1. B3LYP augmented with an empirical dispersion term (B3LYP-D
B3LYP augmented with an empirical dispersion term (B3LYP-D*) as applied to solids
Bartolomeo Civalleri
Theoretical Chemistry Group
Department of Chemistry IFM
& NIS Centre of Excellence
University of Torino
Vallico Sotto July 2009

2. Weak interactions in crystalline solids
Cohesive forces
long-range: electrostatic, induction, dispersion
short-range: exchange repulsion, charge transfer
Weak interactions play an important role in the solid state (see T. Steiner, Angew. Chem. Int. Ed. 41 (2002) 48)
Molecular recognition  crystal packing
Supramolecular chemistry and crystalline engineering
Molecular crystals (polymorphism)
Layered and composite/intercalated materials
Adsorption and reactivity on surfaces
Very important for many properties of interest: structure, interaction energies, vibrational frequencies and thermodynamics, elastic constants, relative stability, …
Vallico Sotto July 2009

3. State of the art in ab initio calculations of MCs
The “standard” ingredients (with many variants) :
a) HF or DFT (Kohn-Sham) Hamiltonians
b) Plane-Wave + Pseudopotentials (no BSSE)
or Localized functions (Gaussian) + All-Electron (BSSE) [CRYSTAL06]
c) Analytic derivatives of energy and other observables (e.g. phonons)
DFT is the most common choice to include electron correlation
Only LDA, GGA and hybrid-GGA (e.g. B3LYP)
methods are routinely available in solid state codes
Present DFT functionals do not account for dispersion energy
BSSE can give binding where there is none.
Vallico Sotto July 2009

4. 3D, 2D and 1D HB molecular crystals
Hydrogen bonded molecular crystals: B3LYP results
3D, 2D and 1D HB molecular crystals
Volume in Å3. Energy in kJ/mol. Deviation from experimental volumes in parentheses.
BSSE corrected cohesive energies are independent from the adopted basis set but they are markedly underestimated
With the large TZP basis set, BSSE is very small but predicted unit cell sizes are largely overestimated
With small basis sets, BSSE artificially compensates the missing dispersion energy
When HB is dominating, B3LYP gives good lattice parameters (not shown)
Vallico Sotto July 2009 4

5. Gaussian BS vs Plane-Wave BS: Urea
PW-PP calculations do not suffer from BSSE
Good agreement between GTFs and PWs
Results are independent of the computational approach (GTFs or PWs)
a is in both cases remarkably overestimated
Errors due to inherently lack of dispersion forces in DFT GTF vs PW basis sets: full comparison for formic acid S. Tosoni, C. Tuma, J. Sauer, B. Civalleri, P. Ugliengo, J. Chem. Phys. 127 (2007)
[1] T. Bucko, J. Hafner, J.G. Angyan, J. Chem. Phys. 122 (2005)
[2] NPD (12 K): S. Swaminathan, B.N. Craven, R.K. McMullan Acta Cryst. B40 (1984) 300
Vallico Sotto July 2009

6. DFT vs vdW forces: new hopes…
How to deal with dispersion interactions in DFT?
New functionals:
vdW-DFT (Langreth, Lundqvist and co-workers)
beyond m-GGA (Perdew’s “Jacob’s Ladder” – fifth rung)
screened Coulomb (or CAM) functionals (Scuseria, Handy, Savin, Hirao, …) - Truhlar’s family (M05, M05-X, M06, …)
Perturbational electron-interaction corrections on top of range-separated hybrids (Savin and coworkers: e.g. Goll et al. PCCP (2005))
- double-hybrids (Grimme JCP 124 (2006) , Head-Gordon JPC-A (2008) )
A pragmatic approach: Wilson-Levy (WL) correlation functional (a-posteriori HF) (T.A. Walsh, PCCP 7 (2005) 403; B. Civalleri et al. CPL 451 (2008) 287)
Empirical corrections by adding a -C6/R6 term (Grimme, Neumann, Yang, Zimmerli, Scoles, …)
A DF model of the dispersion interaction: C6 in terms of exchange-hole dipole moment (Becke-Johnson, JCP 123 (2005) , JCP 124 (2006) , JCP 127 (2007) ) or C6 from MLWFs (Silvestrelli, PRL 100 (2008) )
Dispersion-corrected atom centered pseudo-potentials (U. Rothlisberger and co-workers: e.g. Tapavicza et al. JCTC (2007), G. DiLabio CPL (2008))
Vallico Sotto July 2009

7. Atom-atom additive damped empirical potential of the form -f(R)C6/R6
Empirical –C6/R6 correction: Grimme’s model
S. Grimme, J. Comput. Chem., 2004, 25, 1463 and J. Comput. Chem., 2006, 27, 1787
Atom-atom additive damped empirical potential of the form -f(R)C6/R6
where
s6: scaling factor for each DFT method (s6=1.05 for B3LYP)
C6ij are computed from atomic dispersion coefficients: C6ij = C6i·C6j
Rvdw is the sum of atomic van der Waals radii: Rvdw=Rivdw+Rjvdw
d determines the steepness of the damping function (d=20)
summation over g truncated at 25 Å (estimated error < 0.02 kJ/mol on DE)
Grimme proposed a set of parameters (i.e. C6i and Rivdw) from H to Xe
Total energy is then computed as:
Implemented in CRYSTAL06 for energy and gradients (atoms and cell):
B. Civalleri, C.M. Zicovich-Wilson, et al., CrystEngComm, 2008, DOI: /b715018k
(see supplementary material for erratum)
Vallico Sotto July 2009

8. GRIMME input block E.g.: Urea – B3LYP-D … GRIMME
CRYSTAL
0 0 0
113 5
Optional keywords
END (ENDG)
Basis set
END …
GRIMME 4
Grimme empirical dispersion keyword
s6 (scaling factor) d (steepness) Rcut (cut-off radius, Å)
Nr. of atomic species
Atomic number C6 (Jnm6 mol−1) Rvdw (Å)
Atomic number C6 Rvdw
End of SCF&method input section
Vallico Sotto July 2009

9. GRIMME dataset Parameters available from H to Xe
Rvdw values are derived from the radius of the 0.01 a0−3 electron density contour from ROHF/TZV computations of the atoms in the ground state
C6 coefficients derived from the London formula for dispersion. DFT/PBE0 calculations of atomic ionization potentials Ip and static dipole polarizabilities α. The C6 coefficient for atom i (in Jnm6 mol−1) is then given as (Ip and α in atomic units)
C6i = 0.05NIpi αi
where N has values 2, 10, 18, 36, and 54 for atoms from rows 1–5 of the periodic table
S. Grimme, J. Comput. Chem., 2006, 27, 1787
Suitable for solids?
Vallico Sotto July 2009

10. 14 molecular crystals both dispersion bonded and hydrogen bonded
Tests on a set of selected molecular crystals
14 molecular crystals both dispersion bonded and hydrogen bonded
C6H6
Propane
C2H2
CO2
NH3
Formic acid
Naphthalene
Urotropine
1,4-dichloro-benzene
Experimental sublimation energies at 298K available from published data (estimated error bar: ±4 kJ/mol)
For some of them accurate low temperature structural data from NPD
Formamide
1,4-dicyano-benzene
Succinic anhydride
Boric acid
Urea
Vallico Sotto July 2009 10

11. BSSE corrected cohesive energies vs Experimental data
Cohesive energies: B3LYP vs B3LYP-D Grimme
BSSE corrected cohesive energies vs Experimental data
Exp.: -DE=DH0sub(T)+2RT from data at 298K
Cell fixed geometry optimization of the atomic positions at B3LYP/6-31G(d,p)
B3LYP: MD=54.4
Empirical correction definitely improves cohesive energies
Tendency of B3LYP-D Grimme to overestimate cohesive energy (MD=-6.0 & MAD=8.9) especially for HB molecular crystals
Small basis sets suffer from large BSSE
BSSE corrected data are less basis set dependent
-110 < DE < -25 kJ/mol
Vallico Sotto July 2009 11

12. Grimme’s model: the role of the damping function
The damping function is needed:
to avoid near singularities for small interatomic distances
some short-range correlation effects are already contained in the density functional
However:
crystal packing leads to larger overlap between molecular charge densities
damping function must act to longer-range where the B3LYP method does not contribute to the intermolecular interactions
atomic vdW radii define where the –f(R)C6/R6 contribution becomes dominant
atomic vdW radius for H very important
Strategy: scaling the atomic RvdW
Carbon Rvdw(C)=161 pm
RvdW
From: S. Grimme, J. Comput. Chem. 25 (2004) 1463
See also: P. Jurecka et al. J. Comput. Chem. 28 (2007) 555
Vallico Sotto July 2009 12

13. Determination of the atomic vdW radii scaling factor
Atomic vdW radii (RvdW) were progressively increased to find the best agreement between computed and experimental data
larger scaling for the vdW radius of H (RH)
better balance between dispersion bonded and hydrogen bonded molecular crystals
SRvdW=1.05; SRH=1.30
MD: Mean Deviation; MAD: Mean Absolute Deviation; RMS: Root-Mean-Square Deviation
from experiment (kJ/mol) J. S. Chickos and W. E. Acree, J. Phys. Chem. Ref. Data, 2002, 31, 537
B3LYP-D*
Vallico Sotto July 2009

14. BSSE corrected cohesive energies vs Experimental data
Cohesive energies with B3LYP-D*
BSSE corrected cohesive energies vs Experimental data
Exp.: -DE=DH0sub(T)+2RT from data at 298K
Cell fixed geometry optimization of the atomic positions at B3LYP/6-31G(d,p)
B3LYP-D* gives cohesive energies in excellent agreement with experimental data
MD=2.2 & MAD=6.3
Better balance between hydrogen bonded and dispersion bonded molecular crystals
Vallico Sotto July 2009 14

15. Geometry optimization: B3LYP-D Grimme vs B3LYP-D*
Lattice parameters
TZP basis set suffers from a remarkably small BSSE
B3LYP/TZP largely overestimates lattice parameters
B3LYP-D* lattice parameters are in excellent agreement with experimental data
B3LYP-D (Grimme) gives too short lattice constants
TZP ______
6-31G(d,p)
CO2
NH3
Urea
C2H2
Urotropine
C6H6
Vallico Sotto July 2009 15

16. Interlayer interaction in graphite: B3LYP-D*
Exp.:
a = 2.46 (fixed)
c = 6.71 Å HF
hybrids
BS: 6-31G(d)
BSSE corrected
GGA __
LDA
B3LYP-D*
B3LYP-D* gives results in excellent agreement wrt experiment
At long-range empirical correction correctly decays as -1/R4
Vallico Sotto July 2009 16

17. Interlayer interaction in graphite: B3LYP-D*
BS: 6-31G(d)
Exp.: in Å
a = 2.46
c = 6.70
Opt.:
a = 2.453
c = 6.640
B3LYP-D* (CPC)
Exp.
B3LYP-D* __
B3LYP-D* gives results in excellent agreement wrt experiment
Vallico Sotto July 2009 17

18. Interlayer interaction in graphite: B3LYP-D* long-range
At long-range empirical London-type formula correctly decays as -1/R4
Vallico Sotto July 2009 18

19. CO adsorption on MgO(001)
Distances in Å, interaction energies in kJ/mol, vibrational frequency shifts in cm-1
MgO basis set: Alhrichs’ TVZ; *MgO top-most layer: Alhrichs’ QZVP
B3LYP-MP2 (slab): DE(CPC)=-12.2 kJ/mol M06-HF (cluster): DE(CPC)=-24.6 kJ/mol; Dwh=22 cm-1 CI (cluster): DE(CPC)=-10.5 kJ/mol; Dwh= 19 cm-1
Exp: R. Wichtendahl et al. Surf. Sci. 423, 90 (1999); G. Spoto et al. Prog. Surf. Sci. 76, 71 (2004)
Vallico Sotto July 2009 19

20. Conclusions
Dispersion interactions are crucial and must be taken into account
Grimme’s scheme  Recalibration needed
Useful tool to correct the PES. Electron density is indirectly influenced
It gives results in excellent agreement wrt experiment for cohesive energies and structures.
Lattice modes also well reproduced
A large basis set should be adopted (e.g. TZP) to reduce the BSSE
For molecular crystals:
In perspective:
Work is in progress to test the transferability of B3LYP-D* to alkali halides (e.g. which C6 for Li+, Na+, …?)
C6 from non-empirical models (e.g. Becke-Johnson, Silvestrelli, …)
Vallico Sotto July 2009 20

21. Calculation of vibrational frequencies and tools for their analysis with CRYSTAL06 R. Dovesi (Torino ) L. Valenzano (Torino) C. Zicovich (Cuernavaca) Y. Noël (Paris) F. Pascale (Nancy)
Vallico Sotto July 2009

22. The CRYSTAL code: Quantum-Mechanical, ab-initio, periodic, using a local basis set (“Atomic Orbitals”)

23. CRYSTA06 web site
Vallico Sotto July 2009

24. A few historical references
Formulation and implementation (graphite)
C. Pisani and R. Dovesi Exact exchange Hartree-Fock calculations for periodic systems. I. Illustration of the method. Int. J. Quantum Chem. 17, (1980).
R. Dovesi, C. Pisani and C. Roetti Exact exchange Hartree-Fock calculations for periodic systems. II. Results for graphite and hexagonal boron nitride Int. J. Quantum Chem 17, (1980).
The Coulomb problem: multipolar expansion+Ewald
R. Dovesi, C. Pisani, C. Roetti and V.R. Saunders Treatment of Coulomb interactions in Hartree-Fock calculations of periodic systems. Phys. Rev. B 28, (1983).
V.R. Saunders, C. Freyria-Fava, R. Dovesi, L. Salasco and C. Roetti On the electrostatic potential in crystalline systems where the charge density is expanded in Gaussian functions Mol. Phys. 77, (1992)
V.R. Saunders, C. Freyria-Fava, R. Dovesi and C. Roetti On the electrostatic potential in linear periodic polymers.
Comp. Phys. Comm. 84, (1994)
Towards the hybrids
M. Causà, R. Dovesi, C. Pisani, R. Colle and A. Fortunelli Correlation correction to the Hartree-Fock total energy of solids. Phys. Rev. B 36, (1987).
Vallico Sotto July 2009

25. The periodic model Consistent treatment of Periodicity
3D - Crystalline solids (230 space groups)
2D - Films and surfaces (80 layer groups)
1D – Polymers (75 rod groups)
0D – Molecules (32 point groups)
Infinite sums of particle interactions
Ewald's method
Specific formulæ for 1D, 2D, 3D
Full exploitation of symmetry
in direct space
in reciprocal space
Vallico Sotto July 2009

26. Hamiltonians Restricted and Unrestricted Hartree-Fock Theory
Total and Spin Density Functional Theory
Local functionals [L] and gradient-corrected [G] exchange-correlation functionals
Exchange functionals
Slater [L]
von Barth-Hedin [L]
Becke '88 [G]
Perdew-Wang '91 [G]
Perdew-Burke-Ernzerhof [G]
Correlation functionals
Vosko-Willk-Nusair (VWN5 parameterization) [L]
Perdew-Wang [L]
Perdew-Zunger '81 [L]
von Barth-Hedin [L]
Lee-Yang-Parr [G]
Perdew '86 [G]
Perdew-Wang '91 [G]
Perdew-Burke-Ernzerhof [G]
Hybrid DFT-HF exchange functionals
B3PW, B3LYP (using the VWN5 functional)
User-definable hybrid functionals
Vallico Sotto July 2009

27. The basis set Crystalline orbitals
as linear combinations of Bloch Functions
as linear combinations of Atomic Orbitals
as contractions of Hermite Gaussian functions
Vallico Sotto July 2009

28. Running CRYSTAL2006 Software performance Supported platforms
Memory management: dynamic allocation
Efficient storage of integrals or Direct SCF
Full parallelization (MPI)
Replicated data version
Massive parallel version
up 2048 processors
(soon available)
Supported platforms
Pentium and Athlon based systems with Linux
IBM workstations and clusters with AIX 4.2 or 4.3
SGI workstations and servers
DEC Alpha workstations
HP-UX systems
Sun Solaris
Linux Alpha
Vallico Sotto July 2009

29. The problem of H It is well known that the stretching modes
involving hydrogen atoms are strongly
anharmonic: typically for the O-H stretching
anharmonicity can be as large as 180 cm-1.
However this difficulty is compensated by the full separability of this mode.
Vallico Sotto July 2009

30. Anharmonic correction for hydroxyls
OH stretching is considered as decoupled from any other normal modes
exe=(2 01- 02) / 2 E2
A wide range (0.5 Å) of OH distances must be explored to properly evaluate E1 and E2
02 E1
01 E0
Direct comparison with experiment for fundamental frequency, first overtone and anharmonicity constant
This procedure is automatically implemented in the code
Vallico Sotto July 2009

31. Isolated OH groups in crystals: model structures/1
Edingtonite surface M O H
M=Mg Brucite
M=Ca Portlandite
Chabazite
All calculations with 6-31G(d,p) basis set
Vallico Sotto July 2009

32. B3LYP vs experimental OH frequencies
Vallico Sotto July 2009

33. Is the choice of the Hamiltonian critical?
BRUCITE, Mg(OH)2
Fundamental OH stretching frequencies, cm-1
Experiment
B3LYP
PW91
LDA HF
3654
3663
3480
3325
4070 Δ +9
-174
-329
+416
No hydrogen bond
Experiment
B3LYP
PBE
PBE0
PBE-sol
harmonic
3823
3698
3856
3622
anharmonic
3654
3663
3526
3694
3447 Δ +9
-128
+40
-207
Vallico Sotto July 2009

34. Is the choice of the Hamiltonian critical?
Be(OH)2
Fundamental OD stretching frequencies. All data in cm-1
Experiment
B3LYP
PW91
LDA HF
2566
2468
2213
1757
2902
-98
-353
-809
+336
Hydrogen bonded OH groups
Only B3LYP is in good agreement with experimental free OH frequency
All Hamiltonians are unable to predict shifts due to strong hydrogen bond
The 1D approximation is not appropriated to describe the OH stretching properties in the case of strong interaction of the H atom (as HB). The anharmonic constant is overestimated.
Vallico Sotto July 2009

35. Harmonic frequency in solids with CRYSTAL
Building the Hessian matrix
analytical first derivative
numerical second derivative
Harmonic frequencies at the central zone are obtained by diagonalising the mass weighted Hessian matrix, W
In order to calculate the hamonic frequency we need to build the Hessian matrix.
In a first step, the first derivative is calculated analyticaly and then derivided a second time numerical to obtain the hessian matrix
Harmonic frequencies at the central zone are obtained by diagonalising the mass weighted Hessian matrix, W.
The heavy step of these calculations is the building of the Hessian mat.
You can notice that isotopic substitution can simalated by changing the mass of atoms and rediagonalising W. This is very fast.
Isotopic shift can be
calculated at no cost!
Vallico Sotto July 2009

36. Dependence on the direction of k: limiting cases k→0
The dynamical matrix
The behavior of the phonons of a wave vector k close to the Γ point can be described as follows:
Center-zone phonons:
ANALYTICAL
Dependence on the direction of k: limiting cases k→0
NON ANALYTICAL
*greek indices: atoms in the primitive cell
**latin indices: cartesian coordinates
Vallico Sotto July 2009

37. The analytical part of the dynamical matrix
*Mx= mass of the x atom
**H=Hessian matrix
Vallico Sotto July 2009

38. The Born charges The atomic Born tensors are the key quantities for :
calculation of the IR intensities
calculation of the static dielectric tensor
calculation of the Longitudinal Optical (LO) modes
They are defined, in the cartesian basis, as:
*Ei=component of an applied external field
**μ=cell dipole moment
Vallico Sotto July 2009

39. μ depends on the choice of the cell
BUT
the dipole moment difference between two geometries of the same periodic system (polarization per unit cell) is a defined observable.
The partial second derivatives appearing in the the Born tensors are estimated numerically from the polarizations generated by small atomic displacements (the same as for the second energy derivative)
LOCALIZED WANNIER FUNCTIONS (WF) to compute polarization
Vallico Sotto July 2009

40. Procedure for the polarization derivative calculation
full localization scheme for the equilibrium point → centroids of the resulting WFs
WFs of the central point are projected onto the corresponding occupied manifolds of the distorted structures → centroids of the resulting WFs
difference between the sum of the reference WF centroids at the two geometries
C.M. Zicovich-Wilson, R. Dovesi, V.R. Saunders A general method to obtain well localized Wannier functions for composite energy bands in linear combination of atomic orbital periodic calculations J. Chem. Phys., 115, (2001)
Alternative scheme; through Berry phase
S. Dall’Olio, R. Dovesi, R. Resta
Spontaneous polarization as a Berry phase of the HF wavefunction. Phys, Rev B56, (1997) 9
Vallico Sotto July 2009

41. The non-analytical contribution and the LO modes
Dynamical matrix:
Analytical contribution:
Non-analytical contribution:
Vallico Sotto July 2009

42. Trasverse Optical (TO) modes: the non-analytic part vanishes
K and Zp,m are perpendicular
Longitudinal Optical (TO) modes: the non-analytic part is ≠0
K and Zp,m are parallel
Vallico Sotto July 2009

43. CRYSTAL frequency calculation output
Frequencies, symmetry analysys, IR intensities, IR and Raman activities
Km/mol ?Kesako?
Vallico Sotto July 2009

44. AIMS • Document the numerical stability of the computational process
• Document the accuracy (with respect to experiment, when experiment is accurate)
• Interpret the spectrum and attribute the modes
Vallico Sotto July 2009

45. Garnets: X3Y2(SiO4)3 Space Group: Ia-3d
Name Mg Al
Pyrope Ca
Grossular Fe
Andradite Cr
Uvarovite
Almandine Mn
Spessartine
Space Group: Ia-3d
80 atoms in the primitive cell (240 modes)
Γrid = 3A1g + 5A2g + 8Eg + 14 F1g + 14 F2g A1u + 5 A2u+ 10Eu + 18F1u + 16F2u
17 IR (F1u) and 25 RAMAN (A1g, Eg, F2g) active modes
To illustrate vib freq calc we will use examples from garnet familly.
The generic chemical formula is X3Y2(SiO4)3.
X is a 2+ cations and Y is a 3+cation. Most of our examples are pyrope and grossluar
The eighty atoms in the primitive cell lead to 240 modes but only 17 modes are infrared active (F1u modes)
And 25 Raman modes; 3 modes are translation modes
Vallico Sotto July 2009

46. Silicate garnet grossular structure: Ca3Al2(SiO4)3
tetrahedra O
distorted
dodecahedra
The Garnets structure is cubic.3 types of cations sites are present
As in all silicate, silicon atoms are tetrahedraly bonded to oxygen. Tetrahedra are isolated.
Y3+ cations (as Aluminium in the grossular example) are in a octahedral environement.
X2+ cations (as Calcium in this example) is located in a distorded dodecahedra Al
Cubic Ia-3d
160 atoms in the UC (80 in the primitive)
O general position (48 equivalent)
Ca (24e) Al (16a) Si (24d) site positions
octahedra
Vallico Sotto July 2009

47. The interest for garnets+TM compounds
M.D. Towler, N.L. Allan, N.M. Harrison, V.R. Saunders, W.C. Mackrodt and E. Aprà An ab initio Hartree-Fock study of MnO and NiO. Phys. Rev. B 50, (1994)
R. Dovesi, J.M. Ricart, V.R. Saunders and R. Orlando Superexchange interaction in K2NiF4 . An ab initio Hartree-Fock study J. Phys. Cond. Matter 7, (1995)
Ph. D'Arco, F. Freyria Fava, R. Dovesi and V. R. Saunders Structural and electronic properties of Mg3Al2Si3O12 pyrope garnet: an ab initio study J. Phys.: Cond. Matter 8, (1996)
Vallico Sotto July 2009

48. Symmetry is crucial for solids
R. Dovesi On the role of symmetry in the ab initio Hartree-Fock linear combination of atomic orbitals treatment of periodic systems.
Int. J. Quantum Chem. 29, (1986). INTEGRALS
C. Zicovich-Wilson and R. Dovesi, On the use of symmetry adapted crystalline orbitals in SCF-LCAO periodic calculations. I. The construction of the symmetrized orbitals. Int. J. Quantum Chem. 67, (1998) K SPACE DIAG-IRREPS
C. Zicovich-Wilson and R. Dovesi, On the use of symmetry adapted crystalline orbitals in SCF-LCAO periodic calculations.II. Implementation of the Self-Consistent-Field Scheme and examples Int. J. Quantum Chem. 67, (1998) SYM LABELS TO STATES
R. Dovesi, F. Pascale, C. M. Zicovich-Wilson
The ab inizio calculation of the vibrational spectrum of cristalline compounds; the role of symmetry and related computational aspects.
Beyond standard quantum chemistry: applications from gas to condensed phases
ISBN:
Editor: Ramon Hernandez-Lamoneda (2007) HESSIAN
Vallico Sotto July 2009

49. Hessian construction and Symmetry (Garnet example)
Each SCF+Gradient calculation provides one line of Hik
80 atoms = SCF+G calculations with low (null) symmetry
Point symmetry is used to generate lines of atoms symmetry related
Other symmetries (among x, y, z lines; translational invariance) further reduce the number of required lines
At the end only 9 out of 241SCF+G calculations are required
Vallico Sotto July 2009

50. Cost of the calculations
The 9 SCF+GRAD calculations: Spessartine (open shell).
Elapsed time, in seconds, per point and per processor.
ELAPSED TIME
N points
Sym
SCF
cycles
GRAD
SCF ratio
GRAD ratio
Equil. 48 43
3580
305 1 2 20
14500
3750 4 12 6 18
25400
7300 7 24
Load balancing
Total CPU time
NODE 0 CPU TIME =
NODE 1 CPU TIME =
NODE 2 CPU TIME =
NODE 3 CPU TIME =
79 h = 3.3 days on 16 processors
(Dual Core AMD Opteron 875, 2210 Mhz, 64 bit, shared memory)
Vallico Sotto July 2009

51. Numerical stability of the computational process
DFT integration grid
is the mean absolute deviation of frequencies between 2 values of the indicated option. (in cm-1)
=0.7
Large
Grid
(Rad,Ang)
Standard
(55,434)
Large
(75,974)
XLarge
(99,1454)
XLarge
Standard
=0.6
Standard grid is enough (For the pyrope case)
SCF convergence (total energy, in hartree)
=0.2
Tol∆E=10-10
Tol∆E=10-11
Vallico Sotto July 2009

52. Calculated frequencies stability : Hessian construction number of points in the derivative and step size
dE/dx x
N=2
N : Number of points
N=3
u : Step size
dE/dx
Numerical estimation of d2E/dx2 x u
N=2
=0.1
N=3
u=0.001 Å
=0.4
u=0.003 Å
Vallico Sotto July 2009

53. Basis set effect-pyrope
BSA
BSB
BSC Mg
8-511G(d) - Al
+sp Si
8-631G(d) +d O
8-411G(d)
Description of the three basis sets adopted for the calculation of the vibrational frequencies of pyrope G(d) means that a 8G contraction is used for the 1s shell; a 5G contraction for the 2sp, and a single G for the 3sp and 4sp shells, plus a single G d shell ( =18 AOs per Mg or Al atom).
+sp and +d means that a diffuse sp or d shell has been added to basis set A.
Vallico Sotto July 2009

54. Basis set effect : IR frequencies of Pyrope
Calculated Modes
Exp a)
BSA
BSB
BSC υ Δυ
988 16
970 -2
964 -8
972
913 11
896 -6
890
-12
902
882
865
859
871
691 41
674 24
673 23
~650
594 13
583 2
581 -0
538 3
533
532 -3
535
505 27
484 6
481
478
471
459 4
457
455
428
423 1
422
390 7
383
353 17
349
336
338
334
335 -1
261
260
259
220
216 -5
217 -4
221
193
189
191
195
142 8
140
134
133
121
-13
120
-14
IR-TO modes (F1u) of pyrope as a function of the basis set size. Frequency differences (Δυ) are evaluated with respect to experimental data. υ and Δυ in cm-1.
Why so large differences with exp for this mode? See next slide
a) Hofmeister et. al. Am. Mineral , 418
|Δυ| 5 10 15 20 + →
BSA is to small
BSB and BSC are good
 Let’s use BSB
Vallico Sotto July 2009

55. Calculated Modes (BSB)
Pyrope : IR intensities
Calculated Modes (BSB)
Exp a)
υ cm-1
Δυ cm-1
Calculated Intensity (km/mol)
υ cm-1
970 -2
5715
972
896 -6
5648
902
865
14028
871
674 24 4
~650
583 2
1326
581
533
869
535
484 6
753
478
459
13721
455
423 1
1309
422
383
3552
349 13 85
336
334
6296
260
720
259
216 -5 8
221
189
3330
195
140
134
121
-13
2904
IR-TO modes of pyrope and their intensity. Frequency differences (Δυ) are evaluated with respect to experimental data.
a) Hofmeister et. al. Am. Mineral , 418
When the mode intensity is too small, the mode frequency can not be accurately determined by experiment.
Or sometimes can’t be observed at all! See next slide
Vallico Sotto July 2009

56. Grossular : IR intensities
Calculated Modes
Exp a) υ Δυ
Intensity (km/mol)
903
-11
6652
914
851 -9
3148
860
830
-13
16321
843
627 9
739
618
547 5
740
542
509 4
148
505
481 7
326
474
441 -8
19909
449
424
-6. 88
430
407 - 18
395 -4
9164
399
357 1
162
356
303
751
302
242 -3
1176
245
207 2
322
205
183
939
186
153 -6
293
159
IR-TO modes (F1u) of grossular and their intensity. Frequency differences (Δυ) are evaluated with respect to experimental data.
a) Hofmeister et. al. Am. Mineral , 418
Vallico Sotto July 2009

57. Pyrope raman modes : Calc vs Exp
Calculated Modes
Observed Modes
BSB
Exp. a)
Exp. b)
Exp. c) υ
Δυ a)
1063 -3
1066
1062
930
-15
945
938 -
921 -7
928
925
927
890
-12
902
899
861  -
911(867)
855
-16
871
866
870
654 3
651
648
635
626
604 6
598
565 2
563
562
561
529 4
525
524
514
512
510
511
494
492
490
Frequency differences (Δυ) are evaluated with respect to experimental data of Kolesov, υ and Δυ in cm-1.
in parentheses unpublished results reported by Chaplin et al, Am. Mineral, , 841
The Eg mode at 439 cm-1 and F2g mode at 285 cm-1 reported by Hofmeiser and Chopelas have not been included in the table, because they do not correspond to any calculated frequency. C
a) Kolesov et. al.
Phys. Chem. Min , 142
b) Hofmeister et. al.
Phys. Chem. Min , 503
c) Kolesov et. al.
Phys. Chem. Min , 645
Vallico Sotto July 2009

58. Pyrope raman modes : Calc vs Exp
Calculated Modes
Observed Modes
BSB
Exp. a)
Exp. b)
Exp. c) υ
Δυ a)
383 -0
379
384 4
375
365(379) -
356 -8
364
362
363
353
350
352
337
345
343
320 -2
322
318(342)
309 25
284
342(309)
269  -
272
273
204 -9
213
230
209
211
203
173
208
106
-31
137
127
Frequency differences (Δυ) are evaluated with respect to experimental data of Kolesov, υ and Δυ in cm-1.
The Eg mode at 439 cm-1 and F2g mode at 285 cm-1 reported by Hofmeister and Chopelas have not been included in the table, because they do not correspond to any calculated frequency.
a) Kolesov et. al.
Phys. Chem. Min , 142
b) Hofmeister et. al.
Phys. Chem. Min , 503
c) Kolesov et. al.
Phys. Chem. Min , 645
Vallico Sotto July 2009

59. Transition metal basis set: Mn in spessartine
BSA
BSB
BSC
BSD
(8s)-(6411sp)-(41d)
+ sp
+ d
+ f
Exponent/bohr-2
0.25
0.6 AO
1596
1644
1704
1728
DE/mH --
-6.4
-1.2
-2.2
|D|/cm-1
5.5
2.1
1.1
+sp (+d,+f) means that a diffuse sp (d,f) shell has been added to basis set A.
DE is the energy lowering per transition metal atom.
The trend is similar for the transition metals of the other garnets.
Vallico Sotto July 2009

60. IR-TO frequencies of spessartine
Calc. TO
INT Dn
Exp. TO
106.6
939
-4.6
111.2
137.8
1235
-2.7
140.5
170.0
308
3.0
167
205.4
1469
2.4
203
251.6
548
5.6
246
322.7
2009
6.7
316
356.1
883
350.5
380.7
6015
0.9
379.8
417.5
816
5.5
412
447.8
15594
2.8
445
470.8
1478
9.4
461.4
520.2
252
0.2
520
564.0
1773
6.0
558
639.9
507
9.9
630
852.2
15274
-8.8
861
877.5
4427
-6.5
884
942.8
7134
-3.2
946
IR-TO modes (F1u) of spessartine.
Frequency differences (Δυ) are evaluated with respect to experimental data.
υ and Δυ in cm-1.
EXP Hofmeister and Chopelas, “Vibrational spectoscopy of end-member silicate garnets”, Phys. Chem. Min., 17, (1991).
|Dn| 5 10
Vallico Sotto July 2009

61. IR-LO frequencies of spessartine
Calc. LO
INT Dn
Exp. LO
113.4 38
-1.3
114.7
148.5
115
-1.8
150.3
172.6 44
4.2
168.4
215.6
197
3.2
212.4
254.8 82
5.8
249
328.5
118
8.5
320
358.0 33
6.0
352
395.8
306
12.8
383
419.2 39
5.2
414
601.2
7617
8.2
593
468.7 40
10.7
458
518.3
237
1.3
517
543.9
2625
12.9
531
646.2
1958
638
1039.9
43257
9.9
1030
870.4
386
-0.6
871
913.3
3574
912
IR-LO modes (F1u) of spessartine.
Frequency differences (Δυ) are evaluated with respect to experimental data.
υ and Δυ in cm-1.
EXP Hofmeister and Chopelas, “Vibrational spectoscopy of end-member silicate garnets”, Phys. Chem. Min., 17, (1991).
|Dn| 5 10 15
Vallico Sotto July 2009

62. Spessartine raman modes : Calc vs Exp
Calculated Modes
BSB
Observed Modes
Exp. a)
Exp. b) υ
Δυ a)
F2g
1033 -4
1029
1027
E2g
914 -1
913
A2g
910 -5
905
877 2
879
878
852  - -
892
845 4
849
640
-10
630
628
596
592
5920
588
-15
573
561 -9
552
550
531
522
521
505
500
499
476
475
472
Frequency differences (Δυ) are evaluated with respect to experimental data.
υ and Δυ in cm-1. C
a) Hofmeister & Chopelas, Phys. Chem Min. 1991
b) Kolesov & Geiger, Phys. Chem. Min.1998
Vallico Sotto July 2009

63. Spessartine raman modes : Calc vs Exp
Calculated Modes
BSB
Observed Modes
Exp. a)
Exp. b) υ
Δυ a)
E2g
376 -4
372
F2g
366 -
348 2
350
A2g
342 8
347
320 1
321
318
315 13
302
314
299
-30
269
221
229
195
196
194
165 10
175
163 -1
162
105
Frequency differences (Δυ) are evaluated with respect to experimental data.
υ and Δυ in cm-1.
a) Hofmeister & Chopelas, Phys. Chem Min. 1991
b) Kolesov & Geiger, Phys. Chem. Min.1998
Vallico Sotto July 2009

64. Garnets : Satistics Systems Raman a) Grossular 7.5 3.0 32 Pyrope 7.6
-3.2 31
Andradite
5.3
-5.1 11
Uvarovite
4.6
-0.4 22
Spessartine
6.8
0.6 30
Almandine IR b)
-2.1 13
-0.7
8.5
-8.5 17
4.4
-2.4 12
almandine
6.2
-2.7 33
a) Hofmeister et al 1991
b) Kolesov et al
Statistical analysis of calculated IR and Raman modes of garnets compared with experimental data.
Vallico Sotto July 2009

65. Garnets : Satistics Systems Raman a) and c) Grossular 7.5 3.0 32
Pyrope
7.6 (3.6)
-3.2 (-2.4)
31 (21)
Andradite
5.3
-5.1 11
Uvarovite
4.2
-0.4 22
-2.1 13
4.6
-0.7 IR b)
8.5
-8.5 17
a) Kolesov et. al.
Phys. Chem. Min , 142
b) Hofmeister et. al.
Am. Mineral, , 841
c) Kolesov et al.
Phys. Chem. Min , 645
Statistical analysis of calculated IR and Raman modes of garnets compared to experiment. data.
Vallico Sotto July 2009

66. The isotopic shift
As a tool for the assignement of the modes and for the interpretation of the spectrum.
Each atom at a time
In some cases also infinite mass
Vallico Sotto July 2009

67. Pyrope : 24Mg → 26Mg Dn (cm-1) 100 350 n (cm-1)
Isotopic shift on the vibrational frequencies of pyrope when 26Mg is substituted for 24Mg.
Vallico Sotto July 2009

68. Pyrope : 27Al → 29Al Dn (cm-1) 300 700 n (cm-1)
Isotopic shift on the vibrational frequencies of pyrope when 29Al is substituted for 27Al.
Vallico Sotto July 2009

69. Pyrope : 16O → 18O Dn (cm-1) n (cm-1)
Isotopic shift on the vibrational frequencies of pyrope when 18O is substituted for 16O.
Vallico Sotto July 2009

70. Pyrope : 28Si → 30Si Dn (cm-1) n (cm-1) 1050 850
Isotopic shift on the vibrational frequencies of pyrope when 30Si is substituted for 28Si.
Vallico Sotto July 2009

71. Internal/external modes
Si-O bonds stronger than the others
Modes separated in 2 types:
Internal modes (deformation of the tetrahedra)
External modes (solid tetrahedra)
Vallico Sotto July 2009

72. Isolated tetrahedra modes (internal modes)
Streching
υ1 : Symmetric
υ3 : Asymmetric
Bending
υ2 : Symmetric
υ4 : Asymmetric
Vallico Sotto July 2009

73. Pyrope : Stretching modes
Symmetric stretching υ1
921 cm-1
Asymmetric stretching υ3
890 cm-1 Mg Al Si O
Vallico Sotto July 2009

74. Pyrope : normal modes attribution
υ2 SiO4 bending
476 cm-1
SiO4 rotation
+ Mg translation
200 cm-1
Mainly
Mg translation
117cm-1 Mg Al Si O
Vallico Sotto July 2009

75. The problem of H It is well known that the stretching modes
involving hydrogen atoms are strongly
anharmonic: typically for the O-H stretching
anharmonicity can be as large as 180 cm-1.
However this difficulty is compensated by the full separability of this mode.
Vallico Sotto July 2009

76. Anharmonic correction for hydroxyls
OH stretching is considered as decoupled from any other normal modes
exe=(2 01- 02) / 2 E2
A wide range (0.5 Å) of OH distances must be explored to properly evaluate E1 and E2
02 E1
01 E0
Direct comparison with experiment for fundamental frequency, first overtone and anharmonicity constant
This procedure is automatically implemented in the code
Vallico Sotto July 2009

77. Isolated OH groups in crystals: model structures/1
Edingtonite surface M O H
M=Mg Brucite
M=Ca Portlandite
Chabazite
All calculations with 6-31G(d,p) basis set
Vallico Sotto July 2009

78. B3LYP vs experimental OH frequencies
Vallico Sotto July 2009

79. Is the choice of the Hamiltonian critical?
BRUCITE, Mg(OH)2
Fundamental OH stretching frequencies, cm-1
Experiment
B3LYP
PW91
LDA HF
3654
3663
3480
3325
4070 Δ +9
-174
-329
+416
No hydrogen bond
Experiment
B3LYP
PBE
PBE0
PBE-sol
harmonic
3823
3698
3856
3622
anharmonic
3654
3663
3526
3694
3447 Δ +9
-128
+40
-207
Vallico Sotto July 2009

80. Is the choice of the Hamiltonian critical?
Be(OH)2
Fundamental OD stretching frequencies. All data in cm-1
Experiment
B3LYP
PW91
LDA HF
2566
2468
2213
1757
2902
-98
-353
-809
+336
Hydrogen bonded OH groups
Only B3LYP is in good agreement with experimental free OH frequency
All Hamiltonians are unable to predict shifts due to strong hydrogen bond
The 1D approximation is not appropriated to describe the OH stretching properties in the case of strong interaction of the H atom (as HB). The anharmonic constant is overestimated.
Vallico Sotto July 2009

81. B3LYP frequencies for brucite. A test case
Atomic eigenvectors analysys allows to say which atoms are moving during each normal mode
Isotopic substitutions permit to identify principal atomic contributions to the modes
Comparison between frequencies of the layered bulk structure and a single slab enables to distinguish between interlayer and intralayer interactions
Vallico Sotto July 2009

82. OH stretching modes in brucite Symmetric Stretching Mode (3847 cm-1)
Anti-symmetric
Stretching Mode
(3873 cm-1)
B3LYP coupling is 26 cm-1, experimental 44 cm-1
Does the coupling arise from interlayer or intralayer interactions?
nslab= 3907
nslab= 3912
Slab coupling 5 cm-1
Vallico Sotto July 2009

83. OH bending modes in brucite
Symmetric
Bending Mode
(803 cm-1)
Antisymmetric
Bending Mode
(458 cm-1)
The coupling is very large (344 cm-1)
H---H distance remains inhaltered during antisymetric motion, while protons nearly collide in the symmetric one
nslab= 440
nslab= 463
Slab coupling 23 cm-1
Vallico Sotto July 2009

84. OH stretching in 50% deuterated-brucite
Deuterium
Stretching
(2817 cm-1)
Hydrogen
Stretching
(3860 cm-1)
The two modes are fully decoupled
Compare with 3847 and 3873 cm-1 for the symmetric and antisymmetric modes of the H only compound.
Vallico Sotto July 2009