Electronic tunneling and exchange energy in the Q-dimensional hydrogen-molecule ion

S. Kais, J. D. Morgan Ill, a) and D. FL Herschbach Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 (Received 4 June 199 1; accepted 4 September 199 1)

Dimensional scaling generates an effective potential for the electronic structure of atoms and molecules, but this potential may acquire multiple minima for certain ranges of nuclear charges or geometries that produce symmetry breaking. Tunneling among such minima is akin to resonance among valence bond structures. Here we treat the D-dimensional H,+ molecule ion as a prototype test case. In spheroidal coordinates it offers a separable double-minimum potential and tunneling occurs in only one coordinate; in cylindrical coordinates the potential is nonseparable and tunneling occurs in two coordinates. We determine for both cases the ground state energy splitting AE, as a function of the internuclear distance R. By virtue of exact interdimensional degeneracies, this yields the exchange energy for all pairs of g, u states of the D = 3 molecule that stem from separated atom states with m = I = n - 1, for n = 1 -+ CO. We evaluate AE, by two semiclassical techniques, the asymptotic and instanton methods, and obtain good agreement with exact numerical calculations over a wide range of R. We find that for cylindrical coordinates the instanton path for the tunneling trajectory differs substantially from either a straightline or adiabatic path, but is nearly parabolic. Path integral techniques provide relatively simple means to determine the exact instanton path and contributions from fluctuations around it. Generalizing this approach to treat multielectron tunneling in several degrees of freedom will be feasible if the fluctuation calculations can be made tractable.

I. INTRODUCTION Many diverse phenomena exhibit exponentially small

splittings between quasidegenerate energy levels, produced by quantum mechanical tunneling through barriers separat- ing equivalent potential minima. Molecular physics offers the classic cases of ammonia inversion’ and hindered rota- tion of methyl groups,’ as well as numerous chemical reac- tions.3-7 Likewise, in condensed matter tunneling governs a host of electronic processes. ‘-lo In nuclear physics, tunnel- ing has a major role in radioactive decay and in fusion reac- tions.” In particle physics, gauge field theories involve tun- neling between different field configurations with multiple ground states. l2 Such tunneling processes are often subject to nonseparable potentials so an accurate treatment must include more than one degree of freedom. Because tunneling depends exponentially on physical parameters, perturbative power series expansions cannot be used to calculate directly the small energy splittings. Here we apply semiclassical non- perturbative techniques, particularly the instanton meth- od,13 to determine the ground state energy splitting (or ex- change interaction) AE, (R) for the D-dimensional H+ molecule ion over a wide range of internuclear distance R. As a consequence of exact interdimensional degeneracies, I4 this yields the exchange splitting for all pairs of g, u states of

*’ Also associated with Institute for Theoretical, Atomic, and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138. Regular address: Department of Physics and As- tronomy, University of Delaware, Newark, DE 19716.

the D = 3 molecule that arise from separated atom states withm=Z=n- l,forn= I-CO.

Our aim is to develop and test practical, numerically stable means of treating nonseparable potentials in which tunneling occurs in two or more degrees of freedom. The ultimate goal is to evaluate tunneling of atoms in chemical reactions or electrons in sizable molecules. The instanton method offers a promising approach, but to assess its accura- cy we need to apply it to a nontrivial, well-characterized system. The H,+ molecule ion is particularly suitable for this purpose. Essentially exact numerical calculations14 are available for comparison over a wide range of R. Also, in spheroidal coordinates the double-minimum potential is separable and tunneling occurs in only one coordinate” whereas in cylindrical coordinates the potential is nonsepar- able and tunneling occurs in two coordinates. This offers an opportunity to compare approximation methods for separa- ble and nonseparable versions of the same system.

We evaluate AE, by two semiclassical techniques, the asymptotic and instanton methods. The asymptotic method was introduced and applied to Hc in 1955 by Holstein,16 improved in 1962 by Herring,” and explicated by Landau and Lifshitz.” The method blithely uses an asymptotic ap- proximation for the wave function to evaluate flux through the median plane midway between the nuclei. For the D = 3 molecule this yields the leading term of the exchange split- ting,

AE3 = (4/e)Re- R. (1.1) This simple result, which Herring proved to be asymptoti-

9028 J. Chem. Phys. 95 (12), 15 December 1991 0021-9606/91/249028-14$03.00 0 1991 American Institute of Physics

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tally exact for R + CO, is fairly accurateover a wide range of distance. Higher order terms have been evaluated by Dam- burg and Propin, l9 by Cizek et al.,” and by Tang, Toennies, and Yiu.”

The instanton method also has appropriately multiple parentage. Basic aspects appeared in a 1967 paper by Lange? on multiple-well tunneling in condensed matter physics. The method was independently discovered in 197 1 by MillerZ3 in the context of chemical physics and in 1973 was again independently developed by Banks, Bender, and WU,=~ and later by t’Hooft,25 and by Polyakov26 in quantum field theory. This approach largely overcomes difficulties en- countered in extending Wentzel-Kramers-Brillouin (WKB) techniques to multidimensional systems. As its key feature, the instanton method examines the evolution of a dynamical system in imaginary time, equivalent to motion in real time in an inverted potential ( Y-t - v>. To leading order, the energy splitting is given by

AE=Aexp( --So/k), (1.2) where fi is Plan&s constant and Se is the classical action for motion along the instanton path, the classical trajectory of zero energy between the two maxima in the inverted poten- tial. The pre-exponential factor A involves contributions from fluctuations about the instanton path. It is related to a quantity called the van Vleck factor,27 given by i

A = 18 2So/i9q;ndq:f)J, (1.3)

the determinant of the matrix of second partial derivatives of the action with respect to changes in the initial and final positions q”’ and q (f) for the trajectory. In our application, So for tunneling in two degrees of freedom is evaluated by means of Jacobi’s form of the least action principle.28 This determines the exact instanton path from Euler’s equation of the calculus of variations. For the prefactor A, we use a path integral formulation by Auerbach and Kivelson” to evalu- ate the contribution of trajectories in the vicinity of the in- stanton path. The computations reduce to procedures famil- iar in the analysis of molecular vibrations.29

A chief motivation for our study stems from the dimen- sional scaling approach to electronic structure.30 In zeroth order, corresponding to the D-t CO limit, the Hamiltonian (in a D-scaled space) reduces to an effective electrostatic potential. The minimum of this potential defines a rigid con- figuration of the electrons (the “Lewis structure”). .The first-order term, proportional to l/D, corresponds to har- monic vibratio-ns of the electrons (the “Langmuir vibra- tions”) about the fixed positions attained in the D-+ w limit. Higher-order terms in a l/D perturbation series correspond to anharmonic vibrations. For the ground states of two-elec- tron atoms and H,+ this l/D perturbation series has been evaluated to -30th order.31 Although the series, is an asymptotic expansion, accurate energies for D = 3 can be obtained by suitable summation procedures, In effect, the method provides a multivariable semiclassical algorithim, in which l/D plays the role of Plank’s constant. i :-

The l/D perturbation treatment becomes inadequate, however, when the effective potential W, for the D-, r*) limit has more than a single minimum. Typically, W, ac-

quires multiple minima when the nuclear charges or geomet- rical parameters are varied. For instance, W, for the He atom (Z = 2) ‘has a single minimum, with the electrons equidistant from the nucleus. If the nuclear charge drops below a critical value (Z, = 1.2279 * . * ) , the symmetric con- figuration becomes .a saddle point and W, acquires two equivalent unsymmetrical minima; these have. one electron much closer to the nucleus than the other, as in the H- ion.32 An analogous symmetry breaking transition occurs in W, for H2+ when the internuclear distance R is varied.33 When R is small, Wm has a single minimum, with the elec- tron midway between the nuclei. When R becomes large enough, however, W, has a pair of equivalent minima, with the electron localized on one or the other nucleus. For finite D, tunneling between these double minima becomes promi- nent; it is tantamount to resonance among valence bond structures. Since this produces energy splittings that depend exponentially on D and so vanish more rapidly than any power of l/D, a nonperturbative technique such as the in- stanton method is required to evaluate the tunneling contri- butions.

Section II briefly recapitulates pertinent features of Hc in D’*dimensions, including the W, potential and interdi- mensional degeneracies, and shows that the major D de- pendence of AE, has a simple, generic form. Section III treats tunneling in one degree of freedom. We generalize to D dimensions the asymptotic method and obtain from it a good approximation to AE,. We also formulate the instanton treatment for”a linear path. The analytic result obtained from the separable form of W, in spheroidal coordinates’” is found to coincide at large R with the asymptotic method. In Sec. IV we evaluate explicitly the instanton action and in Sec. V the prefactor for tunneling in two degrees of freedom, using the nonseparable form of W, in cylindrical coordi- nates. Section VI provides comparisons with other semiclas- sical methods and discusses prospects for treating more complex systems. We emphasize an ironic but encouraging aspect. Although electronic tunneling is a quintessential quantum effect, for Hz the energy splittings can be evaluat- ed for a wide range of D by simply using the effective poten- tial for the D- 03 limit, a function exactly calculable from classical electrostatics.

II. THE ti; MOLECULE ION IN PDlMENSlONS TheSchrGdinger equation is readily generalized to an

arbitrary spatial dimensionality D, which denotes the num- ber of Cartesian components comprising any vector. The Laplacian and Jacobian change form but not the potential energy. Resealing the wave function to incorporate the square root of the D-dependent Jacobian volume element casts the Hamiltonian into the same form as for D = 3, with the addition of a centrifugal potential that depends quadrati- cally on D as a parameter,i4 We consider here H$ with “clamped nuclei,” corresponding to the Born-Oppenheimer approximation, for which very accurate numerical solutions at D = 3 are available.34 For general D, solutions can like- wise be obtained by exploiting an exact interdimensional de- generacy: D-+D + 2 is equivalent to m +m + 1, increasing

Kais, Morgan, III, and Herschbach: Electronic tutineling and exchange energy 9029

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by one unit the projection m of the electronic angular mo- mentum on the internuclear axis.14 (We write m = [ml, for simplicity.) Our results pertain to the tunnel effect splitting 5

AE, between the lowest g, u states of the D-dimensional fi E 4Y”lS

-0.5

system. Table I illustrates how, by virtue of the interdimen- 0 .&-I sional degeneracy, AE, for D = 3, 5, 7,... accounts for all F: pairs of g, u states of the D = 3 molecule that stem from E .$ -1.0 separated atom states with m = I = n - 1, for n = 1,2,3 ,... . 2 Figure 1 plots the electronic energies of the lowest few of w these pairs of D = 3 states. In evaluating the D dependence z -1.5 of AE,, we are also determining the m dependence of tun- ;E: neling for all such pairs of states of the D = 3 molecule. ::

Although the H,+ problem for D dimensions is separa- -2.0 ble in spheroidal coordinates,‘4 just as for D = 3, since we want to examine the nonseparable situation, we employ cy- lindrical coordinates defined by

0 1 2 345 10 20 100 00 Scaled R

x1 =z, xk =pF,(n,-,), with p’=Cxi, (2.1) k

for 2 3 indicates the D = 3 pair degenerate with the lowest pair for that D. Further interdimensional degeneracies are specitkd in Ref. 14.

9030 Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy

FIG. 1. Electronic energy curves (omitting nuclear repulsion) for pairs ofg, tl states of Hz+ that stem from separated atom states with m = I = n, - 1, for n, = 1, 2, 3, 4. Data from Refs. 14 and 34. Abscissa scale is 1OR /(R + 5) inordertomap thefullrangeofintemucleardistance.Ordin- ates for n > 1 states have been shifted to coincide with n = 1 asymptote in separated atom limit. Dimension scaled units are used, with distance in I?/Z bohrs and energy in Z’/ti hartrees, where K = CD - 1)/2. By virtue of exact interdimensional degeneracies, increasing the angular momentum projection Irnj by unity is equivalent to increasing D by two. Thus these pairs ofg, u states for the D = 3 system correspond to the n, = 1 states for D’= 3, 5, 7, and 9, respectively, as specified in Table I.

The first term is the scaled centrifugal potential, for m = 0; this contains a D-dependent coefficient,

fD = (D-2)(0-4)/(0- 1)2. (2.4) Note fD --f 1 in the D- CO limit; we shall chiefly use the effec- tive potential W, for this limit. The Coulombic terms are specified by the electron-nucleus distances, given by

r=[p2+(~~RR2)211'2, (2.5) where the + sign pertains for r, and the E__ sign for r,.

Figure 2 shows the effective potential W, (p,z;R) for four values of the scaled internuclear distance. At the united atom limit (R = 0), the potential surface has a single well, but at distances near the equilibrium bond length (at R -2) double minima become prominent. At large R these evolve into a pair of isolated wells in the separated atom limit. The critical point at which the symmetry breaking transition from single to double wells occurs is determined33 from the conditions a W/c+ = 0 and d 2 W/a 2z = 0, both evalu- ated at z = 0. At that point: R, = (27/16)“2 = 1.299 038; PC = (27/32) In = 0.918 559; WC = - 32/27 = - 1.185 185. Throughout the domain R, CR < 00, tun-

neling through the barrier between the two minima occurs. However, as D and hence the effective mass I? increases, tunneling diminishes markedly. In this paper our chief aim is to evaluate the splitting AE,(R) between the lowest two eigenvalues of Eq. (2.2), produced by tunneling in the dou- ble well domain.

Spheroidal coordinates R = (r, + r, )/R and p= Va - r, )/R are related to the cylindrical coordinates by z=R&/2 andp2=R2(R2- l)(l-$)/4. In these

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Kais, Morgan, III, and Herschbach: Electronic tunneling and exchange energy 9031

R=() ;

3 R=2 2

FIG. 2. Effective potential energy surfaces of Hc at D- m limit, for internuclear distance R = 0, 1, 2, 5 bohr units. From calculations of Ref. 33.

coordinates, Eq. (2.2) separates into a pair of equations with effective potentials W, (il;R) and W, (p;R ), treated in de- tail elsewhere. l5

Before undertaking explicit calculations, we can exploit a remarkable consequence of the very simple form of Eq. (2.2). Since I? has the role of an effective mass, and the tunnel effect splitting to leading order is given by Eq. ( 1.2), we anticipate that the dimension dependence of the Sand A factors in AED (R) may resemble that for simple WKB theo- ry. This suggests that the action integral should scale with masslike IpI-[2mlE- MV,[]1’2--m1’2-~andtheprefac- tor like Ipj - 1’2.-m - 1’4-~ - 1’2 To the extent that this . holds, the dimension dependence of the splitting is given by

AE,(R) = K- ‘/2s4(R) exp[ - ~s(R)/fi], (2.6) where A (R ) and S( R ) are now independent of dimension. This predicts that a plot of In [ K”~AE~ ] vs K = (D - 1)/2 at constant R should be linear. Comparison with the exact numerical results of Frantz34 shows that such plots indeed prove to be nearly linear, except at small D and R.

A more accurate scal.ing law can readily be obtained. For the H,’ skates considered here, the D-dimensional ener-

gy splitting in scaled units is related to the D = 3 splitting by the simple transcription14

AE,(R) =gAE,(R/tij. I2.7) This may be combined with the leading term of the asympto- tic expansion for AE, given by Damburg and Propin;” on introducing the scaled units we thereby find

AE,(R) =2[Icc-‘/I’(K)] exp[ - K(R + 1 - ln2R)l. (2.8)

For large D, and hence large K, the bracketed quantity in the preexponential factor becomes simply ( 27~~) - v2 exp ( K) , so the K dependence then agrees exactly with that of Eq. (2.6). Figure 3 shows that plots of ln[ l?(~)pEJIcc- ‘1 vs K at constant R are quite linear even at small D and R.

For comparison with results derived from various ap- proximations, we have evaluated the functions S( R ) and A(R) of Eq. (2.6) by least-squares fits to the data of Fig. 3, and the results are plotted in Figs. 4 and 5. This approximate dimensional scaling, with the action proportional to K and the prefactor to K- ‘12, is expected to hold also for systems

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9032 Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy

n

-16 I 8 1 3 4 5 6 7 8 9 10

D

FIG. 3. Semilogrithmic plot of scaled tunneling splitting AE, vs D for R = 2-10, in accord with Eq. (2.8). Data from Refs. 14 and 34. On upper abcissa are indicated values of the principal quantum number n, for the fJ = 3 separated atom limit, in accord with the interdimensional degener- acies specified in Table I.

with many degrees of freedom since it stems simply from the role of 2 as an effective mass. When this scaling is adequate, we can often simplify the evaluation of AE, (R) by using any convenient D, in particular the D+ (~3 limit.

Ill. TUNNELING IN ONE DEGREE OF FREEDOM

The chief features of both the asymptotic and instanton methods, including aspects that distinguish them from the WKB approximation, can be presented most simply for one degree of freedom. We apply these methods in turn to the H2+ double-minimum potential of Eq. (2.3), and examine particularly how the exchange splitting depends on D and R.

%

1 I I I’ 1 I , 1

3 4 5 6 7,. 8 9 10 R ’

FIG. 4. Dependence of instanton action ,.S, on internuclear distance R, as derived from conventional numerical calculations (-, Ref. 34); asympto- tic approximation [--- -, Rq. (3.14) 1; exact instanton action [W, Eq. (3.29) or Rq. (4.3)]; parabolic approximation [A, Eq. (4.1111; and straight-line approximation [O, Eq. (4. lo), Refs. 3-71. Ordinate in units of pi.

0.5 1 t I I 1 9 I , I

3 4 5 6 7 8 9 10 R

FIG. 5. Dependence of prefactor A on internuclear distance R, as derived from conventional numerical calculations (-, Ref. 34); asymptotic ap- proximation [- - - -, Eq. (3.14) 1; instanton treatment in cylindrical co- ordinates ]W, Eq. (5.1) 1; instanton treatment in spheroidal coordinates [A, Eq. (3.30) ]; and straight-line approximation IO, Eq. (4.10), Refs. 3- 71. Ordinate gives &“A, with K = (D - 1)/2.

A. Asymptotic method for arbitriiry D and large R The derivation outlined by Landau and Lifshitz” is

readily generalized by replacing the x coordinate with a D- dimensional vector: x = (x . I ,. r 2 ,..., xI) ). At large R, the elec- tronic wave functions for the lowest a* and a, levels can be written as

q&,,(x) =2-1’2[~oo(X*,X2,...,XD) It $c( -x1 ,~2,**.JD 11, (3.1)

where x1 = Cl designates the median plane midway between the nuclei and $,, is the wave function of the electron in one of the wells (say that for nucleus ~1, at x1 = R /2). This has the form

$,,(x) =A,[2D-‘~- DRK-Dr(D/2)/r(D)]1’2

Xexp( - r,/K) (3.2) with K F (D - 1)[2 and r, the magnitude of a D-compo- nent vector. Here A, (x, ,R) is a slowly varying function; for R-+co, A, + 1, and I,& is then the wave function for a D- dimensional hydrogen atom. 35 The modifying amplitude A, can be evaluated by requiring &, to satisfy the Schrodinger equation,

-+-j% -+-$+$) I,& = E,r,$,, (3.3) a

where E, = -mtZ 2~ - 2 is the ground-state energy of the hy- drogen atom. This gives

JAo -+ KdX,

(3.4)

where we assume that x *,...,x~

As intended, A, -+ 1 when x1 -+R /2; however, A, =A,,, = 2”e --K/2 on themedian plane where x, = 0. The Holstein-Herring formula ” for the splitting between the g, IC states now involves a (.D-- 1 )-fold integration over the median plane,

AE, = ss s

-*. $o(d$o/dx, )dx, dx,*..dx,. (3.6)

Inserting Eq. (3.2) and transforming to polar coordinates yields

s m

AE, = (~/K)A “, r f-” exp( - 2r,/K)dr, da, R/2

(3.7)

where the integral over the total (D - 1 )-dimensional solid angle is .

da = 2n”/l?(/r). (3.8)

The radial integral gives -m

J r f-’ exp( - 2r,/K)dr@

R/2

= (/r/2)D-‘IyD- ;)e-R’“F,(R/K), (3.9)

where FD (R /K) is a polynomial function defined by D-2

FD (R //r) = c (R /K)“/s! (3.10) .S=O

Here we introduce the dimension-scaled units of Sec. II, with distance in units of K”/Z, and obtain

AE, =N,(D)F,(KR) exp( -KR), (3.11) or

AED.= [No (D)/I’(D 7 l)] (KR)D-2 exp( -kR) (3.12)

if for large R we retain only the leading term in the FD poly- nomial. The normalization factor is .iv,, (D) = sDr(o/2) eXp( - K)/[7T”2r(K)K3]. (3.13) For the 0=3 case (K== l), we have N,(3) =4/e and F3 = 1 + R, so for large R we obtain the asymptotically cor- rect result of Eq. ( 1.1) . A better alternative is that &, (x) could be replaced in Eq. (3.6) by the solution of Schrodinger equation (3.3) at large dimension. This gives a result in ex- act agreement with Eq. (2.8). For large D, we can use Stirl- ing’s formula for the factorials to obtain the action which is given by

S(R)/fi = R - In 2R (3.14) and the prefactor by

A(R) = (2/77)1’2. (3.15) These extremely simple results are included in Figs. 4 and 5.

B. Instanton method for arbitrary D and R We consider motion along a Cartesisan coordinate x,

governed by a double-well potential V(x) with minima at x = f a; the energy zero is chosen so that V( & a) = 0 and units chosen such that the kinetic energy operator is just - id 2/dx2. In its canonical version,13 the instanton method

pertains to potentials adequately approximated as parabolic near the minima, with o = [ VI ( + a) ] *” the frequency for harmonic oscillation about either minimum. The splitting AE of the lowest pair of energy levels has the form of Eq. ( 1.2), to leading order, with the transmission exponent giv- en by

so = I

n [2V(x)] “= dx. (3.16)

This transmiision integral involves the modulus b(x) 1 of the purely imaginary classical momentum, p(x) = dx/dt = {2[0 - V(x)]}“‘, for a nominal particle moving with zero energy in the classically forbidden region. Real dynamics can be restored by replacing time t with it. Then instead of making stationary the action integral

S= I

L(f,x)dt= s

[3(i)=- V(x)]dt (3.17)

for the Lagrangian L (Z&X), we are making stationary

I L(H,x)d( - it) = i

I [j(i)‘- { - V(x))]dt.

(3.18)

This device is equivalent to solving the classical equations of motion for a particle moving in real time with zero energy in an inverted potential, - V(x). The integral So in Eq. (3.16) is just the classical action for the zero energy trajectory con- necting one maximum of the inverted potential to the other. If - V(x) is parabolic near its maxima, then in that vicinity the velocity dx/dt is proportional to - (x & a); according- fY, ln(afx)- -fat and near the maxima Idx/dt 1 -exp( - cot). As t + & CO, the particle approaches the maxima at x = rfr a exponentially slowly. It spends most of its time near these endpoints; thus, relatively speaking, it traverses the region between the maxima in an “instant.” This is why such a trajectory is called an “instanton.”

The great advantage of this view of the transmission exponent So is that it can be immediately generalized to more degrees of freedoms (even to infinitely many degrees of freedom, as with tunneling in quantum field theory36 ) . To effect this generalization, we need only replace the scalar position variable x with a vector, and the scalar displace- ments x f a with magnitudes of the vector displacements. These simple substitutions suffice to determine the exponen- tial factor exp ( - So /fi) , the leading contribution to energy level splittings produced by tunneling for any number of de- grees of freedom. A much more elaborate analysis is re- quired to evaluate the pre-exponential factor. The van Vleck determinant of Eq. ( 1.3) may be recast as

A = Idp:f%3q;i)l (3.19) since the final momentum vector pcf’ is related to the action by pi” = dS /dq, (f). Because the instanton trajectory runs for infinite time, an arbitrarily small initial displacement Sq”’ would yield after infinite time a noninfinitesmal final momentum p(f). Thus, we must first impose large but finite cutoffs + Tat both ends of the time axis and take the limit as Sq”! tends to zero before taking the limit as T-+ CO. This procedure in principle could be used to compute the pre- exponential factor for tunneling in many degrees of freedom.

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9034 Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy

However, it would likely suffer from severe numerical insta- bility, owing to the notorious difficulty of reversibly running trajectories for long times. Therefore we have developed oth- er methods to evaluate the pm-exponential factor in a more numerically stable.manner.

For one degree of freedom, our starting point is the Eu- clidean (imaginary time) version of Feynman’s sum over histories,”

(X/l exP( - HT/49 Ix*) = C exp( - E, T/S) n . .

x (Xfl4 (nIxi>

= N s

exp( -S/A) [dx],

(3.20) where Ixi) and I+) denote the initial and final position ei- genstates and In) the energy eigenstates for the Hamiliton- ian H with eigenvalues En, the time T is positive, N is a normalization factor, S the Euclidean action, and [ dx] de- notes integration over all trajectories x(t) obeying the boundary conditions: x( - T/2) = xi and x( T/2) = xf From Eq. (3.20) we see that the leading term for large T specifies the energy and wave function of the lowest-lying energy eigenstate, (x~I exP( - HT/s9 /Xi> I (X/lP)(QlXi> exp( - E,T/fi).

(3.21) Thus the ground state eigenvalue E,, is projected out by this operation.

To apply these results to the double-well problem, we denote by IL, ) and IRn> the energy eigenstates of the left and right wells. The intervening potential barrier splits the L-R degeneracy so that to leading order in fi the lowest pair of eigenstates is

I Jr > =2-“=(I&,) f I&,)9 (3.22) with corresponding eigenvalues E. & &AE, respectively. If we evaluate for this-pair of states Feynman’s amplitude of Eq. (3.21) with xi and x/ near the bottom of the left and right wells, respectively, for large T we obtain

&I eW-- HT/fi9 Ixi) r(xfIL,)(R,Ixi)exp( - E,,T/+i) sinh (AET/fi).

(3.23) Small overlap terms involving (xi IL,) and (x#~ ) have been dropped. Thus, if we can compute the Feynman ampli-

TABLE II. Instanton results for double-well splittings.

tude, we can determine E. and AE. Coleman*3 has carried out this calculation in detail for the double-well problem at large T with the result (xfl exp( - HT/fi) /xi) = (w/z-F?)“~ exp( - oT/2)

iCsinh[KTexp( -S,/fi)]. (3.24)

Here So is again the instanton action, o is the harmonic oscillation frequency, and k’ is a constant given by K = (S,,/&> ““K, where the quantity g is defmed in terms of determinants representing products of eigenfunctions of the second functional derivative of the action at its station- ary point. Coleman computes x directly, but as he slyly says this is “somewhat tricky” and rather lengthy. In Appendix A we derive a shortcut which gives

(S,)“2~=lim(a-X) exp o X[2V(x)]-“2dx . X--a 1s 0 1

(3.25)

On comparing Eqs. (3.23) and (3.24) we find AE = A exp( - So,%), with

A=2%=2w(&0/n-)~~=(S~)“=~.~ (3.26) This provides, to leading order in fi, an explicit prescription for computing AE from the potential function. Note that, by virtue of Bq. (2.69, the transcription fi++l/~ holds for D scaling to this order.

Table II gives results for three examples, the double- parabola,38 double-cosine,3g and double-quartic4’ oscilla- tors. These are simple enough that the integrals required for So and A can be evaluated analytically. In each case, to lead- ing order in fi, the AE obtained from the instanton method is identical to the known exact result. On comparing with the WKB approximation, we find S, is the same but the A-fac- tors differ, so that

AE( WKB)/AE( exact) = (e/r) “=, (3.27) a ratio inviting transcendental meditation. The error in the WKB approximation has been traced to the use of connect- ing formulas derived by linearizing the potential near the classical turning pointss’ This procedure becomes inade- quate near the.potential minima. The instanton method in effect employs connecting formulas appropriate for a qua- dratic approximation to the potential near the minima. This feature and its tractability in treating more than one degree of freedom are the chief practical advantages of the instanton method.

1.

Potential, V(x) Barrier ht., V, Harmonic freq., CO Action, S&l Prefactor, A

Double parabola

v,[l- (Ixl/a)l’ ;m=2 (2 v, ) “2/c7 2v,/ti 2(fi#vo/7r)“2

Cosine

~v,[1’-ccos(%x/u)l 20’a2/d I’ (rr/2a)(2V,)“’ 4VJiiOJ 4(2liwV,/~)“~

Double quartic

vo I1 - (x/a)‘]’ @o’a’ 2(2VO)“2/U yJ( y,/h) 8(2~oV,/~)‘”

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Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy 9035

The application to HZ+ in spheriodal coordinates in- volves special features, *15 here we only outline the chief re- sults. The R component of the separable potential is mono- tonic, so the double-well potential character appears solely in thep component. In the D-+ 00 limit, the instanton poten- tial is given by

V(x) = v, [ 1 - (x/a>“]2/( 1 -x2)2 (3.28) with x =,u and V, = &[a’/( 1 - a’)]‘. Except for the de- nominator, ( 1 - x2) ‘-2, this function resembles the double- quartic potential of Table II. However, the shape is strongly affected by the constraint 1.x I< 1. This imposes sharp cusps at the minima, especially when a+ 1, as occurs for R large. Both the barrier height, V, = wa3/4, and the harmonic fre- quency, w = 2a/( 1 - a2)2, then grow very large. From Eq. (3.16) the instanton action is given by -.

S, = [2a/(l -a”)] --ln[(l +a)/(1 -a)]. (3.29) This has a different form than the examples of Table II, not simply proportional to the ratio of barrier height to harmon- ic frequency. Nonetheless, Eq; (3.29) gives good agreement with the action evaluated from the exact numerical results, as seen in Fig. 4; indeed, even at large R it does better than the result of Eq. (3.14) obtained from the asymptotic ap- proximation. Also, for a-+0, as occurs when!R approaches the critical R, from above, Eq. (3.29) reduces to the result of Table II for the double-quartic oscillator.

For the prefactor A, the sharp cusps of the H; potential spoil the standard technique of Eq. (3.26) because the in- Stanton path only spends a short time near the classical mini- ma and because the fluctuations about that path diverge while approaching the minima.42 Fortunately, these malad- ies prove to be curable I5 by a suitable resealing of the time variable.43 Other corrections enter because the separation constant differs for the states linked by tunneling and the Jacobian factor in the normalization integral is not separa- ble. l5 Yet a simple result is obtained,

A =4(2tiVo/~,“2/(rZ,~)m, (3.30)

where the electron-nucleus distances r, and r, of Eq. (2.6) are evaluated at the minimum of the effective potential. Again, as seen in Fig. 5, the agreement with the numerical results is good, appreciably better than Eq. (3.15) from the asymptotic approximation.

The cusp problem offers a curious lesson. We will find that, although in cylindrical coordinates the Schrodinger equation is nonseparable and hence involves tunneling in two coordinates, the potential is nicely quadratic near its minima and thereby amenable to the standard instanton methods. In tunneling, as in other devious pursuits, multi- variable pathways can better negotiate around awkward corners.

IV. INSTANTON ACTlONl FOR TWO DEGREES OF FREEDOM

The ground state energy splitting given by the instanton method has the general form of Eq. ( 1.2) regardless of the number of dimensions or degrees of freedom. However, to evaluate the instanton action for a multivariable potential,

so = [2V(x)]‘“dx, I

(4.1) P

we must find a particular zero energy path p for the coordi- nate vector x between the maxima of the inverted potential. The dominant tunneling contributions come from regions near the paths which minimize the instanton action and thus satisfy

6 [2V(x)]“2dS=o, I

(4.2)

where s is the path length, corresponding to ( ds)2 = dxedx. Here we derive, from calculus of variations, an algorithm for the instanton path which minimizes S, for two degrees of freedom. We formulate the method for H2 and determine explicitly the instanton path g and the corresponding action integral.

As seen in Eq. (2.2), the dimension-scaled Schrodinger equation for H2+ takes a simple form in cylindrical coordi- nates with the electron located in the p, z plane. For a fixed internuclear distance R beyond the critical point for symme- try breaking, the instanton path p [p(z>,.z] can be deter- mined by finding p (z) such that it minimizes the integral

%I so =

s F(p,z) [ 1 + (c+/~z)~] 1’2 dz, (4.3)

- %l where F(&z) = [2V(P,z) ] l/2 -and V(p,z) = W(p,z) - W(p, ,z, >, with W the effective potential of Eq. (2.3) for

the D-P CC, limit. The subscript m refers to the potential mini- ma, located at (pm, k z, ), as shown in Fig. 2.

We now invoke a central theorem from the calculus of variations44 pertaining to a functional of the form

s g(p,p,,z)dz, defined on the set of functions p(z) which have continuous first derivatives in the interval ( - z, ,z, ) and satisfy the boundary conditions. A necessary condition for the func- tional to have an extremum for a given functionp(z) is that the function satisfy Euler’s equation,

(4.4)

In our case, g(p, pi, z) is the integrand of Eq. (4.3)) with pZ = dp/dz. Th us, Euler’s equation becomes

$ (1 +p:)“‘-; [ F(p,z) ( 1 + p: ) - “2pZ ] = 0. -I-

(4.5)

On introducing the function F(p, z) and a few algebraic rear- rangements, we obtain a second-order nonlinear differential equation for thep(z) function that determines the instanton path,

d2P -=~[~[l+p~]2-~[l+pilp,J. (4.6) d2

To solve this equation, we convert it to a pair of coupled first- order differential equations,

dy, d “p and’ dy2 dz= d2 ~-&=:Yl. (4.7)

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9036 Kais, Morgan, III, and Herschbach: Electronic tunneling and exchange energy

These can be numerically integrated to obtain y1 = dp/dz and y, = p. After thus determining the instanton path func- tion p(z), the corresponding action integral of Eq. (4.3) is readily computed by evaluating it along this path.

A straight-line approximation for the tunneling path in multivariable nonseparable potentials has been widely used,39 and in many applications it yields a good approxi- mation for the energy splitting. In our case this approxima- tion is obtained by freezing the p coordinate. The corre- sponding splitting AE(p) can then be computed as in Sec. III for astraight-line pathz(p) and averaged overp to obtain the net splitting,

AE(p)l+,,(p-pp,)12dp, (4.8)

where the average employs a ground state harmonic oscilla- tor function,

90 (p -pm 1 = bPhw 1’4 ew[ - b@W @ -pm I”] (4.9)

with K = (D - 1)/2 and wP = [ (a 2 V/ap’) m ] I”. Evalua- tion of the average by steepest descents yields

AE = AE(@ [2wP/j2wP +L~;;(P)]“~

Xexp[ - (qJfd@-p,)2], (4.10)

wherep is the most dominant contribution to Eq. (4.9), the value at the saddle point of the integrand. An analogous procedure can be used for any other postulated form of the tunneling path, such as a parabolic path.

Figure 6 shows contour plots of the effective potential and compares the exact instanton path with the straight-line approximation and with a parabolic approximation. In each case, the parameters are fixed by minimizing the action. We see that in cylindrical coordinates the exact path deviates markedly from the straight line but does closely resemble the parabolic approximation. This result is readily understood

TABLE III. Instanton action S, for Hz’ via several approximations.

as a consequence of the separability in spheroidal coordi- nates.15 Since tunneling occurs in the p coordinate, if we fix the other coordinate at the value corresponding to the poten- tial minimum, ;1= ;1,, we have z = R&&2 and hence

p=iR((il; - 1)1’2[ 1 - (2z/RR,,J2] *‘2, prJR(R; - 1)“2(1.+2-lgx‘L+# -...) (4.11) with x = 2z/Ril,. The expansion variable x never exceeds unity, since IzI 1. By virtue of the small coeffi- cients of the fourth order and higher terms, the path is nearly parabolic.

Table III gives our results for the instanton action, eval- uated for several internuclear distances ranging from R = 3 to 10. The action computed for tunneling in two degrees of freedom (cylindrical coordinates) is seen to agree closely with the analytical result of Eq. (3.29) obtained for one de- gree of freedom (spheroidal coordinates). Figure 4 com- pares the variation with R of the action integral for the exact instanton path and for the two approximations. In the range examined, the action is a nearly linear function of R; relative to the result for the exact instanton path, as R increases the action for the straight-line approximation path becomes too large and that for the parabolic path too small. The agree- ment with the action derived from the numerical calcula- tions3’ is substantially better for the exact instanton path.

V. PATH INTEGRAL EVALUATION OF FLUCTUATION FACTOR

The A factor in the energy splitting, as defined by Eq. ( 1.2) or equivalent expressions, is a measure of the contribu- tion of paths in the vicinity of the instanton path. Evaluating the A factor thus involves solving the equations of motion to obtain the fluctuations around the instanton path. We em- ploy the path decomposition expansion devised by Auerbach and Kivelson, lo a path integral technique which breaks the

R Numerical”

3 1.0416 4 1.7191 5 2.5007 6 3.33 17 7 4.1915 8 5.0709 9 5.9646

10 6.8694

Exact Exact Parabolic cylindricalb

Straight-line spheroidal’ approx.d approx.’

0.9572 0.9569 0.931 1.040 1.7171 1.7171 1.656 1.857 2.528 2.5273 2.441 2.725 3.370 3.3692 3.260 3.624 4.234 4.2336 4.093 . 4.545 5.115 5.1140 4.952 5.418 6.009 6.0077 5.816 6.429 6.913 6.9117 6.716 7.389

“Derived from accurate numerical solution (Ref. 34) by least-squares fit of action to scaled curves of Fig. 3. bFrom numerical integration of Eq. (4.3) along nonseparable instanton path in cylindrical coordinates, as

described in text. ‘From analytic expression of Eq. (3.29), evaluated for separable instanton path in spheroidal coordinates

(Ref. 15). d From numerical integration of Eq. (4.3) along parabolic path of Eq. (4.11), with averaging analogous to J?qs.

(4.8)-(4.10). “From numerical integration of Eq. (4.3) along straight-line path, as described in Eqs. (4.8)-(4.10).

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Kais, Morgan, III, and Herschbach: Electronic tunneling and exchange energy 9037

2.2 R=lO

2.0

1.8

A. 1.6

1.4

1.2

1

1.8f R=S R=5 l.3l - * ’ * ’ * * I

+--- -65 0 0.5 1 I -4 I * -2 * * 0 I I 2 . , 4 ,

I-L 2

R=3 21’7 21

R=3

configuration space into disjoint regions and facilitates solu- tion of the fluctuation equations. The net result for the pre- exponential factor takes the form

s&i = hi(W,W,/K7T) “‘1 (5.1)

where w, and ob are the harmonic frequencies for vibration- al normal modes at the minima of the effective potential and K = (D - 1) /2 is again the dimensional scaling factor intro- duced in Eq. (2.2). The factor A comes from numerical inte- gration of the fluctuation equations. Without recapitulating the derivations of formulas, lo we outline the calculations for H,+ and examine particularly the asymptotic dependence on dimension and internuc1ea.r distance.

The path decomposition method specifies two surfaces, each enclosing one of the potential wells. If the wells have quadratic minima, as holds for H,+ in cylindrical coordi- nates, harmonic oscillator wave functions can be used within the enclosed regions; this brings in the factors involving w, and q, in Eq. (5.1). The computation of the A factor deals

FIG. 6. Instanton paths for tunneling be- tween pairs of minima in etfective potential at D-- CC limit for Hz+ molecule ion with R = 3, 5, and 10 bohr radii. Panekat left show contour maps of effective potential in separable form obtained using spheroidal coordinates (2,~); at right maps for nonse- parable form in cylindrical coordinates (p, z). Heavy dots show exact instanton paths; long dashes indicate parabolic approxima- tion, short dashes straight-line approxima- tion.

with a fluctuation vector q that specifies the deviation of a classical path from the instanton path. By projecting q onto unit vectors II~, and ql, the fluctuations are resolved into components parallel and perpendicular to the instanton path. For the projection onto the plane perpendicular to the instanton path, q = q+qs, the equation of motion is given by

d2q/d2 = [ (fL2 - 3J*)q - Z(dq/dz) l/(i)‘. (5.2)

The various quantities involved can all be determined from the potential function, U(p, z), and the instanton path, p(z), defined in Eq. (4.3). The quantity Cl2 = ql*U*~I is termed the transverse curvature and J = ql, .ql the bending frequency, where U is the matrix comprised of second de- rivatives of the potential function with respect to p and z. These quantities are given by

f-P = (V,/5” - 2v,ip + y$)/P, (5.3) and

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9038 Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy

J= (V./i - V,i>/F=, (5.4) where F = [ 2 V(p,z) ] *” and pZ = dp/dz. Subscripts indi- cate derivatives with respect to coordinates, superior dots with respect to time; all are evaluated on the instanton path. The time derivatives may be recast using the path equation p(z) to obtain

i=dz/dt=F/(l +p;)1’2,

@ = dp/dt = Fp,/ ( 1 + pz” ) “=, and

(5.5)

f=d=z/dt== V, - [pi/(1 +p;)] VP. (5.6) The kinship with Eqs. (4.3)-(4.7) for the instanton actionis evident.

In the classically forbidden region between the potential wells, the fluctuation projection q(z) is evaluated by inte- grating Eq. (5.2) along the instanton path between the de- composition points, from - z, to z, or vice versu.~ This is equivalent in time to proceeding from t = 0 to t = T, defined by T = (3S/dE, whereSis the action along the instanton path between the decomposition points. Accordingly,

=’ T= s

[ ( 1 + p; ) “=/F(p,z) ] dz. (5.7) - 2.3

The initial conditions for the integration to obtain the q(t) function are q( 0) = 1, Q( 0) = qS, where C$ is the velocity pro- jection at the border of either potential well (z = + z, ) . In the harmonic oscillator approximation, this quantity is given by

4 = $3&x; + &vw6x;, (5.8) where x,, xb denote the vibrational normal coordinates, ob- tained in the standard wayz9 from a linear transformation,

x, =zcosy-psiny and xb =zsiny+pcosy, (5.9)

with y the mixing angle, defined by tan 2y = 2 V,/( V, - r”,, ) (5.10)

and determined from the curvatures of the potential near its minima. Thus,

13: = cos’ yV,- + 2 sin y cos y V, + sin2 y VPP (5.11a)

and

wi = sin’ yV, - 2 sin y cos yV, + cos2 y V,. (5.11b)

Once the mixing angle is determined, Eq. (5.8) can be evalu- ated by means of Eqs. (5,9) and (5.5) to obtain the 4 factor.

Finally, after evaluating the q(t) function by numerical integration, the fluctuation factor A of Eq. (5.1) can be com- puted from it and auxilary ingredients by

a=I~q(T)+~(T)+J(T)q,,(T)I-ln X [2l;(O)F( T) 1”’ exp l(w, + wb ) T, (5.12)

where Q( T) = (dq/dz)i, evaluated at z = f z,, and q,, ( T) denotes the “longitudinal” fluctuation, given by

s

T

q, (r> = WT) [J(t)qW/FW ldt (5.13a) 0

h = 2F( +zd) s

[JWq(d (1 +pfY2/ - =d

F2@A ldz. (5.13b) To illustrate the procedure, we sketch the calculations for R = 5. As seen in Fig. 2, for this internuclear distance the effective potential W, (p,z;R) has a pronounced double- well structure. From Eq. (2.3) we find that the minima oc- cur at pm = 0.994 595, z,,, = -& 2.462 25, with W, = - 0.696 85; the curvatures there are V, = 0.9952, Vi0 = - 0.1108, and VPP = 1.0485. The mixing angle y = 38.24” and the normal mode frequencies are w, = 0.9528 and wb I= 1.0657; thus we find 4 = 3.4922. The decomposition points, 2 z, = f 2.021, were chosen to lie close enough to the minima to permit the harmonic approxi- mation; then from Eq. (5.7) we have T= 3.859. Figure 7 shows the corresponding results for q, qll , R, and Jalong the instanton path; with these functions Eq. (5.12) gives A = 0.5407 for the fluctuation factor.

Table IV gives for R = 3-10 values obtained for the di- mension-scaled prefactor, K”~A. The agreement with the ac- curate numerical results is less good than for the action inte- gral (cf. Table III). However, Eq. (5.1) from the instanton

16

.s 12

j8

E4

c 0.8 ‘6 !j b.6 % g 0.4

D-4 0.2

0

Z

FIG. 7. Properties determining prefactor, evaluated along instanton path for R = 5. (a) Transverse fluctuation qand longitudinal fluctuation q,, ; (b) transverse curvature Q and the bending frequency J. The path is symmetric about z = 0.

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I Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy

TABLE IV. Prefactor A for Hz tunneling: 1.1 I I I I I , I 8

Instanton Instanton Straight-line R NumericaP cyl.indricalb spheroidal’ approx.d 1

AE *Lwox AE 0.9 exact

3 0.568 16 0.572 82 0.515 11 0.772 7 4 0.587 11 0.584 49 0.604 10 0.985 6 5 0.592 13 0.608 95 0.653 23 1.106 7 6 0.602 80 0.629 88 0.683 85 1.148 8 I 0.613 16 0.652 58 0.704 57 1.1344 8 0.623 66 0.669 05 0.719 35 ,1.074 0 9 0.632 71 0.678 81 0.730 38 0.979 5

10 0.640 85 0.688 85 0,738 80 0.847 4

*Derived fmm accurate numerical solution (Ref. 34) by least-squares fit to scaled curves of Fig. 3.

bFmm numerical evaluation of fluctuations in cylindrical coordinates, Eq. (5.1).

‘From analytic expression of Bq (3.30), derived using spheroidal coordi- nates (Ref. 15).

dFmm straight-line approximation, as described in Eqs. (4.8)-(4.10).

treatment in two degrees of freedom (cylindrical coordi- nates) proves to be appreciably better than the analytical formula of Eq. (3.30) obtained for one degree of freedom (spheroidal coordinates). As seen in Fig. 5, both instanton versions are much better than the straight-line approxima- tion.

VI. DISCUSSION: PATHS AND PROSPECTS

Previous practical treatments of tunneling in more than one degree of freedom have chiefly employed the uniform semiclassical approximation orthe straight-line approxima- tion. In ‘Table V and Fig. 8 we compare for R = 3-10 our instanton results for AE with the accurate numerical values derived from conventional. quantum calculations and with these usual approximations (all for D = 3 ) . The instanton results again prove to be niuch better than the straight-line approximation. Also, the ‘error in the instanton AE varies much less strongly with the internuclear distance. Thus, the instanton splitting (in either the spheroidal or. cylindrical variant) is accurate to - 5% at R = 3~and - 1% at R = 10, whereas the straight-line result is only .good’ to - 70% at

0.8

0.7 I 2345678 9 10

R

9039

FIG. 8. Comparison of tunneling splittings from tbe asymptotic [Ref. 17 and Bq. ( 1.1) 1, instanton, and uniform semiclassical (Ref. 43) approxima- tions with practically exact numerical result (Ref. 14). The instanton point for R = 2 is from Ref. 15. The AE values all pertain to the mz 1s - cr, 1s splitting for D = 3.

R = 3 and - 10% at R = 10. For the semiclassical approxi- mation, the error is - 12% at R = 3 and actually grows to more than 20% by R = 10.

Some of the error in all of these results is a consequence of computing BE only to leading order in fi. As pointed out by Cizek et a1.,20 in developing a “quasisemiclassical” asymptotic expansion for tunneling, l/R plays the same role as Plan&s constant. Thus, as an indication of the error due to higher order terms, we include in Fig. 8 the ratio of the leading asymptotic approximation, given by the Holstein- Herring result of Eq. ( 1.1)) to the exact numerical result. Indeed, the magnitude of this ratio and its variation with R are fairly similar to our instanton results. This correspon- dence between two quite different methods suggests that the remaining error is largely due to contributions of higher or- der in fi. Even at R - 10, however, both the asymptotic and instanton AE’scliffer from the exact result by - 1%; cur- iously, the asymptotic value is low, the instanton value high. This situation persists even at R = 30, the largest distance for which an accurate numerical AE is available.4’ Hokever, elsewhere” we show that the instanton treatment in spher- oidal coordinates, which can be carried out analytically, in

TABLE V. Tunnel et&t splittings for H$ ; Hartree units. _b

Instanton, Instanton, R Numerical” cylindri&’ spheroidalb

Straight-line Uniform approx.b semiclassical

0.209 477 0.219 940 0.197 834 0.100 534 0.104 967 0.108 487 0.047 128 0.048 606 0.052 175 0.021 325 0.021 661 0.023 536 0.009 322 0.009 458 0.010 218 0.003964 0.004 018 0.004 324’ 0.001651 0.001670 0.001796 0.090 677 0.000 685 0.000 735

0.386 21 0.186 3 0.217 60 0.088 6 0.102 50 0.041 2 0.043 34 0.018 5 0.017 03 . . . 0.006 32 0.003 4 0.002 23 . . . 0.000 74 0.000 5

*From accurate numerical solution (Ref. 34). bFmm Eq. (1.2) using action from Table III and prefactor from Table IV. ’ From uniform semiclassical approximation (Ref. 43).

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9040 Kais, Morgan, III, and Herschbach: Electronic tunneling and exchange energy

fact does agree exactly with the asymptotic result at suffi- ciently large R.

From the perspective of dimensional scaling, it is grati- fying that the use of the effective potential for the D-t CO limit, easily calculable from classical electrostatics, yields such good results for the actual D = 3 system. Since the large-D limit localizes the electron near the potential mini- ma, it is particularly congenial for the instanton method, in which the actual calculations involve only classical mechan- ics. In view of the intrinsically quanta1 character of tunnel- ing and its sensitivity to the potential, it is remarkable that the large D domain can be exploited to treat tunneling in more than one degree of freedom by essentially classical methods. This approach is particularly inviting for treat- ment of tunneling in many-electron systems as well as in chemical reactions.

the action. In this case the integral is dominated by the sta- tionary points of S’, which we denote by 2. In the semiclassi- cal limit ( fi -t 0)) a steepest descent evaluation of the integral yields N(S,/2z-?i)‘” exp(&/fi)

X{det’[ - a2/&’ + V”(X)]}-‘“, L42) where det’ indicates that the zero eigenvalue is to be omitted when computing the determinant. In Vu (x) the primes de- note differentiation with respect to X. The normalization constant is given by N= (w/?rti)1’2exp( --UT/~) [det( -c?~/&‘+w’)]..

(A3)

The feasibility of treating tunneling in more than two degrees of freedom by the procedures exemplified in this study requires means to evaluate the effective potential at large D and to compute from it the instanton action and instanton prefactor dne to fluctuations. The general proce- dure for calculating the large-D potential is already avail- able.46 Likewise, the evaluation of the instanton path and action for the zero-energy trajectory is tractable even for several degrees of freedom. In Appendix B we illustrate a simple means to obtain the leading exponential behavior for a many-electron molecule. The evaluation of the fluctuation factor remains a major roadblock. However, the approach of Sec. V bears enough resemblance to the treatment of molecu- lar vibrations to suggest that Wilson’s s-vector technique2’ might substantially simplify the fluctuation calculations.

On comparing Eq. (A2) with Eq. (Al ) we find K = (So/27n5)‘n~ det( - d2/b't2 + w2)/

det’[ -a’/&” + V”(X)] 1”’

= w(ws,/ln7y2iE (A4) To evaluate the determinants we must construct solutions of

- a2xn/ilt2 + V” (x)X, = 2.,x,. (A51 For i2 = 0 the solution13 is

X, = lim(t+ f ~0)s; “‘dZ/dt

= Kxp( --wit I). (A61 This relation gives

x cot =w

s dx(2V) --‘2

0

= -ln[S;“*F-‘(u-Z)] +O(a---,F) (A7) which on rearrangement yields Eq. (3.25).

ACKNOWLEDGMENTS

We thank Assa Auerbach, Sidney Coleman, Nancy Makri, William Miller, and Barry Simon for enjoyable dis- cussions of various aspects of this work, and gratefully ac- knowledge support received from the National Science Foundation (under Grants No. PHY-8911958 to D.R.H. and No. PHY-8608155 to J.D.M.), the Office of Naval Re- search, and the Venture Research Unit of the British Petro- leum Company. J.D.M. is further grateful for support pro- vided by a National Science Foundation grant for the Institute for Theoretical, Atomic, and Molecular Physics at the Harvard-Smithsonian Center for Astrophysics.

APPENDIX B: LEADING EXPONENTIAL FOR MANY-ELECTRON MOLECULES

With a view toward generalization to multielectron tun- neling, we note here a modest but useful step. It is quite easy to predict the leading exponential behavior of the exchange energy of a diatomic molecule or ion. This requires just a slight generalization of the procedure illustrated in Sec. III using Herring’s formula for the exchange splitting. We need only to know the leading exponential behavior of the corre- sponding atomic wave function (or one-electron density) and how many electrons are involved in the exchange pro- cess.

APPENDIX A: INSTANTON PREFACTOR

Here we derive Eq. (3.25), which accounts for fluctu- ations about the instanton path in the double-well problem. The starting point is Eq. (3.20)) Feynman’s sum over histor- ies. For a single instanton, Coleman13 obtained for the left- hand side

Herring’s formula,

AE= - J

fi -,yV,y dr m = s it* VJJ’ dr, (Bl)

(ddi)‘” exp( - wT/2)KTexp( - S,/fi). (Al) This is seen to correspond to just the leading term in the power series expansion of Eq. (3.24)) which represents the sum over all instantons. In order to determine the quantity K, we compare this single-instanton amplitude to the right- hand side of Eq. (3.20), governed by a functional integral of

involves an integral over the median plane, with x a linear combination of the quasidegenerate eigenfunctions localized on a single atom. To calculate the leading exponential behav- ior, we may approximate ,y by just an unperturbed atomic function. Its long-distance exponential behavior is simply exp[ - (2IP, )“‘,I in atomic units, where IP, is the first ionization potential. The corresponding one-electron den-

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Kais, Morgan, Ill, and Herschbach: Electronic tunneling and exchange energy 9041

sity then behaves asymptotically as exp [ - 2 (21P 1 ) *“r]. Taking the derivative of course yields the same exponential, and evaluation on the mid:plane at r = R /2 yields simply exp [ - (2IP, ) li2R 1. This is appropriate for the exchange of a single electron. For example, for Hc at large R, we have IP, = 1/2n2, so we immediately obtain an exchange split- ting proportional to exp( .- R /n). For the ground state (n = 1 ), this corresponds to Eq. ( 1.1) , the asymptotically exact result.

Another case involving the exchange of a single electron is He,+, in which the electron spends half of its time bound to one He + ion and half bound to the other. For the ground electronic state the relevant binding energy is 0.903 724. * * a.u., so in this case the constant in the exponential is (21P,)1’2= 1.344414***.

This approach can easily be related to an instanton anal- ysis. We take the zero of energy to be the ground-state energy for the electron bound to the atomic ion. In order to tunnel from one atomic ion to the other a large distance R away, the electron must move through an effective potential barrier whose height approaches IP 1 as R + CU. Near each atomic ion this effective potential may have some complicated be- havior, but over most of the intervening region it is nearly equal to IP, . Thus if we invert the effective potential, we find that over most of the region the kinetic energy of a zero- energy classical trajectory is nearly equal to IP, , and ‘the corresponding momentum is nearly (2IP, ) 1’2, from which we obtain

so = s p dqz (21P, )“’ dqz (2Ip, ) 1/2R. WI

It is now obvious what happens in the case of the ex- change of two electrons, as in the singlet-triplet energy split- ting of Hz. In order to exchange the two electrons, both must traverse a barrier of height nearly IP, . This yields an extra factor of 2, so the leading exponential behavior of the ex- change splitting in II, is seen to be simply exp[ - 2(2IP, )“‘R 1, in conformity with the result of Her- ring and Flicker.47

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