Inversion of experimental data to extract intermolecular and

Dec 1, 1993 - 13448 The Journal of Physical Chemistry, Vol. 97,. 1. No. 51, 1993 ... although surprisingly alien to the physical chemistry community, ...
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13447

FEATURE ARTICLE Inversion of Experimental Data To Extract Intermolecular and Intramolecular Potentials Tak-San Bo' and Herschel Rabitz' Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Received: May 25, 1993; In Final Form: September 14, 1993'

We present a general nonlinear inverse method utilizing discrete experimental dara to extract inter- and intramolecular potential energy surfaces. The inverse method is formulated in terms of perturbation expansions of the experimental data upon the functional variations of the underlying potential energy surface-a furictional sensitivity analysis approach. A distinction is drawn between the inverse method and the conventional parameter fitting procedure in that the former treats the potential energy surfaces as continuous functions of the internuclear coordinates, whereas the latter is based on restricted forms with a small number of parameters. The possible numerical instability of molecular nonlinear inverse problems is examined in detail using singclar function expansion analysis and is overcome using the Tikhonov regularization method, which incorporates the a p i o f i smooth properties of the sought-after potential energy surfaces. Numerical studies show that the iterative inversion procedure based on this inverse method is generic, efficient, and stable and is capable of accrlrarely rendering physically acceptable potential energy surfaces for a variety of problems-either spectroscopic or collosional and one-dimensional or multidimensional. An example employing actual laboratory data has been successfully inverted. Application of the method to small polyatomic systems of current interest and improvement of the method by including higher-order sensitivity densities are also discussed.

Introduction Accurate knowledge of inter- and intramolecular potential energy surfaces is essential for the understanding of various molecular processes and chemical reactions.' The determination of potential energy surfaces has come a long wayZd and generally can be pursued by two different approaches as illustrated in the flow chart depicted in Figure 1: One is to perform ab initio quantum chemistry calculations within the Born-Oppenheimer approximation, and the other is to perform inversion of experimental data. Important strides in these endeavors have been made, especially in the past decade, because of recent important advances on many fronts. In particular, the availability of powerful new generations of vectorized and parallel supercomputers,7in conjunction with some truly ingenious recent theoretical developments, has made possible many computationally intensive tasks, including the ab initio calculation of the potential energy surfaces,* the simulation of various microscopic, particularly, spectroscopic and dynamic, properties,g and the simulation of various macro scopic, e.g. thermophysical, properties.I0 Furthermore, the ever improving state of the art of crossed-beam molecular machinesll and high-resolution spectroscopic techniq~esl2-1~ has produced large amounts of very high-quality data that contain direct and precise information on the underlying potential energy surfaces. Nevertheless, the most cutting edge computational quantum chemistry approaches to produce high-accuracy ab initio potential energy surfaces are still limited to relatively small systems of a few atoms and can only be afforded to calculate potential energies at a finite number of discrete geometries, typically around the equilibrium positions. Moreover, ab initio potential energy surfaces often do not possess the accuracy comparable to that needed to explain the modern laser spectroscopic data. On the ~~

~

Abstract published in Advance ACS Absrracrs. December 1 , 1993.

0022-3654/93/2091- 13447%04.00/0

other hand, existing numerical procedures to extract potential energy surfaces from laboratory data are almost exclusively of a parameter fitting nature,IsJ6 except for some simple systems. Direct noniterative inversion methods do exist for various specialized problems, e.g., the Rydberg-Klein-Rees (RKR) methodI7 for diatomic spectral problems, the semiclassicallsJg and quantum mechanical methods20 for elastic atom-atom scattering problems, the modified RKR method21 for van der Waals complex spectral problems, the self-consistent-field (SCF) method22for triatomic rovibrational problems, and the exponentia 1 distorted wave (EDW)23and sudden appr~ximation'~ methods for rotationally inelastic atom-diatom scattering problems. However, except for the diatomic RKR technique (which itself is confined to potentials with a single minimum), these methods have only seen limited applications because of various dynamical constraints imposed on the systems under study and various restrictions imposed on the measured data. At the present time, ab initio quantum computational methods have matured to the point that their ability to produce potential energy sirfaces is mainly limited by the speed of modern computers, whereas the machinery of extracting surfaces from experimental data is still inadequate, despite the large amount of high-quality data awaitirig scrutiny. To prudently determine potential energy surfaces from experimental data, we have to resort to genuine inverse problem theory25 that embraces the following facts. First, experimental data are usually incomplete, erroneous to some degree, or even convoluted-data are typically discrete measurements corresponding to some experimental setups and are contaminated by background noise and averaged due to limited apparatus resolution. Second, any one individual property may not be sensitive to all features of the potential energy surface, and thus measurements of several different properties may be needed to successful inversion. Third, potential energy surfaces generally @ 1993 American Chemical Society

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13448 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

Ho and Rabitz

problem corresponds to thedetermination of the potential function given a set of discrete data (observable measurements). Of paramount importance in dealing with any inverse problem are the fundamental issues of uniqueness, existence, a n d ~ t a b i l i t y . 2 ~ ~ ~ While first principle quantum chemistry calculations can always transport coefficients unambiguously yield one and only one potential energy surface for each molecular system under study, inversion of true experimental data usually does not result in unique solutions. trsnsition probabilities The corresponding inverse problem is usually substantially a o i s czctions line positions underdetermined because the amount of experimental data is always finite. Furthermore, laboratory data errors not only can make the data incompatible such that no inversion solution exists but also can make the inversion unstable. The stability issue is of particular importance when numerical procedures are impleEXPERIMENTAL DATA mented for data inversion. Since inverse problems to determine the inter- and intramolecular potentials from experimental data are typically of a nonlinear nature, it is generally difficult to revolve globally any issue related to the uniqueness and existence Data Inversion of inverse solutions. However, it is possible to generate significant insights relevant to the issue of stability by invoking the inverse method reviewed in this article. It may be also possible to overcome the ambiguity arising from the nonuniqueness of the POTENTIAL ENERGY SURFACE associated inverse problem by introducing apriori restrictions on certain characteristics of the potential energy surfaces, in accord with ab initio or empirical results. ab inifio Calculation8 that we present here for general molecular The inverse problems is intimately related to the functional derivatives (Le. Frkhet derivatives) of the experimental data with respect to the potential function-a functional sensitivity analysis ELECTRONIC INTERACTIONS approach.39 Intensive studies on these quantities for a variety of observables listed in Figure 1 showed that the corresponding forward problems are well-behaved (Le. the forward mapping from the potential function to the data is continuous and Frkhet This method is closely related to thellrewtonwhich is an iterative procedure to obtain Kantorovich inverse solution functions when solving nonlinear inverse problems do not assume fixed functional forms characterized by a small but incorporates a generalized Tikhonov regularization scheme number of parameters, except possibly in the asymptotic region to provide the much needed stability of the solution in the presence Among the inverse methods of large internuclear of the experimental data errors. It is possible to impose highthat are amenable to the posed criteria are the Backus-Gilbert order derivative constraints on potential functions through theory26combined with the Tikhonov regularization m e t h ~ d ~ ’ - ~ * regularization to render physically acceptable smooth solutions. for linear inverse problems and the Backus-Gilbert-Snieder Furthermore, many inherent difficulties arising from the illthe0ry2~ along with the Newton-Kantorovich method30 for posedness of inverse problems dealing with real experimental nonlinear inverse problems. It is particularly important to make data can be explained, and thus untangled, by invoking the wella fundamental distinction between an inverse method and known singular function (i.e. spectral) expansion analysis parameter fitting (estimation) methods: the former treats method.4u7 The latter method is a powerful analytic tool for potential energy surfaces (i.e. the solutions) as functions (of analysis of the Fredholm integral equation of the first kind which internuclear coordinates) whereas the latter is based on fitting forms the core of the inversion method discussed here. chosen functional forms with only afinite number ofparameters. The plan of this paper is as follows. First, we formulate the To be specific, the aim of an inverse method is to obtain the inversion method based on linear functional perturbation theory, potential by considering the experimental data as a functional of the Tikhonov regularization procedure, and the singular function the potential, instead of fitting the data to a constrained potential expansion technique. Then, we numerically study the He Ne form having a few parameters. This viewpoint of data inversion, system using real low-energy elastic scattering data to illustrate although surprisingly alien to the physical chemistry community, each aspect of the inversion algorithm. This is followed by has been widely adopted and has seen much success in many discussions on possible application of the inversion method to other disciplines, including geophysical science,31astronomical problems involving small polyatomic molecules and on possible and optical science.34 It is medical imaging extension of the method to include high-order functional perour intention here to convey the potential usefulness of data turbation terms. Finally, we conclude the paper with a short inversion methods based on general inverse problem theory for summary. the determination of inter- and intramolecular potentials from, especially, the ever growing pool of high-quality spectroscopic and scattering data. Inversion Method At first, we need to draw a distinction between a forward (or The inversion method to be described allows for the simuldirect) problem and an inverse problem. Here the forward taneous use of experimental data from measurements of several problem corresponds to the calculation of various observables, different properties such as given in Figure l.3s-38For convene.g., the scattering cross sections or rotational-vibrational spectra, by solving the governing Schrijdingerequation given the potential ience, we let 0 indicate the set of experimental data 0 * (O,& energy surface of the system under study. In contrast, the inverse where h denotes the hierarchical level in the data of Figure 1 and concentrations nlaxation times

i

II

I

I

I

1

+

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13449

Feature Article i represents the realization of the experimental data at each level taken in decretized form. The sought-after poential energy surface V(R)is a function of a set of coordinates R appropriate for the particular problem. Moreover, we shall adopt a perturbation theory viewpoint-the functional sensitivity analysis app-roach-for the relation between the data 0 and the pptential V(R). In particular, the mapping from the potential V(R)to the (calculated) data O[ Y]-the forward problem-is formulated in terms of the perturbation expansion

For this simple one-dimensional case, eq 2 can be readily transformed into an equivalent expres~ion3~9~~ 6oih

= L m K / l 1 ( R ) f f ) ( R )dR, R2 n 2 0

(4)

where the new kernel K$"](R)can be computed via the following recurrence relation rZr+2G1(R)= -j JoRR'i+ll&ll(R') dR', j = 1,2, ..., n (5) with

#il(R) where SV(k)is an arbitrary y r iation and the real kernels 6 0 i h / SV(@ and bzOih/6V(R)6V(R') are respectively first-order a>d second-order functional derivatives of Oih with respect to V(R). The iterative inversion procedure is based on the following Fredholm integral equation of the first kind,27J8-45946

E

bOih/6V(R)

or

6V(R) = -n J i [ R - RI"-' R"ff)(R') dR' with 6oih= Oih[V+SY]- Oih[VI, after neglecting all higher (12)order terms in eq 1. The high-order terms in eq 1 can also be included in the inversion procedure to improve upon the convergence and to better account for the nonlinear effect. We shall come back to this issue later. In the following text, the quantity oih[V+6v] in eq 1 shall be replaced by the real experimental data, denoted as q h , so that 6oih q h - Oih[Y]. In_princip!e, eq 2 can be solved to improve the potential from V(R)to V(R) 6V(k),and this procedure is repeated until it converges. T i n o v Regularization. The linear inverse problem defined by eq 2 can be extremely unstable (Le., ill-conditioned) because an arbitrary small error (from measurements and linearization) in th_equantity 8 0 i h can produce a large difference in the quantity bV(R),and its solutions may not be meaningful. Consequently, a practical level of stabilization which embodies ceztain quantifiable properties, e.g. smoothness,of the potential V(R)is needed to solve eq 2. Among a host of approaches to overcome this inherent instability, we shall, in the following, adapt a regularization procedure proposed by Tikhonov and Phillip! more than two decades The regularized solution 6V,(R) of eq 2 is defined as the solution which minimizes the following functional

(6)

and the sought-after solution fc")(R)is related to the potential correction 6V(R) as follows

(7b)

+

with r(n 1) being the gamma function and noting thatfcO)(R) = 6V(R). Here we have taken into account the fact that the potential V(R),as well as its derivatives W ( R )1 dV(R)/dRn, usually approaches zero quickly as the internuclear distance R a. As a result, the regularized solution 6V,(R) can now be defined in terms of the nth-order derivative t ) ( R )that minimizes the new functional

-

+

(3) where the quantities 11 ... ;1 and 1) ... ;1 are respectively square norms defined in the data space 0 and the solution space R. Typically, L is a derivativeoperator acting on the solution function 6V(R) and dictating the smoothness of the solution, and cx is the regularizationparameter, a positive real number. Although there appears no general algorithm to optimally evaluate a,its proper choice is important when dealing with the ill-conditionedproblem (2) to balance the magnitude of the residual (the first term on the right-hand side) against the degree of the smoothness (the second term) in eq 3.45-47 The actual implementation of the Tikhonov regularizationneed not strictly follow what is prescribed in eq 3, depending on the type of problem under consideration. This can be illustrated by considering a simple on?-dimensional case in which the multidimensionalpotential V(R)reduces to a one-dimensionalpotential surface V(R),a function of a real variable, say, the internuclear distance R between two inert gas atoms. Many problems of a polyatomic nature are often recast in terms of (radial) coupled equations4 and thus become effectively onedimensionalproblems.

It is instructive to see that the quantity t ) ( R ) satisfies the following conveniently solved Fredholm integral equation of the second kind:

with

The one-dimensionalexample considered here demonstrates how the requirement of a smooth solution (potential) can be implemented efficiently through a simple transformation from the original linear equation (2) for 6V(R)to the new equation (4) for fc")(R).Physically, minimization of either eq 3 or eq 8, assuming that the derivative operator has the form L = d"/dR", yields solutions bV,(R) which are smooth in the sense that the norm 116l(')ll; corresponding to the nth-order derivative 6 c " ) R ) dn6V,(R)/dRnis minimized. However,the transformation leading from eq 2 to eq 4, or from eq 3 to eq 8, further restricts the solution 6Va(R)to the subset of solutions whose derivatives, up to ( n - 1)th order, vanish asymptotically,rather than any solution which would merely minimize the functional (P[6Va]defined in eq 3. For polyatomic systems, it is important to properly choose the coordinatesaccording to the correspondingdynamics under study and the theoretical (or numerical) methods undertaken. In the immediate future triatomic systems provide an important challenge for inversion. The potentials involved in atom-diatom rotationally inelastic excitation processes,5ithe motion of bound rovibrational states of triatomic molecules,s2 or the reactive processes of atom-diatom scattering53can be explicitly expressed

Ho and Rabitz

13450 The Journal of Physical Chemistry, Vol. 97,No. 51, I993

as functions of three Jacobi coordinates (R,r,0). In this case, the potential correction dV(R,r,x), x = cos 0, can be written in terms of Taylor's formula as follows 1

6V(R,r,x) =

r(m)

[O]

Choose a Starting Reference Potential

I

U [l] Solve the Forward Problem

s Rm sr m ( R- Rqm-'(r - r')*' X

1

Oi,,[Vo] ,

U [2] Introduce Experimental Data

-s

Vo(d)

60in I Orh - o$hpO]

U

1

.

r(k)

-I

(x - x?k-16V(m,n,k)(R',r',x?

131 Solve the Linear Equation 60,h = J -6V(d)dd w ) with Tikhonov Rrgularization

where U

6V(m*n,k)(R,r,x) E 6V(R,r,x) (12) aRmar"axk with the proper boundary conditions (reflection symmetry), at x = f l (i.e. 0 = 0 or T ) , 6V"Jv')(R,r,fl) = 0, V 0 Ii Im; 0 I j I n; O I R , r < m (13) Here we have assumed that the potential correction sV(R,r,x) has zero partial derivatives of all orders up to m - 1 and n - 1 with respect to the radial coordinates R and r at R = and r = a,respectively. The expansion in eq 11 for the angular variable x i s taken at x = -1, or equivalently at 0 = T , to make use of the fact that the potential function V(R,r,O) has a reflection symmetry at 0 = T ; cf. eq 13. Although, for simplicity, the reflection symmetry at 0 = 0 is not addressed, it is prudent and straightforward to reformulate eq 11in a form that can take the advantage of both symmetriesat x = f l so that the inverted potential would fully comply with theboundaryconditions(13). After substituting eq 11 into eq 2 and after rearranging integration orders, we can readily rewrite eq 2 in the following form

som~mIClhm,n.k1(R,r,x)6V(m,n,k)(R,r,x) d x d r dR (14) where

K$'*n,kl(R,r,x)

s ss R

E

r ( m ) r(n)r(k) O

r

1

(R'-R)m-l(r'-

O

Equation 14, along with the condition 13, can then be solved to yieldthequantities6Vm~n~o(R,r,-l),l=0, 1, ...,k- l,and6Vcm+sk)(R,r,x), which in turn, through eq 11, render a smooth potential correction GV(R,r,x) of the mth, nth, and kth orders with respect to the coordinates R, r, and x, respectively. The Tikhonov regularization outlined above for the one-dimensional case can be readily implemented to overcome the ill-conditioning problem inherent in the linear integral equation (14). Moreover, it is important to point out that for cases of high, i.e., 22, dimension we can easily incorporate as many physical properties (or symmetries) possessed by the sought-after potential surface, in addition to the asymptotic behaviors at large separations, as possible into the inverse procedure to render physically fully satisfactory potentials. In Figure 2, we sketch how the regularized iterative inversion procedure can be implemented to determine inter- or intramolecular potentials given a set of discrete experimental data. Singular Function Expansion Analysis. To understand the various algorithmic properties, particularly the stability and the

[4] Update the Potential

V , ( d ) = V o ( d )+ N e ( $ )

U [5] Solve the Forward Problem

I

O,h[V,]

U

[B] Check the Convergence : Does {o,hpi.]} m d {O:h} Agree Well ?

No J

\ Yes

rad GO BACK to Step (11

Figure 2. Flow chart illustrating how the inter- and intramolecular potential energy surfacescan be extracted from experimental data using an iterative inversion method based 0," functional sensitivity analysis and Tikhonov regularizationtheory. Vo(R)is thestarting reference potential, oih[Vo]the calculatedobservables, 6&/6 V(R)the functional sensitivity densities, qhthe experimental data, and a the optimal regularization parameter.

regularization,of the inversion procedureoutlinedabove, we need to further examine the Fredholm integral equation of the first kind found in eqs 2 or 4. This can be carried out by resorting to a powerful analytic tool in terms of the singular function expansion t e c h n i q ~ e . 4 ~For ' simplicity, we shall again confine ourselves to the case of a one-dimensional potential V(R); cf. eq 4. The generalization to any multidimensionalcase, e.g. eq 14, is straightforward. Furthermore, we assume that no two data q h are the same, and therefore the kernel K$](R) is of full rank. In terms of the singular function expansion analysis, the kernel #;](I?) in eq 4 defines the following eigenvalue equations:

and

where the singular functions ($I(R)!p = 1, 2, ... , NaJ and singular vectors (u$$!p = 1, 2, ..., N a ] satisfy the following orthonormal relations C u F 1 ( R ) $'(R)R2 dR = aPq

(18)

and

with N a being the total number of experimental data (q,,) under consideration and a,, being the Kronecker delta function.

Feature Article

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13451

Moreover, the completeness relation

together with eq 17, leads to a singular function expansion for the forward kernel @ ( R ) , namely P

in terms of the trio of the singular value, vector and function (pF1,uF1,uF1(R)).The singular values (pF1b= 1, 2, ...,Na]are positive numbers and are usually arranged in descending order pp1 1 pp1 2

... L pN0 Ln1 > 0

(22)

The singular values pF1for p >> 1 may rapidly approach zero when the amount of experimental data N B becomes large; when the condition number pj“]/pE;-the ratio of the largest and smallest singular values-becomes large, the kernel K#](R) becomes ill-conditioned. This is especially true when the same type of data (e.g. belonging to the same hierarchical level in Figure 1 ) are densely distributed. Generally, the smaller the singular value $I, the more oscillatory is the corresponding singular function uF1(R). These fast oscillatory components cause numerical instability and thus have to be removed when attempting the inverse solution of eq 2 and 4. Physically, these small singular values correspond to very small length scale variations in the potential. Regularized Solution of Eq 4. By using the relations 16-21, we can readily expand the solution of eq 9 as follows

in terms of the quantity S O i h and the nth-order regularizedinverse kernel K!i1-’(a;R) defined as

Equations 23 and 24 show how the instability problem manifests itself (either) in the solutions of eq 2, (or) with eq 4 becoming very sensitive to small fluctuations in the quantity 8 0 t h without regularization (a = 0): the high-frequency components in the singular function expansion (24) are disproportionallyweighted by the reciprocalsof their small singuar values. In physical terms, this is a manifestation of the smoothing effect inherent in the forward kernel K!il(R), as seen in the singular function expansion (21), in which the high-frequencycomponents corresponding to very small singular values contribute little. Once the solution &!)(R) is obtained, we can readily compute the correction to the potential SV,(R) according to eq 7b. Moreover, it is interesting to further examine some other properties of the regularized solution j$)(R). First, by substituting eq 4 in eq 23 and by invoking relation 17, we can easily derive a relation between the regularized solutionj$)(R) and the namely, intended true solution p)(R),

e)(R) = ~om3’3,(R,R’)f’”(R’)R” dR’

(25)

where the nth-order regularized resolutionfunction R,(R,R’) is defined as

and can be used as a measure of the resolution of the solution achievable by the input data at the internuclear distance R based on the regularization procedure. It can be readily seen that, due to the filter factor $1’/(p~12 a) in eq 26, the resolution of the solution is a maximum at a = 0; when a very large amount of the data is used, the corresponding resolution kernel R Q X R,(R,R’) may resemble very closely the Dirac delta function 6(R - R’) at a = 0. The resolution is gradually lost as the value of a becomes larger, and the resolution is totally lost at the extreme value a = a. However, high resolution in the solutions may not always be desirable; it can result in undesirable solutionsbecause the data q h will inevitably contain measurement errors and the linearized relation (2), or (41, is only an approximation. Second,we obtain, after substituting eq 23 into eq 4, the relation

+

SOih(a)E JomK/il(R) e ’ ( R ) R 2dR

-SO,,

as a-0 (27) Here we have made use of the following property satisfied by the forward and inverse kernels #il(R) and K$]-’(a,R)

-

as

-

0

(28) obtained from eqs 20 and 24. Equation 27 indicates that to zealously force good agreement between the calculated values b o i h ( ( Y ) , defined in eq 27, and the input values 6 0 i h r given on the left-hand side of eq 4, by choosing a small a value can lead to a very distorted result. This behavior can happen because eq 4 is an approximation (due to the linearization), even when no experimental data error is present. Finally, we can compute the effect of the pure experimental data errors-neglecting the errors from the nonlinear terms in eq 1-on the regularized solutionj$)(R)at each iteration: the mean value of the error in the solution is dihjk

1]1

where ( A q h ) and ( A q h A q k ) are respectively the data error mean value and the data covariance. From eq 24, it is seen that both the error mean value and the covariance of the solution approach zero as the regularization parameter a becomes large, but could become very large as a 0. Thus, a tradeoff exists between the resolution and variance of the extracted potential. A proper choice of the regularizationparameter a at each iteration plays a pivotal role in the iterative inversion procedure. We shall return to this issue as well as the issue of when to stop the iteration-the convergence criterion-in the numerical examples presented below.

-

Numerical Examples A number of model inversion problems have been treated with the method outlined above. Rather than give a broad summary of the results, we shall primarily consider one system, He + Ne, in detail to fully illustrate each aspect of the inversion algorithm. First, we consider real data from low-energy elastic He + Ne scattering in a crossed-beam experiment by Buck et al.5‘ The

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13452 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

relationship between the data Gi and small variations of the nominal potential Vo(R),except well inside the classical forbidden region, Le. R < 2.5 A. It is important to observe that the large angle data, e.g. for 8i > 50°, are mainly sensitive to the inner repulsive wall of the potential, as indicated by the locally pronounced peaks. In contrast, the small-to-medium-angledata, i.e. for 8i < 50°, are sensitive to both the potential minimum region and long-range part of the potential energy surface. This indicates that the data set considered here contains detailed information on the sought-after potential over a wide range of distances and can thus be used for inversion purposes, despite the laboratoryaveraging from smallbeam energy dispersion and finite angular r e s ~ l u t i o n .To ~ ~obtain ~ ~ ~a good quality, smooth potential energy function,we have imposed a 10th (n = 10)-orderderivative constraint via eq 8. The corresponding transformed kernel K:,OJ(R) (see Figure 3b), which describes the functional relaand the quantityflO)(R) defined tionship between the data in eq 7a, is much more smooth as a function of R when compared to its zero-order counterpart Kgl(R). The relationship between the singular values and the singular functionsufo1(R)of the kernel K r l ( R )is displayed in Figure 4a,b, confirming that the singular function indeed becomes more oscillatory as the singular value becomes smaller. Of particular significance is the rapid decrease of the singular value: its magnitude drops quickly from 1.42 to 4 X lo-* for the first 15 singular values and then stays within the range -2.3 X lo-* to -3.4 X for the rest of 73 singular values, implying that only approximately 15 singular functions may actually contribute to the solutionflO)(R)of eq 8. The rapid falloff of the singular values shown in Figure 4a indicates that V(R)may be expressed in terms of a relatively few number of functions. This point is relevant in relation to conventional parameter fitting, where, again, relatively few functions typically will be involved. A key difference is that the present inversion method explicitly identifies the functions in which to expand the potential (or its derivatives) as the singular functions of the corresponding kernel, rather then employing intuition. In Figure 5a, we show the recovered He-Ne potential V(R) (solid curve), along with its upper and lower bounds (dashed curves) of confidence, and the starting reference potential Vi(R) (dotted curve). The recovered potential V(R)can be obtained in terms of the following relation between any two consecutive iterations, say, [k - 11th and kth,

G,

n

w

W I

0

Ea X

-0.4

Figure 3. (a) Functional sensitivity densities STe,/SV(R)for the lowenergy (29.2 meV) He N e elastic scatteringexperimentof Bucket al.54 The calculations have been performed based on the reference potential Vo(R) (see dotted curve in Figure 5a) which is used to start the data inversiondescribed in the text. Apparatus averagingeffectsdue to energy dispersion of the crossed He and N e beams and finite angular resolution of the detector have been properly accounted for. (b) The kernel e o 1 ( R ) ,which is the 10th-order counterpart of the kernel STe,/SV(R), deiicts the relation between the data Gle,and the quantityflO)(R)defined in eq 7a.

+

scattering intensities (denoted as lee) are measured at a finite set of laboratory (Lab) angles 8i, i = i,2, ..., Ne ( ~ 8 8 and ) at the relative collision energy 29.2 meV. Furthermore, the standard deviations(denoted as CQ correspondingto each scattering data point are also availablefrom the measurements. In the following numerical study, we shall assume that the He-Ne potential V(R) for the interatomic distances R 1 10 is exactly known from recent ab initio calculation^.^^ This assumption recognizes the fact that the experimental data are measured only from 6.97' to 79.97' and thus do not contain small enough scattering angles to provide information on the potential at very large distances. We shall use this example primarily to illustrate the utility of the singular functionexpansion analysisand Tikhonov regularization methods for general inverse problems involving real experimental data. The correspondingFredholm integral equation of the first kind, cf. eq 4,49

= JomKgl(R)6V(R)R2dR, i = 1,2, ..., Ne

(31)

where Gi G/cii and ?e,[v] le,[Vl/ci, are respectively the weighted experimentaland calculated intensities at the Lab angle 8i, K t l ( R ) = 6?0,/6V(R),and V(R) is the interatomic potential between He and Ne at the distance R. In Figure 3a we see that the kernel K t l ( R ) ,which is evaluated for the starting reference potential Vo(R), cf. Fig. Sa, displays a detailed structural

V k ( R )= vk-'(R)+ Svk(R) (32) with V ( R ) 3 Vo(R). The upper and lower bounds of confidence-defined as the square-root of the variance-can be evaluated according to the following cumulative relation for the potential variance (Avk(R)Avk(R)) = (Avk-'(R)Avk-'(R)) +

which is obtained by invok ing eqs 7b and 30 and by assuming that the scattering data errors are normally distributed and different iterations are statistically independent from each other. The second term on the right-hand side of eq 33 is evaluated at the kth iteration. It is found that the recovered potential V(R) crosses the zero value at -2.695 A and possesses a minimum of depth -1.76 f 0.02 meV at -3.03 A, compared to the corresponding multipropertyfitting resultsof 2.699 A, 1.827 meV, and 3.029 A, respecti~ely.~~ The slightly smaller well depth of our result may be attributed to the fact that we only use the angular scattering data at one fixed collision energy, but not the much larger amount of the multiproperty data used in ref 54.

The Journal of Physical Chemistry, Vol. 97, No.51, 1993 13453

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. .. .. .. .

0) L-.

I

3

n

5

4

6

internuclear distances R I

I

I

I

7

(A) I

(b)

+

He Ne E = 29.2 meV

Figure 4. (a) Singular values $01 and (b) singular functions vFol ( R ) ,p = 1, 2, ..., 88, of the kernel K r l ( R ) ,shown in Figure 3b.

Nevertheless, we show in Figure 5b that the calculated angular scattering intensities Ie,[ VJ (solid curve) for the recovered potential V(R)agree well with the input experimental data (solid circles); the calculatedintensities Io,[Vo]for the starting reference potential Vo(R) are also given (dotted curve). The results shown in Figure 5a,b have been obtained according to the inversion scheme described in Figure 2. Here, specifically, the regularization parameter a,defined in eq 8, at each iteration has been determined by minimizing the following x2 function with respect to a,namely

which measuresthe goodness of fit of the calculated data (lei[Val), correspondingto the potential Va(R)at an arbitrary value of a, when compared to the experimental data (see Figure 6).58 Furthermore, the inversion procedure was stopped after only two iterations when the corresponding x2 function fell below unity, i.e., from 102.86 to 9.95 to 0.96; the arrows in Figure 6 indicate the optimal a values, Le., 4.36 X 10-6 and 3.63 X 10-11, at the first and second iteractions, respectively. The large flat region around the optimal a value at each iteration may provide further flexibility in choosing a proper regularization parameter without hampering the inversion stability. We should note that while a variety of other approaches for the determination of the regularization parameter a may also do well, the procedure adopted here produces optimal a values that effectively eliminatethe high-

I

I

I

I

I

10

20

30

40

50

Lab-angles

60

0 (deg)

Figures. (a) Interatomicpotentialsof He and Ne: the startingreference potential,dotted curve; the recovered potential,solid curve; and the upper and lower bounds of confidence of the recovered potential, dashed curves. The results here have been obtained based on the 10th-order version of eq 4, and its kernel at the first iteration is given in Figure 3b. The inset shows an expanded view around the potential minimum. (b) Angular distribution of the scattering intensities for the low-energy (29.2 meV) He + N e elastic scattering experiment: experimental data from Buck et al.,54solid circles; calculated data for the starting reference potential, dotted curve; and calculated data for the recovered potential, solid curve.

frequency contribution to the potential and thus yield a reliable, stable inversion. The removal of the high-frequency components associated with the very small singular values of the kernel K:,O1(R)can be readily seen in Figure 7 via the behavior of the resolution kernel Ra(R,R'),which explicitlyreveals how the smallscale structures are averaged at different internuclear distances R. The very small magnitude of Ra(R,R') at small R and R'is a manifestationof the repulsive nature of the interatomicpotential at short distances. It is interesting to observe from the inverse kernel K y l - ' ( R )in Figure 8 that the large-angle data have the most influence on the solutionf(IO)(R)around the potential well region. In contrast, the small-to-medium angle data influence a much wider region, extending from the potential minimum region to large internuclear distances. There exists strong a

Ho and Rabitz

13454 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 125

I

I

I

I

I

I

I

100 n

fl

W

CJ

75

s :

* - 4 4N

3

2k

50

25

0

.'......._._..... c .............................. -14 -12 -10

-8

-6

Figure 8. Regularized inverse kernel @,']-'(a;R),eq 19, corresponding to the forward kernel K r l ( R )shown in Figure 3b. The calculations are carried out for the optimal a at the first iteration discussed in Figure 6.

..." -4

-2

regularization parameter

0

2

4

loglo (cy)

S SPOO A

16

I

I

Figure6. Chi-squarex2function (eq29) illustrates how the regularization parameter a in the Tikhonov regularization procedure is optimized at each iteration, leading to the results shown in Figure4. Only two iterations are needed to obtain converged results: first iteration, solid curve; second iteration, dotted curve. The search for the optimal a at each iteration starts at the rightmost edge of the curve and moves leftward, and the arrows indicate where the optimal a are located.

He-Xe/C(OOOl)

12

8

4

0

-4

-8 9

_. .'

... '.._._.I'

- 12 6

2

p.

w

Figure 7. Resolution kernel BR,(R,R'), eq 21, calculated for the optimal a,i.e. 4.36 X 10-6, at the first iteration discussed in Figure 6.

2

I

I

I

3

4

5

normal distances from surface

6

z

(A)

Figure9. Illustration of full 3-Dinversion results based on using simulated He-Xe/C(0001) elastic scattering experimentaldata (He scattering from a Xe overlayer on graphite) at the heliu? energy 8.8 meV. Displayed here _areHe-Xe/C(OOOl) potentials V(z,R)-z the normal distance from and R the 2-D position vector on the surface-at three lattice sites, Le. the atop (A), bridge (SP),and 3-fold (center, S) sites, as well as the corresponding laterally averaged potentials V t ( z ) . Initial reference potential, dotted curves; desired model potential, solid curves; and recovered potential, dashed curves. The recovered potential is only slightly steeper in the upper portion of the repulsive wall, and its minima are slightly deeper. Moreover, the repulsive walls of the recovered and true potentials pass through zero value at almost the same distances from the surfaces. The inset shows the unit cell of the commensurate (fi X &)R30° xenon monolayer on the basal plane of the graphite(0001) face.

resemblance between the forward kernel $](I?) of Figure 3a and the inverse kernel given in Figure 8; however, the latter does not contain high-frequency oscillations, due to the proper regularization in the inversion procedure. The regularizedinverse kernel K r l - ' ( I ? ) clearly manifests and preserves the close relationship between the scattering data at different angles and the potential and its derivatives at different distances. In addition to the above example that uses real experimental data for inversion, successful applicationsof the inversion method have been made to a variety of simulatedproblems recently. These includethe problems of invertingdiatomic H2 vibration-rotation spectra,38elastic scattering data of two inert gas a t ~ m ~and , ~ ~ -realistic ~ ~ data, exemplified by the elastic He-Ne scattering case, inelastic scattering data of nonadiabaticcurve-crossing systems.59 and for multidimensional potentialenergysurfaces,demonstrated In particular, we have demonstrated (see Figure 9) that the full by the simulated low-energy He-Xe/C(OOOl) diffraction problem. three-dimensional He-Xe/C(OOOl) potential energy surface can The technique is applicable to treating many (small) polyatomic be unambiguouslyrecovered using numerically simulatedspecular systemsof current interest. At the present time, the applications scattering data and three sets of in-plane diffraction data over of this inversion method to the triatomic bound state problems, a wide range of incident polar angles.37 e.g. HCN6M4 and H20,6547 small van der Waals complex problems, e.g. Ar-OH,68-71 Ar-HC1,72 and Ar-HF,73 and the Perspectives for the Future atom-diatom scattering problems, e.g. H+N2,74 are feasible and timely. This comment follows from the fact that the amount of Application to Small Molecules. The success of the above high-qualityexperimentaldata on these systems has been growing inversion method is clearly indicated, especially in dealing with

Feature Article

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13455

steadily, and in addition, a variety of highly efficient theoretical improve the overall convergence rate and, perhaps more importools for solving the relevant SchrGdinger equations have been tantly, can provide a more accurate account of the nonlinear developed recently. effects of the data on the inverted results. Finally, although it is true that to include perturbation terms of sufficiently high With the existence of such a large pool of accurate experimental order in eq 35 can become unmanageable computationally due data, the widely used (least-squares)parameter-fittingprocedures, to the required evaluation of varius high-order functional based on a small number of parameters (usually much fewer sensitivity densities and multidimensional integrals in eqs 36,in than loo), to extract the information of the underlying potential practice, one may likely drop all but the first and second terms energy surface can be quite inadequate. Although the fitting in eq 35. This is quite acceptable for most molecular systems of procedures have worked remarkably well in a number of molecular current concern, especially considering that recent theoretical systems using, especially, highly accurate spectroscopic and numerical tools permit efficient evaluation of the first- and data,64v69*71-73 fixing a potential surface to a specific functional second-order functional sensitivity densities needed for solving form can be rather subjective and thus b i a ~ e d . * ~With - ~ ~ the eqs 36, and state-of-the-art ab initio quantum chemistry mafitting method, each new system must be approached without chineries can readily provide good starting reference potential guidance or systematic logic on how to choose a proper form. energy surfaces for the inversion. Moreover, the fitting procedures presume that thecorrespondence inverse problems are always ouerdetermined (i.e. there are more Conclusion data than the number of parameters), and thus they do not fully make use of the information available in the immense data sets. We have given a succinctaccount of a general nonlinear method On the other hand, the inversion method reviewed here does not for the inversion of discrete experimental data to extract interimpose any parameterized functional form on the sought-after and intramolecular potential energy surfaces. The method is potential energy surface and, more often than not, would only formulatedin terms of functional sensitivityanalysis in conjunction require small computational overhead when compared to the with the Tikhonov regularization theory and the singular function parameter-fitting procedure. For example, a host of advanced expansion method. Particularly, we have made a clear distinction theoretical methods for both bound-state and scattering problems between the inversion method and the usual parameter-fitting are of a variational nature,7680and they require the storage of approaches, in that the former considers the potential energy the full wave function of the system in the calculations. These surfaces as continuous functions of the internuclear coordinates, wave functionsare exactly what are needed to calculate the kernel whereas the latter treat them only in terms of a finite, usually for performaning the inversion outlined here. The full-scale small, number of parameters. The instability of the inverse inversion involving larger molecules, e.g., Ar-H20,*I H ~ - C O Z , ~ ~problems that deal with realistic data was examined analytically and He-NH3,83 will certainlymeet more adversities arisingmainly and resolved by incorporating a priori smoothness properties of from the huge computer costs needed to solve the dynamical the sought-after potential energy surfaces. Numerical studies equations. However, judging from the great strides being made have shown that the inversion method is generic, efficient, and in modern computer technologies and the elegance displayed in stable and is capable of accurately renderingphysically acceptable recent theoretical and numerical endeavors, we may expect that potential energy surfaces. Finally, we have discussed the these difficulties can also be overcome in the near future. feasibility of extending this method to problems involving small High-Order Inversion Methods. The above inversion method polyatomic systems,based on the present status of the experimental is based on the first-order equation (2) and can be further improved techniques and theoretical advances. We have also suggested upon by including higher-order terms in eq 1. For example, on the possible improvement of the inversion method by including the basis of a recent procedure proposed_by Snieder,29-84 we may higher-order functional sensitivity densities in the formulation. assume that the potential correction 6V(R) in eq 2 can be written in terms of a perturbation series Acknowledgment. This work was supported by the US. Department of Energy. We are indebted to Dr. Kevin Perry for SV(R) = V(I)(R) V(2)(R) ... @)(R) ... (35) creating the color picture of Figure 3a. where thejth term W ( R )is associated with thejth-order power References and Notes of thcquantity 6Oih. The governing equations for the functions Yu)(R)can then be obtained by equating terms of the same order (1) Levine, R. D.; Bernstein, R. B. Molecular Reacfion Dynamics and Chemical Reactivity; Oxford University Press: New York, 1987. of the quantity 6oih after we substitute eq 35 into eq 1. The (2) Maitland, G.C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. above manipulation results in a set of linear integral equations Intermolecular Forces: Their Origins and Determination; Clarendon Press: for W'), W 2 )... , Oxford, 198 1. (3) Buck, U.Rev. Mod. Phys. 1974,46,369; Comput. Phys. Rep. 1986,

+

+ +

+

5. 1. (4) Gerber, R. B. Comments At. Mol. Phys. 1985, 17, 65.

shown to second order her e.- It is easyfo see from these equations that the quantities V(I)(R),Y@)(R), ... can be computed sequ.ntially, each adding a higher-ordercorrectionto thequantity 6V(R). Moreover, all thesejinear equations possess the same integral kernel, Le. 6Oih/bV(R),and can be solved subject to the same regularization procedure; they can also be handled more expediently than the original method of Snider. The explicit inclusion of higher-order terms in the inversion procedure can

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