The Energy Expression

We shall now review the potential energy expression (Eq. []]) in some detail and specify the functions that are usually assumed in standard empirical force fields. 

The first term describes the energy change as a bond stretches (and contracts). It is assumed that such an entity as an intrinsic bond exists and is the 

The reference value bo in Eqs. [3] and [4] is not the equilibrium bond length in any particular molecule (except by chance).This is a common misconception, and indeed the value of bo is sometimes taken from an experimentally observed value. This is not valid. Rather, bo is the bond length of the particular bond in a virtual, unperturbed state, that is, in a hypothetical state where the bond exists as an isolated entity, outside the molecule, and not affected by any external forces. In any real molecule, there are forces from neighboring atoms and interactions with other internals, which stretch or compress the bond from its unperturbed value. The more strained the molecule, in general, the farther the bond in the equilibrium molecular structure will be from the reference value bo. An obvious example of this is the classic example of the C —C bond in the strained molecule tri-r rt-butylmethane, whose equilibrium value is 1.61 A, far from the typical reference value of 1. .32-1.53 used in standard force fields. Usually, for most unstrained molecules, bo is close to the equilibrium values. 

For angle bending, a simple harmonic, spring-like representation is usually used for open, unstrained chains  

However, anharmonicity has been found to be nonnegligible even for normal alkanes  

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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 ... 

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. 

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. 

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 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. 

This expression agrees with the previous expression Eq. (25), for neutral unit cells. If the unit cell is not neutral, the energy expression in Eq. (25) should be modified by adding the tenn — nQ/2L - to the right-hand side, where Q is the total charge in the system. The energy expressions (25) and (28) will then agree. 

Note that the factor of 1/2 has disappeared from the energy expression this is because the G matrix itself depends on P, which has to be taken into account. We write SSg in terms of the Hartree—Fock Hamiltonian matrix h, where... 

The sums run over the occupied orbitals note that we have not made any reference to the LCAO approximation. The energy expression is correct for a determinantal wavefunction irrespective of whether the orbitals are of LCAO form or not. 

Examination of the energy expression 17.7 shows that the polarizability components are the second-order energies and a little analysis shows that (for example)... 

Essentially all force field calculations use Cartesian coordinates of the atoms as the variables in the energy expression. To obtain the distance between two atoms one need to calculate... 

The quality of a force field calculation depends on two things how appropriate is the mathematical form of the energy expression, and how accurate are the parameters. If elaborate forms for the individual interaction terms have been chosen, and a large number of experimental data is available for assigning the parameters, the results of a calculation may well be as good as those obtained from experiment, but at a fraction of the cost. This is the case for simple systems such as hydrocarbons. Even a force field with complicated functional forais for each of the energy contributions contains only a handful of parameters when carbon and hydrogen are the only atom types, and experimental data exist for hundreds of such compounds. The parameters can therefore... 

It should be noted that the unoccupied orbitals do not enter the energy expression (3.32), and a rotation between the viitual orbitals can therefore not change the energy. A... 

We have solved this equation approximately, the results being given in Table II in terms of the constant B in the energy expression — Be2/a0p8. 

The results of the approximate solution of this equation, in terms of the constant C in the energy expression — Ce /dap10, are given in Table III. 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

Vibrational spectroscopy has played a very important role in the development of potential functions for molecular mechanics studies of proteins. Force constants which appear in the energy expressions are heavily parameterized from infrared and Raman studies of small model compounds. One approach to the interpretation of vibrational spectra for biopolymers has been a harmonic analysis whereby spectra are fit by geometry and/or force constant changes. There are a number of reasons for developing other approaches. The consistent force field (CFF) type potentials used in computer simulations are meant to model the motions of the atoms over a large ranee of conformations and, implicitly temperatures, without reparameterization. It is also desirable to develop a formalism for interpreting vibrational spectra which takes into account the variation in the conformations of the chromophore and surroundings which occur due to thermal motions. 

The energy expression (4) applies when the orbitals are orthonormal and in seeking a stationary value it is thus necessary to introduce constraints to maintain orthonormality during a variation. When this is done, the orbitals that give a stationary... 

Thus, once we know the various contributions in equation (5-15) we have a grip on the potential Vs which we need to insert into the one-particle equations, which in turn determine the orbitals and hence the ground state density and the ground state energy by employing the energy expression (5-13). It should be noted that Veff already depends on the density (and thus on the orbitals) through the Coulomb term as shown in equation (5-13). Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-electron equations (5-14) also have to be solved iteratively. 

The densities Pi are obtained from a set of degenerate KS wave functions and the w, are the corresponding weights. Without going into details we note that regular density functional theory can be extended to such ensembles. For our problems at hand, we can write down the energy expression as... 

Potential, Ionization—The energy expressed as electron volts (eV) necessary to separate one electron from an atom, resulting in the formation of an ion pair. 

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