The Use of Self-consistent Field Methods to Obtain Atomic Orbitals

8 The Use of Self-consistent Field Methods to Obtain Atomic Orbitals 

In a many-electron atom an approximate wavefunction, 0., can be obtained for the ith electron by solving the following one-electron wave 

The first term on the left-hand side represents the kinetic energy of the electron, and the second term the potential energy, e. is the energy of the electron in the fth orbital. The potential energy term, F(r ), is calculated by assuming that the electrical repulsion resulting from all the other electrons can be represented by a static, spherically symmetric charge distribution. A difficulty with this procedure is that it is impossible to calculate V(r.) until the one-electron wavefunctions of all the other electrons are known. Thus, all the one-electron wavefunctions derived from equation (7.38) are interrelated, which makes the problem difficult to solve. 

In 1928, D. R. Hartree got around this difficulty by adopting the following procedure. First, all the one-electron wavefunctions were estimated, using effective nuclear charges. These wavefunctions were then used to calculate the potential energy term for the first electron. This was substituted into equation (7.38), and an improved wavefunction for electron 1 was calculated. This was then used to calculate the potential energy term for the second electron, and hence to improve its wave-function. This process was repeated for all the electrons until self-consistency was obtained, that is, further repetition of the process did not produce any changes in the wavefunctions. 

Hartree incorporated the Pauli principle by allowing no more than two electrons to be present in each orbital, but the wavefunctions that he used did not involve spin, and were not antisymmetric with respect to interchange of electrons. In 1930, V. Fock modified Hartree s approach by using fully antisymmetric spin orbitals that did not distinguish between electrons. This improved way of calculating atomic orbitals is known as the Hartree Fock self-consistent field (SCF) method. Nowadays, fast computers are used and procedures are followed which allow the one-electron wave equations to be solved simultaneously. 

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