Helium Energy Levels

The electron energy levels for a helium atom demonstrate a number of features of multi-electron atoms.

One electron is presumed to be in the ground state, the 1s state. An electron in an upper state can have spin antiparallel to the ground state electron (S=0, singlet state, parahelium) or parallel to the ground state electron (S=1, triplet state, orthohelium).

Why are the orthohelium states lower? 为什么正态氦态的能量更低？ Comments on helium energy levels 关于氦能级的评论

Orthohelium and Parahelium Energy Levels

In the helium energy level diagram, one electron is presumed to be in the ground state of a helium atom, the 1s state. An electron in an upper state can have spin antiparallel to the ground state electron (S=0, singlet state, parahelium) or parallel to the ground state electron (S=1, triplet state, orthohelium).

It is observed that the orthohelium states are lower in energy than the parahelium states. The explanation for this is:

1. The parallel spins make the spin part of the wavefunction symmetric.
   自旋平行使波函数的自旋部分对称。
2. The total wavefunction for the electrons must be anti-symmetric since they are fermions and must obey the Pauli exclusion principle.
   电子的总波函数必须是反对称的，因为它们是费米子，必须遵守泡利不相容原理。
3. This forces the space part of the wavefunction to be anti-symmetric. The wavefunction for the electrons can be written as the product of the space and spin parts of the wavefunction.
   这迫使波函数的空间部分必须是反对称的。电子的波函数可以写成空间部分和自旋部分的乘积。
4. An anti-symmetric space wavefunction for the two electrons implies a larger average distance between them than a symmetric function of the same type. The probability is the square of the wavefunction, and from a simple functional point of view, the square of an antisymmetric function must go to zero at the origin. So in general, the probability for small separations of the two electrons is smaller than for a symmetric space wavefunction.
   一个反对称的空间波函数对于两个电子意味着它们之间的平均距离比相同类型的对称函数更大。概率是波函数的平方，从简单的函数观点来看，反对称函数的平方在原点处必须趋于零。因此，总体而言，两个电子小距离的概率比对称空间波函数的小。
5. If the electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy.
   如果电子平均更远离核，则地面态电子对核的屏蔽作用会减少，激发态电子因此会更暴露于核。这表明它会更紧密地结合，具有更低的能量。

This effect is sometimes called the "spin-spin interaction" and is addressed by Hund's Rule #1 . It is part of the understanding of the ordering of energy levels in multi-electron atoms.

Helium Energy Levels

The helium ground state consists of two identical 1s electrons. The energy required to remove one of them is the highest ionization energy of any atom in the periodic table: 24.6 electron volts. The energy required to remove the second electron is 54.4 eV, as would be expected by modeling it after the hydrogen energy levels. The He+ ion is just like a hydrogen atom with two units of charge in the nucleus. Since the hydrogenic energy levels depend upon the square of the nuclear charge, the energy of the remaining helium electron should be just 4x(-13.6 eV) = -54.4 eV as observed.

The fact that the second electron is less tightly bound can be interpreted as a shielding effect; the other electron partly shields the second electron from the full charge of the nucleus. Its energy can be used to model the effective shielding as follows.

Another way to view the energy is to say that the repulsion of the electrons contributes a positive potential energy which partially offsets the negative potential energy contributed by the attractive electric force of the nuclear charge. The description of any electron in a multi-electron atom must find a way to characterize the effect of the other electrons on the energy.