Internal Coordinate Quantum Monte Carlo

Very few quantum mechanical problems can be solved exactly; problems in this category include certain types of single oscillators or systems of uncoupled oscillators. To solve the Schrodinger equation for systems of chemical interest (i.e., many-body systems with strongly coupled potential energy surfaces) requires approximate methods. Although many of these methods yield accurate results for small molecules, the computational effort scales so steeply with size as to become intractable for polymer, biological, and other highly interesting systems.

Perhaps the most promising method for solving the Schodinger equation for many body systems is quantum Monte Carlo (QMC). This essentially consists of rewriting the Schrodinger equation as a diffusion equation in imaginary time. This approach has proven very successful in electronic structure and liquid structure problems. Only recently, through the development of internal coordinate QMC (ICQMC), has it become possible to apply this method to polymer and other many-atom chemical systems. The Schrodinger equation can be written which is a diffusion equation with diffusion coefficients Di = h2-

which is a diffusion equation with diffusion coefficients Di = h2-

plus a first order rate term. This can be solved by generating a set of random walkers (also called psi particles) that are diffused according to the diffusion constants and created or annihilated according to the first order rate constant. It can be formally proven that the solution converges to the ground state wavefunction. By using symmetry and orthogonalization methods it is possible to modify this approach to get excited states.

This equation is most suitable for few-body systems. It should be noted that creation/annihilation of random walkers depends only on the total potential energy. In many-body systems this becomes insensitive to individual variations in bond lengths, angles, etc. This means that a larger problem becomes less controllable. The use of importance sampling [19] introduces a level of control that allowed larger systems to be treated. The essential idea of importance sampling is to introduce a guiding function (also called a trial function) ^ and to define $ by ij) = Introducing this trial function and defining = it/h yields

The first and last terms on the right hand side are diffusion and first order terms. The middle term corresponds to what is called drift or quantum force. In every iteration, this term forces each random walker into regions of higher trial wavefunction density, in much the same way that classical forces move atoms in MD simulations. This adds a degree of controllability that allows systems with at least several hundred particles to be treated: the more closely the trial wavefunction resembles the true wavefunction, the faster the calculation will converge.

Since the potential energy surfaces of chemical systems are usually written in terms of internal coordinates such as bend and torsion, it makes sense to write the trial function in terms of internal coordinates also. On physical grounds, we write the trial function as a product

where are the internal coordinates and the are Gaussians. The calculation requires gradients and Laplacians of the internal coordinates. Our expressions for Laplacians are especially important here: because they can be written in terms only of internal coordinates, it is unnecessary to compute and add individual second derivatives. These expressions make it practical to perform ICQMC on modest workstations and high-end PC's.

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