Proton Density from a Path Integral Simulation

Two simulations have been performed for the investigation of quantum effects in ACAC. As mentioned in the above section, there is a ”classical” run and a second one, using the path integral formalism. Both employed the quantum chemical code CPMD [20]. The exchange-correlation functional was BLYP, norm-conserving pseudopotentials of the Goedecker-type have been applied. The wavefunction cutoff was 100 Ry. This is a bit higher than the usual cutoff, but during the MD, the electric field gradients have been computed, which require a higher cutoff [30]. The PIMD used 16 beads. The MD was carried out in the NVT ensemble,

using Nos´e-Hoover-thermostats with a characteristic frequency of 2500 cm−1 and

temperature of 300 K, the time step was 4 a.u., the proton mass has been used for the hydrogen nuclei. The simulation with path integrals has been run up to 5 ps, the classical simulation up to 6 ps. The staging transformation has been employed during the whole run to converge the PIMD simulation. The initial distribution of the beads was generated with a conditional Browian motion path at 500 K

[60]. The ACAC molecule was put into a large box of 10x9x7 ˚A, for simulating

an isolated system. Although the crystal structure would have been available, the influence of collective effects, or electrostatic perturbations by neighboring

5.3 Tunneling Effects in Acetylacetone 79

O

H

O

d

_d

d

!

Figure 5.5: Sketch of the parameters α and δ = (d1 − d2)/dOO, which are used in the following figures. The drawn atoms are representing the H-bond of the considered molecule.

molecules were excluded by this approach. This was done for exploring the pure effect of the proton tunneling.

Since this is only a very small system, the temperature is fluctuating a lot during the simulation, after 50-100 fs the distribution reached a stationary point and so only the first 500 steps were left out of the following analysis of the structure. Still, it was checked explicitly, that the reported results would not be changed qualitatively if only the end of the run was considered.

Before looking at spectroscopic properties, the structure of the molecule based on the simulations will be investigated. The motional behavior of the proton in the hydrogen bond should be affected by the quantum effects. In the classical case, i.e. without nuclear quantum effects, the proton is restricted to one side of the bond, it cannot tunnel through the energy barrier between the two minima of the potential energy surface. In the quantum case, instead, the proton will change from one side to the other, if tunneling is present. Additionally, in this case we expect a non-zero proton density even at the barrier. Two parameters will be shown in detail, both explained in Fig. 5.5. These are the angle α, representing the deviation of the proton from the straight line between the oxygens, and the parameter δ, measuring the difference between the distances of the proton from the two oxygens, normalized to the oxygen-oxygen distance. This means that we do not expect any proton density at δ = 0 in the classical case. It should be noted here, that the term ”proton density” is used for the statistical distribution of the proton over the whole MD, i.e. in each step all corresponding distances are collected and in the end a histogram is made out of these, representing the density of the proton with respect to δ or α. In Fig. 5.6 the relative occurrence of the parameter δ is shown. At first sight, the statistics in the classical case seem

80 5 Nuclear Quantum Effects in Molecular Systems

to be much worse, although the total simulation time is similar. This is due to the fact, that each time step of the PIMD run contains 16 representations of the respective proton, indeed leading to much more data. In the classical case, the value δ = 0 does almost not occur, as expected. Still, the probability is not zero. Also, the proton seems to appear in both minima, although only a single molecule has been simulated. This is due to the fact that at 300 K the kinetic energy of the proton is sufficient to occasionally jump over the barrier and go to the other minimum, which can happen in a simulation of such a strong hydrogen bond, since the barrier is not very high. Also, the non-zero value at δ = 0 can be explained by this argument. The energy that is necessary for overcoming the barrier can be estimated by comparing the energy of the optimized geometry and the transition state with a symmetrical hydrogen bond, which yields ∆E ≈ 0.9 kcal/mol. This value reflects the fact that the hydrogen bond is very short and thus the barrier relatively low. Still, it should be noted that it is known that activation energies, especially in hydrogen bonds, are in general underestimated in DFT. Using this ∆E, the fraction of hydrogens that are able to jump over the barrier can be estimated from a Boltzmann distribution. It amounts to approximately 5% at 300 K. This means, that even in the classical case, using just classical statistical mechanics, the energy barrier should be crossed from time to time, as it happened during the simulation.

In the quantum case the probability of finding the proton at the barrier is much higher than in the classical case. Fig. 5.6 displays the expected behavior, i.e. tunneling between the two minima with a finite proton density for the symmetrical case.

While the parameter δ reveals a pronounced influence by the quantum nature of the nuclei, the angle α, shown in Fig. 5.7, seems not to be affected significantly. Again, statistics are better in the quantum case, but the maximum and the width of the distribution are almost unchanged. Only the width might be larger in the path integral case, but this can also be due to statistics. In both cases, the

maximum appears at α=10◦, leading to a structure similar to Fig. 5.4.

The structural parameters show, that the short hydrogen bond leads to tunneling of the proton if the quantum nature of the nuclei are incorporated. This leads to a

5.3 Tunneling Effects in Acetylacetone 81

very broad distribution of the bond lengths. However, the bending motion of the bonded proton is not affected significantly. In the next sections the consequences for NMR parameters, i.e. the chemical shift and the NQCC, will be presented.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 δ 0 0.01 0.02 0.03 0.04 relative occurrence

(a) Classic Simulation

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 δ 0 0.005 0.01 0.015 relative occurrence

(b) Path Integral Simulation

Figure 5.6: Distribution of the distance parameter δ from a classical (5.6(a)) and a path integral (5.6(b)) simulation. The parameter δ is ex- plained in Fig. 5.5.

82 5 Nuclear Quantum Effects in Molecular Systems 0 5 10 15 20 25 30 35 40 45 50 α [°] 0 0.01 0.02 0.03 0.04 relative occurrence

(a) Classic Simulation

0 5 10 15 20 25 30 35 40 45 50 α [°] 0 0.01 0.02 0.03 0.04 relative occurrence

(b) Path Integral Simulation

Figure 5.7: Distribution of the angle α from a classical (5.7(a)) and a path integral (5.7(b)) simulation. The angle α is shown in Fig. 5.5.

5.3 Tunneling Effects in Acetylacetone 83