General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright
owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
You may not further distribute the material or use it for any profit-making activity or commercial gain
You may freely distribute the URL identifying the publication in the public portal
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from orbit.dtu.dk on: Aug 10, 2022
Multidimensional effects on dissociation of N-2 on Ru(0001)
Diaz, C.; Vincent, J.K.; Krishnamohan, G.P.; Olsen, R.A.; Kroes, G.J.; Honkala, Johanna Karoliina;
Nørskov, Jens Kehlet
Published in:
Physical Review Letters
Link to article, DOI:
10.1103/PhysRevLett.96.096102
Publication date:
2006
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Diaz, C., Vincent, J. K., Krishnamohan, G. P., Olsen, R. A., Kroes, G. J., Honkala, J. K., & Nørskov, J. K. (2006).
Multidimensional effects on dissociation of N-2 on Ru(0001). Physical Review Letters, 96(9), 096102.
https://doi.org/10.1103/PhysRevLett.96.096102
Multidimensional Effects on Dissociation of N
2
on Ru(0001)
C. Dı
´
az,
1,
*
J. K. Vincent,
1,†
G. P. Krishnamohan,
1
R. A. Olsen,
1
G. J. Kroes,
1
K. Honkala,
2,‡
and J. K. Nørskov
2
1
Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands
2
Center for Atomic-Scale Materials Physics, Department of Physics, NanoDTU, Technical University of Denmark,
DK-2800 Lyngby, Denmark
(Received 18 October 2005; published 8 March 2006)
The applicability of the Born-Oppenheimer approximation to molecule-metal surface reactions is
presently a topic of intense debate. We have performed classical trajectory calculations on a prototype
activated dissociation reaction, of N
2
on Ru(0001), using a potential energy surface based on density
functional theory. The computed reaction probabilities are in good agreement with molecular beam
experiments. Comparison to previous calculations shows that the rotation of N
2
and its motion along the
surface affect the reactivity of N
2
much more than nonadiabatic effects.
DOI: 10.1103/PhysRevLett.96.096102 PACS numbers: 68.35.Bs, 68.43.h, 82.65.+r
The Born-Oppenheimer (BO) approximation is a stan-
dard tool of the chemical physicist aiming to compute rates
of chemical reactions. Transition state theory, which is the
standard theory for computing reaction rates for complex
systems, depends on its validity [1]. The BO approxima-
tion has been used successfully in the description of many
gas phase reactions [2]. However, its applicability to
molecule-metal surface reactions has been questioned,
due to the possibility of electron-hole pair excitations.
These reactions are relevant to heterogeneous catalysis
[3,4], which is of enormous relevance: about 90% of the
chemical manufacturing processes employed worldwide
use catalysts in one form or another [5].
Direct evidence for nonadiabatic effects on molecule-
surface scattering comes from experiments showing
electron-hole pair excitation accompanying chemisorption
of atoms and molecules [6], and showing ejection of elec-
trons from low work function metal surfaces accompany-
ing scattering of highly vibrationally excited molecules
with high electron affinity [7]. Also, it has recently been
shown that nonadiabatic [diabatic with [8] or without [9]
couplings] models describe the dissociation of O
2
on
Al(111) well, whereas an adiabatic description fails. It
has even been argued that indirect evidence exists that
nonadiabatic effects decrease the reactivity of N
2
on
Ru(0001) (a low spin molecule with low electron affinity
reacting on a general type transition metal surface) by
more than an order of magnitude [10,11].
The N
2
interaction with Ru(0001) has been intensively
studied [10 –15] because N
2
dissociation is considered the
rate-limiting step in the industrial synthesis of ammonia
over Ru catalysts. Most of the ammonia produced is used
for fertilizers, making ammonia indispensable for our so-
ciety [5]. Recent research [4,16] has shown that Ru steps
are much more important to N
2
dissociation than the
(0001) terraces, but the N
2
Ru0001 system has
emerged as a system of high fundamental interest. It ex-
hibits properties that make it fundamentally different from
the well-studied H
2
=Cu system [17], such that the
N
2
=Ru0001 system can be considered as another proto-
type system of dissociative chemisorption [18]. For
N
2
=Ru0001 the reaction barrier is located much more
in the exit channel than for H
2
=Cu. The value of the intra-
molecular distance at the minimum barrier geometry (r
b
)is
greater than the equilibrium bond distance by 1:3a
0
for
N
2
=Ru0001 (r
b
3:4a
0
) [13] and by 0:8a
0
for H
2
=Cu
(r
b
2:2a
0
) [19]. The barrier V
to reaction is much
higher for N
2
Ru0001 (2eV) [13] than for H
2
Cu (0:5eV) [19]. In contrast to H
2
=Cu [17] comparison
between adiabatic theory and experiment for N
2
reaction
on Ru(0001) has so far presented major discrepancies, for
dissociative chemisorption, associative desorption [11],
and inelastic scattering [18], supporting the idea of a large
influence of nonadiabatic effects.
An unusual feature of reaction of N
2
on Ru(0001) is that
the dissociation probability (S
0
) saturates at a very small
value (10
2
) for incidence energies E
i
V
[10,14]. To
understand the surprisingly low reactivity a combination of
experiment and modeling analysis was applied [11]. The
first model applied was a 2 1Dr; Z; q adiabatic model,
in which, besides the N-N distance (r) and the molecule-
surface distance (Z) [Fig. 1(a)], a coupling to surface
phonons (q) was included. This adiabatic model failed to
reproduce the experimental S
0
, overestimating it by 2
orders of magnitude at high E
i
(Fig. 2). In the second 2
2Dr; Z; q; model an additional coupling to electron-
(
b
)
t
o
p
f
cc
h
c
p
b
r
g
R
u
(
0001
)
NN
NN
ZZ
XX
YY
r
θθ
φφ
(
a
)
60
°°
FIG. 1. (a) Coordinate system used. (b) High symmetry points
of the Ru(0001) surface. White (gray) spheres denote atoms in
the first (second) layer. Atoms in the third (fourth) layer are
directly below the atoms in the first (second) layer.
PRL 96, 096102 (2006)
PHYSICAL REVIEW LETTERS
week ending
10 MARCH 2006
0031-9007=06=96(9)=096102(4)$23.00 096102-1 © 2006 The American Physical Society
hole pair excitation () was introduced leading to much
better agreement with experiment (Fig. 2). However, to
obtain this level of agreement a very large nonadiabatic
coupling was required, i.e., 12 times larger than required
for the description of vibrational damping of O
2
v 1
adsorbed on Pt(111) [11].
Here we show that the description of multidimensional
effects through the inclusion of the other 4 degrees of
freedom (DOFs) of N
2
in the dynamics is essential to
account for the experimental reactivity. A very good quan-
titative description of the reaction dynamics can already be
obtained with a model treating all molecular degrees of
freedom but neglecting electron-hole pair excitation and
phonons. The results and comparison to previous calcula-
tions show that nonadiabatic coupling affects the reaction
of N
2
on Ru(0001) to a much smaller extent than previ-
ously assumed, and provide further justification for the
application of adiabatic models to most molecule-surface
reactions relevant to heterogeneous catalysis.
A detailed description of theoretical methods used in
this work is presented elsewhere [20]. We take the ruthe-
nium surface as frozen. Although N
2
is a heavy molecule
and some energy exchange to phonons could be expected,
experiments on Ru(0001) show S
0
to be independent of
surface temperature (T
s
) in the range of collision energies
here considered [E
i
2:5eV[10] ], justifying our neglect
of the phonons to some extent. The BO approximation is
made, neglecting electron-hole pair excitations. Our model
considers motion in all six DOFs of N
2
[Fig. 1(a)].
We assume that density functional theory (DFT), with
the use of the generalized gradient approximation (GGA)
for the exchange-correlation energy, gives an accurate
description of a molecule-surface reaction if it proceeds
adiabatically [17]. In applying the GGA we have used the
RPBE (revision of Perdew-Burke-Ernzerhof ) functional
[21], which is accurate for molecular chemisorption [21]
and performed well in modeling ammonia production [4].
The ion cores were described using Vanderbilt pseudopo-
tentials [22] (with core cutoff radii of: r
N
c
0:6, r
Ru
c
0:9a
0
) and a plane wave basis set is used for the electronic
orbitals. With the selected number of Ru layers (3) and the
size of the unit cell (2 2), the plane wave energy cutoff
(350 eV), the amount of k points used, and the other
selected input parameters [20], the molecule-surface inter-
action energies are converged to within 0.1 eV of the plane
wave—pseudopotential RPBE results. The DFT calcula-
tions were performed with the
DACAPO code [23].
To obtain a potential energy surface (PES), the DFT data
were interpolated using a modified Shepard (MS) method
[24,25]. This method is here applied to a molecule-surface
reaction with a direct interface to DFT for the first time.
The interpolated PES is given by a weighted series of
second-order Taylor expansions centered on ab initio
data points. The gradients are computed analytically by
DACAPO, and second derivatives from the gradients using
forward differencing.
Reaction probabilities were computed using the quasi-
classical trajectory (QCT) method [26], the initial vibra-
tional energy of the molecule including zero-point energy.
Dissociation is defined to take place whenever r reaches
5:0a
0
with a positive radial velocity. To include the effect
of nozzle temperature (T
n
), reaction probabilities are com-
puted for the N
2
vibrational states v
i
0–2, and the results
weighted assuming the vibrational temperature to equal T
n
.
This implies that P
v
1 for v
i
0 at T
n
700 K, and
0.83, 0.145, 0.025 for v
i
0, 1, and 2, respectively, at
T
n
1850 K. All calculations were performed for normal
incidence, and for the initial rotational state J
i
0, be-
cause the reaction probabilities do not depend significantly
on J
i
[20] for J states with significant population at the
rotational temperatures relevant to the molecular beams
used in the experiments [27].
Figures 3(a) and 3(b) show 2D (r; Z) cuts through the
DFT PES. For N
2
approaching the surface with its bond
parallel to the surface ( 90
) and its center of mass over
a top site [Fig. 3(a)] and halfway between a top site and a
fcc site [Fig. 3(b)], the 2D cuts present very high barriers.
The second geometry is close to the lowest barrier ge-
ometry at X;Y;Z;r;;1:4a
0
;2:20a
0
;2:53a
0
;3:40a
0
;
86:23
;29:33
, X and Y being the Cartesian coordinates
associated with motion parallel to the surface [Fig. 1(a)],
and the barrier height V
being 2.27 eV [the PW91 barrier
was 1.70 eV, but this is very likely too low: PW91 [28] fails
[4] to predict the 0.4 eV barrier found experimentally for
stepped Ru [16], which is accurately described by the
01234
5
6
E
i
(eV)
-6
-5
-4
-3
-2
-1
0
log (S
0
)
6D ad. T
n
=700 K
6D ad. T
n
= 1850 K
2+1D ad. v
i
=0
2+2D non-ad. v
i
=0
Exp. T
n
=700 K
Exp. T
n
=1850 K
Exp. T
n
=1100 K
3
3.5
4
E
i
(eV)
0
1
2
3
4
S
0
X 10
2
FIG. 2 (color). The log of the probability of N
2
dissociation on
Ru(0001) is plotted vs normal translational energy E
i
.
Continuous lines with full symbols represent 6D adiabatic cal-
culations, the dashed line with full symbols represents 2 1D
adiabatic calculations from [11], and the dashed line with open
symbols 2 2D nonadiabatic calculations from [11].
Experimental measurements: full circles from [10]; full green
diamonds from [10]; full red squares from [14]. The inset shows
S
0
times 10
2
vs normal translational energy. Results are shown
for different nozzle temperatures (T
n
).
PRL 96, 096102 (2006)
PHYSICAL REVIEW LETTERS
week ending
10 MARCH 2006
096102-2
RPBE functional [16] ]. Figures 3(c) and 3(d) show that the
N
2
Ru0001 PES exhibits a very large anisotropy and
corrugation near the minimum barrier geometry, much
larger than for a representative H
2
=Cu system [19]. The
N
2
Ru0001 system presents a much narrower bottle-
neck towards dissociation than H
2
=Cu. The effect is largest
for the anisotropy, which is greater for N
2
because its
interatomic distance is larger at the barrier, making it
more difficult to rotate it to a tilted geometry.
The S
0
obtained from the 6D QCT calculations for T
n
700 K for E
i
in the classical regime (E
i
>V
) are in good
agreement with the experimental S
0
for the same T
n
(Fig. 2). Remarkably, inclusion of the four rotational and
parallel translational degrees of freedom with the two
molecular degrees of freedom (r and Z) also considered
in the 2 1D model [11] discussed above lowers the
reaction probability obtained with this model by about 2
orders of magnitude. The reason that the inclusion of the
rotations and parallel translations lowers the reactivity
much more for N
2
=Ru0001 than for H
2
=Cu [17], and
that S
0
is so small for E
i
V
, is that the barrier is a much
narrower bottleneck for N
2
=Ru0001 than for H
2
=Cu,as
discussed above [Figs. 3(c) and 3(d)].
The 6D QCT dynamics overestimates S
0
by about a
factor 3 at E
i
4eV, the agreement being better than
for the nonadiabatic 2 2D model [11] (inset Fig. 2).
This level of agreement between adiabatic theory and
experiment suggests a much smaller role for nonadiabatic
effects than previously assumed on the basis of the com-
parison of experiment to adiabatic 2 1D results [11]. The
remaining disagreement with experiment can be due to the
(i) approximations made in the implementation of the
adiabatic frozen surface model, or to the exclusion of
(ii) phonons and (iii) electron-hole pair excitations from
the model. Concerning (i), the exact exchange-correlation
functional is not yet known, but the use of the GGA with
DFT has allowed the calculation of accurate dissociation
probabilities for H
2
=metal systems [17]. The QCT method
has been shown to provide accurate results for dissocia-
tion of H
2
(v 0 and 1) on Cu(100) [29], H
2
presenting a
much greater challenge to the classical approximation than
N
2
. Concerning (ii), the inclusion of phonons in low-
dimensional models lowers the reactivity [11] as also
found for (iii) electron-hole pair excitations (Fig. 2) [11].
The experiments on laser assisted associative desorption
and vibrationally inelastic scattering referred to above
[18], and our finding that the 6D adiabatic model does
overestimate energy transfer to molecular vibration in
scattering [20], suggest that the discrepancy that remains
between 6D adiabatic theory and experiment (inset Fig. 2)
is in part due to nonadiabatic effects. The finding that
inclusion of phonons also reduces the reaction probability
suggests that the factor 3 discrepancy observed at the
highest E
i
(inset Fig. 2) is an upper bound to the effect
that electron-hole pair excitations might have on the
reactivity.
Our results show that late barrier reactions, like N
2
Ru0001 but also CH
4
Ni111 [30], require a treatment
of all molecular DOFs: the exclusion of part of these can
change the reactivity by 2 orders of magnitude.
Discrepancies of this size between experimental reactivity
and reactivity obtained in low-dimensional simulations
cannot be taken as evidence for nonadiabaticity. Our re-
sults suggest that the BO approximation yields a good
description of nitrogen dissociation at metal surfaces,
even for the high E
i
here considered. Explanations dis-
cussed in detail in Ref. [20] are that N
2
has zero electronic
spin, so that nonadiabatic spin quenching cannot affect the
reactivity [9], and that N
2
has a low electron affinity, so that
the transfer of an electron to the molecule cannot lead to
electronic excitations in the metal [7,8]. For Ru(0001), a
contributing factor may be that the molecular chemisorp-
tion well in front of the barrier is shallow, and that a
reacting molecule is not likely to pass through it [the N
2
orientations at the barrier and in the well differ [12] ], so
that no energy will be lost to electron-hole pair excitation
passing through such a well [6]. More research on reactions
involving N
2
, NO, and O
2
is needed to determine the
importance of these factors, and to establish if and by
how much the large charge rearrangement implied in the
R
u
NN
(
a
)
r
(
a
u
)
(
b
)
R
u
NN
r
(
a
u
)
-30
-15
0
15
30
θ − θ
*
(de
g
.)
0
1
2
3
V - V
*
(eV)
-1
-0.5
0
0.5
1
u - u
*
(au)
0
1
2
(d)(c)
H
2
/Cu(100)
N
2
/Ru(0001)
u
u
FIG. 3. Two-dimensional cuts through the potential energy
surface for (a) the molecule approaching the top site with the
N-N bond parallel to the surface (
i
90
) and (b) the molecule
approaching a site halfway between the top and fcc sites with
i
90
. The potential is for the molecule oriented as indicated
by the inset. The spacing between contour levels is 0.8 eV. The
anisotropy and corrugation of the N
2
=Ru0001 (solid line) and
H
2
=Cu100 [19] (dashed line) potentials near the minimum
barrier is illustrated by plotting their dependence on (c) and
u (d), keeping all other coordinates fixed to the barrier geometry
Q
. Here, u is the coordinate for motion along a straight line
parallel to the surface, such that V varies the least.
PRL 96, 096102 (2006)
PHYSICAL REVIEW LETTERS
week ending
10 MARCH 2006
096102-3
breaking of multiple bonds in -bonded molecules implies
nonadiabaticity [31].
The vibrational efficacy S
0
v0
S
0
v1
S
0
=
E
vib
v 1E
vib
v 0 is a measure of the importance
of molecular vibration for promoting reaction. Here
v
S
0
is the energy required to obtain a reaction probability S
0
for
the molecule initially in the state v, and E
vib
the molecule’s
vibrational energy. The we compute from 6D QCT
results (Fig. 4) for E
i
> 2:3eVand for a range of S
0
values
5 10
4
0:11:6. Thus, in our adiabatic model vi-
brational excitation promotes reaction more efficiently
than increasing E
i
. This is in agreement with an analysis
of the previous experiments [10] for E
i
< 2:55 eV and
T
n
700 and 1850 K (Fig. 2). Taking S
0
T
n
700 K
S
0v0
and S
0
T
n
1850 K1 cS
0v0
cS
0v1
,
with c the fraction of molecules in v 1 at 1850 K
assuming that only v 0 and 1 are populated, we compute
vibrational efficacies greater than 3 from the data. Our
calculations suggest that neglect of the v 2 contribution
[the experiments [10] were done for only 2 T
n
values]
should lead to overestimation of the experimental by
no more than 0.5. Both the present adiabatic theory and
previous experiments [10] thus show a large effect of N
2
vibration on dissociation on Ru(0001), in contrast to a
previous statement [14] that the effect should be less than
for H
2
=Cu ( 0:5).
We have performed QCT calculations on dissociative
chemisorption of N
2
on Ru(0001) based on a DFT adia-
batic PES and treating all six molecular DOFs. The multi-
dimensional effects of N
2
rotations and translations
parallel to the surface dramatically lower the reactivity of
N
2
on Ru(0001), leading to good agreement between adia-
batic theory and experiment, and suggesting a much
smaller role for nonadiabatic effects than previously as-
sumed. The dramatic lowering of the reactivity is due to the
large anisotropy and corrugation that the molecule sees
when approaching the barrier, the N
2
=Ru0001 barrier
presenting a much narrower bottleneck to reaction than
found in the H
2
=Cu prototype system.
Work financially supported by the European
Commission: research training network ‘‘Predicting
Catalysis’’ under Contract No. HPRN-CT-2002-00170.
*Electronic address: c.diaz@chem.leidenuniv.nl
†
Present address: Department of Physical Chemistry,
Uppsala University, Box 579, S-7123 Uppsala, Sweden.
‡
Present address: Nanoscience Center, Department of
Physics, University of Jyva
¨
skyla
¨
, P.O. Box 35, FIN-
40014, Finland.
[1] D. G. Truhlar, B. C. Garrett, and S. J. Klippenstein, J. Phys.
Chem. 100, 12 771 (1996).
[2] D. C. Clary, Science 279, 1879 (1998).
[3] K. Reuter, D. Frenkel, and M. Scheffler, Phys. Rev. Lett.
93, 116105 (2004).
[4] K. Honkala et al., Science 307, 555 (2005).
[5] I. Chorkendorff and J. W. Niemantsverdriet, Concepts of
Modern Catalysis and Kinetics (Wiley-VCH, Weinheim,
2003), student edition.
[6] B. Gergen, H. Nienhaus, W. H. Weinberg, and E. W.
McFarland, Science 294, 2521 (2001).
[7] J. D. White et al. , Nature (London) 433, 503 (2005).
[8] A. Hellman, B. Razaznejad, and B. I. Lundqvist, Phys.
Rev. B 71, 205424 (2005).
[9] J. Behler et al., Phys. Rev. Lett. 94, 036104 (2005).
[10] L. Romm, G. Katz, R. Kosloff, and M. Asscher, J. Phys.
Chem. B 101, 2213 (1997).
[11] L. Diekho
¨
ner et al., J. Chem. Phys. 117, 5018 (2002).
[12] M. J. Murphy, J. F. Skelly, A. Hodgson, and B. Hammer,
J. Chem. Phys. 110, 6954 (1999).
[13] L. Diekho
¨
ner et al., Phys. Rev. Lett. 84, 4906 (2000).
[14] L. Diekho
¨
ner et al., J. Chem. Phys. 115, 9028 (2001).
[15] R. C. Egeberg, J. H. Larsen, and I. Chorkendorff, Phys.
Chem. Chem. Phys. 3, 2007 (2001).
[16] S. Dahl et al., Phys. Rev. Lett. 83, 1814 (1999).
[17] G. J. Kroes et al., Acc. Chem. Res. 35, 193 (2002).
[18] H. Mortensen et al., J. Chem. Phys. 118, 11 200 (2003).
[19] R. A. Olsen et al., J. Chem. Phys. 116, 3841 (2002).
[20] C. Dı
´
az et al.
(to be published).
[21] B. Hammer, L. B. Hansen, and J. K. Nørskov, Phys. Rev. B
59, 7413 (1999).
[22] D. Vanderbilt, Phys. Rev. B 41, R7892 (1990).
[23]
DACAPO, http://dcwww.camp.dtu.dk/campos/Dacapo/.
[24] J. Ischtwan and M. A. Collins, J. Chem. Phys. 100, 8080
(1994).
[25] R. P. A. Bettens and M. A. Collins, J. Chem. Phys. 111,
816 (1999).
[26] M. Karplus, R. N. Porter, and R. D. Sharma, J. Chem.
Phys. 43, 3259 (1965).
[27] A. C. Luntz (private communication).
[28] J. P. Perdew et al., Phys. Rev. B 46, 6671 (1992).
[29] D. A. McCormack and G. J. Kroes, Phys. Chem. Chem.
Phys. 1, 1359 (1999).
[30] R. R. Smith, D. R. Killelea, D. F. DelSesto, and A. L. Utz,
Science 304, 992 (2004).
[31] A. C. Luntz and M. Persson, J. Chem. Phys. 123, 074704
(2005).
234
56
E
i
(eV)
0
0.05
0.1
0.15
0.2
S
0
v
i
=0
v
i
=1
v
i
=2
FIG. 4 (color online). Computed dissociation probability vs in-
cidence energy for several initial vibrational states v
i
and J
i
0.
PRL 96, 096102 (2006)
PHYSICAL REVIEW LETTERS
week ending
10 MARCH 2006
096102-4
![FIG. 2 (color). The log of the probability of N2 dissociation on Ru(0001) is plotted vs normal translational energy Ei. Continuous lines with full symbols represent 6D adiabatic calculations, the dashed line with full symbols represents 2 1D adiabatic calculations from [11], and the dashed line with open symbols 2 2D nonadiabatic calculations from [11]. Experimental measurements: full circles from [10]; full green diamonds from [10]; full red squares from [14]. The inset shows S0 times 102 vs normal translational energy. Results are shown for different nozzle temperatures (Tn).](/figures/fig-2-color-the-log-of-the-probability-of-n2-dissociation-on-1n3v7ees.webp)
![FIG. 3. Two-dimensional cuts through the potential energy surface for (a) the molecule approaching the top site with the N-N bond parallel to the surface ( i 90 ) and (b) the molecule approaching a site halfway between the top and fcc sites with i 90 . The potential is for the molecule oriented as indicated by the inset. The spacing between contour levels is 0.8 eV. The anisotropy and corrugation of the N2=Ru 0001 (solid line) and H2=Cu 100 [19] (dashed line) potentials near the minimum barrier is illustrated by plotting their dependence on (c) and u (d), keeping all other coordinates fixed to the barrier geometry Q . Here, u is the coordinate for motion along a straight line parallel to the surface, such that V varies the least.](/figures/fig-3-two-dimensional-cuts-through-the-potential-energy-9zqc8eto.webp)