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Ideas of Quantum Chemistry P105

Ideas of Quantum Chemistry P105 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | Q. 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. To 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.1 In this way we find immediately that for a single particle the following identity holds S2 Si Sj Si Si - - hSz Q.1 lowering and where S2 S and .S and S- are the lowering and raising operators respectively raising i s s s operators .S .Sx iSy Q.2 - Sx - iSy Q.3 which satisfy the useful relations justifying their names S a 0 S ft h a S-a h ft S-ft 0. 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 i 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 parallel cf. Fig. 1.11 in Chapter 1 . As always the possible projection of the total spin takes one of the values MS 1See Postulate VI in Chapter 1. 1006 Q. SINGLET AND TRIPLET STATES FOR TWO ELECTRONS 1007 -S -S 1 . S i.e. MS 0 for the singlet state and MS -1 0 1 for the triplet state. Now it will be shown that the two-electron spin function a 1 2 - a 2 1 ensures the .