We have selected two molecular systems that are small enough to generate reproducible results in a short amount of time, but complicated enough to demonstrate the power and the applicability of the methods reviewed in the article. These are ethane and propane. In both the cases, it is more difficult to define a map using the first approach discussed earlier, that is, explicitly write down the equations of constraints. Rather, the use of Lie groups of continuous transformations creates a systematic approach to construct Cartesian coordinates from a well defined and physically meaningful parameter space, and this can be generalized and automated to produce linear chains of any dimension.
The process we use to construct the coordinate map for a linear saturated hydrocarbon Cn ,
for ethane (n = 2) and propane (n = 3) is an adaptation of the algorithm outlined in Patriciu et al. The vector space is the Cartesian product of the Ramachandra space of torsions and the inertia ellipsoid for the rotational degrees of freedom. The latter is typically mapped with three Eulerian angles using the convention, whereas Ramachandra's space is typically mapped with n − 1 dihedral angles. We first build a body-fixed Cartesian coordinate representation, and then rotate it into a space-fixed representation, using the following procedure:
- Carbon atom 1 is placed at the origin, .
- Carbon atom 2 is placed along the z-axis, its coordinates are given by the vector where is the unit vector along z, and the carbon–carbon average distance is bohr.
- The position of the other carbon atoms is obtained by rotating and translating the coordinates of carbon 2, where, k varies from 1 to n −2, the angle is the tetrahedral angle, and are the dihedral angles with .
- The coordinates constructed in the previous four steps are translated to the center of mass For ethane and propane, the center of mass is independent of the dihedral angles, and this fact simplifies the analysis.
- The coordinate of each atom are operated upon by the rotations about the three Euler angles, where, clearly, the body-fixed configuration only depends on the torsion angle(s), whereas the space-fixed coordinate set is a function of possibly all n + 2 variables.
Expressions for these tensorial elements can be obtained readily.
DMC simulations of ethane and propane
The symmetry has been used. We verify these expressions with mathematica, and we obtain the same result, although the full simplification algorithm does not yield these expressions directly, rather we obtain expressions containing the masses of the two elements and the parameters rH and rC. Nevertheless, it is straightforward to insert Eqs. (46) and (47) into Eq. (49) through (55) and verify the equivalences.
The top block related to the 2D Ramachandra space, , is diagonal, and the two diagonal elements have the same expression,
where the right-hand side is the same as in ethane, given in Eq. (48). This fact implies that the two torsional degrees of freedom are not coupled with one another geometrically. As in ethane, the rotational degrees of freedom couple geometrically with the torsional ones. In the coupling block , the expressions connecting and the two Euler angles and ψ, are identical to those in Eqs. (50) and (51), and the element connecting with θ vanishes. However, for the second-torsional degree of freedom the coupling elements are more complex and all three Euler angles contribute nonvanishing coupling elements with .
For the ensuing analysis, it is critical to realize that for propane, none of the degrees of freedom are dynamically coupled, as the potential energy for propane is simply a Fourier expansions for the torsions and an external field,
with mhartree as for ethane. The last term on the right-hand side is introduced to mimic the typical level of hindrance to rotation a propane molecule experiences on a surface. As a result, it is possible to gauge directly the effects that geometrical couplings have on the ground state energy and other important physical properties by considering both the sets of degrees of freedom separately, and then, compare the results with a simulation that includes all five degrees of freedom at once. We could have used ethane for this investigation, but we selected propane because the analysis we present here would be very cumbersome to carry out with the method of extended Lagrangian, or with vector spaces. For ethane, the simulation in the torsional space alone provides valuable insight and a convenient mean of comparing our DMC simulations results with a vector space computation of the ground state energy and wavefunction. For ethane, we use the following potential energy model
Figure 2. Convergence of the ground state energy for propane and ethane is shown by graphing the ground state energy estimate versus the step size used in the DMC simulation. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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The metric tensor in the torsional subspaces for ethane and propane is diagonal and independent of the values of the dihedral angles, therefore, a number of higher order terms that enter generally in the expression of the Green function propagator are absent. The homotopy of the space (the ring is a multiply connected set of spaces) and the boundary conditions that the propagator must satisfy, may contribute additional linear terms in general, but these are clearly too small to be observed in ethane and propane. All these conditions create the observed nonlinear behavior of the ground state energy with respect to . Furthermore, systems with larger number of torsions create more complicated expressions for the metric tensor in , and this object need not be constant. Therefore, in general the procedure outlined in this article is expected to converge linearly in a Ramachandra space of arbitrary dimension.
Last, we are able to quantify the excess energy caused by the coupling block from the three ground state energies of propane,
MCPI simulations of propane
For the stochastic evaluation of the Feynman path integral, there are difficulties in using angular variables. For instance, the random series expansion of the Brownian bridge becomes much more complicated to implement into an algorithm when periodic boundary conditions are used on the values of the angles. In trying to use the Euler angle θ in conjunction with Eq. (32), the random variables (the coefficients of the path) must be constrained themselves to values that produce the correct range for the angle θ. Fortunately, we have found a set of coordinates that can map spaces like the ellipsoid of inertia , or the space for the example problem. These are the stereographic projections. The toroid spaces created by the dihedral angles can be easily remapped by defining , as follows,
for all the n − 1 dihedral angles. The transformation of variables from a dihedral angle to the projection is identical to that used for the particle in a ring of unit radius, and for propane, with a mass equal to . The expressions in Eq. (65) follow from straightforward trigonometric identities. The transformation of variables from Euler angles to projections is slightly more involved. One begins by defining a 4D space of quaternions, constrained to the surface of a three-sphere with unit radius, that is, . The conversion map from Euler angles to quaternion coordinates that satisfies this equation of constraint is,
All these equations can be coded readily, and the computation of the metric tensor of propane in is fast. Therefore, the technical problems surrounding Eq. (32) for the MCPI simulations of propane in are eliminated, if we perform our random walk with the stereographic projections and the auxiliary random path coefficients. The results of several MCPI simulations on the pentadimensional propane molecule in an external field are presented in Figure 3.
Figure 3. Reweighted random series simulation of propane in mapped with stereographic projection coordinates. The average energy as a function of is graphed for several values of km [cf. Eqs. (32)-(42)].
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