Explain the origin of single and triplet states in a hydroge

Explain the origin of single and triplet states. in a hydrogen-like atom, all orbitals with the same principle quantum number have degenerate energies This is not the case for many electron atoms. What specific factor causes subshells to reorder in energy for multi-electron atoms? Explain how the layout of the periodic table is related to the Aufbau prmdpte. Write the Slater determinant for a He atom with two electrons. What is the most important reason that a Slater determinant ts helpful m represert.rg the total wave functions for electrons in atomic orbitals?

Solution

Answer to Question 6)

In atoms with more than one electron, interaction between the electrons means that the transition energies are not simply related to differences between the energies of the different energy levels.

The electrons in different orbitals can interact with each other in a range of possible ways, depending on the relative orientations on the electrons spins, and also the nature of the orbitals occupied. In a two electron system, the are two possible types of pairing of the electrons, when in different orbitals, and these are known as singlet and triplet states.

Let us consider the example of Helium.

The ground state electronic configuration is 1s², and the first excited state is 1s¹2s¹. However, whilst the electrons must have paired spins when both occupy the 1s orbital, they need not be when one occupies the 1s orbital and the other occupies the 2s orbital.

Hund\'s rule states that the configuration with the electron spins parallel is at a lower energy than the configuration with paired spins. When the electron spins are paired, their individual spin momenta cancel each other out, and there is no overall spin.

There is only one orientation in which this may be achieved, and hence this configuration is known as a singlet state. When the electron spins are parallel, their individual spin momenta add together to give non-zero total spin, and this can be achieved in three ways. Thus, the state is known as a triplet state.

Answer to Question 7)

In a multi electron system it is the azimuthal quantum number which leads to reordering of energy. The effective charge, Z_eff, decreases with increasing value of , the azmuthial quantum number. This is because electrons in the s-orbital have a greater probability of being near the nucleus than a p-orbital, so the s-orbital is less shielded than ap-orbital. Likewise, a p-orbital is less shielded than a d-orbital.

In a multi-electron atom, the energy of an orbital increases with increasing value of for a given value of n.

In atoms with more than one electron, interaction between the electrons means that the transition energies are not simply related to differences between the energies of the different energy levels. The electrons in different orbitals can interact with each other in a range of possible ways, depending on the relative orientations on the electrons spins, and also the nature of the orbitals occupied. In a two electron system, the are two possible types of pairing of the electrons, when in different orbitals, and these are known as singlet and triplet states. Let us consider the example of Helium. The ground state electronic configuration is 1s2, and the first excited state is 1s12s1. However, whilst the electrons must have paired spins when both occupy the 1s orbital, they need not be when one occupies the 1s orbital and the other occupies the 2s orbital. Hund\'s rule states that the configuration with the electron spins parallel is at a lower energy than the configuration with paired spins. When the electron spins are paired, their individual spin momenta cancel each other out, and there is no overall spin. There is only one orientation in which this may be achieved, and hence this configuration is known as a singlet state. When the electron spins are parallel, their individual spin momenta add together to give non-zero total spin, and this can be achieved in three ways. Thus, the state is known as a triplet state. Answer to Question 7) In a multi electron system it is the azimuthal quantum number which leads to reordering of energy. The effective charge, Zeff, decreases with increasing value of , the azmuthial quantum number. This is because electrons in the s-orbital have a greater probability of being near the nucleus than a p-orbital, so the s-orbital is less shielded than ap-orbital. Likewise, a p-orbital is less shielded than a d-orbital. In a multi-electron atom, the energy of an orbital increases with increasing value of for a given value of n.