Energy Expressions

As mentioned in Section IV, the position of every atom is first expressed in the same coordinate system, and then the calculations of the conformational energies are carried out. The calculations have been basically of two types, depending on whether only the hard-sphere potential or the more complete energy expressions (described in Section V) are used. 

As far as the minimization procedure itself is concerned, allowance is made for the variation of all of the dihedral angles of the backbone and side chains. At present, the only known way to locate the global minimum is to try to find all local minima and choose the one with the lowest energy. 

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The more conventional quantum chemistry methods provide their working equations and energy expressions in temis of one- and two-electron integrals over the final MOs ([Pg.2185]

Since the nuclear coordinates are expanded according to Eq. (5), we can write the derivatives in the kinetic energy expression as... 

Free Energy Function The conformational free energy was estimated by the following energy expression ... 

I Liming now to the numerator in the energy expression (Equation (2.95)), this can be broken do, n into a series of one-electron and two-electron integrals, as for the hydrogen molecule, l ach of these individual integrals has the general form ... 

Two potential energy expressions used for van der Waals interactions are the Fennard-Jones 6/12 potential function or some modification thereof. 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

Inserting these energy expressions above yields ... 

The hydrogenie atom energy expression has no 1-dependenee the 2s and 2p orbitals have exaetly the same energy, as do the 3s, 3p, and 3d orbitals. This degree of degeneraey is only present in one-eleetron atoms and is the result of an additional symmetry (i.e., an additional operator that eommutes with the Hamiltonian) that is not present onee the atom eontains two or more eleetrons. This additional symmetry is diseussed on p. 77 of Atkins. 

The energy expression consists of the sum of simple classical equations. These equations describe various aspects of the molecule, such as bond... 

Stretching, bond bending, torsions, electrostatic interactions, van der Waals forces, and hydrogen bonding. Force fields differ in the number of terms in the energy expression, the complexity of those terms, and the way in which the constants were obtained. Since electrons are not explicitly included, electronic processes cannot be modeled. 

Terms in the energy expression that describe a single aspect of the molecular shape, such as bond stretching, angle bending, ring inversion, or torsional motion, are called valence terms. All force fields have at least one valence term and most have three or more. 

Terms in the energy expression that describe how one motion of the molecule affects another are called cross terms. A cross term commonly used is a stretch-bend term, which describes how equilibrium bond lengths tend to shift as bond angles are changed. Some force fields have no cross terms and may compensate for this by having sophisticated electrostatic functions. The MM4 force field is at the opposite extreme with nine different types of cross terms. 

Molecular mechanics methods are not generally applicable to structures very far from equilibrium, such as transition structures. Calculations that use algebraic expressions to describe the reaction path and transition structure are usually semiclassical algorithms. These calculations use an energy expression fitted to an ah initio potential energy surface for that exact reaction, rather than using the same parameters for every molecule. Semiclassical calculations are discussed further in Chapter 19. 

Molecular dynamics is a simulation of the time-dependent behavior of a molecular system, such as vibrational motion or Brownian motion. It requires a way to compute the energy of the system, most often using a molecular mechanics calculation. This energy expression is used to compute the forces on the atoms for any given geometry. The steps in a molecular dynamics simulation of an equilibrium system are as follows ... 

Compute the forces on each atom from the energy expression. This is usually a molecular mechanics force held designed to be used in dynamical simulations. 

Quantitative energy values are one of the most useful results from computational techniques. In order to develop a reasonable energy expression when two... 

Likewise, a three-layer system could be broken down into small, medium, and large regions, to be computed with low, medium, and high levels of theory (L, M, and // respectively). The energy expression would then be... 

The more recently developed methods define an energy expression for the combined calculation and then use that expression to compute gradients for a geometry optimization. Some of the earlier methods would use a simpler level of theory for the geometry optimization and then add additional energy corrections to a final single point calculation. The current generation is considered to be the superior technique. 

The molecular mechanics force fields available include MM+, OPLS, BIO+, and AMBER. Parameters missing from the force field will be automatically estimated. The user has some control over cutoff distances for various terms in the energy expression. Solvent molecules can be included along with periodic boundary conditions. The molecular mechanics calculations tested ran without difficulties. Biomolecule computational abilities are aided by functions for superimposing molecules, conformation searching, and QSAR descriptor calculation. 

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

For hquid systems these surface energies expressed in mj/m are numerically equivalent to the surface tensions in mN/m(= dyn/cm). If the adhesive is phase 1 and the release coating is phase 2, then the spreading coefficient, S, of 1 on 2 is as given in equation 2. 

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