{"id":249,"date":"2018-11-15T10:31:24","date_gmt":"2018-11-15T10:31:24","guid":{"rendered":"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=249"},"modified":"2019-04-30T08:07:15","modified_gmt":"2019-04-30T08:07:15","slug":"electronic-spectra-of-diatomic-molecules-i","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/chapter\/electronic-spectra-of-diatomic-molecules-i\/","title":{"rendered":"Electronic Spectra of Diatomic Molecules-I"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\nContents:<\/strong>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 <\/strong>The Born-Oppenheimer Approximation<\/strong>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 <\/strong>Classification of Electronic States<\/strong>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 <\/strong>Vibrational Structure of Electronic Transitions<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

The students will be able to learn about the classification of electronic states, vibrational energy levels associated with two electronic states. The vibrational transitions accompanying an electronic transition called vibronic transitions that are divided into progressions and sequences.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n1. The Born-Oppenheimer Approximation<\/strong>\r\n\r\n \r\n

Two or more atoms bind together to form a molecule in such a way that the total energy is lower than the sum of the energy of the constituents. The electrons of the inner shell of atoms forming molecules remain localized about each nucleus, though the valence electrons are distributed throughout the molecule and charge distribution of these electrons provide the binding force. These bonds are ionic or covalent nature and the force experienced by the electrons and nuclei is of comparable intensity. As the nuclei are about two thousand times heavier than the electrons, the motion of the nuclei is much slower than that of electrons. Hence, it is assumed that nuclei occupy fixed position in the atom.<\/p>\r\n \r\n\r\nThe Hamiltonian for the molecule can be written as\r\n\r\n \r\n

H = Tn<\/sub> + Te<\/sub> + Vee<\/sub> + VeN<\/sub> + Vnn<\/sub><\/span>\u2026\u2026\u2026\u2026(a)<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\nWhere Tn= kinetic energy operator of all the nuclei,\r\n\r\nTe<\/span>= kinetic energy operator of all the electrons<\/span>\r\n\r\nVee<\/span>= potential energy operator for coulumbic\u00a0<\/span>interaction between electron<\/span>\u2013 electron<\/span>\r\n\r\nVeN = potential energy operator for coulombic attraction of all electrons and all nuclei<\/span>\r\n\r\nVnn =coulombic repulsion between nucleus- nucleus<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\nIf \u03c8 is an eigenfunction of H then\r\n\r\n<\/div>\r\n
\r\n

H\u03c8(r,R) = E\u03c8 (r,R)\u00a0\u00a0\u00a0\u00a0 \u2026\u2026(b)<\/p>\r\n \r\n

Here r is electronic coordinate and R is internuclear distance The coordinates of the electrons and nuclei cannot be separated out because of term involved in Coulombic interaction between electrons and nuclei. Born and Oppenheimer were able to show that an approximate solution of Eq. (a) can be obtained by first solving the equation for the electrons alone with nuclear fixed positions. Therefore, the operator Tn<\/sub> is neglected assuming nuclear mass is infinite and hence the potential Vnn<\/sub> is constant. The Hamiltonian<\/p>\r\n \r\n

HE<\/sub>= Te<\/sub> + Vee<\/sub> + Ven<\/sub><\/p>\r\n \r\n\r\nand Schr\u00f6dinger wave equation is\r\n

HE<\/sub>\u03c8E<\/sub>(r,R) = EE<\/sub>(R) \u03c8E<\/sub>(r,R)<\/p>\r\n \r\n

However, VeN depends on the position of the nuclei. Ee(R) can be obtained for different values of internuclear distances (from Eq. (b)). The potential energy can be obtained by adding Cnnto Ee(R), So<\/p>\r\n\"\"\r\n

Here Z1<\/sub> and Z2<\/sub> are the atomic numbers of two nuclei 1 and 2, respectively. The value Ee<\/sub> decreases as the nuclei approaches each other because the attraction between electrons and nuclei increases and this is opposed by the repulsion between the two nuclei. There is an equilibrium bond length, that is, there will be a minimum distance at which these two balance. The energy rises sharply with the further decrease in the separation of nuclei. The electronic state is then a bound state. The curve representing the variation of Ee<\/sub> + Vnn<\/sub> is referred to as potential curve. The potential energy curve for each electronic state of the molecule has a different shape. The minima of potential curve falls at different equilibrium\u00a0internuclear distances, in case of stable states. The electronic state is unstable if the potential curve has no minimum. In obtaining potential energy curve, nuclear kinetic energy term is ignored. Thus, V(R) does not depend on the mass of the nuclei. If V(R) is found for a given molecule, it applies to all length within Born \u2013 variation as V(R) is same for H2 , HD and D2 also have same bond length within Born- Oppenheimer approximation.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

The variation of Ee<\/sub> as a function of R provides a part of the potential field in which nuclei move while Vnn<\/sub> provides the other part of this field. The Hamiltonian for the nuclear motion, is<\/p>\r\n \r\n

Hn<\/sub> = Tn<\/sub> + Vnn<\/sub> + Ee\u00a0\u00a0\u00a0<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (e)<\/p>\r\n \r\n\r\nand Schr\u00f6dinger wave equation is\r\n\r\n \r\n

Hn<\/sub>\u03a8 n<\/sub> (R) = (Tn<\/sub> + Vnn<\/sub> + Ee<\/sub>) \u03a8n<\/sub>= En<\/sub>\u03a8n<\/sub>(R)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (f)<\/p>\r\n \r\n

Here \u03a8n<\/sub> is a function of R only and represents the stationary state of the nuclei. The total wavefunction is<\/p>\r\n \r\n

\u03a8 = \u03a8e<\/sub> (r,R) \u03a8n<\/sub>(R)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (g)<\/p>\r\n \r\n\r\nThe total energy E is given by\r\n\r\n \r\n

E = Ee<\/sub>+ En<\/sub><\/p>\r\n \r\n\r\nThe wavefunction \u03a8n<\/sub>, in a first approximation, can be expressed as\r\n\r\n \r\n\r\n\u03a8n= \u03a8v<\/em>\u03a8r\r\n\r\n \r\n\r\nwhere\u03a8v<\/em> is the vibrational eigenfunction of a linear oscillator.\r\n\r\n \r\n\r\nThus to a first approximation\r\n\r\n<\/div>\r\n

\r\n

\u03a8= \u03a8e<\/sub> \u03a8v<\/em><\/sub>\u03a8r<\/sub><\/p>\r\n \r\n

The total energy E of the molecule apart from the translational energy is regarded as the sum of rotational energy Er<\/sub> , vibrational energy Ev<\/sub> , and the electronic energy Ee<\/sub>, that is,<\/p>\r\n \r\n

E = Ee<\/sub>+Ev+ Er<\/sub><\/p>\r\n \r\n

Where Ee<\/sub> is defined as the energy of stable electronic state corresponding to the minimum of V. The minimum of lowest electronic state is chosen as zero point of the energy scale. This choice differs from atom to atom. The total energy in terms of wavenumbers is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

As the electrons move faster than nuclei, the nuclei feel the potential energy of the averaged electronic distribution. This forms the basis of Born- Oppenheimer approximation. However, this approximation breaks down if electronic energy spacings are not large compared to vibrational spacings. This breakdown result in a situation where time scales of nuclear and electronic motions are not separable and it is not possible to separate nuclear and electronic energies.<\/p>\r\n \r\n\r\n2.\u00a0 <\/strong>Classification of Electronic States<\/strong>\r\n\r\n \r\n

In diatomic molecules, the electrons move in strong electrostatic field of nuclei that is considered to be cylindrically symmetric about the bond axis. The orbital angular momentum L<\/strong> precesses very rapidly about the direction of electrostatic field and only axial component of L is a constant of motion. The axial component field is characterized by the quantum number ML where ML = L, L-1, \u2026\u2026\u2026.., -L. As the internuclear field is of electrical nature, therefore energy is not changed by\u00a0the exchange of ML\u2194 \u2013ML. The absolute value of ML is designated by the symbol \u1d27 and \u1d27 =| ML|= 0, 1, 2, 3, \u2026\u2026\u2026..,L. The figure depicts the coupling of <\/span>L<\/strong> about the electric field along the internuclear axis producing the axial component \u1d27<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nThe symbols for the states are:\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n\u03a9\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n\"\"\r\n
\r\n

The quantized components MS of S, about the magnetic field direction, have value \u0127\u00a0 MS. The quantum number MS is designated by the symbol \u2211 that takes the values S, S \u2013 1, S \u2013 2, \u2026\u2026., - S. So, \u2211 can take 2S + 1 values. There is no resultant magnetic field for the electronic state \u2211 (\u22c0 = 0), and hence MS is not defined. These states have only one component, whatever be the multiplicity. Further, spin- orbit coupling can lift the degeneracy of 2S+1 \u22c0 state. The symbol \u03a9 is used to designate the quantum number of z component of total angular momentum(spin plus orbit) in diatomic molecules if coupling between L<\/strong> and S<\/strong> is weak. The quantum number \u03a9 is written as a subscript to the term symbol. Thus the term symbol is 2S+1\u22c0\u03a9 with \u03a9 = \u22c0+S, \u22c0+S \u2013 1 ,\u2026\u2026\u2026., \u22c0-S. As an example, a triplet \u2206state is split by spin orbit coupling into three doubly degenerate levels.<\/p>\r\n \r\n

The Vector addition of \u22c0 + \u2211 producing the total electronic angular momentum \u03a9 is shown in the figure.<\/p>\r\n \r\n

Symmetry properties of electronic functions need to be considered alongwith quantum numbers \u2211, \u22c0 and \u03a9. The reflection operator acing twice in succession on electronic wavefunction must give the original wavefunction, so,<\/p>\r\n\"\"The eigenvalue of 2 is 1. Thus the two eigenvalues of the operator are + 1 and -1. Then<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n\u03a8\u00a0 e+<\/sup>and \u03a8 e-<\/sup>represent the eigenfunction of with eigenvalues + 1 and -1, respectively,\r\n\r\n \r\n

The function \u03a8 +<\/sup>remains unchanged \u00a0while\u00a0 \u00a0\u03a8 - changes sign under reflection operation. The electronic states which are doubly degenerate (\u22c0> 0) may be distinguished as + or -, for example, \u2206+,<\/strong> \u2206- etc. The electronic staste \u2211is not degenerate but can still be classified as \u2211+or \u2211- .Homonuclear diatomic molecules also have inversion symmetry through the midpoint of the bond. With the midpoint taken as origin of the cartesian coordinate system and under inversion operation (xi, yi, zi) \u2192 (-xi, -yi, -zi) of electronic function twice in succession gives the same electronic function then<\/p>\r\n\r\n<\/div>\r\n

\r\n

i2<\/sup> \u03a8e<\/sub>= (+1) \u03a8e<\/sub><\/p>\r\n \r\n

The eigenvalues of operator i are therefore +1 and -1. The wavefunction either remains unchanged or changed sign under inversion. The wavefunction is called even or g (gerade) if it remains unchanged upon inversion. On the other hand if wavefunction changes its sign upon inversion it is called odd or u (ungerade). The symbols g and u are written as subscripts to the term value, like, \u2211g<\/sub>, \u2211u<\/sub>.<\/p>\r\n \r\n\r\nHund\u2019s Rule<\/strong>\r\n\r\n \r\n

(i) The interaction between nuclear rotation and electronic angular momentum is assumed to be weak and also weak spin- orbit coupling. L<\/strong> and S precess independently about the bond axis in diatomic molecules. Along the bond direction the projections of L<\/strong> and S<\/strong>, that is, \u22c0 and \u2211 respectively add together to give \u03a9. So\u00a0 J <\/strong>= R <\/strong>+\u03a9<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nHund\u2019s Rule (i)<\/strong>\r\n\r\n \r\n

(ii) It is assumed that coupling of S and \u22c0 is weak. S<\/strong> couples to the nuclear rotational axis. The magnetic field is also weak and S<\/strong> does not precess about the bond axis. S<\/strong> remains fixed in space. \u22c0is added vectorally to I<\/strong> giving a resultant angular momentum N<\/strong> and N<\/strong> = R<\/strong> + \u22c0 and then N<\/strong> combined with S<\/strong> to give total angular momentum J<\/strong>, So that<\/p>\r\n \r\n

J <\/strong>= N <\/strong>+ S<\/strong><\/p>\r\n\"\"\r\n\r\nHund\u2019s Rule (ii)<\/strong>\r\n\r\n \r\n

Conclusevly, in case of diatomic molecules, S, \u2211, \u22c0 and \u03a9 adequately represent the quantized electronic angular momentum. In an isolated molecule, however, the total angular momentum is due to vectorial coupling of nuclear angular momentum <\/span>I<\/strong>, electronic spin angular momentum <\/span>S<\/strong> and electronic orbital angular momentum <\/span>L<\/strong>.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3. Vibrational Structure of Electronic Transitions<\/strong>\r\n\r\n \r\n\r\nThe total energy E and term value T of the molecule are given by\r\n\r\n \r\n\r\nE = Ee<\/sub>+Ev<\/sub>+ Er\u00a0\u00a0\u00a0<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 and<\/strong>\r\n\r\n\"\"\r\n\r\n \r\n

The wave number of spectral line for a transition from the upper electronic state T1<\/sup>to lower electronic state T2<\/sup> is,<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

The spectral location corresponds to J = 0 \u2192J = 0 rotational transitions. In an electronic transition, there are no strict selection rules on <\/span>v<\/em> and it may take any positive or negative integer. For <\/span>v<\/em> = 0 \u2192 v = 0 transition<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n\"\"\r\n

The figure depicts a set of vibrational energy levels associated with two electronic states. The vibrational transitions accompanying an electronic transition are called vibronic transitions and are further divided into progressions and sequences.<\/p>\r\n\r\n