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FROM THE CURRENT LITERATURE
STATISTICAL FORMULATION OF THE PAULI PRINCIPLE AND CALCULATION OF MANY-ELECTRON ATOMS
The investigations of the prominent Hungarian theoretical physicist P. Gombás in the field of the statistical theory of many-electron systems are widely known from his monograph¹. Recently he has published new results concerning the calculation of many-electron atoms², ³, ⁴.
As is known, fulfillment of the Pauli principle in calculating the wave functions of atoms by the variational method is possible only if the \(\psi\)-functions of all electrons are orthogonal to one another. (In directly solving the Schrödinger equation, this problem is handled in terms of various levels according to filling rules.) The necessity of satisfying a large number of orthogonality conditions greatly complicates calculations for many-electron atoms. Meanwhile, for a whole range of problems it is sufficient to compute only the \(\psi\)-functions of the valence electrons, while the \(\psi\)-functions of the electrons of the atomic core are either already known (for example, from calculations by the Hartree–Fock method) or are of no interest.
As early as 1941 P. Gombás pointed out⁵ the possibility of replacing the Pauli principle for valence electrons, with respect to the electrons of the atomic core, by a certain additional repulsive field acting on the valence electrons and not allowing them to descend to deeper energy levels. The author proceeds from the statistical model of the atom. The electrons of the atomic core are regarded as a degenerate electron gas distributed with density \(\rho\). Then the electrons located in the volume \(dv\) will occupy all energy states up to a certain maximum energy \(u_\mu\). This is a consequence of the Pauli principle. The valence electrons are described by wave functions \(\psi_i\). The minimum energy of the \(i\)-th valence electron is denoted by \(u_i\). Then, in order to place the \(i\)-th valence electron in the volume \(dv\), it is necessary to impart to it the additional energy:
\[ w = \int \psi_i^* [u_\mu(\mathbf r) - u_i(\mathbf r)] \psi_i\, d\tau . \tag{1} \]
This is equivalent to the action on the valence electron of an additional nonclassical repulsive potential:
\[ G_i = -\frac{1}{e}[u_\mu(\mathbf r)-u_i(\mathbf r)] . \tag{2} \]
The Pauli principle for valence electrons, with respect to the completely filled shells of the atomic core, will be taken into account if the electrostatic potential \(V\) is replaced by the modified potential
\[ \Phi_i = V + G_i . \tag{3} \]
Then the wave function of the valence electron may fail to satisfy the orthogonality conditions with respect to the wave functions of the electrons of the atomic core. For the ground states the orthogonality conditions disappear altogether, while for excited states the \(\psi\)-function should be orthogonalized only with the lower states of the valence electrons.
An analytic expression for the potential \(G_i\) was given by Gombás in papers \(^{1,4}\). It was assumed here that all electrons of the atomic core, independently of their \(l\), uniformly fill a sphere of radius \(p_\mu\) in momentum space. In this case
\[ u_\mu=\frac{p_\mu^2}{2m}. \]
According to the author, the minimum energy of a valence electron is
\[ u_i=\frac{1}{2}e^2a_0\,\frac{l_i(l_i+1)}{r_i^2}. \tag{4} \]
(The values \(u_\mu\) and \(u_i\) are given to within a constant in the potential energy \(dv\).) \(u_\mu\) and \(u_i\) can be expressed through the density of the electron gas in the volume \(dv\). As a result the author obtains:
\[ G_i=-\frac{1}{2}(3\pi^2)^{2/3}ea_0\left[\rho^{2/3}(\mathbf r)-\rho_i^{2/3}(\mathbf r)\right], \tag{5} \]
where \(\rho\) is the total density of the electrons of the atomic core at the point \(r\); \(\rho_i\) is the density of the electrons of the atomic core whose energy is less than the minimum energy of the \(l_i\)-electrons (i.e., less than the energy of the \(1s, 2p, 3d\), etc., electrons for \(l_i=0,1,2,\ldots\), respectively).
In a recent paper \(^{2}\) a more accurate calculation of the potential \(G_{l_i}\) is given. In this paper electrons with different \(l\) are considered separately. The sphere of radius \(p_\mu\) in momentum space is divided into coaxial cylindrical sections. In each section points are placed corresponding to electrons with a definite value of \(l\). The maximum energy \(u_\mu\) is thus calculated separately for each value of \(l\). The value of \(u_i\) is taken to be the same as in the preceding work. As a result, for the additional potential a new expression is obtained
\[ G_{l_i}=-\frac{\pi^2 e^2 a_0}{8(2l+1)^2}D_{l_i}^{\,2}-\frac{ea_0}{4r^2}, \tag{6} \]
where
\[ D_{l_i}=4\pi r^2\rho_{l_i}(\mathbf r) \]
and \(\rho_{l_i}\) is the density of the electrons of the atomic core with \(l=l_i\). It should be noted that the calculations of \(G_{l_i}\) in this paper are not completely rigorous.
The idea of replacing the orthogonalization conditions by an additional repulsive potential is very fruitful and substantially facilitates calculations with many-electron systems.
Since the \(\psi\)-function of the valence electron is not orthogonalized with the functions of the electrons of the atomic core, the number of nodes in this function decreases. Consequently, inside the atomic core there is only an average probability distribution for the valence electrons. This is the usual consequence of the statistical method of calculation (it is evident that the function of the valence electron in the region of the atomic core is treated statistically). However, this region is relatively very narrow and does not play an essential role.
Only in one case does this region prove to be significant: in the calculation of the energy. Indeed, the energy of an electron in an atom is given by the exp—
by the expression
\[ E=\frac{\hbar^2}{m}\int_0^\infty \left|\frac{d(rR)}{dr}\right|^2\,dr -\frac{\hbar^2}{2m}l(l+1)\int_0^\infty R^2\,dr -e\int \psi^*V\psi\,d\tau, \tag{7} \]
where \(R(r)\) is the radial part of the \(\psi\)-function.
The region of small \(r\) is inessential for the last two integrals, but is very important for the first integral (which expresses the kinetic energy). Owing to the frequent oscillations in the region of the atomic core, \(\dfrac{d(rR)}{dr}\) assumes such large values there that this region gives the largest contribution to
\[ \int_0^\infty \left|\frac{d(rR)}{dr}\right|^2\,dr \]
(especially for an \(s\)-electron). Meanwhile, this integral is the only positive term in the expression for the energy, since \(V>0\). The Gombás potential gives an additional positive term in the expression for \(E\), namely,
\[ \int G_i\psi_i\psi_i^*\,d\tau \]
and thereby removes the indicated defect.
Therefore the use of the additional Gombás potential is in principle necessary when approximate wave functions with a reduced number of nodes are used for calculating the energy.
In the next two papers\(^{3,4}\) the author gives two applications of the theory developed. In the first paper the energy of the crystal lattice for Cu and K is calculated. Here the additional potential \(G_i\), (6), is also used; it gives a substantial contribution to the total energy (the valence electrons of the metal are regarded as free, i.e., \(\psi_i\) does not depend on the coordinates). The energy per crystal cell, calculated by the author, agrees with the experimental data.
A very important application, in our opinion, of the method of the additional potential is given in the second paper. Since the function of a valence electron need not satisfy any orthogonality condition, it must be nodeless. This is a good theoretical justification for applying Slater-type functions\(^{6}\):
\[ R=Ar^{n^*-1}e^{-\frac{Z-s}{n^*}\,r}. \tag{8} \]
The author calculates the parameters \(\gamma\) for the noble gases by means of the variational method, using the additional potential \(G_i\), (6). The results obtained for \(\gamma\) are in good agreement with the parameters empirically selected by Slater.
L. V.
CITED LITERATURE
- P. Gombás, Statistical Theory of the Atom and Its Applications, IL, § 19, 1951.
- P. Gombás, Acta Physika (Budapest), 1, 285 (1952).
- P. Gombás, ibid., 1, 301 (1952).
- P. Gombás, ibid., 1, 317 (1952).
- P. Gombás, Zeits. f. Phys., 118, 164 (1941).
- J. Slater, Phys. Rev., 36, 57 (1930).