CPHF/KS Applied to Infrared and Raman Intensities and Second Harmonic Generation - Lorenzo Maschio - Unito
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CPHF/KS Applied to Infrared and Raman Intensities and Second Harmonic Generation Lorenzo Maschio lorenzo.maschio@unito.it Dipartimento di Chimica and NIS centre, University of Torino, Italy
Linear Response properties
Slide from Radovan Bast
Linear Response properties
Slide from Radovan Bast
Linear Response properties
Slide from Radovan Bast
An alternative form of the perturbative operator
The total energy of a crystal in an electric field
DIIS for CPHF up to 2nd order
The error vector is defined in reciprocal space
CPHF
2nd-order CPHF
L. Maschio, Theor. Chem. Acc. , 137(4), 60 (2018)
DIIS for CPHF
One layer-MoS2
L. Maschio, Theor. Chem. Acc. , 137(4), 60 (2018)
Infrared and Raman Intensities
IR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
= electric field
= Atomic displacementIR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
In CRYSTAL06 through Wannier functions:
numerical derivatives in direct space
= electric field
= Atomic displacementIR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
In CRYSTAL06 through Wannier functions:
numerical derivatives in direct space
In CRYSTAL09 through Berry Phase:
numerical derivatives in reciprocal space
= electric field
= Atomic displacementIR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
We want analytical derivatives
= electric field
= Atomic displacementIR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
Within Placzeck approximation, Raman tensor elements are defined as:
= electric field
= Atomic displacementIR and non-resonant Raman Intensities
Born Charges (IR intensities): derivative of the dipole moment
Within Placzeck approximation, Raman tensor elements are defined as:
We want analytical derivatives
= electric field
= Atomic displacementExternal electric field in periodic systems
This operator is not consistent with the periodic
boundary conditions, it is not bound and breaks
the translational invariance of the system.External electric field in periodic systems
This operator is not consistent with the periodic
boundary conditions, it is not bound and breaks
the translational invariance of the system.External electric field in periodic systems
This operator is not consistent with the periodic
boundary conditions, it is not bound and breaks
the translational invariance of the system.
Derivative in k: a lot of problems!
We want analytical derivativesWhat must be computed: 1) One CPHF calculation (three directions) 2) One CPHF2 calculation (six directions) 3) Integral gradients and assembly of tensors all at the equilibrium geometry. IR and Raman tensors are built contracting the tensors with the vibrational eigenmodes.
Infrared Intensities The IR intensity of the p-th mode: Reflectivity is calculated from dielectric constant by means of: (θ is the beam incident angle) The dielectric function is obtained with the classical dispersion relation:
Raman intensities, single crystal
Raman Intensities, powder crystal Tensor invariants are obtained averaging the Raman directional intensities
Crystal17 Input: very simple FREQCALC INTENS INTRAMAN INTCPHF END END END
Crystal17 Input: very simple
FREQCALC
INTENS
INTRAMAN
INTCPHF
END
IRSPEC
END
Optional generation of spectra profiles
RAMSPEC
END
END
ENDTheory Vs. Experiment: alpha-quartz
EXP: Handbook of Minerals Raman Spectra database of Lyon ENS
Frequency cm-1Pyrope Mg3Al2Si3O12 Garnets are important rock-forming silicates
Pyrope Mg3Al2Si3O12 : quite a long history
Pyrope Mg3Al2Si3O12
Pyrope Mg3Al2Si3O12
Pyrope Mg3Al2Si3O12
Pyrope Mg3Al2Si3O12
Som general considerations so far Some modes, though Raman active by symmetry considerations, have nearly zero intensity. Assignment of experimental peaks is widely guided by experience
Pyrope Mg3Al2Si3O12 Experimental=Kolesov (2000)
Two other examples
Jadeite NaAlSi2O6
UiO-66Jadeite Experimental spectrum from rruff database
Jadeite
UiO-66
More than 90 Raman-active modes
Exp. Spectrum: S. Bordiga and F. BoninoUiO-66 - better synthesis
UiO-66
UiO-66
References L. Maschio, B. Kirtman, R. Orlando, and M. Rèrat“Ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method“ J. Chem. Phys. 137, 204113 (2012) L. Maschio, B. Kirtman, M. Rèrat, R. Orlando, and R. Dovesi“ Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method I: theory.“ J. Chem. Phys. 139, 164101 (2013) L. Maschio, B. Kirtman, M. Rèrat, R. Orlando, and R. Dovesi“ Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method II: validation and comparison with experiments.“ J. Chem. Phys. 139, 164102 (2013) L. Maschio, B. Kirtman, S. Salustro, C.M.Zicovich-Wilson, R. Orlando, and R. Dovesi“ The Raman spectrum of Pyrope garnet. A quantum mechanical simulation of frequencies, intensities and isotope shifts.“ J. Phys. Chem. A 117 (14), 11464-11471 (2013)
Second Harmonic Generation and Pockels Effect
Second Harmonic Generation and Pockels Effect
Dynamic CPHF Iterative cycle We want to obtain the frequency dependent perturbation matrix
Dynamic CPHF Iterative cycle
Dynamic CPHF Iterative cycle
Dynamic CPHF Iterative cycle
Dynamic CPHF Iterative cycle
2n+1 rule… We get the frequency-dependent hyperpolarizabilty tensor from 1st-order CPHF, but we need 3 frequencies, hence to converge (at most) three CPHF procedures
Dynamic first hyperpolarizability
sum of frequencies must be zero.Dynamic first hyperpolarizability
Second-Harmonic Generation (SHG)
Blue arrow: ordinary (linear) susceptibility
Green arrow: second-harmonic generation
Red arrow: optical rectification.Dynamic first hyperpolarizability
Second-Harmonic Generation (SHG)
Pockels Effect
Pockels Cells are used to rotate
the polarization of a passing beam.Second-Harmonic generation in molecular crystals
SHG imaging is a powerful
experimental tool to find
molecular crystalsThere is also a vibrational contribution! μt and αuv are dipole moment and polarizability components, while ωi and Qi are phonon frequencies and normal modes of the i-th vibration at the Γ-point. The terms ∂μt/∂Qi and ∂αuv/∂Qi are the same tensors needed for the evaluation of infrared and Raman intensities. Straightforwardly computed with keyword BETAVIB in frequency calculation.
Checking the implementation: from 0D to 3D
Checking the implementation: from 0D to 3D
Checking the implementation: from 0D to 3D
One interesting application: MoS2 multi-layer L. Maschio, M. Rérat, B. Kirtman and R. Dovesi, The Journal of Chemical Physics, 143, 244102 (2015)
Another interesting application: Urea and KDP
M. Rérat, L. Maschio, B. Kirtman, B. Civalleri, and R. Dovesi
Journal of Chemical Theory and Computation 12 (1), 107-113 (2016)Urea: SHG tensor components
References Ferrero, M.; Rérat, M.; Kirtman, B. and Dovesi, R. (2008) J. Chem. Phys. 129: 244110. Orlando, R.; Lacivita, V.; Bast, R. and Ruud, K. (2010) J. Chem. Phys. 132: 244106. L. Maschio, M. Rérat, B. Kirtman and R. Dovesi, (2015) J. Chem. Phys. 143, 244102 M. Rérat, L. Maschio, B. Kirtman, B. Civalleri, and R. Dovesi J. Chem. Theory Comput. (2016) 12 (1), 107-113 B. Kirtman, L. Maschio, M. Rèrat, and M. Springborg, in “Frontiers of Quantum Chemistry” (Springer, 2018), pp. 87–115.
Thank you for your attention!
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