Combining the two equivalent classical or density-based wave functions of Eqns. \eqref{eq:h2_17} and \eqref{eq:h2_18} yields the valence bond resonance description of a covalent bond, Eqn. \eqref{eq:h2_25}.⁵

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\begin{equation} \psi = {\phi _a}^{(1)}{\phi _b}^{(2)}\alpha \beta \label{eq:h2_17} \end{equation}
\begin{equation} \psi = {\phi _b}^{(1)}{\phi _a}^{(2)}\alpha \beta \label{eq:h2_18} \end{equation}

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\begin{equation} \psi = {\phi _a}^{(1)}{\phi _b}^{(2)}\alpha \beta + {\phi _b}^{(1)}{\phi _a}^{(2)}\alpha \beta \label{eq:h2_25} \end{equation}

Since the spatial part of \(\psi\) is symmetric with respect to interchange of electrons 1 and 2, the spin portion of the wave function can be made antisymmetric by adding the \(-\beta\alpha\) spin function, Eqn. \eqref{eq:h2_26}.

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\begin{equation} \psi = \left( {{\phi _a}^{(1)}{\phi _b}^{(2)} + {\phi _b}^{(1)}{\phi _a}^{(2)}} \right)\left( {\alpha \beta - \beta \alpha } \right) \label{eq:h2_26} \end{equation}

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The orthogonality of the spin functions \(\alpha\) and \(\beta\) makes the spin orbitals orthogonal without requiring orthogonality of the spatial orbitals. As with Eqns. \eqref{eq:h2_20}-\eqref{eq:h2_21},

i

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\begin{equation} \begin{split} E =& \frac{{\left\langle {{\phi _a}\alpha {\phi _b}\beta \left| H \right|\left( {{\phi _a}\alpha {\phi _b}\beta - {\phi _b}\beta {\phi _a}\alpha } \right)} \right\rangle }}{{\left\langle {{\phi _a}\alpha {\phi _b}\beta \left| {\left( {{\phi _a}\alpha {\phi _b}\beta - {\phi _b}\beta {\phi _a}\alpha } \right)} \right.} \right\rangle }}\\ =& \frac{{\left\langle {{\phi _a}\alpha {\phi _b}\beta \left| H \right|{\phi _a}\alpha {\phi _b}\beta } \right\rangle - \left\langle {{\phi _a}\alpha {\phi _b}\beta \left| H \right|{\phi _b}\beta {\phi _a}\alpha } \right\rangle }}{{\left\langle {{\phi _a}\alpha {\phi _b}\beta \left| {{\phi _a}\alpha {\phi _b}\beta } \right.} \right\rangle - \left\langle {{\phi _a}\alpha {\phi _b}\beta \left| {{\phi _b}\beta {\phi _a}\alpha } \right.} \right\rangle }} \end{split} \label{eq:h2_20} \end{equation}

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\begin{equation} E =\\ \frac{{\left\langle {{\phi _a}{\phi _b}\left| H \right|{\phi _a}{\phi _b}} \right\rangle \left\langle {\alpha \beta \left| {\alpha \beta } \right.} \right\rangle - \left\langle {{\phi _a}{\phi _b}\left| H \right|{\phi _b}{\phi _a}} \right\rangle \left\langle {\alpha \beta \left| {\beta \alpha } \right.} \right\rangle }}{{\left\langle {{\phi _a}{\phi _b}\left| {{\phi _a}{\phi _b}} \right.} \right\rangle \left\langle {\alpha \beta \left| {\alpha \beta } \right.} \right\rangle - \left\langle {{\phi _a}{\phi _b}\left| {{\phi _b}{\phi _a}} \right.} \right\rangle \left\langle {\alpha \beta \left| {\beta \alpha } \right.} \right\rangle }} \label{eq:h2_21} \end{equation}

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terms containing different spin functions will be zero, Eqn. \eqref{eq:h2_27}.

\begin{equation} \left\langle {\alpha \beta \left| {\beta \alpha } \right.} \right\rangle = 0 \label{eq:h2_27} \end{equation}

Thus, for this two electron system, the energy expression can be generated ignoring the spin functions. The energy expression of the wave function of Eqn. \eqref{eq:h2_25} is given in Eqn. \eqref{eq:h2_28}.

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\begin{equation} \begin{split} E =& \frac{{\left\langle {ab + ba\left| H \right|ab + ba} \right\rangle }}{{\left\langle {ab + ba\left| {ab + ba} \right.} \right\rangle }}\\ =& \frac{{\left\langle {ab\left| H \right|ba} \right\rangle + \left\langle {ab\left| H \right|ba} \right\rangle + \left\langle {ba\left| H \right|ab} \right\rangle + \left\langle {ba\left| H \right|ba} \right\rangle }}{{\left\langle {ab\left| {ab} \right.} \right\rangle + \left\langle {ab\left| {ba} \right.} \right\rangle + \left\langle {ba\left| {ab} \right.} \right\rangle + \left\langle {ba\left| {ba} \right.} \right\rangle }} \end{split} \label{eq:h2_28} \end{equation}

In Eqn. \eqref{eq:h2_28} orbital ordering is used to indicate electron number and the shorthand notation \(a\) is used for \(\psi_a\). Factoring the one-electron integrals and supplying labels for the two-electron integrals results in Eqn. \eqref{eq:h2_29}.

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\begin{equation} E = \frac{{2{h_{aa}} + 2{S_{ab}}{h_{ab}} + {J_{ab}} + {K_{ab}}}}{{1 + {S^2}}} + \frac{1}{{{R_{ab}}}} \label{eq:h2_29} \end{equation}

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A physical interpretation of this repulsion arises from classical electrostatics.⁵ We know from classical electrostatics (Gauss' Law) if a charged particle is outside a spherical charged surface, the interaction is as though the sphere were a point charge located at the center of the sphere. If the charged particle is inside the spherical surface, it sees a constant distance-independent potential. As given in Eqn. \eqref{eq:h2+wfxn} the hydrogenic electron density is an exponentially decaying spherical distribution.

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\begin{equation} \phi_a=\sqrt{\frac{\zeta^3}{\pi}}e^{-\zeta r_a} \label{eq:h2+wfxn} \end{equation}

Thus, as the proton penetrates shell upon shell of electron density it progressively becomes less and less distance-dependent approaching \(-\zeta\) at the nucleus. The proton-proton repulsive interaction remains a full 1/R term. The repulsion swamps the attraction at all distances. The rapid increase in energy at roughly 3 Å corresponds to the initial proton penetration of the electron density.

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As with H₂⁺, the attraction is due to the kinetic energy term, and the repulsive potential term is a consequence of placing electron density at the bond midpoint. Comparison of Eqns. \eqref{eq:h2_17} and \eqref{eq:h2_19} for H₂⁺ leads to the suggestion that the binding energy is proportional to \(S^2\).

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\begin{equation} H = {h_a}(1) + {h_a}(2) + {h_b}(1) + {h_b}(2) + \frac{1}{{{r_{12}}}} + \frac{1}{{{R_{ab}}}} \label{eq:h2_19} \end{equation}

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As described in 1928, the bond energy of Eqn. \eqref{eq:h2_29} can be increased to 364 kJ/mol by optimizing \(\zeta\) (\(\zeta_{opt} = 1.166\)).⁶ When \(\zeta\) changes from 1 to 1.166, the atomic terms of Eqn. \eqref{eq:h2_29} rise 72.4 kJ/mol, as the kinetic energy rises 943.9 kJ/mol and the potential drops 871.5 kJ/mol. In addition, at \(R\)=0.741, the pre-resonant term (including atomic terms) rises 28.0 kJ/mol, \(v_{res}\) rises 37.2 kJ/mol, \(t_{res}\) drops 172.4 kJ/mol, and the two-electron resonant term rises 19.2 kJ/mol. This leads to a net increase in the binding energy of 87.9 kJ/mol.

Allowing the exponent to vary and including ionic configurations (not discussed yet) yields a bond energy of 386.2 kJ/mol (\(\zeta_{opt} = 1.193\)).⁷ The binding energy increases to 397.5 kJ/mol (\(\zeta_{opt} = 1.190\)), which is 87% of the total binding energy when polarization is included by adding \(2p_\sigma\) functions. The remaining 13% of the binding energy can be obtained by adding angular correlation and finally including higher order angular terms. Modern bonding models are still attempting to unravel the relative importance of kinetic energy and electron density in chemical bonding.⁸