New Method for Calculating Molecular Orbitals with Application to Cyclic Systems

We consider the molecular orbital problem, i.e., the quantum-mechanical problem of a particle (electron) bound to a configuration of $N$ potentials, when the potentials may overlap. We show why, when the individual potentials are spherically symmetric, it is advantageous to write the wave function $\ensuremath{\Psi}$ in the many-center (LCAO) form, $\ensuremath{\Psi}={\ensuremath{\psi}}^{(1)}({\mathrm{r}}_{1})+\ensuremath{\cdots}{\ensuremath{\psi}}^{(N)}({\mathrm{r}}_{N})$, where ${\mathrm{r}}_{i}$ refers to a coordinate system associated with the $i\mathrm{th}$ potential. The main point of this paper is to show that it is not only advantageous to use this form, but that it is also practical to directly determine the "atomic orbitals" ${\ensuremath{\psi}}^{(1)}\ensuremath{\cdots}{\ensuremath{\psi}}^{(N)}$. To show this we define the ${\ensuremath{\psi}}^{(i)}({\mathrm{r}}_{i})$ in a natural way and then use the Schr\"odinger equation to get a set of coupled integral equations for these functions and to get an analogous set for ${\ensuremath{\varphi}}^{(i)}(\ensuremath{\lambda})$, their Fourier transforms. We expand the ${\ensuremath{\varphi}}^{(i)}(\ensuremath{\lambda})$ in partial waves and make it plausible that the resultant sets of equations can be truncated and that frequently only a small number of partial waves need be retained, so that the equations are practical to solve.To show the spirit of the method we discuss, as a simple application, the ground state of cyclic systems, i.e., of potentials arranged at the vertices of regular polygons. The truncation of the equations then gives an approximate equation for the "$S$-wave part" of the orbitals and an expression which enables us to estimate the higher order partial waves from this $S$-wave part. We present methods for approximately solving the $S$-wave equation, which equation has the advantage of strongly resembling the corresponding one for a single potential, except for a "structure factor" which depends on the configuration. These methods include a variational method which does not involve multicenter integrals, a means of reducing the problem to that of solving the Schr\"odinger equation for an effective potential, and a method involving screened nuclear charge. For Coulomb potentials and for $N=2, 3, 4$ ($N=2$ corresponding to the hydrogen molecule ion) we calculate the $S$-wave part of the orbitals using the latter method; in these calculations arbitrary $N$ is not essentially more difficult than $N=2$.As a further application we use the second of the equations mentioned above to calculate, for $N=2$, the $P$-wave or "polarization" terms in the orbitals. We get good agreement for the form of this part of the wave function with the best variational calculation of this kind, that of Dickinson.Finally, we point out that these two applications are the first steps in an iterative procedure which seems to be promising more generally. Namely, starting from an approximation to an orbital which involves, say, one or more dominant partial waves, it would now appear that one can both calculate the other partial-wave components and correct the initial approximation, and so by iteration arrive at the correct form of the orbital.