Born-Oppenheimer Approximation and Nonadiabatic Effects. Hans Lischka University of Vienna

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1 Bo-Oppeheie Appoxiatio ad Noadiabatic Effects Has Lischa Uivesity of Viea

2 Typical situatio. Fac-Codo excitatio fo the iiu of the goud state. Covetioal dyaics possibly M* ad TS 3. Coical itesectio fuel apid adiatioless tasitio to the goud state 4. Covetioal dyaics i the goud state leadig to P ad/o P F. Beadi M. Olivucci ad M. A. obb Che. Soc. ev

3 Coical Itesectios The itesectio space ca epeset a lage stuctual vaiatio seach fo selected poits e.g. iia o the cossig sea MXS eactio paths dyaics Gaavelli Beadi Olivucci Veve Klei Celai ad obb Faaday Discuss Ceba Beadi Olivucci ad Gaavelli JACS

4 Coical Itesectios Degees of Feedo J. v. Neua ud E. Wige Physi. Zeitschift XXX How ay degees of feedo ae ecessay i ode to achieve degeeacy of two states? Coside a geeal eal syetic atix ad cout the ube of degees of feedo i case of o degeeacy ad with oe degeeacy: Two geoetical paaetes eed to be used i ode to achieve a degeeacy itesectio. 4

5 No-cossig ule Assuptio: Eigefuctios fo all but two states ae ow: ϕ ϕ Ψ 3 Ψ. Ψ c ϕ c ϕ H E H c H H E c Coditio fo degeeacy: H - H 0 H 0 No-cossig ule: Fo a diatoic olecule AB oly oe paaete AB is available. Siultaeous fulfillet of both equatios is ot possible except oe coditio H 0 is fulfilled autoatically by syety. 5 0

6 6 G. Hezbeg ud H.C. Loguett-Higgis Disc. Faaday Soc Lieaized odel close to a coical itesectio 0 c c E x W ly ly E x W y l x x W E ± Double coe

7 Pola coodiates: x cosθ ly si θ x l y Fo the lowe oot: E W x - i.e. c c si θ cosθ cosθ si θ si θ c cos θ c cosθ ta si θ θ Whe Ψ eal the c θ si c Movig oce aoud the oigi θ: 0 π chages the sig of the wave fuctio Bey phase M.V. Bey Poc.. Soc. Lodo A θ cos 0 7

8 Bo-Oppeheie Appoxiatio ad Noadiabatic Couplig Total Hailtoia H Tel TN V Electoic Hailtoia fo fixed uclei H el T el V Electoic Schödige equatio el el H E 0 Total Schödige equatio H E Ψ 0 Expasio of the total wave fuctio Ψ M. Bo K. Huag Dyaical Theoy of Cystal Lattices Claedo Oxfod 956 Appedix VIII 8

9 9 Δ N M T Kietic eegy of the uclei Δ Δ M T N Multiplicatio fo the left with ad itegatio ove electoic coodiates gives: Δ Δ M T

10 0 Fial esult P C E E T el N B P A M C P A P B

11 ca be chose as eal. The 0 d i A The Schödige equatio ca be witte as follows: N P C E U T 0 el B M E U Neglectig the couplig eleets C leads to the adiabatic appoxiatio with 0 E U T N

12 Divisio of iteal coodiates x ad x descibe the coical itesectio x 3 x descibe the eaiig coodiates The plae x x defies the bachig space degeeacy is lifted except at oigi Alog the hypelie i the --diesioal space itesectio space the eegies of the two states ae degeeate x ad x ae always pepedicula to the itesectio space

13 Noadiabatic Couplig at CI level f x Ψ J / xψi f x ci x csf f f x f ci x I J E E J x I c H c CSF Fo E I E J 0 the f ci te will doiate x epesets a uclea coodiate B. H. Legsfield III P. Saxe ad D.. Yaoy J. Phys. Che

14 Itesectio-adapted Coodiates g h E E C t J I x H x C J I I ode to descibe the coe at we eed the gadiets ad the oadiabatic couplig vecto 4

15 s Aalytic epesetatio of the Coe ˆ s G J G I g G J G x g / g g g ; y h / h h Topogaphic paaetes ˆ s ˆ ˆ I h g h / x s x y s y Δgh d g gh d gh h E Δ s x x s y y ± d gh y / gh x y x. 5

16 Miiu o the Cossig Sea Pojected gadiet techique: M. J. Beapa M. A. obb ad H. B. Schlegel Che. Phys. Lett Lagage ultiplie appoach: M.. Maaa ad D.. Yaoy J. Che. Phys