Table 3.
HSMC results for various free energy functionals for four different values of the external force K*
| K=8 | K=20 | K=40 | K = 100 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| HSMC/ nf | -FA | -FB | σA | -FA | -FB | σA | -FA | -FB | σA | -FA | -FB | σA |
| 20,000 | 422.7(5) | 415.1 | 3.4 | 845.3(2) | 836.2 | 2.9 | 1558.7(6) | 1547.4 | 3.7 | 3744.3(6) | 3734.4 | 3.9 |
| 40,000 | 422.4(4) | 415.3 | 2.9 | 845.3(2) | 836.6 | 2.5 | 1558.9(6) | 1547.9 | 3.2 | 3744.4(6) | 3737.1 | 3.2 |
| 80,000 | 422.3(4) | 416.1 | 2.7 | 845.2(2) | 836.3 | 2.2 | 1559.0(6) | 1548.6 | 3.2 | 3744.5(6) | 3737.9 | 2.7 |
| 160,000 | 422.3(4) | 416.9 | 2.7 | 845.2(2) | 837.3 | 2.2 | 1559.0(6) | 1548.4 | 3.2 | 3744.5(6) | 3738.3 | 2.5 |
| -FD -FM | 418.6 | 419.6 | 840.1 | 841.3 | 1551.5 | 1553.7 | 3740.5 | 3741.4 | ||||
| -FQH | 426.9 (2) | 852.4 (2) | 1569.0 (2) | 3750.9 (4) | ||||||||
| -FLS | 431.3 (4) | 856.0 (2) | 1570.6 (3) | 3754.3 (4) | ||||||||
| Eint | −43.5 (2) | −39.5 (4) | −27 (1) | 23.1 (8) | ||||||||
| - Etot | 323.3 (3) | 749.3 (3) | 1466.6 (5) | 3655.6 (3) | ||||||||
FA (eq 12) and FB (eqs 16 and 17) are lower and upper bounds of the free energy, respectively, and σA (eq 14) is the fluctuation of FA. These HSMC results were obtained from samples of 600 conformations. These results are presented only for the smallest bin size, δ=Δαk/60, but for all future sample sizes nf. The results for FM (eq 18) - the average of FA and FB, and for FD (eq 20) - the exact free energy functional, are calculated for δ=Δαk/60 and nf =160,000 only. FQH (eq 21) and FLS (eq 12) are free energies obtained by the quasi-harmonic approximation and the local states method, respectively, and are based on larger samples (see text). The average potential energy, Eint and the total energy, Eint = Eint +Eext of the HSMC samples (in kcal/mol) appear in the bottom rows. All free energies (at T=300 K) are in kcal/mol and are defined up to an additive constant. K is given in kcal/(mol·Å). The statistical error is defined in the caption of Table 2. We estimate the errors in FB and FD to be larger than the corresponding errors in FA at least by a factor of three and two, respectively.