Quantum mechanics potential energy surface

Dr. McLaughl, D.L. Thompson, Ab-initio dynamics HeH+ -h H2 He -h (C2v) classical trajectories using a quantum-mechanical potential-energy surface, /. Chem. Phys. 59 (8) (1973) 4393-4405. 

Waldher, B., Kuta, J., Chen, S., Henson, N., Qark, A.E. ForceFit a code to fit classical force fields to quantum mechanical potential energy surfaces. J. Comp. Chem. 12, 2307—2316 (2010)... 

Exercise C.2 Consider case 3 of Exercise C.lb. Evaluate the gradient and the Hessian at c = 3.3, y = 1.8 and solve the Newton-Raphson equation of Exercise C.la for xi, yi). Comment on this. In practical cases, the quantum mechanical potential energy surface is not quadratic, especially for regions as far away from the neighborhood of the minimum as (jc, y) in this example. 

The general emphasis in force field development is towards transferrable force fields, where the functional form and the values of associated parameters can be used in a wide variety of molecules and crystals. As the parameters are developed empirically, transferability implies a degree of reliability and confidence that the parameters will work for crystals for which they were not specifically parameterised. In a recent development of the so-called tailor-made force field, it was pointed out that for the specific case of crystal structure prediction, the force field does not need to be transferable and that in fact there are some important advantages to having a force field derived specifically for the molecule of interest. Given sufficiently accurate information from quantum mechanical calculations, the tailor-made force field can be obtained by fitting to the quantum mechanical potential energy surface. Neumann defined a number of quantum mechanical data sets which represented both the non-bonded and bonded interactions in the crystal. The parameters of the force field were then optimised to fit these data sets. The quantum mechanical method chosen for the calculations was the DFT(d) method which will be described below. 

Most of the details of the trajectory calculations are the same as described previously. All of the calculations are quasiclassical, i.e,y the vibrational and rotational action variables are restricted to have the correct quantized values at the start of the collision but, except for this and the final-state analysis after the collision, the internuclear motion is purely classical under the influence of the quantum mechanical potential energy surface.The only initial states considered were para-hydrogen states, i.e., states with even rotational quantum numbers. As will be indicated, some runs are for a given initial vibration-rotation state while other runs are for vibration-rotation states selected by Monte Carlo methods from the distribution of quantized states that has the desired temperature. Similarly, the relative translational energy is sometimes fixed for a run while for other runs a random selection is made from a predetermined distribution characteristic of a temperature. In the latter case importance sampling is sometimes used to improve the convergence of the calculated quantities such as the... 

In direct contrast to simulations of physical properties dominated by inter-particle forces, the second problem class (class B) concerns interactions among intramolecular and inter-molecular degrees of freedom in an event which produces a chemical change in the participants. Severely limited by the need for an accurate quantum mechanical potential energy surface, simulations in this area typically involve classical trajectory (i.e., MD) calculations for the simplest chemical reactions... 

U. Burkert and N. Allinger, Molecular Mechanics (ACS, Washington, DC, 1982). M. B. Darkhovskii and A. L. Tchougreeff On unification of quantum chemistry and molecular mechanics. Potential energy surfaces of transition metal complexes as exemplified by spin transition in cis- [Fe(NCS)2(bipy)2], Khim. Fiz. 18, 73-79 (1999) [Chem. Phys. Reports 18, 149 (1999)]. 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

Figure Al.6.10. (a) Schematic representation of the three potential energy surfaces of ICN in the Zewail experiments, (b) Theoretical quantum mechanical simulations for the reaction ICN ICN [I--------------...

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Olsen R A, Philipsen P H T, Baerends E J, Kroes G J and Louvik O M 1997 Direct subsurface adsorption of hydrogen on Pd(111) quantum mechanical calculations on a new two-dimensional potential energy surfaced. Chem. Phys. 106 9286... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

The origin of a torsional barrier can be studied best in simple cases like ethane. Here, rotation about the central carbon-carbon bond results in three staggered and three eclipsed stationary points on the potential energy surface, at least when symmetry considerations are not taken into account. Quantum mechanically, the barrier of rotation is explained by anti-bonding interactions between the hydrogens attached to different carbon atoms. These interactions are small when the conformation of ethane is staggered, and reach a maximum value when the molecule approaches an eclipsed geometry. 

HyperChem can calculate transition structures with either semi-empirical quantum mechanics methods or the ab initio quantum mechanics method. A transition state search finds the maximum energy along a reaction coordinate on a potential energy surface. It locates the first-order saddle point that is, the structure with only one imaginary frequency, having one negative eigenvalue. 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

A fully theoretical calculation of a potential energy surface must be a quantum mechanical calculation, and the mathematical difflculties associated with the method require that approximations be made. The first of these is the Bom-Oppenheimer approximation, which states that it is acceptable to uncouple the electronic and nuclear motions. This is a consequence of the great disparity in the masses of the electron and nuclei. Therefore, the calculation can proceed by fixing the location... 

Consider a reactant molecule in which one atom is replaced by its isotope, for example, protium (H) by deuterium (D) or tritium (T), C by C, etc. The only change that has been made is in the mass of the nucleus, so that to a very good approximation the electronic structures of the two molecules are the same. This means that reaction will take place on the same potential energy surface for both molecules. Nevertheless, isotopic substitution can result in a rate change as a consequence of quantum effects. A rate change resulting from an isotopic substitution is called a kinetic isotope effect. Such effects can provide valuable insights into reaction mechanism. 

Potential energy surfaces are also central to our quantum-mechanical studies, and we are going to meet them again and again in subsequent chapters. Let s start then with Figure 3.1, which shows H2+. We are not going to be concerned with the overall translational motion of the molecule. For simphcity, I have drawn a local axis system with the centre of mass as the origin. By convention, we label the intemuclear axis the z-axis. 

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