Geometry optimisation

In Chapter 2 the sensitivity of solid-state NMR parameters with respect to subtle changes in local geometry was stressed. This indicates that prior to predicting NMR parameters, it is essential to have high confidence in

the structural model(s) being used. Whether structural models have been generated computationally or from experimental data, initially predicted NMR parameters rarely agree well with experiment. This is because the structures often do not reflect an energy minimum on the DFT potential energy surface (PES). In the case where structures have been generated computationally, poor agreement with experiment could stem from the partial modification of an existing structure (e.g., by changing composition, unit cell size or atom positions). Alternatively using low level DFT during the structure generation process could also lead to inconsistencies between predicted and experimental quantities. Structural models based on experimental data are typically constructed from diffraction measurements, meaning they can suffer from their own set of limitations. The accuracy of experimentally determined structures will depend on both the radiation source, e.g., X-ray, synchrotron or neutron radiation, and on whether the sample is in powder or single-crystal form. Additionally, as DFT calculations are typically performed at zero Kelvin, the atom positions or overall unit cell size of structures optimised under DFT can differ from one constructed directly from experimental diffraction measurements, which are often performed closer to room temperature than absolute zero. Given these points, to minimise any discrepancies between DFT-predicted and experimentally determined parameters, it is advisable to perform a geometry optimisation prior to the prediction of quantities such as solid-state NMR parameters. This allows the atomic positions and unit cell vectors to vary to minimise the forces and stresses, and, as a consequence, the total energy. The importance of performing a geometry optimisation calculation prior to the prediction of solid-state NMR parameters is well documented in the literature, with improved agreement between predicted and experimental NMR parameters typically seen.56,58,72–77

Table 3.2 shows the calculated and

experimental isotropic chemical shieldings/shifts for the two

crystallographically distinct 89_{Y sites in Y}

2O3. From this data it can be seen

that the relative shielding difference between the two Y sites is much closer to the relative shift difference seen experimentally following geometry optimisation.

Table 3.2: Calculated and experimental isotropic shielding and shift values for Y2O3,78 _with the magnitude of the difference in isotropic shielding and shift also shown. All calculations were performed using CASTEP 8.0, the GGA PBE functional, an Ecut of 60 Ry and a k-point spacing of 0.04 2π Å–1_{, with ZORA applied for the NMR parameter calculations.}

Y1 site (ppm) Y2 site (ppm) Magnitude of difference (ppm) Calculated (unoptimised) σiso = 2442 σiso = 2380 62 Calculated (optimised) σiso = 2432 σiso = 2388 44 Experimental79 _δ iso = 273 δiso = 314 41

3.8.3 Semi-empirical dispersion correction

The recent introduction of semi-empirical dispersion correction

(SEDC) schemes80 _{into DFT codes has represented another significant}

improvement in the accuracy of quantum chemical calculations, allowing for even closer agreement between experimental and predicted NMR parameters. Dispersion interactions, sometimes referred to as London dispersion forces, arise from long-range electron correlation effects. As such, they represent the attractive r–6 _{term in the 12-6 Lennard-Jones potential}

VLJ =4ε σ r ⎛ ⎝⎜ ⎞⎠⎟ 12 − σ r ⎛ ⎝⎜ ⎞⎠⎟ 6 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ , (3.84)

where ε is the potential energy well depth, σ represents the distance at which the inter-particle potential is zero, and r is the distance between two particles. Failure to consider longer-range attractive interactions can have a significant effect on the accuracy of geometry optimisation, particularly if the crystal packing is strongly dependent on non-covalent interactions such as

hydrogen bonding or π-π stacking interactions. Given their apparent

Table 3.3: The unit cell volume for a series of structures, which were geometry optimised with no SEDC scheme and with the TS SEDC scheme. All calculations were performed using CASTEP 8.0, the GGA PBE functional, an Ecut of 60 Ry and a k-point spacing of 0.04 2π Å–1_. The relative difference in unit cell volumes for the structures optimised in the two separate ways is also given (expressed as a percentage).

Unit cell volume / Å3 Structure Optimised with no

SEDC scheme Optimised with TS SEDC scheme Relative difference (%)a Urea81 _{160.33 144.66 10.83} Al-MIL-5366 _{2008.73 1816.18 10.60} AlPO-53(C)68 _{1238.55 1110.51 11.53} β-Mg2SiO482 555.45 542.50 2.39 β-cristobalite67 _{415.22 412.67 0.62} SnO283 _{75.18 74.50 0.91} Y2O378 1208.84 1176.35 2.76 Y2Sn2O784 1151.56 1128.33 2.06 Y2Ti2O769 _{1046.23 1018.68 2.70}

a _{The relative difference in unit cell volume was determined by [((Vno SEDC – VTS) / VTS) × 100]}

where Vno SEDC and VTS are the unit cell volumes of the structures geometry optimised with no SEDC scheme and with the TS SEDC scheme, respectively.

developed for periodic systems, including the Grimme (D2) and the Tkatchenko and Scheffler (TS) schemes.85,86 _{In these SEDC schemes, an}

isotropic potential is used to describe the dispersion interactions, which at long range can be expressed using a sum of the London potentials over all N atoms in the system,

V=s6 C6,ijRij −6 _, j>i N

∑

∑

where s6 is a scaling factor, C6,ij is a dispersion coefficient between any two

atoms, i and j, separated by interatomic distance Rij. At short range, a

damping function f(R0_ij,R

ij) is applied to the long-range expression in

Equation 3.85. By specifying an interatomic distance cut-off, R_ij0

depends on the van der Waals radii of the atom pairs (i and j), the dampening function reduces the excess dispersion interactions to zero. When this potential is added to the original DFT energy, the dispersion-corrected total energy, Etot can be expressed as

Etot =EDFT+s6 f(SRRij 0 ,Rij) j>i N

∑

∑

where SR is a scaling factor dependent on the EXC functional, used to match

the dampening function to the DFT potential. The factors, s6 and SR, which

scale the dispersion coefficients are parameterised according to the particular SEDC scheme used, i.e., in D2 s6 = 0.75, and in TS SR = 0.94. Table 3.3 shows

the unit cell volumes for a series of structures optimised with no SEDC and with the TS SEDC scheme. It is clear that the decision to omit or include dispersion corrections during the calculation is of more importance for systems such as molecular crystals, which are held together by long-range van der Waals interactions. Applying a SEDC scheme when optimising materials such as aluminophosphates (AlPOs) and metal-organic frameworks (MOFs), systems that exhibit significant structural flexibility, also has a large effect on the resulting unit cell volume. In comparison, the unit cell volume of dense-phase materials, usually held together by strong covalent or ionic interactions, are less dependent on the inclusion of a SEDC scheme during the geometry optimisation. The data in Table 3.3 highlights how the decision whether to include a SEDC scheme or not is highly system- dependent. Any variations in atomic position or unit cell volume can affect the predicted NMR parameters, and it has previously been shown that geometry optimisation using a SEDC scheme such as D2 or TS prior to the calculation of NMR parameters yields better agreement with experimental NMR measurements.87–89