When one of the two 1s electrons in helium is placed in the 2s orbital, two diff

Question

When one of the two 1s electrons in helium is placed in the 2s orbital, two different electronic states are obtained. One is the singlet state, in which the two electrons are indistinguishable and spin-paired: psi_singlet = 1/Squareroot 2 [1s(1)2s(2) + 2s(1)1s(2)] 1/Squareroot 2 [alpha(1) beta(2) - beta(1)alpha(2)] The other is the triplet state, which is really three different electronic states that are degenerate: psi_triplet(+1) = 1/Squareroot 2 [1s(1)2s(2) - 2s(1)1s(2)][alpha(1)alpha(2)] psi_triplet(0) = 1/Squareroot 2 [1s(1)2s(2) - 2s(1)1s(2)] 1/Squareroot 2 [alpha(1)alpha(2) + beta(1) alpha(2)] psi_triplet(-1) = 1/Squareroot 2 [1s(1)2s(2) - 2s(1)1s(2)][beta(1) beta(2)] Remembering the 1s(1) is shorthand for 1s(x_1, y_1, z_1) and 1s(2) is shorthand for 1s(x_2, y_2, z_2), etc., show that the spatial factor of the triplet wavefunctions psi^spatial_triplet = 1/Squareroot 2 [1s(1)2s(2) - 2s(1)1s(2)] vanishes when the two electrons are at the same location. In other words show that this factor goes to zero when x_1=x_2, y_1=y_2, and z_1=z_2. This behavior tells us that the electrons automatically avoid each other in the triplet state. Next, consider the singlet state spatial wavefunction psi^spatial_triplet = 1/Squareroot 2 [1s(1)2s(2) + 2s(1)1s(2)]. Show that this wavefunction does NOT go to zero automatically as the electrons approach the same spatial location. This phenomenon is the fundamental reason why the triplet state lies lower in energy than the singlet state in the 1s2s excited states of helium, and in other systems in which one electron has been excited out of a closed shell atom or molecule.

Explanation / Answer

Note that the triplet spatial wavefunction is zero when the two electrons are at the same position:

r1 = r2 T = 1(1s (r1 )2s (r1 ) 1s (r1 )2s (r1 )) = 0

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whereas the singlet wavefunction is nonzero:

r1 = r S = 1(1s (r1 )2s (r1 ) +1s (r1 )2s (r1 )) = 2 1s (r1 )2s (r1 ) 0

Because the electrons repel each other more when they are close to one another, we therefore expect the singlet to have more electron electron repulsion and a higher energy. This rule turns out to hold quite generally and is called Hund’s rule: for degenerate non interacting states, the configuration with highest spin multiplicity lies lowest in energy. Hund actually has three rules (of which this is the first) concerning the ordering of degenerate non interacting states. So we expect the triplet to be lower.

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