Concluding Remarks

Wavefunctions methods[10] for the chemical bond of large atomic aggregates start with atomic-like orbitals

$$\varphi _{s}=\sum_{ia}c_{ia}^{s}\chi _{a}({\bf r}-{\bf R}_{i})\;\;\;,$$ (56)

which are superpositions of individual atomic-like orbitals $\chi _{a}({\bf r}-{\bf R}_{i})$ localized at the positions ${\bf R}_{i}$ of the atomic nuclei, and attempt to minimize the Hartree-Fock energy functional

$$E_{a} = \sum_{s}(t_{s}^{a}-n_{s}^{a})+\frac{1}{2}\sum_{ss^{\prime }}(d_{ss^{\prime }}^{a}-e_{ss^{\prime }}^{a})+$$ (57)

$$+\frac{1}{2}\sum_{i\neq j}Z_{i}Z_{j}/\left\vert {\bf R}_{i}-{\bf R}_{j}\right\vert \;\;\;,$$

where the matrix elements are the kinetic energy

$$t_{s}^{a}=\int d{\bf r}\varphi _{s}^{*}({\bf p}^{2}/2)\varphi _{s}\;\;\;,$$ (58)

the electron-nuclei attraction

$$n_{s}^{a}=\int d{\bf r}\sum_{i}(Z_{i}/\left\vert {\bf R}_{i}-{\bf r}\right\vert )\varphi _{s}^{*}\varphi _{s}\;\;\;,$$ (59)

and the direct

$$d_{ss^{\prime }}^{a}=\int d{\bf r}d{\bf r}^{\prime }(1/\lef... ... _{s}\cdot \varphi _{s^{\prime }}^{*}\varphi _{s^{\prime }}$$ (60)

and exchange

$$e_{ss^{\prime }}^{a}=\int d{\bf r}d{\bf r}^{\prime }(1/\lef... ...}}^{*}({\bf r}^{\prime })\varphi _{s^{\prime }}(%% {\bf r})$$ (61)

interactions; atomic units are used, and $a$ denotes usually the valence atomic orbitals.

Beside the atomic-like orbitals (56) the molecular-like orbitals[11] $\psi _{s}$ include extended bond-like orbitals $\Phi _{s}$, such that

$$\psi _{s}=\alpha _{s}\varphi _{s}+\beta _{s}\Phi _{s}\;\;,\;\;\alpha _{s}^{2}+\beta _{s}^{2}=1\;\;\;,$$ (62)

as for orthonormal sets of wavefunctions; the density-functionals methods[12] for the chemical bond touch upon this point, especially in connection with the Thomas-Fermi model.[13]

The great disparity between the scale-lengths of the localized atomic-like orbitals $\varphi _{s}$ and the extended bond-like orbitals $%% \Phi _{s}$ provides a certain decoupling of the atomic degrees of freedom from the chemical bond degrees of freedom, up to a density-density interaction originating in the direct Coulomb repulsion; the minimization of the Hartree-Foch energy functional for the molecular-like orbitals $\psi _{s}$ with respect to the $\beta _{s}$-parameters leads to the linear system of equations[14]

$$-A_{s}+\sum_{s^{\prime }}D_{ss^{\prime }}\beta _{s^{\prime }}^{2}=0\;\;\;,$$ (63)

where

$$A_{s}=\varepsilon _{s}^{a}(HF)-\varepsilon _{s}^{b}(HF)+\su... ...{ss^{\prime }}^{b}-\frac{1}{2}d_{ss^{\prime }}^{ab})\;\;\;,$$ (64)

and the matrix $D_{ss^{\prime }}$ is given by

$$D_{ss^{\prime }}=(d_{ss^{\prime }}^{a}-e_{ss^{\prime }}^{a}... ...ime }}^{b}-e_{ss^{\prime }}^{b})-d_{ss^{\prime }}^{ab}\;\;;$$ (65)

the $b$-labelled matrix elements are defined similarly with the atomic-like matrix elements above given by (58)-(61) by using the bond-like orbitals $\Phi _{s}$ instead of the atomic-like orbitals $\varphi _{s}$; and the Hartree-Fock energies are given by

$$\varepsilon _{s}^{a,b}(HF)=t_{s}^{a,b}-n_{s}^{a,b}+\sum_{s^... ...rime }}^{a,b}-\sum_{s^{\prime }}e_{ss^{\prime }}^{a,b}\;\;.$$ (66)

Very likely, the matrix $D_{ss^{\prime }}$ is positive definite, so that $%% A_{s}$ must acquire positive values in order to have solutions to (63); the atomic-like orbitals in the upper valence atomic shells may provide such solutions, leading thus to the chemical bond as described by molecular-like orbitals.

Beside the atomic-like energy functional $E_{a}$ given by (57) one obtains now an additional bond-like energy functional

$$E_{b} = \sum_{s}\beta _{s}^{2}t_{s}^{b}-\frac{1}{2}\sum_{ss^{\prime }}\be... ... \sum_{ss^{\prime }}\beta _{s}^{2}\beta _{s^{\prime }}^{2}d_{ss^{\prime }}^{b}-$$ (67)

$$-\frac{1}{2}\sum_{ss^{\prime }}\beta _{s}^{2}\beta _{s^{\prime }}... ...}^{2}\beta _{s^{\prime }}^{2}(d_{ss^{\prime }}^{a}-e_{ss^{\prime }}^{a})\;\;\;,$$

and an energy

$$\Delta E=-\sum_{s}\beta _{s}^{2}(A_{s}+t_{s}^{b})\;\;;$$ (68)

the bond-like energy functional $E_{b}$ indicates the fractional occupancy $%% \beta _{s}^{2}$ of the bond-like orbitals, and the hamiltonian

$$H_{b} = \sum_{\alpha }{\bf p}_{\alpha }^{2}/2m-\sum_{\alpha }\int d{\bf r... ...\beta }\frac{1}{%% \left\vert {\bf r}_{\alpha }-{\bf r}_{\beta }\right\vert }+$$ (69)

$$+\frac{1}{2}\int d{\bf r}d{\bf r}^{\prime }\cdot \frac{\rho ({\bf... ...\rho (%% {\bf r}^{\prime })}{\left\vert {\bf r}^{\prime }-{\bf r}\right\vert }$$

for the bond-like electrons, where

$$\rho = \sum_{s}\beta _{s}^{2}\varphi _{s}^{*}\varphi _{s}=\sum_{ia;jb}\alpha _{ia;jb}\chi _{ia}^{*}\chi _{jb}=$$ (70)

$$= \sum_{ia;jb}(\sum_{s}\beta _{s}^{2}c_{ia}^{s*}c_{jb}^{s})\chi _{ia}^{*}\chi _{jb}$$

is the density of positive charge left behind in the atomic-like orbitals by the electrons participating in the chemical bond; the hamiltonian (69) describes the interaction between these electrons and their electronic ''holes'' in the ionic cores. One can see that $H_{b}$ given by (69) is in fact the effective hamiltonian given by (3), and the pair-wise distribution $\rho$ given by (70) corresponds to the effective charges

$$z_{ia}^{*}=\alpha _{ia;ia}=\sum_{s}\beta _{s}^{2}\left\vert c_{ia}^{s}\right\vert ^{2}$$ (71)

and $z_{i}^{*}=\sum_{a}z_{ia}^{*}$; and an average occupancy $\beta _{s}^{2}$ equals $z_{ia}^{*}$; neglecting the cross-terms in (70) the density of the ionic cores becomes $\rho =\sum_{ia}\alpha _{ia;ia}\left\vert \chi _{ia}\right\vert ^{2}$ like in (2); or, for point-like charges, $\rho =\sum_{ia}z_{ia}^{*}\delta \left( {\bf r}-{\bf R}_{i}\right) =\sum_{i}z_{i}^{*}\delta \left( {\bf r}-{\bf R}_{i}\right)$, like in (27).

The atomic-like problem as formulated by the energy functional $%% E_{a}$ given by (57) can be solved, in principle, according to the usual practice of the wavefunctions methods;[10] the bond-like hamiltonian $H_{b}$ given by (69) can be treated by the quasi-classical theory of the slightly inhomogeneous electron liquid; the self-consistent solution to equations (63) is then obtained for the parameters $\beta _{s}^{2}$ and, implicitly, for the density $\rho$ of the effective charge of the ionic cores; the actual self-consistent solution must also ensure the minimum value of the total energy functional $%% E_{a}+E_{b}+\Delta E$ with respect to the positions of the atomic nuclei. The chemical bond consists therefore of extended bond-like electronic orbitals of fractional occupancy $\beta _{s}^{2}$ and of localized atomic-like electronic orbitals of fractional occupancy $\alpha _{s}^{2}$ (the atomic-like fractional occupancy may be made to appear explicitly in the atomic-like energy functional $E_{a}$); the total electronic energy ensures the equilibrium of the atomic aggregate with respect to the Coulomb repulsion between the atomic nuclei. The contribution of the atomic-like part of the total energy to the binding energy is a quantal correction with respect to the quasi-classical description, so that, to the first approximation it may be neglected; the unrestricted minimization of the bond-like energy functional requires then a unity occupancy $\beta _{s}^{2}=1$ for the bond-like orbitals, the conservation of the total charge being ensured by the Coulomb interactions in the hamiltonian $H_{b}$ given by (69); within this approximation, one may say, therefore, that the binding energy and the cohesion of the atomic aggregates are given by the theory of the slightly inhomogeneous electron liquid for the bond-like hamiltonian $H_{b}$ given by (69); however, while this procedure is satisfactory for the binding energy and the cohesion, the fractional occupancy must be included for the single-electron properties.