Molecular modelling Born-Oppenheimer approximation

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

For example, the Born-Oppenheimer approximation is ubiquitous. The separation of the electronic and nuclear motion is most often an excellent approximation. However, it is also fundamental to the concept of molecular structure. The model of fixed nuclei surrounded by electrons which accommodate almost instantly any change in the nuclear positions is basic to qualitative and quantitative discussions of molecular structure. 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

Perturbations are defined as deviations in the quantum-number variation of any observable from that predicted by a zero-order molecular structural model based on the Born-Oppenheimer approximation. This section is intended as an outline of the ingredients of molecular structural models. 

The classical dynamics of molecular models is generated by Hamilton s (or Newton s) equations of motion. In the absence of external, time-dependent forces, and within the Born-Oppenheimer approximation, the dynamics of molecular vibrations, rotations, and reactions conserves the total energy . We therefore restrict our attention in the nonlinear dynamics literature to energy-conserving systems, which are technically referred to as Hamiltonian systems. For the purposes of the present discussion, we restrict our attention to Hamiltonian systems with two degrees of freedom ... 

The middle of the twentieth century marked the end of a long period of determining the building blocks of chemistry chemical elements, chemical bonds, and bond angles. The lists of these are not definitely closed, but future changes will be more cosmetic than fundamental. This made it possible to go one step further and begin to rationalize the structure of molecular systems, as well as to foresee the structural features of the compounds to be synthesized. The crucial concept is based on the Born-Oppenheimer approximation and on the theory of chemical bonds and resulted in the spatial structure of molecules. The great power of such an approach was first proved by the construction of the DNA double helix model by Watson and Crick. The first DNA model was built from iron spheres, wires, and tubes. 

L. A. Nafie and T. H. Walnut, Chem. Phys. Lett., 49, 441 (1977). Vibrarional Circular Dichroism Theory A Localized Molecular Orbital Model. T. H. Walnut and L. A. Nafie, ]. Chem. Phys., 67, 1491 (1977). Infrared Absorption and the Born-Oppenheimer Approximation. I. Vibrational Intensity Expression. T. H. Walnut and L. A. Nafie, J. Chem. Phys., 67, 1501 (1977). Infrared Absorption and the Born-Oppenheimer Approximation. II. Vibrational Circular Dichroism. 

Within these approximations, we now have a cluster of particles described in a mass-centered, nonrotating system. How does molecular structure arise out of this At this point the issues we face are common to both relativistic and nonrelativistic quantum chemistry. To introduce the concept of structure, we have to mathematically divide the particle cluster into a nuclear and an electronic part. In solving the electronic part of the equation, the other, nuclear, part is treated as a classical semi-rigid framework that we can adjust parametrically. The technique most commonly used to achieve this separation is the Born-Oppenheimer approximation, which may be regarded as the lowest order of an expansion about a system of infinite nuclear masses. The problems inherent in the application of this approximation are the same both for relativistic and nonrelativistic models. Whether the same measures should also be taken when the approximation starts to break down has not, to our knowledge, been explored. 

Prepare a molecular model with the positions of all nuclei, kept fixed because of the Born-Oppenheimer approximation. 

The QM/MM Hamiltonian can be used to cany out Molecular Dynamics simulations of a complex system. In the case of liquid interfaces, the simulation box contains the solute and solvent molecules and one must apply appropriate periodic boundary conditions. Typically, for air-water interface simulations, we use a cubic box with periodic boundary conditions in the X and Y directions, whereas for liquid/liquid interfaces, we use a rectangle cuboid interface with periodic boundary conditions in the three directions. An example of simulation box for a liquid-liquid interface is illustrated in Fig. 11.1. The solute s wave function is computed on the fly at each time step of the simulation using the terms in the whole Hamiltonian that explicitly depend on the solute s electronic coordinates (the Born-Oppenheimer approximation is assumed in this model). To accelerate the convergence of the wavefunction calculation, the initial guess in the SCF iterative procedure is taken from the previous step in the simulation, or better, using an extrapolated density matrix from the last three or four steps [39]. The forces acting on QM nuclei and on MM centers are evaluated analytically, and the classical equations of motion are solved to obtain a set of new atomic positions and velocities. 

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