The hydrogen molecule ion#

What you need to know

  • We construct trial wavefunction of the molecule from atomic orbitals.

  • Applying variational method we find the two MOs to have distinct spatial profiles with one increasing probability of electron between the two nuclei and the other depleating.

  • These MOs are called bonding and anibonding orbitals. They lower and raise eneryg of molecule relative to two separated atoms.

Constructing MOs from AOs#

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Fig. 90 AOs centered on each H atom in \(H^{+}_2\) molecule#

  • The electronic Schrodinger equation for H\(_2^+\) can be solved exactly because the equation contains only one particle. However, the involved math is very complicated and here we take another simpler but approximate approach. This method will reveal all the important features of chemical bond. An approximate (trial) wavefunction is written as (real functions):

\[{\psi_\pm(\vec{r}_1) = c_11s_A(\vec{r}_1) \pm c_21s_B(\vec{r}_1)}\]

  • where \(1s_A\) and \(1s_B\) are hydrogen atom wavefunctions centered at nucleus A and B, respectively, and \(c_1\) and \(c_2\) are constants. This function is essentially a linear combination of the atomic orbitals (LCAO molecular orbitals).

  • Because the two protons are identical, we must have \(c_1 = c_2 \equiv c\) (also \(c > 0\)).

  • The \(\pm\) notation indicates that two different wavefunctions can be constructed, one with + sign and the other with \(-\) sign. Normalization of the wavefunction requires:

\[{\int{\psi_\pm^*\psi_\pm d\tau} = 1}\]

Overlap integral#

In the following, we consider the wavefunction with a ``+‘’ sign and evaluate the normalization integral (\(S\) = overlap integral, which depends on \(R\)):

\[\begin{split}1 = c^2\int{(1s_A + 1s_B)(1s_A + 1s_B)d \tau}= \\ c^2\int{1s_A^2d \tau} = 1 + c^2{\int{1s_B^2d}\tau}\end{split}\]
\[+ c^2{\int{1s_A1s_Bd\tau}}{= S}+ c^2{\int{1s_B1s_Ad\tau}}{= S}= c^2(2 + 2S)\]

This can be rewritten as:

\[{1 = c^2(2 + 2S) \Rightarrow c = \frac{1}{\sqrt{2 + 2S}}}\]

and the complete ``+‘’ wavefunction is then:

\[{\psi_+ \equiv \psi_g = \frac{1}{\sqrt{2(1 + S)}}(1s_A + 1s_B)}\]

In exactly the same way, we can get the \(-\) wavefunction:

\[{\psi_- \equiv \psi_u = \frac{1}{\sqrt{2(1 - S)}}(1s_A - 1s_B)}\]

Note that the antibonding orbital has \underline{zero} electron density between the nuclei.

Bonding vs antibonding orbitals#

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Fig. 91 Bonding vs anti-bonding wavefunctions (Molecular Orbitals). Show are wavefunctions and probability densities (squares of wavefunctions)#

  • The main feature of a chemical bond is the increased electron density between the nuclei. This identifies the + wavefunction as a bonding orbital and \(-\) as an antibonding orbital.

  • When a molecule has a center of symmetry (here at the half-way between the nuclei), the wavefunction may or may not change sign when it is inverted through the center of symmetry. If the origin is placed at the center of symmetry then we can assign symmetry labels \(g\) and \(u\) to the wavefunctions. If \(\psi(x, y, z) = \psi(-x, -y, -z)\) then the symmetry label is \(g\) (even parity) and for \(\psi(x, y, z) = -\psi(-x, -y, -z)\) we have \(u\) label (odd parity).

  • According to this notation the \(g\) symmetry orbital is the bonding orbital and the \(u\) symmetry corresponds to the antibonding orbital. Later we will see that this is {reversed} for \(\pi\)-orbitals!

  • The overlap integral \(S(R)\) can be evaluated analytically (derivation not shown):

\[{S(R) = e^{-R}\left( 1 + R + \frac{R^3}{3}\right)}\]

  • Note that when \(R = 0\) (i.e. the nuclei overlap), \(S(0) = 1\) (just a check to see that the expression is reasonable).