Ab initio proučavanje neadijabatskih efekata kod malih molekula
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1 Ab initio proučavanje neadijabatskih efekata kod malih molekula Marko Mitić Fakultet za fizičku hemiju, Univerzitet u Beogradu Seminar iz fizike/astrofizike Departman za fiziku, PMF, Novi Sad 15. april Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 1 / 44
2 Uvod "Fizički zakoni na kojima počiva veliki deo fizike i cela hemija su, dakle, potpuno poznati i problem je samo u tome što egzaktna primena tih zakona vodi do jednačina koje su isuviše komplikovane da bi se mogle rešiti." P. A. M. Dirak, Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 2 / 44
3 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 3 / 44
4 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 4 / 44
5 Šredingerova jednačina Molekul = S jezgara (A, B,..., S) N elektrona (α, β,..., N) Talasna funkcija Ψ = Ψ(r A, s A,..., r S, s S, r α, s α,..., r N, s N ) Šredingerova jednačina ĤΨ = EΨ Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 5 / 44
6 Born-Openhajmerova aproksimacija Šredingerova jednačina ĤΨ( r, R) = EΨ( r, R) Nerelativistički hamiltonijan molekula Ĥ = ˆT n + ˆT e + ˆV en + ˆV ee + ˆV nn = 1 2 N S µ=α K=A Z N K r µk + N µ=α ν>µ 1 r µν + S K=A S 1 m K K 1 2 S K=A L>K Z K Z L R KL N µ=α µ Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 6 / 44
7 Born-Openhajmerova aproksimacija Šredingerova jednačina ĤΨ( r, R) = EΨ( r, R) Nerelativistički hamiltonijan molekula Ĥ = ˆTn + ˆT e + ˆV en + ˆV ee + ˆV nn = 1 S 1 K 1 N µ 2 m K 2 K=A µ=α N S Z N K r µk + N 1 S r µν + S µ=α K=A Elektronski hamiltonijan µ=α ν>µ K=A L>K Ĥ e = ˆT e + ˆV en + ˆV ee + ˆV nn Z K Z L R KL Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 6 / 44
8 Born-Openhajmerova aproksimacija Rešavamo elektronsku Šredingerovu jednačinu Ĥ e ψ( r; R) = E el ( R)ψ( r; R) Rešavamo jezgarnu Šredingerovu jednačinu Ψ( r, R) = j Φ j ( R)ψ j ( r; R) [ ˆT n + E el ( R) E]Φ j ( R) + j Λ ij Φ j ( R) = 0 gde je Λ ij = 1 2 S 1 K=A m K [ 2 ψ i ( r; R) K ψ j ( r; R) K + ψ i ( r; R) ψ j ( r; R) ] Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 7 / 44
9 Born-Openhajmerova aproksimacija Zanemarivanje Λ ij nedijagonalni članovi (i j) adijabatska aproksimacija (i j) (i = j) B-O aproksimacija Dobijamo sistem jednačina [ ˆT n + E el ( R)]Φ j ( R) = EΦ j ( R), j = 1, 2,... Narušavanje B-O aproksimacije = neadijabatski efekti i) izbegnuta presecanja potencijalnih površi ii) Jan-Telerov efekat iii) Rener-Telerov efekat Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 8 / 44
10 Metode kvantne hemije "Brute force" metode Semiempirijske metode Ab initio metode Približne metode Perturbacioni račun Varijacioni račun Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 9 / 44
11 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
12 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
13 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
14 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
15 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
16 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
17 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 10 / 44
18 Ab initio metode za rešavanje elektronskog problema Hartree-Fock method (HF) Density functional theory (DFT) (ab initio?) Post-Hartree-Fock methods Møller Plesset perturbation theory (MPn) Configuration interaction (CI) Coupled cluster (CC) Multi-reference methods Multi-configurational self-consistent field (MCSCF, CASSCF) Multi-reference configuration interaction (MRCI) Complete active space perturbation theory (CASPTn) Multi-reference coupled-cluster (MRCC) Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 11 / 44
19 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 12 / 44
20 Mali molekuli The Cologne Database for Molecular Spectroscopy Universität zu Köln, Physikalisches Institut Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 13 / 44
21 Mali molekuli Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 14 / 44
22 Mali molekuli - atmosfera Titana N C C C C N Dicijanoacetilen (C4 N2 ) H C C C C H Diacetilen (C4 H2 ) Cijanoacetilen i cijanodiacetilen opaženi u interstelarnom prostoru, dicijanoacetilen? Atmosfera Titana (N2, CH4,... ) C4 N2 (s) detektovan C4 N2 značajan za meteorologiju Titana X 2 Πu elektronsko stanje katjona C4 N+ 2, ispoljava Rener-Telerov efekat Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 15 / 44
23 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 16 / 44
24 Rener-Telerov efekat - opšta razmatranja Linearni molekuli (elektronska stanja Π,, Φ,... ) C v, D h C S, C 2v, C 2h,... Vibraciono-elektronska (vibronska) sprega (a) V + (b) V + (c) V + V V V ρ ρ ρ Troatomski molekuli Četvoroatomski molekuli Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 17 / 44
25 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 18 / 44
26 Modelni hamiltonijan a) Razmatrano elektronsko stanje je dovoljno odvojeno od drugih elektronskih stanja b) Ravnotežna geometrija molekula je linearna c) Harmonijska aproksimacija d) Sprega između savijajućih i istežućih vibracija je zanemarena e) Rotacija molekula u celini zanemarena f) Korišćene krivolinijske unutrašnje koordinate g) Primenjen operator kinetičke energije jezgara za infinitezimalne savijajuće vibracije h) Primenjen fenomenološki spin-orbitni operator i) Asimptotske (linearne) elektronske talasne funkcije su korišćene za matričnu reprezentaciju operatora kinetičke energije jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 19 / 44
27 Modelni hamiltonijan Molekulski hamiltonijan Elektronski spin-orbitni operator Ĥ = Ĥe + ˆT n + ĤSO Ĥ SO = A SO ˆLz Ŝ z Operator kinetičke energije za savijajuća kretanja jezgara i) krivolinijske unutrašnje koordinate (ρ 1, φ 1, ρ 2, φ 2 ) ii) krivolinijske simetrijske koordinate (ρ C, φ C, ρ T, φ T ) iii) pravolinijske unutrašnje/simetrijske koordinate iv) normalne koordinate Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 20 / 44
28 Modelni hamiltonijan [Perić et al. 2006] Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 21 / 44
29 Modelni hamiltonijan ˆT n u krivolinijskim unutrašnjim koordinatama za dve (dvostruko degenerisane) infinitezimalne savijajuće vibracije ( ˆT n = µ 1 ρ 2 1 ρ 1 ρ 1 ρ 2 1 [ ( + 1 cos(φ 2 φ 1) µ 12 ( + sin(φ 2 φ 1) 2 φ 2 1 ) ( 1 2 2µ 2 2 ρ 1 ρ ρ 1ρ ρ 1 ρ 2 φ 1 ρ 2 ρ 1 φ 2 ρ 2 2 φ 1 φ 2 )] ρ 2 ρ 2 ρ 2 2 ) 2 φ 2 2 ) V = 1 2 k1ρ k2ρ2 2 + k 12ρ 1ρ 2 cos(φ 2 φ 1) Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 22 / 44
30 Elektronske bazne funkcije Vibronska talasna funkcija u obliku Ψ = f 1 ψ 1 + f 2 ψ 2 Izbor elektronskih talasnih funkcija adijabatske talasne funkcije Ĥ e ψ + = V + ψ +, Ĥ e ψ = V ψ Matrični elementi od Ĥel dijagonalni Matrični elementi od ˆT n sadrže članove: ψ + ψ + = ψ ψ = q i q i q i ψ + ψ = ψ + ψ = ψ ψ + q i q i q i B-O aproksimacija? Izbegnuta presecanja; članovi ρ 1 i i ρ 1 i ρ 1 j Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 23 / 44
31 Elektronske bazne funkcije - dijabatizacija Dijabatske elektronske talasne funkcije η 1 = ψ + cos(λτ) ψ sin(λτ) η 2 = ψ + sin(λτ) + ψ cos(λτ) Matrični elementi elektronskog operatora u ovoj bazi E 11 = η 1 Ĥe η 1 = V + cos 2 (Λτ) + V sin 2 (Λτ) E 22 = η 2 Ĥe η 2 = V + sin 2 (Λτ) + V cos 2 (Λτ) E 12 = η 1 Ĥe η 2 = V + V sin(2λτ) = η 2 Ĥe η 1 2 Ugao τ određujemo kao η 2 η 1 = 0, q i τ q i = 1 Λ ψ + ψ q i = 1 Λ ψ ψ + q i Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 24 / 44
32 Elektronske bazne funkcije - dijabatizacija Sledeća transformacija - linearne bazne funkcije ψ 1 = 1 2 (η 1 + iη 2 ) = 1 2 e iλτ (ψ + + iψ ) ψ 2 = 1 2 (η 1 iη 2 ) = 1 2 e iλτ (ψ + iψ ) U linearnoj bazi matrični elementi elektronskog operatora oblika E11 lin = ψ 1 Ĥe ψ 1 = V + + V 2 12 = ψ 1 Ĥe ψ 2 = V + V E lim 2 21 = ψ 2 Ĥe ψ 1 = V + V E lim 2 = ψ 2 Ĥe ψ 2 = E lin 22 e 2iτ Kompleksni matrični elementi "linearnih" el. funkcija po koordinatama jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 25 / 44 e 2iτ
33 Mala odstupanja od linearnosti Asimptotske forme adijabatskih el. talasnih funkcija ψ 0 + = lim ρ 1 0, ρ 2 0 ψ+ = ξ 1 cos[λ(θ τ)] π ψ0 = lim ρ 1 0, ρ 2 0 ψ = ξ 1 sin[λ(θ τ)] π Dijabatske elektronske funkcije η 1 0 = η 2 0 = "Linearne" elektronske funkcije ψ 1 0 = ψ 2 0 = lim η 1 = ξ 1 cos(λθ) ρ 1 0, ρ 2 0 π lim η 2 = ξ 1 sin(λθ) ρ 1 0, ρ 2 0 π lim ψ 1 = ξ 1 e iλθ ρ 1 0, ρ 2 0 2π lim ψ 2 = ξ 1 e iλθ ρ 1 0, ρ 2 0 2π Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 26 / 44
34 Ugao τ U okviru harm. aproksimacije i za Π elektronsko stanje, sledi { } τ = 1 2 arctan ɛ 1ρ 2 1 sin(2φ 1) + ɛ 1ρ 2 2 sin(2φ 2) + 2ɛ 12ρ 1ρ 2 sin(φ 1 + φ 2) ɛ 1ρ 2 1 cos(2φ1) + ɛ1ρ2 2 cos(2φ2) + 2ɛ12ρ1ρ2 cos(φ1 + φ2) Izvodi τ = ρ [ 1 sin(φ 2 φ 1 ) ɛ1 ɛ 12 ρ ɛ 2ɛ 12 ρ ɛ 1ɛ 2 ρ 1 ρ 2 cos(φ 2 φ 1 ) ] ρ 2 (V + V ) 2 τ = ɛ2 2 ρ ɛ2 12 ρ2 1 ρ2 2 + ɛ 12ρ 1 ρ 2 (ɛ 1 ρ ɛ 2ρ 2 2 ) cos(φ 2 φ 1 ) + ɛ 1 ɛ 2 ρ 2 1 ρ2 2 cos 2(φ 2 φ 1 ) φ 2 (V + V ) 2 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 27 / 44
35 S-O matrični elementi Matrični elementi operatora S-O sprege u "linearnoj" asimptotskoj bazi ψ0 1 ĤSO ψ0 1 = ΛΣA SO ψ0 2 ĤSO ψ0 2 = ΛΣA SO = = 0 ψ0 1 ĤSO ψ0 2 ψ0 2 ĤSO ψ0 1 Elektronski matrični elementi su operatori koji treba da deluju na vibracione bazne funkcije Svojstvene funkcije 2D harmonijskog oscilatora Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 28 / 44
36 Kvantni brojevi ˆN z = ˆR z + ˆL z K = l ± Λ Ĵ z = ˆN z + Ŝz P = K + Σ l = υ, υ 2,..., 1 ili 0 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 29 / 44
37 Kompjutacioni detalji Adijabatske energije FV-SA-CASSCF(9,10)/cc-pVQZ Dužine veza konstantne C H r = 2,04 bohr = 1,0795 Å C C R = 2,37 bohr = 1,2542 Å MOLPRO [Werner et al. 2012] Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 30 / 44
38 Parametri koji ulaze u model Presecanja pri planarnim geometrijama [Perić 2006] ρ 2 = uρ 1, Ukupno 5 ab initio računa! u = ɛ12 ± ɛ 2 12 ɛ1ɛ2 ɛ 2 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 31 / 44
39 Model Ab initio φ2 / deg B1/A A1/A B Model Ab initio ρ1/ρ2 10/ /7.5 10/5 10/2.5 10/0.001 E / Eh E / Eh A A /Bu A /Au ρ 2 / deg φ 2 / deg. Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 32 / 44
40 τ / rad 1 ρ τ φ Model Ab initio φ2 / deg ρ 2 / deg Model Ab initio φ2 / deg ρ 2 / deg. Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 33 / 44
41 τ / deg τ / deg Model Ab initio φ2 / deg ρ 2 / deg. φ 2 / deg. Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 34 / 44 0 Model ρ1/ρ2 10/ /7.5 10/5 10/2.5 10/0.001
42 Adibatic Adibatic Diabatic φ2 = 1 deg. (a) V Adibatic Adibatic Diabatic φ2 = 30 deg. (b) V + E / Eh E1 E2 V E / Eh E1 E2 V E E12 E / Eh ρ2 / deg. 5 Adibatic Adibatic Diabatic φ2 = 60 deg. 10 (c) V + E1 E2 V E / Eh ρ2 / deg. E1 5 Adibatic Adibatic Diabatic φ2 = 90 deg. (d) V + 10 V E E12 E ρ2 / deg ρ2 / deg Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 35 / 44
43 Vibronski spektar C 2 H + 2 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 36 / 44
44 Sadržaj izlaganja 1 Kvantno-hemijski problem Šredingerova jednačina Born-Openhajmerova aproksimacija Ab initio metode 2 Mali molekuli 3 Rener-Telerov efekat - opšta razmatranja 4 Model za tretiranje R-T efekta kod četvoroatomskih molekula Modelni hamiltonijan Elektronske bazne funkcije Elektronske bazne funkcije - dijabatizacija Mala odstupanaj od linearnosti Ugao τ S-O matrični elementi Vibronska baza Rezultati 5 Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 37 / 44
45 Model za molekule sa proizvoljnim brojem jezgara Ĥ = Ĥe + ˆT n + ĤSO ( = Ĥe 1 S 2 2 i=1 2 q 2 i + 1 q i q i + 1 q 2 i 2 φ 2 i ) ω i + A SO ˆLz Ŝ z S 2 S 2 τ = 1 ε ij ωi ω j q i q j sin ( ) φ i + φ j 2 arctan i=1 j=1 S 2 S 2 ε ij ωi ω j q i q j sin ( ) φ i + φ j i=1 j=1 Izbegnuta presecanja i pri planarnim i pri neplanarnim geometrijama Model testiran na primeru X 2 Π u stanja molekula C 5 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 38 / 44
46 Vibronski spektar C 5 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 39 / 44
47 Vibronski spektar C 5 (a) [Perić et al. 2008] (b) [Mitić et al. 2016] Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 40 / 44
48 Vibronski spektar C 5 (v1v2v3) (v1v2v3) (v1v2v3) (v1v2v3) 1500 (101) (101) (100) (100) 2100 E/ cm (001) (001) E/ cm (111) (111) K = (000) (000) 1800 ε = 0 A SO = 0 ε = 0 A SO 0 ε 0 A SO 0 ε 0 A SO = 0 ε = 0 A SO = 0 ε = 0 A SO = 0 ε = 0 A SO 0 ε 0 A SO 0 ε 0 A SO = 0 ε = 0 A SO = 0 Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 41 / 44
49 Zaključak Born-Openhajmerova aproksimacija Ab initio metode Mali molekuli Rener-Telerov efekat Model za tretiranje vibronske sprege kod četvoroatomskih molekula Ključan korak - dijabatizacija Kombinacija neke vrste J-T efekta i R-T efekta Model za molekule sa proizvoljnim brojem jezgara Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 42 / 44
50 Radovi M. Perić, S. Jerosimić, M. Mitić, M. Milovanović, R. Ranković, "Underlying theory of a model for the Renner Teller effect in tetra-atomic molecules: X 2 Π u electronic state of C 2 H + 2 ", The Journal of Chemical Physics, 142, (2015), doi: / M. Mitić, R. Ranković, M. Milovanović, S. Jerosimić, M. Perić, "Underlying theory of a model for the Renner-Teller effect in any-atomic linear molecules on example of the X 2 Π u electronic state of C 5 ", Chemical Physics, 464, 55 (2016), doi: /j.chemphys Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 43 / 44
51 Hvala na pažnji! Marko Mitić Ab initio proučavanje neadijabatskih efekata kod malih molekula 44 / 44
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