Consistent Embedding: A Theoretical Framework for Multiscale Modeling

Abstract

A fundamental framework for the undertaking of computational science provides clear distinctions between theory, model, and simulation. Consistent embedding provides a set of principles which when appropriately applied can create multi-scale models that capture the physical behavior of more computationally challenging methods within methods that are more easily computed. The consistent embedding methodology is illustrated within the context of brittle fracture for two serial and one concurrent multi-scale modeling examples. The examples demonstrate how predictive modeling hierarchies can be established.

Introduction

Co-workers and I have previously argued that the fundamental framework for computational science is intrinsically independent of the discipline in which it is applied. [1] In Ref. [1], we argued that process of computational science allows for the clear distinction among theories, models, and simulations. In this framework for the understanding of the computational scientist’s task, theory is taken to be comprised of the axioms and interpretive procedure that construct a mathematical description of the physical world. A model, in our way of thinking, is a chosen physical description of a system or class of systems formulated using the concepts of the theory. One commonly used model is Newtonian dynamics, where the system is modeled as a set of particles that move under the influence of interaction potentials by obeying Newton’s second law. The model can then be computational realized in a simulation using a molecular dynamics computer code where the consequences of the choice of initial conditions and interaction potentials are determined using algorithms and rules. Here an algorithm might solve a differential equation or an eigenvalue problem under the constraining rules applying to boundary condition, number of particles, or a prescribed temperature. In this description of computational science, we see that improvements can be made in theory, model, and/or simulation in an attempt to improve the fidelity of the computed result in comparison with experiment.

With the forgoing framework for understanding the task of the computational scientist, it may be well to confront a particular physical problem as an exemplar. In order to elucidate the interplay among theory, model, and simulation, let us consider the generic problem of mechanical failure. The phenomenon of mechanical failure presents challenges for the computational scientist at a number of temporal and spatial scales, as it involves the breaking of chemical bonds at the atomistic scale, and perhaps the interaction of grains at a somewhat larger length and time scales and the radical reshaping of overall structure at the macroscopic length scales over times that might be from fraction of seconds to days, weeks, or years. If we restrict ourselves to considering only brittle fracture, which is rapid and abrupt, then it seems clear that the fundamental event is the rupturing of chemical bonds. From a theory perspective, one must choose a level of quantum mechanical theory for the description of the evolution of electrons and nuclei in the stress fields that lead to fracture. While a most fundamental approach might be to attempt a solution of the time-dependent Schrodinger equation for all particles, it would quickly become evident that such an approach is computationally too demanding for current computational resources. One might also realize that simplifying the theory so that more common quantum chemical techniques, which rely on the Born-Oppenheimer approximation, and treating the (slower) nuclei as moving on the potential energy surface of the (faster) electrons seems a reasonable starting point for our choice of theory. We could now try to proceed by applying quantum chemical theory to the electrons and Newtonian dynamics to the nuclei to examine the problem of brittle fracture.

Another computational bottleneck would then, no doubt, assail us. The implementation of quantum chemical theory for larger and larger numbers of electrons would quickly become prohibitively expensive. Fortunately, from a theory point of view, only those electrons, and their associated nuclei that are near, that is within a couple of chemical bond lengths or a few Angstoms, of the crack tip, are fully involved in the crack propagation. Atoms, that is, nuclei and their associated electrons, which are more remote are not nearly as perturbed by the crack tip. Hence, one sees the possibility of building a multi-scale model by choosing a set of theories to be applied at various length and time scales as measured from the crack tip. For instance, we might chose Born-Oppenheimer quantum chemistry and Newtonian dynamics in a small region around the crack tip, Newtonian dynamics using atomistic potential in a larger region, and a continuum modeling for the remainder of the structure. Implementing these choices of theory would require a multi-scale model.

Multi-scale modeling is frequently divided into concurrent and serial multi-scale modeling. In serial modeling, inputs into models at one scale are generated by computational simulations at another, typically smaller scale. Here I will present an example of training atomistic potentials for Newtonian dynamics from quantum chemical calculations. Concurrent multi-scale modeling implements different theories at a number of length scales and then joins them in a single simulation hierarchy. The seamless joining of multi-scale models is generally challenging as the interface between models must pass all appropriate information while avoiding the generation of artifacts. Concurrent multi-scale models must also take care to obey conservation laws, like mass conservation or energy conservation, for the full system.

Consistent Embedding

The principle that we call consistent embedding dictates that the information that comprises a model at a larger spatial or longer time scale be compatible with the model used at a smaller or shorter scale. This principle is essential to the development of predictive theory and modeling, as the materials that exist on either side of a multi-scale interface must be physically consistent. For example, if we wish to look at the phenomenon of fracture, then the stress-strain relationship for the model at the shorter length scale should, at the very least, display the same small strain behavior, Young’s modulus, as the model used at the longer length scale. Enforcing this kind of constraint, that the Young’s modulus of the quantum chemical and atomistic models be equivalent, up to some controllable error, is an example of consistent embedding. We can develop other criteria, based on the physical properties being modeled, which improve the likelihood that emergent behavior of larger systems is grounded in the theoretical description of the smaller system.

We have now established the computational science framework in which to do develop a multi-scale theory, model, and simulation for brittle fracture. In this context, we illustrate the implementation on a multi-scale model and simulation based on the principle of consistent embedding. In particular, the choice of quantum chemical theory will be made subject to consistent embedding constraints, where higher level quantum chemical theory will be used to train less computationally demanding semi-empirical quantum chemical forms. Here it is important to note that the form of the quantum chemical Hamiltonian used is known as ‘semi-empirical’, but our serial multi-scale training will be based solely on computed results from correlated calculation. A second illustration of consistent embedding principles in a serial multi-scale model will be provided by the training of atomistic potentials solely from quantum chemical calculations. Finally, concurrent multi-scale modeling within consistent embedding principles will be demonstrated using a pseudo-atom termination scheme to facilitate the transfer of information across a quantum chemical/ classical mechanical interface.

Transfer Hamiltonian

The first example of serial multi-scale modeling that we consider is the training of one less computationally intensive quantum chemical method from a more computational intensive method. The name that has been given to this type of quantum chemical training is the Transfer Hamiltonian. [2] Silica, in particular amorphous silica, is the system whose brittle fracture will serve as the exemplar for this serial multi-scale model. For our higher level of quantum chemical theory we choose a method that includes the effects of electron correlation, known as coupled-cluster theory including single and double excitations (CCSD). This highly accurate level of quantum chemical theory is also, of necessity, very computationally demanding. Hence it is necessary to choose a training molecule which exhibits the chemical bonding of interest in the mechanical failure of amorphous silica, but is limited to a relatively small number of atoms. We choose pyrosilicic acid (H6Si2O7) to create a CCSD training set for the ‘semi-empirical’ Hamiltonian. As seen in Fig. 1, the Si-O bond length is varied through compressions and stretches to generate a training set for the Transfer Hamiltonian. In this case, we have chosen to use a neglect of diatomic differential overlap (NDDO) Hamiltonian as our less computationally demanding quantum chemical model. NDDO is one of a set of approximations collectively referred to as zero differential overlap methods. When computers were much less powerful than they are today, these methods were developed to be computationally tractable and used mathematical forms derived from theory. These forms were parameterized to reproduce certain empirical data, e.g. heats of fusion, for simple molecules and the resulting parameterized Hamiltonians were applied to more complex problems with some success. The empirical parameterization of theoretically derived forms came to be known as ‘semi-empirical’ theory. By choosing to parameterize the NDDO Hamiltonian based on high accuracy, ab initio quantum chemistry, CCSD, we remove the empirical information from the procedure and replace it with a more detailed theoretical model. This substitution of theoretical for empirical information characterizes one possible implementation of the consistent embedding framework for the development of predictive modeling. The training of the Transfer Hamiltonian is accomplished using genetic algorithms, which are tuned to reproduce CCSD forces for the training set of molecular geometries. The choice of training on forces is motivated by our interest in stress-strain relations. In the next section, the implications of this Transfer Hamiltonian will be examined in a somewhat larger system.

Figure 1 Pyrosilicic acid is used to create a training set of forces from CCSD for the Transfer Hamiltonian

Small Strain Potential

Pyrosilicic acid was sufficient for the training of the Transfer Hamiltonian, however, to see the effects of using it, we need a somewhat larger model system. Figure 2 shows a silica nanorod which we use to illustrate the next step in our serial multi-scale model. The nanorod is comprised of 108 atoms with two oxygen atoms for each silicon atom, the same ratio as found in silica. Various deformations are used to build a database of forces for training a small strain potential based on Transfer Hamiltonian calculations. Two ionic silica potentials, referred to by their authors’ initials, have found wide use in recent years, BKS [3] and TTAM [4, 5]. These potentials have the same general form and we have chosen to use this form for the parameterization of a small strain potential from Transfer Hamiltonian force data. Again, the parameterization is accomplished using a genetic algorithm. As shown in Table 1, the small strain potential reproduces the Young’s modulus obtained by the quantum chemical Hamiltonian to within a few percent for uniaxial stain along the long axis of the nanorod, while equilibrium bond lengths and bond angles are reproduced to within 2.5%. Details of the construction of the small strain potential can be found elsewhere. [6]

Figure 2 A silica nanorod comprised of 108 atoms, oxygen is green and silicon is gray

Table 1: Young’s modulus comparison among potential and Transfer Hamiltonian

Concurrent Multi-scale Modeling

As a last example, we turn to the topic of concurrent multi-scale modeling. Dealing with the details of the interface is essential in this style of multi-scale modeling and as our target properties relate to stress-strain relations, we must concern ourselves with forces on either side of the interface. However, brittle fracture occurs by the rupture for chemical bonds, so it is also essential that the character of the chemical bonding of the system be preserved as well. The small strain potential presented in the previous section assures that the forces across a quantum chemical/classical mechanical interface are in good agreement for small strains. This has been confirmed by the agreement of the Young’s modulus and equilibrium configurations. In this section, we consider the interaction between the classical mechanical part of the system and the part describe by quantum chemistry. We choose to represent the effect of the remainder of the nanorod in the quantum chemical regime by a truncation scheme which we call pseudoatoms. These pseudoatoms replace oxygen atoms in the full system that serve as the interface between the two styles of treatment. For a typical fracture problem, the quantum chemical treatment would be focused around the crack tip, where the most strained chemical bonds are found.

Psuedoatoms are trained using the pyrosilicic acid molecule shown in Fig. 1. Fig. 3 illustrates that the full system as seen from the quantum chemical viewpoint. In the case of the Transfer Hamiltonian, a fluorine atom has been reparameterized to preserve the equilibrium bond lengths and electron distribution in the remainder of the molecule. More detailed studies of the effect of pseudoatoms in the context of both the Transfer Hamiltonian and Density Functional Theory have been presented elsewhere. [7]

Figure 3 Pseudoatoms (labeled Modified F) are trained to reproduce local effects in the electron density

Conclusions

The distinction among the roles of theory, model, and simulation provides us with insight into ways one might improve our descriptions of the physical world. Using the conceptual framework of consistent embedding, we are able to pose sharp questions with quantifiable answers that can allow us to assess the quality of serial and current multi-scale models and their simulations. Illustrations of serial multi-scale modeling shown here, indicate that less computationally intensive quantum chemical methods can be developed that reflect the quality of more computationally intensive quantum chemical methods to a few percent for chosen properties, for brittle fracture we have concerned ourselves with forces. Further, small strain potentials can be trained, using forms available in the literature, to capture the behavior of quantum chemical methods. Finally, a strategy for concurrent multi-scale modeling has been provided that allows a system to be separated into classical and quantum domains while preserving the fidelity of each to the full system. These ingredients are essential to a predictive modeling capability.