Anharmonic energy in periodic systems
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1 Anharmonic energy in periodic systems Bartomeu Monserrat University of Cambridge Electronic Structure Discussion Group 13 March 213
2 Vibrational properties overview Harmonic phonons are a very good approximation. Anharmonic phonons: light elements, hydrogen bonds, high temperatures,... Interacting phonons: Phase stability. Thermal expansion. Electron-phonon coupling: Temperature dependence of band gaps. Superconductivity.
3 Outline Theoretical background Born-Oppenheimer and harmonic approximations Vibrational self-consistent field Results Conclusions
4 Outline Theoretical background Born-Oppenheimer and harmonic approximations Vibrational self-consistent field Results Conclusions
5 Condensed matter Hamiltonian Ĥ = 1 2 i 1 2 α 2 2m i α α r i r j i i j i α Z α r i r α α β α Z α Z β r α r β
6 Born-Oppenheimer approximation Trial wavefunction: Ψ m ({r i }, {r α }) = ψ m ({r i }; {r α })Φ({r α }) (1) Equations of motion: Ĥ e ψ m ({r i }; {r α }) = ɛ m ({r α })ψ m ({r i }; {r α }) Ĥ vib Φ({r α }) = EΦ({r α }) Decoupled Hamiltonians: Ĥ e = 1 2 i r i i i r j j i i + 1 Z α Z β 2 r α r β Ĥ vib = 1 2 α α β α 1 m α 2 α + ɛ m ({r α }) α Z α r i r α
7 Harmonic approximation Vibrational Hamiltonian in {r α } (or {u α }): Ĥ vib = 1 2 R p,α 1 m α 2 pα R p,α;r p,β Normal mode analysis: {u pα } {q ks } u pα Φ pα;p βu p β u pα;i = q ks = 1 q ks e ik Rp w ks;iα N m α 1 N k,s R p,α,i mα u pα;i e ik Rp w ks;iα Vibrational Hamiltonian in {q ks }: Ĥ vib = ( 1 2 k,s 2 q 2 ks ω2 ks q2 ks )
8 Principal axes approximation to the BO energy surface V ({q ks }) = V ()+ k,s V ks (q ks )+ 1 2 Static lattice DFT total energy V ks;k s (q ks, q k s )+ k,s k,s DFT total energy along frozen independent phonon DFT total energy along frozen coupled phonons
9 Vibrational self-consistent field equations Phonon Schrödinger equation: k,s q 2 ks + V ({q ks }) Φ({q ks }) = EΦ({q ks }) Ground state ansatz: Φ({q ks }) = k,s φ ks(q ks ) Self-consistent equations: ( 1 2 ) 2 qks 2 + V ks (q ks ) φ ks (q ks ) = λ ks φ ks (q ks ) V ks (q ks ) = φ k s (q k s ) V ({q k s }) φ k s (q k s ) k,s k,s
10 Diamond independent phonon term (I) V ({q ks }) = V ()+ k,s V ks (q ks )+ 1 2 V ks;k s (q ks, q k s )+ k,s k,s Energy (ev) Diamond Anharmonic Harmonic q (a.u.)
11 Diamond independent phonon term (II).2 Anharmonic Harmonic.15 φ(q) q (a.u.)
12 Diamond coupled phonons term V ({q ks }) = V ()+ k,s V ks (q ks )+ 1 2 V ks;k s (q ks, q k s )+ k,s k,s q 2 (a.u.) Energy (ev) q 2 (a.u.) q 1 (a.u.) q 1 (a.u.)
13 LiH independent phonon term (I) V ({q ks }) = V ()+ k,s V ks (q ks )+ 1 2 V ks;k s (q ks, q k s )+ k,s k,s LiH Anharmonic Harmonic Energy (ev) q (a.u.)
14 LiH independent phonon term (II).2.15 Anharmonic Harmonic φ(q) q (a.u.)
15 LiH coupled phonons term V ({q ks }) = V ()+ k,s V ks (q ks )+ 1 2 V ks;k s (q ks, q k s )+ k,s k,s q 2 (a.u.) Energy (ev) q 2 (a.u.) q 1 (a.u.) q 1 (a.u.)
16 Outline Theoretical background Born-Oppenheimer and harmonic approximations Vibrational self-consistent field Results Conclusions
17 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond Number of normal modes 8
18 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 H 7 Li Number of normal modes 8
19 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 H 7 Li 2 H 7 Li Number of normal modes 8
20 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 7 H Li 2 7 H Li Graphene Number of normal modes 8
21 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 7 H Li 2 7 H Li Graphene Graphane Number of normal modes 8
22 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 7 H Li 2 7 H Li Graphene Graphane Hydrogen-III (15 GPa) Number of normal modes 8
23 Anharmonic ZPE correction Anharmonic correction (mev/atom) Diamond 1 7 H Li 2 7 H Li Graphene Graphane Hydrogen-III (15 GPa) Hydrogen-III (3 GPa) Number of normal modes 8
24 Outline Theoretical background Born-Oppenheimer and harmonic approximations Vibrational self-consistent field Results Conclusions
25 Conclusions Summary: Principal axes approximation to the Born-Oppenheimer energy surface. VSCF method for the solution of the vibrational equation. Examples of anharmonic energy correction. Extended framework (see future talk): Phonon expectation values for general phonon-dependent operators. Electron-phonon interactions, stress tensor, mean atomic positions, hyperfine tensor,...
26 Acknowledgements: Prof Richard J. Needs Dr Neil D. Drummond TCM group EPSRC References: B. Monserrat, N.D. Drummond, R.J. Needs, arxiv:
27 Outline Additional Material
28 Additional Material: diamond phonon dispersion ω nq (ev).8.4 Γ X L W Γ
29 Additional Material: LiH phonon dispersion.12 1 H 7 Li 2 H 7 Li ω nq (ev).8.4 Γ X L W Γ
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