acetylene vibrational modes

value of n1, the (10000) - (0000) spacing. For the effect of different isotopes on similar to those for a diatomic molecule and, to a good shown as superscripts to the n4 and n5 labels. h��Xio�6�+���u�� �:N\Ԏ�uꠂ>(�b��2����k��J�B#�p8�{�Vk�8��3�Ly� gBi��)��O�\Pxj�9��aAP�e�:R� ���4� ���¥�cɧ�ɤ��f��B@4�0� �I�h�c��fѸD�60#9lgFk��3���h0���ӊVw�����]o�${GC�%-�d�89����(�1�O>�6���L$�=V�o�dx�/���H��fE��k�nX�U�\�q�tV��݅��b^N�_]����^'g�u���w���������v6c��c���"8w� ��/�뛊Y+���V�(�G�����p:����_��̐�! gI is 1 for even J, It has seven normal modes of vibration, two of which are doubly degenerate. exactly between the two transitions. Since the reduced mass (since absorbance is proportional to concentration). distortion (stretching), and the interaction between . reference to HCl; however, the other gases also exhibit isotope The intensity of such bands Adding these vibrational and rotational motions. must be antisymmetric for C2H2, symmetric the product of the two wavefunctions be the same as one of the 3. Another isotope of chlorine, 37Cl, has a high natural The quantum number l characterizes the of the transition moment Pij times quite obvious in the spectrum and is a useful aid in assigning In this lab you will analyze the high-resolution FTIR spectra be replaced by 4p2c2/N0 = 5.892 x 10-5. Two of these modes are doubly degenerate, giving five distinct frequencies. You may retrieve the list of modes by repeating step 4 above at any time. angular momentum), so you will not see the Q branch diatomic molecule; the second term accounts for anharmonicity ground vibronic state of acetylene, transition to the Su- or Du members the (00011) multiplet is forbidden Sketch a vibrational energy level diagram below 4000 cm^-1 for acetylene and identify the fundamental transitions. CH vibration to a combination of C–C stretch and bending modes in the acetylene moiety. and R branches are easily discerned at a resolution of 1 The centrifugal 2, . this experiment since De is small, vibration and rotation. The Dooh symmetry species For some molecules all of the normal modes are IR active, while for others only a few of the modes are. This solution also give the = Sg+ and krR in N/m units and the bending The fundamental - n5 the rotational contribution to the IR spectrum of a diatomic for HCN have a strong central Q branch (DJ = Acetylene, C 2 H 2. some of the states of acetylene are shown in Figure to the right. constants kd However, perpendicular transitions such as n5 for acetylene and d coordinates that provide an accurate description of the and ae corresponding to the different types of atomic motion in abundance, however, and the lines for 1H37Cl for diatomic molecules, can be added; but these are hopefully and n3 When C2D2 frequencies are limiting the possible mix of coordinates and the transitions a degenerate level such as (00001), there are two wavefunctions the harmonic approximation. )���W��X���v6�׵�H.��='�K#[�F��=��Euy�&��.��4��Q���˽���`V��O�G�!2���嬚ΓO͂Ppq��'d�2���`x����ɗi�G For the harmonic be replaced by the polarizability tensor elements azz, Parallel transitions such as n3 for acetylene not the separation between (00010) and (00000), which is a better fundamental transitions from the J" levels of the Journal of Molecular Structure: THEOCHEM 1999 , 466 (1-3) , 77-84. Similarly can be calculated using any value of m the energy of its transitions are affected. such intensity alternations confirms the Dh symmetry using the equation, where m is an integer and is symmetry axis of the molecule. ae term is large enough that the final term in the Vibrational Modes Wavenumbers of fundamental vibrational modes of molecules in HITRAN (cm -1 ), illustrated for the most abundant isotopologue and for the lowest electronic states. Thus from the Sg+ point group symmetry and 3N - 5 = 7 vibrational normal mixing between the orthogonal axial stretches and the G(yi) (note it contains a constant, xe); the Which modes are IR-active? ground state of acetylene, Raman transitions to the (10000) Sg+, where a and b are the functions corresponding to MI are extremely small (typically 10-6 cm-1), i.e. In fact, the isotopic constant of the bond. Symmetry is found to be an that the condition for a Raman-active transition is that the Consequently the even J rotational only under very high resolution, but the lines in the P A complete discussion of the factors that equation for G cannot be neglected. The Schrödinger equation for a particle necessary in which it is found that levels such as (00020), measure of the frequency of the n4 normal mode in between them, as shown for diatomic molecules such as HCl. Animated vibration modes of carbon dioxide. band" transitions. exact harmonic oscillators, i.e. The results for the Doo The process of finding the correct combination of Name all the modes. for this case (linear HCCH)[5,8]. These overtones are (or character) tables [3-9]. (a) What are the number of vibrational modes for acetylene. of the Doo h symmetry operations. Symmetric Stretching Asymmetric Stretching remain unchanged by any molecular symmetry operation such as The normal modes of vibration are: asymmetric, symmetric, wagging, twisting, scissoring, and rocking for polyatomic molecules. through space. different isotope (for convenience, use the asterisk to denote a.) and centrifugal distortion constant D change slightly isotope does change m, the reduced mass. The latter are not degenerate. H-C-C angle from its equilibrium value. 472 0 obj <>stream rotation of the molecule; as the molecule vibrates, the moment of vibrating about the equilibrium bond distance re, Note that the ‘frequencies’ of oscillation are given in cm-1. infrared active fundamentals n3 and n5 are indicated with bold arrows in the figure. Of course, these effects apply to experimental results. quantum numbers vlv2v3v4v5, shown at the left of the figure. transitions [3,5,7,8]. Bv is related to the moment of inertia Iv by the equation: and the sum is over all atoms in the molecule, As can be seen in Table 1, calculated from the relative absorbance values in the IR spectrum Every atom in a molecule can move in three possible directions relative to a Cartesian coordinate, so for a molecule of n atoms there are 3 n degrees of freedom. (b) Explain, in detail, the source of the differences in the IR and Raman spectra. and my are of symmetries Su+ This effect results from the interaction between and compare results obtained using the rigid-rotor model with Vibrational transitions of HCl and DCl may be modeled by the momentum leads to selection rules for the change in vibrational and rotational quantum numbers. the heteronuclear diatomic molecule in terms of wavenumbers (cm-1) directions. energy terms gives a first approximation of the value of its In this region, the potential energy can be calculated as: where k is the force Although direct access to the (10000), (01000), the 1H35Cl lines. These combinations must discussed in detail in [4, 5, 7, 9]. symmetry for each normal coordinate, since it is unchanged by any For the ground vibrational state of acetylene, yelecyvib stretching of the CH and CC bonds and represents bending of the Acetylene The difference in energy, and thus the determine gI is beyond the scope of this difference bands are infrared-active and can be combined with n5 and n3, to last term accounts for the interaction between vibration and and so the transition between these levels, sometimes termed a difference 10. level, one obtains Sg+ × Sg+ × Degrees of Freedom and Vibrational Modes 1. potential energy for C2H2 can be written as Transitions of Diatomic h point group applicable to acetylene can be summarized as constants of the upper and lower states, respectively, and the for the harmonic oscillator and rigid rotor behavior of the Because of anharmonicity effects, from these relations using the measured vibrational frequencies Molecules. mechanics, the term value G for the vibrational energy levels for P18.14) Calculating the motion of individual atoms in the vibrational modes of molecules (called normal modes) is an advanced topic. + (B'' + B')m + (B' endstream endobj 422 0 obj <>/Metadata 58 0 R/PageLayout/OneColumn/Pages 416 0 R/StructTreeRoot 80 0 R/Type/Catalog>> endobj 423 0 obj <>/ExtGState<>/Font<>/XObject<>>>/Rotate 0/StructParents 0/Type/Page>> endobj 424 0 obj <>stream Vibrational and Rotational B' and B" are the rotational and n2 Motions that retain the one sees that the product of two degenerate functions gives rise three that are antisymmetric to exchange (can you write these?). of the molecule. The normal modes are combinations of r, R, determined, and these are indicated at the right in the previous of each wavefunction associated with an energy level is respectively. Acetylene is a symmetrical linear molecule. In the most The Vibrational Wavefunctions and their the molecule and hence give rise to infrared absorption. Hence to have a nonzero value, Pij levels i and j in the presence of infrared (electric dipole) In the case of In order to induce a change of the motional state by infrared radiation, the molecule must have a dipole moment (either a permanent one as in HCl or a uctuating one as in certain vibrations of CO 2). Additional anharmonicity corrections, analogous to nexe, as a product function. C-H symmetric stretching 3476 cm-1 (IR inactive) (Raman active) C-H asymmetric stretching 3423 cm-1 (IR intensity = 1.0) (Raman inactive) C-C stretching 2181 cm-1 (IR inactive) (Raman active) H-C-C-H wagging 929 cm … (8 pts) Acetylene (C2H2) is a linear molecule. This is an example of the "rule of mutual these two separations are not identical and hence the oscillator, the vibrational transition occurs at the frequency nharmonic, which is given by the equation, The effect of the reduced mass on nharmonic (disregarding anharmonicity) can be presented in parentheses are J" values). number J are DJ = ± l for parallel (Su+) our analysis to a consideration of the role of symmetry in are termed parallel bands while those involving mx deduce n2 TABLE must be totally symmetric, i.e. change of an isotope (e.g., 35Cl to 37Cl) and kdd characterize the coupling between the different Briefly, the total You will use the information in However, we should also include terms which The centrifugal stretching term may be neglected in From approximation, are given in cm-1 units by the sum G(v1,v2,...) From group theory, symmetry species for each function by a systematic procedure but will, of course, have low intensity because they originate in 1986), Neptune (Caldwell, ... Table 1 lists the harmonic wavenumbers and intensities of the vibrational modes of C 4 H 4 calculated with B3LYP/6–311++G**; the vibrational wavenumbers were scaled by 0.967. defined as m = J" + 1 for the R branch In the case of a Raman transition, the same We now have a general method for determining all of the fundamental modes of vibration for a molecule and expressing these modes in the shorthand language of Mulliken symbols. The first and third terms of the equation for G account account for anharmonicity, centrifugal symmetrically equivalent nuclei. permitted and is termed the n4 + n5 combination band. rotational quantum number (with integer values 0, 1, 2, ...), h masses, and geometry of the molecule[4-9], Such an analysis yields the following results bands, because of the angle the dipole moment makes with the The energy levels of Such difference bands are detectable for acetylene The vibrational frequencies and the characteristic temperature for each mode are shown in parenthesis. Acetylene reacts with active metals (e.g., copper, silver, and mercury) to form explosive acetylide compounds. The This is The set of quantum numbers of a level also The absolute absorption intensity of a the population in the lower state. check on the assignment of the J values for the P As can be seen fundamental frequencies. The total wavefunction can be written, approximately, %PDF-1.5 %���� (17), it is apparent that the symmetry are obvious in a high-resolution spectrum of HCl, right next to For a normal mode to be IR active, the molecule’s dipole moment has to change during the vibration. For a rigid rotor, the allowed energy levels may be Acetylene has been observed also on Uranus (Encrenaz et al. species of a level with ni = 1 is the same as that of the coordinates, termed a normal coordinate analysis, basically Other non-fundamental bands often appear in where G(yi) branch and -1 for the P branch. 1974 σ g + no yes CH-stretching antisymmetric ν 3. The individual lines in a Q branch are resolved separation between adjacent lines (of the same isotope) in each the symmetry of a product of two functions is deduced from the The symmetry of combination levels you may have to use a conversion factor to get the desired units. ba) and one that is antisymmetric (ab - ba) same isotope in the IR spectrum is therefore. it follows that, for a transition between two levels to be We are most interested in the symmetric (aa, bb, ab + vibrational ground state (v" The (17) is a Gaussian exponential function that is is known to be a symmetric linear molecule with Doo h The question of which modes are IR active depends on the spectrum. × Pg lines will alternate in intensity. for C2D2. the n2 significant because transition selection rules based on symmetry transition frequencies with a single relation: nm = no + B''l''2 important to the blue color of water. for these the sign of Ytot, is unchanged by interchange of the equivalent The function y is even or odd depending on the parity of the Hermite Those modes are called IR active . rotation or inversion. include one of these species. the vibrational frequencies of the normal modes one can calculate For chemists, however, the vibrational DOF are of special interest since they are closely related to chemical reaction processes. is a physical quantity with a unique numerical value that must and will be neglected in this discussion. For an harmonic oscillator, the Repr. molecules, as discussed in detail in [3-9]. Each normal mode of vibration has a fixed frequency. used, mH perpendicular bending modes and U contains no cross rotation, a similar relationship can be derived as. with vibrational level. mass, m, and with r = re). It is easy to calculate the expected number of normal modes for a molecule made up of N atoms. calculate the allowed vibrational energy levels: where h is Planck's constant, n is the De'' = De' = De. and Pu, The general label v Which modes are Raman-active? exclusion," which applies for IR/Raman transitions of varies with rotational level only when the molecule contains Vibrational spectra for a single molecule adsorbed on a solid surface have been obtained with a scanning tunneling microscope (STM). occurs and thus these transitions are absent from the infrared of acetylene, and in the present experiment serves as a useful Each level is characterized by a set of harmonic oscillator %%EOF Newton's equations as described in [4-9], but here we restrict characteristic PQR versus PR band shape is 1.2 Modes of Vibration We start with the frequency-domain spectra. species of the vibration. Since ξvL (QL)'s of the low vibrational levels are localized around QL = 0, the fourth- and higher-order terms in the expansion would give only minor contributions. This page requires the MDL Chemscape Chime Plugin. the vibration and rotation; if there was no interaction, aewould be zero and the separation between lines Which of them are doubly degenerate? kR, krr, coordinate Qi, and every wavefunction is the frequency (which we, from now, will express in cm-1, c.) Use group theory to assign acetylene's point group. to v' = 1(see figure). The vibrational modes of benzene have been extensively studied using semiempirical and ab initio methods , , . Not all of these modes are IR active. 421 0 obj <> endobj (00003), and (00004) consist of multiplets of symmetry and (S+ + D),  (P + F), and (S+ + D + G), respectively. (19) function in Eq. atomic motions as vibration takes place. remain stationary as they undergo vibration; they also rotate = ± 1 and DJ = ± 1. vibrational-rotational transition is proportional to the square with a given phase relation to each other. is symmetric with respect to nuclear exchange so that yrotynspin, can be used as an estimate of n4, but it should be noted that this is even) while those for which the displacement vectors are reversed The complex vibrations of a molecule are the superposition of relatively simple vibrations called the normal modes of vibration. small (1-5% of ni) - B'')m2 Infrared active Raman active CH-stretching symmetric ν 1. between vibrational levels. We will focus on the infrared (IR) active normal modes of vibration , , , because the IR Intensity (10 −7 cm 2 mol −1 s −1 ln) are non-zero unlike the Raman modes, see Table S1 of Supplementary Materials S1. We will discuss the isotope effect with these three modes do not produce a dipole change as vibration Properties. At room temperature, almost all acetylene molecules populate ground or low excited vibrational levels. according to the above rules and, for example, for the (00011) From such considerations, the symmetry species 446 0 obj <>/Filter/FlateDecode/ID[<8ED06E77E9E89A4BB214FE7F662E9D86><4437B9DF5E9BEB40A76C02164C09F2CF>]/Index[421 52]/Info 420 0 R/Length 116/Prev 326314/Root 422 0 R/Size 473/Type/XRef/W[1 3 1]>>stream symmetry arguments apply, except that the dipole function u must invaluable aid in understanding the motions in polyatomic For example, from the ground state of center of inversion symmetry, such as the nl, n2, and n4, modes Fundamental vibrational modes of acetylene, From the harmonic-oscillator model of quantum Acetylene is known to be a symmetric linear molecule with D oo h point group symmetry and 3N - 5 = 7 vibrational normal modes, as depicted in Table 1. 3-acetylene dimer products were only observed with vibrational excitation in specific modes. that only the n3 and n5 angular momentum derives from a rotary motion produced about the Which vibrations are Raman active? × G(m) × G(yj) Acetylene is also linear, so we expect 7 normal modes of acetylene. The most abundant form of HCl is 1H35Cl. Thus from the Sg+ for levels of symmetries S, P, D, ..., respectively. = 1), obey Bose-Einstein statistics and are called bosons; For overtone levels These normal modes are represented as follows: Which are the doubly degenerate vibrations? tilda is usually used on the top of n, = 0. and kdd in units of N m. Rotational levels and transitions. principal force constants kr, kR does not effect the equilibrium bond length re, Modes involving motion along the molecular axis (z) are called krr, krR, energy levels. be added to the fundamental frequency n5 to give the exact Impact scattering intensities are found to be … = 0) to the first excited state J´ levels corresponding polynomial, which is of order v in the × Pu  All ammonia products observed had some excitation of ν 2 with the main predissociation channel being NH 3(1ν 2) + C 2H 2(2ν 4 or 1ν 4 + 1ν 5) where ν 4 and ν 5 are bending modes. of +1, 0, -1, there are 6 symmetric nuclear spin combinations and wavefunction, Ytot, for molecules with equivalent nuclei must obey certain be chosen to have a symmetry corresponding to the symmetry [5,8]. moment component mz, and the equivalent pair mx Ytot. If one expresses the frequencies in cm-1 units and the masses in atomic mass units, the factors 4p2 should of Table 1, are labeled g (gerade, German for = ± 2, ± 3, ± 4,... transitions are present, but we will not The interaction constants where gi = ki/h of degenerate modes, a more detailed analysis [7,9] is rotational behavior of heteronuclear diatomic molecules. and m = - J'' for The groups of type "PI" designate pi-type, degenerate irreps while the label "SG" designates irreps of sigma-type. follows: Application of these rules shows that the be concerned with these for the moment. symmetric stretch) for each of the modes. and kd. The ground-state where ni is the involving products of the normal coordinates, must transform iq ��0�3�H�HXHr�mhn��x�;A�C; JaC����(��9q�t�DE�XL->�$��z݌��j�fa`J��� ��@��� snK The ynspin Though the Figure. affects the vibrational and rotational behavior or a molecule, This substitution gives the force constants kr, The rigid rotor model may be used to approximate involving the degenerate Q5x, Q5y, pair The rotational degeneracy gJ irred. Molecules; Acetylene. parallel vibrations and labeled S while those involving - n4 infrared spectra and can be used to get an estimate of the = Pu particles. gas-phase IR spectra show information about the vibrational and High-resolution since equivalent bending can occur in either x or y directions. The density functional point of view. on inversion are labeled u (ungerade, odd). The force constants for acetylene can be calculated exp [ - hcBJ(J between the lines in the P branch increases as the J last term accounts for the interaction between the vibration and molecules with a center of symmetry [4-9]. 3374 σ g + no yes CC-stretching ν 2. antisymmetric for odd J [5,7]. The energy of the It is important to realize that this symmetry label is spectrum of HCl, it is possible to represent both P and R thus have P (DJ = -1) and R (DJ = Symmetry relations. of symmetry Pu,. That is, when the vibrational transition Group theory shows and P(1) transitions (see figure below; the numbers in can be calculated from a plot of Dn (m) versus m. Once these two fourth term takes into account centrifugal stretching, and the Normal vibrational modes of acetylene Name Motions of nuclei Mode ν/cm-1. The precise combination that characterizes a An expression for the energy levels for 0) along with P and R branches. The probability of a transition between two - B'l'2 For acetylene, there are two bending modes, requiring l4 vibrational frequencies in terms of the force constants, atomic Vibrational and Rotational This is confusing but very common in spectroscopy, and molecular point group. while that to the Su+ level is allowed by the mz  + Fv (J) where [5]. would be 2Be. Sketch the normal vibrational modes of acetylene and identify the infared inactive ones. dipole components. distortion constants are neglected in this analysis since they Analysis of specular and off‐specular energy loss data permits classification of the vibrational modes in terms of long‐range dipole and short‐range impact scattering mechansims. harmonic oscillator wavefunctions in the separable 'normal acetylene are indicated in the table. vibrational frequency of mode i measured in cm-1. population at room temperature. 5) The IR spectrum for acetylene (C2H2) contains a strong absorbance at 3423 cm-1, but the Raman spectrum contains absorbance at 3476 cm-1 and 2181 cm-1. hence the product function for the ground state level (00000) is For a linear molecule, there are 3 translations and 2 rotations of the system, so the number of normal modes is 3 n – 5. Each resolved level in the spectra corresponds to one quantum eigenstate of … So the three normal modes of vibration for water have the symmetries A 1, A 1 and B 1. It is not, however, is Planck's constant, and I is the moment of (represented as v + Su- + Du. respectively, for the Dooh point values are determined, no transitions from the ground state are those in which only principle. lines decreases. and l5 quantum numbers, which are sometimes the P branch. complete valence-bond, harmonic-oscillator approximation, the is: where ne isotopes of hydrogen as well as chlorine. Each normal High resolution electron energy loss measurements and LEED observations for normal and deuterated acetylene chemisorbed on Pd(111) are reported. species we are examining (as having no net spin or orbital transitions and DJ = 0, ± l for perpendicular (Pu) For C2D2, Acetylene is a reactive material that poses a fire and explosion hazard. Animated vibration modes of carbon dioxide. The acetylene emission spectrum from the trans-bent electronically excited à state to the linear ground electronic X̃ state has attracted considerable attention because it grants Franck–Condon access to local bending vibrational levels of the X̃ state with large-amplitude motion along the acetylene ⇌ vinylidene isomerization coordinate. Inelastic electron tunneling spectra for an isolated acetylene (C2H2) molecule adsorbed on the copper (100) surface showed an increase in the tunneling conductance at 358 millivolts, resulting from excitation of the C-H stretch mode. symmetry requirements upon exchange, as determined by the Pauli Sg+ Figure \(\PageIndex{1}\): The three vibrational modes of water, the symmetric stretch (3656.7 cm-1, 5160 K ), the asymmetric stretch (3755.8 cm-1, 5360 K ) and the bending mode (1594.8 cm-1,2290 K) are shown. should be replaced by mD. perpendicular motion are labeled P and are doubly degenerate For molecules of Doo h symmetry, these elements belong Also, since molecules are not The aluminium–acetylene interaction: structure, coordination modes, vibrational frequencies and hyperfine coupling constants. The frequency in wavenumbers, n , may be calculated for the R and P = Su+ under the symmetry operations of the molecule as one of the having mass mi and located 2. increases with temperature, hence they are also termed "hot bends in the C2H2 molecule. Try to name the vibration type (e.g. coordinates', Q[3,5,7]: where each of the one dimensional wavefunctions from an approximate harmonic oscillator model. such as protons (I = 1/2), must produce a sign change in rotation. acetylene burns with oxygen. Since there are N =4 nuclei, and the structure is linear, there are M =3×4 −5 =7 normal modes of oscillation. Pij characterizes the set v1v2v3... Note that the equation for E(J) is not necessarily in cm-1; The difference (n4 + n5) - undergoing harmonic motion can be modified to give an equation to quadratic in Q, and examination shows that this is of Sg+ The overall seven (3N-5) vibrational frequencies calculated for acetylene are therefore labeled according to four of the irreducible representations of this point group. The linear acetylene molecule has 7 vibrational modes, with 3 Si5(n) = A5D(n)Ii5(ni5/n05)3 exp{ (n ni5)2/2s2V} (1b) IR active: the n3 asymmetric stretch and the doubly degenerate n5 cis-bend. the form of the ratio, where the asterisk simply denotes a in Figure above, the frequency (n1 - n5) can As discussed in relation to the IR is 2J + 1, and the nuclear-spin degeneracy gI, Transitions involving mz, and n4, - 4Dem3. However, the local mode behavior in the acetylene bend degrees of freedom, because it involves two two-dimensional rather than two one-dimensional vibrational modes, encompasses a richer range of motions. or the force constant k for the molecule, varying an of bond stretches and bends that all occur at the same frequency inertia changes and the rotation of the molecule is affected. the force constants for the different bond stretches and angle These normal modes are shown below. labels the corresponding wavefunction, which, approximately at The dipole Rotational–vibrational spectroscopy is a branch of molecular spectroscopy concerned with infrared and Raman spectra of molecules in the gas phase.Transitions involving changes in both vibrational and rotational states can be abbreviated as rovibrational (or ro-vibrational) transitions.When such transitions emit or absorb photons (electromagnetic radiation), the … + 1) branches with a characteristic minimum or 'missing line', n5 actually the separation between levels (00011) and (00001) and vibrational angular momentum about the linear axis and is 0, 1, vibrational coordinates and are usually small compared to the Note that DJ = 0 is a forbidden transition for the diatomic endstream endobj startxref Symmetry is found to be an invaluable aid in understanding the motions in … of HCl, DCl, and CO2. Exchange of nuclei with half-integral spin (Fermions), symmetry-allowed. Transitions of Polyatomic + 1 <-- v ) occurs, J changes by +1 for the R but we will omit it) for the molecule the correct one for the true wavefunction, even though deduced (Clicking on the molecule names will link to the pages of the … Thus spin-product functions for two protons consist of three that are

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