The inspiration for this post came from this paper by J. D. Dunitz and T. K. Ha titled:
By considering the hydrogen molecule, H
In this post we answer this question, for I believe, the first time.
Hydrogen Molecular Ion
The hydrogen molecular ion, H
e⁻ / \ / \ p⁺ --- p⁺
H
The complete Schrödinger equation cannot be solved exactly but by invoking the Born-Oppenheimer approximation the electronic Schrödinger equation can be solved exactly using confocal-elliptic coordinates. This allows for complete separation of the Hamiltonian resulting in two coupled differential equations solvable using a recurrence relation.
With the advancement in computer power, we can obtain very accurate energies and wave functions to these molecular ions without assuming the Born Oppenheimer approximation i.e. including the motion of the nuclei explicitly in the calculation. I have previously published work which accomplishes this and in this post we will use these very high accuracy wavefunctions to explore the effect of nuclear charge on the bond length in hydrogen molecular-like ions with different nuclear charges.
How to Calculate Bond Length?
The bond length of H
The expectation value corresponds to the average value so can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood. Something we found in the paper cited above is that the calculated expectation values of the proton-proton distance did not agree well with the experimental value, whilst the most probable value did. Have a look at the following table which lists the calculated values of
Most probable | ||
---|---|---|
Calculated | 2.063 913 867 | 1.989 685 |
Experiment | 1.987 99 | 1.987 99 |
The calculated value of the most probable
Method
A brief overview of the method used to numerically solve the Schrodinger equation is now given, for a more indepth discussion consult this review paper we published.
Assume the following wavefunction form, consisting of a triple orthogonal set of Laguerre polynomials
where
are perimetric coordinates, linear combinations of inter-particle coordinateswhere
are non-linear variational parameters.Substitute the wavefunction form into the time independent, non-relativistic Schrödinger equation
Substitute the Laguerre recursion relations
Which leads to a 57-term recursion relation
Collapsing the triple index,
down results inWhich we solve as a generalized eigenvalue problem
where
is the eigenvector (wavefunction).
Modest 2856-term wavefunctions were calculated for hydrogen molecular-like systems with nuclear charges in the range
Particle Densities and the Most Probable Bond Distance
Once the wavefunctions,
which characterizes the spatial distribution of particle
with,
A unique and really interesting property of the Dirac delta function is the sifting property, defined as
and gives the Dirac delta function a sense of measure. It measures the value of
In integral form this calculation looks like
The calculation itself uses parallelized numerical integration in C++. I will describe how to do this in a future post.
Expectation Value of
We described above how to calculate the particle density in order to find the most probable bond distance, but how do we calculate the expectation value? This is done using the following
where coordinate transformations are applied to convert the inter-particle coordinates,
The calculation itself can use the same parallelized numerical integration code mentioned above but can also be solved using the Laguerre recursion relations. This topic, given its depth and tangential nature, might find its place in a future blog post, where a more in-depth exploration can be undertaken.
Results
The following table shows the calculated expectation value (average value) and most probable value of
Nuclear charge / Z | ||
---|---|---|
0.4 | 2.296 558 484 | 2.260 134 |
0.5 | 2.158 214 212 | 2.125 930 |
0.6 | 2.078 038 239 | 2.048 665 |
0.7 | 2.035 824 320 | 2.007 950 |
0.8 | 2.021 769 795 | 1.994 954 |
0.9 | 2.031 411 074 | 2.004 474 |
1.0 | 2.063 913 867 | 2.035 328 |
1.1 | 2.122 205 373 | 2.092 229 |
1.2 | 2.215 691 763 | 2.185 469 |
1.21 | 2.228 968 027 | 2.202 792 |
1.22 | 2.241 708 265 | 2.212 782 |
1.23 | 2.255 127 463 | 2.227 767 |
1.24 | 2.269 180 307 | - |
1.241 | 2.270 636 718 | - |
1.242 | 2.272 101 112 | - |
1.243 | 42.561 145 435 | - |
1.3 | 141.428 426 211 | - |
Note that in the particle densities a spherical average was applied,
Note The values for the maximum positions in the particle densities were calculated through interpolation of the particle density data points using a spline fitting. This is so we do not need to calculate thousands of data points, instead requring fewer and can ‘fill in’ the gaps using a good quality spline fit and differentiate this function to find the maximum. The plot below shows the expectation value and most probable bond lengths against nuclear charge.
The bond length can be seen to decrease as the nuclear charge increases between
By just considering the two protons the result seems counter-intuitive, but the presence of the electron explains the physics behind this behaviour. The electron has a charge of -1 so when the nuclear charges of the protons are < 1 the electron dominates the electrostatic interactions in the system. Even though the mass of the proton is ~ 1836 times greater than the mass of the electron, it is able to localize the protons to be closer together via its charge influence. As the charge of the protons increases to
To finish off lets have a look at what the particle density does as the nuclear charge increases, which has been visualised in the following animation. The vertical dotted line represents the most probable bond length (with spherical average) for the hydrogen molecular ion, H
This is a really nice way to visualise how the particle density changes with increasing nuclear charge:
We can see that when the charge of the protons is
the particle density profile is wider which signifies a greater uncertainty in the bond length as it has a greater probability of exisiting over a wider space, .As the charge increases further, the particle density profile becomes narrower and taller which signifies the uncertainty in bond length is decreasing: the increased charge is stabilising the system.
At the end of the animation it looks like the plots dissapear but they do not. At
there is a dramatic change in the particle density as the system becomes unbound into a hydrogen atom and a lone electron which means the density becomes massively diffuse over all space. The scales change by orders of magnitude and so it collapses to the axis in the above animation which is just visible.
Conclusions
We have shown that the bond length in H
References
[1] Taken from NIST