Quantum Chemistry of Solids / 22-LCAO Calculations of Perfect-crystal Properties

9.3 Total Energy and Related Observables in LCAO Methods for Solids

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from them the values of the force-constant matrices and hence the dynamical matrix and phonon-dispersion curves. The majority of calculations were performed based on DFT PW method, for example to study phonons in the rutile structure, [672]. The calculation of the vibrational frequencies of crystals was implemented in the LCAO CRYSTAL06 code (see www.crystal.unito.it) and applied to di erent oxides: quartz SiO2 [669], corundum Al2O3 [673], calcite CaCO3 [674].

In the LCAO approximation frequencies at Γ are evaluated in the direct method in the same way as for molecules [669]: a set SCF calculations of the unit cell are performed at the equilibrium geometry and incrementing each of the nuclear coordinates in turn by u (use of symmetry can reduce the number of the required calculations). Second-order energy derivatives are evaluated numerically. Obtaining frequencies at wavevector symmetry points di erent from Γ would imply the construction of appropriate supercells (see Chap. 3). These supercells are chosen in the same way as was discussed in the electronic-structure calculations due to the one-to one correspondence between the supercell choice and the set of k-points equivalent to the Γ point in the reduced BZ. A finite range of interaction in the lattice sum (9.99) is assumed, usually inside the supercell chosen (compare with the cyclic-cluster model in the electronicstructure calculations). In the case of ionic compounds, long-range Coulomb e ects due to coherent displacement of the crystal nuclei are neglected, as a consequence of imposing the periodic boundary conditions [669]. Therefore, Wij (0) needs to be corrected for obtaining the longitudinal optical (LO) modes [675]. For this reason, in some cases only transverse optical (TO) parts of the phonon spectrum are calculated as is done in the combined DFT PW-DFT LCAO lattice dynamics study of TiO2 rutile [676]. The phonon frequencies computed in [676] for optimized crystal structure are reported in Table 9.33 and compared with experimental data.

The LDA frequencies are in excellent agreement with the experimental frequencies, especially if compared with the frequencies measured at low temperature (T 4 K), when these data are available. The deviation between the LDA and experimental frequencies is 13 cm−1 at most, and is often much smaller than that. For instance, the deviation drops to no more than 2 cm−1 for the two sti est modes, B2g and A1g , and it remains small also for several of the softer modes. Both PBE and PW91 results are much less satisfactory. With the exception of the B1g mode, the GGA functionals systematically underestimate the LDA and measured frequencies. It is found in [676] that this discrepancy between LDA and GGA results is mostly due to the di erence in the equilibrium lattice parameters at zero pressure predicted by the functionals. The LDA frequency is the highest because LDA predicts the smallest equilibrium volume, and the PBE equilibrium volume is large enough to lead to an imaginary frequency.

All the results discussed were obtained within the plane-wave pseudopotential implementation of DFT. For all functionals, calculations were repeated for the equilibrium geometries, bulk moduli, and energy profiles along the ferroelectric TO A2u mode with the all-electron LCAO scheme. Apart from slight quantitative di erences, in all cases the all-electron LCAO calculations agree well with the plane-wave, pseudopotential results, confirming their independence of the particular numerical scheme used to implement DFT. We note that the hybrid HF-DFT LCAO calculations could in principle give better agreement with experimental data for phonon frequencies of rutile as was shown in the B3LYP LCAO calculations of the vibrational spectrum

408 9 LCAO Calculations of Perfect-crystal Properties

Table 9.33. Calculated and measured frequencies (in cm−1) relative to the Γ -point of bulk TiO2 rutile ( [676]).*

Mode	PBE	PW91	LDA	Neutrons	IR and Raman
					Raman
B2g	774.3	780.9	824.7	825	827
A1g	565.9	572.0	611.6	610	612 (611)
Eg (2)	429.2	434.2	463.2	445	447 (455)
B1g	154.2	151.7	137.0	142	143 (143)
					Silent
A2g	423.6	425.4	421.7	not found	—
B2u	357.5	363.4	393.0	406	—
B1u	79.2	99.2	104.0	113	—
					Infrared
Eu (TO)	468.6	472.0	488.4	494	500
Eu (TO)	353.5	357.1	383.9 not found		388
Eu (TO)	124.0	127.4	191.4	189	183
A2u (TO) 186.3		47.1	154.4	173 (142)	167 (144)

The double degeneracy of the Eg mode is indicated by the label (2). The references

to the experimental neutron-scattering and the infrared (IR) and Raman data are given in [676]. When available, the low-temperature (T 4 K) experimental

frequencies are reported in brackets

of calcite CaCO3, [674]. In this case the mean absolute error is less than 12 cm−1 (frequencies range from 100 to 1600 cm−1).

As we have seen in this chapter, the HF and DFT LCAO methods, implemented in the computer code CRYSTAL, present e cient computational schemes for the study of di erent properties of periodic systems. Real-solid properties in many cases depend on the point defects, destroying the translational symmetry. In the next chapter we discuss the models used for defective crystals study and give examples of the LCAO calculations of defective crystals.