vallico sotto july 20091 bartolomeo civalleri theoretical chemistry group department of chemistry...

Vallico Sotto July 2009 1

Bartolomeo CivalleriTheoretical Chemistry GroupDepartment of Chemistry IFM& NIS Centre of Excellence

University of [email protected]

B3LYP augmented with an empirical dispersion term (B3LYP-D*) as

applied to solids


Weak interactions in crystalline solidsWeak interactions in crystalline solids

• Cohesive forces

• long-range: electrostatic, induction, dispersiondispersion

• 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 structure, interaction energies, vibrational frequenciesenergies, vibrational frequencies and thermodynamics, elastic constants, relative stability, …


State of the art in ab initio calculations of MCsState of the art in ab initio calculations of MCs

Only LDA, GGA and hybrid-GGA (e.g. B3LYP)hybrid-GGA (e.g. B3LYP) methods are routinely available in solid state codes

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 (BSSEBSSE) [CRYSTAL06]

c) Analytic derivatives of energy and other observables (e.g. phonons)

DFT is the most common choice to include electron correlation

Present DFT functionals do not account for dispersion energy

BSSE can give binding where there is none.


Hydrogen bonded molecular crystals: B3LYP resultsHydrogen bonded molecular crystals: B3LYP results

Urea Boric acid Formic acid Basis set 6-31G(d,p) TZP 6-31G(d,p) TZP 6-31G(d,p) TZP

Volume 150.8

(3.9%) 160.7

(10.7%) 273.3 (4.0%)

351.6 (33.7%)

207.1 (7.3%)

257.5 (33.4%)

E(CPC) [BSSE]

-66.1 [32.8]

-67.9 [6.8]

-72.2 [37.4]

-75.0 [4.7]

-38.2 [26.0]

-40.0 [4.9]

E(exp.) -103.6 -106.7 -65.0

• 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 energyBSSE artificially compensates the missing dispersion energy• When HB is dominating, B3LYP gives good lattice parameters (not shown)

Volume in Å3. Energy in kJ/mol. Deviation from experimental volumes in parentheses.

3D, 2D and 1D 3D, 2D and 1D HB molecular HB molecular

crystalscrystals


DFT vs vdW forces: new hopes…DFT vs vdW forces: new hopes…

How to deal with dispersion interactions in DFT?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) 034108, 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) 024101, JCP 124 (2006) 014104, JCP 127 (2007) 154108) or C6 from MLWFs (Silvestrelli, PRL 100 (2008) 053002)

• Dispersion-corrected atom centered pseudo-potentials (U. Rothlisberger and co-workers: e.g. Tapavicza et al. JCTC (2007), G. DiLabio CPL (2008))


Empirical –CEmpirical –C66/R/R66 correction: Grimme’s model correction: Grimme’s model

ij

Disp dmp ijij ij

CE s f R

R6

6 , 6,

' ( )gg g

ij vdwdmp ij d R Rf R

e ,, / 1

1

1 gg

B3LYP-D B3LYP DispE =E +E

• s6: scaling factor for each DFT method (s6=1.05 for B3LYP)• C6

ij are computed from atomic dispersion coefficients: C6ij = C6

i·C6j

• Rvdw is the sum of atomic van der Waals radii: Rvdw=Rivdw+Rj

vdw

• d determines the steepness of the damping function (d=20)• summation over g truncated at 25 Å (estimated error < 0.02 kJ/mol on E)• Grimme proposed a set of parameters (i.e. C6

i and Rivdw) from H to Xe

Atom-atom additive damped empirical potential of the form -f(R)CAtom-atom additive damped empirical potential of the form -f(R)C66/R/R66

Total energy is then computed as:

Implemented in CRYSTAL06 for energy and gradients (atoms and cell):

S. Grimme, J. Comput. Chem., 2004, 25, 1463 and J. Comput. Chem., 2006, 27, 1787

where

B. Civalleri, C.M. Zicovich-Wilson, et al., CrystEngComm, 2008, DOI: 10.1039/b715018k(see supplementary material for erratum)


Grimme empirical dispersion keywords6 (scaling factor) d (steepness) Rcut (cut-off radius, Å)Nr. of atomic speciesAtomic number C6 (Jnm6 mol−1) Rvdw (Å)Atomic number C6 Rvdw

Atomic number C6 Rvdw

Atomic number C6 Rvdw

End of SCF&method input section

GGRIMMERIMME input block input block

Urea CRYSTAL0 0 01135.565 4.68456 0.0000 0.5000 0.32608 0.0000 0.5000 0.59537 0.1459 0.6459 0.17661 0.2575 0.7575 0.28271 0.1441 0.6441-0.0380 Optional keywordsEND (ENDG)Basis setEND

……GRIMMEGRIMME1.05 20. 25.1.05 20. 25.441 0.14 1.001 1 0.14 1.001 6 1.75 1.4526 1.75 1.4527 1.23 1.3977 1.23 1.3978 0.70 1.3428 0.70 1.342……ENDEND

E.g.: Urea – B3LYP-D

ijRcut

Disp dmp ijij ij

CE s f R

R6

6 , 6,

' ( )gg g


e ,, / 1

1

1 gg

ij jiC C C6 6 6 jivdw vdw vdwR R R


GGRIMMERIMME datasetdataset

S. Grimme, J. Comput. Chem., 2006, 27, 1787

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

Parameters available from H to Xe

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 = 0.05NIp

i αi

where N has values 2, 10, 18, 36, and 54 for atoms from rows 1–5 of the periodic table

Suitable for solids?


Naphthalene

Tests on a set of selected molecular crystalsTests on a set of selected molecular crystals

• 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

14 molecular crystals both dispersion bonded and hydrogen bonded14 molecular crystals both dispersion bonded and hydrogen bonded

Boric acid

CO2 NH3 Formic acid

Urea

C2H2

UrotropineFormamide

Succinic anhydride

C6H6

1,4-dicyano-benzene

Propane

1,4-dichloro-benzene


-140

-120

-100

-80

-60

-40

-20

0

20

40

-140 -120 -100 -80 -60 -40 -20 0 20 40

B3LYP-D GrimmeB3LYPExp.

Experimental lattice energy (kJ/mol)

BS

SE

cor

rect

ed c

ohe

sive

ene

rgy

(kJ/

mol

) Cohesive energies: B3LYP vs B3LYP-D GrimmeCohesive energies: B3LYP vs B3LYP-D Grimme

• 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

BSSE corrected cohesive energies vs Experimental dataBSSE corrected cohesive energies vs Experimental data

Cell fixed geometry optimization of the atomic

positions at B3LYP/6-31G(d,p)

Exp.: -E=H0sub(T)+2RT from data at 298K

-110 < E < -25 kJ/mol


Grimme’s model: the role of the damping functionGrimme’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 RStrategy: scaling the atomic RvdWvdW


e ,, / 1

1

1 gg

From: S. Grimme, J. Comput. Chem. 25 (2004) 1463

i jvdW vdW vdWR R R

RvdW

CarbonRvdw(C)=161 pm

See also: P. Jurecka et al. J. Comput. Chem. 28 (2007) 555


Determination of the atomic vdW radii scaling factorDetermination of the atomic vdW radii scaling factor

MDMD:: Mean Deviation; MADMAD:: Mean Absolute Deviation; RMSRMS:: Root-Mean-Square Deviation from experiment (kJ/mol)

J. S. Chickos and W. E. Acree, J. Phys. Chem. Ref. Data, 2002, 31, 537

• ss66=1.00=1.00

• 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

• SSRvdWRvdW=1.05; =1.05; SSRHRH=1.30=1.30

B3LYP-D*

-10.00

-5.00

0.00

5.00

10.00

15.00

MD

RMS

MAD


-140

-120

-100

-80

-60

-40

-20

0

20

40

-140 -120 -100 -80 -60 -40 -20 0 20 40

B3LYPB3LYP-D GrimmeB3LYP-D* pwExp.

Experimental lattice energy (kJ/mol)

BS

SE

cor

rect

ed c

ohe

sive

ene

rgy

(kJ/

mol

) Cohesive energies with B3LYP-D*Cohesive energies with B3LYP-D*

• 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

BSSE corrected cohesive energies vs Experimental dataBSSE corrected cohesive energies vs Experimental data

Cell fixed geometry optimization of the atomic

positions at B3LYP/6-31G(d,p)

Exp.: -E=H0sub(T)+2RT from data at 298K


-10

-5

0

5

10

15

NH3 C2H2 CO2 Urotropine Urea C6H6

B3LYP-D* pwB3LYP-D GrimmeB3LYP

Me

an %

de

via

tion

fro

m e

xper

imen

t

Geometry optimization: B3LYP-D Grimme vs B3LYP-D*Geometry optimization: B3LYP-D Grimme vs B3LYP-D*

6-31G(d,p) - - - -6-31G(d,p) - - - -

TZP ______TZP ______

Lattice parametersLattice 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 constantsCO2NH3 UreaC2H2 Urotropine C6H6


B3LYP-D* (CPC)

-15

0

15

30

45

5.5 6.0 6.5 7.0 7.5 8.0 8.5

PBE0(CPC)B3LYP(CPC)X3LYP(CPC)PW91(CPC)PBE(CPC)SVWN(CPC)HF(CPC)

c (Å)

Inte

ract

ion

en

erg

y (k

J/m

ol)

Interlayer interaction in graphite: B3LYP-D*Interlayer interaction in graphite: B3LYP-D*

Exp.: a = 2.46 (fixed) c = 6.71 Å

HF

LDA

GGA

hybridsBSSE

corrected

BS: 6-31G(d)

B3LYP-D* gives results in excellent agreement wrt experimentAt long-range empirical correction correctly decays as -1/R4

__

B3LYP-D*


-30

-25

-20

-15

-10

-5

5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2

c (Å)

Inte

ract

ion

ener

gy (

kJ/m

ol)

B3LYP-D*

B3LYP-D* (CPC) Exp.

Interlayer interaction in graphite: B3LYP-D*Interlayer interaction in graphite: B3LYP-D*

Exp.: in Å a = 2.46c = 6.70

Opt.:a = 2.453c = 6.640

BS: 6-31G(d)

B3LYP-D* gives results in excellent agreement wrt experiment

__


-100

-80

-60

-40

-20

0

0 5 10 15 20

fit: Edisp

=A/R4.04

Computed dispersion correction

Interlayer distance (Å)

Edi

sp (

meV

/ato

m)

Interlayer interaction in graphite: B3LYP-D* long-rangeInterlayer interaction in graphite: B3LYP-D* long-range

At long-range empirical London-type formula correctly decays as -1/R4

1/R4


CO adsorption on MgO(001)CO adsorption on MgO(001)

MgO basis set: Alhrichs’ TVZ; *MgO top-most layer: Alhrichs’ QZVP

Exp: R. Wichtendahl et al. Surf. Sci. 423, 90 (1999); G. Spoto et al. Prog. Surf. Sci. 76, 71 (2004)

1x2 B3LYP B3LYP-D B3LYP-D* Exp. CO bs TZ2P TZ2P TZ2P QZ2P QZ2P*

d(Mg...CO) 2.578 2.512 2.489 2.478 2.488

E(CPC) [BSSE]

-1.6 [7.2]

-17.9 [6.8]

-16.4 [7.3]

-16.0 [8.0]

-16.2 [4.5] 16.3

H0(0) 0.93 -15.3 -13.7 -13.3 -13.5 12.6

h 22 24 24 24 26 15

B3LYP-MP2 (slab): E(CPC)=-12.2 kJ/molM06-HF (cluster): E(CPC)=-24.6 kJ/mol; h=22 cm-1

CI (cluster): E(CPC)=-10.5 kJ/mol; h= 19 cm-1

Distances in Å, interaction energies in kJ/mol, vibrational frequency shifts in cm -1


ConclusionsConclusions

In perspective: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, …)

• 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:For molecular crystals:

Dispersion interactions are crucial and must be taken into accountDispersion interactions are crucial and must be taken into account

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

The CRYSTAL code:

Quantum-Mechanical, ab-initio, periodic,

using a local basis set (“Atomic Orbitals”)


www.crystal.unito.it www.crystal.unito.it

CRYSTA06 web site


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, 501-516 (1980).

• R. Dovesi, C. Pisani and C. RoettiExact exchange Hartree-Fock calculations for periodic systems.II. Results for graphite and hexagonal boron nitrideInt. J. Quantum Chem 17, 517-529 (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, 5781-5792 (1983).

• V.R. Saunders, C. Freyria-Fava, R. Dovesi, L. Salasco and C. RoettiOn the electrostatic potential in crystalline systems where the charge density is expanded in Gaussian functions. Mol. Phys. 77, 629-665 (1992)

• V.R. Saunders, C. Freyria-Fava, R. Dovesi and C. RoettiOn the electrostatic potential in linear periodic polymers.

Comp. Phys. Comm. 84, 156-172 (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, 891-897 (1987).


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


Hamiltonians

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]

•Restricted and Unrestricted Hartree-Fock Theory

•Total and Spin Density Functional Theory

Local functionals [L] and gradient-corrected [G]exchange-correlation functionals

Hybrid DFT-HF exchange functionals B3PW, B3LYP (using the VWN5 functional) User-definable hybrid functionals


The basis set

m

mmn a ),(),( rkrk k

g

gk grrk )(),( mi

m e

Crystalline orbitals

as linear combinations of Bloch Functions

as linear combinations of Atomic Orbitals

as contractions of Hermite Gaussian functions

)()( grgr mm c


Running CRYSTAL2006

Software performance• 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


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.


E2

E1

E0

02

01

exe=(2 01- 02) / 2

Anharmonic correction for hydroxylsAnharmonic correction for hydroxyls

OH stretching is OH stretching is considered as decoupled considered as decoupled from any other normal from any other normal

modesmodes


modesmodes

A wide range (0.5 Å) of OH A wide range (0.5 Å) of OH distances must be distances must be

explored to properly explored to properly evaluate Eevaluate E11 and E and E22



Direct comparison with Direct comparison with experiment for experiment for

fundamental frequency, fundamental frequency, first overtone and first overtone and

anharmonicity constant anharmonicity constant



anharmonicity constant anharmonicity constant This procedure is automatically This procedure is automatically

implemented in the codeimplemented in the code


Isolated OH groups in crystals: model structures/1Isolated OH groups in crystals: model structures/1

MMOO

HH

M=MgM=Mg BruciteBruciteM=CaM=Ca PortlanditePortlandite

Edingtonite surfaceEdingtonite surface

ChabaziteChabazite

All calculations All calculations with 6-31G(d,p) with 6-31G(d,p)

basis setbasis set


basis setbasis set


B3LYP vs experimental OH frequenciesB3LYP vs experimental OH frequencies

System 01 Raman 01 IR

Brucite Calc 3663 3694

Exp 3654 3698

Portlandite Calc 3637 3650

Exp 3620 3645

Edingtonite Calc -- 3742

Exp -- 3747

Chabazite Calc -- 3648

Exp 3603


Is the choice of the Hamiltonian critical?Is the choice of the Hamiltonian critical?

Experiment B3LYP PW91 LDA HF

3654 3663 3480 3325 4070

Δ +9 -174 -329 +416

Experiment B3LYP PBE PBE0 PBE-sol

harmonic 3823 3698 3856 3622

anharmonic 3654 3663 3526 3694 3447

Δ +9 -128 +40 -207

BRUCITE, Mg(OH)BRUCITE, Mg(OH)22

No hydrogen bond

Fundamental OH stretching frequencies, cm-1



Hydrogen bonded OH Hydrogen bonded OH groupsgroups



2566 2468 2213 1757 2902

-98 -353 -809 +336

Fundamental OD stretching frequencies. All data in cm-1

•Only B3LYP is in good agreement with experimental free OH Only B3LYP is in good agreement with experimental free OH frequencyfrequency

•All Hamiltonians are unable to predict shifts due to strong All Hamiltonians are unable to predict shifts due to strong hydrogen bondhydrogen bond

•The 1D approximation is not appropriated to describe the OH The 1D approximation is not appropriated to describe the OH stretching properties in the case of strong interaction of the H stretching properties in the case of strong interaction of the H

atom (as HB). The anharmonic constant is overestimated.atom (as HB). The anharmonic constant is overestimated.





Be(OH)Be(OH)22


Harmonic frequency in solids with CRYSTALHarmonic frequency in solids with CRYSTAL

0

( 0)G

ijij G

i j

HW k

M M

Harmonic frequencies at the Harmonic frequencies at the central zone are obtained by central zone are obtained by

diagonalising the mass weighted diagonalising the mass weighted Hessian matrix, WHessian matrix, W

Building the Hessian matrixBuilding the Hessian matrixBuilding the Hessian matrixBuilding the Hessian matrix

jj

Vv

u

analytical first derivativeanalytical first derivativeanalytical first derivativeanalytical first derivative

0

(0,...., ,...) (0,...., ,...)

2j j i j i

jii i

v v u v uH

u u

numerical second derivativenumerical second derivativenumerical second derivativenumerical second derivative

Isotopic shift can beIsotopic shift can becalculated at no cost!calculated at no cost!

Isotopic shift can beIsotopic shift can becalculated at no cost!calculated at no cost!


The dynamical matrix

,, ,( 0) ( 0) ( 0)i j

NAi j i jW W W

k k k

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


The analytical part of the dynamical matrix

, ,

1( 0)i j i jW H

M M

k

*Mx= mass of the x atom

**H=Hessian matrix


The Born chargesThe 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:

*ij i

j i j

VZ

u E u

*Ei=component of an applied external field

**μ=cell dipole moment


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 WANNIER FUNCTIONS (WF) to compute polarization


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. SaundersA general method to obtain well localized Wannier functions for composite energy bands in linear combination of atomic orbital periodic calculationsJ. Chem. Phys., 115, 9708-9719 (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, 10105 (1997)


The non-analytical contribution and the LO modesThe non-analytical contribution and the LO modes

,, ,( 0) ( 0) ( 0)i j

NAi j i jW W W

k k k

, ,

1( 0)i j i jW H

M M

k

Dynamical matrixDynamical matrix:

Analytical contribution:

Non-analytical contribution:

* *

,

4( 0)

m mi n njm nNA

i j

m mn nmn

k Z k Z

W

M M k k

k


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


CRYSTAL frequency calculation outputCRYSTAL frequency calculation output

Frequencies, symmetry analysys, IR intensities, IR and Raman activitiesFrequencies, symmetry analysys, IR intensities, IR and Raman activitiesFrequencies, symmetry analysys, IR intensities, IR and Raman activitiesFrequencies, symmetry analysys, IR intensities, IR and Raman activities


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


GarnetsGarnets: X: X33YY22(SiO(SiO44))33

Space Group: Ia-3d

80 atoms in the primitive cell (240 modes)

Γrid = 3A1g + 5A2g + 8Eg + 14 F1g + 14 F2g + 5A1u + 5 A2u+ 10Eu + 18F1u + 16F2u

17 IR (F1u) and 25 RAMAN (A1g, Eg, F2g) active modes

X Y Name

Mg Al Pyrope

Ca Al Grossular

Ca Fe Andradite

Ca Cr Uvarovite

Fe Al Almandine

Mn Al Spessartine


Silicate garnet grossular structure: Ca3Al2(SiO4)3

Ca

Al

O

Si

O

O

•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

distorted dodecahedra

distorted dodecahedra

tetrahedratetrahedra

octahedraoctahedra


The interest for garnets+TM compounds

• M.D. Towler, N.L. Allan, N.M. Harrison, V.R. Saunders, W.C. Mackrodt and E. ApràAn ab initio Hartree-Fock study of MnO and NiO.Phys. Rev. B 50, 5041-5054 (1994)

• R. Dovesi, J.M. Ricart, V.R. Saunders and R. OrlandoSuperexchange interaction in K2NiF4 . An ab initio Hartree-Fock study J. Phys. Cond. Matter 7, 7997-8007 (1995)

• Ph. D'Arco, F. Freyria Fava, R. Dovesi and V. R. SaundersStructural and electronic properties of Mg3Al2Si3O12 pyrope garnet: an ab initio studyJ. Phys.: Cond. Matter 8, 8815-8828 (1996)


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, 1755-1774 (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, 299-309 (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 examplesInt. J. Quantum Chem. 67, 311-320 (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: 978-81-7895-293-2 Editor: Ramon Hernandez-Lamoneda (2007) HESSIAN


1. Point symmetry is used to generate lines of atoms symmetry related

2. 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

Hessian construction and Symmetry(Garnet example)

Each SCF+Gradient calculation provides one line of Hik

80 atoms = 240+1 SCF+G calculations with low (null) symmetry


Cost of the calculationsCost of the calculations

ELAPSED TIME

N points SymSCF

cyclesSCF GRAD

SCF ratio

GRAD ratio

Equil. 48 43 3580 305 1 1

2 2 20 14500 3750 4 12

6 1 18 25400 7300 7 24

The 9 SCF+GRAD calculations: Spessartine (open shell).

Elapsed time, in seconds, per point and per processor.

Load balancing

NODE 0 CPU TIME = 286054.450

NODE 1 CPU TIME = 284862.080

NODE 2 CPU TIME = 285570.430

NODE 3 CPU TIME = 285803.140

... ... ... ... ... ...

Total CPU time

79 h = 3.3 days on 16 processors

(Dual Core AMD Opteron 875, 2210 Mhz,

64 bit, shared memory)


Numerical stability of the computational process

•DFT integration grid

Standard

LargeXLarge

=0.7

=0.6

Standard grid is enough (For the pyrope case)

Grid (Rad,Ang)

Standard (55,434)

Large (75,974)

XLarge (99,1454)

•SCF convergence (total energy, in hartree)

Tol∆E=10-10 =0.2

Tol∆E=10-11

is the mean absolute deviation of frequencies between 2 values of the indicated option. (in cm-1)


Calculated frequencies stability : Hessian constructionnumber of points in the derivative and step size

Numerical estimation of d2E/dx2

dE/dx

x

N=2

dE/dx

x

N=3

u

u

u

N : Number of points

u : Step size

N=2=0.1

N=3 u=0.001 Å=0.4

u=0.003 Å


BSA BSB BSC

Mg 8-511G(d) - -

Al 8-511G(d) +sp -

Si 8-631G(d) +sp +d

O 8-411G(d) - +d

Description of the three basis sets adopted for the calculation of the

vibrational frequencies of pyrope. 8-511G(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 (1+4+4+4+5=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.

Basis set effect-pyropeBasis set effect-pyrope


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.

Calculated Modes Exp a)

BSA BSB BSC

υ Δυ υ Δυ υ Δυ 988 16 970 -2 964 -8 972

913 11 896 -6 890 -12 902

882 11 865 -6 859 -12 871

691 41 674 24 673 23 ~650

594 13 583 2 581 -0 581

538 3 533 -2 532 -3 535

505 27 484 6 481 3 478

471 16 459 4 457 2 455

428 6 423 1 423 1 422

390 7 383 0 383 -0 383

353 17 349 13 349 13 336

338 2 334 -2 335 -1 336

261 2 260 1 259 -0 259

220 -1 216 -5 217 -4 221

193 -2 189 -6 191 -4 195

142 8 140 6 140 6 134

133 -1 121 -13 120 -14 134

a) Hofmeister et. al. Am. Mineral. 1996. 81, 418

Basis set effect : IR frequencies of PyropeBasis set effect : IR frequencies of Pyrope

|Δυ| 0 5 10 15 20 +

→

•BSA is to small

•BSB and BSC are good Let’s use BSB

Why so large differences with exp for this mode?See next slide


Calculated Modes (BSB) Exp a)

υ cm-1 Δυ cm-1Calculated Intensity

(km/mol) υ cm-1

970 -2 5715 972

896 -6 5648 902

865 -6 14028 871

674 24 4 ~650583 2 1326 581

533 -2 869 535

484 6 753 478

459 4 13721 455

423 1 1309 422

383 0 3552 383

349 13 85 336

334 -2 6296 336

260 1 720 259

216 -5 8 221

189 -6 3330 195

140 6 24 134

121 -13 2904 134

IR-TO modes of pyrope and their intensity. Frequency differences (Δυ) are evaluated with respect to experimental data.


Pyrope : IR intensitiesPyrope : IR intensities

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


IR-TO modes (F1u) of grossular and their intensity. Frequency differences (Δυ) are evaluated with respect to experimental data.


Calculated Modes Exp a)

υ Δυ Intensity (km/mol) υ 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 1 751 302

242 -3 1176 245

207 2 322 205

183 -3 939 186

153 -6 293 159

Grossular : IR intensitiesGrossular : IR intensities


Frequency differences (Δυ) are evaluated with respect to experimental data of Kolesov, 1998. υ and Δυ in cm-1.

in parentheses unpublished results reported by Chaplin et al, Am. Mineral, 1998. 83, 841

Calculated Modes Observed Modes

BSB Exp. a) Exp. b) Exp. c)

υ Δυ a) υ υ υ1063 -3 1066 1062 1066

930 -15 945 938 -

921 -7 928 925 927

890 -12 902 899 -

861 - - 911(867) -

855 -16 871 866 870

654 3 651 648 648

635 - - 626 -

604 6 598 598 -

565 2 563 562 561

529 4 525 524 -

514 2 512 510 511

494 2 492 490 492

a) Kolesov et. al.

Phys. Chem. Min. 1998. 25, 142

b) Hofmeister et. al.

Phys. Chem. Min. 1991. 17, 503

c) Kolesov et. al.

Phys. Chem. Min. 2000. 27, 645

Pyrope raman modes : Calc vs ExpPyrope raman modes : Calc vs Exp

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.


Frequency differences (Δυ) are evaluated with respect to experimental data of Kolesov, 1998. υ and Δυ in cm-1.

a) Kolesov et. al.

Phys. Chem. Min. 1998. 25, 142

b) Hofmeister et. al.

Phys. Chem. Min. 1991. 17, 503

c) Kolesov et. al.

Phys. Chem. Min. 2000. 27, 645

Pyrope raman modes : Calc vs ExpPyrope raman modes : Calc vs Exp

Calculated Modes Observed Modes

BSB Exp. a) Exp. b) Exp. c)

υ Δυ a) υ υ υ383 -0 383 379 384

379 4 375 365(379) -

356 -8 364 362 363

353 -0 353 350 352

337 -8 345 - 343

320 -2 322 318(342) 320

309 25 284 342(309) -

269 - - 272 273

204 -9 213 230 209

209 -2 211 203 -

173 - - 208 -

106 -31 137 - 127

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.


BSA BSB BSC BSD

(8s)-(6411sp)-(41d) + sp + d + f

Exponent/bohr-2 0.5 0.5 0.25 0.25 0.6

AO 1596 1644 1704 1728

E/mH -- -6.4 -1.2 -2.2

|/cm-1 -- 5.5 2.1 1.1

Transition metal basis set: Mn in spessartineTransition metal basis set: Mn in spessartine

+sp (+d,+f) means that a diffuse sp (d,f) shell has been added to basis set A.

The trend is similar for the transition metals of the other garnets.

E is the energy lowering per transition metal atom.


IR-TO frequencies of spessartineIR-TO frequencies of spessartine

Calc. TO INT 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 5.6 350.5

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

0 5 10

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, 503-526 (1991).


IR-LO frequencies of spessartineIR-LO frequencies of spessartine

Calc. LO INT 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 8.2 638

1039.9 43257

9.9 1030

870.4 386 -0.6 871

913.3 3574 1.3 912

0 5 10

IR-LO modes (F1u) of spessartine.



EXP Hofmeister and Chopelas, “Vibrational spectoscopy of end-member silicate garnets”, Phys. Chem. Min., 17, 503-526 (1991).

15




Calculated Modes

BSB

Observed Modes

Exp. a) Exp. b)

υ Δυ a) υ υF2g 1033 -4 1029 1027E2g 914 -1 913 913A2g 910 -5 905 905F2g 877 2 879 878E2g 852 - - 892F2g 845 4 849 849F2g 640 -10 630 628E2g 596 -4 592 5920F2g 588 -15 573 573A2g 561 -9 552 550E2g 531 -9 522 521F2g 505 -5 500 499F2g 476 -1 475 472

a) Hofmeister & Chopelas, Phys. Chem Min. 1991

b) Kolesov & Geiger, Phys. Chem. Min.1998

Spessartine raman modes : Calc vs ExpSpessartine raman modes : Calc vs Exp




Spessartine raman modes : Calc vs ExpSpessartine raman modes : Calc vs Exp

Calculated Modes

BSB

Observed Modes

Exp. a) Exp. b)

υ Δυ a) υ υE2g 376 -4 372 372F2g 366 - - -F2g 348 2 350 350A2g 342 8 350 347E2g 320 1 321 318F2g 315 13 302 314E2g 299 -30 269 -F2g 221 0 221 229F2g 195 1 196 194F2g 165 10 175 163E2g 163 -1 162 162F2g 105 - - -

a) Hofmeister & Chopelas, Phys. Chem Min. 1991

b) Kolesov & Geiger, Phys. Chem. Min.1998


Statistical analysis of calculated IR and Raman modes of garnets compared with experimental data.

Systems

Ramana)

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

IRb)

Grossular 7.5 -2.1 13

Pyrope 4.6 -0.7 13

Andradite 8.5 -8.5 17

Uvarovite

Spessartine 4.4 -2.4 12

almandine 6.2 -2.7 33

Garnets : SatisticsGarnets : Satistics

max

a) Hofmeister et al 1991

b) Kolesov et al. 1998


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


(cm-1)

(cm-1)100 350

Pyrope : Pyrope : 2424Mg Mg →→ 2626MgMg

Isotopic shift on the vibrational frequencies of pyrope when 26Mg is substituted for 24Mg.


Isotopic shift on the vibrational frequencies of pyrope when 29Al is substituted for 27Al.

(cm-1)

(cm-1)300 700

Pyrope : Pyrope : 2727Al Al →→ 2929AlAl


Isotopic shift on the vibrational frequencies of pyrope when 18O is substituted for 16O.

(cm-1)

(cm-1)

Pyrope : Pyrope : 1616O O →→ 1818OO

Vallico Sotto July 2009 70Isotopic shift on the vibrational frequencies of pyrope when 30Si is substituted for 28Si.

(cm-1)

(cm-1)

850 1050

Pyrope : Pyrope : 2828Si Si →→ 3030SiSi


Si-O bonds stronger than the others

Modes separated in 2 types:

• Internal modes (deformation of the tetrahedra)

• External modes (solid tetrahedra)

Internal/external modesInternal/external modes


Isolated tetrahedra modes (internal modes)Isolated tetrahedra modes (internal modes)

Streching

υ1 : Symmetricυ1 : Symmetric

υ3 : Asymmetricυ3 : Asymmetric

Bending

υ2 : Symmetricυ2 : Symmetric

υ4 : Asymmetricυ4 : Asymmetric


Pyrope : Stretching modesPyrope : Stretching modes

Symmetric stretching υ1

921 cm-1

Symmetric stretching υ1

921 cm-1

Asymmetric stretching υ3

890 cm-1

Asymmetric stretching υ3

890 cm-1

Mg Al Si O


Pyrope : normal modes attributionPyrope : normal modes attribution

υ2 SiO4 bending

476 cm-1

υ2 SiO4 bending

476 cm-1

SiO4 rotation

+ Mg translation200 cm-1

SiO4 rotation

+ Mg translation200 cm-1

Mg Al Si O

Mainly Mg translation

117cm-1

Mainly Mg translation

117cm-1


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.


E2

E1

E0

02

01

exe=(2 01- 02) / 2

Anharmonic correction for hydroxylsAnharmonic correction for hydroxyls


modesmodes


modesmodes







anharmonicity constant anharmonicity constant



anharmonicity constant anharmonicity constant This procedure is automatically This procedure is automatically

implemented in the codeimplemented in the code


Isolated OH groups in crystals: model structures/1Isolated OH groups in crystals: model structures/1

MMOO

HH

M=MgM=Mg BruciteBruciteM=CaM=Ca PortlanditePortlandite

Edingtonite surfaceEdingtonite surface

ChabaziteChabazite


basis setbasis set


basis setbasis set


B3LYP vs experimental OH frequenciesB3LYP vs experimental OH frequencies

System 01 Raman 01 IR

Brucite Calc 3663 3694

Exp 3654 3698

Portlandite Calc 3637 3650

Exp 3620 3645

Edingtonite Calc -- 3742

Exp -- 3747

Chabazite Calc -- 3648

Exp 3603




3654 3663 3480 3325 4070

Δ +9 -174 -329 +416

Experiment B3LYP PBE PBE0 PBE-sol

harmonic 3823 3698 3856 3622

anharmonic 3654 3663 3526 3694 3447

Δ +9 -128 +40 -207

BRUCITE, Mg(OH)BRUCITE, Mg(OH)22

No hydrogen bond

Fundamental OH stretching frequencies, cm-1






2566 2468 2213 1757 2902

-98 -353 -809 +336

Fundamental OD stretching frequencies. All data in cm-1









Be(OH)Be(OH)22


B3LYP frequencies for brucite. A test caseB3LYP frequencies for brucite. A test case

Atomic eigenvectors analysys allows to say which atoms Atomic eigenvectors analysys allows to say which atoms are moving during each normal modeare moving during each normal mode

Atomic eigenvectors analysys allows to say which atoms Atomic eigenvectors analysys allows to say which atoms are moving during each normal modeare moving during each normal mode

Isotopic substitutions permit to identify principal atomic Isotopic substitutions permit to identify principal atomic contributions to the modescontributions to the modes

Isotopic substitutions permit to identify principal atomic Isotopic substitutions permit to identify principal atomic contributions to the modescontributions to the modes

Comparison between Comparison between frequencies of the frequencies of the

layered bulk structure layered bulk structure and a single slab enables and a single slab enables

to distinguish between to distinguish between interlayerinterlayer and and intralayerintralayer

interactionsinteractions

Comparison between Comparison between frequencies of the frequencies of the

layered bulk structure layered bulk structure and a single slab enables and a single slab enables

to distinguish between to distinguish between interlayerinterlayer and and intralayerintralayer

interactionsinteractions


OH stretching modes in bruciteOH stretching modes in brucite

B3LYP coupling is B3LYP coupling is 26 cm26 cm-1-1, ,

experimental 44 experimental 44 cmcm-1-1

Does the coupling Does the coupling arise from arise from

interlayerinterlayer or or intralayerintralayer

interactions? interactions?

Does the coupling Does the coupling arise from arise from

interlayerinterlayer or or intralayerintralayer

interactions? interactions?

slabslab= = 39123912

slabslab= = 39073907

Slab coupling 5 cmSlab coupling 5 cm-1-1

Symmetric

Stretching Mode

(3847 cm-1)

Anti-symmetric

Stretching Mode

(3873 cm-1)


OH bending modes in bruciteOH bending modes in brucite

The coupling is The coupling is very large (344 very large (344

cmcm-1-1))H---H distance H---H distance

remains inhaltered remains inhaltered during during

antisymetric antisymetric motion, while motion, while protons nearly protons nearly collide in the collide in the

symmetric one symmetric one

H---H distance H---H distance remains inhaltered remains inhaltered

during during antisymetric antisymetric motion, while motion, while protons nearly protons nearly collide in the collide in the

symmetric one symmetric one slabslab= 440= 440

slabslab= 463= 463

Slab coupling 23 cmSlab coupling 23 cm-1-1

Symmetric

Bending Mode

(803 cm-1)

Antisymmetric

Bending Mode

(458 cm-1)


OH stretching in 50% deuterated-bruciteOH stretching in 50% deuterated-brucite

The two modes The two modes are fully are fully

decoupleddecoupled

Compare with Compare with 3847 and 3873 3847 and 3873

cm-1 for the for the symmetric and symmetric and antisymmetric antisymmetric modes of the H modes of the H only compound.only compound.

Compare with Compare with 3847 and 3873 3847 and 3873

cm-1 for the for the symmetric and symmetric and antisymmetric antisymmetric modes of the H modes of the H only compound.only compound.

Deuterium

Stretching

(2817 cm-1)

Hydrogen

Stretching

(3860 cm-1)

vallico sotto july 20091 bartolomeo civalleri theoretical chemistry group department of chemistry...

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