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.