Selection rules hydrogen atom

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

The hydrogenolyaia of cyclopropane rings (C—C bond cleavage) has been described on p, 105. In syntheses of complex molecules reductive cleavage of alcohols, epoxides, and enol ethers of 5-keto esters are the most important examples, and some selectivity rules will be given. Primary alcohols are converted into tosylates much faster than secondary alcohols. The tosylate group is substituted by hydrogen upon treatment with LiAlH (W. Zorbach, 1961). Epoxides are also easily opened by LiAlH. The hydride ion attacks the less hindered carbon atom of the epoxide (H.B. Henhest, 1956). The reduction of sterically hindered enol ethers of 9-keto esters with lithium in ammonia leads to the a,/S-unsaturated ester and subsequently to the saturated ester in reasonable yields (R.M. Coates, 1970). Tributyltin hydride reduces halides to hydrocarbons stereoselectively in a free-radical chain reaction (L.W. Menapace, 1964) and reacts only slowly with C 0 and C—C double bonds (W.T. Brady, 1970 H.G. Kuivila, 1968). 

So far we have considered only hydrogen, helium, the alkali metals and the alkaline earth metals but the selection rules and general principles encountered can be extended quite straightforwardly to any other atom. 

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

As a second example of the determination of selection rules from the properties of special functions, consider the hydrogen atom. At any given instant the dipole moment is ft = er, where r describes the position of the electron with respect to the proton and e is the electronic charge. The wavefiinctions for the hydrogen atom are given by... 

Verify the selection rules for the hydrogen atom as given in the last paragraph of Section 12.3.3. 

Since n is not a whole number, we can conclude that the hydrogen atom does not orbit at a radius of 4.00 A (i.e., such an orbit is forbidden by selection rules). 

Let us see the migration of a hydrogen atom and how the selection rules have been put forward to explain the formation of an imaginary transition state in a sigmatropic reaction. 

So it is the number of electrons and not the number of atoms which determines the selection rule. Therefore, the selection rules for hydrogen migration in thermal sigmatropic shifts can be summarized as follows as given in the table ... 

Combining these results, we have the hydrogen-atom electric-dipole selection rules ... 

For the hydrogen atom, the parity is determined by /. (See Section 1.17.) The selection rule that A/= 1 is in agreement with the rule that parity changes in electric-dipole transitions. 

The ene reaction has proved to be particularly powerful in synthesis when carried out intramolecularly. The usual increase in rate for an intramolecular reaction allows relatively unreactive partners to combine. Thus the diene 6.13 gives largely (14 1) the cis disubstituted cyclopentane 6.15 by way of a transition structure 6.14. It is important to recognize that the selective formation of the ci j-disubstituted cyclopentane has nothing to do with the rules for pericyclic reactions. It is a consequence of the lower energy when the trimethylene chain spans the two double bonds in such a way as to leave the hydrogen atoms on the same side of the folded bicyclic structure. This... 

This is the first example we will encounter of selection rules for allowed transitions. Physically, the selection rules arise because the electric field needs a dipole moment in order to interact with the atom, and only these specific changes in the quantum numbers create a dipole moment. More generally, the selection rules for absorption in a hydrogen atom are... 

A hydrogen atom with its electron in a 2p orbital will decay back down to the Is orbitral in approximately 1 nanosecond, giving off a photon with A = 121 nm (determined by the energy difference between the two states). On the other hand, an electron in a 2s state is stuck (we call 2s a metastable level), since emission to the only lower state (1. s ) is forbidden by the Al = 1 selection rule. On average, it takes about 100 ms for the electron to get back down to the ground state from 2s. 

According to the selection rules, one-photon absorption occurs only if the change in angular momentum (change in L) is +1 or -1 (Al = 1, A/ = 0, 1 (0 o 0 not allowed), AL = 0, 1, AS = 0) (Al is according to the hydrogenic atom model, whereas AL is for multielectron atoms). The selection rules allow transition in one-photon absorption only to the p states from the s ground state as a result only even-to-odd parity is allowed. 

Eormation of a Cyclic Transition State Structure Erontier Orbital Approach Some Examples of Hydrogen Shifts Migrations in Cyclopropane rings Migrations of Atoms or Groups other than Hydrogen Selection Rules... 

The scattering is purely kinematic there is no interaction of the neutron with the electrons. Thus, there are no selection rules, and all modes are allowed. Modes such as torsions, out-of-plane bends, and skeletal deformations often give intense IINS features, because a small angular motion of the atom to which the hydrogen is... 

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