Classical mechanics molecular properties

We will in this section consider the mathematical structure for computational procedures when calculating molecular properties of a quantum mechanical subsystem coupled to a classical subsystem. Molecular properties of the quantum subsystem are obtained when considering the interactions between the externally applied time-dependent electromagnetic field and the molecular subsystem in contact with a structured environment such as an aerosol particle. Therefore, we need to study the time evolution of the expectation value of an operator A and we express that as... 

Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

In addition to the described above methods, there are computational QM-MM (quantum mechanics-classic mechanics) methods in progress of development. They allow prediction and understanding of solvatochromism and fluorescence characteristics of dyes that are situated in various molecular structures changing electrical properties on nanoscale. Their electronic transitions and according microscopic structures are calculated using QM coupled to the point charges with Coulombic potentials. It is very important that in typical QM-MM simulations, no dielectric constant is involved Orientational dielectric effects come naturally from reorientation and translation of the elements of the system on the pathway of attaining the equilibrium. Dynamics of such complex systems as proteins embedded in natural environment may be revealed with femtosecond time resolution. In more detail, this topic is analyzed in this volume [76]. 

The development of theoretical chemistry ceased at about 1930. The last significant contributions came from the first of the modern theoretical physicists, who have long since lost interest in the subject. It is not uncommon today, to hear prominent chemists explain how chemistry is an experimental science, adequately practiced without any need of quantum mechanics or the theories of relativity. Chemical thermodynamics is routinely rehashed in the terminology and concepts of the late nineteenth century. The formulation of chemical reaction and kinetic theories take scant account of statistical mechanics and non-equilibrium thermodynamics. Theories of molecular structure are entirely classical and molecular cohesion is commonly analyzed in terms of isolated bonds. Holistic effects and emergent properties that could... 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

Molecular mechanics lies conceptually between quantum mechanics and classical mechanics, in that data obtained from quantum mechanical calculations are incorporated into a theoretical framework established by the classical equations of motion. The Bom-Oppenheimer approximation, used in quantum mechanics, states that Schrddinger s equation can be separated into a part that describes the motion of electrons and a part that describes the motion of nuclei, and that these can be treated independently. Quantum mechanics is concerned with the properties of electrons molecular mechanics is concerned with the nuclei, while electrons are treated in a classical electrostatic manner. 

The molecular mechanics (MM) or force field method is an empirical method based on classical mechanics and adjustable parameters. It has the disadvantage of being limited in its application to certain kinds of compounds for which the required parameters have been determined (experimentally or by theoretical calculations). Its advantage is a considerably shorter computation time in comparison with other procedures having the same purpose. This method has been shown to be very reliable and efficient in determing molecular geometries, energies, and other properties for a wide variety of compounds. 

Molecular Mechanics. Molecular mechanics (MM), or empirical force field methods (EFF), are so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a structure to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical eneigy, but a classical mechanical one). The sum of the components is called the force field eneigy, or steric energy, which also routinely includes the electrostatic eneigy components. Typically, the steric energy is expressed as... 

This chapter introduces the core concepts of what is now called classical physics (mechanics, electricity, magnetism, and properties of waves). Today we think of classical physics as a special case in a more general framework which would include relativistic effects (for particles with velocities which approach the speed of light) and quantum effects, which are needed for a complete description of atomic behavior. Nonetheless, we will find that this classical perspective (with a few minor corrections) serves as an excellent starting point for understanding many atomic and molecular properties. 

From its inception, the combined Quantum Mechanics/Molecular Mechanics (QM/MM) method [1-3] has played an important roll in the explicit modeling of solvent [4], Whereas Molecular Mechanics (MM) methods on their own are generally only able to describe the effect of solvent on classical properties, QM/MM methods allow one to examine the effect of the solvent on solute properties that require a quantum mechanical (QM) description. In most cases, the solute, sometimes together with a few solvent molecules, is treated at the QM level of theory. The solvent molecules, except for those included in the QM region, are then treated with an MM force field. The resulting potential can be explored using Monte Carlo (MC) or Molecular Dynamics (MD) simulations. Besides the modeling of solvent, QM/MM methods have been particularly successful in the study of biochemical systems [5] and catalysis [6],... 

The MCSCF/CM response method provide procedures for obtaining frequency-dependent molecular properties when investigating a molecule coupled to a structured environment and the basis is achieved by treating the quantum mechanical subsystem on a quantum mechanical level and the structured environment as a classical subsystem described by a molecular mechanics force field. The important interactions between the two subsystems are included directly in the optimized wave function. 

---