Spin, angular momentum multiplicity

Singlet and triplet states refer to those states that have a total electron spin quantum number equal to 0 and 1, respectively. The spin multiplicity calculated as 25 - - 1, where S corresponds to the spin angular momentum, corresponds to 1 and 3 for the singlet and triplet states, respectively. 

For a given total electron spin quantum number (S), the multiplicity is the number of possible orientations of the spin angular momentum for the same spatial electronic wavefunction. Thus, the multiphcity equals 25 -F 1. For... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

The dynamic state is defined by the values of certain observables associated with orbilal and spin motions of the electrons and with vibration and rotation of [lie nuclei, and also by symmetry properties of the corresponding stationary-state wave functions. Except when heavy nuclei ate present, the total electron spin angular momentum of a molecule is separately conserved with magnitude Sh. and molecular slates are classified as singlet, doublet, triplet., . according to the value of the multiplicity (25 + I). This is shown by a prefix superscript lo the term symbol, as in atoms. 

In section 7.4.2 we dealt with perhaps the simplest contribution to the effective Hamiltonian for a particular electronic state, that of the rotational kinetic energy. We now turn our attention to contributions which are only slightly more complicated, the so-called fine structure interactions involving the electron spin angular momentum S. Obviously for S to be non-zero, the molecule must be in an open shell electronic state with a general multiplicity (2S +1). 

Here, as elsewhere, the subscripts i and a stand for electrons and nuclei respectively. The factor (S Si)/S(S + 1) is used to project the contribution from each open shell electron i onto the total spin angular momentum S. We remind ourselves that the effective Hamiltonian is constructed to operate within an electronic state with a given multiplicity (2S +1). 

Multiplicity Spin Multiplicity) The number of possible orientations, calculated as 2S -L 1, of the spin angular momentum corresponding to a given total spin quantum number (S), for the same spatial electronic wavefunction. A state of singlet multiplicity has S = 0 and 2S -i- 1 = 1. A doublet state has S = 1/2, 2S -i- 1 = 2, etc. Note that when S > L (the total orbital angular momentum quantum number) there are only 21 -t 1 orientations of total angular momentum possible. 

Strictly speaking, however, the spin angular momentum and its components are not constants of motion in nonlinear molecules, and the classification of states by multiplicity is therefore only approximate. Spin-orbit coupling is the most important of the terms in the Hamiltonian that cause a mixing of zero-order pure multiplicity states. The interaction between the spin angular momentum of an electron and the orbital angular momentum of the same electron causes the presence of a minor term in the Hamiltonian, which may be written as... 

The discovery of this interaction led to the postulation that an atomic nucleus possesses a spin-angular momentum represented by the spin angular momentum vector h, where I is the nuclear spin and h is Planck s constant, h, divided by Itt. It has been found experimentally that I is an odd integer multiple of for nuclei of odd atomic mass numbers (isotope number), zero for nuclei of even atomic mass numbers and even nuclear charges (atomic number), and an integer for nuclei of even atomic mass numbers and odd nuclear charges. The nuclei that we are concerned with here, H, C, and F, have an I of 5. 

The superscripts denote the number of electrons in each atomic orbital. Electrons must be paired (i.e., of equal and opposite spin) in atomic orbitals containing two electrons. Paired electrons do not contribute to the atomic or molecular spin angular momentum. Hund s rule of maximum multiplicity states that in the lowest energy configuration, the electrons must be spread over as many available orbitals of equal energy as possible, in order to maximize the spin multiplicity. Since three 2p orbitals are available in which 4 electrons must be distributed, the lowest electronic state therefore has two unpaired electrons in each of two 2p orbitals. Each unpaired electron contributes to the spin angular momentum. Thus 2S + 1 = 3 for the oxygen atom, and the term symbol is 3P. 

The vector points from nucleus a to electron i, and the vector r- points from electron j to electron i Pj is the linear momentum vector operator of electron i, and Sj is the spin angular momentum vector operator of electron i. The presence of electron spin operators in allows it to change the spin part of an electron wave function on which it operates, and hence the operator is able to mix wave functions of different spin multiplicities. The spin-orbit coupling term contains a one-electron part and a two-electron part H<, and its mathematical form accounts for the origin of the term spin-orbit coupling. 

The multiplicity of the state is 2S + 1, the number of possible values of the z component of the spin angular momentum. 

For a state with the total spin angular momentum 5=1, the spin multiplicity Ms is three (there are three Ms values 1, 0, —1). The spin multiplicity is equal to 25 + 1. The following examples should make the use of the symbols clear for Mi = 4, 5= 1/2, the symbol is G for = 2, 5 = 3/2, the symbol is D for Mi = 0, 5 = 1, the symbol is 5. In speaking of states with spin multiplicities of 1, 2, 3, 4,. .. we call them singlets, doublets, triplets, quartets,. . ., respectively. Thus, the three example states above should be called doublet G, quartet D, and triplet 5, respectively. 

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