SINGLET AND TRIPLET STATES FOR TWO ELECTRONS

An angular momentum is a vector, and this pertains also to spin angular momenta (see Chapter 1). The spin angular momentum of a certain number of elementary particles is the sum of their spin vectors. Tb obtain the total spin vector, we therefore have to add the x components of the spins of the particles, similarly for the y and z components, and to construct the total vector from them. Then we might be interested in the corresponding spin operators. These operators will be created using Pauli matrices.  

In this way we find immediately that, for a single particle, the following identity holds 

For any stationary state the wave function is an eigenfunction of the square of the total spin operator and of the z-component of the total spin operator. The one and two-electron cases are the only ones for which the total wave function is the product of space and spin parts. 

The maximum projection of the electron spin on the z axis is equal to (in a.u.). Hence, the maximum projection for the total spin of two electrons is equal to 1. This means that in this case only two spin states are possible the singlet state corresponding to S = 0 and the triplet state with S = 1 (see Postulate V). In the singlet state the two electronic spins are opposite ( pairing of electrons ), while in the triplet state the spin vectors are parafiel (cf. Fig. 1.11 in Chapter 1). As always, the possible projection of the total spin takes one of the values Ms — 

Now it will be shown that the two-electron spin function a(l)j8(2) — a(2)j8(l) ensures the singlet state. First, let us construct the square of the total spin of the two electrons  

---

NH . In a theoretical study with the aim of interpreting Hund s rule for molecular systems, SCF calculations were carried out for the low-lying singlet and triplet states of two-electron, first-row hydride cations including NH [9]. 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

Let us consider e.g. a two-orbital two-electron model system with the orbitals a and b which can be understood as notation for one-dimensional irreducible representations of the point group of a TMC. In this case it is easy to see that the corresponding singlet and triplet states and (T = B, S = 0,1) are given correspondingly by ... 

IR bands for the both states. For example, singlet and triplet states of 2-naphthyl-(methoxycarbonyl)carbenes (NMCs, 17) show very different IR spectra (Scheme 9.4) (1590, 1625, 1640 cm for NMC and 1660 cm for NMC). Both states are observable in this case. In the singlet state, the carbomethoxy group assumes a conformation perpendicular to the naphthylcarbene plane to avoid destabilization of the empty carbenic 2p atomic orbital by the electron-withdrawing carbonyl group, while in the triplet, the methoxycarbonyl group is in the same plane to delocalize an unpaired electron. For this reason, there is a barrier between the two states and hence both of them are observable under these conditions. 

---