static-limit value of the corresponding response \n", + "function {cite}`tbphklbamjsmcpl393`, such that\n", + "\n", + "$$\n", + "G' \\propto \\langle\\langle \\hat{p}_\\alpha;\\hat{l}_\\beta \\rangle\\rangle_\\omega -\n", + "\\langle\\langle \\hat{p}_\\alpha;\\hat{l}_\\beta \\rangle\\rangle_0.\n", + "$$\n", + "\n", + "Since there are no empirical rules relating the stereochemistry of \n", + "a molecule to the observed sign of the optical rotation, an \n", + "important area of application for optical rotation calculations \n", + "would be the combined theoretical and experimental determination \n", + "of absolute configuration. However, this is a difficult task due \n", + "in part to the large variations with geometry in the optical \n", + "rotation of conformationally flexible molecules, and in part \n", + "because of the small magnitude of $^{\\mathrm {ORD}}\\beta$. These \n", + "factors make the sign of the optical rotation hard to determine \n", + "with confidence. One way to increase the predictive power of the \n", + "calculations is to change the frequency of the incident light to \n", + "shorter wavelengths. The magnitude of the optical rotation \n", + "increases dramatically as the frequency approaches that of a \n", + "resonance. Thus, if both theory and experiment could determine the \n", + "optical rotation at frequencies closer to electronic excitation\n", + "energies, \n", + "theoretical calculations could provide a much more reliable proof \n", + "of the absolute configuration of the molecule {cite}`egrgvrzcrjacs126`." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/_sources/docs/spec_prop/tpa.ipynb b/_sources/docs/spec_prop/tpa.ipynb index c4e2b532b..0e26139fd 100644 --- a/_sources/docs/spec_prop/tpa.ipynb +++ b/_sources/docs/spec_prop/tpa.ipynb @@ -5,6 +5,7 @@ "id": "dbae2d4c-6517-490f-8809-4e2033c17bf3", "metadata": { "editable": true, + "jp-MarkdownHeadingCollapsed": true, "slideshow": { "slide_type": "" }, @@ -155,7 +156,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.12" + "version": "3.11.5" } }, "nbformat": 4, diff --git a/_sources/docs/spec_prop/tpcd.ipynb b/_sources/docs/spec_prop/tpcd.ipynb new file mode 100644 index 000000000..23d350697 --- /dev/null +++ b/_sources/docs/spec_prop/tpcd.ipynb @@ -0,0 +1,140 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "000ed0cb-057c-4f19-aa9e-1921ffc80db0", + "metadata": {}, + "source": [ + "# Two-photon circular dichroism" + ] + }, + { + "cell_type": "markdown", + "id": "a3a52668-d863-454a-88ef-f212c0d80d92", + "metadata": {}, + "source": [ + "Two-photon circular dichroism {cite}`Tinoco` arises in chiral systems due to the differential\n", + "absorption of two photons, of which at least one is circularly polarized {cite}`Hernandez`. In\n", + "this sense it can be seen as the nonlinear extension of ECD. The\n", + "observable, the anisotropy of the two-photon transition strength,\n", + "is proportional to the two-photon rotatory\n", + "strength:\n", + "\n", + "\\begin{eqnarray}\n", + "^nR^{\\rm TPCD}(\\omega) & = & -b_1 [{\\mathcal{B}}_1(\\omega)]_{0n} - b_2\n", + "[{\\mathcal{B}}_2(\\omega)]_{0n} -b_3 [{\\mathcal{B}}_3(\\omega)]_{0n}\n", + "\\end{eqnarray}\n", + "\n", + "where $b_1$, $b_2$ and $b_3$ are numbers,\n", + "combinations of the analogous polarization and set-up-related coefficients \n", + "$F$, $G$ and $H$ given for\n", + "two-photon absorption. The molecule-related\n", + "parameters ${\\mathcal{B}}_1$, ${\\mathcal{B}}_2$ and\n", + "${\\mathcal{B}}_3$ take the form:\n", + "\n", + "\\begin{eqnarray}\n", + "[{\\mathcal{B}}_1 (\\omega)]_{0n} & = & \\frac{1}{\\omega ^3} \n", + "{\\mathcal{M}}_{\\rho \\sigma }^{{\\rm {p}},0n} (\\omega)\n", + "{\\mathcal{P}}_{\\rho \\sigma}^{{\\rm p}*,0n}(\\omega), \\\\\n", + "[{\\mathcal{B}}_2 (\\omega)] _{0n} & = & \\frac{1}{2 \\omega ^3} \n", + "{{\\mathcal{T}}_{\\rho \\sigma }^{ + ,0n}} (\\omega) {\\mathcal{P}}_{\\rho\n", + "\\sigma }^{{\\rm p}*,0n}(\\omega), \\\\\n", + "[{\\mathcal{B}}_3 (\\omega)] _{0n} & = & \\frac{1}{\\omega ^3} \n", + "{{\\mathcal{M}}_{\\rho \\rho }^{{\\rm p},0n}} (\\omega)\n", + "{\\mathcal{P}}_{\\sigma \\sigma}^{{\\rm p}*,0n}(\\omega),\n", + "\\end{eqnarray}\n", + "\n", + "and they are therefore appropriate contractions of generalized\n", + "two-photon second-rank tensors.\n", + "Indeed, these tensors are defined (for the general case of two\n", + "photons of different frequency) as follows:\n", + "\n", + "\\begin{eqnarray}\n", + "{\\mathcal{P}}_{\\alpha \\beta}^{{\\rm p},0n}(\\omega_1,\\omega_2) & = & \\frac{1}{\\hbar} \\sum_{m} \\Big \\{ \\frac{(\\hat{\\mu}^{\\rm p}_{\\alpha})_{0m} (\\hat{\\mu}^{\\rm p}_{\\beta})_{mn} } {\\omega_{m0} - \\omega_1}\n", + "+ \\frac{(\\hat{\\mu}^{\\rm p}_{\\beta})_{0m} (\\hat{\\mu}^{\\rm\n", + "p}_{\\alpha})_{mn} } {\\omega_{m0} - \\omega_2} \\Big \\}, \\\\\n", + "{\\mathcal{M}}_{\\alpha \\beta }^{{\\rm p},0n}(\\omega_1,\\omega_2) & = & \\frac{1}{\\hbar} \\sum_{m} \\Big \\{ \\frac{(\\hat{\\mu}^{\\rm p}_{\\alpha})_{0m} (\\hat{m}_{\\beta})_{mn} } {\\omega_{m0} - \\omega_1}\n", + "+ \\frac{(\\hat{m}_{\\beta})_{0m} (\\hat{\\mu}^{\\rm p}_{\\alpha})_{mn} }\n", + "{\\omega_{m0} - \\omega_2} \\Big \\}, \\\\\n", + "{\\mathcal{T}}_{\\alpha \\beta }^{+ ,0n}(\\omega_1,\\omega_2) & = &\n", + "\\frac{1}{\\hbar} \\varepsilon_{\\beta \\rho \\sigma} \\sum_{m} \\Big \\{\n", + "%\\frac{(\\hat{T}_{\\alpha \\rho}^+)_{0m} (\\hat{\\mu}_{\\sigma})_{mn} } {\\omega_{m0} -\n", + "%\\omega_1} + \\frac{(\\hat{\\mu}_{\\sigma})_{0m} (\\hat{T}_{\\alpha \\rho}^+)_{mn} }\n", + "\\frac{(\\hat{T}_{\\alpha \\rho}^+)_{0m} (\\hat{\\mu}^{\\rm p}_{\\sigma})_{mn}} {\\omega_{m0} -\n", + "\\omega_1} + \\frac{(\\hat{\\mu}^{\\rm p}_{\\sigma})_{0m} (\\hat{T}_{\\alpha \\rho}^+)_{mn} }\n", + "{\\omega_{m0} - \\omega_2} \\Big \\},\n", + "\\end{eqnarray}\n", + "\n", + "where the velocity form of the dipole\n", + "operator\n", + "\n", + "$$\n", + "\\hat{\\mu}^{\\rm p}_\\alpha = - \\frac{e}{m_e} \\sum_i {\\hat\n", + "p}_{i\\alpha}.\n", + "$$\n", + "\n", + "and that of the quadrupole operator (mixed form)\n", + "\n", + "$$\n", + "\\hat{T}_{\\alpha \\beta}^+ = - \\frac{e}{m_e} \\sum_i ({\\hat\n", + "r}_{i\\alpha} {\\hat p}_{i\\beta} + {\\hat\n", + "p}_{i\\alpha} {\\hat r}_{i\\beta}).\n", + "$$\n", + "\n", + "Within the formalism of response theory, the second-rank\n", + "tensors of interest are obtained from the single residues of\n", + "appropriate quadratic response functions. The quadratic response functions of\n", + "relevance for two-photon circular dichroism are\n", + "\n", + "\\begin{eqnarray}\n", + "\\langle \\langle \\hat{\\mu}^{\\rm p}_\\alpha;\\hat{\\mu}^{\\rm p}_\\beta,V^{\\omega_n} \n", + "\\rangle \\rangle_{\\omega_1,\\omega_2} & \\Rightarrow & {\\mathcal{P}}^{{\\rm p},0n}_{\\alpha\n", + "\\beta}(\\omega_1,\\omega_2), \\\\\n", + "\\langle \\langle \\hat{\\mu}^{\\rm p}_\\alpha;\\hat{m}_\\beta,V^{\\omega_n} \n", + "\\rangle \\rangle_{\\omega_1,\\omega_2} & \\Rightarrow & {\\mathcal{M}}^{{\\rm p},0n}_{\\alpha\n", + "\\beta}(\\omega_1,\\omega_2), \\\\\n", + "\\varepsilon_{\\beta \\rho \\sigma} \\langle \\langle\n", + "%\\hat{T}^+_{\\alpha\\rho};\\hat{\\mu}_\\sigma,V^{\\omega_n} \\rangle\n", + "\\hat{T}^+_{\\alpha\\rho};\\hat{\\mu}^{\\rm p}_\\sigma,V^{\\omega_n} \\rangle\n", + "\\rangle_{\\omega_1,\\omega_2} & \\Rightarrow & \\mathcal{T}^{+,0n}_{\\alpha\n", + "\\beta}(\\omega_1,\\omega_2),\n", + "\\end{eqnarray}\n", + "\n", + "where $V^{\\omega_n}$ is an arbitrary operator (corresponding to the excitation vector to the state $n$).\n", + "Single residues of quadratic response functions are efficiently\n", + "and accurately computed nowadays with a number of wave function\n", + "models. Nevertheless, DFT has been used almost exclusively \n", + "in the few theoretical studies of two-photon circular dichroism that were published." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8960fc97-d795-48b1-8b75-8a9ad1f24c5a", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/_sources/docs/spec_prop/dichroism.ipynb b/_sources/docs/spec_prop/tpor.ipynb similarity index 66% rename from _sources/docs/spec_prop/dichroism.ipynb rename to _sources/docs/spec_prop/tpor.ipynb index ce5faa482..0dbd47d70 100644 --- a/_sources/docs/spec_prop/dichroism.ipynb +++ b/_sources/docs/spec_prop/tpor.ipynb @@ -2,17 +2,17 @@ "cells": [ { "cell_type": "markdown", - "id": "938884bc-61c0-4eb9-95f6-e7f5ed3918dc", - "metadata": { - "editable": true, - "slideshow": { - "slide_type": "" - }, - "tags": [] - }, + "id": "da8bf99c-c71e-4ac3-b44e-5374ecaad793", + "metadata": {}, "source": [ - "# Dichroism" + "# Two-photon optical rotation" ] + }, + { + "cell_type": "markdown", + "id": "56d34367-c7c0-4593-b87a-f22110c520dc", + "metadata": {}, + "source": [] } ], "metadata": { @@ -31,7 +31,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.12" + "version": "3.11.5" } }, "nbformat": 4, diff --git a/_sources/docs/spec_prop/uv_vis.ipynb b/_sources/docs/spec_prop/uv_vis.ipynb new file mode 100644 index 000000000..69efbb1d9 --- /dev/null +++ b/_sources/docs/spec_prop/uv_vis.ipynb @@ -0,0 +1,91 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "04c0a70d-f428-4696-a635-df3ee4b90aac", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "" + }, + "tags": [ + "remove-output" + ] + }, + "source": [ + "# UV/vis absorption" + ] + }, + { + "cell_type": "markdown", + "id": "a48f1b57-f5e0-4d33-98f4-170bf692956d", + "metadata": {}, + "source": [ + "UV/vis spectroscopy is used to measure the amount of wavelengths of UV or visible light that is absorbed by or transmitted through a sample in comparison to a reference or blank sample. It is a property feeling the effect of the composition of the sample composition, and possibly able to provide information on the content and concentration of the sample.\n", + "\n", + "Humans are able to see a spectrum of visible light, from approximately 380 nm, violet, to 780 nm, red. UV light corresponds to wavelengths shorter than that of visible light, and goes to approximately 100 nm. Therefore, light can be described by its wavelength, which can be useful In UV-Vis spectroscopy one analyzes or identify substances by locating the specific wavelengths corresponding to maximum absorbance.\n", + "\n", + "A spectrophotometer is made with a light source, which passes first though a monochromator and/or through various types of filters, used to separate light keeping only photons within a narrow selection of wavelengths. Then the light beam passes through to the sample, where it is partly absorbed. Often after the monocromator the original beam is split (using appropriate setups) so that part of the light might go through the sample of interest, and the other through the reference sample. The beam(s) then hit a detector (a photomultiplyer tube, or a photodiode) which converts light into readable electronic signals." + ] + }, + { + "cell_type": "markdown", + "id": "96a6cf4f-0a61-4889-b0d1-bacd1d713d21", + "metadata": {}, + "source": [ + "\n", + "```{figure} ../../img/spec_prop/spectrometer.png\n", + "---\n", + "name: spectrometer\n", + "width: 600px\n", + "align: center\n", + "---\n", + "```" + ] + }, + { + "cell_type": "markdown", + "id": "0d435155-87f9-4c03-a2fd-a817b96c4e91", + "metadata": {}, + "source": [ + "From the point of view of the theoretical scientist trying to interprete and rationalize then effect, the most common used approximation is the Beer-Lambert law, stating that the intensity of light passing through an absorbing medium decreases exponentially. The transmitted intensity $I$ is related to the incident intensity $I_0$ via the relationship\n", + "\n", + "$$\n", + "\\log (\\frac{I_0}{I}) = A = \\epsilon_0 c L\n", + "$$\n", + "\n", + "where the absorbance $A$ is shown to be connected to the concentration of the absorber ($c$), to the length of the optical path within the sample ($L$) and to the extinction coefficient, or molar absorptivity $\\epsilon$, a fundamental property of the absorbing species in the specific environment, depending on the wavelength of the radiation and on the experimental parameters (pressure, temperature).\n", + "The intensity of light being proportional to the square of the amplitude of the electromagnetic field, in turn related to the refraction index $n$, the link between $\\epsilon$ and $n$ is immediate. Both quantities, governing the behavior of light in refractive and absorptive regions of the eletromagnetic spectra, can be expanded in powers of the light intensity, since they also depend on it. We distinguish *linear* absorbance (that described by the first, intensity-independent, term of these power expansions) from the *nonlinear* absorbance, which is intensity-dependent, becoming more and more relevant as laser sources of stronger intensities are brought into play. In the following we will therefore distinguish *linear*, or one-photon, absorption (OPA) - or more genarally optical processes - from the *nonlinear*, multi-photon absorption (MPA). " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d4d37dce-0862-4c8e-b8ed-760564c9052d", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.8" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/docs/basics_notebook.html b/docs/basics_notebook.html index 1c2342e58..71bdd8d58 100644 --- a/docs/basics_notebook.html +++ b/docs/basics_notebook.html @@ -72,7 +72,7 @@ - + @@ -307,17 +307,23 @@
  • Experiment to theory