In order to understand why is the kinetic energy attractive, lets first consider the overlap of two functions\eqref{eq:two_center_overlap} plotted in the top panel in Figure 1

\begin{equation} {s_{ab}} = \left\langle {{\phi _a}} \right|\left. {{\phi _b}} \right\rangle = = \int\limits_{ - \infty }^\infty {\int\limits_\infty ^\infty {\int\limits_\infty ^\infty {dxdydz{\phi _a}{\phi _b}} } } \label{eq:two_center_overlap} \end{equation}

If we just look at the \(x\) component we have the product of two exponentials, plotted in the second panel. This product is summed (integrated) from negative to positive infinity. As the functions come closer together the product gets larger and the summation (integration) gets larger. The

Let us examine the two-center kinetic energy term, Eqn. \eqref{eq:two_center_kin}, in more detail.

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\begin{equation} t_{ab} = \left\langle {{\phi _a}\left| { - \frac{1}{2}{\nabla ^2}} \right|{\phi _b}} \right\rangle \label{eq:two_center_kin} \end{equation}

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Expanding the \(x\) component of the Laplacian, \(\nabla^2\), yields Eqn. \eqref{eq:del_dx_portion}.

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\begin{equation} \left\langle {{\phi _a}\left| {\frac{{{\partial ^2}}}{{\partial {x^2}}}} \right|{\phi _b}} \right\rangle = \int\limits_{ - \infty }^\infty {\int\limits_{-\infty} ^\infty {\int\limits_{-\infty} ^\infty {dxdydz{\phi _a}*\frac{{{\partial ^2}{\phi _b}}}{{\partial {x^2}}}} } } \label{eq:del_dx_portion} \end{equation}

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The chain rule \((\int u dv = uv - \int v du)\) can be used to rearrange Eqn. \eqref{eq:del_dx_portion} into Eqn. \eqref{eq:del_dx_expand}, and since \(\phi_a\) goes to \(0\) as \(x\) goes to \(\infty\), the first two terms vanish and we are left with Eqn. \eqref{eq:two_center_kin_simp}.

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\begin{equation} \begin{split} \left\langle {{\phi _a}\left| {\frac{{{\partial ^2}}}{{\partial {x^2}}}} \right|{\phi _b}} \right\rangle =& \int\limits_{ - \infty }^\infty {\int\limits_{-\infty} ^\infty {dydz{\phi _a}*\frac{{\partial {\phi _b}}}{{\partial x}}}}\\ &{} - \int\limits_{ - \infty }^\infty {\int\limits_{-\infty} ^\infty {dydz{\phi _a}*\frac{{\partial {\phi _b}}}{{\partial x}}}}\\ &{} - \int\limits_{ - \infty }^\infty {\int\limits_{-\infty} ^\infty {\int\limits_{-\infty} ^\infty {dxdydz\frac{{\partial {\phi _a}}}{{\partial x}}*\frac{{\partial {\phi _b}}}{{\partial x}}} } } \end{split} \label{eq:del_dx_expand} \end{equation}
\begin{equation} \left\langle {{\phi _a}\left| { - \frac{1}{2}{\nabla ^2}} \right|{\phi _b}} \right\rangle = \frac{1}{2}\left\langle {\nabla {\phi _a} \bullet \nabla {\phi _b}} \right\rangle \label{eq:two_center_kin_simp} \end{equation}

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The dot product of the gradients (slopes) of the wavefunction is the key to bonding. The gradients of the two functions are plotted in the third panel. When the slopes of the two functions are of opposite sign the dot product will contribute negatively to the integral whereas with the functions are of the same sign they will add to the integral, see the fourth panel.This negative integrand lowers the energy, providing the bond.

Overlap (S) = 0.858; Kinetic Energy (tab) = 0.307;

Distance between centers