The Force Field Energy

2 Force Field Methods

2.2 The Force Field Energy

The force field energy is written as a sum of terms, each describing the energy required for distorting a molecule in a specific fashion.

(2.1) E_str is the energy function for stretching a bond between two atoms,E_bendrepresents the energy required for bending an angle, E_tors is the torsional energy for rotation around a bond,E_vdw and E_el describe the non-bonded atom–atom interactions, and finally E_crossdescribes coupling between the first three terms.

E_FF=E_str+E_bend+E_tors+E_vdw+E_el+E_cross

torsional stretch

non-bond bend

Figure 2.1 Illustration of the fundamental force field energy terms

Given such an energy function of the nuclear coordinates, geometries and relative energies can be calculated by optimization. Stable molecules correspond to minima on the potential energy surface, and they can be located by minimizing E_FFas a function of the nuclear coordinates. Conformational transitions can be described by locating

transition structure on the E_FFsurface. Exactly how such a multi-dimensional function optimization may be carried out is described in Chapter 12.

2.2.1 The stretch energy

E_stris the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a “natural”, or “equilib- rium”, bond length,R₀. Terminating the expansion at second order gives eq. (2.2).

(2.2) The derivatives are evaluated at R=R₀and the E(0) term is normally set to zero, since this is just the zero point for the energy scale. The second term is zero as the expan- sion is around the equilibrium value. In its simplest form the stretch energy can thus be written as eq. (2.3).

(2.3) Here k^ABis the “force constant” for the A—B bond. This is the form of a harmonic oscillator, with the potential being quadratic in the displacement from the minimum.

The harmonic form is the simplest possible, and sufficient for determining most equi- librium geometries. There are certain strained and crowded systems where the results from a harmonic approximation are significantly different from experimental values, and if the force field should be able to reproduce features such as vibrational fre- quencies, the functional form for E_strmust be improved. The straightforward approach is to include more terms in the Taylor expansion.

(2.4) This of course has a price: more parameters have to be assigned.

Polynomial expansions of the stretch energy do not have the correct limiting behav- iour. The cubic anharmonicity constant k₃is normally negative, and if the Taylor expan- sion is terminated at third order, the energy will go toward −∞for long bond lengths.

Minimization of the energy with such an expression can cause the molecule to fly apart if a poor starting geometry is chosen. The quartic constant k₄is normally positive and the energy will go toward +∞for long bond lengths if the Taylor series is terminated at fourth order. The correct limiting behaviour for a bond stretched to infinity is that the energy should converge towards the dissociation energy. A simple function that satisfies this criterion is the Morse potential.⁴

(2.5)

Here D is the dissociation energy and a is related to the force constant. The Morse function reproduces the actual behaviour quite accurately over a wide range of dis- tances, as seen in Figure 2.2.There are, however, some difficulties with the Morse poten- tial in actual applications. For long bond lengths the restoring force is quite small.

Distorted structures, which may either be a poor starting geometry or one that devel- ops during a simulation, will therefore display a slow convergence towards the

E R D e

k D

Morse( )∆ =

(

− ^∆R

)

= 1 −

2

a 2

a

E_str

(

∆R^AB

)

=k^AB2

(

∆R^AB

)

+k^AB

(

∆R^AB

)

+k^AB

(

∆R^AB

)

+

2 3

3 4

4 L

E_str

(

R^AB−R₀^AB

)

=k^AB

(

R^AB−R^AB0

)

²=k^AB

(

∆R^AB

)

²

E R R E E

R R R E

R R R

str AB AB d AB AB AB AB

d

d

− d

(

⁰

)

^{= ( )+}

(

^{− 0}

)

^{+ 2}2

(

⁻ 0

)

0 1 2

2

equilibrium bond length. For minimization purposes and simulations at ambient tem- peratures (e.g. 300 K) it is sufficient that the potential is reasonably accurate up to

~40 kJ/mol above the minimum (the average kinetic energy is 3.7 kJ/mol at 300 K). In this energy range there is little difference between a Morse potential and a Taylor expansion, and most force fields therefore employ a simple polynomial for the stretch energy. The number of parameters is often reduced by taking the cubic, quartic, etc., constants as a predetermined fraction of the harmonic force constant. A popular method is to require that the nth-order derivative at R₀matches the corresponding derivative of the Morse potential. For a fourth-order expansion this leads to the following expression.

(2.6) The aconstant is the same as that appearing in the Morse function, but may be taken as a fitting parameter. An alternative method for introducing anharmonicity is to use the harmonic form in eq. (2.3) but allow the force constant to depend on the bond distance.⁵

Figure 2.2 compares the performance of various functional forms for the stretch energy in CH₄. The “exact” form is taken from electronic structure calculations ([8,8]- CASSCF/aug-cc-pVTZ). The simple harmonic approximation (P2) is seen to be accu- rate to about ±0.1 Å from the equilibrium geometry and the quartic approximation (P4) up to ±0.3 Å. The Morse potential reproduces the real curve quite accurately up to an elongation of 0.8 Å, and becomes exact again in the dissociation limit.

For the large majority of systems, including simulations, the only important chemi- cal region is within ~40 kJ/mol of the bottom of the curve. In this region, a fourth-order polynomial is essentially indistinguishable from either a Morse or the exact curve, as shown in Figure 2.3, and even a simple harmonic approximation does a quite good job.

E_str

(

∆R^AB

)

=k^AB2

(

∆R^AB

)

²

[

¹−^a

(

∆R^AB

)

+^127a2

(

∆R^AB

)

²

]

0 100 200 300 400

–0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 1.0

"Exact"

P2 P4 Morse

∆R_CH (Å)

Energy (kJ/mol)

Figure 2.2 The stretch energy for CH4

Until now, we have used two different words for the R₀parameter, the “natural” or the “equilibrium” bond length. The latter is slightly misleading. The R₀parameter is notthe equilibrium bond length for any molecule! Instead it is the parameter which, when used to calculate the minimum energy structure of a molecule, will produce a geometry having the experimental equilibrium bond length. If there were only one stretch energy in the whole force field energy expression (i.e. a diatomic molecule),R₀ would be the equilibrium bond length. However, in a polyatomic molecule the other terms in the force field energy will usually produce a minimum energy structure with bond lengths slightly longer than R₀.R₀is the hypothetical bond length if no other terms are included, and the word “natural” bond length is a better description of this parameter than “equilibrium” bond length. Essentially all molecules have bond lengths that deviate very little from their “natural” values, typically by less than 0.03 Å. For this reason a simple harmonic is usually sufficient for reproducing experimental geometries.

For each bond type, i.e. a bond between two atom types A and B, there are at least two parameters to be determined,k^ABand R₀^AB. The higher order expansions, and the Morse potential, have one additional parameter (aor D) that needs to be determined.

2.2.2 The bending energy

E_bendis the energy required for bending an angle formed by three atoms A—B—C, where there is a bond between A and B, and between B and C. Similarly to E_str,E_bend is usually expanded as a Taylor series around a “natural” bond angle and terminated at second order, giving the harmonic approximation.

0 10 20 30 40

–0.2 –0.1 0.0 0.1 0.2

"Exact"

P2 P4 Morse

Energy (kJ/mol)

∆R_CH (Å) Figure 2.3 The stretch energy for CH4

(2.7) While the simple harmonic expansion is adequate for most applications, there may be cases where higher accuracy is required. The next improvement is to include a third- order term, analogous to Estr. This can give a very good description over a large range of angles, as illustrated in Figure 2.4 for CH4. The “exact” form is again taken from electronic structure calculations (MP2/aug-cc-pVTZ). The simple harmonic approxi- mation (P2) is seen to be accurate to about ±30° from the equilibrium geometry and the cubic approximation (P3) up to ±70°. Higher order terms are often included in order also to reproduce vibrational frequencies. Analogous to Estr, the higher order force constants are often taken as a fixed fraction of the harmonic constant. The con- stants beyond third order can rarely be assigned values with high confidence owing to insufficient experimental information. Fixing the higher order constant in terms of the harmonic constant of course reduces the quality of the fit. While a third-order poly- nomial is capable of reproducing the actual curve very accurately if the cubic constant is fitted independently, the assumption that it is a fixed fraction (independent of the atom type) of the harmonic constant deteriorates the fit, but it still represent an improvement relative to a simple harmonic approximation.

Ebend

(

qABC−q⁰ABC

)

⁼kABC

(

qABC⁻qABC⁰

)

²

0 50 100 150 200 250 300 350 400

40 60 80 100 120 140 160 180

"Exact"

P2 P3

Energy (kJ/mol)

q_HCH (°) Figure 2.4 The bending energy for CH4

In the chemically important region below ~40 kJ/mol above the bottom of the energy curve, a second-order expansion is normally sufficient.

Angles where the central atom is di- or trivalent (ethers, alcohols, sulfides, amines and enamines) present a special problem. In these cases, an angle of 180° corresponds to an energy maximum, i.e. the derivative of the energy with respect to the angle should be zero and the second derivative should be negative. This may be enforced by suit- able boundary conditions on Taylor expansions of at least order three. A third-order

polynomial fixes the barrier for linearity in terms of the harmonic force constant and the equilibrium angle (∆E^≠=k(q−q0)²/6). A fourth-order polynomial enables an inde- pendent fit of the barrier to linearity, but such constrained polynomial fittings are rarely done. Instead, the bending function is taken to be identical for all atom types, for example a fourth-order polynomial with cubic and quartic constants as a fixed fraction of the harmonic constant.

These features are illustrated for H₂O in Figure 2.5, where the “exact” form is taken from a parametric fit to a large amount of spectroscopic data.⁶The simple harmonic approximation (P2) is seen to be accurate to about ±20° from the equilibrium geome- try and the cubic approximation (P3) up to ±40°. Enforcing the cubic polynomial to have a zero derivative at 180° (P3′) gives a qualitatively correct behaviour, but reduces the overall fit, although it is still better than a simple harmonic approximation.

0 50 100 150 200

60 80 100 120 140 160 180

"Exact"

P2 P3 P3'

q_HOH

Energy (kJ/mol)

Figure 2.5 The bending energy for H2O

Although such refinements over a simple harmonic potential clearly improve the overall performance, they have little advantage in the chemically important region up to ~40 kJ/mol above the minimum. As for the stretch energy term, the energy cost for bending is so large that most molecules only deviate a few degrees from their natural bond angles. This again indicates that including only the harmonic term is adequate for most applications.

As noted above, special atom types are often defined for small rings, owing to the very different equilibrium angles for such rings. In cyclopropane, for example, the carbons are formally sp³-hybridized, but have equilibrium CCC angles of 60°, in con- trast to 110° in an acyclic system. With a low-order polynomial for the bend energy, the energy cost for such a deformation is large. For cyclobutane, for example,E_bendwill dominate the total energy and cause the calculated structure to be planar, in contrast to the puckered geometry found experimentally.

For each combination of three atom types, A, B and C, there are at least two bending parameters to be determined,k^ABCand q0ABC.

2.2.3 The out-of-plane bending energy

If the central B atom in the angle ABC is sp²-hybridized, there is a significant energy penalty associated with making the centre pyramidal, since the four atoms prefer to be located in a plane. If the four atoms are exactly in a plane, the sum of the three angles with B as the central atom should be exactly 360°, however, a quite large pyra- midalization may be achieved without seriously distorting any of these three angles.

Taking the bond distances to 1.5 Å, and moving the central atom 0.2 Å out of the plane, only reduces the angle sum to 354.8° (i.e. only a 1.7° decrease per angle). The corre- sponding out-of-plane angle,c, is 7.7° for this case. Very large force constants must be used if the ABC, ABD and CBD angle distortions are to reflect the energy cost asso- ciated with the pyramidalization. This would have the consequence that the in-plane angle deformations for a planar structure would become unrealistically stiff. Thus a special out-of-plane energy bendterm (E_oop) is usually added, while the in-plane angles (ABC, ABD and CBD) are treated as in the general case above.E_oopmay be written as a harmonic term in the angle c(the equilibrium angle for a planar structure is zero) or as a quadratic function in the distance d, as given in eq. (2.8) and shown in Figure 2.6.

(2.8) Such energy terms may also be used for increasing the inversion barrier in sp³- hybridized atoms (i.e. an extra energy penalty for being planar), and E_oopis also some- times called E_inv. Inversion barriers are in most cases (e.g. in amines, NR₃) adequately modelled without an explicit E_invterm, the barrier arising naturally from the increase in bond angles upon inversion. The energy cost for non-planarity of sp²-hybridized atoms may also be accounted for by an “improper” torsional energy, as described in Section 2.2.4.

For each sp²-hybridized atom there is one additional out-of-plane force constant to be determined,k^B.

E_oop( )c =k^Bc² or E_oop( )d =k d^{B 2}

Figure 2.6 Out-of-plane variable definitions

2.2.4 The torsional energy

E_torsdescribes part of the energy change associated with rotation around a B—C bond in a four-atom sequence A—B—C—D, where A—B, B—C and C—D are bonded.

Looking down the B—C bond, the torsional angle is defined as the angle formed by the A—B and C—D bonds as shown in Figure 2.7. The angle wmay be taken to be in the range [0°,360°] or [−180°,180°].

The torsional energy is fundamentally different from E_strand E_bendin three aspects:

(1) A rotational barrier has contributions from both the non-bonded (van der Waals and electrostatic) terms, as well as the torsional energy, and the torsional param- eters are therefore intimately coupled to the non-bonded parameters.

(2) The torsional energy function must be periodic in the angle w: if the bond is rotated 360° the energy should return to the same value.

(3) The cost in energy for distorting a molecule by rotation around a bond is often low, i.e. large deviations from the minimum energy structure may occur, and a Taylor expansion in wis therefore not a good idea.

To encompass the periodicity,E_torsis written as a Fourier series.

(2.9) The n=1 term describes a rotation that is periodic by 360°, the n=2 term is periodic by 180°, the n=3 term is periodic by 120°, and so on. The V_nconstants determine the size of the barrier for rotation around the B—C bond. Depending on the situation, some of these V_nconstants may be zero. In ethane, for example, the most stable con- formation is one where the hydrogens are staggered relative to each other, while the eclipsed conformation represents an energy maximum. As the three hydrogens at each end are identical, it is clear that there are three energetically equivalent staggered, and three equivalent eclipsed, conformations. The rotational energy profile must therefore have three minima and three maxima. In the Fourier series only those terms that have n=3, 6, 9, etc., can therefore have V_nconstants different from zero.

For rotation around single bonds in substituted systems, other terms may be neces- sary. In the butane molecule, for example, there are still three minima, but the two gauche (torsional angle ~±60°) and anti (torsional angle ~180°) conformations now have different energies. The barriers separating the two gaucheand the gauche and anti conformations are also of different height. This may be introduced by adding a term corresponding to n=1.

For the ethylene molecule, the rotation around the C=C bond must be periodic by 180°, and thus only n=2, 4, etc., terms can enter. The energy cost for rotation around a double bond is of course much higher than that for rotation around a single bond in

E Vn n

n

tors( )w = ( )w

∑

= ^cos 1

Figure 2.7 Torsional angle definition

ethane, which would be reflected in a larger value of the V₂ constant. For rotation around the C=C bond in a molecule such as 2-butene, there would again be a large V₂constant, analogous to ethylene, but in addition there are now two different orien- tations of the two methyl groups relative to each other,cisand trans. The full rotation is periodic with a period of 360°, with deep energy minima at 0° and 180°, but slightly different energies of these two minima. This energy difference would show up as a V₁ constant, i.e. the V₂constant essentially determines the barrier and location of the minima for rotation around the C=C bond, and the V₁constant determines the energy difference between the cisand transisomers.

Molecules that are composed of atoms having a maximum valence of four (essen- tially all organic molecules) are with a few exceptions found to have rotational pro- files showing at most three minima. The first three terms in the Fourier series in eq.

(2.9) are sufficient for qualitatively reproducing such profiles. Force fields that are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos(3w) and double bonds only cos(2w)).

Systems with bulky substituents on sp³-hybridized atoms are often found to have four minima, the anticonformation being split into two minima with torsional angles of approximately ±170°. Other systems, notably polyfluoroalkanes, also split the gauche minima into two, often called gauche(angle of approximately ±50°) and ortho(angle of approximately ±90°) conformations, creating a rotational profile with six minima.⁷ Rotations around a bond connecting sp³- and sp²-hybridized atoms (such as CH₃NO₂) also display profiles with six minima.⁸These exceptions from the regular three minima rotational profile around single bonds are caused by repulsive and attractive van der Waals interactions, and can still be modelled by having only terms up to cos(3w) in the torsional energy expression. Higher order terms may be included to modify the detailed shape of the profile, and a few force fields employ terms with n = 4 and 6.

Cases where higher order terms probably are necessary are rotation around bonds to octahedral coordinated metals, such as Ru(pyridine)₆or a dinuclear complex such as Cl₄Mo–MoCl₄. Here the rotation is periodic by 90° and thus requires a cos(4w) term.

It is customary to shift the zero point of the potential by adding a factor of one to each term. Most rotational profiles resemble either the ethane or ethylene examples above, and a popular expression for the torsional energy is given in eq. (2.10).

(2.10)

The +and −signs are chosen such that the one-fold rotational term has a minimum for an angle of 180°, the two-fold rotational term has minima for angles of 0° and 180°, and the three-fold rotational term has minima for angles of 60°, 180° and 300° (−60°).

The factor ¹/2is included such that the V_iparameters directly give the height of the barrier if only one term is present. A more general form for eq. (2.10) includes a phase factor, i.e. cos(nw−t), but for the most common cases of t=0° or 180°, it is completely equivalent to eq. (2.10). Figure 2.8 illustrates the functional behaviour of the three indi- vidual terms in eq. (2.10).

The V_i parameters may also be negative, which corresponds to changing the minima on the rotational energy profile to maxima, and vice versa. Most commonly

E V

V V

tors ABCD ABCD ABCD

ABCD ABCD

ABCD ABCD

w w

w w

( )

=

[

+

( ) ]

+

[

−

( ) ]

+

[

+

( ) ]

1 2 1

1 2 2 1 2 3

1

1 2

1 3

cos cos cos