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\begin{document}

\title{Nodal Surfaces of Wave Functions of Many-Electron Systems}



\author[1]{Evgenii Smolenskii}%
\author[1]{Anton Ryzhov}%
\author[1]{Irina Chuvaeva}%
\author[1]{Oleg Eliseev}%
\affil[1]{Zelinsky Institute of Organic Chemistry RAS}%


\vspace{-1em}



  
  \date{May 23, 2024}


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\selectlanguage{english}
\begin{abstract}
Possible theoretical basis for the hypothesis of prohibition of some
electron configurations in atoms and molecules presented by one of the
authors earlier is considered. Existence of such prohibited
configurations follows from the wave function antisymmetry of atoms and
molecules. Our hypothesis is as follows: the configurations, in which
the potentials of any two electrons with the same spins are equal, are
forbidden. If the hypothesis comes true, it will be possible to find the
nodal surfaces of many-electron wave functions for atoms and molecules
analytically a priori.%
\end{abstract}%



\sloppy


\textbf{Nodal Surfaces of Wave Functions of Many-Electron Systems}

\textbf{Eugenii A. Smolenskii,} 11\emph{N. D. Zelinsky Institute of
Organic Chemistry Russian Academy of Sciences, Leninsky Prospect, 47,
Moscow, Russian Federation, 119991} \textbf{Anton N. Ryzhov, Irina V.
Chuvaeva, Oleg L. Eliseev}

Correspondence to: Eugenii A. Smolenskii
(E-mail:\emph{smolensk@ioc.ac.ru} )

\textbf{Introduction}

The role of antisymmetry requirement of wave functions in the structure
of many-electron system configurational space (atoms and molecules,
including their excited states {[}7{]}) is shown in {[}1-6{]}. One of
the consequences of this requirement is prohibition for electrons to
take some spaces in physical space with zero value of the wave function.
These positions of electrons determine some nodal surfaces in
configurational space, which divide the space to \emph{N} ! (\emph{N} !
= 1[?]2[?]3[?]\ldots{}[?]\emph{N} ) equivalent areas for atoms and molecules
with\emph{N} electrons. The equivalence means that, in case we receive
the solution of Schr\selectlanguage{ngerman}ödinger equation (\selectlanguage{greek}Ψ \selectlanguage{english}function) for one such area, the
solution for the rest of them is received simply by permutation of the
electron indices {[}8{]}. In the same article {[}8{]} it is shown that
some unknown fundamental physical forces called antisymmetry forces by
us must exist. Specifically they forbid to electrons to take the
positions mentioned above and characterized usually by some symmetry. In
paper {[}9{]} the hypothesis was offered that the wave function turns to
zero for configurations with equal potentials of any two electrons with
the same spins. This hypothesis was proved with calculations for He, Li
and Be atoms and for molecules H\textsubscript{2} in triplet state and
LiH {[}10, 11{]}. In the present paper same foundation of our hypothesis
is given.

\textbf{Thought experiment}

Let us have a thought experiment. At first, we fixate some configuration
for atom or molecule with \emph{N} electrons, choose two electrons with
same spins and give them indices ``1'' and ``2''. Let the wave function
for this configuration have some value \selectlanguage{greek}Ψ\selectlanguage{english}0 and \selectlanguage{greek}Ψ\selectlanguage{english}0 \textgreater{} 0 for
definiteness (the sign of \selectlanguage{greek}Ψ\selectlanguage{english}0 does not matter). Then the wave function
must change sign and turn to --\selectlanguage{greek}Ψ\selectlanguage{english}0 when we change places of electrons
indices ``1'' and ``2''. We take with fingers or tweezers electrons
``1'' and ``2'' and move them in some arbitrary way changing their
places. The other electrons and atom nuclei stay still. In this
situation electron ``1'' runs some trajectory and electron ``2'' runs
another trajectory in physical space. The two trajectories in
three-dimensional physical space can be described by one line in
six-dimensional configuration space named by us the interchange
trajectory. The spins can be ignored because they coincide.

It turns out that any choice of the interchange trajectory gives \selectlanguage{greek}Ψ\selectlanguage{english}=0 at
some moment. (For example, we can move electron ``1'' far away in atomic
scale, electron ``2'' put on its place and then choose some intricate
way to move electron ``1'' to the former place of electron ``2'' to
avoid symmetrical situations). It follows from continuity of the wave
functions and the well-known mathematical analysis theorem that proves a
continuous function determined on some segment and having opposite signs
on its ends turn to zero at least at one point of the segment. We
consider the interchange trajectory as such segment. There can be more
than one zero points but their number will be odd necessarily. The nodal
surfaces of the wave function in configurational space consist of these
points.

\textbf{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 \emph{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) \emph{N} is the number of electrons in atom \selectlanguage{russian}А, i.e. the
charge of the nucleus, \(r_{A,i}\) is the distance between
electron with index \emph{i} and the nucleus, \(\ r_{\text{ik}}\) is the
distance from this electron to the electron with index \emph{k} . The
summation is being carried for all k from 1 to \emph{N} except~\emph{i}
.

Let's consider function \selectlanguage{greek}Φ\selectlanguage{english}=\(u_{1}-u_{2}\), 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 \selectlanguage{greek}Ψ \selectlanguage{english}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 \selectlanguage{greek}Φ \selectlanguage{english}turns to zero exists. The meaning of our hypothesis {[}9{]}
is as follows: for the points with \selectlanguage{greek}Φ\selectlanguage{english}=0 \selectlanguage{greek}Ψ\selectlanguage{english}=0 too, i.e. nodal points of
these functions on the interchange trajectory are the same. This means
that the set of \selectlanguage{greek}Φ \selectlanguage{english}nodal surfaces is a subset of the \selectlanguage{greek}Ψ \selectlanguage{english}nodal surfaces
set. Function \selectlanguage{greek}Ψ \selectlanguage{english}turns to zero not only for the configurations with \selectlanguage{greek}Φ\selectlanguage{english}=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 \selectlanguage{greek}Ψ \selectlanguage{english}and \selectlanguage{greek}Φ
\selectlanguage{english}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.

\textbf{Conclusions}

In conclusion, we can point out that in case our hypothesis is proved by
calculation for complex atoms and molecules it will give us a
possibility to carry quant mechanic calculations on a new high level
because then we can determine the boundaries of the
above-mentioned\emph{N} ! equivalent areas analytically \emph{a priori}
. In fact, we suggest a new quant chemistry as it was mentioned in
{[}8{]}.

P.S. It should be admitted that our thought experiment is difficult to
carry in physical space for we have to pass through the zero point of
wave function. This is forbidden by the forces considered in {[}8{]} and
responsible for antisymmetry of wave functions. Figuratively speaking,
we don't know presently if we have enough energy to overcome the forces
of antisimmetry and transfer our electron through this point. This
experiment is easier to carry if the coordinates of other electrons are
being changed. Of course, we don't know how electrons in atoms and
molecules move. Moreover, we can't know this in principle if the
uncertainty principle of Heisenberg {[}12{]} is valid. By the way, in
the same paper {[}12{]} Heisenberg for the first time used the term
``electron trajectory'' and considered several thought experiments with
electrons. Electrons don't stay still without doubt. In Stern and
Gerlach experiments trajectories of electrons were observed but the
electrons were free and not connected with atoms nuclei. However, we are
sure that science does not stand still and is discovering new and
unknown laws of the material world, so the mysterious properties of
electrons will be explained together with value and range of action of
above-mentioned antisimmetry forces in comparison with other fundamental
forces.

\textbf{Acknowledgments}

The authors are grateful to the Program of Development of Scientific
Schools of ZIOC RAS.

\textbf{Keywords:} wave functions, Pauli principle, wave functions
antisymmetry, Schr\selectlanguage{ngerman}ödinger equation.

\textbf{References and Notes}

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