J . Phys. Chem. 1984, 88, 4887-4891
4887
Approximate Single-Valued Representatlons of Muitivalued Potential Energy Surfaces J. N. Murrell* and S . Carter School of Chemistry and Molecular Sciences, University of Sussex, Brighton, BNl 9QJ. U.K. (Received: January 18, 1984)
A new method is proposed for constructing approximate single-valued representations of multivalued potential energy surfaces subject to the restriction that for each chemically distinguishable dissociation limit there is only one set of molecular states. The method is computationally simpler and more accurate than methods previously used. The method has been applied to the ground-state surface of H 2 0 with the restricted dissociation limits H2(X,'Z,+) + O(ID) and OH(X,211) + H(2S), and the resulting function optimized to the vibrational frequencies of HzO. An analysis of the ground-state surface of H202 shows that the method can be extended to tetraatomic systems.
Introduction In a series of papers',2 we have developed a general strategy for constructing functions which represent the potential energy surfaces of triatomic molecules, and in other papers3s4the method has been applied to tetraatomic systems. The method has been shown to be applicable to all types of surfaces: attractive and repulsive, those having one or several minima, and for single-valued or many-valued surfaces. The first paper in this series1 was on the ground-state surface of HzO. It described a method for obtaining a potential function which satisfied the dissociation limits of the surface and the spectroscopic parameters of the equilibrium molecule. However, the H 2 0surface is not strictly single-valued,a point that was noted at the time but not treated thoroughly until a later paper,5 and in this first paper a method was given for approximating such surfaces by single-valued functions. In the present paper we return to this topic but propose an alternative solution which we believe to be superior to that first given because it is computationally simpler and more generally applicable. We should perhaps stress that the main impetus for our work stems from our wish to study the dynamics of more complicated systems than H 2 0 , and it is in that context that we now introduce the problem in more detail. The first step in our procedure is to specify the fragments which are produced on dissociation according to the restrictions of symmetry and spin conservation contained in the Wigner-Witmer rulesS6 We illustrate this with the example of the singlet ground-state surface of H202. If we restrict the dissociation channels to those that have just two fragments, then the limits are as follows: 20H(X,2rI) H02(g,2A") + H(%)
It can be seen that for two of these dissociation limits there are two channels because it is not possible, by spin conservation, to produce both fragments in their ground states. The actual channel that is reached at these limits will depend on the geometry of the ~
~~
~
(1) K. S. Sorbie and J. N. Murrell Mol. Phys., 29, 1387 (1975). (2) S.Carter, I. M. Mills, J. N. Murrell, and A. J. C. Varandas, Mol. Phys., 45, 1053 (1982). (3) S. Carter, I. M. Mills, and J. N. Murrell, Mol. Phys., 39, 455 (1980). (4) S. Carter, I. M. Mills, and J. N. Murrell, Mol. Phys., 41, 191 (1980). (5) J. N. Murrell, S. Carter, I. M. Mills, and M. F. Guest, Mol. Phvs.. 42, 605 (1981). ( 6 ) E. P. Wigner and E. E. Witmer, Z. Phys., 51, 859 (1928).
0022-3654/84/2088-4887$01 .SO10
+
fragments. For example, in the H2 O2limit, if the H-H distance is short and the 0-0 distance long, then channel i will have the lower potential energy; if the reverse is true, channel ii will be lower. From the above analysis it appears that the singlet ground-state surface of H,02 is two-valued and as such can be represented by the eigenvalues of a 2 X 2 matrix. We have shown that our method can be applied to such problems5but not without difficulty, particularly if there is information about only one sheet of the surface. However, as we now show, the H202problem is even more complex than this. It was pointed out in our first paper' that the ground state of H,O has two dissociation channels for removing either an oxygen or hydrogen atom. V.
vi. vii.
... v111.
-
H ~ O ( % , I A ~ ) H ~ ( X , ' B ~ ++o('D) )
-
-
H2(b,3z.,+) + o ( ~ P )
+ H(%) OH(A,?2+) + H(%) OH(X,Q)
(2) Channels v and vi are determined by spin conservation and vii and viii by the conservation of spatial symmetry. The ground state of H 2 0 correlates with a '2+state in C,,, and this_cannot dissociate by channel vii. It follows that as the H,O(X,'A,) state appears in only one of the dissociation channels of (I), the complete singlet ground-state surface of H202must be three-valued. There are other indications of this; both of the O2 states appearing in (1) dissociate to two 3Poxygen atoms. Hence, at least one other O2 state must be invoked to produce the O('D) which appears in channels iii and v. A complete three-valued surface would be represented by the eigenvalues of a 3 X 3 matrix, and although it would be, in principle, possible to parameterize the elements of such a matrix to points on the surface, this is not a task that we consider feasible in the present state of knowledge. Moreover, even if this was achieved the resulting potential functions would be computationally very expensive for dynamical calculations. To run classical trajectories, for example, requires the rapid evaluation of the potential and its derivatives along the trajectory, and that would be expensive if at each point a 3 X 3 matrix had to be diagonalized. There is therefore a strong interest in obtaining an approximate single-valued representation of such a surface. The solution to the water problem which was proposed by Sorbie andMurrel1' was a single-valued surface in which there was a cusp in the two-body term for the H-H interaction. This was canceled by a similar cusp in the three-body term in the interaction region. However, it is not possible by this method to retain perfect smoothing (analyticity) for finite H2-0 separations but nonanalyticity at infinite separations. In practice, the resulting function was proved, by the analysis of vibrational eigenvalues, to be extremely good for regions around the potential minimum, but the imperfect smoothing at large H2-0 separations does lead to 0 1984 American Chemical Society
4888 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
Murrell and Carter TABLE I: Parameters of the H20Potential Whose Functional Form Is Given by Expressions 7, 8, and 9
Diatomic Potentials DJeV a,/A-'
r.lA
OH(X?II) HH(X,'Z,+)
2 H + Ot'D)
+---
4.6211 4.7472
0.9696 0.7414
4.507 3.961
a,lA-2
a*/A-'
4.884 4.064
3.795 3.574
-jI "
Comparison of the Sorbie-Murre11 (full line) and the Schinke-Lester (dotted line) H-H two-body potentials of H20. R is the Figure 1.
H-H distance.
difficulties for trajectory calc~lations.~A further criticism of the Sorbie-Murre11 potential is that although it satisfied the asymptotic limits v and vi (of (2)), it did not satisfy the two H OH channels vii and viji, and hence the conical intersection of 2 and II surfaces, which oqurs for some linear configurations of the molecule, was not reproduced. Schinke and Lesters used h o t h e r method to produce a single-valued H 2 0 surface. They made a compromise between channels v and vi of (2) by taking the Hz two-body term to be the H2(X,'B,+) potential plus the O('D) energy for R" < 1.1 A, and for R" 3 1.1 %I they took an exponential function
+
A ~xP(-~RHH)
(3)
with parameters A and a chosen to have continuity in the potential, and its first derivative at R" = 1.1 A. The resulting two-body potential therefore extrapolates to the energy of the dissociation limit, 2H(2S) O(3P), chosen as zero. A comparison of the Sorbie-Murre11 and Schnike-Lester potentials is given in Figure l. While the Schinke-Lester has the merit of being analytic, it gives an incorrect representation of the H2 + 0 limit above the u = 1 state of H2 and would therefore be inadequate for some dynamical problems. It needs to be stressed that any single-valued analytic representation of the H 2 0 surface is an approximation, and the aim must be to minimize the effects of this for the dynamical problems of interest. There are clearly many compromises between the two-body potentials shown in Figure 1 which would be superior to either, and it is this aspect which we now explore.
Atomic Potential Vo(])= 1.958 eV f = Il2[1 - tanh (0.9509(3p3 - p1 - p2))] pl/A = RoH - 0.9572 p2/A = Ron! - 0.9572 P ~ / A= R"' - 1.5139
+
A Single-Valued H20Potential The method we develop in this paper involves the use of switching functions; these are analytic functions that provide a smooth transition between specified functional behavior in different regions of space. The most commonly employed switching function is
L ( x ) = y2(1 - tanh (ax/2))
(4)
which approaches 1 as x approaches --m and approaches 0, as exp(-ax), as x approaches -m. As this function has already been employed in our method's2 to impose the dissociation limits on the potential, it is very suitable for the extension we propose. The only problem to be solved is the form of the variable x . Our objective must be to choose x so that different one- or two-body terms are switched on in different regions of space, consistent with the specified asymptotic limits. By definition, a single-valued surface has only one set of fragment states for each chemically distinguishable asymptotic (7) K. S. Sorbie and J. N. Murrell, Mol. Phys., 31, 905 (1976). (8) R. Schinke and W. A. Lester, J . Chem. Phys., 72, 3754 (1980). (9) D. F. Smith and J. Overend, Spectrochim. Acta, Part A , 28A, 471 (1972).
limit. Hence, to obtain a single-valued function for the ground state of water, we must select from scheme 2 one channel from v and vi and one from vii and viii. We do this by an energy criterion. Firstly, we note that as the H2(b,3&+) state is repulsive, the energy of this channel must be above the energy of 2H(2S) O(,P), and any classical trajectory which exited on channel vi would produce these atomic products. However, in most experimental situations there is likely to be insufficient energy to do this. Secondly, we note that as it requires at least 4.1 eV more energy to produce OH(A,zB+) than OH(X,211), the former is generally of no interest in dynamical studies. We therefore devise a single-valued potential which satisfies the restricted dissociation scheme
+
V.
H20(%,'A1)
-
vii.
-
H,(X,'B,+) +O('D)
OH(X,211)
+ H(2S)
(5)
and as the OH(X,211)state dissociates to ground-statate atoms, it is necessary to employ a switching function for the one-body term which allows (the energy of O('D)) to be present in the limit v but absent in vii. We now examine the variable x = nP3 - PI
- P2
(6)
where p I = R, - R,O are the displacements of the internuclear distances from a reference structure (R3 the H-H distance, R l and Rzthe 0-H distances). If n 3 2, x takes the limit --m for dissociation to H2 + 0 and +-m for dissociation to O H + H. For the three-atom limit 2H 0, x has no unique limit for all space. This can be seen by examining linear configurations H-H-0 for which R1 = R2 R,,and as R2 and R, are independent variables,
+
+
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4889
Multivalued Potential Energy Surfaces
Y:
(
H----0
CONTOUR =-loo00 CONTOUR = 1.00
)
2.5-
I ANG
1 EV INTERVALS EV
200-
H 1.5-
0
I
H 100-
.5-
1.0
j
X:
(
0----H
)
I
1.5
I 2.0
I
2.5
3.0
I ANG
Figure 2. Contours for bond stretching of H20with a constant bond angle of 104.S0 (contour 1 -10.0 eV, intervals 1.0 eV).
(6) can take any value in the three-atom limit. However, if n is large, then in most space of the three-atom limit x will approach +a,and as the O(lD) energy should be switched off in this limit, a high value of n has some advantages. We choose the value n = 3 for our analysis. For the origin of the switching function we take the equilibrium configuration of H20, the same origin used by Sorbie and Murrell' for their three-body term. There is no overriding factor for this choice, but it is obviously convenient for analysis to take all variables from the same reference. We therefore write the onebody switching function
P in (9). For any values of the range parameters it is possible to obtain coefficients of a quadratic polynomial which reproduce the energy, geometry, and harmonic force field of the equilibrium H 2 0 molecule. Surfaces thus obtained can then be scanned over the whole space to see if they had any unphysical behavior, e.g. other minima or maxima for which there is no physical evidence. This was done for variations of a , y l = y2, and y3 in the range 1-4 A-1, and the most satisfactory overall otential was jud ed to be that with cy = 1.5 .&-I, y l = y2 = 2.6 i - l , and y3= 1.5 $-I. The next step was to calculate, by the variation method,I0 the vibration frequencies of this best surface and to compare them with experimental frequencies. The difference between observed f = Y2(1 - tanh [(3P3 - PI - pz)(a/2)1) (7) and calculated frequencies can now be minimized by extending the polynomial to a full quartic in the three variables pi and and the single-valued representation of the H 2 0 surface is, in the optimizing these coefficients and the value of a by using the many-body expansion procedure described in an earlier paper.2 Further optimization VHHO = VO(')f(Rl,R27R3)+ VOHC2)(R1)+ vOH(2)(R2) + of the yi is in principle also possible by the same criterion but was VHHc2)(R3) + VHHO(~)(R~,RZ~R~) (8) not found to give any significant improvement. Table I gives the parameters of the best potential that we have fl*)is a two-body term that goes to zero at the dissociation limit obtained, and Table I1 compares the calculated and observed of the relevant diatomic. F3) is a three-body term that is zero vibration frequencies on which the optimization has been achieved. in all atom-plus-diatom dissociation limits. This is written as a The agreement between observed and calculated values is very product of a polynomial P(pi) and functions such as (4) which similar to that obtained with the Sorbie-Murre11 potential.' impose the required asymptotic limit Figures 2-4 show views of the surface, and a comparison with f13) = P(pJ i,IT (1 - tanh (piyi/2)) (9) the lower sheet of the two-valued representation5 can be made. =1,3 The only significant difference is in the region of linear configwhere pi are displacements from the equilibrium configuration. urations, both H-H-0 and H-0-H, for which the 2-II concial Optimization of the H 2 0 Potential The parameters in the potential (8) to be optimized are the range parameters cy and yi and the coefficients of the polynomial
(10) R. J. Whitehead and N. C. Handy, J . Mol. Specrrosc., 55, 356 (1975).
4890 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
205
0
I I I H-H I
WR
I R
Murre11 and Carter Irl
TI \ I l l \ \ 1 1 1 R 1.9
II
Y-AXIS/EV
200 20
ANGLE BETWEEN X AND Y AXES = 90 DEG
19
l o 5 18 17 16
100
15
CONTOUR =-loo00 CONTOUR = a50
1 EV INTERVALS EV
16
.5
13
2
00
165
I
X: ( H----H
)
265
26 0
c 2
3
/ ANG
Figure 3. Contours for the C2, insertion of O('D) into H2 (controur 1 -10.0 eV, intervals 0.5 eV).
intersections exist. These, of course, cannot be reproduced by our single-valued representation.
A Preliminary Discussion of the H20z Surface We present a brief discussion of the Hz02surface in order to show that the method we are proposing for single-valued representations of many-valued surfaces can be generalized to more complicated systems than H20. From scheme 1 we select only the lowest energy channels for products in their equilibrium configurations; we select i and not ii, iii and not iv. Thus, for each of the asymptotic regions with molecular fragments we have only one electronically distinguishable product, and these are described in the following scheme: i. ii. iii.
iv.
H202(%,'A)
-
-
20H(X,211)
H2(X,'Zg+) + Oz(a,'A,)
-
+ O('D) H02(%,2A") + H(%) HzO(R,'Al)
(10)
A surface produced according to this restriction would, for example, allow one to study the dynamics of the collision of H H0, to give products in channels i, ii, and iii of (10). Of the two triatomic species mentioned in (lo), we have already discussed H 2 0 , and the other, HO,, presents no problem as it dissociates to ground-state fragments at all asymptotic limits according to the scheme
+
H02(%,zA")
-
-
H(%)
+
+ 02(X,3Z;)
o(3~) OH(X,~II)
(11)
TABLE II: Comparison of Observed and Variationally Calculated Vibration Wavenumbers (cm-') of the H,O Surface"
DZO
H20 level
calcd
obsdb
calcd
obsdb
100 010 00 1 200 020 002 110 101 01 1 300 030 003 210 120 20 1 102 02 1 012 111 130 03 1 013 21 1 121
3649 1596 3162 1176 3150 7453 5249 7245 5336 10817 4641 11044 8772 6197 10596 11102 6874 9005 8819 8282 8359 12581 12161 10344
3657 1595 3756 7201 3151 7445 5235 7250 5331 (10858) 4667 11032 8762 6775 10613 (11102) 6871 9000 8807 8274 8374 12565 12151 10329
2669 1174 2792 5279 2335 5541 3844 5375 3956
267 1 1178 2788 5292 (2356) (5576) (3849) 5374 3956
5107
5105
6538
6533
"Bracketed values are not observed frequencies but our own upper bound estimates. Reference 9.
The configuration of the HZO2species is defined by values of the six internuclear distances. The two H,O potentials will be
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4891
Multivalued Potential Energy Surfaces
3nO-
2n5-
2.0-
105ha
'1.0-
n5-
1
no-
-
-20 0
0 H
-1n 0
n0
20 0
n o
A T O M M O V i N G AROUND 0----H
O---- H BOND L E N G T H =
n97 ANG
CONTOUR l=-lOnOO E V CONTOUR I N T E R V A L S = X AND
Y
n50 E V
AXES/ANGSTROMS
Figure 4. Contours for H moving around OH (ROH= 0.97
A) (contour 1 -10.0 eV, intervals 0.5 eV).
VHHO(R1,R2,R3), already written as (8), and an equivalent function V H H O ( R ~ , R ~ The , R ~ HOz ) . potential is"
Q
l0 0
c
satisfies these requirements. The one-body term, being the energy of the oxygen atom, must also be associated with a switching function. The O(lD) energy VH00(R19R586) = must only appear in the dissociation scheme lOiii, which is at the V O H ( ~ ) (+R V ~O ) H ( ~ ) ( & ) V O O ( ~ ) ( R ~V)H O O ( ~ ) ( R I , R ~ , Rlimits ~ ) R , = R2 = R6 = m and R4 = R5 = R6 = m, but it must (12) in these limits appear both for the free oxygen atom and for the oxygen atom in the H 2 0 fragment; in the latter it must have the where the two-body terms are for the diatomic states contained same form as the switching functionfin (8). We therefore write in (1 1 ) . There is an equivalent function I/Hoo(R~,R&). the oxygen atom component of the H202 potential as We note that the only two-body term for the OH species that appears in (8) and (12) is for the 211 state and that this is the same state that occurs in the H202dissociation scheme (1Oi). Likewise, the only H2 state that appears in (8) and satisfies (1Oii) is l Z g + . Thus, these two-body terms present no ambiguity in the H202 potential. For 02,however, we have two conditions to satisfy. The two-body potential must represent the 3Z; state in (12) but the lAg state in the limit lOiii. To achieve this, we write the O2 contribution to the H202potential in the form gVo0,'~'(R6)+ (1 - g ) V00b(~)(R6) (13) where a and b label the 32; and IA, potentials, respectively, and g is a function of the variables Ri which takes the limits 0 or 1 in the asymptotic regions. In all but three asymptotic regions both Vooa(2)and Vmb@)are zero; hence, we need to concern ourselves only with the behavior of g in these three. In one of them (scheme lOii), R 1 = R2 = R4 = R5 = m, and we require g to approach zero in this limit. In the other two (scheme lOv), we have either R2 = R3 = R4 =. m or R , = R3 = R5 = m, and in these g must approach unity. Using the variable Y = P1 + P2 the function
+ P4 + P5 - 4P3
Pi
= R, - R,O
(14)
