# When can a Computer Simulation act as Substitute for an Experiment? A Case-Study from Chemisty

Johannes Kästner and Eckhart Arnold

### 3.2 The Role of Quantum Mechanics as Comprehensive Background Theory

In their daily work, theoretical chemists are not concerned with the physical theories upon which their simulations are based. They rather focus on the design, selection, and justification of models, approximations, and algorithms which allow to apply the background theory to specific chemical reactions at an affordable computational cost and with sufficient accuracy. Although the physical background theories are usually taken for granted without further question, a few words on their epistemic role are in order, because understanding the epistemic role that the background theories play for these simulations is important for their philosophical justification.

It is an important condition for the kind of simulations that are done in theoretical chemistry that they can rely on background theories that are well-approved and uncontested within the range of application. We can speak of theories that fulfill this condition as “comprehensive theories”. With comprehensive theories we mean theories that correctly describe all causally relevant factors for a well-defined range of phenomena. Or, to put it in simpler words, everything that happens within this range of phenomena happens according to the theory. For such a theory to be well-approved and uncontested, three conditions must hold:

1. The theory has been empirically confirmed in many important instances.

2. The theory has not been disconfirmed in any instances. If any anomalies (i.e., contradictions of the theory with empirical facts and, thus, possible candidates for falsification of the theory) have occurred, then the sub-range of phenomena for which anomalies are to be expected can at least clearly be delineated.

3. Any alternative theory (i.e., a different theory that fully or partly covers the same range of phenomena) has identical consequences as the comprehensive theory within the overlap region of their respective ranges of phenomena and within reasonable bounds of precision.

The theory of quantum mechanics meets these requirements for the description of chemical reactions. Anything that can happen in a chemical reaction is - at least in principle - covered by quantum mechanics. Quantum mechanics can be formulated in different ways. For example, the Schrödinger picture is equivalent to the Heisenberg picture and to Feynman's path integral method (Jen 98). These formulations may be regarded as alternative theories as defined above.

In principle, quantum electrodynamics, i.e. quantum mechanics in a formulation which takes effects of the theory of special relativity into account, should be more accurately used as the “comprehensive” background theory. Then quantum mechanics can already be seen as the first approximation to quantum electrodynamics. An even cruder approximation is to use molecular dynamics as a comprehensive theory for the behaviour of molecules. Its accuracy may be sufficient in cases where no changes in the electronic structure occur. Thus, in these cases it may count as a comprehensive theory even though it is not the most fundamental theory. The range of phenomena it covers is then delineated by the non-occurrence of changes in the electronic structure.

Slightly simplifying, one could also say that a theory is comprehensive for a certain range of phenomena if we can safely expect it to produce results of sufficient accuracy within this range of phenomena. It should be noted that this is not circular, because we need to be sure beforehand (i.e. be able to “safely expect”) that it produces results with sufficient accuracy. It should also be noted that the property of being comprehensive as we understand it here is relational to particular classes or ranges of phenomena. Thus, when speaking of a theory as being comprehensive in this sense we do not mean to say that it is the most general theory about a certain subject matter in the sense in which, say, the theory of relativity is the most general theory about space, time and gravity.

Unfortunately, it is only some areas of some sciences where we really have comprehensive theories. But if we do, it has important epistemological consequences for the validation of models and simulations. Generally speaking, the existence of a comprehensive theory increases the trustworthiness of our models or simulations and it eases the burden of validation. For if a simulation is based on a comprehensive theory then the question whether the simulation's results are valid is reduced to the question whether the approximations and modelling techniques are passable. In case these can sufficiently be justified theoretically, an additional empirical validation of the simulation is not necessary any more, because we assume the theory to be correct and to describe the phenomenon in question comprehensively.

Contrast this with the situation when there is no comprehensive theory. In this case, even if we could justify all approximations and simulation techniques theoretically, we would still need direct empirical validation.[6] For, unless our simulation was confirmed by empirical validation we would not know whether the theoretical assumptions about the simulated phenomena hold true in a particular application case or not.

This situation frequently occurs, for example, in agent-based simulations in economics and other social sciences. Agent-based simulations cannot rely on any uncontested theory, because, typically, there exists either no theory at all for the phenomena simulated by agent-based simulations, in which case these simulations must rely on ad-hoc assumptions. Or there exist different and competing theories, in which case it is hard to justify the choice of a particular one of them without direct validation. Or the theories, like utility theory, are too sparse and have too little content to serve as a comprehensive theory. The contrast that exists between economic theory and physics in this respect is often overlooked, but it has been pointed out very clearly in Cartwright (2009, p.\ 48/49). It is all the more unfortunate, therefore, that proper validation is not yet common practice in the field of agent-based-simulations (Heath et al. 2009).

In science and engineering the favorable case occurs more often in which an uncontested and well-approved background theory does indeed exist. For example, simulations of chemical processes such as the $H_2$-formation simulation described below can rely on quantum mechanics as a comprehensive theory. As we shall see, this greatly reduces the need for direct empirical validation of their results and makes it possible to employ them as experiment surrogates.

[6] By direct validation we mean validation of the simulation set-up with an experiment that closely mimics the simulation set-up.