Molecular Structure and Quantum Mechanics

Our other possible locus for emergence in chemistry is molecular structure. Are molecules anything more than systems of charged particles, interacting according to the laws of quantum mechanics? I seem to be in a minority of one in giving that question an affirmative answer, and certainly so when I defend the idea that molecular structure is strongly emergent. Yet I think the unanimity on the other side is baseless, as I have argued elsewhere (see Hendry 2006a, 2010a, 2010b). Because this is familiar territory, I will give these issues a fairly brisk treatment here.

Textbooks of physical chemistry often present the application of quantum mechanics to chemistry as a process that begins with the writing down of a Schrodinger equation for an isolated molecule, purely in terms of the electrons and nuclei present. The aim is to solve the equation and thereby explain the characteristic structures of molecules, which chemists have used to explain the chemical behaviour of substances since the 1860s. When it appeared on the scene in the mid-1920s, quantum mechanics was widely expected to provide a complete account of chemistry. Just a few years later, Paul Dirac famously wrote:

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

(Dirac 1929: 714)

Here, non-relativistic quantum mechanics is assumed to be a sort of “theory of everything” for the motions of electrons and nuclei, and therefore for any molecule. Physicists and philosophers who use that phrase usually mean a theory that could—in principle—explain everything that happens in a system to which it is applied, to the extent that it can be explained. Think of Newton’s laws applied to the planetary motions: natural philosophers since Newton’s time have imagined a God’s-eye-view application of his laws which could be used to predict all future planetary positions, if only we had accurate enough access to their current positions and momenta, plus large enough computers to cope with very detailed and accurate mathematical models of the solar system. A more formal way to put this is to say that in a multi-dimensional configuration space representing the dynamical state of the solar system, the laws governing planetary motions uniquely specify its future evolution, given only its current state. The question of whether molecular structure is strongly emergent is, I think, best understood as the question of whether we have good reasons to think that, from a God’s-eye- view, non-relativistic quantum mechanics is a “theory of everything” in this sense, or whether some looser relationship between the dynamics and the evolution of the system is better supported.

The problem raised by Dirac is that for any chemical system bigger than a hydrogen atom, the Schrodinger equation, the central equation of this theory of everything, is insoluble analytically. This means that approximations must be introduced: known falsehoods that will affect the calculations in well-understood ways. For molecules, this means the Born-Oppenheimer or ‘small oscillation’ approximation. It is worthwhile separating this into two separate moves. First, nuclear and electronic motions are considered as separate (even though electrons and nuclei are known to interact), yielding an overall wavefunction that is a product of nuclear and electronic wavefunctions. In the second step, the nuclei are then assumed to be at rest, on account of their much higher masses, and therefore slower motion. The problem of calculating the wavefunction for the electrons can now be addressed on its own, and the molecule’s energy calculated from that. In the last twenty years or so this problem has increasingly been addressed through density functional theory (DFT), in which the aim is to calculate the electron density, rather than the molecular orbitals of yore. The electronic energy can be calculated for a few nuclear configurations near the (empirically given) equilibrium configuration, and the fact that it is the equilibrium configuration is thus explained, after a fashion: it is the local minimum in a particular region of the potential-energy surface. The problem is that it is explained in a way that seems to undermine the status of non-relativistic quantum mechanics as a theory of everything for molecules, and therefore for chemistry.

Brian Sutcliffe and Guy Woolley (2012) argue that the Born-Oppenheimer approximation should not be called an approximation, because it fundamentally alters important mathematical properties of the equations and their solutions. Sutcliffe and Woolley raise two difficulties, concerning isomers and symmetry properties. Turning to isomers first, the Schrodinger equation for a molecule is fully determined once the nuclei and electrons present are enumerated. This means that isomers, such as ethanol (CH 3 CH2OH) and dimethyl ether (СНзОСН3), mentioned earlier, will share the same Schrodinger equation. But the Born-Oppenheimer equations for ethanol and dimethyl ether are quite different. In applying the Born-Oppenheimer approximation we have moved straight from an equation that applies to both of these molecules to a different equation that applies to just one of them and not the other. How, from a mathematical point of view, did we do that? By putting in by hand the parameters that specify an important difference between the two cases: the nuclear positions. We have explained the geometrical configuration of an ethanol molecule as a local minimum on a particular potential-energy surface. We have explained the geometrical configuration of a dimethyl ether molecule as a local minimum on its particular potential-energy surface. But can we say we have a “theory of everything” that encompasses all of this? We have an equation that, in a sense, allows both possibilities. But that is a very weak way of being a theory of everything. We cannot say that it determines the different possibilities to arise when they do arise. Do we have good reasons to say that quantum mechanics is any more than this, unless supplemented with structural insights from chemistry?

The second problem concerns symmetry. For good physical reasons, the only force appearing in molecular Schrodinger equations is the electrostatic or Coulomb force: other forces are negligible at the relevant scales. But the Coulomb force has spherical symmetry. How, from this slim basis, do we get the great variety of different symmetry properties (chiral (asymmetrical), cylindrical, hexagonal and many more) exhibited by real molecules? In practice the lower symmetries are introduced as part of the Born-Oppenheimer approximation. Surely ‘approximation’ is a misnomer for a procedure that changes the symmetry properties of the problem, introducing the specific symmetry properties we need to understand the behaviour of each kind of molecule on a case-by-case basis.

I now turn to the strong emergentist interpretation of this situation, which I offer not because I am committed to it, but to establish the plausibility of an alternative to ontological reductionism, and its strong interpretation of what it is to be a “theory of everything.” The strong emergentist sees the role of quantum mechanics as much closer to that of thermodynamics: its universal laws deepen our understanding of the behaviour of the systems to which we apply it, but it cannot explain everything. Thermodynamics must always be applied in tandem with other information about the system. Likewise, the Schrodinger equation provides an important framework for studying molecules, because it encompasses all the possibilities, but for that very reason it is implausible to see it as fully specifying the dynamical behaviour of every kind of molecule, given only the charges and masses of the constituent particles. It is too abstract on its own, and too far removed from the particular structures we study in chemistry. It allows too many other, un-chemical possibilities, and we have no general account of the different classes of solutions it does allow, or of the relationships between them. So, instead we simply assume that the known structures exist, and explore the energetic landscape around them to provide an understanding of their dynamical behaviour.

Prospects for Strong Emergence in Chemistry 155

 
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