Improvements on the approximation

The LCAO approximation employed above leads to only a qualitatively correct description of chemical bonding in the H$_2^+$ molecule ion. The bond length and binding energy are quantitatively off from the exact values. In order to see how to improve the appproximation, let us look at some of the limits of the LCAO approximation.

First in the limit $R\longrightarrow 0$, the two protons are in approximately the same spatial location (exactly, strictly speaking, but this is unphysical), so that the wave function should have the same spatial dependence as that of a He$^+$ wave function (the neutrons in an actual He nucleus have no significant effect on the electronic wave function). The ground state energy of He$^+$ should be $E_0=-Z^2e^2/2a_0 = -2e^2/a_0$. However, in the LCAO approximation employed above, the energy $E_- - e^2/R$ (the nuclear repulsion term must be subtracted off because it becomes both infinite and unphysical if we fuse the two protons together into a single ``nucleus'') becomes

$$E_- - {e^2 \over R} \longrightarrow -{3e^2 \over 2a_0}$$

Thus, this limit is not properly represented in the current LCAO scheme. In addition, the form of the wave function $\psi_-({\bf r})$ does not reduce to a He$^+$ wave function when $R\longrightarrow 0$ but rather reduces to

$$\psi_-({\bf r})\longrightarrow \left({1 \over \pi a_0^3}\right)^{1/2} e^{-r/a_0}$$

i.e., just a $1s$ orbital of hydrogen. In order to remedy this situation, one might imagine introducing $Z$, the atomic number, as a variational parameter into the LCAO approximation. That is, one could introduce atomic orbitals of the form

$$\langle {\bf r}\vert\psi_1(Z)\rangle = \psi_1({\bf r};Z) = \left({Z^3 \over \pi a_0^3}\right)^{1/2}e^{-Z\left\vert{\bf r} + {R \over 2}\hat{{\bf z}}\right\vert/a_0}$$

$$\langle {\bf r}\vert\psi_2(Z)\rangle = \psi_2({\bf r};Z) = \left({Z^3 \over \pi a_0^3}\right)^{1/2}e^{-Z\left\vert{\bf r} - {R \over 2}\hat{{\bf z}}\right\vert/a_0}$$

and then construct a modified LCAO scheme with a trial wave function of the form

$$\vert\psi\rangle = C_1\vert\psi_1(Z)\rangle + C_2\vert\psi_2(Z)\rangle$$

The bond length and binding energy obtained in this case are 2.00 $a_0$ and 2.35 eV, respectively. While the bond length is now correct, the energy is still off from the exact value, which illustrates that the geometry tends to converge faster than the energy.

In the limit of $R$ large but not infinite, the approximation of spherical atomic orbitals is also not entirely correct. The reason for this can be seen by the fact that a positive charge in the vicinity of a hydrogen atom will produce an electric field that causes a distortion of the electronic charge distribution about the hydrogen nucleus, as the figure below illustrates:

図 4:

This polarization effect can be included in the trial wavefunction by mixing in a little bit of a $p$-orbital. For example, if we constructed a trial wavefunction of the form

$$\vert\psi\rangle = C_1\vert\chi_1\rangle + C_2\vert\chi_2\rangle$$

where

$$\langle {\bf r}\vert\chi_1\rangle = \chi_1({\bf r}) = \psi^2_{1s}({\bf r}_1) + \sigma \psi^2_{2p}({\bf r}_1)$$

where $\sigma$ is a mixing coefficient that is treated as a variational paraleter, then, with a similar definition for $\vert\chi_2\rangle$, a a variational scheme that includes the polarization effect is obtained.

In order to include both limits, one would then introduce the $Z$ dependence into the $\chi$ orbitals above and use as a trial wavefunction

$$\vert\psi\rangle = C_1\vert\chi_1(Z)\rangle + C_2\vert\chi_2(Z')\rangle$$

which allows for an asymmetry due to the presence of $Z$ and $Z'$. Thus, treating $C_1$, $C_2$, $Z$, $Z'$, and $\sigma$ as variational parameters, a bond length of 2.00 $a_0$ and binding energy of $2.73 eV$ are obtained, which are in very good agreement with the exact results.