Where does strain energy come from?

(According to Molecular Mechanics)

The following table comes from Molecular Mechanics calculations on different conformations of butane (executed using the computer program Chem3DPro). The conformations are energy-minimized anti and gauche (note that the C-C-C-C angle in minimized gauche is not 60), and the energy maxima (which we'll soon be calling passes, or transition structures, or transition states) between them.

In the second set of entries ("Relative to Anti") the values for the lowest energy (anti) conformation is subtracted to show how the others differ from it. It is worth looking at how the strain energy is distributed. There is always a give and take among different types of distortion - one can reduce van der Waals repulsion by bending or stretching bonds, for example. At first it seems curious that the order of strain energy (highlighted for gauche) is commonly torsion > bending > stretching, since the force constants are such that the amount of strain energy for a given amount of atomic motion is in the opposite order (easier to twist than to bend than to stretch). This makes sense when you think twice, because the stiffness of the higher force constants suppresses motion and most energy get stored in "softer" modes of distortion. [If this seems unnatural to you click here for an explanation.]

Below the two summary tables are the structure of the fully eclipsed conformation and a table of its individual strain contributions.

Name

anti

gauche

eclipsed

fully eclipsed

(anticlinal)

(synclinal)

C-C-C-C Angle

180

64.04

119.8

0

Stretch

0.1542

0.1725

0.1956

0.2301

Bend

0.2977

0.6064

0.4744

1.4262

Stretch-Bend

0.0543

0.0741

0.078

0.1258

Torsion

0.0073

0.4323

2.3682

2.3168

Non-1,4 VDW

-0.4086

-0.3496

-0.3151

-0.0523

1,4 VDW

2.0688

2.1022

2.7256

2.8492

Total

2.1737

3.0379

5.5266

6.8957

Relative to Anti

Stretch

0

0.0183

0.0414

0.0759

Bend

0

0.3087

0.1767

1.1285

Stretch-Bend

0

0.0198

0.0237

0.0715

Torsion

0

0.425

2.3609

2.3095

Non-1,4 VDW

0

0.059

0.0935

0.3563

1,4 VDW

0

0.0334

0.6568

0.7804

Total

0

0.8642

3.3529

4.722

Here is a tabulation steric energy components for the fully eclipsed conformation shown (very slightly different from the model used for the composite table above) followed by the individual contributions to each components. Columns headed with K(*) are force constants for the * distortion.  Within each component the individual energies are sorted to be in decreasing order, so it is useful to look at the first few (which are Highlighted) to see which atoms they are coming from.

Stretch

0.2292

Bend

1.41

Stretch-Bend

0.1253

Torsion

2.3169

Non-1,4 VDW

-0.034

1,4 VDW

2.8495

Total

6.8968

Stretch

Length

R(0) preferred

K(S)

Energy

C(2)-C(7)

1.543

1.523

4.4

0.122

central bond

C(1)-C(2)

1.534

1.523

4.4

0.040

C(7)-C(11)

1.534

1.523

4.4

0.040

C(2)-H(8)

1.117

1.113

4.6

0.007

C(7)-H(9)

1.117

1.113

4.6

0.007

C(2)-H(6)

1.117

1.113

4.6

0.007

C(7)-H(10)

1.117

1.113

4.6

0.007

C(1)-H(5)

1.115

1.113

4.6

0.001

C(11)-H(12)

1.115

1.113

4.6

0.001

C(1)-H(3)

1.113

1.113

4.6

0.000

C(1)-H(4)

1.113

1.113

4.6

0.000

C(11)-H(13)

1.113

1.113

4.6

0.000

C(11)-H(14)

1.113

1.113

4.6

0.000

1,4 van der Waals

R

RV preferred

K(V) 1,4

Energy

H(8),H(9)

2.275

3

0.047

0.692

eclipsed Hs

These strains have

H(6),H(10)

2.276

3

0.047

0.690

been reduced by

C(1),C(11)

2.908

3.8

0.044

0.402

eclipsed Cs

stretching/bending

H(9),H(12)

2.484

3

0.047

0.227

H(5),H(8)

2.486

3

0.047

0.224

H(5),H(6)

2.486

3

0.047

0.224

H(10),H(12)

2.488

3

0.047

0.222

H(10),H(13)

2.531

3

0.047

0.166

H(4),H(6)

2.532

3

0.047

0.164

H(3),H(8)

2.533

3

0.047

0.164

H(9),H(14)

2.534

3

0.047

0.162

C(2),H(14)

2.915

3.34

0.046

0.029

C(7),H(3)

2.917

3.34

0.046

0.028

C(7),H(4)

2.918

3.34

0.046

0.028

C(2),H(13)

2.919

3.34

0.046

0.027

H(6),H(9)

2.89

3

0.047

-0.034

H(8),H(10)

2.89

3

0.047

-0.034

C(2),H(12)

3.566

3.34

0.046

-0.051

C(7),H(5)

3.566

3.34

0.046

-0.051

C(1),H(9)

3.358

3.34

0.046

-0.053

C(1),H(10)

3.359

3.34

0.046

-0.053

C(11),H(6)

3.358

3.34

0.046

-0.053

C(11),H(8)

3.359

3.34

0.046

-0.053

H(3),H(6)

3.104

3

0.047

-0.054

H(4),H(8)

3.104

3

0.047

-0.054

H(9),H(13)

3.104

3

0.047

-0.054

H(10),H(14)

3.105

3

0.047

-0.054

Non 1,4

R

R preferred

K(V) non 1,4

Energy

H(3),H(14)

2.367

3

0.047

0.196

Hs on opposing

H(4),H(13)

2.376

3

0.047

0.183

methyl groups

C(1),H(14)

2.848

3.34

0.046

0.052

(Also reduced by

C(11),H(3)

2.852

3.34

0.046

0.049

stretching/bending)

C(11),H(4)

2.853

3.34

0.046

0.049

C(1),H(13)

2.857

3.34

0.046

0.047

H(5),H(12)

5.126

3

0.047

-0.005

H(5),H(9)

4.224

3

0.047

-0.016

H(5),H(10)

4.225

3

0.047

-0.016

H(6),H(12)

4.225

3

0.047

-0.016

H(8),H(12)

4.223

3

0.047

-0.016

H(5),H(13)

3.924

3

0.047

-0.023

H(4),H(12)

3.921

3

0.047

-0.023

H(3),H(12)

3.918

3

0.047

-0.023

H(5),H(14)

3.915

3

0.047

-0.023

H(8),H(13)

3.88

3

0.047

-0.026

H(3),H(10)

3.877

3

0.047

-0.026

H(4),H(9)

3.877

3

0.047

-0.026

H(6),H(14)

3.875

3

0.047

-0.026

C(1),H(12)

4.018

3.34

0.046

-0.034

C(11),H(5)

4.018

3.34

0.046

-0.034

H(4),H(10)

3.437

3

0.047

-0.042

H(6),H(13)

3.438

3

0.047

-0.042

H(3),H(9)

3.436

3

0.047

-0.042

H(8),H(14)

3.436

3

0.047

-0.042

H(4),H(14)

2.978

3

0.047

-0.054

H(3),H(13)

2.988

3

0.047

-0.054

Angle bend

Angle

preferred

K(B)

EBend

K(St-Bend)

ESB

C(2)-C(7)-C(11)

116.408

109.5

0.45

0.471

0.12

0.0652

C(1)-C(2)-C(7)

116.408

109.5

0.45

0.471

0.12

0.0652

H(6)-C(2)-H(8)

105.65

109.4

0.32

0.099

H(10)-C(7)-H(9)

105.646

109.4

0.32

0.099

H(12)-C(11)-H(13)

106.961

109

0.32

0.029

H(4)-C(1)-H(5)

106.964

109

0.32

0.029

H(3)-C(1)-H(5)

106.982

109

0.32

0.029

H(12)-C(11)-H(14)

106.984

109

0.32

0.029

C(7)-C(11)-H(14)

111.609

110

0.36

0.020

0.09

0.0042

C(2)-C(1)-H(4)

111.602

110

0.36

0.020

0.09

0.0042

C(2)-C(1)-H(3)

111.588

110

0.36

0.020

0.09

0.0042

C(7)-C(11)-H(13)

111.582

110

0.36

0.020

0.09

0.0042

C(11)-C(7)-H(9)

107.977

109.41

0.36

0.016

0.09

-0.0051

C(1)-C(2)-H(6)

107.984

109.41

0.36

0.016

0.09

-0.0051

C(1)-C(2)-H(8)

107.998

109.41

0.36

0.016

0.09

-0.005

C(11)-C(7)-H(10)

108.003

109.41

0.36

0.016

0.09

-0.005

C(2)-C(1)-H(5)

110.755

110

0.36

0.005

0.09

0.0022

C(7)-C(11)-H(12)

110.754

110

0.36

0.005

0.09

0.0022

C(2)-C(7)-H(9)

109.117

109.41

0.36

0.001

0.09

-0.0016

C(7)-C(2)-H(6)

109.129

109.41

0.36

0.001

0.09

-0.0016

C(7)-C(2)-H(8)

109.143

109.41

0.36

0.001

0.09

-0.0015

H(3)-C(1)-H(4)

108.732

109

0.32

0.001

C(2)-C(7)-H(10)

109.157

109.41

0.36

0.001

0.09

-0.0014

H(13)-C(11)-H(14)

108.732

109

0.32

0.001

Torsions

angle

V1

V2

V3

E torsion

C(1)-C(2)-C(7)-C(11)

0

0.2

0.27

0.093

0.293

All these

C(1)-C(2)-C(7)-H(9)

-122.469

0

0

0.267

0.266

values

H(6)-C(2)-C(7)-C(11)

-122.484

0

0

0.267

0.266

involve

C(1)-C(2)-C(7)-H(10)

122.535

0

0

0.267

0.266

torsion

H(8)-C(2)-C(7)-C(11)

122.518

0

0

0.267

0.266

about the

H(6)-C(2)-C(7)-H(10)

0

0

0

0.237

0.237

central (C2-C7)

H(8)-C(2)-C(7)-H(9)

0

0

0

0.237

0.237

bond.

H(6)-C(2)-C(7)-H(9)

115.05

0

0

0.237

0.233

They sum to

H(8)-C(2)-C(7)-H(10)

-114.95

0

0

0.237

0.233

2.3 kcal

H(9)-C(7)-C(11)-H(13)

-175.683

0

0

0.237

0.003

H(4)-C(1)-C(2)-H(8)

175.923

0

0

0.237

0.003

H(3)-C(1)-C(2)-H(6)

-176.009

0

0

0.237

0.003

H(10)-C(7)-C(11)-H(14)

176.241

0

0

0.237

0.002

H(9)-C(7)-C(11)-H(12)

-56.638

0

0

0.237

0.002

H(5)-C(1)-C(2)-H(8)

56.864

0

0

0.237

0.002

H(5)-C(1)-C(2)-H(6)

-56.941

0

0

0.237

0.002

H(10)-C(7)-C(11)-H(12)

57.165

0

0

0.237

0.001

H(9)-C(7)-C(11)-H(14)

62.442

0

0

0.237

0.001

H(3)-C(1)-C(2)-H(8)

-62.2

0

0

0.237

0.001

H(4)-C(1)-C(2)-H(6)

62.115

0

0

0.237

0.001

H(10)-C(7)-C(11)-H(13)

-61.873

0

0

0.237

0.001

C(2)-C(7)-C(11)-H(13)

61.26

0

0

0.267

0.000

H(3)-C(1)-C(2)-C(7)

60.913

0

0

0.267

0.000

H(4)-C(1)-C(2)-C(7)

-60.966

0

0

0.267

0.000

C(2)-C(7)-C(11)-H(14)

-60.62

0

0

0.267

0.000

H(5)-C(1)-C(2)-C(7)

180

0

0

0.267

0.000

C(2)-C(7)-C(11)-H(12)

-179.684

0

0

0.267

0.000


If the concept of storing more energy in the easier distortion seems funny to you, think of hooking a rubber band to a suspension spring of a Mack truck and pulling on the end. The rubber band would stretch and the spring wouldn't, so all the energy would go into the rubber band.
If you want to be more concrete and quantitative, think of it this way:

The force on a spring depends on how much it is stretched (dx). The energy stored in the spring depends on the square of the distortion (this is why the potential for an harmonic oscillator is a parabola). Where k is the spring constant,

F = -k dx ; E = 1/2 k (dx)2

When two springs interact, as when you hook a weak spring ( k2) to a strong spring ( k1 = 8 x k2) and pull on the two ends, the force on the two springs is the same, so the weak one stretches more (here 8 times as much):

F2 = (-)F1 means that k2 dx2 = k1 dx1 means that dx2 = 8 dx1

This means that the weaker spring stores 8 times as much energy as the stronger spring (k is 8 times smaller, but dx2 is 64 times larger).

In summary: since the forces must balance, more energy goes into the weaker spring.

copyright 2001 J. M. McBride