S-1
Supporting information for:
Photodissociation of LiFH and NaFH van der Waals complexes: A semiclassical trajectory study
Ahren W. Jasper, Michael D. Hack, Arindam Chakraborty, Donald G. Truhlar, and Piotr Piecuch
Date of preparation of the supporting information: June 13, 2001
Contents of the supporting information: Page 1. Functional form of the model LiFH potential energy matrix S-2
1.1. LiFH U11 surface S-2
1.2. LiFH U22 surface S-4
1.3. LiFH U12 coupling surface S-8 2. Functional form of the model NaFH-B potential energy matrix S-9
2.1. NaFH-B U12 coupling surface S-9
2.2. NaFH-B U11 and U22 surfaces S-10 S-2
1. Functional form of the model LiFH potential energy matrix
1.1. LiFH U11 surface The lowest-energy quasidiabatic surface in the Li + FH arrangement is labeled
U11 and is described by the sum of three diatomic terms,
e U11(R) S1(R) S2 (rHF ) S3 (R) DHF , (1) where the LiH diatomic S1(R) is a dressed state (i.e., it represents Li–H interactions in the close presence of F, not in isolation) that is taken as purely repulsive, the HF diatomic
S2(rHF) is an attractive Morse-like curve, and the LiF diatomic S3 (R) is another dressed state and is taken to be a shallow Morse curve. This simple functional form is appropriate to describe U11 which is relatively devoid of features, having only a small van der Waals well and otherwise being repulsive along the Li + HF coordinate. The use of dressed states for the Li–H and Li–F interactions is necessary because the LiH and LiF diatomic limits are energetically inaccessible for this quasidiabatic surface at the energies we are interested in studying.
The LiH diatomic curve for the U11 matrix element is a combination of two repulsive curves and is given by
a c a S1(R) S1 (rLiH ) [S1 (rLiH ) S1 (rLiH )]1(R) , (2)
a a a 0 S1 (rLiH ) D1 exp[1 (rLiH rLiH )], (3)
c c c 0 S1 (rLiH ) D1 exp[1 (rLiH rLiH )], (4) where 1(R) is a switching function given by
1 1 r1 1 1(R) tanh( ) , (5) 2 2 1
r1 = rLiH – rLiF + rHF. (6) S-3
The HF potential curve S2 was fit to experimental Rydberg-Klein-Rees (RKR) data for the HF molecule;1 the functional form is a Morse curve with a range parameter that depends on the HF bond length,
e S2 (rHF ) DHF X 2 (rHF )[X 2 (rHF ) 2], (7) where
e X 2 (rHF ) exp[2 (rHF )(rHF rHF )] , (8) and
0 0 2 2 (rHF ) b1 b2 (rHF rHF ) b3(rHF rHF ) . (9)
The LiF diatomic curve for the U11 matrix element is a combination of two shallow Morse curves and is given by
a c a S3(R) S3 (S3 S3 )3(R) , (10)
a a a a S3 (rLiF ) D3 X 3 (rLiF )[X 3 (rLiF ) 2], (11)
c c c c S3 (rLiF ) D3 X 3 (rLiF )[X 3 (rLiF ) 2] , (12)
a a a X 3 (rLiF ) exp[3 (rLiF rLiF )] , (13)
c c c X 3 (rLiF ) exp[3(rLiF rLiF )] , (14)
1 1 r3 3 3(R) tanh , (15) 2 2 3
r3 = rLiH – rLiF + rHF. (16)
The values of the parameters used in the U11 potential matrix element are given in Table I. S-4
1.2. LiFH U22 surface
2 The U22 potential energy matrix element was fit to a modified London-Eyring-
Polanyi-Sato (LEPS)3,4 form,
U 22 (R) J1(R) J 2 (R) J3 (R) 1 2 e * (17) W (R) Z(R) DHF ELi , 2 where 2 2 W (R) (K2 (R) K1(R)) (K1(R) K3(R)) . (18) 2 (K3 (R) K2 (R)) .
The functions J and K ( = 1, 2, and 3) are given by
1 J (R) (S (R) T (R)) , (19) 2 1 K (R) (S (R) T (R)) , (20) 2 and
Z(R) c2a exp[c2bW (R) c2c (rHF rLiH rLiF )] . (21) is a necessary to remove a cusp that would otherwise occur in Eq. (17) when W(R) goes through zero. The LEPS function Eq. (17) allows the global potential energy surface for the triatomic system to be expressible as a function of diatomic terms, specifically, three singlet terms (S1, S2, and S3) and three triplet terms (T1, T2, and T3).
The LiH singlet curve is a combination of two different curves, one of which is present at all geometries and one of which is turned on when the LiH diatom is interacting with the F atom,
a c S1(R) S1 (rLiF ) S1 (rLiF )1(R) , (22)
a 2 S1 (rLiH ) cS1,1X S1(rLiH ) cS1,2 X S1(rLiH ) , (23)
a e X S1(rLiH ) exp[ S1(rLiH rLiH )], (24) S-5
c c e S1 (rLiH ) exp[S1(rLiH rLiH )], (25)
1 1 r1 1 1(R) tanh( ) , (26) 2 2 1
r1 = rLiH – rLiF + rHF. (27)
The LiH triplet is a modified anti-Morse curve,
2 T1(R) cT1,1XT1(rLiH ) cT1,2 XT1(rLiH ) , (28)
e X T1(rLiH ) exp[T1(rLiH rLiH )] . (29)
The HF singlet S2, is similar to the form used for the U11 potential matrix element,
mod S2 (R) DS2 (R)X 2 (rHF )[X 2 (rHF ) 2] , (30) where
1 1 r mod e c 1 LiF S2 DS2 (R) DHF DS2 (1 cos ) tanh . (31) 2 2 2 S2 which allows the depth of the Morse curve to vary as a function of for short LiF distances. The function X2(rHF) is defined by Eq. (8).
The HF triplet potential is a linear combination of two repulsive curves,
1 T (R) T 0 (r ) T180 (r ) (1 cos ) , (32) 2 2 HF 2 HF 2
0 0 2 0 T2 (rHF ) cT 2,1XT 2 (rHF ) cT 2,2 XT 2 (rHF ) , (33)
180 180 2 180 T2 (rHF ) cT 2,3 X T 2 (rHF ) cT 2,4 X T 2 (rHF ) , (34)
0 0 0 X T 2 (rHF ) exp[T 2 (rHF )(rHF rT 2 )], (35)
180 180 0 X T 2 (rHF ) exp[T 2 (rHF )(rHF rT 2 )] . (36) S-6
The LiF potential curve has a long-range tail arising from the contribution of the Li+ + F– state. This long range character is not present in other parts of the potential surface, for example, in the Li + HF entrance channel. Therefore, it is necessary to allow the singlet describing the LiF interaction to change as a function of the other two internuclear distances. This is accomplished by defining the LiF singlet as
c a c S3(R) S3 (R) [S3 (R) S3 (R)] S3 (R) . (37)
a The asymptotic term, S3 (R) , is given by
a e * a a S3 (R) (DLiF ELi )X S3 (R)[X S3 (R) 2], (38) where
a e X S3 (R) exp[S3 (R)(rLiF rLiF )] , (39) and
r n (r ) LiF i LiF (R) (R) S3 f n , (40) r (R) LiF f
f (R) i (rLiF ) G(R)( ff i (rLiF )) , (41)
1 1 rHF G 1 G(R) tanh 1 cos , (42) 2 2 G 2
b r (r ) b 2 LiF i LiF 1 2 . (43) b3 b4rLiF
The somewhat complicated definition of the Morse range parameter defined by Eqs. (40)–(42) stems from practical considerations. We wish to use the LiFH surface in quantum mechanical calculations; these calculations are much easier to carry out when the diabatic coupling vanishes at all three dissociation limits. It was found, however, that when we caused the diabatic coupling to decay in the LiF + H channel, the analytical fit S-7 no longer qualitatively agreed with the ab initio data, owing to the diabatic coupling being involved in defining the shape of the excited adiabatic surface. In order to somewhat alleviate this shortcoming, the adjustments defined in Eqs. (37)–(43) above were introduced.
The form of the LiF diatomic used in the Li + HF channel is given by
c c * c c S3 (R) [DS3 (R) ELi ]X S3(R)[X S3 (R) 2] , (44)
1 Dc Dc,0 (r ) Dc,180 (r ) (1 cos) , (45) S3 S3 HF S3 HF 2
c c e X S3 (R) exp[ S3 (rLiF rLiF )] , (46)
1 c c,0 (r ) c,180 (r ) (1 cos ) . (47) S3 S3 HF S3 HF 2
The asymptotic and close forms of the LiF singlet are joined together with a switching function,
1 1 g(R) S3(R) tanh , (48) 2 2 S3 (R)
e g(R) (rHF rHF )cos S3(R) (S3 (R) rLiF )sin S3 (R) , (49)
1 (R) 0 180 (r ) (1 cos ) , (50) S3 S3 S3 HF 2
1 (R) 0 180 (r ) (1 cos ) , (51) S3 S3 S3 HF 2
1 (R) 0 180 (r ) (1 cos ) . (52) S3 S3 S3 HF 2
The LiF triplet potential is a modified anti-Morse curve,
1 T (R) T 0 (r ) T 180 (r ) (1 cos ) , 3 3 HF 3 HF 2 (53)
0 0 2 0 T3 (rLiF ) cT 3,1(rLiF )X T 3(rLiF ) cT 3,2 (rLiF )X T 3(rLiF ) , (54) S-8
180 180 2 180 T3 (rLiF ) cT 3,3(rLiF )X T 3 (rLiF ) cT 3.4 (rLiF )X T 3 (rLiF ) , (55)
0 0 e X T 3(rLiF ) exp[T 3(rLiF )(rLiF rLiF )], (56)
180 180 e X T 3 (rHF ) exp[T 3 (rHF )(rHF rLiF )]. (57)
The values of the parameters used in the U22 potential matrix element are given in Table II.
1.3. LiFH U12 coupling surface The off-diagonal electronic potential energy surface is described by
0 1 1 rHF 12 U12 (R) U12 (R) tanh , (58) 2 2 12 0 where U12 (R) is a physically motivated functional form that is caused to approach zero for large values of rHF where the excited-state potential energy surface becomes energetically inaccessible. Nonadiabatic transitions in this regions are unimportant and eliminating the off-diagonal coupling in these regions greatly reduces the expense of accurate quantum mechanical dynamics calculations. Of the three diatomic asymptotes, there is only electronic coupling in the LiH and LiF arrangements. The two electronic states that we treat arise from the Li(2s) and Li(2p) atomic orbitals, and the coupling between these vanishes as Li is separated from the HF diatom. It is therefore natural to treat the diabatic coupling in the full system as arising from diatomic terms in the LiF and LiH bond distances, 6 rLiH rLiH U LiH (rLiH ) g1 exp 6 1 , (59) r 0 r 0 LiH LiH 8 rLiF rLiF U LiF (rLiF ) g3 exp 8 1 . (60) r 0 r 0 LiF LiF S-9
These are functions which are zero when the diatomic distances are zero, increase in magnitude to a maximum of g at r = 0 (X = H, F), and then decrease in magnitude i LiX rLiX at a rate determined by the power on the pre-exponential factor. We expect the magnitude of each of these terms to be reduced by the approach of the remaining atom to the diatom, and we use the following form to accomplish this: 0 U (R) U LiH (rLiH )HF,1(rHF )LiF (rLiF ) 12 (61) U LiF (rLiF )HF,2 (rHF )LiH (rLiH ).
The reduction functions HF,1, LiF, HF,2, and LiH are given by
1 1 rHF HF,1 HF,1(rHF ) tanh , (62) 2 2 HF,1
1 1 rLiF LiF LiF (rLiF ) tanh , (63) 2 2 LiF
1 1 rHF HF,2 HF,2 (rHF ) tanh , (64) 2 2 HF,2
1 1 rLiH LiH LiH (rLiH ) tanh . (65) 2 2 LiH
The values of the parameters used in the U22 potential matrix element are given in Table III.
2. NaFH-B potential energy surface We have recently presented the details of an analytic two-state NaFH quasidiabatic energy matrix.5 The fit was successfully used to reproduce the experimentally observed photoabsorption spectrum of the Na...FH van der Waals complex,6 indicating that the fit is very accurate in the interaction region. The NaFH fit has two undesirable asymptotic features that we correct here: 1) the diabatic coupling
U12 does not vanish in the NaF + H asymptote, and 2) the dissociation energy of the HF diatomic in the Na + HF asymptote is 0.35 eV below the experimental value. In this Appendix, we present an improved NaFH fit (called NaFH-B) which has the desired asymptotic forms and is obtained by modifying the original NaFH fit (called NaFH-A). S-10
Specifically, the diabatic coupling is cut off for large HF internuclear bond distances, and an asymptotic correction function is added to each of the diagonal diabats.
2.1. NaFH U12 coupling surface The coupling between the two lowest energy states of the NaFH system does not vanish in the NaF + H asymptote. Asymptotic coupling complicates several aspects of our semiclassical and quantum mechanical simulations. We note that this coupling is not important in determining the dynamics of the NaFH system at reasonable energies due to the high energy of the upper electronic state in this asymptote. The functional form of the diabatic coupling surface for the improved NaFH-B fit, is given by B A U12 (R) U12 (R) 12 (rHF ) , (66) A 5 where U12 is the diabatic coupling function presented previously for the NaFH-A fit.
12 (RHF ) is a cutoff function in the HF bond direction and has the form
1 1 rHF r12 12 (rHF ) tanh , (67) 2 2 12 where r12 = 3.5 a0 and 12 = 0.5 a0. The cutoff function does not significantly change the value of the diabatic coupling near the line of avoided crossings.
2.2. NaFH U11 and U22 surfaces
B B The diagonal diabats U11 and U 22 for NaFH-B are obtained by adding correction functions to the original NaFH diabats, B A U jj (R) U jj (R) F j (R) , (68)
A 5 where j = 1 or 2 and U jj are the surfaces presented previously. We require that the functions Fj correct the asymptotic diatomic energy curves, minimize the change to the interaction region, and smoothly introduce the correction into the surfaces. A simple
3 choice for Fj that satisfies these three requirements is the difference of two LEPS functions S-11
B A F j (R) L j (R) L j (R) , (69)
A where L j is a LEPS function with asymptotic forms that are exactly equal to the
B asymptotic forms of the NaFH-A fit, and L j is a LEPS function with the desired (i.e., experimental) asymptotic forms. The LEPS function can be written x x L j (R) Q, j (r ) 1/ 2 (70) 2 J x (r ) 1 J x (r )J x (r ) D x , , j 2 , j , j e,HF where
Q x (r ) 1 [S x (r ) T x (r )], (71) , j 2 , j , j
J x (r ) 1 [S x (r ) T x (r )], (72) , j 2 , j , j x x = ‘A’ or ‘B’, the summations run over = HF, NaF, or NaH, and De,HF sets the zero of energy. A The functional forms and parameters used for the singlets S,i are equal to the
5 singlets that appear in the NaFH-A fit. The U11 diabat in the NaF + H and NaH + F
B asymptotes and the U22 diabat in the NaH + F asymptote are not corrected, i.e., S,i =
A S,i for (,i) = (NaH,1), (NaH,2), and (NaF,1). (Note that because the diabatic coupling is cut off in the HF bond direction, the lower diabatic surface in the NaF + H asymptote
(U22) must be re-fit, such that the lower adiabat is left unchanged asymptotically.) The
B B B B remaining singlets required for Li are SHF,1, SHF,2 , and SNaF,2 . These singlets have the following functional forms:
B B B B 2 B SHF,1(rHF ) De,HF 1 exp[ HF (rHF )(rHF re,HF )] De,HF , (73)
S B (r ) 1 1 B (r ) S B (r ) E B , (74) HF,2 HF 2 HF,2 HF HF,1 HF s p S-12
B B B B 2 B SNaF,2 (rNaF ) De,NaF 1 exp[ NaF (rNaF )(rNaF re,NaF )] De,NaF , (75) where B B B B B B 2 HF (rHF ) c1,HF c2,HF (rHF c4,HF ) c3,HF (rHF c4,HF ) , (76)
B c4,NaF r c B NaF 2,NaF B c3,NaF B (r ) cB NaF NaF 1,NaF B , (77) c4,NaF r c B NaF 1,NaF B c3,NaF
B B B B HF,2 (rHF ) tanh HF[SHF,1(rHF ) Es p ]. (78) The parameters for these functions are given in Table IV. The HF singlet fit is based on experimental RKR data.1 Experimental data for the NaF curve7 is available up to ~0.5 eV. During the fitting procedure for the NaF curve, the dissociation energy was not allowed to vary, and the electronic structure data used in the original fit were included above 0.5 eV to make sure that the corrected NaF curve was qualitatively correct above 0.5 eV. The triplet functions in Eqs. (71) and (72) do not affect the asymptotic forms of the LEPS equations, but they are important in determining the character of the interaction region. In order to provide the correction functions Fj with flexibility, we introduced
x adjustable parameters into the triplet functions. For F1, one adjustable parameter w,1 was introduced which weights the entire triplet by a constant, i.e., x x T,1(r ) w,1t,1(r ) , (79)
x where the set of w,1 are listed in Table V. (Note that weighting the triplet functions by a single parameter is exactly equivalent to introducing Sato parameters2 in the LEPS
x equation.) The form of tHF,1 is an anti-Morse curve with a constant range parameter,
t (r ) 1 Dt 2exp[ t (r r t )] exp[2 t (r r t )] HF,1 HF 2 e,HF e,HF HF e,HF e,HF HF e,HF , (80) S-13
t t t where De,HF , re,HF , and e,HF are listed in Table V. The functional forms of the
5 collinear geometry triplets that appear in the previous fit were used for tNaF,1 and tNaH,1 .
x For F2, the TNaH,2 triplets were not parameterized and were set equal to the
5 x x collinear triplets that appear previously. The functional forms of the THF,2 and TNaF,2 triplets are anti-Morse curves with two adjustable parameters, i.e.,
T x (r ) 1 Dt 2 w1, x exp[ t (r r t )] w2, x exp[2 t (r r t )] (81) ,2 2 e, ,2 e, ,2 e, k,x where the set of w,2 (k = 1,2) and the anti-Morse curve parameters are given in Table V. x 1,x 2,x The values of the adjustable parameters w,1, w,2 , and w,2 were obtained
8 using a genetic algorithm such that the magnitudes of F1 and F2 were minimized in the interaction region. From Table VI, we see that the correction functions do not significantly change the properties of the NaFH surface at the minima of the van der Waals and exciplex wells. Note that the mean unsigned deviation from the electronic structure data for the original fit is 0.02–0.03 eV. S-14
TABLE I. Values of the parameters used in the LiFH U11 potential energy function.
Parameter Value Parameter Value
a 0 D1 12.90323 eV rHF 2.1042 a0 c a D1 7.09677 eV D3 0.14286 eV a 1 c 1 1.73333 a 0 D3 0.25 eV c 1 a 1 1 1.36 a 0 3 1.22667 a 0 0 c 1 rLiH 1.53333 a0 3 1.92857 a 0 1.07143 a a 3.66667 a 1 0 rLiF 0 0.6 a c 2.48571 a 1 0 rLiF 0 1 0.4 3 0.4 a0 1 b1 1.1622 a 0 3 0.89333 a0 2 b2 –0.025647 a 0 3 0.7333 b 0.059062 3 e 1.733 a 3 a 0 rHF 0 e DHF 6.122 eV S-15
TABLE II. Values of the parameters used in the LiFH U22 potential energy function.
Parameter Value Parameter Value c 3.5 eV e 2.9553 a 2a rLiF 0 –2 1 c2b 0.27362 eV ff 0.25333 a0 c 0.15 1 3.87097 11/ n 2c a 0 a 0 cS1,1 4.32258 eV n 8 cS1,2 7.06452 eV G 1.41935 a0 a 1 S1 1.36 a 0 G 3.77419 a0 c 1 1 S1 0.90667 a 0 b1 0.064076 a 0 e rLiH 1.2 a0 b2 103.57 2 1 0.72 a0 b3 4.6498 a 0 1 0.5 a0 b4 7.0489 1.0 e 1.733 a 1 rHF 0 c 1.64516 eV c,0 5.25806 eV T1,1 DS3 c 10.38710 eV c,180 2.58065 eV T1,2 DS3 2.10667 1 c,0 0.91613 1 T1 a 0 S3 a0 e 6.122 eV c,180 0.95484 1 DHF S3 a0 c 0 DS2 0.26 eV S3 0.51613 a0 2.38095 a 180 1.33333 a S2 0 S3 0 0.5 a 0 0.09333 rad. S2 0 S3 c 1.26667 eV 180 0.18 rad. T2,1 S3 c 16.06667 eV 0 0.49032 a T2,2 S3 0 c 11.61290 eV 180 0.31613 a T2,3 S3 0 cT2,4 15.51613 eV cT3,1 0.38710 eV 0 1 T 2 2.10667 a 0 cT3,2 1.80645 eV 180 1 T 2 1.73333 a 0 cT3,3 0.51613 eV e DLiF 5.909 eV cT3,4 0.51613 eV E * 1.848 eV 0 0.89333 1 Li T3 a0 180 1 T3 0.80 a 0 S-16
TABLE III. Values of the parameters used in the LiFH U12 potential energy function.
Parameter Value Parameter Value
12 3.87097 a0 HF,2 1.45161 a0 12 0.45806 a0 HF,1 1.75806 a0 g1 1.27742 eV HF,2 0.98387 a0 g3 0.48 eV LiF 2.17742 a0 0 rLiH 2.5873 a0 LiF 0.56452 a0 0 rLiF 3.47619 a0 LiH 4.27097 a0 HF,1 1.15484 a0 LiH 2.5 a0 S-17
TABLE IV: Singlet correction function parameters for NaFH-B.
Parameter Value Parameter Value B B De,HF 6.122 eV De,NaF 4.94 eV B B re,HF 1.733 a0 re,NaF 3.6395 a0 B 1 B 1 c1,HF 1.1622 a 0 c1,NaF 0.32453 a 0 B 2 B c2,HF –0.025647 a 0 c2,NaF 1.5102 B 3 B c3,HF 0.059062 a 0 c3,NaF 3.0938 a 0 B B c4,HF 2.1042 a 0 c4,NaF 1.7107 B 1 Es p 2.097338 eV HF 15. a 0 S-18
TABLE V: Triplet correction function parameters for NaFH-B. Parameter Value Parameter Value t B De,HF 5.77096 eV wNaH,1 3.5875 t A re,HF 1.733 a0 wNaH,1 3.5875 t 1 1,B e,HF 1.2669 a 0 wHF,2 5.54 t 2,B De,NaF 4.49 eV wHF,2 4.15 t 1,A re,NaF 3.6395 a0 wHF,2 5.125 t 1 2,A e,NaF 0.696141 a 0 wHF,2 5.125 B 1,B wHF,1 2.913 wNaF,2 9.01 A 2,B wHF,1 3.1183 wNaF,2 1.92 B 1,A wNaF,1 2.0626 wNaF,2 8.8 A 2,A wNaF,1 2.0528 wNaF,2 1.62 S-19
TABLE VI: Location and energy of the van der Waals and exciplex wells for the previous (A) and improved (B) NaFH fits.
vdW well exciplex NaFH-A NaFH-B NaFH-A NaFH-B rHF (a0) 1.738 1.739 1.795 1.827 RNaF (a0) 4.671 4.684 4.304 4.167 (deg) 117.8 117.8 115.1 116.9 V1 (eV) -0.0761 -0.0737 -0.0312 0.0172 V2 (eV) 1.624 1.684 1.571 1.600 V2-V1 (eV) 1.700 1.758 1.602 1.583 S-20
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