MOLECULAR POTENTIAL ENERGY QUANTUM MECHANICAL PROBLEM

MOLECULAR POTENTIAL ENERGY QUANTUM MECHANICAL PROBLEM 

In all the contexts of molecular modeling reviewed briefly above, it was taken for granted that the quantities on the right hand side of the above equations - the potential energies of molecular systems in their corresponding electronic states considered as functions of the system variables Um(q) = Um(R ay(q)) i.e. the PES - exist and 

The procedure begins with writing down the quantum mechanical Hamiltonian for a molecular system (electrons + nuclei) in the coordinate space  

In these expressions written with use of so-called atomic units (elementary charge, electron mass and Planck constant are all equal to unity) RQs stand as previously for the spatial coordinates of the nuclei of atoms composing the system r) s for the spatial coordinates of electrons Mas are the nuclear masses Zas are the nuclear charges in the units of elementary charge. The meaning of the different contributions is as follows Te and Tn are respectively the electronic and nuclear kinetic energy operators, Vne is the operator of the Coulomb potential energy of attraction of electrons to nuclei, Vee is that of repulsion between electrons, and Vnn that of repulsion between the nuclei. Summations over a and ft extend to all nuclei in the (model) system and those over i and j to all electrons in it. 

The variables describing electrons and nuclei are termed electronic and nuclear. For the majority of problems which arise in chemistry, the nuclear variables can be thought to be the Cartesian coordinates of the nuclei in the physical three-dimensional space. Of course the nuclei are in fact inherently quantum objects which manifest in such characteristics as nuclear spins - additional variables describing internal states of nuclei, which do not have any classical analog. However these latter variables enter into play relatively rarely. For example, when the NMR, ESR or Mossbauer experiments are discussed or in exotic problems like that of the ortho-para dihydrogen conversion. In a more common setting, such as the one represented by the 

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MOLECULAR POTENTIAL ENERGY QUANTUM MECHANICAL PROBLEM... 

For application in the quantum mechanical problem of determining the atomic or molecular electronic structure we need the potential energy function V r) = often simply called the potential for brevity. 

The calculation of potential energy surfaces of interacting atoms is a complicated quantum-mechanical problem. It was solved for very simple systems only (see [132, 134, 241, 322, 413]). Consequently, along with the ab initio calculations there are may semiempirical methods based on theoretical correlations between readily measurable molecular parameters. Moreover, direct models of the potential surfaces are widely used [243]. 

If the reaction cross section, Q e, U j is known, the rate constant for the corresponding chemical reaction can be calculated from (8.6) and (8.11). To interpret the details of molecular beam experiments even more information is needed. The hard-sphere model was obviously far too naive to give reliable estimates of the reaction cross section. Improvement, by considering the dynamics of reaction across a realistic potential-energy surface such as that described in Section 9.2, is a formidable quantum mechanical problem. As already mentioned it has not been solved, except for low-energy H + Hg chemical reaction. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

The applications of NN to solvent extraction, reported in section 16.4.6.2., suffer from an essential limitation in that they do not apply to processes of quantum nature therefore they are not able to describe metal complexes in extraction systems on the microscopic level. In fact, the networks can describe only the pure state of simplest quantum systems, without superposition of states. Neural networks that indirectly take into account quantum effects have already been applied to chemical problems. For example, the combination of quantum mechanical molecular electrostatic potential surfaces with neural networks makes it possible to predict the bonding energy for bioactive molecules with enzyme targets. Computational NN were employed to identify the quantum mechanical features of the... 

Turning to molecular physics, we note first papers by Ya.B. which are close to the problem of phase transition. We begin with the theory of interaction of an atom with a metal (11). By applying quantum-mechanical perturbation theory to the interaction of the virtual dipole moment of an atom with conducting electrons of the metal, the dependence on distance of the attractive force of the atom to the surface is obtained. The calculation led to a slow, r2, law for the potential energy decay with distance. This paper was published in 1935, and for many years remained essentially the only one devoted to the subject. 

Computational approaches to potential energy may be divided into two broad categories quantum mechanics (Hehre et al, 1986) and molecular mechanics (Berkert and Allinger, 1982). The basis for this division depends on the incorporation of the Schrodinger equation or its matrix equivalent. It is now widely recognized that both methods reinforce one another in an attempt to understand chemical and biological behavior at the molecular level. From a purely practical standpoint, the complexity of the problem, time constraints, computer size, and other limiting factors typically determine which method is feasible. 

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