An Alternative Generating Function Method

Another method of introducing Legendre Polynomials is through the dipole moment generating function. Since this method is very important in Quantum Mechanical computations concerning poly-electronic atoms and molecules, it is worth our attention. When one considers the Hamiltonian of the Helium Atom's electrons (for example), one has

$$-\frac{Ze^2}{r_1} - \frac{Ze^2}{r_2} + \frac{e^2}{r_{12}}$$ (4)

where r₁₂ is the distance between electron 1 and electron 2, i.e., it is the electron-electron repulsion term. We examine this term in this discussion. We can write this electron-electron repulsion term as

$$\frac{1}{r_{12}} = \frac{1}{\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos \theta}}$$ (5)

where r₁ and r₂ are the distances from the nucleus to electrons 1 and 2 respectively. $\theta$ is the angle between the vectors from the nucleus to electron 1 and electron 2. It is required that we do this is two domains, one in which r₁>r₂ and one in which r₂>r₁. This is done for convergence reasons (vide infra). For the former case, we define

$$\zeta = \frac{r_2}{r_1}$$

so that $\zeta < 1$< 1$">, and Equation 5 becomes

$$\frac{1}{r_{12}} = \frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \cos \theta}}$$ (6)

which we now expand in a power series in $\cos \theta$ (which will converge while $\zeta < 1$< 1$">). We have

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \cos \theta}} =$$ (7)

$$\frac{1}{r_{1}} \left ( 1 + \frac{1}{1!} \left . \frac{d\left ( \... ...d \cos \theta ^2} \right \vert _{\cos \theta =0} \cos^2\theta + \cdots \right )$$ (8)

It is customary to change notation from $\cos \theta$ to $\mu$, so

$$\frac{1}{r_{12}} = \frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}}$$ (9)

which we now expand in a power series in $\mu$ (which will converge while $\zeta < 1$< 1$">). We have

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} = \frac... ... \zeta^2 - 2 \zeta \mu}} \right ) }{d \mu}\right \vert _{\mu = 1} \mu+ \right .$$

$$\left . \frac{1}{2!} \left . \frac{d^2\left ( \frac{1}{\sqrt{1 + ... ... 2 \zeta \mu}} \right ) }{d \mu^2} \right \vert _{\mu=1}\mu^2 + \cdots \right )$$ (10)

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} =$$

$$\frac{1}{r_{1}} \left (1 + \frac{1}{1!} \left . \frac{-1}{2} (1 +... ... 2 \zeta \mu}} \right ) }{d \mu^2} \right \vert _{\mu=1}\mu^2 + \cdots \right )$$ (11)

which is

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} =$$

$$\frac{1}{r_{1}} \left (1 - \frac{1}{2} (1 + \zeta^2 - 2 \zeta )^{-\frac{3}{2}} \left ( -2 \zeta \right ) \right . \mu+$$

$$\left . \frac{1}{2!} \left . \frac{d\left ( \frac{-1}{2} (1 + \ze... ...2 \zeta \right ) \right . }{d \mu} \right \vert _{\mu=1}\mu^2 + \cdots \right )$$ (12)

i.e.,

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} =$$

$$\frac{1}{r_{1}} \left (1 - \frac{1}{2} (1 + \zeta^2 - 2 \zeta )^{-\frac{3}{2}} \left ( -2 \zeta \right ) \right . \mu+$$

$$\left . \frac{1}{2!} \left . \frac{d\left ( (1 + \zeta^2 - 2 \zet... ...( \zeta \right ) \right ) }{d \mu} \right \vert _{\mu=1}\mu^2 + \cdots \right )$$ (13)

which is, after the second differentiation

 

$$\frac{1}{r_{1}}\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} =$$

$$\frac{1}{r_{1}} \left (1 - \frac{1}{2} (1 + \zeta^2 - 2 \zeta )^{-\frac{3}{2}} \left ( -2 \zeta \right ) \right . \mu+$$

$$\left . \frac{1}{2!} \left . \left ( -\frac{3}{2} (1 + \zeta^2 - ... ...c{5}{2}} (-2 \zeta) \zeta \right ) \right \vert _{\mu=1}\mu^2 + \cdots \right .$$ (14)

Dropping the r₁ term, we have

 

$$\frac{1}{\sqrt{1 + \zeta^2 - 2 \zeta \mu}} =$$

$$\left (1 - \frac{1}{2} (1 + \zeta^2 - 2 \zeta )^{-\frac{3}{2}} -2 \zeta \right ) \mu+$$

$$\frac{1}{2!} \left ( +2\frac{3}{2} (1 + \zeta^2 - 2 \zeta )^{-\frac{5}{2}} \zeta^2 \right ) + \cdots$$ (15)