Because the difference of energy between rotational levels is in the microwave region (1-10 cm-1) rotational spectroscopy is commonly called microwave spectroscopy. For an oscillatory or a rotational motion of a pendulum, one end stream ~����D� The wave functions for the rigid rotor model are found from solving the time-independent Schrödinger Equation: $\hat{H}=\dfrac{-\hbar}{2\mu}\nabla^2+V(r) \label{2.2}$. Looking back, B and l are inversely related. h��(NX(W�Y#lC�s�����[d��(!�,�8�:�졂c��Z�x�Xa � �b}�[S�)I!0yν������Il��d ��.�y������u&�NN_ kL��D��9@q:�\���ul �S�x �^�/yG���-̨��:ҙ��i� o�b�����3�KzF"4����w����( H��G��aC30Ũ�6�"31d'k�i�p�s���I���fp3 ��\*� �5W���lsd9���W��A����O�� ��G�/����^}�N�AQu��( ��rs���bS�lY�n3m ̳\Bt�/�u! To convert from units of energy to wave numbers simply divide by h and c, where c is the speed of light in cm/s (c=2.998e10 cm/s). /Length 4926 �a'Cn�w�R�m� k�UBOB�ؖ�|�+�X�an�@��N��f�R��&�O��� �u�)܂��=3���U-�W��~W| �AȨ��B��]X>6-׎�4���u�]_�= ��.�mE�X7�t[q�h�����t>��x92$�x������$���*�J�Qy����i�=�w/����J��=�d��;>@��r'4_�}y(&S?pU���>QE�t�I���F�^I��!ٞy����@-�����B|��^NO"�-�69�����=�Yi7tq 40. Click Get Books and find your favorite books in the online library. For a free diatomic molecule the Hamiltonian can be anticipated from the classical rotational kinetic energy and the energy eigenvalues can be anticipated from the nature of angular momentum. The rotational constant is dependent on the vibrational level: $\tilde{B}_{v}=\tilde{B}-\tilde{\alpha}\left(v+\dfrac{1}{2}\right)$. ��"Hz�-��˅ZΙ#�=�2r9�u�� ΁(�{��}:��!8�G�QUoށ�L�d�����?���b�F_�S!���J�Uic�{H Researchers have been interested in knowing what Godzilla uses as the fuel source for his fire breathing. Title: Rotational and vibrational spectroscopy 1 Rotational and vibrational spectroscopy. Internal rotations. Due to the dipole requirement, molecules such as HF and HCl have pure rotational spectra and molecules such as H2 and N2 are rotationally inactive. �g���_�-7e?��Ia��?/҄�h��"��,�{21I�Z��.�y{��'���T�t �������a �=�t���;9R�tX��(R����T-���ܙ����"�e����:��9H�=���n�B� 4���陚$J�����Ai;pPY��[\�S��bW�����y�u�x�~�O}�'7p�V��PzŻ�i�����R����An!ۨ�I�h�(RF�X�����c�o_��%j����y�t��@'Ϝ� �>s��3�����&a�l��BC�Pd�J�����~�-�|�6���l�S���Z�,cr�Q��7��%^g~Y�hx����,�s��;t��d~�;��$x$�3 f��M�؊� �,�"�J�rC�� ��Pj*�.��R��o�(�9��&��� ���Oj@���K����ŧcqX�,\&��L6��u!��h�GB^�Kf���B�H�T�Aq��b/�wg����r������CS��ĆUfa�É Where $$\tilde{\alpha}$$ is the anharmonicity correction and $$v$$ is the vibrational level. The radiation energy absorbed in IR region brings about the simultaneous change in the rotational and vibrational energies of the molecule. Set the Schrödinger Equation equal to zero: $\dfrac{\sin{\theta}}{\Theta\left(\theta\right)}\dfrac{d}{d\theta}\left(\sin{\theta}\dfrac{d\Theta}{d\theta}\right)+\beta\sin^2\theta+\dfrac{1}{\Phi\left(\phi\right)}\dfrac{d^2\Phi}{d\phi^2}=0$. 42. Why is Rotational Spectroscopy important? Rotational spectroscopy. Missed the LibreFest? �6{,�F~$��x%āR)-�m"ˇ��2��,�s�Hg�[�� What is the potential energy of the Rigid-Rotor? $\tilde{\nu}=\left[\tilde{w}\left(\dfrac{3}{2}\right)+\tilde{B}_{1}\left(J+1\right)\left(J+2\right)\right]-\left[\tilde{w}\left(\dfrac{1}{2}\right)+\tilde{B}_{0}J\left(J+1\right)\right]$, $\tilde{\nu}=\tilde{w}+\left(\tilde{B}_{1}-\tilde{B}_{0}\right)J^2+\left(3\tilde{B}_{1}-\tilde{B}_{0}\right)J+2\tilde{B}_{1}$, $\tilde{\nu}=\left[\tilde{w}\left(\dfrac{3}{2}\right)+\tilde{B}_{1}\left(J-1\right)J\right]-\left[\tilde{w}\left(\dfrac{1}{2}\right)+\tilde{B}_{0}J\left(J+1\right)\right]$, $\tilde{\nu}=\tilde{w}+\left(\tilde{B}_{1}-\tilde{B}_{0}\right)J^2-\left(\tilde{B}_{1}+\tilde{B}_{0}\right)J$. Peaks are identified by branch, though the forbidden Q branch is not shown as a peak. However, in our introductory view of spectroscopy we will simplify the picture as much as possible. Step 2: Because the terms containing $$\Theta\left(\theta\right)$$ are equal to the terms containing $$\Phi\left(\phi\right)$$ they must equal the same constant in order to be defined for all values: $\dfrac{\sin{\theta}}{\Theta\left(\theta\right)}\dfrac{d}{d\theta}\left(\sin{\theta}\dfrac{d\Theta}{d\theta}\right)+\beta\sin^2\theta=m^2$, $\dfrac{1}{\Phi\left(\phi\right)}\dfrac{d^2\Phi}{d\phi^2}=-m^2$. Polyatomic molecules. Vibrational Partition Function Vibrational Temperature 21 4.1. %���� The Hamiltonian Operator can now be written: $\hat{H}=\hat{T}=\dfrac{-\hbar^2}{2\mu{l^2}}\left[\dfrac{1}{\sin{\theta}}\dfrac{\partial}{\partial{\theta}}\left(\sin{\theta}\dfrac{\partial}{\partial{\theta}}\right)+\dfrac{1}{\sin{\theta}}\dfrac{\partial^2}{\partial{\phi^2}}\right]\label{2.5}$. In addition to having pure rotational spectra diatomic molecules have rotational spectra associated with their vibrational spectra. Therefore, when we attempt to solve for the energy we are lead to the Schrödinger Equation. �/�jx�����}u d�ى�:ycDj���C� Abstract. Some examples. The system can be simplified using the concept of reduced mass which allows it to be treated as one rotating body. The distance between the masses, or the bond length, (l) can be considered fixed because the level of vibration in the bond is small compared to the bond length. The computed ro-vibrational energy levels of diatomic molecules are now examined. ld�Lm.�6�J�_6 ��W vա]ՙf��3�6[�]bS[q�Xl� x��[Ys�H�~����Pu�����3ڙnw�53�a�"!�$!�l��߼ as the intersection of $$R_1$$ and $$R_2$$) with a frequency of rotation of $$\nu_{rot}$$ given in radians per second. The rotation of a diatomic molecule can be described by the rigid rotor model. -1. Including the rotation-vibration interaction the spectra can be predicted. Diatomics. 86 Spectroscopy ch.5 Replacing the first two terms by( ̅ ¢ ¢¢)the wave number of an electronic vibrational transition = ¢ ¢¢ + ¢ ¢ ¢ + − ¢¢ ¢¢ ¢¢ + ̅ ¢ ¢¢ could by any one of the (0,0), (1,0) ,(2,0) ----- The selection rule for J depends on the type of electronic transition. The energy of the transition must be equivalent to the energy of the photon of light absorbed given by: $$E=h\nu$$. �VI�\���p�N��Ŵ}������x�J�@nc��0�U!����*�T���DB�>J+� O�*��d��V��(~�Q@$��JI�J�V�S��T�>��/�쮲.��E�f��'{!�^���-. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. 3 represents the trend of a diatomic molecule’s vibrational-rotational spectra. the kinetic energy can now be written as: $T=\dfrac{M_{1}R_{1}^2+M_{2}R_{2}^2}{2}\omega.$. Define symmetric top and spherical top and give examples of it. The J-1 transitions, shown by the red lines in Figure $$\PageIndex{3}$$, are lower in energy than the pure vibrational transition and form the P-branch. The rotational spectra of non-polar molecules cannot be observed by those methods, but can be observed and measured by Raman spectroscopy. >> The order of magnitude differs greatly between the two with the rotational transitions having energy proportional to 1-10 cm-1 (microwave radiation) and the vibrational transitions having energy proportional to 100-3,000 cm-1 (infrared radiation). Physical Biochemistry, November 2004 ; Dr Ardan Patwardhan, a.patwardhan_at_ic.ac.uk,Dept. The correction for the centrifugal distortion may be found through perturbation theory: $E_{J}=\tilde{B}J(J+1)-\tilde{D}J^2(J+1)^2.$. The theory of rotational spectroscopy depends upon an understanding of the quantum mechanics of angular momentum. The orientation of the masses is completely described by $$\theta$$ and $$\phi$$ and in the absence of electric or magnetic fields the energy is independent of orientation. is the reduced mass, $$\mu$$. If we assume that the vibrational and rotational energies can be treated independently, the total energy of a diatomic molecule (ignoring its electronic energy which will be constant during a ro-vibrational transition) is simply the sum of its rotational and vibrational energies, as shown in equation 8, which combines equation 1 and equation 4. Spectroscopy 1: rotational and vibrational spectra The vibrations of diatomic molecules Molecular vibrations Consider a typical potential energy curve for a diatomic molecule. This contrasts vibrational spectra which have only one fundamental peak for each vibrational mode. Vibrational spectroscopy. Rotational spectroscopy is therefore referred to as microwave spectroscopy. Fig. This model can be further simplified using the concept of reduced mass which allows the problem to be treated as a single body system. Rotational–vibrational spectroscopy: | |Rotational–vibrational spectroscopy| is a branch of molecular |spectroscopy| concerned w... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Microwave spectroscopy For diatomic molecules the rotational constants for all but the very lightest ones lie in the range of 1–200 gigahertz (GH z). Schrödinger equation for vibrational motion. Vibrational and Rotational Spectroscopy of Diatomic Molecules Spectroscopy is an important tool in the study of atoms and molecules, giving us an understanding of their quantized energy levels. Vibrational Spectroscopy Selection rules. where $$\nabla^2$$ is the Laplacian Operator and can be expressed in either Cartesian coordinates: $\nabla^2=\dfrac{\partial^2}{\partial{x^2}}+\dfrac{\partial^2}{\partial{y^2}}+\dfrac{\partial^2}{\partial{z^2}} \label{2.3}$, $\nabla^2=\dfrac{1}{r^2}\dfrac{\partial}{\partial{r}}\left(r^2\dfrac{\partial}{\partial{r}}\right)+\dfrac{1}{r^2\sin{\theta}}\dfrac{\partial}{\partial{\theta}}\left(\sin{\theta}\dfrac{\partial}{\partial{\theta}}\right)+\dfrac{1}{r^2\sin^2{\theta}}\dfrac{\partial^2}{\partial{\phi}} \label{2.4}$. Diatomic Molecules Species θ vib [K] θ rot [K] O 2 2270 2.1 N 2 3390 2.9 NO 2740 2.5 Cl 2 808 0.351 kT hc kT hc Q e vib 2 1 exp exp 1 Choose reference (zero) energy at v=0, so G e v 1 1 exp Because $$\tilde{B}$$ is a function of $$I$$ and therefore a function of $$l$$ (bond length), so $$l$$ can be readily solved for: $l=\sqrt{\dfrac{h}{8\pi^2{c}\tilde{B}\mu}}.$. $$R_1$$ and $$R_2$$ are vectors to $$m_1$$ and $$m_2 These energy levels can only be solved for analytically in the case of the hydrogen atom; for more complex molecules we must use approximation methods to derive Polyatomic molecules. Harmonic Oscillator Vibrational State Diatomic Molecule Rotational State Energy Eigenvalue These keywords were added by machine and not by the authors. In regions close to Re (at the minimum) the potential energy can be approximated by parabola: € V= 1 2 kx2 x = R - R e k – the force constant of the bond. In spectroscopy it is customary to represent energy in wave numbers (cm-1), in this notation B is written as \(\tilde{B}$$. << Watch the recordings here on Youtube! We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Fig.13.1. This causes the potential energy portion of the Hamiltonian to be zero. We will first take up rotational spectroscopy of diatomic molecules. /Filter /FlateDecode The spectra for rotational transitions of molecules is typically in the microwave region of the electromagnetic spectrum. This process is experimental and the keywords may be updated as the learning algorithm improves. The Schrödinger Equation can be solved using separation of variables. Energy states of real diatomic molecules. For a diatomic molecule the energy difference between rotational levels (J to J+1) is given by: $E_{J+1}-E_{J}=B(J+1)(J+2)-BJ(J=1)=2B(J+1)$. Step 3: Solving for $$\Phi$$ is fairly simple and yields: $\Phi\left(\phi\right)=\dfrac{1}{\sqrt{2\pi}}e^{im\phi}$. In the high resolution HCl rotation-vibration spectrum the splitting of the P-branch and R-branch is clearly visible. In the context of the rigid rotor where there is a natural center (rotation around the COM) the wave functions are best described in spherical coordinates. N���d��b��t"�΋I#��� with the Angular Momentum Operator being defined: $\hat{L}=-\hbar^2\left[\dfrac{1}{\sin{\theta}}\dfrac{\partial}{\partial{\theta}}\left(\sin{\theta}\dfrac{\partial}{\partial{\theta}}\right)+\dfrac{1}{\sin{\theta}}\dfrac{\partial^2}{\partial{\phi^2}}\right]$, $\dfrac{-\hbar^2}{2I}\left[\dfrac{1}{\sin{\theta}}\dfrac{\partial}{\partial{\theta}}\left(\sin{\theta}\dfrac{\partial}{\partial{\theta}}\right)+\dfrac{1}{\sin{\theta}}\dfrac{\partial^2}{\partial{\phi^2}}\right]Y\left(\theta,\phi\right)=EY\left(\theta,\phi\right) \label{2.6}$. Rotational Spectroscopy Of Diatomic Molecules. ;@ޮPު[����Z�����$�Lj�m� m��3r2��6uudO���%��:�bŗU�`*$_W3�h���1v��'' �%B������F:�˞�q�� the kinetic energy can be further simplified: The moment of inertia can be rewritten by plugging in for $$R_1$$ and $$R_2$$: $I=\dfrac{M_{1}M_{2}}{M_{1}+M_{2}}l^2,$. Have questions or comments? Rotational Spectra of diatomics. Due to the small spacing between rotational levels high resolution spectrophotometers are required to distinguish the rotational transitions. From pure rotational spectra of molecules we can obtain: 1. bond lengths 2. atomic masses 3. isotopic abundances 4. temperature Important in Astrophysics: Temperature and composition of interstellar medium Diatomic molecules found in interstellar gas: H 2, OH, SO, SiO, SiS, NO, NS, (From Eisbergand Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (1985)) 10x10-21) Estimated rotational energies vs. quantum number j, for O 2 8 The diatomic molecule can serve as an example of how the determined moments of inertia can be used to calculate bond lengths. Vibrational and Rotational Transitions of Diatomic Molecules High-resolution gas-phase IR spectra show information about the vibrational and rotational behavior of heteronuclear diatomic molecules. The distance between the two masses is fixed. The diagram shows the coordinate system for a reduced particle. Effect of anharmonicity. The moment of inertia and the system are now solely defined by a single mass, $$\mu$$, and a single length, $$l$$: Another important concept when dealing with rotating systems is the the angular momentum defined by: $$L=I\omega$$, $T=\dfrac{I\omega^2}{2}=\dfrac{I^2\omega^2}{2I}=\dfrac{L^2}{2I}$. This chapter is mainly concerned with the dynamical properties of diatomic molecules in rare-gas crystals. In wave numbers $$\tilde{B}=\dfrac{h}{8\pi{cI}}$$. Classify the following molecules based on moment of inertia.H 2O,HCl,C 6H6,BF 3 41. The system can be entirely described by the fixed distance between the two masses instead of their individual radii of rotation. Combining the energy of the rotational levels, $$\tilde{E}_{J}=\tilde{B}J(J+1)$$, with the vibrational levels, $$\tilde{E}_{v}=\tilde{w}\left(v+1/2\right)$$, yields the total energy of the respective rotation-vibration levels: $\tilde{E}_{v,J}=\tilde{w} \left(v+1/2\right)+\tilde{B}J(J+1)$. 5 0 obj Due to the relationship between the rotational constant and bond length: $\tilde{B}=\dfrac{h}{8\pi^2{c}\mu{l^2}}$. ��#;�S�)�A�bCI�QJ�/�X���/��Z��@;;H�e����)�C"(+�jf&SQ���L�hvU�%�ߋCV��Bj쑫{�%����m��M��$����t�-�_�u�VG&d.9ۗ��ɖ�y ���! How would deuterium substitution effect the pure rotational spectrum of HCl. The dumbbell has two masses set at a fixed distance from one another and spins around its center of mass (COM). Figure $$\PageIndex{2}$$: predicts the rotational spectra of a diatomic molecule to have several peaks spaced by $$2 \tilde{B}$$. Let $$Y\left(\theta,\phi\right)=\Theta\left(\theta\right)\Phi\left(\phi\right)$$, and substitute: $$\beta=\dfrac{2IE}{\hbar^2}$$. Legal. Rotational transitions are on the order of 1-10 cm-1, while vibrational transitions are on the order of 1000 cm-1. Because $$\tilde{B}_{1}<\tilde{B}_{0}$$, as J increases: The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This is an example of the Born-Oppenheimer approximation, and is equivalent to assuming that the combined rotational-vibrational energy of the molecule is simply the sum of the separate energies. Download full Rotational Spectroscopy Of Diatomic Molecules Book or read online anytime anywhere, Available in PDF, ePub and Kindle. The classical vibrational frequency νis related to the reduced mass μ[= m1m2/(m1 + m2)] and the force constant k by 6.1 Diatomic molecules ν= (1/2π)[k/μ]1/2 Vibrational term values in unit of wavenumber are given where the vibrational quantum number v = 0, 1, 2, … hc Ev = G(v) = ω(v + ½) Chapter 6. assume, as a first approximation, that the rotational and vibrational motions of the diatomic molecule are independent of each other. Derive the Schrodinger Equation for the Rigid-Rotor. Similar to most quantum mechanical systems our model can be completely described by its wave function. When a molecule is irradiated with photons of light it may absorb the radiation and undergo an energy transition. A diatomic molecule consists of two masses bound together. Relationships between the radii of rotation and bond length are derived from the COM given by: where l is the sum of the two radii of rotation: Through simple algebra both radii can be found in terms of their masses and bond length: The kinetic energy of the system, $$T$$, is sum of the kinetic energy for each mass: $T=\dfrac{M_{1}v_{1}^2+M_{2}v_{2}^2}{2},$. This causes the terms in the Laplacian containing $$\dfrac{\partial}{\partial{r}}$$ to be zero. The change in the bond length from the equilibrium bond length is the vibrational coordinate for a diat omic molecule. Rotational spectroscopy is sometimes referred to as pure rotati… Rotational energies of a diatomic molecule (not linear with j) 2 1 2 j j I E j Quantum mechanical formulation of the rotational energy. {��yx����]fF�G֧�&89=�ni&>�3�cRlN�8t@���hC�P�m�%��E�� �����^F�@��YR���# However, the reader will also find a concise description of the most important results in spectroscopy and of the corresponding theoretical ideas. The difference in energy between the J+1 transitions and J-1 transitions causes splitting of vibrational spectra into two branches. �w4 ��j��S�V\��Z X'��ې\�����VS��L�&���0�Hq�}tɫ7�����8�Qb��e���g���(N��f ���٧g����u8Ŕh�C�w�{�xU=���I�¬W�i_���}�����w��r�o���)�����4���M&g�8���U� ��Q��䢩#,��O��)ڱᯤg]&��)�C;�m�p�./�B�"�'Q 6H������ѥS4�3F% �4��� �����s�����ds�jA�)��U��Pzo?FO��A�/��\���%����z�{plF�$�$pr2 [�]�u���Z���[p�#��MJ�,�#���g���vnach��9O��i�Ƙ^�8h{�4hK�B��~��b�o�����ܪE'6�6@��d>2! Raman effect. What is the equation of rotational … Explain the variation of intensities of spectral transitions in vibrational- electronic spectra of diatomic molecule. In spectroscopy, one studies the transitions between the energy levels associated with the internal motion of atoms and molecules and concentrates on a problem of reduced dimen- sionality3 k− 3: The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. Dr.Abdulhadi Kadhim. Recall the Rigid-Rotor assumption that the bond length between two atoms in a diatomic molecule is fixed. Following the selection rule, $$\Delta{J}=J\pm{1}$$, Figure 3. shows all of the allowed transitions for the first three rotational states, where J" is the initial state and J' is the final state. In addition to having pure rotational spectra diatomic molecules have rotational spectra associated with their vibrational spectra. @ �Xg��_W 0�XM���I� ���~�c�1)H��L!$v�6E-�R��)0U 1� ���k�F3a��^+a���Y��Y!Տ�Ju�"%K���j�� As molecules are excited to higher rotational energies they spin at a faster rate. Solving for $$\theta$$ is considerably more complicated but gives the quantized result: where $$J$$ is the rotational level with $$J=0, 1, 2,...$$. The frequency of a rotational transition is given approximately by ν = 2 B (J + 1), and so molecular rotational spectra will exhibit absorption lines in … The order of magnitude differs greatly between the two with the rotational transitions having energy proportional to 1-10 cm -1 (microwave radiation) and the vibrational transitions having energy proportional to 100-3,000 cm -1 (infrared radiation). Sketch qualitatively rotational-vibrational spectrum of a diatomic. Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. 6Vª�I�J���>���]�X�>4u��H\��#at����_d23��(L�Zp��Ⱉ�U�� ���#91��D̡hn$�g���0a:̤�ϨN��"�&�~$Ȧ9 k�~$��h��S|i+J#0oV�F;�w���$#LJ�_�"=܆I � X��+�P럷9=�qȘ��8�ײLX����������.��E�9@�ǚ�6� ~* ڂ��!E�( Z����6�q=���B��sʦ� �d�m�CUg��3�=�M#@�'�ۓd�H���;����r���@�̻�R��B�z�%����#߁��M�\$ϼ;���&2�5��������CA�:�c;���I �8�����4sbPYm&�~�Gt�~z�Rb�w/���?�/�c�˿���޿���["=��a/:�3�pVt�����9B���sI Studies on the residue showed that the fuel, Compound G, is a diatomic molecule and has a reduced mass of 1.615x10. Therefore there is a gap between the P-branch and R-branch, known as the q branch. Written to be the definitive text on the rotational spectroscopy of diatomic molecules, this book develops the theory behind the energy levels of diatomic molecules and then summarises the many experimental methods used to study their spectra in the gaseous state. Changes in the orientation correspond to rotation of the molecule, and changes in the length correspond to vibration. • The Molecular Spectra can be divided into three Spectral ranges corresponding to the different types of the transitions between the molecular energy states :- SPECTRA REGION STATES OBSERVED IN Rotational Spectra Microwave or far infrared with λ = 0.1mm to 1cm Separated by Small energy intervals Heteronuclear diatomic Molecules (HCl,CO).. Vibrational Spectra Infrared Region with … Step 4: The energy is quantized by expressing in terms of $$\beta$$: Step 5: Using the rotational constant, $$B=\dfrac{\hbar^2}{2I}$$, the energy is further simplified: $$E=BJ(J+1)$$. Molecule and has a reduced mass which allows the problem to be zero vibrational energy levels to. Info @ libretexts.org or check out our status page at https: //status.libretexts.org,.. B } =\dfrac { h } { 8\pi { cI } } )... 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