15846
J. Phys. Chem. 1995, 99, 15846-15857
Potential Energy Surfaces for the Low-Lying 2A” States of HO2 via a Multivalued Double Many-Body Expansion: Modeling Basic Attributes A. J. C. Varandas* and A. I. Voronin Departamento de Quimica, Uniuersidade de Coimbra, 3049 Coimbra Codex, Portugal Received: May 5, 1995; In Final Form: August 3, 1 9 9 9
A previously reported extension of the double many-body expansion formalism has been used to model the main features of the ground and excited :!Affstates of HO:!. The approach uses accurate two-body extended Hartree-Fock approximate correlation energy curves, while introducing three-body effects through the dressing of the diatomic states that arise in the diatomics-in-molecules treatment of the potential matrix. Although minimal atomic basis sets have been employed, a correct description of the complete manifold of states has been obtained for the title system. Thus, the potential energy surfaces of the present work may be used for dynamics calculations, which would be valuable to test their reliability.
1. Introduction
the ground state data of the triatomic state, the potential energy surfaces obtained for the triatomic excited states are in error The hydroperoxyl radical is known to be the single most particularly at the atom-diatom dissociation limits. Moreover, important species in combustion processes.’ This radical is also the ground (and excited) state diatomic potentials are based on an intermediate for many chemical reactions such as those simple functional forms, and hence, as the authors have pointed occurring in atmospheric chemistry, destruction of ozone, and out,* they cannot describe accurately vibrational excited states photochemical air pollution. Knowledge of its potential energy beyond the first two or three lowest ones. surfaces are fundamental for studying the dynamics and kinetics Recently, we have suggestedz7 an extension of the DMBE of the elementary reactions in which it is involved (e.g., refs method to multivalued potential energy surfaces which uses also 2-5). the traditional diatomics-in-molecules formalism to set up the Many analytical representations of the ground state potential structure of the potential matrix. The novelty of our approach energy surface for the title system have been proposed, using is that the diatomic states are dressed with n-body energy terms, both empirical and semiempirical approaches (refs 6-9 and while spin-recoupling and orientational effects are treated as in references therein). Of them, the most popular in recent years the DIM formalism. The only numerical applicationz7 of the has been DMBE IV.6 This was obtained from the semiempirical DMBE method in this form has been to NOz, with the potential double many-body expansionI0,l1(DMBE) method on the basis energy surface for the corresponding ground electronic state of the assumption that the molecular potential is single-valued being found in good agreement with available spectroscopic and and using for the calibration procedure accurate ab i n i t i ~ ’ ~ - ’ ~ ab initio data. In this work, we use it to describe the basic and s p e c t r o s ~ o p i c data. ~ ~ ~ ’ Although ~ the results of dynamics topological features of the A” potential energy surfaces for the calculations have shown that DMBE IV accounts for most title system. features of the r e a c t i ~ n ~ - H(%) ~ ~ ’ ~ -02(3Z,) ~~ OH(211) The paper is organized as follows. In section 2, we describe O(3P) and its r e v e r ~ e , ~ f ’it, ~cannot ~ - ~ ~obviously describe the the general structure of the DIM potential matrix for HOZ(~A”). conical intersections occurring at some regions of the molecular Section 3 provides a description of the involved diatomic configuration space. In fact, it is well established from the potential energy curves within the DMBE approach. The results available ab initio calculations13~’4 that such intersections exist including the final multivalued DMBE potential energy surface for CzVand D,, confgurations. A correct representation of these are presented in section 4. The major conclusions are in section potential surface crossings may therefore be crucial for accurate 5. molecular dynamics calculations, while presenting a challenge in the modeling of global potential energy surfaces. To our 2. Structure of the DIM Potential Matrix knowledge, the only reported24 H02 potential energy surface which shows these topological features is that of Kendrick and The Wigner-Witmer rules applied to the title system give Pack.8 These authors have used the diatomics-in-moleculeszs~26 (DIM) formalism, while treating many of the potential energy curves for the excited states of the diatomic fragments as adjustable functions to fit existing ab initio data. Using a total of 94 parameters, they have been able to provide a least-squares fit to the most recent ab initio energies of Walch et with an average root mean squared error which is probably within Because the ground state hydroperoxyl radical (2A’f)correlates the accuracy of the calculated points at many regions of the with the ground states of both the reactants and products, the HO:! potential energy surface. However, because they have used corresponding potential energy surface is usually considered to the diatomic excited state curves as disposable functions to fit be single-valued. However, recent ab initio calculations have @Abstract published in Advance ACS Absrrucrs, September 15, 1995. shown that intersections may occur at some regions of the
+
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0022-365419512099-15846$09.0010
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0 1995 American Chemical Society
Low-Lying 2A" States of HO2
J. Phys. Chem., Vol. 99, No. 43, 1995 15847
molecule configuration space, and indeed two such conical intersections have been localized,I2 one along the path for C2" insertion of H into 0 2 and the other for H approaching collinearly to 0 2 . Note that in the work of Walch et al.I3 the barriers appearing prior to the collinear and C2, saddle points should truly correspond to 2x-/211and 2B1/2A2crossings. The characterization of such attributes as barriers may therefore be somewhat misleading, since their rounded shape is due to the single-valuedpolynomial representation used in that work. Note also that such crossings are dictated by symmetry arguments and hence become avoided intersections at less symmetrical geometries (Cs). Our aim in this work is therefore to provide a realistic description of the relevant set of potential energy surfaces for the low-lying 2A'r states of the HO2 radical. Such treatment requires the inclusion of both covalent and ionic states, as suggested by the following evidence. First, ab initio calculations by Metz and Lievin28have shown that the coefficient of the ionic contribution to the zeroth-order MC SCF wave function for H02(2A") as a function of the OH bond distance has a dominating role near the equilibrium geometry of the triatomic molecule. Indeed, similar evidence has been provided by ab initio calculation^^^.^^ for the OH radical, which have shown that at small distances the ionic configuration dominates over the covalent one. For example, in the case of the 211state, the configuration changes from lc?2c?3dl&~r~ (ionic) to 1c?2c?3a4a141ny (covalent) in the vicinity of R 3 ao. Similarly, for the A2Z+ state, it changes from 3alJt' to lst'4a. These changes have also been shown to play a key role on properties of the 211 and 2Z+ states of the OH radical such as the spin-orbit splitting constant29A(R) and dipole moment30 as a function of the intemuclear distance. Complementary theoretical evidence on the importance of the ionic contributions may come from the Mulliken population analysis for similar systems such as HF2.31 Second, there is experimental data supporting such an ionic nature of the wave function near the equilibrium H02 structure. The first piece of data comes from the fundamental vibrational frequencies of the hydroperoxyl radical. The 0-0 vibrational fundamental has been m e a s ~ r e d ' ~to%be ~ ~v3- = ~ ~1101 cm-', which compares well with that of the 0 2 - ion,35 Le., 1 4 0 2 3 = 1089 cm-'; note that the fundamental vibrational frequency [ ~ ( O Z=) 1580.2 cm-'1 for the ground-state neutral oxygen molecule differs significantly from this value. Related spectroscopic evidence is provided by the equilibrium bond distances: Roo(HO2) = 2.5143 m,15.16 Re(02-) = 2.532 Q , Re(02) ~ ~ = 2.2818 Ro~(H02)= 1.8346 a0,15,16 and R,(OH) = 1.8344 Further corroborating information comes from vibrational stretching force constants. For example, the 0-0 stretching force constant in H02 has which should be been estimated6 to be 0.3774 hartree compared with the values 0.359 and 0.756 hartree m-2for 0 2 - 33 and 0 2 , 3 8 respectively. According to the above considerations, the DIM potential matrix must assume the form
-
H=
[zc21
where I&, Hi, and H,, are submatrices of the full potential matrix H and the indices c and i stand for the covalent and ionic contributions. Each of these submatrices will be examined in the following subsections. 2.1. Covalent Part. Our main goal in this section is to provide a physically motivated form for modeling the covalent part of the H02(2A") potential energy surfaces. For simplicity,
we reduce the atomic basis set to a minimal one. 'Thus, the present calculations consider only the following eight configurations from atoms in ground states
q3 = AP,3P,2Sx1 q4= AP,3P,2Sx2
(3)
where X I and x 2 are the appropriate doublet spin functions obtained by coupling the 2S hydrogen atom with the 3P oxygen atoms and A i s the antisymmetrizing operator. In the following, the two oxygen atoms will be denoted a and b, and the hydrogen atom c. Similarly, the involved intemuclear separations will be denoted R1 = Rac,R2 = Rh, and R3 = Rab, while the included angles are 81 = Lcab and 82 = Lcba; the two oxygen atoms are considered to lie along the z-axis. In addition, it is assumed that the basis set (eq 3) corresponds to the spin coupling of the diatomic fragment ab. According to the DIM p r o c e d ~ r e ,the ~ ~Hamiltonian .~~ of the triatomic system is written as
where 5%b, %, and ZC are the Hamiltonian operators for the corresponding diatomic fragments and, similarly, Z, and 5% are the atomic Hamiltonians. Thus, the Hamiltonian matrix for the triatomic system assumes the form
a,
where the boldface notation is used to indicate matrix quantities; e.g., Hab is the Hamiltonian matrix for fragment ab, and Ha is the diagonal energy matrix for atom a (similarly for the other fragments). Note that the matrices R ~ cand Rc rotate the b and c atomic states to follow the bc and ac axis, while Th and Tacare the appropriate spin-recoupling matrices that transform the basis set in eq 3 to the spin eigenfunctions of the bc and ac fragments. The Rh rotation matrix has the following elements: (R&i = -cos 82 ( i = 3, 4, 7, 8), (R& = 1 ( i = 1, 2, 5 , 6), (Rbc)3,7 = &)4,io = sin 82, and ( R d i j = -(Rbc)j,i (i f j ) . Similarly, one has for Rac: (Rac)j,i = cos 81 (i = 1, 2,5, 6),
and
In addition, the Hat,matrix is given by
Varandas and Voronin the submatrices a and b are defined by
3ng- 3nu (9) 'ng+'n,0 3ng-3nu0 3na+ 3nu
-1 0 2
3ng+3n,0
lng-lnu0
0 and
'Ag+'% 0 'Ag-'% 0 3 A g + 0 3Ag -1 0 2 'Ag-Ix 0 'Ag+lX0 0 3Ag-3q 0 3Ag
(10)
+
I
( q M , L 2 M 2G:$s,s2Ms2 ~nlLIMl SlMSI)l1n2L2M2's2Ms2)(1 3)
where Jz has the meaning previously assigned, (ZI,LI,SI) are the usual orbital and spin angular momenta symbols for atom 1 (similarly for atom 2), S and A are the total spin angular momentum and the projection of the total angular momentum L along the molecule internuclear axis, and Gk>,L2M2and G~$S,S2M,2 denote Clebsh-Gordan coefficients. 2.2. Ionic Part. The ionic polyatomic basis functions of 2A't symmetry include the following ten configurations 2
3
2
1
3
The matrix elements of Habcan now be conveniently evaluated using the set of functions 4, (j = 9-18), since {4,} are the wave functions for the ionic states of 0 2 - . Thus, the ionic submatrix Hab may be written as
where Hab is the Hamiltonian matrix of fragment ab evaluated in terms of the {&} wave functions. 2.2.2. Hbc Matrix. The matrix elements of the operator 9 & can be more conveniently calculated using directly the wave functions {qj} (j = 9-18). After rotation on the angle n 02, we obtain for the ionic submatrix Hb, the usual DIM expression
where H ~ = c (32-,31T+, 31T-, 2Z+, 21T-, 211+, 311-, 3C-, 2X+, 211+)is a diagonal matrix and Rk is the rotation matrix with elements Rii = -cos 02 (i = 9, 11, 12, 14-18), R I O , ~=OR13.13 = 1, R X ,=~ -~ R11,16 = R12.18 = -R14,17 = sin 02, and Rij = -Rj,i ( i f j ) ; all undefined elements should be taken as zero. The spin-recoupling matrix is in this case the unit one. The ionic-covalent matrix elements connecting the ionic and covalent parts in the full Hamiltonian matrix H ~ ccan be calculated directly in terms of {qj} (j = 1-18). The nonzero matrix elements are the following
1
q9= A P , P, sx; qlo= A P y Py sx; qll= A 2P , 3P, 1sx; q I 2= AP,2Pz1Sx;
2.2.3. Ha, Matrix. The matrix elements of the operator can be obtained as for Zc.One gets VI7
= AP,2Pz1Sx;
VI8
= -&P,2Px1Sx;
(14)
where 3P, and 2Pp are the spatial functions of atoms 0 and 0-, ' S is the spatial function of H+,and the first, second, and third atomic states refer always to atoms a, b, and c, respectively. The doublet spin functions for the two oxygen atoms are now denoted x{ and xi. 2.2.1. Hab Matrix. The wave functions $9-7)18 can be expressed in terms of the ionic Z,n, and A states. We obtain the following matrix relation
II,=M$ where
(15)
3 = {vi} (i = 9-18), $ = (A+, Z,: g,A-, Z;, q ,
n;, l-Ii),and Zt2 = 1/&[2Zil,, f 2q1,2] (corresponding expressions apply to llt2 and A*). In turn, the transformation matrix assumes the form M = (a, a, b, b), where
I$,
IT:,
zc
where Hac= (2Z+, 211+,3Z-, 3rI+,31T2Z+, , 211+,31T, 3Z-) is a diagonal matrix and R, is the rotation matrix with elements Rii = cos 81 (i = 9, 11, 12, 14-18), RIO,IO = R13.13 = 1, R9.17 = - R I I , I=~ R12.17= -R14,18 = sin 81, and Rij = -Rj,i ( i rj);all undefined Rij matrix elements should be considered as zero. As for Hk, the spin-recoupling matrix is the unit matrix. The nonzero matrix elements of Hk are then the following
Low-Lying 2A" States of HOz
J. Phys. Chem., Vol. 99, No. 43, 1995 15849 potential energy curves has been represented by the form
= ./zH;fl, = H1;13 =
&&,
(24)
In summary, in addition to the diatomic potential curves arising in the covalent submatrix, we also need potentials for 10 diagonal diabatic elements of the ionic submatrix Hab, namely 2 2 - 2 - 2 - 2 Zg2, q,, q ngi, 'nul, 'ng2, 'nu29 'A,, and 'Au. Moreover, we must gave potential energy curves for four ionic states, %i, 'C;, 311i, and 'Z;, and the matrix elements of the ionic-covalent interaction q;.
xi9
where the parameters D and ai have been obtained as described elsewhere.47 For the asymptotic exchange energy of 0 2 and OH we have used the theoretical data reported in the literature.42,47In both cases, it assumes the form A
E::"'
+
= -k"( 1 C i i i R i )exp( -7R)
(29)
i= 1
3. Analytical Potentials for the Diatomic Fragments
where d, iii, and are theoretical parameters related to the asymptotic behavior of the wave function through some molecular integrals.42 Thus, the EHFACE2U potential energy curves of 0 2 and OH assume the form of model III in ref 38; Le., the EHF part of the potential energy curve leads to the correct exchange energy at asymptotic distances. As pointed out above, the damping function xexc(R) accounts for charge overlap effects at short distances. Since this damping function = VEHF(R) + Vdc(R) (25) is generally not known, it has been a p ~ r o x i m a t e dby ~ ~that of the leading term in the long-range Coulombic energy (see eq where VEHFis the extended Hartree-Fock energy and Vdc the 31). Accordingly, for 0 2 , such a damping function has been dynamical correlation which includes the long-range dispersion represented by x ~ ( R )although , the long-range quadrupoleenergy. In some cases, a simpler version39(EHFACE2) of this quadrupole electrostatic interaction vanishes for some of the model has been utilized. Note that, for convenience, v d c electronic states considered in the present work. Similarly, we includesI0 also the long-range electrostatic and induction enerhave used 26(R) for OH, since the leading Coulombic term gies appearing in the EHF energy due to the similarity of their corresponds always to the induced dipole-induced dipole functional forms. Thus, the EHF energy term is given by38 dispersion interaction. Of course, one requires that y(m) > 7 3 in order to ensure that Ti:"' is the dominant term at asymptotic v,, = -DR-'(I &' +) exp[-y(r)r] X ~ ~ , ( R ) ~ ; : ~ ( Rdistances.38 ) This has been warranted through a trial-and-error i= 1 procedure. Except for the parameters referring to the triplet (26) states of 0 2 , and the 2,411and 294Z- states of OH that have been where given e l s e ~ h e r e ,all ~ ~other . ~ ~parameters for 0 2 are reported in Tables 1 and 2. (27) Y = Y o [ l + Y1 tanh(y2r)I The dynamical correlation is approximated by
A fundamental requirement in the DIM treatment of the title system is to have the potential energy curves for the relevant electronic states of 02 and OH. Except for the excited states of OH, for which the available information is limited, all electronic states have been represented by the extended HartreeFock approximate correlation energy38(EHFACE2U) model
+
+
FA"'
and is the leading c o n t r i b u t i ~ n ~of- ~the ~ exchange energy at asymptotic distances, the damping function of which will be represented by xexc. In addition, r = R - Re is the displacement coordinate from the equilibrium geometry Re and D, y;, and ai (i = 1-3) are parameters that can be determined from a fit to ab initio or experimental data, as described elsewhere.38 Specifically, we have used RKR turning points to obtain the least-squares parameters in the potential energy curves for the X3Z,, CA,, and a'A, states of 0 2 , as well as for the X 2 n curve of OH. For all other 0 2 states, the EHF term in eq 26 has been obtained using ab initio energies for the calibration procedure; except for O2(C1q-), for which we have used the data of Partridge et ~ l . , "all ~ ab initio energies referring to the 3rIg,311u,In,, and Ill,,electronic states of 0 2 have been taken from the work by Saxon and Liu.44 Note that the 311g and Illgadiabatic potentials show in reality a clear avoided crossing with excited states in the vicinity of R 3 UO, which we did not consider, since only the upper states cannot be described at the minimal basis set level adopted in the current work. In tum, the potential energy curve for the 1% state of OH has been adjusted to the ab initio CI calculations of Langhoff et ~ l . , "while ~ those for the a4Z- and 1411 states have been obtained from the 122- ab initio curve of Langhoff et al.@ and the 12Z--14Z- and 12Z--14n splittings reported by Easson and Pryce,& respectively. However, due to the unavailability of sufficient input data to define the van der Waals minimum of these excited states of OH, the EHF part of their
-
v,,, = -CCJ,(R)R-"
(30)
n
where the damping functions assume the
and A, and B, are the auxiliary functions
where ai and /3i (i = 0, 1) are universal parameters (dimensionless): a0 = 25.9528, al = 1.1868, PO = 15.7381, and /3, = 0.097 29. In turn,p is a scaling parameter defined by p = (5.5 1.25Ro), Ro = 2((rh)1'2 (ri)1'2) is the L ~ R parameter, o ~ ~ ~ and ( I ; ) is the expectation value of the squared radii for the ' ~ electrostatic ~~~ outermost electrons in atom X. As u ~ u a l , the and induction damping functions have been approximated by those for the dispersion energy simply by using the appropriate powers of n. Thus, it remains to describe the procedure used in obtaining the dispersion coefficients. For 0 2 , the c,5 dispersion coefficients have been taken from Kellyso for the various ML values of the two oxygen atoms. In turn, we have used for the 2Z- and 211 states of OH the values = 13.26 hartrees a: and CF = 11.47 hartrees a:, which have been
+
+
15850 J. Phys. Chem., Vol. 99, No. 43, 1995
Varandas and Voronin
TABLE 1: Values of the Coefficients in the Extended Hartree-Fock Part of the EHFACEZU and EHFACE2 Potentials of Singlet 02 and 0 2 - , Respectively system 0 2
102Dlhartrees
;$
'l-4 'n. 022n,l 2n", 2x-
zq 2AU
2A, (I
allno-'
a2/ao-2
a31ai3
rdai'
Y'
Yduo- I
28.101 860 4.870 300 0.735 691 0.093 978
0.228 350 4.629 370 1.158 601 0.766 785
-0.167 931 6.050 491 0.389 142 0.178 674
0.272 389 3.633 806 0.048 010 0.015 263
1.830 093 3.897 190 2.072 045 1.870 569
4.464 204 1.949 742 1.895 689 1.672 984
0.010 980 0.131 998 0.027 326 0.010 667
19.313 042 5.396 124 9.729 353 -3.506 654 15.382 968 3697.218 969
2.241 673 3.741 569 2.866 546 -0.144 259 2.277 120 1.330 762
-0.571 395 2.618 663 2.400 454 2.865 696 1.401 924
0.454 887 0.823 020 0.854 138 0.664 495 0.359 557
1.015 366 1.736 072 2.078 422 1.446 006 1.779 567
rU-5
N
N
N
For the 2A, state of
02-,
N
the data refers to eq 44.
N
N
nr
nr nr
N N
N N
N
nr
= not relevant.
TABLE 2: Values of R,, Asymptotic Exchange Parameters A, 61,and 62,and Long-Range Dispersion Coefficients of 02-Used in the Present Work" system
Rdao
Aihartrees
IAE
2.2970 2.8743 6.2016 6.2016
4.7562 8.1954 2.03055 2.03055
'< In"
iil/ail -0.217 -0.126 -0.300 -1.738
611 290 41 1 445
ii2/ai2
Cshartrees ai
10-1 x Chartrees a:
-0.246 100 -0.142 824 1.152 890 1.152 890
-0.935
1.820 1.806 1.733 1.733
rU-5
3.74
nr
For all states, the values of the asymptotic exchange parameters Ci and 7 are 0.5 and 2.0 a i ' , respectively. In addition, Ro = 5.6617 a0 assumes the same value for all states of 0 2 . The values of CSand c10have been calculated from c6 using eq 34. N = not relevant.
0.1 c
w
\
3
0.0
-0.1
-0.2
2
4 R / a,
--0.2
6
0
2
4
6
8
1
0
R/ a,
Figure 1. Potential energy curves for the triplet (solid lines) and singlet (dotted lines) states of 0 2 . Also shown by the open circles are the RKR data52for the ground state triplet (open circles) and lowest singlet (open diamonds) curves. calculated as described elsewhere.'" In all cases the values of the cg and C ~dispersion O coefficients have been estimated using the universal c o r r e l a t i ~ n ~ ~ J ~ (34) where Ks = 1, ~ 1 =0 1.31, and a = 1.54 are universal parameters; all expectation values necessary to evaluate ROhave been taken from the tabulations of D e ~ c l a u x . Thus, ~ ~ we assume that eq 34 is valid also for excited molecular electronic states; note that Ro depends only on the expectation values of the atoms. The complete 0 2 and OH potential energy curves are shown in Figures 1 and 2 and compared with existing RKR52,53data. As already point out in section 2, the ground-state OH(X211) has strong ionic character. Predominance of the ionic structure at short internuclear distances is suggested from theoretical
Figure 2. Potential energy curves for the 2,411(solid line) and 2,4X (dotted lines) states of OH. Also shown by the open circles are the RKR datas3for the ground state curve. c a l c ~ l a t i o n sand ~ ~the ~ ~fact ~ ~ ~that ~ OH has a large dipole moment with the polarization O-H+.54 Moreover, the form of the dipole moment function, increasing with R before passing through a maximum, has been attributeds5 to a large ionic contribution that decreases sharply as R increases. In the DIM approach one should therefore treat the diabatic covalent and ionic tenns separately. This will be discussed next for both the X211 and 122+states of OH, although only the ionic component of the latter is relevant for the present work. As follows from the ab initio c a l c ~ l a t i o n sfor ~ ~ OH, the adiabatic X211, 2211, and 3211 curves may be viewed as the eigenvalues of the 3 x 3 potential matrix
WII
w12
Wic (35)
Wjj(R) (j = 1-3),
Wi,, and
qcare
in an obvious cor-
J. Phys. Chem., Vol. 99, No. 43, 1995 15851
Low-Lying 2A" States of HO2 respondence with the diabatic X211, 2211, 3211, and ionicconvalent coupling terms. Since the W22 and corresponding adiabatic 2211state curves correlate asymptotically with O('D) H(2S), they cannot be described using the basis set of eqs 3 and 14. For consistency, we have therefore considered the simplified matrix
+
0.4 0.3 c
w
0.2
\
where Y,(R) (j = 1, 2) denote effective diabatic potentials that correlate asymptotically with O(3P) H and O-(2P) H+, and Kc is the associated ionic-covalent coupling term. As has also been shown in previous ab initio w0rk,2~*~O the adiabatic X211 state changes its principal configuration from covalent (la22a23a40[lZ+]1n3) to ionic (la22a23a21n3)at R 3 m. Thus, the diabatic V , (j = 1, 2 ) and coupling Vi2 terms can be defined from the requirement that the diagonalization of the matrix in eq 36 should reproduce the adiabatic X211 and 3211 curves. After the usual procedure, one obtains the following coupling between the diabatic and adiabatic Pll and 3211terms
+
>
+
0.0
-
+ V22(R)= w: 32n+ V i l ( R )= 0; X211
0.1
-0.1
-0.2
0; X2n
(38)
0.5
(39)
0.4
0.2
0; = tanh(yR")
(40)
0.1
a; = 1 - tanh(qR")
(41)
0.0
+
-
12
+
where wi(R) ( k = 1, 2) are the weights of the covalent and ionic configurations. Their dependence on the interatomic distance R has been determined by de Vivie et and has been approximated in this work by the switching functions
- -
8
Figure 3. Potential energy curves for the adiabatic (dashed lines) and diabatic (solid lines) *II states of OH. The corresponding coupling term is also shown by the dotted line. Indicated in the right top comer is the 0- H+ dissociation limit.
(37)
where 7;1 is an adjustable parameter chosen from the requirement that the ionic and covalent characters have an equal value at R, = 2.9 a g . 2 9 3 5 6 We found it convenient to use n = 4, which gives 9/aO4 = In 3/(2R:). In turn, the 3211 curve has been represented using the EHFACE2 model without worrying about the normalization of the kinetic field; the relevant numerical parameters have been given elsewhere.47 As seen from Figures 3 and 4, the diabatic V I1 (R) and V22(R)potential curves obtained from eqs 37-39 join smoothly the 3*ll and X211 states at small and large values of R, respectively. As expected, the ioniccovalent coupling term has a maximum at R 2.9 and decreases to zero both at R 0 and R 00. A completely similar phenomenon takes place for the A 2 F , B2Z+, and C2Z+ potential curves of OH. Accordingly, the corresponding diabatic states VI i(A2Z+) and VI1 (B2Z+) that correlate with A2Z+ and C 2 F (0- H+)at R = have been determined from expressions analogous to those in eqs 37-39 but involving the A2Z+ and C2Z+ states rather than X211 and 3211. In this case the EHFACE2U curve for the A2Z+ state has been obtained using RKR data,57while that for C2Zf was represented by a simpler EHFACE2 model that has been calibrated using ab initio data;56as before, the weighting factors have been represented by eqs 40 and 41. Figure 6 illustrates the resulting potential energy curves. As expected, the behavior is quite similar to that found previously for the X211 and 32FI states. We now turn to the diatomic states of 0 2 - , namely X211,,, 2111g2, 2111u1, 211u2, 2X9;, 'Xi, 'q,, 2q2, 2Ag,and 2Au. They have been modeled differently depending on whether they correspond
4
R/ a,
0: 3211
V1,(R)= (32n- x2n)wiO2
-
-
0
0.3
,..
. / ... .. .. ..... .. .. .
-0.1
_ _
-0.2
0
4
8
12
R / a, Figure 4. As in Figure 3 but for the *2-states of OH.
to bound or unbound state curves, but in all cases the calibration was done using the ab inito MC SCF data of Michels et Accordingly, the bound state curves have been represented using the EHFACE2 model, with the parameters in the EHF part being calibrated from the published theoretical force field data (Re, we,o d e x ,G, , and B e ) . In turn,the long range part of the potential energy curves has assumed the form of eq 30, with the coefficients being gathered in Table 3. Specifically, they have been determined as follows. The long-range charge-quadrupole electrostatic coefficients have been obtained from C3 = Q0/2, where Q represents the unit negative charge of the 0-ion and 0 is the permanent quadrupole moment5*of the oxygen atom. Similarly, the quadrupole-quadrupole electrostatic coefficients have been calculated following the procedure of Zygelman et ~ 1 with . the ~ assumptionm ~ that the expectation values of the square radial coordinate are the same for both the 0 and 0species. The induction coefficients have in turn been obtained using C4 = Q d 2 , where Q has the meaning previously assigned and a is the static polarizability of the oxygen atom in its a(lI) or n(l) states. Moreover, the cfjdispersion coefficients have been obtained from the London foxmula
Varandas and Voronin
15852 J. Phys. Chem., Vol. 99, No. 43, 1995
TABLE 3: Values of the Long-Range Coefficient@Used for 02-
Cd Cd Cd Cd state R, hartrees a i hartrees a: hartrees ai hartrees a: 3.7412 18.199 48 X2n,, 2.534 12 0.45 2.5955
'l-42
'nu,
2.628 27
's
4.1196
2nu2
c W
\
5
3.76055
-O'l -0.2
'A,
t
'J
2
4
6
8
1
R/ a, ---e-,
._.._
_.(
0.5
0.3 0.2
0.1
\
0.0 -0.1
-0.2
-0.3 -0.4
0 -0.935 -0.935
9.453 85 18.199 48 9.453 85 9.099 74 18.907 69 9.099 74 18.907 69 18.907 69 18.907 69
been calculated from
states i and j
AJhartrees
b,,la,'
X2n,,and 211sz
8.372 82 9.466 49 1.355 12 56.129 75
1.047 23 1.152 43 0.740 14 1.793 64
'XIU, and 211u2 'Zi,and '2and
*q '
