New Models for Aqueous Systems: Construction of Vibrational Wave Functions for use in Monte Carlo Simulations.

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1 New Models for Aqueous Systems: Construction of Vibrational Wave Functions for use in Monte Carlo Simulations. Maria A. Gomez and Lawrence R. Pratt T-12 and CNLS Theoretical Division Los Alamos National Laboratory Los Alamos, NM There has been great interest recently in simulation of proton exchange events in liquid water. The obvious limitations of rigid water models have led to a long and continuing history of flexible and dissociative water models for simulation of aqueous systems. One idea for improving the fidelity of simulations of chemistry in aqueous solution is to consider the model from the perspective of the quantum mechanical distribution of internal coordinates. For example, an accurate and compact wave function might be used as a model in a simulation to sample configurations of the system. Here we investigate constructing simple intramolecular wave functions for aqueous species. In particular, we describe some new Monte Carlo techniques for construction of compact wave functions for the internal atomic motion of the D 3 O + ion. Proton Transfer Water, water everywhere, Nor any drop to drink. Samuel Coleridge, Rhyme of the Ancient Mariner. From acid/base chemistry in water to synthetic proton conductors to photosynthesis, proton transfer events play a pivotal role. Proton conduction though polymer membranes, zeolites, ceramics, aluminas, and metal oxides has many potential applications to fuel cells, catalysis, steam electrolysis, solid state gas sensors, and solid state batteries. Without doubt, proton conduction is a crucial phenomenon spanning a wide variety of areas [1]. Despite its importance, even the simplest reaction, proton transfer in water, is not completely understood. Bernal and Fowler originally proposed the existence of a connected hydrogen bond pathway such as seen in Fig. 1. An excess proton on one side of this pathway causes a sequence of proton shifts and results in an excess proton in the other side of the chain.

2 Fig. 1. A proton on one side of the chain causes a sequence of proton shifts. Today, although the general idea is accepted the details are still debated. Does the proton transfer occur via a hydronium or a Zundel or an Eigen cation? (See Fig. 2.) Just what species is involved is important because it determines energy barriers and hence pathways to reactions. Studies suggest that a definitive answer to these question needs to include quantum dynamical treatment of the proton [2, 3]. Fig. 2. Various forms of a solvated proton. Modeling of Chemistry in Water Flexible and dissociative models for simulation of liquid water have often been used in classical statistical mechanical studies of aqueous materials. They present technical advantages for carrying out simulations and are of the essence where dissociation of water molecules is necessary to the chemistry being studied. However, since studies suggest that quantum mechanics plays a non-negligible role in

3 proton transfer [2, 3], a simple way to include the effects of quantum mechanics is desired. Our idea is to attempt to produce an accurate, compact wave function that might be incorporated into simulation tools to sample configurations of the system. This idea is the basis of the present contribution. Of course, such wave functions should be of interest for their own sake. One background point of conceptual interest: the vast majority of simulation calculations assume classical statistical mechanics. Simulations may be carried out on a potential generated by electronic structure theory but atoms move as point particles classically on this surface, i.e., vibrational excitations are generally not considered. By and large, these quantum mechanical effects are incorporated only implicitly through empirical parameterizations of potential energy functions and through constraints such as rigid approximations to internal degrees of freedom. Hydrogen however has a very small mass and so this point picture is not completely appropriate. Schrödinger s equation instead suggests that we look at probability distributions of the atoms and follow these in space. These probability distributions are the square of wave functions. Methods like Diffusion Monte Carlo (DMC) [4] find wave functions, i.e., eigenfunctions of Schrödinger s equation. The ground state wave function is the appropriate probability distribution at zero temperature. (See Fig. 3.) Fig. 3. Methods used to study molecular systems and the temperatures at which they are valid. At any given temperature only some of the eigenstates of the Schrödinger equation are accessible. Path integral [5] approaches incorporate only the relevant states. In this case, each atom is replaced by a path and statistical averages are obtained from distributions of paths at some temperature. At high temperatures, the classical limit is reached and only a single point is needed for the path. At low temperatures, many points are needed and the calculation becomes computationally expensive. It would be nice to have a less expensive technique just for these types of situations. (See Fig. 3.)

4 Goal We would like to do Monte Carlo simulations that sample from a simple probability distribution. Classically, molecules move on the thermally accessible regions of a potential energy surface. Their distribution on this surface is the Boltzmann distribution,, where V(x) is the potential. Quantum mechanically, however, their distribution depends on thermally accessible quantum states. For some systems, sampling from just a few vibrational wave functions ( relevant quantum mechanics. ) is sufficient to capture the Our goal is to determine the complication and accuracy in constructing wave functions that might be transferable and useful in studies of chemistry in water. We seek high accuracy in compact wave functions for the ground state and a few excited states of solutes that raised quantum mechanical questions. Model Wave Functions for Atomic Motion Instead of using the Schrödinger equation to obtain wave functions, it is convenient to use its corresponding integral equation. In this format, a new quantity is introduced namely, the density matrix or the propagator. It is easy to see that the wave functions are eigenfunctions of the density matrix. Sethia, Sanyal, and Singh [6] developed a useful method for obtaining the natural orbitals of the density matrix in 1-D. If the density matrix were known, its natural orbitals could be obtained from the propagator equation by following Mark Kac s advice: be wise, discretize! In fact, the short time or high temperature asymptotic limit of the density matrix is known analytically. At short time or high temperature, it is a great approximation to the propagator. As a result, this asymptotic limit has been a cornerstone in methods such as DMC. Here too, using this asymptotic limit yields numerically exact orbitals except for a time step error. For higher dimensional problems, we can exploit the analytically known asymptotic density matrix to produce the necessary reasonable pair orbitals after tracing out all other degrees of freedom. Basically, we will focus on a particular bond, OD, and trace over the rest of the molecule. Tracing is essentially performing integrals via Monte Carlo. After tracing over the rest of the molecule, we end up with a reduced density matrix that can be diagonalized to obtain pair functions for the ground and a few excited

5 states of the OD vibration. Similarly, pair functions for DD vibrations can be found. Our wave functions will be comprised of a product of OD pair functions and a product of DD pair functions. It is interesting to draw analogies with electronic structure. Viewing this problem from a Hartree-Fock perspective. The oxygen and deuteriums play the roles of nucleus and electrons respectively. The single particle orbitals are the natural OD orbitals. Finally, the DD functions play the role of introducing pair correlation effects. Later, we will consider how to include 3 and 4 body correlations. Accuracy of Pair-Product Wave Functions Using the variational principle and some distribution theory, we develop a measure of how well optimized are our pair wave functions? First, we would like to find the extremum in the energy. Simple manipulations of that extremum and expressing the wave function as an exponential of pair interactions lead to this stationary requirement. Note that this stationary requirement is satisfied for the exact ground state function since the local energy is spatially constant. This stationary requirement essentially removes correlation between the local energy and the two particle joint density operator. Any deviation from this stationary requirement is defined as the error of the wave function. To get a fractional error, we divide by the pair distribution.

6 Figure 4 shows the fractional error in the OD pair distribution. Notice, that the errors are smaller than 1.5 %. Comparing the fractional error with the pair distribution, we see that the largest errors are in the tails of the orbitals. Fig. 4. The OD pair distribution and its fractional error. Correlations Beyond Pairs The pair product wave functions are suitable to use within variational Monte Carlo, including excited states, and density matrix Monte Carlo calculations [7, 8]. Together with the pair product wave functions, the traditional variational theorem permits identification of wave function features with significant potential for further optimization. Such variables are principal candidates for construction of explicitly correlated wave functions. Consideration of several natural possibilities for such structural variables identified the the vector triple product u=r OD1 * (r OD2 X r OD3 ) as more correlated with the local energy of the pair product wave function than any other combination considered for the D 3 O + ion. Fig. 5. The wave function distribution of this vector triple product (solid line) and the optimization deficit (dashed line).

7 Note that there is significant correlation at u=0. This suggests adding a product of the form to the pair product wave function. The pair product wave function with moving and scaling parameters times this new multiplicative many-body factor was optimized using Variational Monte Carlo (VMC). The energies obtained are about as good as the VMC result with mixing of 9 states. The new distribution and optimization deficit for the OSS potential are shown in Fig. 6. Fig. 6.The correlated, optimized wave function distribution of u is shown by the solid line. The new optimization deficit is the dashed line. Using pair product wave functions with this correlation piece as the basis in a McMillan VMC calculation with 9 functions reduced the ground state energy errors to 5-6%. Conclusion A pair product wave function for D 3 O + has been obtained from an approximate density matrix. This pair product wave function captures two body interactions fairly well. VMC includes correlation by mixing in excited states. However, it takes about 99 states to get to an error of 4-6%. Density Matrix Monte Carlo gets accurate energies with 9 wave functions. However, it does not provide a simple wave function. A compact ground state function is obtained by using the variational principle to identify the most significant many-body terms and including it directly into the wave function. For the D 3 O + ion the most significant such variable was the vector triple product, u=r OD1 * (r OD2 X r OD3 ). Variational Monte Carlo with 9 of these correlated functions yields a ground state wave function with an error of 5-6% in the zero point energy.

8 Future Directions Having constructed wave functions for small molecules, we can now explore how these might be transferable and useful in studies of proton transfer. Ideally, we would like to do Monte Carlo simulations that include the quantum mechanics of small molecules present in larger systems while sampling from a simple probability distribution. This Research Highlights is based on the paper: M. A. Gomez and L. R. Pratt, Construction of simulation wave functions for aqueous species: D 3 O +, J. Chem. Phys. 109, 8783 (1998). References 1. P. Colomban. Proton Conductors: Solids, Membranes and Gels: Materials and Devices (Cambridge University Press, Great Britain, 1992). 2. M. Pavese, S. Chawla, J. Lobaugh, and G. A. Voth. J. Chem. Phys. 107, 7428 (1997). 3. M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello. Science 275, 817 (1997). 4. B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, River Edge, NJ, 1994). 5. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., New York, 1965). 6. A. Sethia, S. Sanyal, and Y. Singh, J. Chem. Phys. 93, 7268 (1990). 7. D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988). 8. B. Bernu, D. M. Ceperley, and W. A. Lester, Jr., J. Chem. Phys. 93, 552 (1990).