# How large are atomic positional/angular fluctuations within a molecule?

I have seen videos of thermal fluctuations/oscillations of atoms that have covalent bonds with each other, and they are usually quite dramatic. I'm curious how strong the fluctuations really are of a typical molecule at room temperature.

In case my question isn't clear:

• Let $$\alpha(t)$$ be the angle between the two hydrogen atoms in $$H_2O$$. Let $$d\alpha=max_t(\alpha(t))-min_t(\alpha(t))$$ be the measure of how much the angle oscillates. What is $$d\alpha$$ at room temperature, roughly?
• Similarly, let $$ds = max_t(d(t)) - min_t(d(t))$$ be the measure of how much the distance between two covalently bonded carbon atoms oscillates. What is $$ds$$ at room temperature, roughly?

Summary: At room temperature, the H$$_2$$O bond angle has thermal fluctuations of order $$10^{-2}$$ degrees. And for $$C_2$$ at room temperature, the bond length fluctuates on the order of $$2\cdot 10^{-4}$$ pm.

You can treat this problem rather generally as a simple harmonic oscillators so long as the amplitude fluctuations are sufficiently small compared to the spacing between atoms and nothing significantly nonlinear is going on, which is true at the relatively low temperature given by OP.

Granted the harmonicity condition holds, there is a general relationship between a harmonic oscillator's frequency, temperature, and displacement $$x$$:

$$\delta x= \sqrt{\frac{\hbar}{2 M \omega} (2n+1)}$$

Where $$\omega$$ is the vibrational frequency, $$M$$ is a particular mass factor, $$n$$ is the number of vibrational quanta which depends on temperature via a Bose distribution $$n=1/(e^{\hbar \omega/k_BT}-1)$$ where $$k_B T$$ is the thermal energy.

Technically, the mass factor $$M$$ complicates things and you need to do proper linear algebra to convert eigenmodes to displacement vectors, but in the case of water where the oxygen mass is much larger than hydrogen and the bending mode is most relevant, you can ignore the Oxygen motion, focus solely on the bending mode (see image below), and set $$M$$ to the hydrogen mass. More detailed and correct vibrational maths are described here.

The bending mode energy of water is about 2304 Kelvin (or 1654 cm$$^{-1}$$), which gives an $$n\approx 0.0002$$ at room temperature of 300 Kelvin. If we want to understand thermal vibrations of the angle, we should ignore the zero point motion of $$\delta x$$ when $$n=0$$, which is just from Heisenberg uncertainty.

We will set aside the $$n=0$$ component which carries no temperature dependence, so we can Taylor expand to get $$\delta x$$'s dependence on $$n$$

$$\delta x_{thermal} \approx n \cdot \sqrt{\frac{\hbar}{2 M \omega}}$$

And $$\sqrt{\frac{\hbar}{2 M \omega}} \approx 7 pm$$ for Hydrogen's bending mode, which should be compared to the bond length of about 96 picometers.

Then we can compute the ratio of the thermal length change to the zero temperature bond length to get an order of magnitude estimate for the angle change.

$$\delta x_{thermal} \approx 0.001 pm \implies d\theta \approx \frac{\delta x_{thermal}}{\Delta x_{Bond}}=1.4\cdot 10^{-2} \, mrad$$

So we would expect thermal changes to the bond angle at room temperature of order 10 microradians, or about one hundredth of a degree.

We can do a similar computation for a $$C_2$$ molecule, although it is unstable at room temperature apparently. There, the vibrational frequency is 2630 Kelvin (1827 cm$$^{-1}$$), giving an order of magnitude $$\delta x$$ from the formula earlier of $$2\cdot 10^{-4}$$ pm

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TLDR: the fluctuations in bond lengths and angles will depend on several factors, including the identity of the molecules, their temperature, the material phase, etc. But the fluctuations are typically quite modest unless a bond actually dissociates, which happens infrequently outside of chemical reactions at normal temperatures.

To quantify this, I will give some simulation data from protonated water clusters taken from my paper on vibrational spectroscopy of small molecules. The two clusters used for these data are shown below. The simulations use ab initio density functional theory to obtain potential energy gradients used to propagate the motions of the various atoms in the system. To make the graphs below I sampled roughly 50,000-100,000 frames from each simulation depending on the amount of data available on my personal computer. Feel free to reach out to me via email from my faculty page if you have any trouble accessing the paper. While these data will have some slight artifacts that differ from pure water or water in the condensed phase since the simulations are implicitly gas-phase and the system includes an extra proton, they do give a reasonable starting point for what to expect for water.

First off, I will give some data explicitly relevant to the gas phase. The protonated water cluster H$$^+$$(H$$_2$$O)$$_4$$ allows us to learn about the distribution of bond lengths in waters exposed to vacuum (which is essentially like water vapor) and a protonated water involved in hydrogen bonding. The data below span several different temperatures simulated using canonical sampling through velocity rescaling. What we immediately see is that there is a shoulder in both the angular and bond length distributions (more pronounced for angles at low temperatures) corresponding to the protonated core water. This core typically has slightly longer bond lengths and bond angles than a non-protonated water. Involving a water (the protonated water, in this case) in hydrogen bonds typically increases its average bond length from about 0.97 Angstroms to about 1 Angstrom at 50K, with the distribution getting steadily wider for both molecules with increasing temperature. For reference, a lone water molecule has an equilibrium bond length of about 0.96 Angstroms. Since the distribution of bond lengths is approximately Gaussian, the full width at half the maximum (FWHM) approximately obeys FWHM$$\approx 2.355 \sigma$$ where $$\sigma$$ is the standard deviation of the distribution. The FWHM has the following approximate trend for the bond length data, It is clear from this data that the distribution of the bond lengths and angles is highly dependent on temperature, though the range of values is typically not huge even at reasonably high temperatures. For instance, the FWHM suggests that the bond lengths fluctuate with a standard deviation of only $$\sigma\approx 0.02$$ Angstroms at 300K.

Finally, let's see what happens in a situation that more closely mirrors the condensed phase. The protonated water cluster H$$^+$$(H$$_2$$O)$$_{21}$$ has a large number of waters involved in a complex network of hydrogen bond donation and acceptance, which means that its radial and angular distributions will be more like those of water (or in this case ice). The simulation at 50K shows that the bond lengths are typically higher than those of the corresponding 4 water cluster that more closely mimics the gas phase. The average bond length increases to about 1 Angstrom, which is more typical of condensed phase water. The FWHM of the bond length data is about 0.1 Angstroms, leading to a standard deviation of 0.042 Angstroms. These are incredibly modest fluctuations, but also highly sensitive to changes in temperature like before.

While there is much more that could be said about the distributions of bond length and angles in water, hopefully this is enough raw data that you can extract some of the heuristic information that you need.

I suggest a simpler calculation for a hydrogen molecule, which can be easily generalized to other cases.

A hydrogen molecule has a single vibrational mode. We can start with Hamiltonian: $$H=\frac{p_1^2}{2m}+ \frac{p_2^2}{2m} + \frac{k(x_1-x_2)^2}{2},$$ where $$m$$ is the mass of a hydrogen atom. For simplicity (but without loss of generality) I consider only the axis along the bond connecting the two hydrogen atoms. I also assume linear regime - although at high temperatures the potential cannot be treated anymore as quadratic, and nonlinear effects play a role.

Let us switch to the center-of-mass coordinates $$X= \frac{x_1+x_2}{2}, x=x_1-x_2,\\ P=p_1+p_2, p=p_1-p_2$$ We now have $$H=\frac{P^2}{2M} + \frac{p^2}{2\mu} + \frac{kx^2}{2},$$ where $$M=2m$$ is the total mass of the molecule, whereas $$\mu=m/2$$ is the reduced mass of the effective oscillator describing the vibrational mode. The frequency of oscillations is $$\omega=\sqrt{k/\mu}$$, so that we have $$k=\mu\omega^2$$. Furthermore, at maximum displacement, $$A$$, all the vibrational energy is the potential energy, i.e., $$\mu\omega^2A^2/2$$.

On the other hand, equipartition theorem suggests that for a molecule in a gas the mean energy in every degree of freedom is $$k_BT/2$$, that is $$\frac{\mu\omega^2A^2}{2}=\frac{k_BT}{2}\Rightarrow A=\sqrt{\frac{k_B T}{\mu\omega^2}}$$ The hydrogen molecule vibrational frequency (known from its spectrum) is $$8.25\times 10^{14}s^{-1}$$, hydrogen mass and Boltzmann constant are known, so we can do our calculation.

Wikipedia provides a ready-made visualization for another covalent molecule - HCl, from which one can read numbers:

• Could you state your final answer for the standard deviation/variation at the end of your answer, and the termperature for which it applies, or the formula in terms of temperature? Commented Jul 5 at 6:32