Chapter 2
The Concept of the Potential Energy Surface
Everything should be made as simple as possible, but not simpler.
Albert Einstein
Abstract The potential energy surface (PES) is a central concept in computational
chemistry. A PES is the relationship – mathematical or graphical – between
the energy of a molecule (or a collection of molecules) and its geometry. The
Born–Oppenheimer approximation says that in a molecule the nuclei are essentially
stationary compared to the electrons. This is one of the cornerstones of computational
chemistry because it makes the concept of molecular shape (geometry)
meaningful, makes possible the concept of a PES, and simplifies the application
of the Schr€odinger equation to molecules by allowing us to focus on the electronic
energy and add in the nuclear repulsion energy later; this third point, very important
in practical molecular computations, is elaborated on in Chapter 5. Geometry
optimization and transition state optimization are explained.
2.1 Perspective
We begin a more detailed look at computational chemistry with the potential energy
surface (PES) because this is central to the subject. Many important concepts that
might appear to be mathematically challenging can be grasped intuitively with the
insight provided by the idea of the PES [1].
Consider a diatomic molecule AB. In some ways a molecule behaves like balls
(atoms) held together by springs (chemical bonds); in fact, this simple picture is the
basis of the important method molecular mechanics, discussed in Chapter 3. If we
take a macroscopic balls-and-spring model of our diatomic molecule in its normal
geometry (the equilibrium geometry), grasp the “atoms” and distort the model by
stretching or compressing the “bonds”, we increase the potential energy of the
molecular model (Fig. 2.1). The stretched or compressed spring possesses energy,
by definition, since we moved a force through a distance to distort it. Since the
E.G. Lewars, Computational Chemistry,
DOI 10.1007/978-90-481-3862-3_2, # Springer ScienceþBusiness Media B.V. 2011
9
model is motionless while we hold it at the new geometry, this energy is not kinetic
and so is by default potential (“depending on position”). The graph of potential
energy against bond length is an example of a potential energy surface. A line
is a one-dimensional “surface”; we will soon see an example of a more familiar
two-dimensional surface rather than the line of Fig. 2.1.
Real molecules behave similarly to, but differ from our macroscopic model in
two relevant ways:
1. They vibrate incessantly (as we would expect from Heisenberg’s uncertainty
principle: a stationary molecule would have an exactly defined momentum and
position) about the equilibrium bond length, so that they always possess kinetic
energy (T) and/or potential energy (V): as the bond length passes through the
equilibrium length, V ¼ 0, while at the limit of the vibrational amplitude, T ¼ 0;
at all other positions both T and V are nonzero. The fact that a molecule is never
actually stationary with zero kinetic energy (it always has zero point energy;
Section 2.5) is usually shown on potential energy/bond length diagrams by drawing
a series of lines above the bottom of the curve (Fig. 2.2) to indicate the
possible amounts of vibrational energy the molecule can have (the vibrational
levels it can occupy). A molecule never sits at the bottom of the curve, but rather
occupies one of the vibrational levels, and in a collection of molecules the levels
are populated according to their spacing and the temperature [2]. We will
usually ignore the vibrational levels and consider molecules to rest on the actual
potential energy curves or (see below) surfaces.
2. Near the equilibrium bond length qe the potential energy/bond length curve
for a macroscopic balls-and-spring model or a real molecule is described
fairly well by a quadratic equation, that of the simple harmonic oscillator
ðE ¼ ð1=2Þ K ðq À qeÞ2
, where k is the force constant of the spring). However,
the potential energy deviates from the quadratic (q2
) curve as we move away
from qe (Fig. 2.2). The deviations from molecular reality represented by this
anharmonicity are not important to our discussion.
energy
0
bond length, q
qe
Fig. 2.1 The potential
energy surface for a diatomic
molecule. The potential
energy increases if the bond
length q is stretched or
compressed away from its
equilibrium value qe. The
potential energy at qe (zero
distortion of the bond length)
has been chosen here as the
zero of energy
10 2 The Concept of the Potential Energy Surface
Figure 2.1 represents a one-dimensional PES in the two-dimensional graph of
E vs. q. A diatomic molecule AB has only one geometric parameter for us to vary,
the bond length qAB. Suppose we have a molecule with more than one geometric
parameter, for example water: the geometry is defined by two bond lengths and a
bond angle. If we reasonably content ourselves with allowing the two bond lengths
to be the same, i.e. if we limit ourselves to C2v symmetry (two planes of symmetry
and a two-fold symmetry axis; see Section 2.6) then the PES for this triatomic
molecule is a graph of E versus two geometric parameters, q1 ¼ the O–H bond
length, and q2 ¼ the H–O–H bond angle (Fig. 2.3). Figure 2.3 represents a twodimensional
PES (a normal surface is a 2-D object) in the three-dimensional graph;
we could make an actual 3-D model of this drawing of a 3-D graph of E versus
q1 and q2.
We can go beyond water and consider a triatomic molecule of lower symmetry,
such as HOF, hypofluorous acid. This has three geometric parameters, the H–O and
O–F lengths and the H–O–F angle. To construct a Cartesian PES graph for HOF
analogous to that for H2O would require us to plot E vs. q1 ¼ H–O, q2 ¼ O–F, and
q3 ¼ angle H–O–F. We would need four mutually perpendicular axes (for E, q1, q2,
q3, Fig. 2.4), and since such a four-dimensional graph cannot be constructed in our
three-dimensional space we cannot accurately draw it. The HOF PES is a 3-D
“surface” of more than two dimensions in 4-D space: it is a hypersurface, and
potential energy surfaces are sometimes called potential energy hypersurfaces.
Despite the problem of drawing a hypersurface, we can define the equation E ¼ f
(q1, q2, q3) as the potential energy surface for HOF, where f is the function that
describes how E varies with the q’s, and treat the hypersurface mathematically. For
example, in the AB diatomic molecule PES (a line) of Fig. 2.1 the minimum
energy
0
quadratic curve
.
.
.
vibrational levels
true molecular
potential energy
curve
bond length, q
qe
Fig. 2.2 Actual molecules do not sit still at the bottom of the potential energy curve, but instead
occupy vibrational levels. Also, only near qe, the equilibrium bond length, does the quadratic curve
approximate the true potential energy curve
2.1 Perspective 11
potential energy geometry is the point at which dE/dq ¼ 0. On the H2O PES
(Fig. 2.3) the minimum energy geometry is defined by the point Pm, corresponding
to the equilibrium values of q1 and q2; at this point dE/dq1 ¼ dE/dq2 ¼ 0. Although
hypersurfaces cannot be faithfully rendered pictorially, it is very useful to a
computational chemist to develop an intuitive understanding of them. This can be
gained with the aid of diagrams like Figs. 2.1 and 2.3, where we content ourselves
with a line or a two-dimensional surface, in effect using a slice of a multidimensional
diagram. This can be understood by analogy: Fig. 2.5 shows how 2-D slices
angle
H H
O
O
H H
energy
.
Pmin
q1 = O H bond length
q1 = 0.958 Å
q2 = 104.5°
q2 =
Fig. 2.3 The H2O potential energy surface. The point Pmin corresponds to the minimum-energy
geometry for the three atoms, i.e. to the equilibrium geometry of the water molecule
energy
q3
q2
q1
Fig. 2.4 To plot energy
against three geometric
parameters in a Cartesian
coordinate system we would
need four mutually
perpendicular axes. Such a
coordinate system cannot be
actually constructed in our
three-dimensional space.
However, we can work with
such coordinate systems, and
the potential energy surfaces
in them, mathematically
12 2 The Concept of the Potential Energy Surface
can be made of the 3-D diagram for water. The slice could be made holding one or
the other of the two geometric parameters constant, or it could involve both of them,
giving a diagram in which the geometry axis is a composite of more than one
geometric parameter. Analogously, we can take a 3-D slice of the hypersurface for
HOF (Fig. 2.6) or even a more complex molecule and use an E versus q1, q2
diagram to represent the PES; we could even use a simple 2D diagram, with q
representing one, two or all of the geometric parameters. We shall see that these 2D
and particularly 3D graphs preserve qualitative and even quantitative features of the
mathematically rigorous but unvisualizable E ¼ f(q1, q2, . . . qn) n-dimensional
hypersurface.
2.2 Stationary Points
Potential energy surfaces are important because they aid us in visualizing and understanding
the relationship between potential energy and molecular geometry, and in
understanding how computational chemistry programs locate and characterize structures
angle
H H
O
O
H Henergy
slice parallel to bond length axis
energy
energy
bond length
2D surface
1D "surface"
bond angle
1D "surface"
slice parallel to
angle axis
q2 =
q1 = O H bond length
Fig. 2.5 Slices through a 2D potential energy surface give 1D surfaces. A slice that is parallel to
neither axis would give a plot of geometry versus a composite of bond angle and bond length, a
kind of average geometry
2.2 Stationary Points 13
of interest. Among the main tasks of computational chemistry are to determine the
structure and energy of molecules and of the transition states involved in chemical
reactions: our “structures of interest” are molecules and the transition states linking
them. Consider the reaction
O
O O
transition state
reaction (2.1)
O
O O
isoozone
+
O
–
O O
ozone
A priori, it seems reasonable that ozone might have an isomer (call it isoozone)
and that the two could interconvert by a transition state as shown in Reaction (2.1).
We can depict this process on a PES. The potential energy E must be plotted against
only two geometric parameters, the bond length (we may reasonably assume that
the two O–O bonds of ozone are equivalent, and that these bond lengths remain
equal throughout the reaction) and the O–O–O bond angle. Figure 2.7 shows the
PES for Reaction (2.1), as calculated by the AM1 semiempirical method (Chapter
6; the AM1 method is unsuitable for quantitative treatment of this problem, but the
potential energy surface shown makes the point), and shows how a 2D slice from
H
O
F
energy
Pmin
q2 = O F bond length
q1 = O H bond length
Fig. 2.6 A potential energy surface (PES) for HOF. Here the HOF angle is not shown. This
picture could represent one of two possibilities: the angle might be the same (some constant,
reasonable value) for every calculated point on the surface; this would be an unrelaxed or rigid
PES. Alternatively, for each calculated point the geometry might be that for the best angle
corresponding to the other two parameters, i.e. the geometry for each calculated point might be
fully optimized (Section 2.4); this would be a relaxed PES
14 2 The Concept of the Potential Energy Surface
this 3D diagram gives the energy/reaction coordinate type of diagram commonly
used by chemists. The slice goes along the lowest-energy path connecting ozone,
isoozone and the transition state, that is, along the reaction coordinate, and the
horizontal axis (the reaction coordinate) of the 2D diagram is a composite of O–O
bond length and O–O–O angle. In most discussions this horizontal axis is left
quantitatively undefined; qualitatively, the reaction coordinate represents the
O
O
OO
O O
O
O O
energy
transition state
2.0
60°
120.9°
global minimum
OZONE
1.060
80
100
120
140
160
KJmol–1
1.3161.160
relative minimum
iSOOZONE
O-O-O angle, degrees
0-0,Å
intrinsic reaction coordinate (IRC)
Fig. 2.7 The ozone/isoozone potential energy surface (calculated by the AM1 method; Chapter
6), a 2D surface in a 3D diagram. The dashed line on the surface is the reaction coordinate
(intrinsic reaction coordinate, IRC). A slice through the reaction coordinate gives a 1D “surface” in
a 2D diagram. The diagram is not meant to be quantitatively accurate
2.2 Stationary Points 15
progress of the reaction. The three species of interest, ozone, isoozone, and the
transition state linking these two, are called stationary points. A stationary point on
a PES is a point at which the surface is flat, i.e. parallel to the horizontal line
corresponding to the one geometric parameter (or to the plane corresponding to two
geometric parameters, or to the hyperplane corresponding to more than two geometric
parameters). A marble placed on a stationary point will remain balanced, i.e.
stationary (in principle; for a transition state the balancing would have to be
exquisite indeed). At any other point on a potential surface the marble will roll
toward a region of lower potential energy.
Mathematically, a stationary point is one at which the first derivative of the
potential energy with respect to each geometric parameter is zero1
:
@ E
@ q1
¼
@ E
@ q2
¼ Á Á Á ¼ 0 (*2.1)
Partial derivatives, ∂E/∂q, are written here rather than dE/dq, to emphasize that
each derivative is with respect to just one of the variables q of which E is a function.
Stationary points that correspond to actual molecules with a finite lifetime (in
contrast to transition states, which exist only for an instant), like ozone or isoozone,
are minima, or energy minima: each occupies the lowest-energy point in its region
of the PES, and any small change in the geometry increases the energy, as indicated
in Fig. 2.7. Ozone is a global minimum, since it is the lowest-energy minimum on
the whole PES, while isoozone is a relative minimum, a minimum compared only to
nearby points on the surface. The lowest-energy pathway linking the two minima,
the reaction coordinate or intrinsic reaction coordinate (IRC; dashed line in
Fig. 2.7) is the path that would be followed by a molecule in going from one
minimum to another should it acquire just enough energy to overcome the activation
barrier, pass through the transition state, and reach the other minimum. Not all
reacting molecules follow the IRC exactly: a molecule with sufficient energy can
stray outside the IRC to some extent [3].
Inspection of Fig. 2.7 shows that the transition state linking the two minima
represents a maximum along the direction of the IRC, but along all other directions
it is a minimum. This is a characteristic of a saddle-shaped surface, and the
transition state is called a saddle point (Fig. 2.8). The saddle point lies at the
“center” of the saddle-shaped region and is, like a minimum, a stationary point,
since the PES at that point is parallel to the plane defined by the geometry parameter
axes: we can see that a marble placed (precisely) there will balance. Mathematically,
minima and saddle points differ in that although both are stationary points
(they have zero first derivatives; Eq. 2.1), a minimum is a minimum in all directions,
but a saddle point is a maximum along the reaction coordinate and a
minimum in all other directions (examine Fig. 2.8). Recalling that minima and
maxima can be distinguished by their second derivatives, we can write:
1
Equations marked with an asterisk are those which should be memorized.
16 2 The Concept of the Potential Energy Surface
For a minimum
@2
E
@q2
> 0 (*2.2)
for all q.
For a transition state
@2
E
@q2
> 0 (*2.3)
for all q, except along the reaction coordinate, and
@2
E
@q2
< 0 (*2.4)
along the reaction coordinate.
The distinction is sometimes made between a transition state and a transition
structure [4]. Strictly speaking, a transition state is a thermodynamic concept, the
species an ensemble of which are in a kind of equilibrium with the reactants in
Eyring’s2
transition-state theory [5]. Since equilibrium constants are determined by
free energy differences, the transition structure, within the strict use of the term, is a
free energy maximum along the reaction coordinate (in so far as a single species can
minimum
transition state
transition state
region
reaction coordinate
energy
Fig. 2.8 A transition state or saddle point and a minimum. At both the transition state and the
minimum ∂E/∂q ¼ 0 for all geometric coordinates q (along all directions). At the transition state
∂E2
/∂q2
< 0 for q ¼ the reaction coordinate and > 0 for all other q (along all other directions). At a
minimum ∂E2
/∂q2
> 0 for all q (along all directions)
2
Henry Eyring, American chemist. Born Colonia Juara´rez, Mexico, 1901. Ph.D. University of
California, Berkeley, 1927. Professor Princeton, University of Utah. Known for his work on the
theory of reaction rates and on potential energy surfaces. Died Salt Lake City, Utah, 1981.
2.2 Stationary Points 17
be considered representative of the ensemble). This species is also often (but not
always [5]) also called an activated complex. A transition structure, in strict usage,
is the saddle point (Fig. 2.8) on a theoretically calculated (e.g. Fig. 2.7) PES.
Normally such a surface is drawn through a set of points each of which represents
the enthalpy of a molecular species at a certain geometry; recall that free energy
differs from enthalpy by temperature times entropy. The transition structure is thus
a saddle point on an enthalpy surface. However, the energy of each of the calculated
points does not normally include the vibrational energy, and even at 0 K a molecule
has such energy (zero point energy: Fig. 2.2, and Section 2.5). The usual calculated
PES is thus a hypothetical, physically unrealistic surface in that it neglects vibrational
energy, but it should qualitatively, and even semiquantitatively, resemble the
vibrationally-corrected one since in considering relative enthalpies ZPEs at least
roughly cancel. In accurate work ZPEs are calculated for stationary points and
added to the “frozen-nuclei” energy of the species at the bottom of the reaction
coordinate curve in an attempt to give improved relative energies which represent
enthalpy differences at 0 K (and thus, at this temperature where entropy is zero, free
energy differences also; Fig. 2.19). It is also possible to calculate enthalpy and
entropy differences, and thus free energy differences, at, say, room temperature
(Section 5.5.2). Many chemists do not routinely distinguish between the two terms,
and in this book the commoner term, transition state, is used. Unless indicated
otherwise, it will mean a calculated geometry, the saddle point on a hypothetical
vibrational-energy-free PES.
The geometric parameter corresponding to the reaction coordinate is usually a
composite of several parameters (bond lengths, angles and dihedrals), although for
some reactions one two may predominate. In Fig. 2.7, the reaction coordinate is a
composite of the O–O bond length and the O–O–O bond angle.
A saddle point, the point on a PES where the second derivative of energy with
respect to one and only geometric coordinate (possibly a composite coordinate) is
negative, corresponds to a transition state. Some PES’s have points where the
second derivative of energy with respect to more than one coordinate is negative;
these are higher-order saddle points or hilltops: for example, a second-order saddle
point is a point on the PES which is a maximum along two paths connecting
stationary points. The propane PES, Fig. 2.9, provides examples of a minimum, a
transition state and a hilltop – a second-order saddle point in this case. Figure 2.10
shows the three stationary points in more detail. The “doubly-eclipsed” conformation
(Fig. 2.10a) in which there is eclipsing as viewed along the C1–C2 and the
C3–C2 bonds (the dihedral angles are 0
viewed along these bonds) is a secondorder
saddle point because single bonds do not like to eclipse single bonds and
rotation about the C1–C2 and the C3–C2 bonds will remove this eclipsing: there are
two possible directions along the PES which lead, without a barrier, to lower-energy
regions, i.e. changing the H–C1/C2–C3 dihedral and changing the H–C3/C2–C1
dihedral. Changing one of these leads to a “singly-eclipsed” conformation
(Fig. 2.10b) with only one offending eclipsing CH3–CH2 arrangement, and this is
a first-order saddle point, since there is now only one direction along the PES which
leads to relief of the eclipsing interactions (rotation around C3–C2). This route
18 2 The Concept of the Potential Energy Surface
gives a conformation C which has no eclipsing interactions and is therefore a
minimum. There are no lower-energy structures on the C3H8 PES and so C is the
global minimum.
The geometry of propane depends on more than just two dihedral angles, of
course; there are several bond lengths and bond angles and the potential energy will
vary with changes in all of them. Figure 2.9 was calculated by varying only the
dihedral angles associated with the C1–C2–C3–C4 bonds, keeping the other
geometrical parameters the same as they are in the all-staggered conformation. If
at every point on the dihedral/dihedral grid all the other parameters (bond lengths
and angles) had been optimized (adjusted to give the lowest possible energy, for
that particular calculational method; Section 2.4), the result would have been a
relaxed PES. In Fig. 2.9 this was not done, but because bond lengths and angles
change only slightly with changes in dihedral angles the PES would not be altered
much, while the time required for the calculation (for the potential energy surface
scan) would have been greater. Figure 2.9 is a nonrelaxed or rigid PES, albeit not
very different, in this case, from a relaxed one.
Chemistry is essentially the study of the stationary points on potential energy
surfaces: in studying more or less stable molecules we focus on minima, and
in investigating chemical reactions we study the passage of a molecule from a
A, hilltop
B, transition state
C, minimum
400
300
200
200
300
400
500
100
100
0
0
–100 –100
H--C
1--C
2--C
3 dihedral
H--C3--C2--C1dihedral
B
A
C
Fig. 2.9 The propane potential energy surface as the two HCCC dihedrals are varied (calculated
by the AM1 method, Chapter 6). Bond lengths and angles were not optimized as the dihedrals
were varied, so this is not a relaxed PES; however, changes in bond lengths and angles from
one propane conformation to another are small, and the relaxed PES should be very similar to
this one
2.2 Stationary Points 19
minimum through a transition state to another minimum. There are four known
forces in nature: the gravitational force, the strong and the weak nuclear forces, and
the electromagnetic force. Celestial mechanics studies the motion of stars and
planets under the influence of the gravitational force and nuclear physics studies
the behaviour of subatomic particles subject to the nuclear forces. Chemistry is
concerned with aggregates of nuclei and electrons (with molecules) held together
by the electromagnetic force, and with the shuffling of nuclei, followed by their
C1
C2
C3
.
CH3
C1
C2
C3
C1
C2
C3
C1
C2
C3
.
C3
C2
C1
.
C1
C2
C3
.
C3
C2
C1
.
CH3
C2
C3
C1
.
C2
C3
C1
total of 6 eclipsing interactions
total of 3 eclipsing interactions no eclipsing interactions 3 eclipsing interactions
(CH/ CC, CH/CH, CH/CH)
no eclipsing interactions no eclipsing interactionsno eclipsing interactions
sawhorse drawings Newman projections
hiltop
transition state
minimum
a
c
b
CH3
3 eclipsing interactions
(CH/CC, CH/CH, CH/CH)
3 eclipsing interactions
(CH/ CC, CH/CH, CH/ CH)
CH3
CH3CH3
Fig. 2.10 The stationary points on the propane potential energy surface. Hydrogens at the end of
CH bonds are omitted for clarity
20 2 The Concept of the Potential Energy Surface
obedient retinue of electrons, around a potential energy surface under the influence
of this force (with chemical reactions).
The concept of the chemical potential energy surface apparently originated with
R. Marcelin [6]: in a dissertation-long paper (111 pages) he laid the groundwork for
transition-state theory 20 years before the much better-known work of Eyring [5,7].
The importance of Marcelin’s work is acknowledged by Rudolph Marcus in his
Nobel Prize (1992) speech, where he refers to “. . .Marcelin’s classic 1915 theory
which came within one small step of the transition state theory of 1935.” The paper
was published the year after the death of the author, who seems to have died in
World War I, as indicated by the footnote “Tue´ a` l’ennemi en sept 1914”. The first
potential energy surface was calculated in 1931 by Eyring and Polanyi,3
using a
mixture of experiment and theory [8].
The potential energy surface for a chemical reaction has just been presented as a
saddle-shaped region holding a transition state which connects wells containing
reactant(s) and products(s) (which species we call the reactant and which the
product is inconsequential here). This picture is immensely useful, and may well
apply to the great majority of reactions. However, for some reactions it is deficient.
Carpenter has shown that in some cases a reactive intermediate does not tarry in a
PES well and then proceed to react. Rather it appears to scoot over a plateau-shaped
region of the PES, retaining a memory (“dynamical information”) of the atomic
motions it acquired when it was formed. When this happens there are two (say)
intermediates with the same crass geometry, but different atomic motions, leading
to different products. The details are subtle, and the interested reader is commended
to the relevant literature [9].
2.3 The Born–Oppenheimer Approximation
A potential energy surface is a plot of the energy of a collection of nuclei and
electrons against the geometric coordinates of the nuclei – essentially a plot of
molecular energy versus molecular geometry (or it may be regarded as the mathematical
equation that gives the energy as a function of the nuclear coordinates). The
nature (minimum, saddle point or neither) of each point was discussed in terms of
the response of the energy (first and second derivatives) to changes in nuclear
coordinates. But if a molecule is a collection of nuclei and electrons why plot
energy versus nuclear coordinates – why not against electron coordinates? In other
words, why are nuclear coordinates the parameters that define molecular geometry?
The answer to this question lies in the Born–Oppenheimer approximation.
3
Michael Polanyi, Hungarian-British chemist, economist, and philosopher. Born Budapest 1891.
Doctor of medicine 1913, Ph.D. University of Budapest, 1917. Researcher Kaiser-Wilhelm
Institute, Berlin, 1920–1933. Professor of chemistry, Manchester, 1933–1948; of social studies,
Manchester, 1948–1958. Professor Oxford, 1958–1976. Best known for book “Personal
Knowledge”, 1958. Died Northampton, England, 1976.
2.3 The Born–Oppenheimer Approximation 21
Born4
and Oppenheimer5
showed in 1927 [10] that to a very good approximation
the nuclei in a molecule are stationary with respect to the electrons. This is a
qualitative expression of the principle; mathematically, the approximation states
that the Schr€odinger equation (Chapter 4) for a molecule may be separated into an
electronic and a nuclear equation. One consequence of this is that all (!) we have to
do to calculate the energy of a molecule is to solve the electronic Schr€odinger
equation and then add the electronic energy to the internuclear repulsion (this latter
quantity is trivial to calculate) to get the total internal energy (see Section 4.4.1). A
deeper consequence of the Born–Oppenheimer approximation is that a molecule
has a shape.
The nuclei see the electrons as a smeared-out cloud of negative charge which
binds them in fixed relative positions (because of the mutual attraction between
electrons and nuclei in the internuclear region) and which defines the (somewhat
fuzzy) surface [11] of the molecule (see Fig. 2.11). Because of the rapid motion of
the electrons compared to the nuclei the “permanent” geometric parameters of the
molecule are the nuclear coordinates. The energy (and the other properties) of a
molecule is a function of the electron coordinates (E ¼ C(x, y, z of each electron);
Section 5.2), but depends only parametrically on the nuclear coordinates, i.e. for
each geometry 1, 2, . . . there is a particular energy: E1 ¼ C1(x, y, z. . .), E2 ¼ C2 (x,
y, z. . .); cf. xn
, which is a function of x but depends only parametrically on the
particular n.
r1
r2
r3
a1
a2
Fig. 2.11 The nuclei in a molecule see a time-averaged electron cloud. The nuclei vibrate about
equilibrium points which define the molecular geometry; this geometry can be expressed simply as
the nuclear Cartesian coordinates, or alternatively as bond lengths and angles r and a here) and
dihedrals, i.e. as internal coordinates. As far as size goes, the experimentally determined van der
Waals surface encloses about 98% of the electron density of a molecule
4
Max Born, German-British physicist. Born in Breslau (now Wroclaw, Poland), 1882, died in
G€ottingen, 1970. Professor Berlin, Cambridge, Edinburgh. Nobel Prize, 1954. One of the founders
of quantum mechanics, originator of the probability interpretation of the (square of the) wavefunction
(Chapter 4).
5
J. Robert Oppenheimer, American physicist. Born in New York, 1904, died in Princeton 1967.
Professor California Institute of Technology. Fermi award for nuclear research, 1963. Important
contributions to nuclear physics. Director of the Manhattan Project 1943–1945. Victimized as a
security risk by senator Joseph McCarthy’s Un-American Activities Committee in 1954. Central
figure of the eponymous PBS TV series (Oppenheimer: Sam Waterston).
22 2 The Concept of the Potential Energy Surface
Actually, the nuclei are not stationary, but execute vibrations of small amplitude
about equilibrium positions; it is these equilibrium positions that we mean by the
“fixed” nuclear positions. It is only because it is meaningful to speak of (almost)
fixed nuclear coordinates that the concepts of molecular geometry or shape and of
the PES are valid [12]. The nuclei are much more sluggish than the electrons
because they are much more massive (a hydrogen nucleus is about 2,000 more
massive than an electron).
Consider the molecule H3
+
, made up of three protons and two electrons. Ab
initio calculations assign it the geometry shown in Fig. 2.12. The equilibrium
positions of the nuclei (the protons) lie at the corners of an equilateral triangle
and H3
þ
has a definite shape. But suppose the protons were replaced by positrons,
which have the same mass as electrons. The distinction between nuclei and electrons,
which in molecules rests on mass and not on some kind of charge chauvinism,
would vanish. We would have a quivering cloud of flitting particles to which a
shape could not be assigned on a macroscopic time scale.
A calculated PES, which we might call a Born–Oppenheimer surface, is normally
the set of points representing the geometries, and the corresponding energies,
of a collection of atomic nuclei; the electrons are taken into account in the calculations
as needed to assign charge and multiplicity (multiplicity is connected with the
number of unpaired electrons). Each point corresponds to a set of stationary nuclei,
and in this sense the surface is somewhat unrealistic (see Section 2.5).
2.4 Geometry Optimization
The characterization (the “location” or “locating”) of a stationary point on a PES,
that is, demonstrating that the point in question exists and calculating its geometry
and energy, is a geometry optimization. The stationary point of interest might be a
minimum, a transition state, or, occasionally, a higher-order saddle point. Locating
a minimum is often called an energy minimization or simply a minimization, and
0.851 Å 0.851 Å
0.851 Å
H
H H
–
–
++
+
The H3
+ cation: 3 protons, 2 electrons
Definite geometry
make the masses of the
nuclei and electrons equal
No definite geometry
Fig. 2.12 A molecule has a definite shape because unlike the electrons, the nuclei are (relatively)
stationary (since they are much more massive). If the masses of the nuclei and the electrons could
be made equal, the distinction in lethargy would be lost, and the molecular geometry would
dissolve
2.4 Geometry Optimization 23
locating a transition state is often referred to specifically as a transition state
optimization. Geometry optimizations are done by starting with an input structure
that is believed to resemble (the closer the better) the desired stationary point and
submitting this plausible structure to a computer algorithm that systematically
changes the geometry until it has found a stationary point. The curvature of the
PES at the stationary point, i.e. the second derivatives of energy with respect to the
geometric parameters (Section 2.2) may then be determined (Section 2.5) to
characterize the structure as a minimum or as some kind of saddle point.
Let us consider a problem that arose in connection with an experimental study.
Propanone (acetone) was subjected to ionization followed by neutralization of the
radical cation, and the products were frozen in an inert matrix and studied by IR
spectroscopy [13]. The spectrum of the mixture suggested the presence of the enol
isomer of propanone, 1-propen-2-ol (Reaction 2.2):
C
Reaction 2
O
C
OH
H3C H2CCH3 CH3
To confirm (or refute) this the IR spectrum of the enol might be calculated (see
Section 2.5 and the discussions of the calculation of IR spectra in subsequent
chapters). But which conformer should one choose for the calculation? Rotation
about the C–O and C–C bonds creates six plausible stationary points (Fig. 2.13),
O O
O O
O
2 3
45
6
H
H
H
H
H
H
H
H
HH
H H
H
H H
O
1 H
H
H
H
H
H
H
H
H
Fig. 2.13 The plausible
stationary points on the
propenol potential energy
surface. A PES scan
(Fig. 2.14) indicated that 1 is
the global minimum and 4 is a
relative minimum, while
2 and 3 are transition states
and 5 and 6 are hilltops. AM1
calculations gave relative
energies for 1, 2, 3 and 4 of 0,
0.6, 14 and 6.5 kJ molÀ1
,
respectively (5 and 6 were not
optimized). The arrows
represent one-step (rotation
about one bond) conversion
of one species into another
24 2 The Concept of the Potential Energy Surface
and a PES scan (Fig. 2.14) indicated that there are indeed six such species.
Examination of this PES shows that the global minimum is structure 1 and that
there is a relative minimum corresponding to structure 4. Geometry optimization
starting from an input structure resembling 1 gave a minimum corresponding to 1,
while optimization starting from a structure resembling 4 gave another, higherenergy
minimum, resembling 4. Transition-state optimizations starting from appropriate
structures yielded the transition states 2 and 3. These stationary points were
all characterized as minima or transition states by second-derivative calculations
(Section 2.5) (the species 5 and 6 were not located). The calculated IR spectrum of 1
(using the ab initio HF/6–31G* method – Chapter 5) was in excellent agreement
with the observed spectrum of the putative propenol.
This illustrates a general principle: the optimized structure one obtains is that
closest in geometry on the PES to the input structure (Fig. 2.15). To be sure we have
found a global minimum we must (except for very simple or very rigid molecules)
search a potential energy surface (there are algorithms that will do this and locate
the various minima). Of course we may not be interested in the global minimum; for
example, if we wish to study the cyclic isomer of ozone (Section 2.2) we will use as
Fig. 2.14 The 1-propen-2-ol potential energy surface (calculated by the AM1 method) (see
Fig. 2.13)
2.4 Geometry Optimization 25
input an equilateral triangle structure, probably with bond lengths about those of an
O–O single bond.
In the propenol example, the PES scan suggested that to obtain the global
minimum we should start with an input structure resembling 1, but the exact values
of the various bond lengths and angles were unknown (the exact values of even the
dihedrals was not known with certainty, although general chemical knowledge
made H–O–C–C ¼ H–C-C¼C ¼ 0
seem plausible). The actual creation of input
structures is usually done nowadays with an interactive mouse-driven program,
in much the same spirit that one constructs plastic models or draws structures
on paper. An older alternative is to specify the geometry by defining the various
bond lengths, angles and dihedrals, i.e. by using a so-called Z-matrix (internal
coordinates).
To move along the PES from the input structure to the nearest minimum is
obviously trivial on the one-dimensional PES of a diatomic molecule: one simply
changes the bond length till that corresponding to the lowest energy is found.
On any other surface, efficient geometry optimization requires a sophisticated
algorithm. One would like to know in which direction to move, and how far in
that direction (Fig. 2.16). It is not possible, in general, to go from the input structure
to the proximate minimum in just one step, but modern geometry optimization
algorithms commonly reach the minimum within about ten steps, given a reasonable
input geometry. The most widely-used algorithms for geometry optimization
[14] use the first and second derivatives of the energy with respect to the geometric
parameters. To get a feel for how this works, consider the simple case of a
one-dimensional PES, as for a diatomic molecule (Fig. 2.17). The input structure
is at the point Pi(Ei, qi) and the proximate minimum, corresponding to the
optimized structure being sought, is at the point Po(Eo, qo). Before the optimization
energy
geometry
TS
B
A
B′
A′
several steps
several steps
Fig. 2.15 Geometry optimization to a minimum gives the minimum closest to the input structure.
The input structure A0
is moved toward the minimum A, and B0
toward B. To locate a transition
state a special algorithm is usually used: this moves the initial structure A0
toward the transition
state TS. Optimization to each of the stationary points would probably actually require several
steps (see Fig. 2.16)
26 2 The Concept of the Potential Energy Surface
energy
geometry
geometry
optimized structure
.
input structure
Fig. 2.16 An efficient optimization algorithm knows approximately in which direction to move
and how far to step, in an attempt to reach the optimized structure in relatively few (commonly
about five to ten) steps
bond length, q
0
qe
E
E–E0 = k(q–q0)2
Input structure
Pi(Ei, qi)
Equilibrium (optimized) structure
Po(E0, q0)
Fig. 2.17 The potential energy of a diatomic molecule near the equilibrium geometry is approximately
a quadratic function of the bond length. Given an input structure (i.e. given the bond length
qi), a simple algorithm would enable the bond length of the optimized structure to be found in one
step, if the function were strictly quadratic
2.4 Geometry Optimization 27
has been carried out the values of Eo and qo are of course unknown. If we assume
that near a minimum the potential energy is a quadratic function of q, which is a
fairly good approximation, then
E À Eo ¼ k ðq À qoÞ2
(2.5)
At the input point ðdE=dqÞi ¼ 2kðqi À qoÞ (2.6)
At all points d2
E=dq2
¼ 2k ð¼ force constantÞ (2.7)
From Eqs: ð2:6Þ and ð2:7Þ; ðdE=dqÞi ¼ ðd2
E=dq2
Þ ðqi À qoÞ (2.8)
and qo ¼ qi À ðdE=dqÞi=ðd2
E=dq2
Þ (2.9)
Equation 2.9 shows that if we know (dE/dq)i, the slope or gradient of the PES
at the point of the initial structure, (d2
E/dq2
), the curvature of the PES (which for
a quadratic curve E(q) is independent of q) and qi, the initial geometry, we
can calculate qo, the optimized geometry. The second derivative of potential energy
with respect to geometric displacement is the force constant for motion along
that geometric coordinate; as we will see later, this is an important concept in
connection with calculating vibrational spectra.
For multidimensional PES’s, i.e. for almost all real cases, far more sophisticated
algorithms are used, and several steps are needed since the curvature is not exactly
quadratic. The first step results in a new point on the PES that is (probably) closer to
the minimum than was the initial structure. This new point then serves as the initial
point for a second step toward the minimum, etc. Nevertheless, most modern
geometry optimization methods do depend on calculating the first and second
derivatives of the energy at the point on the PES corresponding to the input
structure. Since the PES is not strictly quadratic, the second derivatives vary from
point to point and are updated as the optimization proceeds.
In the illustration of an optimization algorithm using a diatomic molecule,
Eq. 2.9 referred to the calculation of first and second derivatives with respect to
bond length, which latter is an internal coordinate (inside the molecule). Optimizations
are actually commonly done using Cartesian coordinates x, y, z. Consider
the optimization of a triatomic molecule like HOF in a Cartesian coordinate
system. Each of the three atoms has an x, y and z coordinate, giving nine geometric
parameters, q1, q2, . . . , q9; the PES would be a nine-dimensional hypersurface on
a 10D graph. We need the first and second derivatives of E with respect to each of
the nine q’s, and these derivatives are manipulated as matrices. Matrices are
discussed in Section 4.3.3; here we need only know that a matrix is a rectangular
array of numbers that can be manipulated mathematically, and that they provide a
convenient way of handling sets of linear equations. The first-derivative matrix,
the gradient matrix, for the input structure can be written as a column matrix
28 2 The Concept of the Potential Energy Surface
gi ¼
@E=@q1ð Þi
@E=@q2ð Þi
..
.
@E=@q9ð Þi
0
B
B
B
@
1
C
C
C
A
(2.10)
and the second-derivative matrix, the force constant matrix, is
H ¼
@2
E=@q1q1 @2
E=@q1q1 Á Á Á @2
E=@q1q9
@2
E=@q2q1
..
.
@2
E=@q2q2 Á Á Á
..
.
Á Á Á
@2
E=@q2q9
..
.
@2
E=@q9q1 @2
E=@q9q2 Á Á Á @2
E=@q9q9
0
B
B
B
B
@
1
C
C
C
C
A
(2.11)
The force constant matrix is called the Hessian.6
The Hessian is particularly
important, not only for geometry optimization, but also for the characterization of
stationary points as minima, transition states or hilltops, and for the calculation of
IR spectra (Section 2.5). In the Hessian ∂2
E/∂q1q2 ¼ ∂2
E/∂q2q1, as is true for all
well-behaved functions, but this systematic notation is preferable: the first subscript
refers to the row and the second to the column. The geometry coordinate matrices
for the initial and optimized structures are
qi ¼
qi1
qi2
..
.
qi9
0
B
B
B
@
1
C
C
C
A
(2.12)
and
qo ¼
qo1
qo2
..
.
qo9
0
B
B
B
@
1
C
C
C
A
(2.13)
The matrix equation for the general case can be shown to be:
qo ¼ qi À HÀ1
gi (2.14)
which is analogous to Eq. 2.9 for the optimization of a diatomic molecule, which
could be written
qo ¼ qi À ðd2
E=dq2
ÞÀ1
ðdE=dqÞi
6
Ludwig Otto Hesse, 1811–1874, German mathematician.
2.4 Geometry Optimization 29
For n atoms we have 3n Cartesians; qo, qi and gi are 3n  1 column matrices and
H is a 3n  3n square matrix; multiplication by the inverse of H rather than division
by H is used because matrix division is not defined. Equation 2.14 shows that for an
efficient geometry optimization we need an initial structure (for qi), initial gradients
(for gi) and second derivatives (for H). With an initial “guess” for the geometry (for
example from a model-building program followed by molecular mechanics) as
input, gradients can be readily calculated analytically (from the derivatives of the
molecular orbitals and the derivatives of certain integrals). An approximate initial
Hessian is often calculated from molecular mechanics (Chapter 3). Since the PES is
not really exactly quadratic, the first step does not take us all the way to the
optimized geometry, corresponding to the matrix qo. Rather, we arrive at q1, the
first calculated geometry; using this geometry a new gradient matrix and a new
Hessian are calculated (the gradients are calculated analytically and the second
derivatives are updated using the changes in the gradients – see below). Using q1
and the new gradient and Hessian matrices a new approximate geometry matrix q2
is calculated. The process is continued until the geometry and/or the gradients (or
with some programs possibly the energy) have ceased to change appreciably.
As the optimization proceeds the Hessian is updated by approximating each
second derivative as a ratio of finite increments:
@2
E
@qi@qj
%
Dð@E=@qjÞ
Dqi
(2.15)
i.e. as the change in the gradient divided by the change in geometry, on going from the
previous structure to the latest one. Analytic calculation of second derivatives is
relatively time-consuming and is not routinely done for each point along the optimization
sequence, in contrast to analytic calculation of gradients. A fast lower-level
optimization, for a minimum or a transition state, usually provides a good Hessian and
geometry for input to a higher-level optimization [15]. Finding a transition state (i.e.
optimizing an input structure to a transition state structure) is a more challenging
computational problem than finding a minimum, as the characteristics of the PES at
the former are more complicated than at a minimum: at the transition state the surface
is a maximum in one direction and a minimum in all others, rather than simply a
minimum in all directions. Nevertheless, modifications of the minimum-search algorithm
enable transitions states to be located, albeit often with less ease than minima.
2.5 Stationary Points and Normal-Mode Vibrations – Zero
Point Energy
Once a stationary point has been found by geometry optimization, it is usually
desirable to check whether it is a minimum, a transition state, or a hilltop. This is
done by calculating the vibrational frequencies. Such a calculation involves finding
the normal-mode frequencies; these are the simplest vibrations of the molecule,
30 2 The Concept of the Potential Energy Surface
which, in combination, can be considered to result in the actual, complex vibrations
that a real molecule undergoes. In a normal-mode vibration all the atoms move in
phase with the same frequency: they all reach their maximum and minimum
displacements and their equilibrium positions at the same moment. The other
vibrations of the molecule are combinations of these simple vibrations. Essentially,
a normal-modes calculation is a calculation of the infrared spectrum, although the
experimental spectrum is likely to contain extra bands resulting from interactions
among normal-mode vibrations.
A nonlinear molecule with n atoms has 3n À 6 normal modes: the motion of each
atom can be described by three vectors, along the x, y, and z axes of a Cartesian
coordinate system; after removing the three vectors describing the translational
motion of the molecule as a whole (the translation of its center of mass) and the
three vectors describing the rotation of the molecule (around the three principal
axes needed to describe rotation for a three-dimensional object of general geometry),
we are left with 3n À 6 independent vibrational motions. Arranging these in
appropriate combinations gives 3n À 6 normal modes. A linear molecule has 3n À 5
normal modes, since we need subtract only three translational and two rotational
vectors, as rotation about the molecular axis does not produce a recognizable
change in the nuclear array. So water has 3n À 6 ¼ 3(3) À 6 ¼ 3 normal modes,
and HCN has 3n À 5 ¼ 3(3) À 5 ¼ 4 normal modes. For water (Fig. 2.18) mode 1 is
a bending mode (the H–O–H angle decreases and increases), mode 2 is a symmetric
stretching mode (both O–H bonds stretch and contract simultaneously) and mode 3
is an asymmetric stretching mode (as the O–H1 bond stretches the O–H2 bond
contracts, and vice versa). At any moment an actual molecule of water will be
undergoing a complicated stretching/bending motion, but this motion can be considered
to be a combination of the three simple normal-mode motions.
Consider a diatomic molecule A–B; the normal-mode frequency (there is only
one for a diatomic, of course) is given by [16]:
en ¼
1
2pc
k
m
 1=2
(*2.16)
where ~n ¼ vibrational “frequency”, actually wavenumber, in cmÀ1
; from deference
to convention we use cmÀ1
although the cm is not an SI unit, and so the other units
will also be non-SI; ~n signifies the number of wavelengths that will fit into one cm.
The symbol n is the Greek letter nu, which resembles an angular vee; en could be
O
H H
O
H H
O
H H
1595cm–1
bend
3652cm–1
symmetric stretch
3756cm–1
asymmetric stretch
Fig. 2.18 The normal-mode vibrations of water. The arrows indicate the directions in which the
atoms move; on reaching the maximum amplitude these directions are reversed
2.5 Stationary Points and Normal-Mode Vibrations – Zero Point Energy 31
read “nu tilde”; n, “nu bar”, has been used less frequently. c ¼ velocity of light, k ¼
force constant for the vibration, m ¼ reduced mass of the molecule ¼ (mAmB)/(mA +
mB); mA and mB are the masses of A and B.
The force constant k of a vibrational mode is a measure of the “stiffness” of the
molecule toward that vibrational mode – the harder it is to stretch or bend the
molecule in the manner of that mode, the bigger is that force constant (for a
diatomic molecule k simply corresponds to the stiffness of the one bond). The
fact that the frequency of a vibrational mode is related to the force constant for the
mode suggests that it might be possible to calculate the normal-mode frequencies of
a molecule, that is, the directions and frequencies of the atomic motions, from its
force constant matrix (its Hessian). This is indeed possible: matrix diagonalization
of the Hessian gives the directional characteristics (which way the atoms are
moving), and the force constants themselves, for the vibrations. Matrix diagonalization
(Section 4.3.3) is a process in which a square matrix A is decomposed into
three square matrices, P, D, and PÀ1
: A ¼ PDPÀ1
. D is a diagonal matrix: as with k
in Eq. 2.17 all its off-diagonal elements are zero. P is a premultiplying matrix and
PÀ1
is the inverse of P. When matrix algebra is applied to physical problems, the
diagonal row elements of D are the magnitudes of some physical quantity, and each
column of P is a set of coordinates which give a direction associated with that
physical quantity. These ideas are made more concrete in the discussion accompanying
Eq. 2.17, which shows the diagonalization of the Hessian matrix for a
triatomic molecule, e.g. H2O.
H ¼
@2
E=@q1q1 @2
E=@q1q2 Á Á Á @2
E=@q1q9
@2
E=@q2q1 @2
E=@q2q2 Á Á Á @2
E=@q2q9
..
. ..
.
Á Á Á ..
.
@2
E=@q9q1 @2
E=@q9q2 Á Á Á @2
E=@q9q9
0
B
B
B
@
1
C
C
C
A
¼
q11 q12 Á Á Á q19
q21 q22 Á Á Á q29
..
.
q91 q92 Á Á Á q99
0
B
B
B
B
@
1
C
C
C
C
A
k1 0 Á Á Á 0
0 k2 Á Á Á 0
..
.
0 0 Á Á Á k9
0
B
B
B
B
@
1
C
C
C
C
A
PÀ1
P k
(2.17)
Equation 2.17 is of the form A ¼ PDPÀ1
. The 9 Â 9 Hessian for a triatomic
molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization
into a P matrix whose columns are “direction vectors” for the vibrations
whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P
and the corresponding k1, k2 and k3 of k refer to translational motion of the
molecule (motion of the whole molecule from one place to another in space);
these three “force constants” are nearly zero. Columns 4, 5 and 6 of P and the
corresponding k4, k5 and k6 of k refer to rotational motion about the three principal
32 2 The Concept of the Potential Energy Surface
axes of rotation, and are also nearly zero. Columns 7, 8 and 9 of P and the
corresponding k7, k8 and k9 of k are the direction vectors and force constants,
respectively, for the normal-mode vibrations: k7, k8 and k9 refer to vibrational
modes 1, 2 and 3, while the seventh, eighth, and nineth columns of P are composed
of the x, y and z components of vectors for motion of the three atoms in mode 1
(column 7), mode 2 (column 8), and mode 3 (column 9). “Mass-weighting” the
force constants, i.e. taking into account the effect of the masses of the atoms (cf.
Eq. 2.16 for the simple case of a diatomic molecule), gives the vibrational frequencies.
The P matrix is the eigenvector matrix and the k matrix is the eigenvalue
matrix from diagonalization of the Hessian H. “Eigen” is a German prefix meaning
“appropriate, suitable, actual” and is used in this context to denote mathematically
appropriate entities for the solution of a matrix equation. Thus the directions of the
normal-mode frequencies are the eigenvectors, and their magnitudes are the massweighted
eigenvalues, of the Hessian.
Vibrational frequencies are calculated to obtain IR spectra, to characterize
stationary points, and to obtain zero point energies (below). The calculation of
meaningful frequencies is valid only at a stationary point and only using the same
method that was used to optimize to that stationary point (for example an ab initio
method with a particular correlation level and basis set – see Chapter 5). This is
because (1) the use of second derivatives as force constants presupposes that the
PES is quadratically curved along each geometric coordinate q (Fig. 2.2) but it is
only near a stationary point that this is true, and (2) use of a method other than that
used to obtain the stationary point presupposes that the PES’s of the two methods
are parallel (that they have the same curvature) at the stationary point. Of course,
“provisional” force constants at nonstationary points are used in the optimization
process, as the Hessian is updated from step to step. Calculated IR frequencies are
usually somewhat too high, but (at least for ab initio and density functional
calculations) can be brought into reasonable agreement with experiment by multiplying
them by an empirically determined factor, commonly about 0.9 [17] (see the
discussion of frequencies in Chapters 5–7).
A minimum on the PES has all the normal-mode force constants (all the
eigenvalues of the Hessian) positive: for each vibrational mode there is a restoring
force, like that of a spring. As the atoms execute the motion, the force pulls and
slows them till they move in the opposite direction; each vibration is periodic, over
and over. The species corresponding to the minimum sits in a well and vibrates
forever (or until it acquires enough energy to react). For a transition state, however,
one of the vibrations, that along the reaction coordinate, is different: motion of the
atoms corresponding to this mode takes the transition state toward the product or
toward the reactant, without a restoring force. This one “vibration” is not a periodic
motion but rather takes the species through the transition state geometry on a oneway
journey. Now, the force constant is the first derivative of the gradient or slope
(the derivative of the first derivative); examination of Fig. 2.8 shows that along the
reaction coordinate the surface slopes downward, so the force constant for this
mode is negative. A transition state (a first-order saddle point) has one and only one
negative normal-mode force constant (one negative eigenvalue of the Hessian).
2.5 Stationary Points and Normal-Mode Vibrations – Zero Point Energy 33
Since a frequency calculation involves taking the square root of a force constant
(Eq. 2.16), and the square root of a negative number is an imaginary number, a
transition state has one imaginary frequency, corresponding to the reaction coordinate.
In general an nth-order saddle point (an nth-order hilltop) has n negative
normal-mode force constants and so n imaginary frequencies, corresponding to
motion from one stationary point of some kind to another.
A stationary point could of course be characterized just from the number of
negative force constants, but the mass-weighting requires much less time than
calculating the force constants, and the frequencies themselves are often wanted
anyway, for example for comparison with experiment. In practice one usually
checks the nature of a stationary point by calculating the frequencies and seeing
how many imaginary frequencies are present; a minimum has none, a transition
state one, and a hilltop more than one. If one is seeking a particular transition state
the criteria to be satisfied are:
1. It should look right. The structure of a transition state should lie somewhere
between that of the reactants and the products; for example, the transition state
for the unimolecular isomerization of HCN to HNC shows an H bonded to both
C and N by an unusually long bond, and the CN bond length is in-between that of
HCN and HNC.
2. It must have one and only one imaginary frequency (some programs indicate
this as a negative frequency, e.g. À1,900 cmÀ1
instead of the correct 1,900i
(i ¼
ffip
(À1)).
3. The imaginary frequency must correspond to the reaction coordinate. This is
usually clear from animation of the frequency (the motion, stretching, bending,
twisting, corresponding to a frequency may be visualized with a variety of
programs). For example, the transition state for the unimolecular isomerization
of HCN to HNC shows an imaginary frequency which when animated clearly
shows the H migrating between the C and the N. Should it not be clear from
animation which two species the transition state connects, one may resort to an
intrinsic reaction coordinate (IRC) calculation [18]. This procedure follows the
transition state downhill along the IRC (Section 2.2), generating a series of
structures along the path to the reactant or product. Usually it is clear where the
transition state is going without following it all the way to a stationary point.
4. The energy of the transition state must be higher than that of the two species it
connects.
Besides indicating the IR spectrum and providing a check on the nature of
stationary points, the calculation of vibrational frequencies also provides the
zero-point energy (ZPE; most programs will calculate this automatically as part
of a frequency job). The ZPE is the energy a molecule has even at absolute zero
(Fig. 2.2), as a consequence of the fact that even at this temperature it still vibrates
[2]. The ZPE of a species is usually not small compared to activation energies or
reaction energies, but ZPEs tend to cancel out when these energies are calculated
(by subtraction), since for a given reaction the ZPE of the reactant, transition state
and product tend to be roughly the same. However, for accurate work the ZPE
34 2 The Concept of the Potential Energy Surface
should be added to the “total” (electronic þ nuclear repulsion) energies of species
and the ZPE-corrected energies should then be compared (Fig. 2.19). Like the
frequencies, the ZPE is usually corrected by multiplying it by an empirical factor;
this is sometimes the same as the frequency correction factor, but slightly different
factors have been recommended [17].
The Hessian that results from a geometry optimization was built up in steps from
one geometry to the next, approximating second derivatives from the changes in
gradients (Eq. 2.15). This Hessian is not accurate enough for the calculation of
H C N
HC N
H
C N
47.22
0
219
52.2
0
202
49.7
raw ab initio energy
ZPE
corrected ab initio energy
–92.87520
0.01798
–92.85722
–92.79195
0.01161
–92.78034
–92.85533
0.01705
–92.83828
ZPE
0.01798 × 2626
using the raw energies
Reaction profile
using the ZPE-corrected energies
energy
reaction coordinate
44.77
30.49
Fig. 2.19 Correcting relative energies for zero-point energy (ZPE). These are ab initio HF/6-
31G* (Chapter 5) results for the HCN ! HNC reaction. The corrections are most simply made by
adding the ZPE to the raw energy (in energy units called Hartrees or atomic units), to get the
corrected energies. Using corrected or uncorrected energies, relative energies are obtained by
setting the energy of one species (usually that of lowest energy) equal to zero. Finally, energy
differences in Hartrees were multiplied by 2,626 to get kJ molÀ1
. The ZPEs are also shown here in
kJ molÀ1
, just to emphasize that they are not small compared to reaction energies or activation
energies, but tend to cancel; for accurate work ZPE-corrected energies should be used
2.5 Stationary Points and Normal-Mode Vibrations – Zero Point Energy 35
frequencies and ZPE’s. The calculation of an accurate Hessian for a stationary point
can be done analytically or numerically. Accurate numerical evaluation approximates
the second derivative as in Eq. 2.15, but instead of D(∂V/∂q) and Dq being
taken from optimization iteration steps, they are obtained by changing the position
of each atom of the optimized structure slightly (Dq ¼ about 0.01 A˚ ) and calculating
analytically the change in the gradient at each geometry; subtraction gives
D(∂V/∂q). This can be done for a change in one direction only for each atom
(method of forward differences) or more accurately by going in two directions
around the equilibrium position and averaging the gradient change (method of
central differences). Analytical calculation of ab initio frequencies is much faster
than numerical evaluation, but demands on computer hard drive space may make
numerical calculation the only recourse at high ab initio levels (Chapter 5).
2.6 Symmetry
Symmetry is important in theoretical chemistry (and even more so in theoretical
physics), but our interest in it here is bounded by modest considerations: we want to
see why symmetry is relevant to setting up a calculation and interpreting the results,
and to make sense of terms like C2v, Cs, etc., which are used in various places in this
book. Excellent expositions of symmetry are given by, for example, Atkins [19] and
Levine [20].
The symmetry of a molecule is most easily described by using one of the
standard designations like C2v, Cs. These are called point groups (Schoenflies
point groups) because when symmetry operations (below) are carried out on a
molecule (on any object) with symmetry, at least one point is left unchanged. The
classification is according to the presence of symmetry elements and corresponding
symmetry operations. The main symmetry elements are mirror planes (symmetry
planes), symmetry axes, and an inversion center; other symmetry elements are the
entire object, and an improper rotation axis. The operation corresponding to a
mirror plane is reflection in that plane, the operation corresponding to a symmetry
axis is rotation about that axis, and the operation corresponding to an inversion
center is moving each point in the molecule along a straight line to that center then
moving it further, along the line, an equal distance beyond the center. The “entire
object” element corresponds to doing nothing (a null operation); in common
parlance an object with only this symmetry element would be said to have no
symmetry. The improper rotation axis corresponds to rotation followed by a
reflection through a plane perpendicular to that rotation axis. We are concerned
mainly with the first three symmetry elements. The examples below are shown in
Fig. 2.20.
C1 A molecule with no symmetry elements at all is said to belong to the group C1
(to have “C1 symmetry”). The only symmetry operation such a molecule permits is
the null operation – this is the only operation that leaves it unmoved. An example is
CHBrClF, with a so-called asymmetric atom; in fact, most molecules have no
36 2 The Concept of the Potential Energy Surface
H
H
H
H
O
O
O
O
HH
H
H
H H
HH
H2
C C CH2H H
H
H
H
CH
H
H
buckminsterfullerene, C60
The bonds are of two kinds,
with lengths ca. 1.39 and 1.46Å
F
F
Cooh
D2d
As for D2
(above), plus two dihedral mirror
planes: the two planes that contain the HCH groups
bisect the two axes that are shown as dashed lines
(the third axis passes through CCC).
Td
S
F
F
F
F
F
F
Oh
I
O O
H
F
O
H F
O
H
CO2
H
HO2
C
HO
OH
H
H
Ci
Inversion center ( )
F
F
H
H
H
B
O
OO
H
H
CooV
H
C
N
N
H
H
H
C3
axis
C2
axis
C2 axis
H
F
Cl
Br
C
C1
No symmetry elements
Cs
The three atoms lie in a plane
O O
H
H
C2
Twofold rotation axis
C
H
HH
H
pyramidane
C4v
D2h
As for D2
(above), plus a mirror plane
D2
A C2
axis perpendicular to the ring plane,
and two C2 axes perpendicular to that axis
C2h
C2
axis (perpendicular to the
molecular plane) and a horizontal
mirror plane (the molecular plane)
C3h
C3
axis and horizontal
mirror plane
C3V C3 axis and three
mirror planes
C2v C2 axis and two mirror planes passing
through this axis, one the molecular plane, the
other at right angles to the molecular plane
H
Fig. 2.20 Examples of molecules with various symmetry elements (belonging to various point
groups)
2.6 Symmetry 37
symmetry – just think of steroids, alkaloids, proteins, most drugs. Note that a
molecule does not need an “asymmetric atom” to have C1 symmetry: HOOF in
the conformation shown is C1 (has no symmetry).
Cs A molecule with only a mirror plane belongs to the group Cs. Example: HOF.
Reflection in this plane leaves the molecule apparently unmoved.
C2 A molecule with only a C2 axis belongs to the group C2. Example: H2O2 in
the conformation shown. Rotation about this axis through 360
gives the same
orientation twice. Similarly C3, C4, etc. are possible.
C2v A molecule with two mirror planes whose intersection forms a C2 axis
belongs to the C2v group. Example: H2O. Similarly NH3 is C3v, pyramidane is C4v,
and HCN is C1v.
Ci A molecule with only an inversion center (center of symmetry) belongs to the
group Ci. Example: meso-tartaric acid in the conformation shown. Moving any
point in the molecule along a straight line to this center, then continuing on an equal
distance leaves the molecule apparently unchanged.
C2h A molecule with a C2 axis and a mirror plane horizontal to this axis is C2h (a
C2h object will also perforce have an inversion center). Example: (E)-1,2-difluoroethene.
Similarly B(OH)3 is C3h.
D2 A molecule with a C2 axis and two more C2 axes, perpendicular to that axis,
has D2 symmetry. Example: the tetrahydroxycyclobutadiene shown. Similarly, a
molecule with a C3 axis (the principal axis) and three other perpendicular C2 axes is
D3.
D2h A molecule with a C2 axis and two perpendicular C2 axes (as for D2 above),
plus a mirror plane is D2h. Examples: ethene, cyclobutadiene. Similarly, a C3 axis
(the principal axis), three perpendicular C2 axes and a mirror plane horizontal to the
principal axis confer D3h symmetry, as in the cyclopropenyl cation. Similarly,
benzene is D6h, and F2 is D1h.
D2d A molecule is D2d if it has a C2 axis and two perpendicular C2 axes (as for D2
above), plus two “dihedral” mirror planes; these are mirror planes that bisect two C2
axes (in general, that bisect the C2 axes perpendicular to the principal axis).
Example: allene (propadiene). Staggered ethane is D3d (it has D3 symmetry elements
plus three dihedral mirror planes. Dnd symmetry can be hard to spot.
Molecules belonging to the cubic point groups can, in some sense, be fitted
symmetrically inside a cube. The commonest of these are Td, Oh and I; they will be
simply exemplified:
Td This is tetrahedral symmetry. Example: CH4.
Oh This might be considered “cubic symmetry”. Example: cubane, SF6.
I Also called icosahedral symmetry. Example: buckminsterfullerene.
Less-common groups are S4, and the cubic groups T, Th (dodecahedrane is Th)
and O (see [19,20]). Atkins [19] and Levine [20] give flow charts which
make it relatively simple to assign a molecule to its point group, and Atkins
provides pictures of objects of various symmetries which often make it possible
to assign a point group without having to examine the molecule for its symmetry
elements.
38 2 The Concept of the Potential Energy Surface
We saw above that most molecules have no symmetry. So why is a knowledge of
symmetry important in chemistry? Symmetry considerations are essential in the
theory of molecular electronic (UV) spectroscopy and sometimes in analyzing in
detail molecular wavefunctions (Chapter 4), but for us the reasons are more
pragmatic. A calculation run on a molecule whose input structure has the exact
symmetry that the molecule should have will tend to be faster and will yield a
“better” (see below) geometry than one run on an approximate structure, however
close this may be to the exact one. Input molecular structures for a calculation are
usually created with an interactive graphical program and a computer mouse: atoms
are assembled into molecules much as with a model kit, or the molecule might be
drawn on the computer screen. If the molecule has symmetry (if it is not is not C1)
this can be imposed by optimizing the geometry with molecular mechanics
(Chapter 3). Now consider water: we would of course normally input the H2O
molecule with its exact equilibrium C2v symmetry, but we could also alter the input
structure slightly making the symmetry Cs (three atoms must lie in a plane). The C2v
structure has two degrees of freedom: a bond length (the two bonds are the same
length) and a bond angle. The Cs structure has three degrees of freedom: two bond
lengths and a bond angle. The optimization algorithm has more variables to cope
with in the case of the lower-symmetry structure.
What do we mean by a better geometry? Although a successful geometry
optimization will give essentially the same geometry from a slightly distorted
input structure as from one with the perfect symmetry of the molecule in question,
corresponding bond lengths and angles (e.g. the four C–H bonds and the two HCH
angles of ethene) will not be exactly the same. This can confuse an analysis of the
geometry, and carries over into the calculation of other properties like, say, charges
on atoms – corresponding atoms should have exactly the same charges. Thus both
esthetic and practical considerations encourage us to aim for the exact symmetry
that the molecule should possess.
2.7 Summary
The potential energy surface (PES) is a central concept in computational chemistry.
A PES is the relationship – mathematical or graphical – between the energy of a
molecule (or a collection of molecules) and its geometry.
Stationary points on a PES are points where ∂E/∂q ¼ 0 for all q, where
q is a geometric parameter. The stationary points of chemical interest are
minima (∂2
E/∂qiqj > 0 for all q) and transition states or first-order saddle points;
∂2
E/∂qiqj < 0 for one q, along the reaction coordinate (intrinsic reaction coordinate,
IRC), and > 0 for all other q. Chemistry is the study of PES stationary points
and the pathways connecting them.
The Born–Oppenheimer approximation says that in a molecule the nuclei are
essentially stationary compared to the electrons. This is one of the cornerstones
of computational chemistry because it makes the concept of molecular shape
2.7 Summary 39
(geometry) meaningful, makes possible the concept of a PES, and simplifies the
application of the Schr€odinger equation to molecules by allowing us to focus on the
electronic energy and add in the nuclear repulsion energy later; this third point, very
important in practical molecular computations, is elaborated on in Chapter 5.
Geometry optimization is the process of starting with an input structure “guess”
and finding a stationary point on the PES. The stationary point found will normally
be the one closest to the input structure, not necessarily the global minimum. A
transition state optimization usually requires a special algorithm, since it is more
demanding than that required to find a minimum. Modern optimization algorithms
use analytic first derivatives and (usually numerical) second derivatives.
It is usually wise to check that a stationary point is the desired species
(a minimum or a transition state) by calculating its vibrational spectrum (its
normal-mode vibrations). The algorithm for this works by calculating an accurate
Hessian (force constant matrix) and diagonalizing it to give a matrix with the
“direction vectors” of the normal modes, and a diagonal matrix with the force
constants of these modes. A procedure of “mass-weighting” the force constants
gives the normal-mode vibrational frequencies. For a minimum all the vibrations
are real, while a transition state has one imaginary vibration, corresponding to
motion along the reaction coordinate. The criteria for a transition state are appearance,
the presence of one imaginary frequency corresponding to the reaction
coordinate, and an energy above that of the reactant and the product. Besides
serving to characterize the stationary point, calculation of the vibrational frequencies
enables one to predict an IR spectrum and provides the zero-point energy
(ZPE). The ZPE is needed for accurate comparisons of the energies of isomeric
species. The accurate Hessian required for calculation of frequencies and ZPE’s can
be obtained either numerically or analytically (faster, but much more demanding of
hard drive space).
References
1. (a) Shaik SS, Schlegel HB, Wolfe S (1992) Theoretical aspects of physical organic chemistry:
the SN2 mechanism. Wiley, New York. See particularly Introduction and chapters 1 and 2. (b)
Marcus RA (1992) Science 256:1523. (c) For a very abstract and mathematical but interesting
treatment, see Mezey PG (1987) Potential energy hypersurfaces. Elsevier, New York. (d)
Steinfeld JI, Francisco JS, Hase WL (1999) Chemical kinetics and dynamics, 2nd edn.
Prentice Hall, Upper Saddle River, NJ
2. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ,
section 4.3
3. Shaik SS, Schlegel HB, Wolfe S (1992) Theoretical aspects of physical organic chemistry: the
SN2 mechanism. Wiley, New York, pp 50–51
4. Houk KN, Li Y, Evanseck JD (1992) Angew Chem Int Ed Engl 31:682
5. Atkins P (1998) Physical chemistry, 6th edn. Freeman, New York, pp 830–844
6. Marcelin R (1915) Ann Phys 3:152. Potential energy surface, p 158
7. Eyring H (1935) J Chem Phys 3:107
8. Eyring H, Polanyi M (1931) Z Phys Chem B12:279
40 2 The Concept of the Potential Energy Surface
9. (a) Carpenter BK (1992) Acc Chem Res 25:520. (b) Carpenter BK (1997) Am Sci March–
April:138. (c) Carpenter BK (1998) Angew Chem Int Ed 37:3341. (d) Reyes MB,
Carpenter BK (2000) J Am Chem Soc 122:10163. (e) Reyes MB, Lobkovsky EB, Carpenter
BK (2002) J Am Chem Soc 124:641. (f) Nummela J, Carpenter BK (2002) J Am Chem Soc
124:8512. (g) Carpenter BK (2003) J Phys Org Chem 16:858. (h) Litovitz AE, Keresztes I,
Carpenter BK (2008) J Am Chem Soc 130:12085
10. Born M, Oppenheimer JR (1927) Ann Phys 84:457
11. A standard molecular surface, corresponding to the size as determined experimentally (e.g. by
X-ray diffraction) encloses about 98 per cent of the electron density. See e.g. Bader RFW,
Carroll MT, Cheeseman MT, Chang, C (1987) J Am Chem Soc 109:7968
12. (a) For some rarefied but interesting ideas about molecular shape see Mezey PG (1993) Shape
in chemistry. VCH, New York. (b) An antimatter molecule lacking definite shape: Surko CM
(2007) Nature 449:153. (c) The Cl + H2 reaction: Foreword: Bowman JL (2008) Science
319:40; Garand E, Zhou J, Manolopoulos DE, Alexander MH, Neumark DM (2008) Science
319:72; Erratum 320:612. (d) Baer M (2006) Beyond Born–Haber. Wiley, Hoboken, NJ
13. Zhang XK, Parnis JM, Lewars EG, March RE (1997) Can J Chem 75:276
14. See e.g. Cramer C (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester,
UK, Section 2.4.1
15. Hehre WJ (1995) Practical strategies for electronic structure calculations. Wavefunction Inc.,
Irvine, CA, p 9
16. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, p 65
17. Scott AP, Radom L (1996) J Phys Chem 100:16502
18. Foresman JB, Frisch Æ (1996) Exploring chemistry with electronic structure methods, 2nd
edn. Gaussian Inc., Pittsburgh, PA, pp 173–211
19. Atkins P (1998) Physical chemistry, 6th edn. Freeman, New York, chapter 15
20. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ,
chapter 12
Added in press:
21. Kraka E, Cremer D (2010) Review of computational approaches to the potential energy
surface and some new twists, the unified reaction valley approach URVA. Acc Chem Res
43:591–601
References 41
Easier Questions
1. What is a potential energy surface (give the two viewpoints)?
2. Explain the difference between a relaxed PES and a rigid PES.
3. What is a stationary point? What kinds of stationary points are of interest to
chemists, and how do they differ?
4. What is a reaction coordinate?
5. Show with a sketch why it is not correct to say that a transition state is a
maximum on a PES.
6. What is the Born–Oppenheimer approximation, and why is it important?
7. Explain, for a reaction A ! B, how the potential energy change on a PES is
related to the enthalpy change of the reaction. What would be the problem with
calculating a free energy/geometry surface?
Hint: Vibrational frequencies are normally calculated only for stationary
points.
8. What is geometry optimization? Why is this process for transition states (often
called transition state optimization) more challenging than for minima?
9. What is a Hessian? What uses does it have in computational chemistry?
10. Why is it usually good practice to calculate vibrational frequencies where practical,
although this often takes considerably longer than geometry optimization?
Harder Questions
1. The Born–Oppenheimer principle is often said to be a prerequisite for the
concept of a potential energy surface. Yet the idea of a potential energy surface
(Marcelin 1915) predates the Born–Oppenheimer principle (1927). Discuss.
2. How high would you have to lift a mole of water for its gravitational potential
energy to be equivalent to the energy needed to dissociate it completely into
hydroxyl radicals and hydrogen atoms? The strength of the O–H bond is about
400 kJ molÀ1
; the gravitational acceleration g at the Earth’s surface (and out to
hundreds of kilometres) is about 10 m sÀ2
. What does this indicate about the
role of gravity in chemistry?
3. If gravity plays no role in chemistry, why are vibrational frequencies different
for, say, C–H and C–D bonds?
4. We assumed that the two bond lengths of water are equal. Must an acyclic
molecule AB2 have equal A–B bond lengths? What about a cyclic molecule
AB2?
5. Why are chemists but rarely interested in finding and characterizing secondorder
and higher saddle points (hilltops)?
6. What kind(s) of stationary points do you think a second-order saddle point
connects?
42 2 The Concept of the Potential Energy Surface
7. If a species has one calculated frequency very close to 0 cmÀ1
what does that
tell you about the (calculated) potential energy surface in that region?
8. The ZPE of many molecules is greater than the energy needed to break a bond;
for example, the ZPE of hexane is about 530 kJ molÀ1
, while the strength of a
C–C or a C–H bond is only about 400 kJ molÀ1
. Why then do such molecules
not spontaneously decompose?
9. Only certain parts of a potential energy surface are chemically interesting:
some regions are flat and featureless, while yet other parts rise steeply and are
thus energetically inaccessible. Explain.
10. Consider two potential energy surfaces for the HCN ⇌ HNC reaction: A, a
plot of energy versus the H–C bond length, and B, a plot of energy versus the
HNC angle. Recalling that HNC is the higher-energy species, sketch qualitatively
the diagrams for A and B.
Harder Questions 43
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