Theoretical Justification of Madelung's Rule
D. Pan Wong California State University Fullerton, CA 92634
The empirical rule discovered by Madelung' for the sequence of filling electron shells of neutral atoms in the Periodic Table consisted of two parts:
1s
/ /
(1)When considering consecutive neutral atoms, the electron shells fill up in the order of the quantum numher sum (n I ) . (2) For electrons in states of equal (n 0 , the order of filling goes with
increasing n.
+
+
/
The extraordinary success of Madelung's rule is evidenced by its application in almost aII general chemistry textbooks in its graphical form, showing the order of filling of electrons with the given atomic number (see Fig. 1). None of the books attempted to give theoretical or physical meaning to these peculiar 45' lines. I'iiny the Fermi-Thomas statistical model with the Tirtr nrwn,ximate solution. Klechkwski%we the first theoretical jistification for part'l of ~ a d e l u n g ' srule. A different approach, but with the same result is presented here for both parts of Madelung's rule plus some applications. Historical Background In 1926, after Bohr's triumphant calculation on the hydrogen atom came Schrhdinger's famed wave equation and Max Born's probability interpretation of the wave. The subsequent major theoretical work on atoms may be the statistical treatment of atomic structures independently suggested by Fermi and Thomas3 Thomas and Fermi assumed that the atomic electrons hehave like free electron gas obeying Ferm-Dirac statistics. The kinetic enerev and the unnormalized wave function of a free elwtron gas acted un hy no force and honded by an infinitttly hieh vort,ntinl cubic wnll, a:'. are the same ns the Schrigdinper soiu$ons for the particle in8-dimensional box.
Assuming the electrons would not be ahle to escape to infinity, they can possess kinetic energy up to the maximum kinetic energy Eo, which is equal to the potential energy. At this point, the system has reached statistical equilibrium. The volume of this constant energy sphere, (4/3)ar3, has radius equal to [8ma2Eolh2]'/2. The numher of electrons with enerev less than Eo in the positive cells = (2 electrons/cell) (%) (~01ume"ofthe sohere)
/
Figure 1. Device tor remembering order in which electrons are added to
atoms.
Number of electrons - 8r Po3 (1) Volume 3 h3 Substituting into eqn. (I), Maximum kinetic energy = Potential energy, gives n =
8n n=[2meV(r)]3/2 (2) 3h3 Since the assumption of continuous charge distribution removed the concept of shells, Fermi and Thomas were ahle to apply the well-known Poisson equation of electrostatistics relating potential energy with charge density.
where p = charge density which is defined as Numbei of electrons p = (electron charge) Unit volume p = (-eNn) (4) This is to say that the charge density of electrons a t distance r from the nucleus is determined by the maximum kinetic energy of the free electron gas. Substitute eqn. (4) into eqn. (3) to obtain
I
1
Where
Equation (5) is the basic equation of statisticalmethod. It has been extended and applied on molecules and metals.
Po = maximum momentum of the electron in this volume; =v'%z V = volume = a3 The specification uf 2 rlectruns per cell is a modification of itatistical mechanics to include t he Pauli exclusion principle. Hence, the expression for the maximum numher ofk~ectrons with Po per unit volume, n,is
Madelung, E., "Mathematische Hilfsmittel des Physikers," 3rd Ed., Springer, Berlin, 1936, p. 359. Klechkowski, V. M. Zh. Ersperim. i Teor. Fiz., 41.465 (1962). [Transl. Souiet Physics J. Expt, and Theor. Phys. 14,334 (1962)l. Fermi, E., Mem. Accad. Lineei 6,602 (1927)Thomas L. H., Proc.
714 1 Journal of Chemical Education
'
Cambridge Phil. Soe. 23,542 (1927) or Collected Papers (Univ. of Chicago Press, 1962).
Since V ( r )is function of r alone, one can transform eqn. (5) into universal differential equation of Thomas and Fermi by substituting
." .
Where i = angular momentum = PI.i If we want to know the numher of electrons with angular momentum L between L and L + dL, we must modify eqn. (8) to specify the spherical shell with r between r and r d r (See Fig. 2).
+
Volume = (orea) (thickness) (10 =
me2 8.
= ( 4 ~ 9( d ) r)
Thus, the number of electrons with angular momentum L between L and L dL
and using atomic units, h = m = e = ao = 1. The transformed second-order differential equation is
x3%)xu(%)= 7 x1
+
= N ( L ) dl,
(7) = (Number of
Where the original boundary conditions
electronslunit vol) (volume)
rV(r)-Oasr-m
are changed to
=*
Ze Ze V(r)-T~(~)
Substitute into eqn. (9):
orx(x)=lasx-0
rV(r)
-
( x ) ($)x(z) = Zex(x)- 0 .
.
orx(r)-Oasx-m
Equation (7) can only be solved numerically. Interestingly, Fermi was able to use the potential function given in eqn. (6) to calculate the numher of s, p, or d electrons in an atom with given atomic number. Fermi derived the relationship as follows. Since the actual momentum vector of free electron gas may he anvwhere inside the centered momentum-soace sohere of radius Po, Figure 2 is constructed to determine the total momentum space volume. Total ualurne = (area) ( d P l ) = (circumferenceo f cylinder) (height) ( ~ P L ) = ( 2 x 4 ( h )( d P d = ( 2 x P l ) [2 d P o z - PL2] ( d P 1 ) = ~ R P \/Po2 L -P I 2d P l
+
=
Unit volume
- ( 2 ) 4 r P LdPo2 - P h3
~
z.
NiW) = E I ( lI2) Lhl+ (h) ~ r ' d \ / 2 m e V ( r-) ( 1 + %)2h2dl h" II r ..
- ,
Fermi evaluated the inteyrnl expression in eqn. (10) numrrirally and vbrained theorettr~drrsults in fairl\.rood - . ilrreement with the empirical values.
The numher of electrons per unit volume with momentum d P L with two electrons of opposite between PL and P I spin per cell 2 electronslcell] (Total Vol) N
one obtains the total number of electrons whose orbital angular momentum number is 1 in atom with atomic numher
* r~2 P
'7
Analytical Solution to Fermi-Thomas Model The problem with Fermi's eqn. (10) is the difficulty in ohtaining the numerical solution of the integral and the obscurity of its physical meaning. A simple and elegant analytical equation can be derived for N , by substituting the analytical potential energy equation by Tietz into eqn. (10). Tietz4 found ~
Ze V ( r )= - ~ r
( 7 =)
Ze
(11)
?(I+?/
where c = (a/8)213 and b is defined in eqn. (6).Tietz's equation is not an exact solution. But this approximate solution is close to the numerical solution of Fermi as shown in Figure 3. Upon substituting Tietz's V(r), eqn. (10) will be transformed into
where
and the integral limits rl, rz equal to the two positive solutions of the integrand.
Figure 2. Momentumpace sphere of radius Po
'Tietz, T. J Chem. Phys. 23,1167 (1965) Volume 56. Number 11. November 1979 / 715
Figure 3. Comparison of Fermi's numerically integrated solution with Tietz's approximate solution.
I
- T . F Calculation 0
+
J
Exp. D a t a Eqn. 14
Figure 4. Number of electrons in a given wbshell, t in atom Z versus atomic number 2. Comparing actual data against ThomarFermicalculation using eqn. ( l o ) and eqn. (14) derived in this paper.
1 where t = r+q 1.2
?r
=
rl
arcsin 1= 2
c2
2B Equation (12) can be solved by first simplifying R=d+fr+gr2
where
After pages of simple substitution and evaluation of the integrand at the limits r l and r2, everything miraculously cancelled out to give NdZ) = (21
d = -B
+ 1)
I-
2
+@
(1
(13)
Substitute back the original values of A, B, b, and c, the neat looking analytical equation emerges.
~=A-%B
g = - g2 -
bZ then apply a series of helpful internal solutions:
N[(Z) = 2(21 + 1){(6Z)'I3- (21 + 1)) (14) Comparison of eqn. (14) with Fermi's solution, eqn. (10) and experimental results for 1 = 0 and 1 are given in Figure 4. Similar results can he obtained for 1 = 2 or 3, hut they are not given in Figure 4 for simplicity. Eauation (14) . ..rives almost the same result as e m . (10). exrtyt onr can e\.aluate eqn. (14) one thousand times faster. inteeration of ~ n I 10) . indicated that F
