**2.1. Voronoi Integration**

**2.1.4. Advantages over Wannier Localization**

After we had developed the Voronoi integration approach to computing
molecu-lar electromagnetic moments, one of the first questions was if these moments are
suitable to predict vibrational spectra. To investigate this, we used a simulation
tra-jectory of liquid methanol to compute infrared and Raman spectra from molecular
electric moments based on either Wannier localization or Voronoi integration.^{MB23}
The results are presented in Figure 2.1.4.

**Figure 2.1.4.:**Comparison of predicted infrared(top panel) and Raman (bottom
panel)spectra of liquid methanol, using Wannier localization(dashed
lines)and Voronoi integration(solid lines)to compute the molecular
electric moments.^{MB23}

It can be seen that apart from minor differences in the intensities, both the infrared and the Raman spectrum are almost identically predicted by the two approaches.

The differences result from the fact that the Wannier localization enforces strictly neutral molecules(as Wannier centers bear integer charge), while the Voronoi integra-tion allows for a certain amount of charge transfer between the molecules. It can be concluded that molecular electromagnetic moments based on Voronoi integration are well suitable to predict bulk phase vibrational spectra.

2.1. Voronoi Integration

**Timing and Convergence**

As described in Section 1.4, all known methods to perform a Wannier localization of a periodic system are iterative, which means that they converge towards the desired solution, but actually never reach the exact solution. Unfortunately, it can not even be guaranteed that the used algorithms always converge. In other words, it can happen that the localization procedure fails for a particular frame along a simulation, so that the electromagnetic moments are missing for that frame. Pre-dicting vibrational spectra relies on computing time-correlation functions, where a missing frame would be highly problematic.

**Figure 2.1.5.:**Logarithmic frame times for a standard BOMD simulation (black
curve)and a BOMD with Wannier localization(red curve)of a liquid
phase system with≈1000 atoms.

To give an example, consider Figure 2.1.5, where the frame times of a standard
BOMD simulation with CP2k^{212–214}(black curve)are compared to those with
addi-tional Wannier localization in each step(red curve). The system is in the liquid phase
and contains 936 atoms(cubic cell size≈20Angstrom). Please note the logarithmic
vertical axis. The average frame time of the standard BOMD is 47.9 s, while the
frame time with additional Wannier localization is 139.3 s on average. This means
that 65% of the total computer time are spent for the localization step, which is
certainly not satisfactory. Apart from that, the frame times are highly irregular
with Wannier localization. The reason is that CP2k uses the very efficient “crazy
angle” algorithm for the localization by default. If this algorithm does not converge,
iterative Jacobi diagonalization^{211} is employed as a fallback. The latter is slightly
more robust but considerably slower, so that frame times of several thousand
sec-onds can be observed if the fallback is activated. But even if considering only the

frames for which the fast “crazy angle” algorithm did converge, the time for the localization alone is still≈30 s per frame, which is still a considerable amount of the total computer time.

Our Voronoi integration approach, in contrast, is not an iterative method and does not need to converge. This means that there do not exist cases in which no electromagnetic moments can be obtained for a particular frame. Furthermore, our approach is significantly faster than the Wannier localization—it requires only 2.0 s per frame for the same system on a single CPU core. Therefore, more than a factor 2 of total computer time can be saved by utilizing Voronoi integration instead of Wannier localization for each frame of the simulation.

**Issues with Aromatic Systems**

Another disadvantage of the Wannier localization approach are certain issues with
aromatic systems. Please consider Figure 2.1.6, where we have predicted the
in-frared spectrum of liquid benzene based on Wannier localization(dashed line)and
Voronoi integration(solid line).^{MB23}In the Wannier-based spectrum, artificial peaks
appear between 1200 and 1350 cm^{−}^{1}. These peaks are neither present in the
experi-mental spectrum, nor if the spectrum is predicted via total cell dipole moment or
Voronoi integration.

**Figure 2.1.6.:**Predicted infrared spectrum for liquid benzene based on Wannier
localization (dashed line) and Voronoi integration(solid line). The
Wannier-based spectrum possesses artificial peaks between 1200
and 1350 cm^{−1}.^{MB23}

After some investigation, we were able to identify the cause of these artificial peaks.

When performing a Wannier localization of benzene, the aromatic electrons need to

2.1. Voronoi Integration

be localized, so that an alternating single bond/double bond pattern results—see the left panel of Figure 2.1.7. When considering all vibrational normal modes of benzene, one of them deforms the molecule towards cyclohexatriene(see right panel of Figure 2.1.7). As this mode possesses an inversion symmetry, it cannot alter the total dipole moment of the molecule, and is therefore invisible in infrared spec-troscopy. However, if this mode is active, the six ring bonds no longer have identical bond lengths, and there appear preferred positions where the Wannier centers of the aromatic electrons should be localized. As a result, the single bond/double bond pattern of the aromatic electrons flips with the frequency of this vibration.

Due to numerical inaccuracies(an iterative localization can never be fully converged), this leads to a small jump in the molecular dipole moment, which appears as an artificial peak in the infrared spectrum at the frequency of the cyclohexatriene de-formation mode which should be invisible.

**Figure 2.1.7.:**Wannier localization for one benzene molecule leads to an
alter-nating single bond/double bond pattern(left panel); normal mode
of benzene which deforms the molecule towards cyclohexatriene
(right panel).^{MB23}

The situation becomes even worse if one tries to compute a Raman spectrum based on molecular polarizabilities resulting from Wannier localization and external field finite differences. Depending on the direction of the external field, the preferred localization of single bonds and double bonds in the ring differs, so that different such patterns can occur in the two calculations for the finite differences. This introduces an amount of noise which is actually so large that the Raman spectrum completely vanishes in the noise. For an example, see Figure 2.1.8, where the Raman spectrum of liquid benzene was predicted based on Wannier localization(dashed curve) and on Voronoi integration (solid line). As described above, the Wannier-based spectrum consists almost exclusively of noise, while the Voronoi-Wannier-based result gives a good prediction of the Raman spectrum.

**Figure 2.1.8.:**Predicted Raman spectrum for liquid benzene based on Wannier
localization (dashed line) and Voronoi integration (solid line).^{MB23}
The Wannier-based prediction shows only noise.

**Computing Higher Multipoles**

Wannier localization is frequently used in the literature to compute molecular
electric dipole moments. However, certain types of vibrational spectroscopy(such
as Raman optical activity—see Section 2.8)require also the electric quadrupole
mo-ment.^{68}To the best of our knowledge, it is not possible to compute higher molecular
electric multipoles via Wannier localization. Based on the Voronoi integration, on
the other hand, this is not an issue, as shown in Equation 2.8.1.

**Conclusions**

To conclude this section, we find the following four advantages of Voronoi integra-tion over Wannier localizaintegra-tion:

• Voronoi integration requires considerably less computer time for medium-sized and large systems. More than a factor of 2 can be saved.

• Iterative Wannier localization is not guaranteed to converge at all; Voronoi integration is non-iterative and always yields results.

• Wannier localization has severe issues with aromatic systems(artificial bands in the infrared spectrum, large amounts of noise in the Raman spectrum), while Voronoi integration has no problems with such systems.

• In contrast to Wannier localization, Voronoi integration can also compute higher multipole moments such as the electric quadrupole tensor, which is, e. g., required to predict ROA spectra.