Original hypothesis
Let’s go back to our hypothesis [9]. Every electron in atom A has
some definite value of potential for any configuration of electrons. The
value of potential for electron with number i can be expressed by
the following formula:
\(u_{i}=\frac{N}{r_{A,i}}-\sum_{k\neq i}\frac{1}{r_{\text{ik}}}\)(1)
In formula (1) N is the number of electrons in atom А, i.e. the
charge of the nucleus, \(r_{A,i}\) is the distance between electron with
index i and the nucleus, \(\ r_{\text{ik}}\) is the distance from
this electron to the electron with index k . The summation is
being carried for all k from 1 to N except i .
Let’s consider function Φ=\(u_{1}-u_{2}\), i.e. the potential
difference of electrons “1” and “2” that have the same spins as was
pointed before. This function is defined for any point of the
interchange trajectory of electrons “1” and “2” and has the same
quality as the wave function Ψ itself, i.e. at the ends of the
interchange trajectory it has opposite signs according to formula (1).
It means that inside the interchange trajectory at least one point where
function Φ turns to zero exists. The meaning of our hypothesis [9]
is as follows: for the points with Φ=0 Ψ=0 too, i.e. nodal points of
these functions on the interchange trajectory are the same. This means
that the set of Φ nodal surfaces is a subset of the Ψ nodal surfaces
set. Function Ψ turns to zero not only for the configurations with Φ=0,
i.e. \(u_{1}\)=\(u_{2}\), but for all the cases \(u_{i}\)=\(u_{j}\) if
spins of electrons “i” and “j” are the same. Let’s consider the
interchange trajectory of electrons “1” and “2” with arbitrary
coordinates in case when the coordinates of other electrons and nuclei
are fixed. Let’s the initial distance between the electrons “1” and
“2” is decreasing and approaching to zero. Then distances between the
zero points of Ψ and Φ turn to zero too (even if the distances were not
the same initially), i.e. these points coincide in the limit. This, of
course, is not a proof of our hypothesis but a suggestion giving some
hope that it is valid.