ICD - THE DECAY WIDTHS II

Transcription

1 ICD - THE DECAY WIDTHS II BEYOND WIGNER-WEISSKOPF Přemysl Kolorenč Institute of Theoretical Physics, Charles University in Prague ICD Summer School Bad Honnef September 1 st 5 th P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

2 Decay widths Beyond 1 st order Complex absorbing potential tame the divergent Siegert state to fit into the L 2 space Fano-ADC discrete state revisited time independent, non-perturbative approach P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

3 Complex absorbing potential Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

4 Complex absorbing potential CAP Example V (r) = 2r 2 e r E res = i Solution of SE at real and complex energy (H E) Ψ E = 0 E R/C Introduce CAP W = iη(r r C ) 2 exponential damping on a length scale 1/ 4 η P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

5 Complex absorbing potential CAP Example V (r) = 2r 2 e r E res = i Solution of SE at real and complex energy (H E) Ψ E = 0 E R/C Introduce CAP W = iη(r r C ) 2 exponential damping on a length scale 1/ 4 η P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

6 Complex absorbing potential CAP Eigenvalues of Ĥ(η) V (r) = 7.5r 2 e r + iη(r r c ) 2 E res1 = i E res2 = i η = η = Riss and Meyer, J. Phys. B 26, 4503 (1993) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

7 Complex absorbing potential η-trajectory of the resonance pole Best approximation E(η) to Siegert energy E(η = 0) minimization of E(η) E(0) = η de(η) dη + O(η2 ) Reflection properties of CAP: Riss and Meyer, J. Chem. Phys. 105, 1409 (1996) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

8 Complex absorbing potential η-trajectory of the resonance pole Best approximation E(η) to Siegert energy E(η = 0) minimization of E(η) E(0) = η de(η) dη + O(η2 ) Reflection properties of CAP: Riss and Meyer, J. Chem. Phys. 105, 1409 (1996) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

9 Complex absorbing potential Complex symmetric eigenvalue problem Symmetric bilinear form instead of usual scalar product (φ ψ) φ(x)ψ(x) d 3 x Possible incompletness of the spectrum of nonhermitian operator Ĥ(η) = Ĥ + iηŵ occurs only for isolated points of η not a real issue for numerical calculations Diagonalization techniques and search for the stabilization points complex Lanczos algorithm parallel filter diagonalization Santra and Cederbaum, Phys. Rep. 368, 1 (2002) Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge, 2011 P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

10 Complex absorbing potential CAP in ab initio many-body methods Simple choice W (x, c, n) = 3 i=1 ( x i c i ) n, x i > c i efficient matrix elements evaluation in Guassian basis sets minimum reflection considerations irrelevant due to low basis quality elimination of Φ N 0 perturbation: (ϕ p W ϕ q) = 0 for p or q occupied CAP/CI W = 1h 2h1p 1h 0 0 2h1p 0 W aa δ ii δ jj CAP/ADC(3) Santra and Cederbaum, Phys. Rep. 368 (2002) Vaval and Cederbaum, J. Chem. Phys. 126, (2007) CAP/EOM-CC Ghosh, Pal and Vaval, J. Chem. Phys. 139, (2013); Mol. Phys. 112, 669 (2014) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

11 Complex absorbing potential Decay widths Beyond 1 st order Fano-ADC discrete state revisited time-independent, non-perturbative approach P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

12 Fano-Feshbach theory Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

13 Fano-Feshbach theory Discrete state in continuum example Potential scattering V (r) = 10r 2 e r E res = i Discrete state in continuum character of exact wave function (E R) Ψ E = a(e) φ d + b(e, ɛ) χ ɛ dɛ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

14 Fano-Feshbach theory Discrete state in continuum example Potential scattering V (r) = 10r 2 e r E res = i Discrete state in continuum character of exact wave function (E R) Ψ E = a(e) φ d + b(e, ɛ) χ ɛ dɛ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

15 Fano-Feshbach theory Discrete state in continuum example Potential scattering V (r) = 10r 2 e r E res = i Discrete state in continuum character of exact wave function (E R) Ψ E = a(e) φ d + b(e, ɛ) χ ɛ dɛ a 2 (E) : a 2 (E) Γ(E) (E E d (E)) 2 + Γ 2 (E)/ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

16 Fano-Feshbach theory Projection operator approach Separation of the Hilbert space resonance and background continuum H = Q P Q = φ d φ d P = χ ɛ χ ɛ dɛ PQ = 0 Key requirement P asymptotically complete Hamiltonian lim PΨ E(r) = lim Ψ E (r) = φ d (r) L 2 r r H = H QQ + H PP + H QP + H PQ, H QP = QHP,... Schrödinger equation (E H) Ψ E = 0 (E R) Feshbach, Ann. Phys. 19, 287 (1962); Domcke, Phys. Rep. 208, 97 (1991) Čížek, Ph.D. thesis, Charles University in Prague (1999) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

17 Fano-Feshbach theory Projection operator approach Separation of the Hilbert space resonance and background continuum H = Q P Q = φ d φ d P = χ ɛ χ ɛ dɛ PQ = 0 Key requirement P asymptotically complete Hamiltonian lim PΨ E(r) = lim Ψ E (r) = φ d (r) L 2 r r H = H QQ + H PP + H QP + H PQ, H QP = QHP,... Schrödinger equation (E H) Ψ E = 0 (E R) Feshbach, Ann. Phys. 19, 287 (1962); Domcke, Phys. Rep. 208, 97 (1991) Čížek, Ph.D. thesis, Charles University in Prague (1999) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

18 Fano-Feshbach theory Solution in the Q-space Projected Schrödinger equation (E H PP )P Ψ E = H PQ Q Ψ E (1) (E H QQ )Q Ψ E = H QP P Ψ E (2) (1) together with (E H PP ) χ E = 0 gives P Ψ E = χ E + (E H PP + iη) 1 H PQ Q Ψ E (3) (3) + (2) yields equation for the Q-projection of Ψ E (E H QQ )Q Ψ E = H QP χ E + H QP (E H PP + iη) 1 H PQ Q Ψ E (4) Solution in Q-space Q Ψ E = [E H QQ H QP (E H PP + iη) 1 H PQ ] 1 H QP χ E P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

19 Fano-Feshbach theory Solution in the Q-space Projected Schrödinger equation (E H PP )P Ψ E = H PQ Q Ψ E (1) (E H QQ )Q Ψ E = H QP P Ψ E (2) (1) together with (E H PP ) χ E = 0 gives P Ψ E = χ E + (E H PP + iη) 1 H PQ Q Ψ E (3) (3) + (2) yields equation for the Q-projection of Ψ E (E H QQ )Q Ψ E = H QP χ E + H QP (E H PP + iη) 1 H PQ Q Ψ E (4) Solution in Q-space Q Ψ E = [E H QQ H QP (E H PP + iη) 1 H PQ ] 1 H QP χ E P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

20 Fano-Feshbach theory Solution in the Q-space Projected Schrödinger equation (E H PP )P Ψ E = H PQ Q Ψ E (1) (E H QQ )Q Ψ E = H QP P Ψ E (2) (1) together with (E H PP ) χ E = 0 gives P Ψ E = χ E + (E H PP + iη) 1 H PQ Q Ψ E (3) (3) + (2) yields equation for the Q-projection of Ψ E (E H QQ )Q Ψ E = H QP χ E + H QP (E H PP + iη) 1 H PQ Q Ψ E (4) Solution in Q-space Q Ψ E = [E H QQ H QP (E H PP + iη) 1 H PQ ] 1 H QP χ E P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

21 Fano-Feshbach theory Solution in the Q-space Projected Schrödinger equation (E H PP )P Ψ E = H PQ Q Ψ E (1) (E H QQ )Q Ψ E = H QP P Ψ E (2) (1) together with (E H PP ) χ E = 0 gives P Ψ E = χ E + (E H PP + iη) 1 H PQ Q Ψ E (3) (3) + (2) yields equation for the Q-projection of Ψ E (E H QQ )Q Ψ E = H QP χ E + H QP (E H PP + iη) 1 H PQ Q Ψ E (4) Solution in Q-space Q Ψ E = [E H QQ H QP (E H PP + iη) 1 H PQ ] 1 H QP χ E P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

22 Fano-Feshbach theory Complex level shift function F(E) QHP(E PHP + iη) 1 PHQ evaluation of the operator P(E PHP + iη) 1 P = dɛ dɛ χ ɛ χ ɛ (E PHP + iη) 1 χ ɛ χ ɛ = dɛ dɛ χ ɛ χ ɛ χ ɛ (E ɛ + iη) 1 χ ɛ = dɛ dɛ χ ɛ (E ɛ + iη) 1 δ(ɛ ɛ ) χ ɛ 1 applying the usual lim η 0 + x+iη = P 1 x iπδ(x) = P ɛ dɛ dɛ χ ɛ χ ɛ E ɛ δ(ɛ ɛ ) iπ dɛ dɛ χ ɛ χ ɛ δ(ɛ ɛ )δ(e ɛ) = P dɛ χ ɛ χ ɛ iπ χ E χ E E ɛ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

23 Fano-Feshbach theory Complex level shift function F(E) QHP(E PHP + iη) 1 PHQ evaluation of the operator P(E PHP + iη) 1 P = dɛ dɛ χ ɛ χ ɛ (E PHP + iη) 1 χ ɛ χ ɛ = dɛ dɛ χ ɛ χ ɛ χ ɛ (E ɛ + iη) 1 χ ɛ = dɛ dɛ χ ɛ (E ɛ + iη) 1 δ(ɛ ɛ ) χ ɛ 1 applying the usual lim η 0 + x+iη = P 1 x iπδ(x) = P ɛ dɛ dɛ χ ɛ χ ɛ E ɛ δ(ɛ ɛ ) iπ dɛ dɛ χ ɛ χ ɛ δ(ɛ ɛ )δ(e ɛ) = P dɛ χ ɛ χ ɛ iπ χ E χ E E ɛ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

24 Fano-Feshbach theory Complex level shift function F(E) QHP(E PHP + iη) 1 PHQ The only nonzero element of F(E) is φ d F(E) φ d = P Energy dependent level shift and decay width dɛ φ d QHP χ ɛ χ ɛ PHQ φ d E ɛ iπ φ d QHP χ E χ E PHQ φ d φ d F(E) φ d = (E) i 2 Γ(E) Γ(E) 2π V de 2 V dɛ = φ d H χ ɛ Vdɛ 2 (E) P E ɛ dɛ = 1 2π P Γ(ɛ) E ɛ dɛ P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

25 Fano-Feshbach theory Interpretation as decay width Ψ E = a(e) φ d + b(e, ɛ) χ ɛ dɛ φ d χ ɛ = 0 gives Q Ψ E = a(e) φ d = [E QHQ F (E)] 1 QHP χ E Resonance profile of a(e) φ d Q Ψ E = a(e) = V de E E d (E) + i 2 Γ(E) = a(e) 2 = 1 Γ(E) 2π (E E d (E)) 2 + Γ 2 (E)/4 Fano, PRA 124, 1866 (1961) Howat, Åberg and Goscinski, J. Phys. B 9, 1575 (1978) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

26 Fano-Feshbach theory Interpretation as decay width Ψ E = a(e) φ d + b(e, ɛ) χ ɛ dɛ φ d χ ɛ = 0 gives Q Ψ E = a(e) φ d = [E QHQ F (E)] 1 QHP χ E Resonance profile of a(e) φ d Q Ψ E = a(e) = V de E E d (E) + i 2 Γ(E) = a(e) 2 = 1 Γ(E) 2π (E E d (E)) 2 + Γ 2 (E)/4 Fano, PRA 124, 1866 (1961) Howat, Åberg and Goscinski, J. Phys. B 9, 1575 (1978) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

27 Fano-Feshbach theory Two-potential formula for the T -matrix S fi = δ fi 2πiδ(E f E i )T fi T -matrix T fi = i V f background scattering and resonance T -matrix T (E, E) = T bg (E, E) + T res (E, E) = k H PP K χ ɛ + χ ɛ H PQ Ψ E resonance term T res (E, E) = χ E H PQ (E H QQ F (E)) 1 H QP χ E Siegert energy poles of the resonance term of T -matrix z res = E d + (z res ) i 2 Γ(z res) = E res i 2 Γ E d i 2 Γ(E d) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

28 Fano-Feshbach theory Two-potential formula for the T -matrix S fi = δ fi 2πiδ(E f E i )T fi T -matrix T fi = i V f background scattering and resonance T -matrix T (E, E) = T bg (E, E) + T res (E, E) = k H PP K χ ɛ + χ ɛ H PQ Ψ E resonance term T res (E, E) = χ E H PQ (E H QQ F (E)) 1 H QP χ E Siegert energy poles of the resonance term of T -matrix z res = E d + (z res ) i 2 Γ(z res) = E res i 2 Γ E d i 2 Γ(E d) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

29 Fano-Feshbach theory Discrete state and Γ(E) example V (r) = 5r 2 e r 0.8 φ d φ d1 wave function amplitude φ d2 φ d3 Γ φ d2 φ d r Energy Kolorenč, Ph.D. thesis, Charles University in Prague (2005) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

30 Fano-Feshbach theory Fano-ADC-Stieltejs method overview Fano-Feshbach theory of resonances exact continuum wave function discrete state in continuum character N c Ψ E = a(e) φ d + b β (E, ɛ) χ β,ɛ dɛ β=1 φ d L 2 discrete state (not a Hamiltonian eigenstate) χ β,ɛ background continuum χ β,ɛ χ β,ɛ = δ β βδ(ɛ ɛ ) decay width of the resonance represented by φ d Γ(ɛ) = 2π β φ d H χ β,ɛ 2 ADC representation of the many-body wave functions background continuum discretized χ β,ɛ, ɛ R χ i, i N, χ i χ j = δ ij Stieltjes imaging spectral moments µ k of Γ(ɛ) correctly reproduced by {ɛ i, χ i } µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ 2 dɛ 2π ɛ k i φ d H χ i 2 β i P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

31 Fano-Feshbach theory Fano-ADC-Stieltejs method overview Fano-Feshbach theory of resonances exact continuum wave function discrete state in continuum character N c Ψ E = a(e) φ d + b β (E, ɛ) χ β,ɛ dɛ β=1 φ d L 2 discrete state (not a Hamiltonian eigenstate) χ β,ɛ background continuum χ β,ɛ χ β,ɛ = δ β βδ(ɛ ɛ ) decay width of the resonance represented by φ d Γ(ɛ) = 2π β φ d H χ β,ɛ 2 ADC representation of the many-body wave functions background continuum discretized χ β,ɛ, ɛ R χ i, i N, χ i χ j = δ ij Stieltjes imaging spectral moments µ k of Γ(ɛ) correctly reproduced by {ɛ i, χ i } µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ 2 dɛ 2π ɛ k i φ d H χ i 2 β i P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

32 Fano-Feshbach theory Fano-ADC-Stieltejs method overview Fano-Feshbach theory of resonances exact continuum wave function discrete state in continuum character N c Ψ E = a(e) φ d + b β (E, ɛ) χ β,ɛ dɛ β=1 φ d L 2 discrete state (not a Hamiltonian eigenstate) χ β,ɛ background continuum χ β,ɛ χ β,ɛ = δ β βδ(ɛ ɛ ) decay width of the resonance represented by φ d Γ(ɛ) = 2π β φ d H χ β,ɛ 2 ADC representation of the many-body wave functions background continuum discretized χ β,ɛ, ɛ R χ i, i N, χ i χ j = δ ij Stieltjes imaging spectral moments µ k of Γ(ɛ) correctly reproduced by {ɛ i, χ i } µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ 2 dɛ 2π ɛ k i φ d H χ i 2 β i P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

33 ADC Algebraic Diagrammatic Construction Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

34 ADC Algebraic Diagrammatic Construction ADC Intermediate state representation of GF G(ω) = f (ω1 K C) 1 f 1 perturbation theoretically corrected ( correlated ) ground state Φ N 0 2 correlated excited states (not orthogonal!) Ψ 0 J = ĈJ Φ N 0 {ĈJ} = {ĉ i, ĉ aĉiĉ j,... } [J] = 1 (one-hole), 2 (two-hole one-particle),... 3 precursor states Gramm-Schmidt orthogonalization to lower exc. classes Ψ # J = Ψ0 J K [K ]<[J] Ψ K Ψ K Ψ 0 J 4 intermediate states symmetric orthogonalization within the exc. class Ψ J = J [J ]=[J] can be still classified as 1h, 2h1p,... Ψ # J (ρ # 1/2 ) J J ρ # IJ = Ψ# I Ψ # J Schirmer, PRA 43, 4647 (1991); Mertins, Schirmer, PRA 53, 2140 (1996) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

35 ADC Algebraic Diagrammatic Construction ADC(2)x for the 1p-Green s Function 1h 2h1p H ADC = K + C = 1h ɛ i δ ii + C (1) i,i + C (2) i,i C (1) i,a i j 2h1p C (1) aij,i ( ɛ i ɛ j + ɛ a )δ ii δ jj δ aa +C (1) aij,a i j representation of φ d, χ ɛ in the basis of intermediate states initial state selected eigenstate of QH ADC Q of 1h character final states all eigenstates of PH ADC P of 2h1p character Decay of doubly ionized systems (ICD after Auger,... ) ADC(2)x for 2p-Green s function 2h and 3h1p excitation classes,... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

36 ADC & Fano partitioning Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

37 ADC & Fano partitioning Fano partitioning scheme I (heteronuclear clusters) molecular orbitals (MOs) localized either on atom A or atom B discrete state φ d φ d = i Y i ĉ i Φ N 0 + aij i, j A Y a ij ĉ aĉiĉ j Φ N 0 final ( continuum ) states χ ɛ χ ɛ = i Y i ĉ i Φ N 0 + aij Y a ij ĉ aĉiĉ j Φ N 0 i, j A, B Y i 2 1 i no interatomic correlation in initial state Averbukh and Cederbaum, JCP 123, (2005) Kolorenč et al., JCP 129, (2008) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

38 ADC & Fano partitioning Fano partitioning scheme II (homonuclear clusters) scheme I not applicable MOs delocalized due to inversion symmetry solution localization example: 2h1p intermediate states derived from 2h singlet: 1 Φ i,j,a = 1 2 (ĉ a ĉi ĉ j ĉ a ĉi ĉ j ) Φ 0 1 diagonalize 2 2 Hamiltonian submatrix for {1 Φ i,j,a, 1 Φ i,j,a } a is selected particle orbital {i, i }, {j, j } are gerade-ungerade pairs higher-energy eigenstate: one-site character (both holes on a single atom) lower-energy eigenstate: two-site character (each hole on a different atom) 2 one-site states contribute to the discrete state expansion (Q-space), two-site" to the continuum of final states (P-space) generalization to 2p-GF possible, but not straightforward Averbukh and Cederbaum, JCP 125, (2006) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

39 ADC & Fano partitioning Fano partitioning scheme II (homonuclear clusters) scheme I not applicable MOs delocalized due to inversion symmetry solution localization example: 2h1p intermediate states derived from 2h singlet: 1 Φ i,j,a = 1 2 (ĉ a ĉi ĉ j ĉ a ĉi ĉ j ) Φ 0 1 diagonalize 2 2 Hamiltonian submatrix for {1 Φ i,j,a, 1 Φ i,j,a } a is selected particle orbital {i, i }, {j, j } are gerade-ungerade pairs higher-energy eigenstate: one-site character (both holes on a single atom) lower-energy eigenstate: two-site character (each hole on a different atom) 2 one-site states contribute to the discrete state expansion (Q-space), two-site" to the continuum of final states (P-space) generalization to 2p-GF possible, but not straightforward Averbukh and Cederbaum, JCP 125, (2006) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

40 ADC & Fano partitioning Fano partitioning scheme II (homonuclear clusters) scheme I not applicable MOs delocalized due to inversion symmetry solution localization example: 2h1p intermediate states derived from 2h singlet: 1 Φ i,j,a = 1 2 (ĉ a ĉi ĉ j ĉ a ĉi ĉ j ) Φ 0 1 diagonalize 2 2 Hamiltonian submatrix for {1 Φ i,j,a, 1 Φ i,j,a } a is selected particle orbital {i, i }, {j, j } are gerade-ungerade pairs higher-energy eigenstate: one-site character (both holes on a single atom) lower-energy eigenstate: two-site character (each hole on a different atom) 2 one-site states contribute to the discrete state expansion (Q-space), two-site" to the continuum of final states (P-space) generalization to 2p-GF possible, but not straightforward Averbukh and Cederbaum, JCP 125, (2006) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

41 ADC & Fano partitioning Fano partitioning scheme III (universal) generalization of scheme II: diagonalization of the whole 2h1p ADC blocks characterized by the particle orbital p resulting eigenvalues reflect the structure of DI spectrum allows association of the eigenstates with open or closed channels P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

42 ADC & Fano partitioning Heteronuclear dimers Ne 2+ (2s 1 2p 1 )Ar 1 Σ [S.I] 1 Σ [S.III] Π [S.I] 1 Π [S.III] Γ(R) [mev] 10 1 R eq R [Å] P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

43 ADC & Fano partitioning Homonuclear dimers ICD after Auger in Ne 2 1 Σg [S.II] 1 Σg [S.III] Πu [S.II] 1 Πu [S.III] Γ(R) [mev] 10 0 R eq R [Å] P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

44 Stieltejs-Chebychev moment theory Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

45 Stieltejs-Chebychev moment theory Stieltjes imaging motivation description of continuum in L 2 basis no direct use of the formula Γ(ɛ) = 2π φ d H χ β,ɛ 2 β spectrum discretized {ɛ, χ ɛ } {ɛ i, χ i } incorrect boundary condition, normalization to unity χ i χ j = δ ij δ(ɛ i ɛ j ) completness χ β,ɛ χ β,ɛ dɛ χ i χ i β i spectral moments k 0 used for convergence reasons µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ χ β,ɛ H φ d dɛ β 2π i ɛ k i φ d H χ i χ i H φ d = 2π i ɛ k i φ d H χ i 2 = i ɛ k i γ i Carravetta and Ågren, PRA 35, 1022 (1987),... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

46 Stieltejs-Chebychev moment theory Stieltjes imaging motivation description of continuum in L 2 basis no direct use of the formula Γ(ɛ) = 2π φ d H χ β,ɛ 2 β spectrum discretized {ɛ, χ ɛ } {ɛ i, χ i } incorrect boundary condition, normalization to unity χ i χ j = δ ij δ(ɛ i ɛ j ) completness χ β,ɛ χ β,ɛ dɛ χ i χ i β i spectral moments k 0 used for convergence reasons µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ χ β,ɛ H φ d dɛ β 2π i ɛ k i φ d H χ i χ i H φ d = 2π i ɛ k i φ d H χ i 2 = i ɛ k i γ i Carravetta and Ågren, PRA 35, 1022 (1987),... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

47 Stieltejs-Chebychev moment theory Stieltjes imaging motivation description of continuum in L 2 basis no direct use of the formula Γ(ɛ) = 2π φ d H χ β,ɛ 2 β spectrum discretized {ɛ, χ ɛ } {ɛ i, χ i } incorrect boundary condition, normalization to unity χ i χ j = δ ij δ(ɛ i ɛ j ) completness χ β,ɛ χ β,ɛ dɛ χ i χ i β i spectral moments k 0 used for convergence reasons µ k = ɛ k Γ(ɛ) dɛ = 2π ɛ k φ d H χ β,ɛ χ β,ɛ H φ d dɛ β 2π i ɛ k i φ d H χ i χ i H φ d = 2π i ɛ k i φ d H χ i 2 = i ɛ k i γ i Carravetta and Ågren, PRA 35, 1022 (1987),... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

48 Stieltjes imaging Stieltejs-Chebychev moment theory Goal: find Gaussian quadrature associated with unknown weight function Γ(ɛ) µ k = ɛ k Γ(ɛ) dɛ = n S j (ɛ (n S) j ) k w (n S) j, k = 0,... 2n S 1 1 recurence formula for polynomials orthogonal with respect to Γ(ɛ) can be expressed in terms of {ɛ i, γ i } Γ(ɛ)Q n (1/ɛ)Q m (1/ɛ) dɛ = N n δ nm 2 abscissas 1/ɛ (n S) j roots of Q ns (1/ɛ) (diagonalization of tridiagonal matrix) 3 weights w (n S) j w (n S) j = N m ns 1 m=0 Q2 m(1/ɛ (n S) j ) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

49 Stieltjes imaging Stieltejs-Chebychev moment theory Goal: find Gaussian quadrature associated with unknown weight function Γ(ɛ) µ k = ɛ k Γ(ɛ) dɛ = n S j (ɛ (n S) j ) k w (n S) j, k = 0,... 2n S 1 1 recurence formula for polynomials orthogonal with respect to Γ(ɛ) can be expressed in terms of {ɛ i, γ i } Γ(ɛ)Q n (1/ɛ)Q m (1/ɛ) dɛ = N n δ nm 2 abscissas 1/ɛ (n S) j roots of Q ns (1/ɛ) (diagonalization of tridiagonal matrix) 3 weights w (n S) j w (n S) j = N m ns 1 m=0 Q2 m(1/ɛ (n S) j ) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

50 Stieltjes imaging 4 Stieltejs distribution Stieltejs-Chebychev moment theory F(ɛ) = F (n S) (ɛ) = ɛ q j=1 ɛ min Γ(ɛ ) dɛ F (n S) (ɛ) w (n S) j ɛ (n S) q < ɛ < ɛ (n S) q+1 F (n S) (ɛ (n S) i ) = 1 2 [F (n S) (ɛ (n S) i 0) + F (n S) (ɛ (n S) i + 0)] 5 desired approximation to the width function Stieltjes derivative Γ (n S) (ɛ) = 1 2 w (n S) q+1 + w (n S) q ɛ (n S) q+1 ɛ(n S) q ɛ (n S) q < ɛ < ɛ (n S) q+1 6 effectively yields Γ(ɛ) at n S 1 discrete points using 2n S lowest µ k Müller-Plathe and Diercksen, PRA 40, 696 (1989),... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

51 Stieltjes imaging 4 Stieltejs distribution Stieltejs-Chebychev moment theory F(ɛ) = F (n S) (ɛ) = ɛ q j=1 ɛ min Γ(ɛ ) dɛ F (n S) (ɛ) w (n S) j ɛ (n S) q < ɛ < ɛ (n S) q+1 F (n S) (ɛ (n S) i ) = 1 2 [F (n S) (ɛ (n S) i 0) + F (n S) (ɛ (n S) i + 0)] 5 desired approximation to the width function Stieltjes derivative Γ (n S) (ɛ) = 1 2 w (n S) q+1 + w (n S) q ɛ (n S) q+1 ɛ(n S) q ɛ (n S) q < ɛ < ɛ (n S) q+1 6 effectively yields Γ(ɛ) at n S 1 discrete points using 2n S lowest µ k Müller-Plathe and Diercksen, PRA 40, 696 (1989),... P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

52 Stieltejs-Chebychev moment theory Stieltejs imaging output example ɛ (n S+1) q < ɛ (n S) q < ɛ (n S+1) q+1 P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

53 Stieltejs-Chebychev moment theory Direct expansion approach basis of known orthogonal polynomials L n (1/x) Votavová, Bc. thesis, Charles University in Prague (2014) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

54 Comparison with CAP methods Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

55 Comparison with CAP methods ICD widths Fano-ADC vs. CAP van der Waals clusters Ne 2 ( 2 Σ + u, R = 3.2 Å) CAP/CI (d-aug-cc-pvdz + 3s3p3d) 10.3 mev JCP 115, 6853 (2001) CAP/ADC (d-aug-cc-pvqz) 7.0 mev JCP 126, (2007) Fano-ADC (d-aug-cc-pvqz + 5s5p5d) 7.0 ± 0.3 mev Hydrogen-bonded systems (HF) 2 donor acceptor E R (ev) Γ (mev) E R (ev) Γ (mev) CAP/CI (aug-cc-pvqz) CAP/EOM-CC (aug-cc-pvtz) Fano-ADC (d-aug-cc-pvqz) Santra et al., Chem. Phys. Lett. 303, 413 (1999) Ghosh et al., Mol. Phys. 112, 669 (2014) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

56 Comparison with CAP methods ICD widths Fano-ADC vs. CAP van der Waals clusters Ne 2 ( 2 Σ + u, R = 3.2 Å) CAP/CI (d-aug-cc-pvdz + 3s3p3d) 10.3 mev JCP 115, 6853 (2001) CAP/ADC (d-aug-cc-pvqz) 7.0 mev JCP 126, (2007) Fano-ADC (d-aug-cc-pvqz + 5s5p5d) 7.0 ± 0.3 mev Hydrogen-bonded systems (HF) 2 donor acceptor E R (ev) Γ (mev) E R (ev) Γ (mev) CAP/CI (aug-cc-pvqz) CAP/EOM-CC (aug-cc-pvtz) Fano-ADC (d-aug-cc-pvqz) Santra et al., Chem. Phys. Lett. 303, 413 (1999) Ghosh et al., Mol. Phys. 112, 669 (2014) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

57 Comparison with CAP methods Ne clusters Fano-ADC vs. Wigner-Weisskopf 100 Fano-ADC Wigner-Weisskopf exp: bulk 80 Γ [mev] exp: surface # of Ne atoms exp: Öhrwall et al., PRL 93, (2004) WW: Santra et al., Phys. Rev. B 64, (2001) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

58 Appendix partial decay widths, computational details Outline 1 Complex absorbing potential 2 Fano-Feshbach theory 3 ADC Algebraic Diagrammatic Construction 4 ADC & Fano partitioning 5 Stieltejs-Chebychev moment theory 6 Comparison with CAP methods 7 Appendix partial decay widths, computational details P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

59 Appendix partial decay widths, computational details Partial decay widths in Fano-ADC rigorously unattainable in L 2 theory channels only defined asymptotically estimation possible using suitable L 2 channel projectors P β 1 evaluate channel-specific couplings γ β,i = 2π φ d HP β χ i 2 2 apply Stieltjes imaging procedure on the sets {ɛ i, γ β,i } Γ β 3 normalize by independently evaluated total width Γ Γ β Γ β α Γ α Γ reliability depends on how well the channel projectors can be defined in terms of localized wave functions Cacelli, Caravetta and Moccia, Mol. Phys. 59, 385 (1986) Averbukh and Cederbaum, J. Chem. Phys. 123, (2005) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

60 Appendix partial decay widths, computational details Partial decay widths in Fano-ADC rigorously unattainable in L 2 theory channels only defined asymptotically estimation possible using suitable L 2 channel projectors P β 1 evaluate channel-specific couplings γ β,i = 2π φ d HP β χ i 2 2 apply Stieltjes imaging procedure on the sets {ɛ i, γ β,i } Γ β 3 normalize by independently evaluated total width Γ Γ β Γ β α Γ α Γ reliability depends on how well the channel projectors can be defined in terms of localized wave functions Cacelli, Caravetta and Moccia, Mol. Phys. 59, 385 (1986) Averbukh and Cederbaum, J. Chem. Phys. 123, (2005) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

61 Appendix partial decay widths, computational details Diagonalization procedures Initial state selected eigenstate of QHQ Davidson diagonalization convergence of individual eigenvectors J. Comp. Phys. 17, 87 (1975); SIAM Rev. 42, 267 (2000) JADAMILU library Comp. Phys. Commun. 177, 951 (2007) Final states all eigenstates of PHP full diagonalization band Lanczos diagonalization possible representation of the matrix in Krylov space {q i, Hq i, H 2 q i,... } diagonalization of comparatively small band matrix T conserves spectral moments of the matrix [Gokhberg, Ph.D. thesis (2008)] starting block must contain PHQ φ d vital for convergence χ i H φ d couplings available as elements of (short) eigenvectors of T Stieltjes-Lanczos: J. Chem. Phys. 134, (2011); 139, (2013); 140, (2014) Parlett, The Symmetric Eigenvalue Problem, SIAM (1998) P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42

62 Summary Appendix partial decay widths, computational details Fano-ADC-Stieltjes computationally efficient tool for calculating decay widths stable over many orders of magnitude applicable when Q P separation possible in terms of L 2 basis van der Waals clusters (orbital localization achievable) Auger decay (large energetic gap) problems in hydrogen-bonded clusters? results may be sensitive to the discrete state definition estimation of partial rates available Complex absorbing potential applicable to wider range of problems usable with different ab initio methods non-hermitian eigenvalue problem, questionable with Gaussian basis NO BLACK BOX METHODS! P. Kolorenč (Charles University in Prague) ICD widths II September 4th, / 42