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⁠,

η (old) \to \sqrt{β/2} η (here)

⁠, and

ξ (old) \to \sqrt{β/2} ξ (here)

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The problem is similar to that of determining the relationship between the molecular wave functions and the localized basis sets in quantum chemistry. However, the quantum chemistry problem is simply solved (e.g., by MO-LCAO approximation) because the localized basis functions are roughly factorizable in terms of atomic orbitals [Hartree-Fock-Ruthaan approximation; R. G. Parr, The Quantum Theory of Molecular Electronic Structure, (W. A. Benjamin, New York, 1964)]. This is not true for complicated multidimensional systems where the localized conformational macrostates are not factorizable in terms of the individual atomic coordinates.

48.

For an arbitrary nonequilibrium state

Ψ

⁠,

J

has support everywhere. However, only the current generated by the components of

Ψ

which lie within the macrostate subspace contribute to the intermacrostate transition rates since, using Eqs. (9c) and (26) and integration-by-parts, it is easy to show that

{∫∇w}_{α} \cdot J {[Ψ] dR=∫∇w}_{α} \cdot J [P_{Φ} Ψ] dR

⁠, where

P_{Φ}

is the projection operator defined in Eq. (24). The current generated by the eigenfunction components that are outside the subspace corresponds to fast relaxation of the probability distribution within, not between, macrostates.

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50.

Consider the eigenfunction expansion of

Φ_{β}

(see Ref. 16): The use of hard window functions would correspond to including all excited eigenfunctions of

H

in

Φ_{β},

resulting in an unbounded spectrum of relaxation rates which would lead to a singularity in the rate at

t=0

⁠.

51.

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52.

A master equation similar to Eq. (2) can be written for the

p_{α}^{occ}

using the occupation probability transition matrix

Γ^{occ}

⁠. The time-dependent generalization of Eq. (17) implies that

tr Γ^{occ} = tr Γ^{obs}

⁠, so the minimum decay condition can equivalently be imposed in either the occupation or observation representation.

53.

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54.

See Ref. 16 for further discussion. If

H

is invariant under a continuous symmetry group (e.g., rigid body translation and rotation), we will only be interested in group-invariant macrostates; therefore the sums in Eq. (24) can be restricted to group- invariant eigenfunctions, and degeneracies associated with irreducible representations of the group of rank

>1

will not enter. If

H

is invariant under a discrete group of transformations, the natural requirement that all group-transformation-related macrostates be included in the set of macrostates (i.e., that the macrostates form a basis for a representation of the group) implies that the macrostate subspace will include either all or none of the degenerate eigenfunctions belonging to an irreducible representation of the group. Thus, there will be no symmetry-induced ambiguity in the subspace specification.

55.

This variational equation for

σ

is equivalent to the Rayleigh-Ritz variational equation for the first excited eigenfunction; see Ref. 16.

56.

The

\det W

term ensures that the node is located between the two macrostates; the denominator will become small if

σ

is varied so much that the nodal surface does not separate the macrostates. However, this does not occur during normal numerical variation.

57.

The solution of Eq. (33) must be bounded for this to be possible. This will be true even when the conformation space is unbounded as long as the ensemble is confined by a potential which grows faster than

R^{2}

as

R→∞

⁠. When this is so, all low-lying eigenfunctions (and hence all functions in the macrostate subspace) will behave like

ψ_{0}

as

R→∞

[A. Ulitsky and D. Shalloway (in preparation)]. Thus, by Eq. (15), the window and transition functions will go to constants.

58.

Note that

| \erf (0.83x)− \tanh (x)|<0.02     ∀x

⁠. Thus the functions provide essentially equivalent parameterizations for the variational condition.

59.

For a system with

N

degrees-of-freedom, both the nodal surface and the transition “plane” are

(N−1)

-dimensional manifolds.

60.

Since

η

is a vector, Eqs. (40) represents

N+1

conditions where

N

is the number of degrees-of-freedom.

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62.

Kramers (Ref. 61) one-dimensional result, which assumes that the macrostate is confined by a harmonic potential, is easily generalized to the one-dimensional form of Eq. (45) which involves

p_{X}

and

Z

instead of the characteristic harmonic frequency.

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68.

This method (see Ref. 66) was chosen because it uses only the objective function and does not require derivatives which are relatively expensive to accurately compute by Monte Carlo integration.

69.

The integrals in the right-hand-sides of Eqs. (47) were computed to 3% accuracy as determined by the variance (Ref. 65).

70.

The integrals were computed to 0.5% accuracy. The convergence condition was

{|ΔH}_{XY} {/H}_{XY} |<0.03

⁠.

71.

We used

δ_{η} {=δ}_{ξ} =0.03

⁠.

72.

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73.

r_{t}^{η}

⁠, and hence

{ΔV}_{η},

can be kept fixed when performing this variation, since the dependence on it cancels in the computation of

γ_{1}^{c} .

74.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, 2nd ed. (Springer, New York, 1997).

75.

The reduced distributions were confined to a symmetric box of length 10. This gave

γ_{1}

to 1% accuracy.

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D. Shalloway (in preparation) .

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83.

The individual transition rates add because the transition regions do not overlap. This is evident just from visual inspection of the transition region conformation shown in the middle of Fig. 4(A): This conformation is adequately separated (on the scale set by

K_{t})

from the transition region conformations resulting from the replacement of atoms 3 and 4 with other pairs of adjacent atoms.

84.

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85.

As an additional check, transition rates were also calculated using the mean first-passage time evaluated from Langevin dynamics trajectories. Similar results were obtained.

86.

N=3

when calculating

{Δχ}^{2}

using Eq. (49b) since only the coordinates of one atom enter Eq. (61).

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90.

In the isotropic-friction Brownian dynamics limit we relate the VTST variables in Ref. 88 to the MVM variables by the equivalences

{γ≡(D β)}^{−1}

⁠,

(q,z)≡R

⁠,

ρ_{0} ≡[\sqrt{2} {/(β D |η|}^{2})] ξ

⁠,

(C,D)≡[\sqrt{2} /(β D |η|)]η̂

⁠, and

λ^{‡} {≡β D η}^{2}

⁠, where the VTST (MVM) variables are on the left (right). The planar VTST variational equations (Eqs. 4.19, 4.30 and 4.21 of Ref. 88) are then equivalent to the simplified planar Gaussian Eqs. (44) and the computed transition rates are equal up to the approximation

p_{X} {≈∫}_{−∞}^{0} \exp [−βw(q)] dq/∫ \exp [−βw(q)] dq

⁠, where

w(q)

is the potential of mean force along an ad hoc preselected direction and

q=0

is the maximizer of

w(q)

in the barrier region.

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96.

We expect that the minimum uncertainty condition will impose the stronger conditions

w_{γ} ≈0

⁠, for

α≠β

⁠, and

w_{α} {+w}_{β} ≈1

⁠, but this has not been proven.

97.

It is possible that, as in the two-macrostate case, the minimum uncertainty condition will force the

w_{α}

to approach zero or one, but this has not been proven.

98.

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