1 Adiabatic and diabatic representations

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1 1 Adiabatic ad diabatic represetatios 1.1 Bor-Oppeheimer approximatio The time-idepedet Schrödiger equatio for both electroic ad uclear degrees of freedom is Ĥ Ψ(r, R) = E Ψ(r, R), (1) where the full molecular Hamiltoia reads (i atomic uits, where h = e = m e = 1/(4πɛ 0 ) = 1) Ĥ = ˆTe + ˆVe + ˆTN + ˆVN + ˆVeN (2) = e,i + 1 r i j>i i r j 1 2 N,i + Z i Z j 2 M i i R j>i i R j Z j r ij i R j. (3) We use {r, e } to refer to electro coordiates ad mometa ad {R, N } for uclear coordiates ad mometa. Z i are the uclear charges ad M i are the masses of the uclei i. (both beig expressed as multiples of the electro charge ad electro mass, respectively). Equatio (2) provides a shorthad otatio for the five terms i Eq. (3), amely electro kietic eergy, electro-electro repulsio, uclear kietic eergy, uclear-uclear repulsio ad electro-uclear attractio. By defiig the electroic Hamiltoia Ĥ e = ˆT e + ˆV e + ˆV en, (4) we get the eige-eergies E el (R) from the electroic Schrödiger equatio: We ow isert the wavefuctio described below ito Eq. (1), we get Ĥ e Ψ(r, R) = E el (R) Ψ(r, R). (5) Ψ(r, R) = ψ(r; R) χ(r), (6) Ĥ(r, R) ψ(r; R) χ(r) = ( ˆT N + E el (R) + ˆV N ) ψ(r; R) χ(r) (7) = E ψ(r; R)χ(R). (8) The Laplacia i the uclear kietic eergy operator of the full wavefuctio ca be rewritte as: 2 N,iΨ(r, R) = N,i [ N,i ψ(r; R) χ(r)] = N,i [χ(r) N,i ψ(r; R) + ψ(r; R) N,i χ(r)] = N,i χ(r) N,i ψ(r; R) + χ(r) 2 N,i ψ(r; R) + N,i ψ(r; R) N,i χ(r) + ψ(r; R) 2 N,i χ(r) = ψ(r; R) 2 N,i χ(r) + 2 N,i ψ(r; R) N,i χ(r) +χ(r) 2 N,i ψ(r; R). (9) 1

2 As a result, Eq. (7) ca be writte as: Ĥ(r, R) ψ(r; R) χ(r) = (E el (R) + ˆV N ) ψ(r; R) χ(r) 1 [ψ(r; R) 2 N,i χ(r) 2 M i i +2 N,i ψ(r; R) N,i χ(r) + 2 N,i ψ(r; R) χ(r)]. (10) These last three terms i Eq. (10) arise from the uclear kietic eergy operator actig o the full electroic-uclear wavefuctio. While the first of these three terms represets the uclear kietic eergy, the other two terms cotai derivatives of the electroic wave fuctio with respect to the uclear coordiates. As these derivatives carry the iverse masses of the uclei as prefactors their relative weight is usually quite small ad we ca eglect them. If we eglect these terms (which is kow as the Bor-Oppeheimer approximatio), we ed up with a time-idepedet Schrödiger equatio for the uclei: (ote that the ˆT N below is the operator actig oly o the uclear coordiates) Ĥ(r, R) χ(r) = ( ˆT N + E el (R) + ˆV N (R)) χ(r) = E χ(r). (11) However, the derivative terms ca become large whe potetial eergy surfaces cross. I such regios of the potetial, small chages i the uclear geometry ca cause sigificat chages of the character of the electroic wave fuctio. I a time-domai perspective, this meas that the electroic dyamics evolve o a time scale comparable to the uclear dyamics as the eergies of the electroic states approach each other i the crossig regio of potetial eergy surfaces. Thus the electroic level spacigs become comparable to typical vibratioal eergy-level spacigs. 1.2 Adiabatic represetatio We ow substitute Ψ(r; R) = ψ (r; R) χ (R), (12) i the time-idepedet Schrödiger equatio ad project oto ψ m (r; R) : ψ m (r; R) Ĥ(r; R) withi which ψ (r; R) χ (R) = = ψ m (r; R) Ĥ(r; R) ψ (r; R) }{{} χ (R) H m (R) χ (R), (13) H m (R) χ (R) = ( [ ˆT N + E el (R) + ˆV N (R)]δ m + ) 1 (2 T (1) m,i 2 M (R) N,i + T (2) m,i ) χ (R), i i = E χ m (R) with the idetities: (14) T (2) m,i ψ m 2 N,iψ, T (1) m,i ψ m N,i ψ. (15) 2

3 Here the braket otatio refers to itegratio over the electroic coordiates oly. The differetiatio works similar as i Eq. (9). We ow simplify the vectors T m (1) by evaluatig ψ m ψ ad takig the electroic wave fuctios to be orthoormal at all uclear geometries, i.e. ψ m ψ = δ m. Thus N,i ψ m ψ = N,i ψ m ψ + ψ m N,i ψ = 0, (16) N,i ψ m ψ = ψ m N,i ψ. (17) If we choose real-valued electroic wavefuctios, it follows from the above equatio that T (1),i = 0. If we ow keep the diagoal elemets of Eq. (14) oly, we ed up with a expressio similar to Eq. (11) that cotais a additioal term T (2) ( ˆT N + E el (R) + ˆV N (R) + i,i : ) 1 T (2),i 2 M i χ (R) = E χ (R). (18) The derivative terms T (2),i give a additioal cotributio to the eergy. However, they do ot give rise to a couplig betwee differet electroic states, which is cotaied i the off-diagoal elemets i Eq. (14). The difficulty to calculate these off-diagoal elemets is the mai reaso to oly employ the adiabatic represetatio wheever the off-diagoal elemets are sufficietly small. 1.3 Diabatic represetatio We isert the followig asatz for the wavefuctio Ψ(r, R) = ψ (r; R 0 ) χ (0) (R). (19) i the time-idepedet Schrödiger equatio (usig Eq. 3 o the Exercise sheet): ψ (r; R 0 ) ( ˆT N + ˆV N )χ (0) (R) + χ (0) (R) Ĥe ψ (r; R 0 ) = E ψ (r; R 0 ) χ (0) (R). (20) Note that the kietic eergy operator of the uclei ˆTN is ot actig o the electroic wavefuctios ψ (r; R 0 ) aymore, as those are defied for a fixed set of uclear coordiates R 0. We ow project oto ψ m (r; R 0 ) : where [( ˆT N + ˆV N )δ m + H e,m ]χ (0) (R) = E χ (0) m (R), (21) H e,m = ψ m (r; R 0 ) Ĥe(r; R) ψ (r; R 0 ), (22) with Ĥe beig the electroic Hamiltoia defied i Eq. (4). As the electroic wavefuctios are defied for a fixed uclear geometry R 0 they are eigefuctios of the electroic Hamiltoia at a fixed uclear geometry, i.e. Ĥ e (r; R 0 ) ψ (r; R 0 ) = E el (R 0 ) ψ (r; R 0 ). (23) 3

4 Therefore we rewrite the electroic Hamiltoia usig ˆV ˆV e + ˆV en where we have used Ĥ e (r; R) = ˆT e (r) + ˆV (r; R) = Ĥe(r; R 0 ) ˆV (r; R 0 ) + ˆV (r; R), (24) Ĥ e (r; R 0 ) ˆV (r; R 0 ) = ˆT e (r) + ˆV e (r) + ˆV en (r; R 0 ) [ ˆV e (r) + ˆV en (r; R 0 )] = }{{} ˆT e (r), (25) 0 ad ote that ˆT e (r; R 0 ) = ˆT e (r), ˆV e (r; R 0 ) = ˆV e (r). Now we ca simplify the matrix elemets H e,m : We use the defiitio H e,m = ψ m (r; R 0 ) Ĥe(r; R 0 ) ˆV (r; R 0 ) + ˆV (r; R) ψ (r; R 0 ) = E el (R 0 ) δ m + ψ m (r; R 0 ) ˆV (r, R) ˆV (r, R 0 ) ψ (r; R 0 ). (26) U m = ψ m (r; R 0 ) ˆV (r, R) ˆV (r, R 0 ) ψ (r; R 0 ), (27) as give i the task ad isert Eq. (26) i Eq. (21), which yields: [ { ˆT N + ˆV ] N + E el (R 0 )}δ m + U m χ (0) = E χ (0) m, (28) or i the other otatio (Eq i the book of Taor), where ˆT = ˆT N + ˆV N + E el (R 0 ) [ ] ˆT δ m + U m χ (0) = E χ (0) m. (29) This expressio is ofte refered to as the time-idepedet Schrödiger equatio i a diabatic basis. More specifically, our choice of the basis i Eq. (19) is ofte called a crude adiabatic basis. Compare Eq. (28) with the adiabatic form of the time-idepedet Schrödiger equatio i Eq. (14). Both describe the same physics, but istead of derivative coupligs (T m (1) ad T m) (2) betwee the potetial eergy surfaces of the differet electroic states, we ow deal with coordiate coupligs U m. The diagoal elemets of U m are referred to as diabatic potetials, where the off-diagoal elemets itroduce the coupligs betwee the diabatic surfaces. The diagoalizatio of U the gives the adiabatic potetial eergy surfaces. The drawback of this diabatic basis is that the electroic wave fuctio has to be calculated at may differet geometries R 0. This results i a much larger umber of basis fuctios required for a accurate calculatio as compared to the adiabatic basis (Eq. (12)). Note: I may textbooks the time-idepedet Schrödiger equatio i a crude adiabatic basis is writte as [ { ˆT ] N + E el (R 0 )}δ m + U m χ (0) = E χ (0) m, (30) where U m = ψ m (r; R 0 ) ˆV (r, R) ˆV (r, R 0 ) ψ (r; R 0 ), (31) with ˆV ˆV e + ˆV en + ˆV N, i.e. the uclear-uclear repulsio is cotaied i ˆV. This has the advatage that all Coulomb potetial terms are grouped together i the U m elemets, which clarifies its role as represetatio of a potetial eergy surface. 4

5 2 Avoided crossigs of potetial eergy curves of diatomic molecules We cosider two potetial eergy curves of electroic states belogig to the same symmetry of a diatomic molecule. We determie the eigevalues of the potetial matrix ( ) U11 U U = 12. (32) U 21 U 22 By diagoalizig this matrix, we arrive to: (U 11 V ) (U 22 V ) U 12 2 = 0. (33) The eigevalues represet the adiabatic potetial eergy curves takig the couplig betwee the two diabatic curves due to the off-diagoal elemets i U ito accout are: V ± = U 11 + U 22 2 The separatio betwee the two curves is cosequetly ± 1 2 (U11 U 22 ) U (34) V = V + V = (U 11 U 22 ) U 12 2, (35) which has to be zero at the crossig poit of the two curves. Both terms uder the square root are larger tha or equal to zero because the diagoal elemets U 11 ad U 22 are real-valued. Thus the separatio betwee the two curves ca oly be zero whe both terms uder the square root are simultaeously equal to zero. Hece U 11 (R) = U 22 (R), (36) U 12 (R) = 0. (37) As we are dealig with a diatomic molecule, we have oly oe degree of freedom, amely the iteruclear separatio R. Strictly speakig, oe would have to evaluate Eq. 31 to obtai U ij (R) the the solutio of the above costraits will be apparet. However, sice solutios of Eq. 31 are rather difficult, we ofte assume U ij (R) arbitrary fuctios. This meas we have two liear 1D equatios that have to be satisfied by oe parameter oly, which is ulikely. If the two states have differet symmetries, the off-diagoal couplig elemets are always equal to zero. Therefore, i oe dimesio the potetial eergy curves associated with states of the same symmetry geerally do ot cross. As a example V + ad V from Eq. (34) are plotted i Fig. 1 for o-zero off-diagoal elemets U 12 (R) = U 21 (R). Clearly, the two potetial eergy curves do ot cross. This is a geeral pheomeo ad ofte referred to as avoided crossig, which meas that potetial eergy curves of the same electroic symmetry of diatomic molecules do ot cross, uless the eergies are degeerate ad the couplig betwee both curves is zero (which is uphysical, as couplig betwee two potetial eergy curves geerally occurs if they approach each other). If we have more tha oe free parameter, the coditios (36) ad (37) for curve crossigs ca be fulfilled simultaeously. I two dimesios, there is geerally a sigle or a discrete set of poits where the curves itersect, the so-called coical itersectio or seams of coical itersectios. For N > 2 dimesios the itersectio becomes N 2 dimesioal. The simplest case of a coical itersectio for a two-dimesioal system is ivestigated i the ext task. 5

6 V Advaced kietics Solutio 6 April 8, V+ V- U11 U22 U U11 = A + R U22 = A - R U12 = R/4 + 2 A = R Figure 1: Illustratio of a avoided crossig. The adiabatic potetials are plotted accordig to Eq. (34) as a fuctio of the iteruclear separatio R. The parameters used i Eq. (34) are give i the figure. All parameters are dimesioless. 6

7 3 Coical itersectios of potetial eergy surfaces of polyatomic molecules We cosider a model potetial matrix ( ) x + h R1 l R U = 2. (38) l R 2 x h R 1 This matrix represets a molecule with two idepedet coordiates R 1 ad R 2. x, l ad h are costats. All matrix elemets vary liearly with either R 1 or R 2, which is a reasoable assumptio if we are oly iterested i the behavior of the potetial eergy surfaces ear the itersectio (the first-order Taylor expasio is applied). The eigevalues of Eq. (38) ca be foud similarly as for Eq. (32): ad thus are (x + h R 1 V ) (x h R 1 V ) (l R 2 ) 2 = 0, (39) V ± (R 1, R 2 ) = x ± (h R 1 ) 2 + (l R 2 ) 2. (40) The correspodig potetial eergy surfaces as fuctio of R 1 ad R 2 are plotted i Fig. 2 for the parameters specified i the figure. Both surfaces are coes that itersect at their vertex. Thus the itersectio poit is referred to as coical itersectio. As already discussed i the previous task this shows that for two (ad more) dimesios, potetial eergy surfaces geerally cross cotrary to the case of diatomic molecules, where avoided crossigs occur. Coical itersectios betwee a electroically excited ad the groud state of a molecule play a importat role i radiatioless deexcitatio mechaisms. For istace, DNA is stable uder UV-radiatio due to the rapid radiatioless populatio trasfer from excited states to the groud state through coical itersectios. 7

8 Figure 2: Illustratio of a coical itersectio. The adiabatic potetials are plotted accordig to Eq. (40) as a fuctio of the iteruclear separatio R 1 ad R 2. The parameters used i Eq. (40) are give i the figure. All parameters are dimesioless. 8