Most philosophical debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of structures be predicted from quantum theory without assuming their existence or invoking classical concepts? I will argue that the answer is yes, contrary to much of the philosophical literature, which relies heavily on the widespread use of the Born-Oppenheimer approximation (BOA) in quantum chemistry calculations. However, the fact that these arguments for irreducibility are weak does not mean that emergence (defined in terms of novelty) is not central to chemistry.
In a previous post, I discussed recent work showing how the BOA is not necessary for quantum chemistry and that molecular structure can be defined independently of it.
However, since the BOA plays a central role in the philosophical arguments, it is worth reviewing what it is and what it does and does not assume or mean.
In 1927, Born and Oppenheimer introduced an approximation to allow the solution of the full quantum equations for electrons interacting with charged nuclei. Without the BOA, much of theoretical chemistry and solid-state physics would be incredibly difficult in practice. The approximation is based on the separation of time and energy scales associated with electronic and nuclear motion. It leads to the concept of potential energy surfaces for electronic states. They define an effective theory for the dynamics of the atomic nuclei in a molecule or solid.
The full Hamiltonian (given earlier) can be denoted by
where the first term is the kinetic energy operator for the nuclei. In the Born-Oppenheimer approximation (BOA) the full wavefunction is written as a product of a nuclear wavefunction and an electronic wavefunction.
Substituting this in the eigenvalue equation for the full Hamiltonian leads to separate eigenvalue equations for the electronic and nuclear wavefunctions, assuming terms depending on gradients with respect to R of the electronic terms can be neglected.
In the first equation, the nuclear co-ordinates appear as parameters not as operators. This is central to the philosophical debates.
The second equation can be viewed as an effective Hamiltonian for the nuclear degrees of freedom. The function E_e(R) defines the potential energy surface of the molecule.
In the BOA the nuclear probability distribution defined above is
As discussed in the earlier post, the structure of many molecules can be defined in terms of the value of R at which the probability is maximum.
I make four points about the BOA that are relevant to philosophical debates about whether molecular structure is predictable in a logically consistent manner from quantum theory.
1. The BOA does treat the nuclear degrees of freedom quantum mechanically. They are described by the nuclear wavefunction Phi(R), which is determined by the second eigenvalue equation. Consequently, the BOA does not violate Heisenberg’s uncertainty principle, contrary to some claims in the philosophy literature.
2. The BOA is not ad hoc. Corrections to it can be calculated and have been for many molecules. These corrections are typically small, being of order (me/Mi)^1/2. Exceptions, such as near conical intersections (where the potential energy surfaces for two electronic states touch) are well-known and well-studied.
3. For most small molecules, the results of BOA calculations compare favourably with wave-functions obtained from solutions of the full quantum Hamiltonian. When there are differences, they are largely small quantitative differences. When the differences are qualitative, they have largely been anticipated from knowledge of the limitations of the BOA.
4. The Born-Oppenheimer approximation is an example of a general approach to quantum mechanics problems, discussed by Migdal. Consider a system composed of two subsystems that have dynamics on two vastly different time scales, termed fast and slow. The effects of the fast system on the slow system can be treated by adding a potential energy term to the Hamiltonian operator of the slow system.
In forthcoming posts, I will discuss quantum justifications for classical descriptions of nuclear dynamics on potential energy surfaces and then discuss philosophers' views about the BOA and molecular structure.
No comments:
Post a Comment