-
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.
Kais, Morgan, III, and Herschbach: Electronic tunneling and
exchange energy 9033
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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
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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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