Application of SCF-SI Tlheory to Polyatomic Molecules
The Journal of Physical Chemistry, Vol. 83, No. 8, 7979 905
In the first example A is a linear molecule and B and Xt are three-~dimensiionalrotors. Therefore, Au = 4, and eq 16 becomes the classical expression for pJ(Et - E,), the density of vibrational states of four vibrations with geo-
References and Notes
metric mean frequency given by the fourth root of C,/C;. For a linear niolecule (i.e., two-dimensional) CzD= l/crBl, where (T is the symmetry number and B1is the rotational constant. For a three-dimensional rotor C3D = [ ~ / a ' B ~ ] l / ~ , where IT is thie symmetry number and B is the geometric mean of the three principle rotational constants. Making these substitutions into eq 16 gives
where CT is the product of the symmetry numbers of the individual reactants." The density of vibrational states, p4t(Et - E,), can be calculated from semiclassical expressiion~l~ or by direct count.23 In the second example, A, B, and XI are all three-dimensional rotors. In this case Au = 5. By extracting the term (7r/P)ll2from the ratio C,'/C,, eq A1 can be rewritten as liZ
(5)
cvt( E , - E:,)'i2 c,r(9/2)
Equation A3 is the classical expression for dp15,1(E, Eo)/dEt wheice /1'5,1(h't - E,) is the density of vibrational-rotational states of five vibrations with geometric mean frequency given by the fifth root of Cv/C; and of a one-dirnensional rotation with rotational constant given by Et. Equation 16 now becomes
where B1 andl B2 are the geometric imean rotational constants of the reactants. The quantity dp/dEt can be calculated directly from most semiclassical expressions for P.
(1) See, for exampie, M. A. Fluendy and K. P Lawley, "Chemical Applications of Molecular Beam Scattering", Chapman and Hall, London, 1973; Adv. Mass Spectrom., 7 (1976). (2) See, for example, D. L. Bunker, "Methods in Computational Physics", Academic Press, New York, 1971. (3) See, for example, J. C. Poianyi, Acc. Chem. Res., 5, 161 (1972), and references therein. (4) P. Pechukas, "Statistical Approximations in Collision Theory" in "Dynamics of Molecular Collisions", W. H. Miller, Ed., Plenum Press, New York, 1976. See also E. Pollack and P. Pechukas, J . Chem. Phys., 69, 1218 (1978). (5) R. A. Marcus, J . Chem. fhys., 45, 2630 (1966). Preliminary work can be found in R. A. Marcus, J. Chem. fhys., 45, 2138 (1866). (6) R. A. Marcus, J . Chem. fhys., 46, 959 (1967). (7) R. A. Marcus, J. Chem. Phys., 82, 1372 (1975). (8) G. Worry and R. A. Marcus, J . Chem. Phys., 67, 1636 (1877). (9) See, for example, the chapter by W. J. Chesnavich, T. Su, arid M. T. Bowers in "Kinetics of Ion-Molecule Reactions", P. Ausloos, Ed., Plenum Press, New York, 1979. (10) R. J. Cotter, R. W, Rozett, and W. S. Koski, J . Chem. Phys., 57, 4100 (1972). The deconvolution procedure they used is the method given by Chantry, J . Chem. Phys., 55, 2746 (1971). (11) K. T. Alben, A. Auerbach, W. M. Ollison, J. Weiner, and R. J. Cross, Jr., J . Am. Chem. Soc., 100, 3274 (1978). (12) K. Morokuma, B. C. Eu, and M. Karplus, J . Chem. Phys., 51, 5193 (1969). (13) Compare, for example, eq 10 of ref 5. (14) W. Forst, "Theory of Unimolecular Reactions", Academic Press, New York, 1973, Chapter 6. (15) Write da = Pabdb, and make the substitution L h = ~~.v,,,b. (16) H. Eyring, J. 0. Hirschfeider, and H. S. Taylor, J . Chem. Phys., 4, 479 (1936); E. Vogt and H. Wannier, Phys. Rev., 95, 1190 (1!354); G. Gioumousis and D. P. Stevenson, J . Chem. Phys., 29, 294 (1958). (17) A preliminaryaccount of this work has been given by W. J. Chesnavich and M. T. Bowers in "Gas Phase Ion Chemistry", M. T. Bowers, Ed., Academic Press, New York, 1979. (18) J. C. Light, Discuss. Faraday Soc., No. 44, 14 (1967), and references therein. For a review of recent advances see ref 17. (19) W. J. Chesnavich and M. T. Bowers, J. Chem. Phys., 68, 901 (1978). For related approaches see, for example, B. C. Eu and W. S. Liu, ibid., 63, 592 (1975); T. M. Mayer, B. E. Wilcomb, and R. B. Bernstein, ibid., 67, 3507 (1977), and D. M. Manos and J. M. Parson, ibid , 69, 231 (1978). (20) See, for exampie, ref 11 and references therein. (21) Reference 5. Note that if A v = 0, then eq 16 becomes eq 24 (with, perhaps, some constant factors included for changes in rotational constants and vibrational frequencies). When eq 24 is inserted into eq 17, the line of centers expression is obtained. (22) D. R. Coulson, J. Am. Chem. SOC.,100, 2992 (1978); E. Pollack and P. Pechukas, ibid., 100, 2984 (1978). (23) S. E. Stein and B. S. Rabinovitch, J. Chem. Phys., 58, 2438 (1972).
Application of SCF-SI Theory to Vibrational Motion in Polyatomic Molecules Joel M. Bowman,* Kurt Christoffe1,t and Frank Tobid Deparlment of Chemistry, Illinois Institute of Technology, Chicago, Illinois 606 16 (Received October 23, 1978) Publication costs assisted by the National Science Foundation
The self-consistentfield and state interaction methods are formulated for coupled vibrational motion in polyatomic molecules. This general approach should have many applications. Two model systems are considered. The vibration in nonbending COZ is studied and the calculated energies are compared to those obtained by use of other approximate methods. A model isomerization reaction in which motion along the reaction coordiinate is coupled to a harmonic bond is studied with the self-consistent field method. The dependence of the rate of isomerization on coupling strength and excitation in the harmonic bond is focused upon and easily understood within the self-consistent field framework.
I. Introduction In its role in polyatomic vibratiorial spectroscopy, anharmonic coupling has traditionally been viewed as a perturbation. However, in unimolecular reaction rate *Alfred P. Sloan Fellow. t I.I.T. Fellow. Science Applications, Dayton, Ohio 45432,
0022-3654/79/2083-0905$01 .OO/O
theory, notably RRKM theory,',' its character is seen as Paramount. Clearly, there exists a challenge to theorists to develop methods which Can handle the coupling range spanned by these two viewpoints in a unified and Physically meaningful way. This challenge is made especially acute in light of Bunker and co-workers' pioneering classical trajectory studies of unimolecular reactions." In this work, phase space "bottlenecks" due to the details of 0 1979 American
Chemical Society
906
J. M. Bowman, K. Christoffel, and F. Tobin
The Journal of Physical Chemistry, Vol. 83, No. 8, 7979
the dynamics were observed. Molecular beam,4 multiphoton,j and single photon6 experiments have also greatly stimulated recent trajectory Many of the issues relating to unimolecular reactions are also of importance in polyatomic collision dynamics. Several classical trajectory studies of energy transfer in triatom systems,s11 and one explicitly demonstrating the importance of anharmonic coupling on energy transfer in C02,12have been reported recently. A quantal study of the importance of anharmonicity in mode-to-mode energy transfer in COz is in progress in this laboratory and results of the harmonic system have already been reported.13 A general quantal approach which we have developed is the description of the vibrational states of polyatomic molecules employing the self-consistent field (SCF) approach.14 This approach was also developed previously, by Sprandel and Kern, working independently.lj Like the well-known theory in electronic structure calculations, the total wave function is written as a product of mode-type, single “particle” (phonon) functions, and then variationally optimized. Thus, this picture retains a physically transparent framework of independent motions (like Hartree-Fock theory) while obtaining optimum accuracy. In addition, as we will demonstrate, the SCF basis is a very good one in which to represent the “exact” quantal wave function. These two aspects of the SCF method are illustrated in the present paper. First, we review the SCF method and show how it can be combined with state interaction (SI) theory to efficiently obtain “exact” quantal eigenvalues and eigenfunctions for coupled polyatomic vibrational motion. Application is made to nonbending COz and the results are compared to other approximate methods. Second, the SCF theory is applied to a model system consisting of motion in a double well coupled to a harmonic oscillator. One-dimensional double well potentials have long been used in the analysis of microwave spectra.16 We expect that this model will be of use in the description of unimolecular isomerizations, particularly isomerizations induced by laser excitation of a normal mode other than the reactive mode. The effects of coupling between motion along the “reaction” coordinate and bound harmonic motion transverse to the reaction coordinate are investigated in this work. The SCF theory is particularly well-suited to such a study since the separable form of the SCF wave function leads to new and meaningful physical insight into the effects of such coupling on the isomerization rate.
11. Self-Consistent Field a n d S t a t e Interaction Theory SCF Theory. Let the total Hamiltonian for the vibrational motion of an m-mode polyatomic molecule be given by m
where hop(i)represents the separable part of Hopfor the ith mode and V , is a coupling potential depending on all of the normal coordinates. Consider now the variationally best simple product wave function
The mode functions 4L(QJsatisfy the following coupled integro-differential equations14 [hopO’)+ ( n 4 n , l V J n 4 n z )- ~ n , l 4 n , ( Q I ) = 0 (2.3) 1 $1
1 fl
j=l,m
#:f:z,,,,,nm
is normalized to unity, as usual. The total energy is given by
The coupled equations represented by (2.3) are one dimensional, so they can be solved numerically, self-consistently. The procedure is to start the iteration with a zero-order guess for the &(QJ given, for example, by
[hop(i)- &:)l~L?(QJ
=0
(2.5)
i = 1,m The 4 :;’ are used to evaluate the effective coupling potentials and a new set of $nt is calculated by solving (2.3). These first iterates 4;;)are then inserted into (2.3) to generate second iterates 4;’’. The process continues until the eigenvalues E,, and the iunctions &(QJ are unchanged upon further iteration. (Of course, in acutal computations, a convergence criterion based on a desired or machineimposed precision is adopted.) The numerical method employed in solving the eigenvalue equation (eq 2.3) for a given iteration is based on a finite difference boundary value meth0d.l’ The finite difference approximation to the second derivative operators transforms eq 2.3 into a matrix equation. The matrix is Householder transformed to tridiagonal form and the desired eigenvalue and eigenvector are obtained by Sturm sequence bisection and inverse i t e r a t i ~ n . ’ ~All of the calculations reported were done on the IIT Prime 400 computer in double precision (approximately 12 significant digits). S I Theory. The SCF functions form a very good basis in which to express the exact wave function. Thus, in terms of the SCF basis the exact wave function $K(Q) is given by
$tCF(Q)
(2.6) The resulting matrix Schroedinger secular equation is
(H-ES)C=O
(2.7)
where HIJ
=
($?CFIHopl#?CF)
(2.8a)
and SIJ =
($PCFI$jCF)
(2.8b)
After diagonalization of the overlap matrix S, the numerical solution of eq 2.7 proceeded as before with the Householder transformation of the H matrix to tridiagonal form. Again, the eigenvectors and eigenvalues were obtained by Sturm sequence bisection and inverse iteration. In some applications of the SCF-SI method, the “exact” wave function corresponding to a given SCF state, $EcF, is sought. In order to limit the number of SCF states employed in the SI expansion for $K (eq 2.6), only those SCF states which interact strongly with $ECF are included. The matrix element H K L is a measure of this interaction, and we consider only those SCF states, #FcF, for which IffKL[1 H, where H is some threshold energy. This approach is illustrated in the next section, and the “exactness” of the expansion is examined as a function of the value chosen for H . 111. Application to C O z In this section we present SCF and SI calculations of the energies for nonbending COz. The potential employed is the one given by Suzakawa, Thompson, and Wolfsberg.lo
The Journal of Physical Chemistry, Vol. 83,
Application of SCF-SI Theory to Polyatomic Molecules
TABLE I: CIO, Nonbending Potential and Coordinate Systems V m ( p i ) = D e [ -l exp(-api)]*;i= 1, 2a D e ___--- 0,200399 hartree 01 = 1.60235 (bohr radius)-] F R = 0.081452 hartree (bohr radius)-* Bond Displacement to Normal Coordinate Transformation P ] =: WQS i- X Q A p2 w Q s - XQA w = 1/2(2/m0)"2
a
No. 8, 1979 907
TABLE 11: State Interaction Convergence of the Energy of the Ground State of Nonbending CO, no. of
states
H , hartree ,mixed 5.0(-4)a 1.0(-4) 5.0(-6) 5.0(-7) 5.0(- 1 2 ) a
E S I , hartree 0.8524550(-2) 0.8504954(-2) 0.8504688(- 2) 0.8504686(-2) 0.8504686(-2)
1 5 17 32 50
ESCF - E S I ,
hartree
0.0 1.9595(-5) 1.9862(-5) 1.9864(-5) 1.9864(- 5)
( - n ) denotes 10.".
TABLE 111: Exact Quantum (SI) and Approximate Energies (in eV) for Ground and Excited States of
Reference 10.
Nonbending CO,
(This potential was also previously used to describe unperturbed C 0 2 in recent trajectory studies of energy t r a n ~ f e r . ~ J ~ J For ' ) nonbending COz, this potential is expressed in terms of bond displacement coordinates, p1 and p2: (3.1) V(P1,P2) = Vm(P1) + vm(p2) -k FRPlPZ The potentials V, are simple Morse functions; their parameters andl F R are given in Table I. In terms of the normal coordimates Qs and &A, defined in Table I, the total Hamiltonian is given by
where V(QS,QA)= 13,[1 - exp(-auQs - ~ x Q A ) ] ' + D,[1 - exp(-awQs -t. axQA)]' + FR(u2Qs2- x ~ Q A ~ (3.3) )
nS,nA
SI
SCF
HOa
PTb
SCc
00 10 01 20
0.2314 0.3965 0.5212 0.5606 0.6826 0.7238 0.8057 0.8431 0.8861 0.9636 1.0027 1.0475 1.0851 1.1208 1.1615 1.2081
0.2320 0.3986 0.5228 0.5642 0.6886 0.7288 0.8079 0.8535 0.8926 0.9730 1.0174 1.0554 1.0873 1.1371 1.1804 1.2173
0.2325 0.4004 0.5294 0.5683 0.6974 0.7362 0.8264 0.8653 0.9041 0.9943 1.0332 1.0720 1.1233 1.1623 1.2011 1.2399
0.2320 0.3971 0.5218 0.5612
0.2308 0.3961 0.52121 0.5607
-
11 30 02 21 40 12 31 50 03 22 41 60
a Uncoupled harmonic oscillator. Second-order quantal perturbation theory (ref 12b). Semiclassical (ref
12b).
The coupled equations for the SCF mode wave functions are, from ( 2 . 3 )
C-l/z(a2/aQB,2) + (d)nAIV(Qs,QA)Id)nA)- c,,ld),,(Qs)
9.0
=0
r
(3.4a) 70
[-1/2((12/@!~2)+ (hslV(Q~,Q~)Id)n,bf n ~ l d ) n ~ ( Q A= ) 0 (3.4b) The expression for E :: is given by eq 2.4. The iteration procedure was initiated by approximating V as two uncoupled quadratic plotentials and, therefore, d)ko) and $Lo) are simply harmonic oscillator wave functions. %h'e SC# states considered here took approximately 20 iterations to converge according to the following convergence criteria: ( I ~ ~ ( -Q~J ~ ( Q JII 10-5 )
(3.5a)
I
v
60
"0 x
50
m w 40 I U
2
W
30 20
(3.5b)
IO
SI wave functions and energies were obtained for a given
00
l(~(") n,
- c(n-l))/t(n)l n, n, 5
i = S, A
reference SCF state as described in the previous section. An illustration of the sensitivity of the SI energy of the ground state to the choice of the threshold energy H is given in Table 11. With just a five-state mixing ( H = 5.0 X hartree), 98.6% of the correlation energy, Le., EsCF - EsI is accounted for This rapid convergence is illustrated graphically in Figure 1for the (nS,nA)states (0,O) and (0,3). The total correlation energy is reached for both states a t approximately the same threshold energy H. Finally, in Table 111 a comparison is made of energies calculated in four approxirnation methods with converged SI energies. The semiclawical (SC) and second-order quantal perturbation theory (PT) calculations were performed by Schatz and Mulloney.lZb The perturbation calculations are for a coupled quartic potential derived from the full Suzakawa-Thompson-Wolfsberg patential.1° Except for
__
0
2
4
6
8
1 0 1 2
14
-LOG H Figure 1. Energy difference between the self-consistent field (SCF) and exact state interaction (SI)energies for two vibrational states of nonbending GO, as a function of the negative logarithm of the threshold parameter, H, defined in the text.
the ground state, the four SC energies are in excellent agreement with the exact SI energies. The second-order perturbation energies are in better agreement with the SI energies than are the SCF energies for the states indicated. This suggests that the normal harmonic modes are not very strongly coupled. Further evidence for this is the fact that the corresponding harmonic energies are typically in error by only 3-5%. Additional comparisons between SCF and second-order perturbation calculations would be instructive.
TABLE IV: Potential Parameters for the TwoDimensional Double Wella
1.60 I
I
0.060503 hartree/(bohr radius)2 b = 0.008453 hartree/(bohr r a d i ~ s ) ~ c = 1.1954 (bohr radius)-2 V, = 0.056241 hartree w 0 = 0.015001 (atomic time units).’ a = 6/bohr radius n=2 rn = 4665.664 electron mass a=
a
J. M. Bowman, K. Christoffel, and F. Tobin
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
908
I
I
I
Reference 19.
The SCF-SI method has been shown to be quite efficient for calculating “exact” quantal eigenvalues and eigenfunctions for nonbending C02. We believe that the use of both the SCF basis and the threshold criterion for constructing the Hamiltonian matrix are responsible for this efficiency.
IV. Self-consistent Field Study of an Isomerization Reaction The simplest model for an isomerization reaction is motion in a double well potential along the reaction coordinate. The isomerization rate is then the rate of passage from one well to the other. Other physical processes can also be represented as motion in a double well, e.g., diffusion of hydrogen in metals.18 A more sophisticated model is one in which motion along the reaction coordinate is coupled to other internal degrees of freedom. Such a model has recently been considered by Garrett.lga In terms of x,the reaction coordinate, and y , an internal vibrational coordinate, this potential is given by V(X,Y) = V,,(X)
+ 1/mu,2(x)Y2
(4.1)
where VDw(x)is a symmetric double well potential. This coupled-mode isomerization model is well-suited to an SCF formulation. In our SCF formulation the total potential V(x,y) is written as the sum of two single-coordinate potentials and a coupling potential: V(x,y) = vDW(x) + v H O ( Y ) + vC!(x,y) (4.2) where VD,(x) = l/,ux2 + 1/2bx4 + VI exp(-cx2)
(4.2a)
VHO(Y) = 1/muo2y2
(4.2b)
and VC = V,o(y)[X exp(-(ax)”)][X exp(-(ax)n) - 21
(4.2~)
The parameters defining the potential V(x,y) are given in Table IV. The strength of the coupling is determined by a coupling parameter X. Values of X in the range from 0 to 0.3 are considered,,with X = 0 representing the uncoupled problem. The effect of this coupling is a relaxation of the vibrational frequency in the y coordinate near the saddle point. This allows more energy to be available to promote the motion along the reaction coordinate. The separable product wave function is given by +n,n, = +nx(x)Xn>(Y) (4.3) and the following Fock equations determine the singlecoordinate functions &(x) and xn,(y): [(-1/2m)(a2/6x2) + V D , ( X ) + (XnyIVCIXny)- ~ n z 1 4 n x ( x )= 0 (4.4a) [(-1/2m)(’32/6y2) + VHOb) + (@n,IVcI4n,)
-
~ n , I ~ n , ( y )=
0 (4.4b)
These equations are solved numerically using the iterative
-1.60
-0.80
0.00
0.80
1.60
X Figure 2. The first six double well enerav eiaenvalues n, = 0-5. for the uncoupled, h = 0, potential. Doublgw& potential for h = 0 as a function of the double well reaction coordinate, x(au).
SCF procedure to obtain the one-dimensional eigenvalues, the single-coordinate functions, and the total energies as described in section 11. The coordinates x and y were sampled over the range from -2.00 to 2.00 au at 1101 grid points. The eigenvalues obtained were considered to be converged when the relative difference between eigenvalues from two successive iterations was less than or equal to The single-coordinate functions were considered to be converged if the absolute difference in the function from successive iterations at every grid point was less than or equal to 10-j. These convergence criteria were satisfied after 4 to 46 iterations depending upon the SCF state of interest. This involves the use of approximately 1-20 min of CPU time on the Prime system for the calculation of a single SCF state. As a test of the accuracy of our technique, we compared the SCF energies for the uncoupled harmonic oscillator to the exact harmonic oscillator energies for quantum numbers 0 through 7 . The relative difference between the SCF eigenvalues and the analytical energies was never greater than 4.0 X The SCF energies calculated for the uncoupled problem (A = 0) are summarized in Figure 2, which is an energy level diagram for the six lowest energy states in the double well. As expected, the states occur in pairs consising of a symmetric state and an antisymmetric state of slightly higher energy. The quantity of interest when a simple rate theory is applied to this model isomerization is the energy diifference, A, between the states in these pairs. In Table V the values for ASCF obtained by taking the difference in total energy between In,,n, = 0) and Jnz’,ny= 0) are shown for the two lowest pairs of double well states at four values of A. The exact quantum results, AEQ, given in Table V for comparison were obtained by Garrettlg8for the same potential. Garrett also employed various semiclassical techniques to obtain A. The SCF procedure yields the same splittings as the exact quantum calculations for the uncoupled problem. For X = 0.1, a case corresponding to weak coupling, our SCF values of 4 still compare favorably to the exact quantum results. For the more strongly coupled problems ( A = 0.2 and X = 0.3) the simple SCF procedure produces less accurate results, but, in all cases considered in Table V, the SCF procedure proves superior to all of the semiclassical techniques employed by Garrett. It is hoped that future calculations using state mixing in the SCF-SI technique will provide us with more accurate splittings, energies, and single-coordinate functions for the more highly coupled cases. Looking a t Table V, one immediately notices that as X increases, the energy difference 4 between members of
0 8
l"F==--l
I
-240
-40
00
-20
X
;.:I -400
035
-
-5
f-
'z
,
The Journal of Physical Chemistry, Vol. 83, No. 8, 7979 909
Application of SCF-SI Theory to Polyatomic: Molecules
-600 -40
X
2 U -
1.60
:j
-
NX
x Flgure 3. Self-consistent field total energies as a function of coupling parameter, A. Each set of energies corresponds to the fixed value of nu (the harmonic oscillaltor quantum number) indicated.
state pairs, also increases. It is instructive to investigate the manner in which this increased splitting arises. In Figure 3 we have plotted the total energy of 24 states as a function of the coupling parameter X. Each graph of Figure 3 shows the energy of the six lowest lying double well states a t a fixed ny as a function of A. This figure displays two important features of the coupled problem. First, it is evildent that the total energy of all states is lowered as a result of' the interaction that accompanies coupling of the two coordinates. The other striking feature of Figure 3 is the greater interaction shown by symmetric (n, = 0, 2 , 4 ) slates compared to their antisymmetric (n, = 1, 3, 5) partners. These trends in energy with X can be most easily understood by a consideration of the SCF effective potential. The energy of the SCF state Inx,ny)is given by
+ eny - (+n,XnylVCl+n,Xny)
(4.5)
where t n l and f,, are the eigenvalues given in eq 4.4a and 4.4b and Vc is defined according t o eq 4 . 2 ~ .The matrix element, (&xn IVcl+,,,xn ), in eq 4.5, is called the interaction energy. 6-1 Figure bwe show the effective potential, (xn,(Y)lVCIXn,(,Y)),denoted by Veff(x;ny)for ny = 0, 1, 2, 3 and X = 0.3. The Veffshown are typical of the effective potentials for id1 values of n3 and A; only the magnitude of Veffvaries with the state of interest. Clearly there is a sharp minimum in Veff(x;n,) a t x I- 0. This behavior results from the form of w,(x) o ~ ~ (=x )wo[l - X exp(-a2x2)]
-owl
-T I
-
-240 4 0
-20
20
00
X
I
1.0.3
q,Iff0
enr
>-
I
I
=
-040
xnyl
225-
En,,ny
Ni
Figure 4. Effective potentials ( V,Ixn ) as a function of x(au) for ny = 0-3, n, = 0, and coupling paramefkr, X = 0.3.
kY
220
000
(4.6)
3%1.20
-40
T
-20
00
20
40
X
Figure 5. Harmonic bond frequency w y as a function of the double well reaction coordinate x(au).
which also possess a spiked minimum at x = 0 as shown in Figure 5. Thus, the interaction energy is always negative and the SCF energies for the coupled problem are correspondingly lower than the respective uncoupled energies. The behavior of Veff(x;n,) also accounts for the observation that the energies of even n, states decrease more rapidly with increasing X than do the odd n, states. This follows since the uncoupled (and SCF) even n, states have nonzero amplitude a t x = 0 where Veff(x;n,) is largest, whereas the odd n, states have a node there (as shown in Figure 6). Therefore, we expect that the magnitude of the interaction energy integral would be larger for even n, states than for odd n, states. Since Veff(x;n,) is everywhere negative or zero, the interaction energies are all negative, and as a result symmetric states are lowered in energy more than their antisymmetric partners. It can be easily shown,20for states in which the energy contained in the double well coordinate is well below the barrier height, that one can represent a wave packet initially localized in one of the wells by a suitable linear combination of product wave functions, $nz,ny. If the time evolution of this localized wave packet is followed, it is seen that the wave packet will oscillate between the two wells
910
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
J. M. Bowman, K. Christoffel, and F. Tobin 1.00 nx =2
n,=O
1.20
I
0.50
r
2
I
c
8
3
-
L
5 0.00
0.80
4
5 040
-
OW
A
-050
1
J
1
-100
-40
-2.0
X
2.0
00
I
-4.0
-2.0
d I
0.0
I
2.0
1
X
I.0C
1.00
n =4 0.50
nx:5
I60
0.50
5
0.00
a
5
-0.50
-0.50
-100
-0.80
-4.0
-2.0
0.0
2.0
‘
-1.00
X X Figure 6. Uncoupled double well eigenfunctions for n, = 0-5 as a function of the double well reaction coordinate x(au).
TABLE V: Energy Differences Double Well
0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.3 a
0,0;1,0 2,0;3,0 0,0;1,0 2,0;3,0 0,0;1,0 2,0;3,0 0,0;1,0 2,0;3,0
0.83 36.4 0.96 44.4 1.10 54.9 1.27 68.4
A
for the Coupled
0.83 36.4 0.95 44.2 1.08 53.6 1.22 64.7
0.00 0.00 1.04 0.45 1.82 2.37 3.94 5.41
Exact quantum results from ref 19a.
with a frequency w / i r where w A / h . Since this motion from one well to the other is “isomerization”, in our model, U / T is then the isomerization rate. In the present model it is of interest to see how the splittings (and hence the isomerization rate) vary with excitation of the harmonic bond for a fixed coupling A. In Figure 7 we have shown how Aol and A23 vary with ny for X = 0.1. Here An,,,,’ is defined as the difference in total energy between the states Inx/,vy)and In,.,ny). The dotted line in each of the graphs indicates the value of A when X = 0, where the modes are uncoupled and therefore the energy difference is independent of the state of the harmonic oscillator. When X = 0.1 there is weak coupling and the energy difference increases monotonically with ny for 0 I ny I7. This indicates that A, and hence the rate of isomerization, increases as ny and the energy of the system
increase. It is interesting to note that the total energy of the coupled system is substantially greater than the classical barrier height of the uncoupled double well potential for all of the states shown in Figure 7. For the uncoupled case, however, most of the total energy is trapped in the harmonic bond. The isomerization rate in the uncoupled system is quite small in comparison to the coupled systems, since the energies of the uncoupled double well states are below the classical barrier height and there is no means for the energy in the harmonic bond to reach the double well motion. For the coupled case shown in Figure 7 , some of the energy in the harmonic bond can be transferred to the double well to promote the isomerization. Let us now consider a first-order perturbation argument which allows us to represent the splitting A as a function of ny for a fixed X. We replace &z(x) and xn (y) by their zero-order, Le., A = 0, approximations, &/)(xf and xi!(y). Thus, for example, the first-order splitting 4;) is 1
= %,=I ( 0 ) + &O) n3 -
%,=O (0)
-
ny
+ ( Q l o ’ x ~ ~ ) ~ v c-~ ~ ~ o ’ x ~ ~ (oi(,O~X~~~lVcl~BO~X!$ (4.7)
We can rewrite eq 4.7 as
ah:’
= A(!,)
+ ( $lo)lV&;ny) Ido) ) - ( do)l V%(w,) Ido) ) (4.8)
where the zero-order splitting A#! is simply the difference between the zero-order double well eigenvalues, ci$l -
Application of SCF-SI Tlheory to Polyatomic Molecules 800
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The Journal
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t
1
a" ,
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200
-20
00
20
40
20
4
X
0
U 0
1
"1
L 2
3
4
5
6
7
"Y
! 20
40r-
1-
'""
I
nyi'
I
30
040
O M
-40
-20
, 00
20
0 -40
-20
00
X
Figure 8. Ground state double well eigenfunctions as a function of the double well reaction coordinate, x(au), for X = 0.3 and nyas indicated.
x.00 0
1
2
3
4
5
6
7
5 Figure 7. Energy splittings between the first pair of states, A,,, and the second pair of states, as a function of the harmonic oscillator quantum number, nu for X = 0.1. Open circles are the self-consistent field results, dashed gives the uncoupled, X 0, results, and solid line the first-order perturbation theory results.
c ~ o ~ o .For X
imuch less than one, for example X = 0.1,
@)(x;ny)can be written to first order in A as (cf. eq 4 . 2 ~ )
This equation can be simplified to read V!$)(:e;ny) = -X(n, + '/z)hwoexp(-a2x2)
(4.10)
Thus, A& is
A# = Ah!)
+ A(ny+ l/z)hwoCol
(4.11)
where col= ((P6°)lexp(-a23c2)1~60)) - ( ~1°)lexp(-a2x2)1~10)) (4.12) From our previous argument concerning the magnitude of the effective potential for even n, and odd n, states, we know that the first term on the right-hand side of eq 4.12 is larger than the second term, and thus Col is a positive quantity. The conclusion from first-order perturbation analysis is that A,,l (and A23) should vary linearly with both nyand A. As shown in Figure 7, this prediction is in rough qualitative agreement with the accurate SCF results for A01 for nSy< 3 ,and for A23 for ny < 1. The deviations grow rapidly for ny greater than these specified valuesaZ1 Consider now the more strongly coupled cases, X = 0.2 and X = 0.3. As X increases the interaction energy of symmetric states becomes so great as to produce entirely
new state pairings. A t large X values the interaction potential is so great that the total potential can no longer be described simply as a double well potential. The change in the interaction potential is best reflected by the change in the nature of the single-coordinate function & ( x ) . For A = 0.3 the single coordinate functions, &,(x), for (nx,ny) pairs (O,O), (O,l), (0,2), and (0,3) are shown in Figure 8. Compare these to the single-coordinate function for the uncoupled problem (Figure 6). Clearly for ny > 2 the character of the states has been drastically altered. The significance of A according to the simple theory discussed above is obviously lost as localized wave packets can no longer be constructed in a simple manner. Apparently a more general treatment of the wave packets is required. Future calculations of isomerization reactions are planned based on the SCF and SCF-SI theories. These will include a study of branching probabilities when more than one stable product can be formed, multiphoton induced isomerization, and tests of approximate theories of isomerization reactions, e.g., RRKM theory.
V. Summary and Conclusions The self-consistent field theory of anharmonic coupling of polyatomic vibrations together with state interaction provides an efficient algorithm for obtaining "exact" quantal eigenfunctions and eigenvalues for both low lying and highly excited vibrational states. This was demonstrated for the first 16 states of nonbending COz in the present paper and appears to hold even for states approaching the dissociation limit.22 Self-consistent field theory has been applied successfully to a prototype isomerization reaction. The rate of isomerization in this model is a sensitive function of the coupling to, and the energy content of, a harmonic bond coupled to motion along the "reaction" coordinate. In addition to providing insight into this dependence the SCF basis can easily be used to obtain the "exact" stationary states describing isomerization reactions. These states can
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The Journal of Physical Chemistry, VoL 83,
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then be used in a realistic time-dependent study of the detailed isomerization rates. The SCF or "exact" states should find application in studies of multiphoton or single photon induced isomerizations.
Acknowledgment. One of the authors (K.C.) thanks Dr. Judith Grobe Sachs for her aid in the final preparation of this manuscript. This work was supported by the National Science Foundation. References and Notes (1) P. J. Robinson and K. A. Holbrook, "Unimolecular Reactions", Wiley, London, 1972. (2) For a recent review, see W. L. Hase in "Modern Theoretical Chemistry", W. H. Miller, Ed., Vol. 11, Plenum Press, New York, 1976, Chapter 3. (3) (a) D. L. Bunker, J . Chem. Phys., 40, 1946 (1964): (b) D. L. Bunker and M. Pattengill, ibid., 48, 772 (1968): (c) H. H. Harris and D. L. Bunker, Chem. Phys. Lett., 11, 433 (1971): (d) D. L. Bunker, J . Chem. Phys., 57, 332 (1972). (4) K. Shobatake, Y. T. Lee, and S. A. Rice, J . Chem. Phys., 59, 6104 (1973), and references cited therein. (5) See, for example, M. J. Coggiola, P. A. Schulz, Y. T. Lee, and Y. R. Shen, Phys. Rev. Lett., 38, 17 (1977), and references cited therein. (6) K. V. Reddy and M. J. Berry, "Dye Laser Induced Photochemistry", presented at the 175th National Meeting of the American Chemical
Su et al. Society, Anaheim, CA, Mar 12-17, 1978, No. 151. (7) J. D. McDonald and R. A. Marcus, J. Chem. Phys., 65, 2180 (1976). (8) C. A. Parr, A. Kuppermann. and R. N. Porter, J . Chem. Phys., 66, 2914 (1977). (9) N. Sathyamurthy and L. M. Raff, J . Chem. Phys., 66, 2191 (1977). (IO) H. H. Suzakawa, Jr., M. Wolfsberg, and D. L. Thompson, J , Chern. Phys., 68, 455 (1978). (11) A. J. Stace and J. N. Murrell, J . Chem. Phys., 68, 3028 (1978). (12) (a) G. C. Schatz and M. D. Moser, J . Chem. Phys., 68, 1992 (1978); (b) G. C. Schatz and T. Mulloney, J . Phys. Chem., this issue. (13) J. M. Bowman and S. C. Leasure, Chem. Phys. Lett., 56, 183 (1978). (14) J. M. Bowman, J . Chem. Phys., 68, 608 (1978). (15) G. D. Carney, L. I. Sprandel, and C. W. Kern, Adv. Chem. Phys., 37, 305 (1978). (16) J. 8. Coon, N. W. Naugie, and R. D. McKenzie, J . Mol. Spectrosc., 20, 107 (1966). (17) J. H. Wilkinson, "The Algebraic Eigenvalue Problem", Clarendon Press, Oxford, 1965. (18) See, for example, J. H. Weiner, J . Chem. Phys., 68, 2492 (1978), and references cited therein. (19) (a) E. C. Garrett, Ph.D. Thesis, University of California, Berkeley, 1977, unpublished; (b) J. D. Swalen and J. A. Ibers, J . Chem. Phys., 36, 1914 (1962). (20) E. Merzbacher, "Quantum Mechanics", 2nd ed, Wiley, New York, 1970, p 70. (21) The second-order correction to A will add a term which varies quadratically with nu. This would improve the qualitative agreement with the SCF result. (22) F. Tobin and J. M. Bowman, unpublished.
A Fourier Transform Infrared Kinetic Study of HOC1 and Its Absolute Integrated Infrared Band Intensities Fu Su, Jack G. Calvert," Charlet R. Lindley, William M. Uselman, and John H. Shaw Departments of Chemistry and Physics, The Ohio State University, Columbus, Ohio 43210 (Received October 16, 1978) Publication costs assisted by the US. Environmental Protection Agency
Long-path, Fourier transform infrared spectroscopy has been employed in the kinetic study of the products of the photolysis of dilute C12-03-H2mixtures in excess O2 and N2 in experiments at 25 f 3 "C and 700 torr total pressure. The initial rates of formation of the products of the reaction, HC1, HOCl, and H20z,and O3 loss were studied as a function of the ratio of reactants, [H2]/[03],over the range 0.10 X 103-6.8 X lo3. The match of these experimental data with computer-generated rate data employing a rather complete reaction set was used to test the mechanism and refine some of the rate constant estimates. From these data, the absolute extinction coefficients for the three fundamental bands of HOCl were derived. The integrated band intensities for the u l , u2, and u3 absorption regions were estimated to be 2.3 X lo2, 3.0 X lo2, and 4.3 X 10' cm-2 atm-l, respectively.
Introduction Molina and Rowland' first suggested in 1974 the potential role of the chlorofluoromethanes and other chlorine-containing compounds in perturbing significantly the natural stratospheric ozone levels. Interest has grown since that time in the quantitative evaluation of the extent of this perturbation. In theory one anticipates that the measurable depletion of ozone should occur slowly, some years after C1-compound injection, so only indirect observations related to the proposed reaction mechanism can be made to test the 03-depletionhypothesis today. In this regard there has been active research related to both the kinetics of the intermediates and the elementary processes involved as well as the experimental measurements of the atmospheric abundance of the chlorine-containing species presumed to be involved in the O3 depletion mechanism (Cl, C10, HC1, C1N02, C10N02,etc.).2 The comparison of 0022-3654/79/2083-09 12$0 1.OO/O
the measured and the predicted levels of each species provides a significant, although indirect, test of the hypothesis. It has been suggested that HOCl might form in the stratosphere through the occurrence of the homogeneous reaction 12* and conceivably on aerosol surfaces through H 0 2 + OC1- O2 HOCl (1) C10N02 + H 2 0 (+ aerosol) HOCl + HON02 (+ aerosol) (2) the heterogeneous reaction 2.6 Recent kinetic studies favor HOCl involvement in that they show K1 to be large, about 3.8 X cm3 molecule-l s-1.7-9 However, recent measurements of the ultraviolet spectrum of HOCl suggest a rapid photolysis of HOCl will occur in the stratosphere; this leads to the conclusion that the HOCl will not provide a significant sink for inert chlorine in the stratosphere."
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0 1979 American
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