The Breit-Pauli Approximation

As in the one-electron case, it is possible (see for example Bethe and Salpeter 1957 and Armstrong and Feneuille 1974), to derive relativistic corrections to the non-relativistic many-electron Hamiltonian. The method is based on approximations that include all effects to order $(Z\alpha)^2$. The first approximation is the use of the Breit Hamiltonian described in the previous section. The second is the Pauli limit for the radial functions. The approximation is therefore called the Breit-Pauli approximation.

If we start with the radial functions, we need to consider two limits. First, the non-relativistic approach, where the small component is neglected and the large is substituted for the Schrödinger solution, P_S, viz

   $$P_D(r)\approx P_S(r) \quad Q_D(r)\approx 0$$

This is referred to as the first Pauli limit. Corrections to this   limit, to the power $Z\alpha$, are given by the second Pauli limit, and can be written as

$$P_D(r)\approx P_S(r)-(Z\alpha)P_1(r)\quad Q_D(r)\approx (Z\alpha)Q_1(r)$$ (199)

where P₁ and Q₁ are functions, independent of Z and $\alpha$, which can be expressed in terms of P_S (Bethe and Salpeter 1957, Armstrong and Feneuille 1974).

To derive all operators that include relativistic effects to order $(Z\alpha)^2$ we combine the Dirac-Coulomb Hamiltonian DiracCoulomb with the second Pauli limit Pauli2, and the Breit operators Breit with the first Pauli limit Pauli1. The following terms will appear:

1.

The kinetic energy term in DiracCoulomb along with the second Pauli limit give the mass correction: 

$$H_{MC}=-\frac{\alpha^2}{8}\sum_i\bnabla_i^4$$ (200)

2.

The nuclear potential, $\sum_iZ/r_i$, and the second Pauli limit result in two corrections, the one-body Darwin term  

$$H_{D1}=-\frac{\alpha^2Z}{8}\sum_i\bnabla_i^2\left(\frac{1}{r_i}\right)$$ (201)

and the nuclear spin-orbit term 

$$H_{SO}=\frac{\alpha^2Z}{2}\sum_i\frac{1}{r_i^3}\bi{l}_i\bdot\bi{s}_i$$ (202)

The former is the correction to the potential energy from retardation effects (Jackson 1962), while the latter describes the magnetic interaction between the spin and orbital angular momenta of one electron.

3.

The instantaneous Coulomb interaction, $\sum_{i\neq j}\frac{1}{r_{ij}}$, and the second Pauli limit give the corrections of similar type, but in two-particle form. These are the two-body Darwin term:  

$$H_{D2}=-\frac{\alpha^2}{4}\sum_{i<j}\bnabla_i^2\left(\frac{1}{r_{ij}}\right)$$ (203)

and the spin-other-orbit interaction: 

$$H_{SOO}=-\frac{\alpha^2}{2}\sum_{i<j}\frac{\bi{r}_{ij}\times\bi{p}_i} {r_{ij}^3}\left(\bi{s}_i+2\bi{s}_j\right)$$ (204)

H_SOO contains two parts, both the correction to the spin-orbit interaction due to the modification of the central field, and the interactions between the orbital angular momentum of electron iwith the spin of electron j. In this form it also contains some corrections arising from the Breit interaction.

4.

The Breit interaction Breit and the first Pauli limit provide three extra terms. The orbit-orbit interaction  

$$H_{OO}=-\frac{\alpha^2}{2}\sum_{i<j}\left[\frac{\bi{p}_i\bdot... ...t(\bi{r}_{ij}\bdot\bi{p}_i\right)\bi{p}_j} {r_{ij}^3}\right]$$ (205)

is the magnetic interaction between the orbital angular momenta of two different electrons. The spin-spin interaction  

$$H_{SS}=\alpha^2\sum_{i<j}\frac{1}{r_{ij}}\left[\bi{s}_i\bdot\... ..._ij\right)\left(\bi{s}_j\bi{r}_{ij}\right)} {r_{ij}^3}\right]$$ (206)

corresponds to the magnetic interaction between two electron spins. Finally, the electron spin-spin contact term has the form  

$$H_{SSC}=-\frac{8\pi\alpha^3}{3}\sum_{i<j}\left(\bi{s}_i\bdot\bi{s}_j\right) \delta\left(\bi{r}_i\bdot\bi{r}_j\right).$$ (207)

and accounts for the interaction of the spin magnetic moments of two electrons occupying the same position in space.