Dataset on the piezo-spectroscopic behaviour of hydroxylapatite: Effect of mechanical stress on the Raman and Infrared vibrational bands from ab initio quantum mechanical simulations.

This article reports data on the vibrational (Raman and Infrared) behavior of hydroxylapatite [OHAp, Ca10(PO4)6(OH)2, space group P63] under mechanical stress, which were discussed in details in the work of Ulian and Valdrè (2017) [1]. The dataset has been obtained by ab initio quantum mechanical means, by employing Density Functional Theory methods, in particular the B3LYP hybrid functional, all-electron Gaussian-type orbitals basis sets and a correction to take into account the effects of dispersive forces.

Specifications Table

Value of the data

Geometries of hydroxylapatite [OHAp, Ca10(PO4)6(OH)2, space group P63] at both equilibrium and stressed conditions.

Vibrational analysis (Raman and IR) of each hydroxylapatite geometry.

Vibrational spectra (Raman and IR) of each hydroxylapatite geometry up to 4000 cm−1, which could be employed for comparison with experimental data.

Results obtained at the Density Functional Theory (DFT) level, employing hybrid B3LYP functional and including a correction to take into account the contribution of dispersive forces.

Data

Hydroxylapatite geometry at equilibrium and under mechanical stress

Equilibrium and deformed (strained) OHAp models were realized and geometrically optimized, and the stress for each deformation was calculated according to stress-strain formulations.

Hydroxylapatite (OHAp, s.g. P63) was optimized to take into account the effect of dispersive force contribution in the final unit cell and internal geometries (Table 1). Then, it was deformed according to the three symmetry-independent strains related to the P63 space group [1]:where ε1 and ε3 are normal strains (uniaxial) perpendicular to the (100) and (001) surfaces of the hexagonal unit cell and ε4 is a shear stress parallel to the (001) surface. The unit cell data (lattice parameters and atomic coordinates) for ε1, ε3 and ε4 deformed OHAp are reported in Table 2, Table 3, Table 4, respectively. In Eq. (1), δ represents a multiplicative factor used to control the compressive (δ > 0) or tensile (δ < 0) deformation of the OHAp unit cell. Four unit cell configurations (two in expansion, δ = − 0.04 and δ = − 0.02, and two in compression, δ = + 0.02 and δ = + 0.04) were geometrically optimized for each considered strain (ε, ε, ε), resulting in twelve deformed structures of OHAp. In the case of normal strain (ε1 and ε3), the unit cell was expanded/contracted by ± 4% and ± 2%, with resulting applied stress in the range ± 9 GPa. Symmetry analysis conducted on the deformed geometries revealed that for strain ε1 the OHAp unit cell belongs to space group P21, for strain ε3 to P63 and ε4 to space group P1 (absence of symmetry).

Table 1

Simulated OHAp lattice parameters (a, b and c reported in Å; α, β and γ in degrees) and internal coordinates of each irreducible atom (relative to s.g. P63) at equilibrium.

Equilibrium
a		9.38593	α		90
b		9.38593	β		90
c		6.87087	γ		120
Ca1	X	0.333333	O2	X	− 0.412019
	Y	− 0.333333		Y	0.465624
	Z	− 0.000114		Z	0.244387
Ca2	X	− 0.333333	O3	X	0.334650
	Y	0.333333		Y	0.251336
	Z	− 0.001445		Z	0.071593
Ca3	X	0.247056	O4	X	− 0.347179
	Y	− 0.004869		Y	− 0.256809
	Z	0.249398		Z	− 0.065903
P1	X	0.396585	O5	X	0.000000
	Y	0.366658		Y	0.000000
	Z	0.250675		Z	− 0.211631
O1	X	0.324502	H1	X	0.000000
	Y	0.483345		Y	0.000000
	Z	0.252703		Z	− 0.070371

Table 2

Simulated OHAp lattice parameters (a, b and c reported in Å; α, β and γ in degrees) and internal coordinates of each irreducible atom (relative to s.g. P21) under the effect of strain ε1.

		δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04			δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04
a		9.10581	9.2455	9.52707	9.66888	α		90.00	90.00	90.00	90.00
b		9.38593	9.38593	9.38593	9.38593	β		90.00	90.00	90.00	90.00
c		6.87087	6.87087	6.87087	6.87087	γ		120.50	120.50	119.51	119.04
Ca1	X	0.334810	0.334094	0.332927	0.332281	O4	X	− 0.407217	− 0.409670	− 0.414299	− 0.416287
	Y	− 0.331032	− 0.332457	− 0.333257	− 0.333138		Y	0.464545	0.465670	0.464858	0.463486
	Z	− 0.002772	− 0.001538	0.000396	0.001281		Z	0.240659	0.242976	0.245895	0.247193
Ca2	X	− 0.335426	− 0.334529	− 0.332699	− 0.331698	O5	X	− 0.469999	− 0.467559	− 0.464408	− 0.463039
	Y	0.331427	0.332667	0.333309	0.333293		Y	0.117859	0.120143	0.123902	0.125989
	Z	− 0.002140	− 0.001585	− 0.001351	− 0.001216		Z	0.239863	0.242299	0.245715	0.247234
Ca3	X	0.238849	0.243491	0.249691	0.252164	O6	X	− 0.120787	− 0.121688	− 0.123114	− 0.123577
	Y	− 0.012941	− 0.008877	− 0.001104	0.002777		Y	0.410566	0.411251	0.412787	0.413645
	Z	0.247401	0.248632	0.249881	0.250231		Z	0.240694	0.242877	0.245898	0.247425
Ca4	X	0.012074	0.008724	0.000618	− 0.004550	O7	X	0.326445	0.331460	0.337021	0.338766
	Y	0.254700	0.253625	0.250064	0.247442		Y	0.246297	0.249132	0.253683	0.255873
	Z	0.248849	0.249276	0.249949	0.250410		Z	0.073655	0.072288	0.071044	0.070760
Ca5	X	− 0.256111	− 0.253600	− 0.250597	− 0.249725	O8	X	− 0.245517	− 0.248248	− 0.254588	− 0.258257
	Y	− 0.248862	− 0.247339	− 0.247559	− 0.248625		Y	0.085670	0.084471	0.082113	0.080969
	Z	0.249720	0.249546	0.249725	0.249876		Z	0.069842	0.070934	0.072006	0.072041
P1	X	0.395158	0.395861	0.397267	0.398207	O9	X	− 0.082900	− 0.083337	− 0.082818	− 0.082408
	Y	0.364127	0.365397	0.368077	0.369622		Y	− 0.333158	− 0.334300	− 0.334646	− 0.334680
	Z	0.250976	0.250771	0.250699	0.250741		Z	0.073071	0.072278	0.071116	0.070702
P2	X	− 0.364316	− 0.365317	− 0.368288	− 0.370176	O10	X	− 0.346732	− 0.347075	− 0.346556	− 0.345796
	Y	0.030767	0.030307	0.029306	0.028805		Y	− 0.255203	− 0.255916	− 0.257720	− 0.258815
	Z	0.248902	0.249928	0.251094	0.251384		Z	− 0.064935	− 0.065518	− 0.066304	− 0.066697
P3	X	− 0.029861	− 0.029834	− 0.029849	− 0.029788	O11	X	0.253804	0.255077	0.259281	0.261861
	Y	− 0.397445	− 0.396998	− 0.396220	− 0.395826		Y	− 0.095908	− 0.092927	− 0.087972	− 0.085534
	Z	0.250235	0.250525	0.250926	0.251265		Z	− 0.066095	− 0.065884	− 0.065922	− 0.066167
O1	X	0.324735	0.323687	0.326660	0.330157	O12	X	0.093661	0.091860	0.088316	0.086403
	Y	0.482800	0.482319	0.485632	0.488963		Y	0.352673	0.349651	0.344521	0.342029
	Z	0.254065	0.253318	0.252327	0.251940		Z	− 0.065452	− 0.065756	− 0.065818	− 0.065559
O2	X	− 0.479400	− 0.481514	− 0.484890	− 0.486650	O13	X	0.004396	0.001682	− 0.000994	− 0.001575
	Y	− 0.160573	− 0.159795	− 0.158046	− 0.156986		Y	0.000806	0.000320	− 0.000271	− 0.000473
	Z	0.252610	0.252714	0.252693	0.252461		Z	− 0.203916	− 0.208469	− 0.214420	− 0.217080
O3	X	0.166247	0.162538	0.155331	0.151899	H1	X	0.002633	0.000802	− 0.000841	− 0.001210
	Y	− 0.319183	− 0.321969	− 0.327414	− 0.330061		Y	− 0.000019	− 0.000395	0.000372	0.001256
	Z	0.252679	0.252709	0.252524	0.252451		Z	− 0.062913	− 0.067321	− 0.073074	− 0.075674

Table 3

Simulated OHAp lattice parameters (a, b and c reported in Å; α, β and γ in degrees) and internal coordinates of each irreducible atom (relative to s.g. P63) under the effect of strain ε3.

		δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04			δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04
a		9.38593	9.38593	9.38593	9.38593	α		90.00	90.00	90.00	90.00
b		9.38593	9.38593	9.38593	9.38593	β		90.00	90.00	90.00	90.00
c		6.59603	6.73345	7.00828	7.1457	γ		120.00	120.00	120.00	120.00
Ca1	X	0.333333	0.333333	0.333333	0.333333	O2	X	− 0.409655	− 0.410986	− 0.412928	− 0.413543
	Y	− 0.333333	− 0.333333	− 0.333333	− 0.333333		Y	0.466622	0.465970	0.465637	0.465740
	Z	− 0.000031	− 0.000278	− 0.000904	− 0.001605		Z	0.244320	0.244309	0.243946	0.243187
Ca2	X	− 0.333333	− 0.333333	− 0.333333	− 0.333333	O3	X	0.337166	0.335544	0.333634	0.332652
	Y	0.333333	0.333333	0.333333	0.333333		Y	0.252233	0.251681	0.250917	0.250429
	Z	− 0.003014	− 0.002237	− 0.000716	− 0.000374		Z	0.066124	0.068984	0.074434	0.077408
Ca3	X	0.247451	0.247081	0.246898	0.246772	O4	X	− 0.348993	− 0.347836	− 0.347275	− 0.347891
	Y	0.000046	− 0.002840	− 0.006431	− 0.007450		Y	− 0.257121	− 0.256932	− 0.256962	− 0.257259
	Z	0.248920	0.249246	0.249385	0.249155		Z	− 0.061829	− 0.063824	− 0.067928	− 0.069991
P1	X	0.399551	0.397883	0.395473	0.394719	O5	X	0.000000	0.000000	0.000000	0.000000
	Y	0.369103	0.367792	0.365772	0.365044		Y	0.000000	0.000000	0.000000	0.000000
	Z	0.250104	0.250441	0.250902	0.251102		Z	− 0.208776	− 0.210548	− 0.211885	− 0.211451
O1	X	0.329540	0.326911	0.322026	0.319982	H1	X	0.000000	0.000000	0.000000	0.000000
	Y	0.486221	0.484797	0.481844	0.480488		Y	0.000000	0.000000	0.000000	0.000000
	Z	0.251860	0.252352	0.253216	0.253807		Z	− 0.061677	− 0.066456	− 0.073360	− 0.075575

Table 4

Simulated OHAp lattice parameters (a, b and c reported in Å; α, β and γ in degrees) and internal coordinates of each irreducible atom (relative to s.g. P1) under the effect of strain ε4.

		δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04			δ = + 0.04	δ = + 0.02	δ = + 0.02	δ = + 0.04
a		9.38781	9.38640	9.38640	9.38781	α		94.58	92.29	92.29	94.58
b		9.39344	9.38781	9.38781	9.39344	β		87.71	88.85	88.85	87.71
c		6.87636	6.87224	6.87224	6.87636	γ		120.02	120.01	120.01	120.02
Ca1	X	0.317777	0.325394	0.345306	0.350730	O7	X	− 0.424147	− 0.416872	− 0.414115	− 0.421470
	Y	− 0.335664	− 0.339535	− 0.318267	− 0.308686		Y	0.460737	0.464704	0.464319	0.459810
	Z	− 0.004382	− 0.000749	0.000061	0.003185		Z	0.220911	0.236204	0.253665	0.272171
Ca2	X	− 0.350739	− 0.345318	− 0.325344	− 0.317811	O8	X	0.421491	0.414110	0.416883	0.424165
	Y	0.308678	0.318245	0.339541	0.335575		Y	− 0.459857	− 0.464346	− 0.464694	− 0.460757
	Z	− 0.496843	− 0.499870	0.499292	0.495590		Z	− 0.227839	− 0.246542	− 0.263921	− 0.279192
Ca3	X	− 0.316706	− 0.322659	− 0.347882	− 0.353545	O9	X	− 0.471770	− 0.467099	− 0.471200	− 0.475702
	Y	0.339178	0.343603	0.315378	0.307537		Y	0.114190	0.120651	0.116920	0.111324
	Z	0.003451	− 0.000631	− 0.000976	− 0.003025		Z	0.225423	0.226771	0.263951	0.268135
Ca4	X	0.353566	0.347874	0.322603	0.316635	O10	X	− 0.121780	− 0.122116	− 0.122702	− 0.122305
	Y	− 0.307562	− 0.315382	− 0.343604	− 0.339244		Y	0.416154	0.414081	0.416816	0.419510
	Z	0.497034	0.498979	0.499305	− 0.496550		Z	0.280222	0.268411	0.221001	0.212080
Ca5	X	0.238172	0.244317	0.240684	0.235374	O11	X	0.122281	0.122688	0.122093	0.121789
	Y	− 0.007066	− 0.005546	− 0.007633	− 0.008280		Y	− 0.419475	− 0.416855	− 0.414034	− 0.416116
	Z	0.263151	0.255414	0.243789	0.236597		Z	− 0.287908	− 0.279002	− 0.231674	− 0.219790
Ca6	X	− 0.235385	− 0.240627	− 0.244328	− 0.238180	O12	X	0.475653	0.471260	0.467054	0.471780
	Y	0.008264	0.007666	0.005526	0.007063		Y	− 0.111372	− 0.116880	− 0.120669	− 0.114178
	Z	− 0.263314	− 0.256208	− 0.244574	− 0.236824		Z	− 0.231836	− 0.236129	− 0.273243	− 0.274632
Ca7	X	0.011899	0.008172	0.004476	0.007596	O13	X	0.294097	0.314600	0.348475	0.357871
	Y	0.255380	0.254440	0.255109	0.256278		Y	0.232824	0.243480	0.256793	0.264575
	Z	0.245777	0.250077	0.248823	0.252624		Z	0.092206	0.083082	0.062531	0.058012
Ca8	X	− 0.255123	− 0.252422	− 0.253419	− 0.257677	O14	X	− 0.357847	− 0.348406	− 0.314463	− 0.293989
	Y	− 0.251298	− 0.247303	− 0.247645	− 0.252167		Y	− 0.264600	− 0.256753	− 0.243430	− 0.232812
	Z	0.238718	0.243382	0.256264	0.261076		Z	− 0.441962	− 0.437439	− 0.416837	− 0.407743
Ca9	X	0.257719	0.253400	0.252409	0.255165	O15	X	− 0.239997	− 0.241474	− 0.265029	− 0.269416
	Y	0.252215	0.247583	0.247319	0.251278		Y	0.089710	0.085247	0.085724	0.085911
	Z	− 0.238915	− 0.243751	− 0.256653	− 0.261272		Z	0.067441	0.071672	0.071015	0.074323
Ca10	X	− 0.007549	− 0.004481	− 0.008189	− 0.011967	O16	X	− 0.091499	− 0.090552	− 0.070627	− 0.068819
	Y	− 0.256235	− 0.255130	− 0.254376	− 0.255382		Y	− 0.367644	− 0.357511	− 0.305906	− 0.295881
	Z	− 0.247446	− 0.251170	− 0.249952	− 0.254167		Z	0.065286	0.064135	0.081394	0.080736
P1	X	0.387784	0.392693	0.394568	0.389046	O17	X	0.068879	0.070548	0.090518	0.091524
	Y	0.363307	0.365549	0.365743	0.363954		Y	0.295981	0.305844	0.357462	0.367689
	Z	0.259845	0.257975	0.243578	0.241845		Z	− 0.419296	− 0.418581	− 0.435836	− 0.434678
P2	X	− 0.389017	− 0.394562	− 0.392695	− 0.387777	O18	X	0.269374	0.265004	0.241449	0.239960
	Y	− 0.363976	− 0.365739	− 0.365571	− 0.363317		Y	− 0.085944	− 0.085783	− 0.085235	− 0.089713
	Z	− 0.258135	− 0.256412	− 0.242030	− 0.240183		Z	− 0.425671	− 0.428976	− 0.428328	− 0.432568
P3	X	− 0.367292	− 0.366735	− 0.368260	− 0.368541	O19	X	− 0.302262	− 0.328259	− 0.357389	− 0.364360
	Y	0.028048	0.030259	0.028740	0.027573		Y	− 0.236288	− 0.248801	− 0.261053	− 0.266613
	Z	0.235070	0.241021	0.260623	0.267007		Z	− 0.086547	− 0.075973	− 0.058820	− 0.056118
P4	X	− 0.028723	− 0.028859	− 0.028823	− 0.028992	O20	X	0.364312	0.357338	0.328103	0.302313
	Y	− 0.394558	− 0.395317	− 0.393824	− 0.392640		Y	0.266572	0.260987	0.248735	0.236280
	Z	0.257796	0.253098	0.247908	0.243674		Z	0.443890	0.441155	0.423940	0.413440
P5	X	0.028992	0.028805	0.028865	0.028743	O21	X	0.245316	0.247494	0.269866	0.274226
	Y	0.392671	0.393793	0.395355	0.394586		Y	− 0.095802	− 0.092378	− 0.090265	− 0.087929
	Z	− 0.256317	− 0.252053	− 0.246906	− 0.242159		Z	− 0.062120	− 0.065850	− 0.065471	− 0.068438
P6	X	0.368504	0.368268	0.366731	0.367270	O22	X	0.094696	0.094350	0.079329	0.075152
	Y	− 0.027606	− 0.028760	− 0.030237	− 0.028078		Y	0.373387	0.365363	0.320241	0.307422
	Z	− 0.232990	− 0.239389	− 0.258990	− 0.264961		Z	− 0.061900	− 0.060497	− 0.074635	− 0.074407
O1	X	0.316110	0.320362	0.322261	0.318653	O23	X	− 0.075131	− 0.079222	− 0.094332	− 0.094689
	Y	0.480524	0.482163	0.482437	0.482438		Y	− 0.307374	− 0.320080	− 0.365381	− 0.373399
	Z	0.280226	0.270527	0.235184	0.225712		Z	0.425550	0.425321	0.439479	0.438105
O2	X	− 0.318623	− 0.322250	− 0.320379	− 0.316106	O24	X	− 0.274193	− 0.269836	− 0.247486	− 0.245369
	Y	− 0.482462	− 0.482432	− 0.482195	− 0.480534		Y	0.087944	0.090310	0.092335	0.095788
	Z	− 0.274270	− 0.264797	− 0.229445	− 0.219720		Z	0.431558	0.434516	0.434159	0.437889
O3	X	− 0.477362	− 0.481229	− 0.479654	− 0.476029	O25	X	0.012018	0.004040	0.002550	0.009398
	Y	− 0.162376	− 0.159121	− 0.160975	− 0.162981		Y	0.004631	− 0.000335	− 0.002154	− 0.000222
	Z	0.226760	0.238412	0.266527	0.277438		Z	− 0.218148	− 0.213933	− 0.210326	− 0.209714
O4	X	0.159946	0.159757	0.159432	0.159360	O26	X	− 0.009426	− 0.002549	− 0.004045	− 0.012063
	Y	− 0.321765	− 0.323181	− 0.324593	− 0.322395		Y	0.000251	0.002210	0.000317	− 0.004566
	Z	0.251920	0.248974	0.254990	0.251836		Z	0.290258	0.289557	0.285960	0.281846
O5	X	− 0.159362	− 0.159447	− 0.159762	− 0.159926	H1	X	0.000821	− 0.000434	0.004726	0.011511
	Y	0.322395	0.324573	0.323147	0.321769		Y	0.001723	− 0.003302	0.001300	− 0.001104
	Z	− 0.248142	− 0.244909	− 0.250952	− 0.248073		Z	− 0.077412	− 0.072863	− 0.069308	− 0.068463
O6	X	0.476043	0.479636	0.481252	0.477346	H2	X	− 0.011516	− 0.004702	0.000488	− 0.000875
	Y	0.162940	0.160961	0.159139	0.162351		Y	0.001220	− 0.001243	0.003345	− 0.001712
	Z	− 0.222613	− 0.233535	− 0.261582	− 0.273231		Z	0.431514	0.430576	0.427022	0.422587

Vibrational frequencies

For each optimized hydroxylapatite model, both at equilibrium (δ = + 0.00) and at strained configurations (δ = ± 0.02 and δ = ± 0.04), vibrational frequencies (Raman and IR) were obtained by means of finite displacements method. In the OHAp unit cell, there are 44 atoms, resulting in 132 degrees of freedom (129 with vibrational character) [1]. The calculated frequencies for ε1, ε3 and ε4 deformed OHAp are reported in Tables S1–S3 (Supplementary material), respectively. To aid the comparison between the peak positions, for the strains ε1 and ε4 the normal modes of the non-deformed hydroxylapatite were calculated with the symmetry of the strained cells (P21 and P1, respectively).

Vibrational intensities

Vibrational intensities were calculated within the Placzek approximation (Raman, partial derivatives of the polarizability tensor with respect to atomic positions) [3], [4], [5] and analytically through Coupled-Perturbed Kohn–Sham approach (Infrared) [6], [7], [8], [9]. The intensity of each normal mode, IR and Raman, was calculated for each OHAp model and reported in the Supplementary material section (Tables S1–S3).

Theoretical design, materials, and methods

The data here presented was obtained by first principle simulations on periodic systems, using the CRYSTAL14 code [10], which implements the Hartree–Fock and Kohn–Sham self-consistent field method.

Basis set

Multielectron wave functions are constructed as an antisymmetrized product (Slater determinant) of monoelectronic crystalline orbitals (CO) that are linear combination of local functions (atomic orbitals, AO) centred on each atom in the system. In turn, atomic orbitals (basis set) are linear combinations of Gaussian-type functions (GTF). The all-electron basis sets employed in the present simulations were chosen among previously adopted ones for [2], [11], [12], [13].

Hamiltonian and computational parameters

The Becke [14] three-parameter (B3LYP) hybrid exchange functional in combination with the gradient-corrected correlation functional of Lee et al. [15] has been adopted for all calculations. The exchange–correlation contribution is performed over a grid of points and is the result of a numerical integration of the electron density and its gradient. The adopted pruned grid is given by 75 points and 974 angular points (XLGRID) and obtained from The Gauss–Legendre quadrature and Lebedev schemes [16]. The tolerance thresholds that control accuracy of the Coulomb and exchange integrals were set to 10−7 and 10−16, respectively [10]. The Hamiltonian matrix has been diagonalized using a shrinking factor that leads to 12 reciprocal lattice points (k-points). The convergence on total energy was reached when the difference between the energy of two subsequent self-consistent field cycles was less than 10−8 Hartree.

Van der Waals (dispersive) forces were included with the (DFT+D2 scheme [17], which adds the following contribution to the calculated DFT energy:

The summation over all atom pairs ij and g lattice vectors excludes the self- interaction contribution (i = j) for every g. The parameters C represent the dispersion coefficient for the atom i, R is the interatomic distance between atom i in the reference cell and atom j in the neighbouring cells at distance |g| and s is a functional-dependent scaling factor. The function f is used to dump the energy correction to avoid double counting of short-range contributions to the energy and depends on the sum of atomic van der Waals radii and on a steepness parameter (d = 20). Due to the molecular nature of the DFT+D2 scheme, which tends to overestimate cohesive energy in solid crystals, the original B3LYP+D parameters where modified, setting s to 1, R(H) to 1.30 and the heavier atom van der Waals radii were scaled by a factor 1.05 (B3LYP-D* approach) [18], [19], [20], [21], [22], [23].

Vibrational calculations

In periodic systems and within the harmonic approximation, the phonon frequencies at Γ point are evaluated diagonalising the central zone (k = 0) mass-weighted Hessian matrix:

is the second derivative of the electronic and nuclear repulsion energy E evaluated at equilibrium u 0 with respect to the displacement of atom A in cell 0 () and displacement of atom B in cell G () from their equilibrium position , :

In CRYSTAL14, the calculation of the Hessian at equilibrium is made by the analytical evaluation of the energy first derivatives, of E with respect to the atomic displacements:while second derivatives at u = 0 (where all first derivatives are zero) are calculated numerically using a "two-point" formula:

The Hessian matrix eigenvalues provide the normal harmonic frequencies ω and it is obtained with 3N + 1 SCF and gradient calculation.

Subject area	Physics
More specific subject area	Vibrational spectroscopy (Raman and IR) of biomaterials
Type of data	Tables
How data was acquired	Quantum mechanical simulations at the Density Functional Theory (DFT)/B3LYP level of theory, including dispersive forces contributions (CRYSTAL14 code)
Data format	Raw, analyzed
Theoretical factors	Starting geometry taken from previous DFT simulations[2].
Theoretical features	Quantum mechanical simulations conducted using Density Functional Theory, B3LYP functional and Gaussian-type orbitals basis sets.
	Inclusion of dispersive forces contribution via DFT-D2 scheme, corrected for the B3LYP functional (B3LYP-D* approach).
	Geometry optimization of the unit cell with and without applied strains.
Data source location	Bologna, P. Porta San Donato 1, Italy
Data accessibility	Data is displayed within this article.
Related research article	This Data in Brief article is submitted as a companion paper to: Ulian, G. & Valdrè, G. (2017) Effect of mechanical stress on the Raman and Infrared bands of hydroxylapatite: a quantum mechanical first principle investigation. Journal of the Mechanical Behavior of Biomedical Materials, in press.